b,transcriber's notes_/b3 ,the text of this book has been preserved as closely as practicable to its original form4 ,however1 the author used some unusual symbols1 and ,i have taken the liberty of using ,unicode characters with similar appearance 7 7 as substitutes1 disregarding their official meaning and aware that they might not display on all devices4 ,an archaic symbol used by the author to indicate the mathematical #8factorial' function has been replaced by the modern equivalent1 viz4 6 ,unusual placements of various sub- and superscripted symbols remain as in the original text4 ,in this plain-text version1 italic text is denoted by 99asterisks992 superscripted and subscripted characters are enclosed within curly brackets preceded by a caret 77 or underline 77 respectively4 ,inconsistencies of punctuation have been corrected silently1 but inconsistent spellings such as 99,roemer1 ,rmer1 ,rmer99 have not been altered4 ,a list of corrected spellings is appended at the end of the book4 ,footnotes have been renumbered consecutively and positioned below the relevant paragraphs4 ,a missing footnote marker has been inserted on p4#gea after tracking down the original document4 ,a missing negative symbol has been added to an exponent in a formula on p4#cbg4 ,there is a misleading calculation on p4#aid and the table that follows1 regarding progressive powers of two3 77#bb7b7b is equivalent to 7#af7b which equals #bef not #fe1cef as stated1 but #baf does equal #fe1cef4 sic has been inserted on p4#agi alongside a statement that the alphabet contains #bd letters2 however1 the statement may well be correct given that it was written in #agjd by a ,flemish author and the language is not specified4 ,a large arithmetic triangle on p4#ahd cannot be displayed optimally in this plain-text version for lack of space ,,the ,,principles ,,of ,,science4 ,illustration ,illustration3 ,,the ,,logical ,,machine4 ,,the ,,principles ,,of ,,science3 99,a ,,treatise ,,on ,,logic99 99,,and99 99,,scientific ,,method499 ,,by ,w4 ,,stanley ,,jevons1 ,,ll4,d4 7,,edinb471 ,m4,a4 7,,lond471 ,f4,r4,s4 ,london3 ,,macmillan ,,and ,,co4 #ahhc4 99,the ,right of ,translation and ,reproduction is ,reserved499 ,,london3 ,r4 ,clay1 ,sons1 ,taylor1 ,printers1 ,,bread ,,street ,,hill4 ,stereotyped ,edition4 ,,preface 99,,to ,,the ,,first ,,edition994 ,it may be truly asserted that the rapid progress of the physical sciences during the last three centuries has not been accompanied by a corresponding advance in the theory of reasoning4 ,physicists speak familiarly of ,scientific ,method1 but they could not readily describe what they mean by that expression4 ,profoundly engaged in the study of particular classes of natural phenomena1 they are usually too much engrossed in the immense and ever-accumulating details of their special sciences to generalise upon the methods of reasoning which they unconsciously employ4 ,yet few will deny that these methods of reasoning ought to be studied1 especially by those who endeavour to introduce scientific order into less successful and methodical branches of knowledge4 ,the application of ,scientific ,method cannot be restricted to the sphere of lifeless objects4 ,we must sooner or later have strict sciences of those mental and social phenomena1 which1 if comparison be possible1 are of more interest to us than purely material phenomena4 ,but it is the proper course of reasoning to proceed from the known to the unknown--from the evident to the obscure--from the material and palpable to the subtle and refined4 ,the physical sciences may therefore be properly made the practice-ground of the reasoning powers1 because they furnish us with a great body-of precise and successful investigations4 ,in these sciences we meet with happy instances of unquestionable deductive reasoning1 of extensive generalisation1 of happy prediction1 of satisfactory verification1 of nice calculation of probabilities4 ,we can note how the slightest analogical clue has been followed up to a glorious discovery1 how a rash generalisation has at length been exposed1 or a conclusive 99experimentum crucis99 has decided the long-continued strife between two rival theories4 ,in following out my design of detecting the general methods of inductive investigation1 ,i have found that the more elaborate and interesting processes of quantitative induction have their necessary foundation in the simpler science of ,formal ,logic4 ,the earlier1 and probably by far the least attractive part of this work1 consists1 therefore1 in a statement of the so-called ,fundamental ,laws of ,thought1 and of the all-important ,principle of ,substitution1 of which1 as ,i think1 all reasoning is a development4 ,the whole procedure of inductive inquiry1 in its most complex cases1 is foreshadowed in the combinational view of ,logic1 which arises directly from these fundamental principles4 ,incidentally ,i have described the mechanical arrangements by which the use of the important form called the ,logical ,alphabet1 and the whole working of the combinational system of ,formal ,logic1 may be rendered evident to the eye1 and easy to the mind and hand4 ,the study both of ,formal ,logic and of the ,theory of ,probabilities has led me to adopt the opinion that there is no such thing as a distinct method of induction as contrasted with deduction1 but that induction is simply an inverse employment of deduction4 ,within the last century a reaction has been setting in against the purely empirical procedure of ,francis ,bacon1 and physicists have learnt to advocate the use of hypotheses4 ,i take the extreme view of holding that ,francis ,bacon1 although he correctly insisted upon constant reference to experience1 had no correct notions as to the logical method by which from particular facts we educe laws of nature4 ,i endeavour to show that hypothetical anticipation of nature is an essential part of inductive inquiry1 and that it is the ,newtonian method of deductive reasoning combined with elaborate experimental verification1 which has led to all the great triumphs of scientific research4 ,in attempting to give an explanation of this view of ,scientific ,method1 ,i have first to show that the sciences of number and quantity repose upon and spring from the simpler and more general science of ,logic4 ,the ,theory of ,probability1 which enables us to estimate and calculate quantities of knowledge1 is then described1 and especial attention is drawn to the ,inverse ,method of ,probabilities1 which involves1 as ,i conceive1 the true principle of inductive procedure4 ,no inductive conclusions are more than probable1 and ,i adopt the opinion that the theory of probability is an essential part of logical method1 so that the logical value of every inductive result must be determined consciously or unconsciously1 according to the principles of the inverse method of probability4 ,the phenomena of nature are commonly manifested in quantities of time1 space1 force1 energy1 c41 and the observation1 measurement1 and analysis of the various quantitative conditions or results involved1 even in a simple experiment1 demand much employment of systematic procedure4 ,i devote a book1 therefore1 to a simple and general description of the devices by which exact measurement is effected1 errors eliminated1 a probable mean result attained1 and the probable error of that mean ascertained4 ,i then proceed to the principal1 and probably the most interesting1 subject of the book1 illustrating successively the conditions and precautions requisite for accurate observation1 for successful experiment1 and for the sure detection of the quantitative laws of nature4 ,as it is impossible to comprehend aright the value of quantitative laws without constantly bearing in mind the degree of quantitative approximation to the truth probably attained1 ,i have devoted a special chapter to the ,theory of ,approximation1 and however imperfectly ,i may have treated this subject1 ,i must look upon it as a very essential part of a work on ,scientific ,method4 ,it then remains to illustrate the sound use of hypothesis1 to distinguish between the portions of knowledge which we owe to empirical observation1 to accidental discovery1 or to scientific prediction4 ,interesting questions arise concerning the accordance of quantitative theories and experiments1 and ,i point out how the successive verification of an hypothesis by distinct methods of experiment yields conclusions approximating to but never attaining certainty4 ,additional illustrations of the general procedure of inductive investigations are given in a chapter on the ,character of the ,experimentalist1 in which ,i endeavour to show1 moreover1 that the inverse use of deduction was really the logical method of such great masters of experimental inquiry as ,newton1 ,huyghens1 and ,faraday4 ,in treating ,generalisation and ,analogy1 ,i consider the precautions requisite in inferring from one case to another1 or from one part of the universe to another part2 the validity of all such inferences resting ultimately upon the inverse method of probabilities4 ,the treatment of ,exceptional ,phenomena appeared to afford an interesting subject for a further chapter illustrating the various modes in which an outstanding fact may eventually be explained4 ,the formal part of the book closes with the subject of ,classification1 which is1 however1 very inadequately treated4 ,i have1 in fact1 almost restricted myself to showing that all classification is fundamentally carried out upon the principles of ,formal ,logic and the ,logical ,alphabet described at the outset4 ,in certain concluding remarks ,i have expressed the conviction which the study of ,logic has by degrees forced upon my mind1 that serious misconceptions are entertained by some scientific men as to the logical value of our knowledge of nature4 ,we have heard much of what has been aptly called the ,reign of ,law1 and the necessity and uniformity of natural forces has been not uncommonly interpreted as involving the non-existence of an intelligent and benevolent ,power1 capable of interfering with the course of natural events4 ,fears have been expressed that the progress of ,scientific ,method must therefore result in dissipating the fondest beliefs of the human heart4 ,even the #8,utility of ,religion' is seriously proposed as a subject of discussion4 ,it seemed to be not out of place in a work on ,scientific ,method to allude to the ultimate results and limits of that method4 ,i fear that ,i have very imperfectly succeeded in expressing my strong conviction that before a rigorous logical scrutiny the ,reign of ,law will prove to be an unverified hypothesis1 the ,uniformity of ,nature an ambiguous expression1 the certainty of our scientific inferences to a great extent a delusion4 ,the value of science is of course very high1 while the conclusions are kept well within the limits of the data on which they are founded1 but it is pointed out that our experience is of the most limited character compared with what there is to learn1 while our mental powers seem to fall infinitely short of the task of comprehending and explaining fully the nature of any one object4 ,i draw the conclusion that we must interpret the results of ,scientific ,method in an affirmative sense only4 ,ours must be a truly positive philosophy1 not that false negative philosophy which1 building on a few material facts1 presumes to assert that it has compassed the bounds of existence1 while it nevertheless ignores the most unquestionable phenomena of the human mind and feelings4 ,it is approximately certain that in freely employing illustrations drawn from many different sciences1 ,i have frequently fallen into errors of detail4 ,in this respect ,i must throw myself upon the indulgence of the reader1 who will bear in mind1 as ,i hope1 that the scientific facts are generally mentioned purely for the purpose of illustration1 so that inaccuracies of detail will not in the majority of cases affect the truth of the general principles illustrated4 99,december #ae1 #ahgc499 ,,preface 99,,to ,,the ,,second ,,edition994 ,few alterations of importance have been made in preparing this second edition4 ,nevertheless1 advantage has been taken of the opportunity to revise very carefully both the language and the matter of the book4 ,correspondents and critics having pointed out inaccuracies of more or less importance in the first edition1 suitable corrections and emendations have been made4 ,i am under obligations to ,mr4 ,c4 ,j4 ,monro1 ,m4,a41 of ,barnet1 and to ,mr4 ,w4 ,h4 ,brewer1 ,m4,a41 one of ,her ,majesty's ,inspectors of ,schools1 for numerous corrections4 ,among several additions which have been made to the text1 ,i may mention the abstract 7p4 #adc7 of ,professor ,clifford's remarkable investigation into the number of types of compound statement involving four classes of objects4 ,this inquiry carries forward the inverse logical problem described in the preceding sections4 ,again1 the need of some better logical method than the old ,barbara ,celarent1 c41 is strikingly shown by ,mr4 ,venn's logical problem1 described at p4 #ij4 ,a great number of candidates in logic and philosophy were tested by ,mr4 ,venn with this problem1 which1 though simple in reality1 was solved by very few of those who were ignorant of ,boole's ,logic4 ,other evidence could be adduced by ,mr4 ,venn of the need for some better means of logical training4 ,to enable the logical student to test his skill in the solution of inductive logical problems1 ,i have given 7p4 #abg7 a series of ten problems graduated in difficulty4 ,to prevent misapprehension1 it should be mentioned that1 throughout this edition1 ,i have substituted the name 99,logical ,alphabet99 for 99,logical ,abecedarium991 the name applied in the first edition to the exhaustive series of logical combinations represented in terms of 99,a991 99,b991 99,c991 99,d99 7p4 #id74 ,it was objected by some readers that 99,abecedarium99 is a long and unfamiliar name4 ,to the chapter on ,units and ,standards of ,measurement1 ,i have added two sections1 one 7p4 #cbe7 containing a brief statement of the ,theory of ,dimensions1 and the other 7p4 #cai7 discussing ,professor ,clerk ,maxwell's very original suggestion of a ,natural ,system of ,standards for the measurement of space and time1 depending upon the length and rapidity of waves of light4 ,in my description of the ,logical ,machine in the 99,philosophical ,transactions99 7vol4 #afj1 p4 #dih71 ,i said--8,it is rarely indeed that any invention is made without some anticipation being sooner or later discovered2 but up to the present time ,i am totally unaware of even a single previous attempt to devise or construct a machine which should perform the operations of logical inference2 and it is only1 ,i believe1 in the satirical writings of ,swift that an allusion to an actual reasoning machine is to be found40 ,before the paper was printed1 however1 ,i was able to refer 7p4 #eah7 to the ingenious designs of the late ,mr4 ,alfred ,smee as attempts to represent thought mechanically4 ,mr4 ,smee's machines indeed were never constructed1 and1 if constructed1 would not have performed actual logical inference4 ,it has now just come to light1 however1 that the celebrated ,lord ,stanhope actually did construct a mechanical device1 capable of representing syllogistic inferences in a concrete form4 ,it appears that logic was one of the favourite studies of this truly original and ingenious nobleman4 ,there remain fragments of a logical work1 printed by the ,earl at his own press1 which show that he had arrived1 before the year #ahjj1 at the principle of the quantified predicate4 ,he puts forward this principle in the most explicit manner1 and proposes to employ it throughout his syllogistic system4 ,moreover1 he converts negative propositions into affirmative ones1 and represents these by means of the copula 8is identic with40 ,thus he anticipated1 probably by the force of his own unaided insight1 the main points of the logical method originated in the works of ,george ,bentham and ,george ,boole1 and developed in this work4 ,stanhope1 indeed1 has no claim to priority of discovery1 because he seems never to have published his logical writings1 although they were put into print4 ,there is no trace of them in the ,british ,museum ,library1 nor in any other library or logical work1 so far as ,i am aware4 ,both the papers and the logical contrivance have been placed by the present ,earl ,stanhope in the hands of the ,rev4 ,robert ,harley1 ,f4,r4,s41 who will1 ,i hope1 soon publish a description of them4#a #a ,since the above was written ,mr4 ,harley has read an account of ,stanhope's logical remains at the ,dublin ,meeting 7#ahgh7 of the ,british ,association4 ,the paper will be printed in 99,mind994 7,note added ,november1 #ahgh47 ,by the kindness of ,mr4 ,harley1 ,i have been able to examine ,stanhope's logical contrivance1 called by him the ,demonstrator4 ,it consists of a square piece of bay-wood with a square depression in the centre1 across which two slides can be pushed1 one being a piece of red glass1 and the other consisting of wood coloured gray4 ,the extent to which each of these slides is pushed in is indicated by scales and figures along the edges of the aperture1 and the simple rule of inference adopted by ,stanhope is3 8,to the gray add the red and subtract the 99holon9910 meaning by holon 77 the whole width of the aperture4 ,this rule of inference is a curious anticipation of ,de ,morgan's numerically definite syllogism 7see below1 p4 #afh71 and of inferences founded on what ,hamilton called 8,ultra-total distribution40 ,another curious point about ,stanhope's device is1 that one slide can be drawn out and pushed in again at right angles to the other1 and the overlapping part of the slides then represents the probability of a conclusion1 derived from two premises of which the probabilities are respectively represented by the projecting parts of the slides4 ,thus it appears that ,stanhope had studied the logic of probability as well as that of certainty1 here again anticipating1 however obscurely1 the recent progress of logical science4 ,it will be seen1 however1 that between ,stanhope's ,demonstrator and my ,logical ,machine there is no resemblance beyond the fact that they both perform logical inference4 ,in the first edition ,i inserted a section 7vol4 i4 p4 #be71 on 8,anticipations of the ,principle of ,substitution10 and ,i have reprinted that section unchanged in this edition 7p4 #ba74 ,i remark therein that1 8,in such a subject as logic it is hardly possible to put forth any opinions which have not been in some degree previously entertained4 ,the germ at least of every doctrine will be found in earlier writings1 and novelty must arise chiefly in the mode of harmonising and developing ideas40 ,i point out1 as ,professor ,t4 ,m4 ,lindsay had previously done1 that ,beneke had employed the name and principle of substitution1 and that doctrines closely approximating to substitution were stated by the ,port ,royal ,logicians more than #bjj years ago4 ,i have not been at all surprised to learn1 however1 that other logicians have more or less distinctly stated this principle of substitution during the last two centuries4 ,as my friend and successor at ,owens ,college1 ,professor ,adamson1 has discovered1 this principle can be traced back to no less a philosopher than ,leibnitz4 ,the remarkable tract of ,leibnitz1#b entitled 8,non inelegans ,specimen ,demonstrandi in ,abstractis10 commences at once with a definition corresponding to the principle3-- 8,eadem sunt quorum unum potest substitui alteri salva veritate4 ,si sint 99,a99 et 99,b991 et 99,a99 ingrediatur aliquam propositionem veram1 et ibi in aliquo loco ipsius 99,a99 pro ipso substituendo 99,b99 fiat nova propositio que itidem vera1 idque semper succedat in quacunque tali propositione1 99,a99 et 99,b99 dicuntur esse eadem2 et contra1 si eadem sint 99,a99 et 99,b991 procedet substitutio quam dixi40 #b ,leibnitii 99,opera ,philosophica qu extant994 ,erdmann1 ,pars ,i4 ,berolini1 #ahdj1 p4 #id4 ,leibnitz1 then1 explicitly adopts the principle of substitution1 but he puts it in the form of a definition1 saying that those things are the same which can be substituted one for the other1 without affecting the truth of the proposition4 ,it is only after having thus tested the sameness of things that we can turn round and say that 99,a99 and 99,b991 being the same1 may be substituted one for the other4 ,it would seem as if we were here in a vicious circle2 for we are not allowed to substitute 99,a99 for 99,b991 unless we have ascertained by trial that the result is a true proposition4 ,the difficulty does not seem to be removed by ,leibnitz' proviso1 8idque semper succedat in quacunque tali propositione40 ,how can we learn that because 99,a99 and 99,b99 may be mutually substituted in some propositions1 they may therefore be substituted in others2 and what is the criterion of likeness of propositions expressed in the word 8tali08 ,whether the principle of substitution is to be regarded as a postulate1 an axiom1 or a definition1 is just one of those fundamental questions which it seems impossible to settle in the present position of philosophy1 but this uncertainty will not prevent our making a considerable step in logical science4 ,leibnitz proceeds to establish in the form of a theorem what is usually taken as an axiom1 thus 799,opera991 p4 #ie73 8,theorema ,i4 ,qu sunt eadem uni tertio1 eadem sunt inter se4 ,si 99,a99 99,b99 et 99,b99 99,c991 erit 99,a99 99,c994 ,nam si in propositione 99,a99 99,b99 7vera ea hypothesi7 substituitur 99,c99 in locum 99,b99 7quod facere licet per ,def4 ,i4 quia 99,b99 99,c99 ex hypothesi7 fiet 99,a99 99,c994 ,q4 ,e4 ,dem40 ,thus ,leibnitz precisely anticipates the mode of treating inference with two simple identities described at p4 #ea of this work4 ,even the mathematical axiom that #8equals added to equals make equals1' is deduced from the principle of substitution4 ,at p4 #ie of ,erdmann's edition1 we find3 8,si eidem addantur coincidentia fiunt coincidentia4 ,si 99,a99 99,b991 erit 99,a99 99,c99 99,b99 99,c994 ,nam si in propositione 99,a99 99,c99 99,a99 99,c99 7qu est vera per se7 pro 99,a99 semel substituas 99,b99 7quod facere licet per ,def4 ,i4 quia 99,a99 99,b997 fiet 99,a99 99,c99 99,b99 99,c99 ,q4 ,e4 ,dem40 ,this is unquestionably the mode of deducing the several axioms of mathematical reasoning from the higher axiom of substitution1 which is explained in the section on mathematical inference 7p4 #afb7 in this work1 and which had been previously stated in my 99,substitution of ,similars991 p4 #af4 ,there are one or two other brief tracts in which ,leibnitz anticipates the modern views of logic4 ,thus in the eighteenth tract in ,erdmann's edition 7p4 #ib71 called 8,fundamenta ,calculi ,ratiocinatoris01 he says3 8,inter ea quorum unum alteri substitui potest1 salvis calculi legibus1 dicetur esse quipollentiam40 ,there is evidence1 also1 that he had arrived at the quantification of the predicate1 and that he fully understood the reduction of the universal affirmative proposition to the form of an equation1 which is the key to an improved view of logic4 ,thus1 in the tract entitled 8,difficultates ,qudam ,logic10#c he says3 8,omne 99,a99 est 99,b992 id est quivalent 99,,ab99 et 99,a991 seu 99,a99 non 99,b99 est non-ens40 #c ,erdmann1 p4 #ajb4 ,it is curious to find1 too1 that ,leibnitz was fully acquainted with the ,laws of ,commutativeness and 8,simplicity0 7as ,i have called the second law7 attaching to logical symbols4 ,in the 8,addenda ad ,specimen ,calculi ,universalis0 we read as follows4#d 8,transpositio literarum in eodem termino nihil mutat1 ut 99ab99 coincidet cum 99ba991 seu animal rationale et rationale animal40 8,repetitio ejusdem liter in eodem termino est inutilis1 ut 99b99 est 99aa992 vel 99bb99 est 99a992 homo est animal animal1 vel homo homo est animal4 ,sufficit enim dici 99a99 est 99b991 seu homo est animal40 #d ,ibid4 p4 #ih4 ,comparing this with what is stated in ,boole's 99,mathematical ,analysis of ,logic991 pp4 #ag-ah1 in his 99,laws of ,thought991 p4 #bi1 or in this work1 pp4 #cb-ce1 we find that ,leibnitz had arrived two centuries ago at a clear perception of the bases of logical notation4 ,when ,boole pointed out that1 in logic1 99xx99 99x991 this seemed to mathematicians to be a paradox1 or in any case a wholly new discovery2 but here we have it plainly stated by ,leibnitz4 ,the reader must not assume1 however1 that because ,leibnitz correctly apprehended the fundamental principles of logic1 he left nothing for modern logicians to do4 ,on the contrary1 ,leibnitz obtained no useful results from his definition of substitution4 ,when he proceeds to explain the syllogism1 as in the paper on 8,definitiones ,logic10#e he gives up substitution altogether1 and falls back upon the notion of inclusion of class in class1 saying1 8,includens includentis est includens inclusi1 seu si 99,a99 includit 99,b99 et 99,b99 includit 99,c991 etiam 99,a99 includet 99,c9940 ,he proceeds to make out certain rules of the syllogism involving the distinction of subject and predicate1 and in no important respect better than the old rules of the syllogism4 ,leibnitz' logical tracts are1 in fact1 little more than brief memoranda of investigations which seem never to have been followed out4 ,they remain as evidence of his wonderful sagacity1 but it would be difficult to show that they have had any influence on the progress of logical science in recent times4 #e ,erdmann1 p4 #ajj4 ,i should like to explain how it happened that these logical writings of ,leibnitz were unknown to me1 until within the last twelve months4 ,i am so slow a reader of ,latin books1 indeed1 that my overlooking a few pages of ,leibnitz' works would not have been in any case surprising4 ,but the fact is that the copy of ,leibnitz' works of which ,i made occasional use1 was one of the edition of ,dutens1 contained in ,owens ,college ,library4 ,the logical tracts in question were not printed in that edition1 and with one exception1 they remained in manuscript in the ,royal ,library at ,hanover1 until edited by ,erdmann1 in #ahci-dj4 ,the tract 8,difficultates ,qudam ,logic10 though not known to ,dutens1 was published by ,raspe in #agfe1 in his collection called 99uvres ,philosophiques de feu ,mr4 ,leibnitz992 but this work had not come to my notice1 nor does the tract in question seem to contain any explicit statement of the principle of substitution4 ,it is1 ,i presume1 the comparatively recent publication of ,leibnitz' most remarkable logical tracts which explains the apparent ignorance of logicians as regards their contents and importance4 ,the most learned logicians1 such as ,hamilton and ,ueberweg1 ignore ,leibnitz' principle of substitution4 ,in the ,appendix to the fourth volume of ,hamilton's 99,lectures on ,metaphysics and ,logic991 is given an elaborate compendium of the views of logical writers concerning the ultimate basis of deductive reasoning4 ,leibnitz is briefly noticed on p4 #cai1 but without any hint of substitution4 ,he is here quoted as saying1 8,what are the same with the same third1 are the same with each other2 that is1 if 99,a99 be the same with 99,b991 and 99,c99 be the same with 99,b991 it is necessary that 99,a99 and 99,c99 should also be the same with one another4 ,for this principle flows immediately from the principle of contradiction1 and is the ground and basis of all logic2 if that fail1 there is no longer any way of reasoning with certainty40 ,this view of the matter seems to be inconsistent with that which he adopted in his posthumous tract4 ,dr4 ,thomson1 indeed1 was acquainted with ,leibnitz' tracts1 and refers to them in his 99,outline of the ,necessary ,laws of ,thought994 ,he calls them valuable2 nevertheless1 he seems to have missed the really valuable point2 for in making two brief quotations1#f he omits all mention of the principle of substitution4 #f ,fifth ,edition1 #ahfj1 p4 #aeh4 ,ueberweg is probably considered the best authority concerning the history of logic1 and in his well-known 99,system of ,logic and ,history of ,logical ,doctrines991#g he gives some account of the principle of substitution1 especially as it is implicitly stated in the 99,port ,royal ,logic994 ,but he omits all reference to ,leibnitz in this connection1 nor does he elsewhere1 so far as ,i can find1 supply the omission4 ,his ,english editor1 ,professor ,t4 ,m4 ,lindsay1 in referring to my 99,substitution of ,similars991 points out how ,i was anticipated by ,beneke2 but he also ignores ,leibnitz4 ,it is thus apparent that the most learned logicians1 even when writing especially on the history of logic1 displayed ignorance of ,leibnitz' most valuable logical writings4 #g ,section #abj4 ,it has been recently pointed out to me1 however1 that the ,rev4 ,robert ,harley did draw attention1 at the ,nottingham ,meeting of the ,british ,association1 in #ahff1 to ,leibnitz' anticipations of ,boole's laws of logical notation1#h and ,i am informed that ,boole1 about a year after the publication of his 99,laws of ,thought991 was made acquainted with these anticipations by ,r4 ,leslie ,ellis4 #h ,see his 8,remarks on ,boole's ,mathematical ,analysis of ,logic40 99,report of the #cfth ,meeting of the ,british ,association1 ,transactions of the ,sections991 pp4 #c-f4 ,there seems to have been at least one other ,german logician who discovered1 or adopted1 the principle of substitution4 ,reusch1 in his 99,systema ,logicum991 published in #agcd1 laboured to give a broader basis to the 99,dictum de ,omni et ,nullo994 ,he argues1 that 8the whole business of ordinary reasoning is accomplished by the substitution of ideas in place of the subject or predicate of the fundamental proposition4 ,this some call the 99equation of thoughts9940 ,but1 in the hands of ,reusch1 substitution does not seem to lead to simplicity1 since it has to be carried on according to the rules of ,equipollence1 ,reciprocation1 ,subordination1 and ,co-ordination4#i ,reusch is elsewhere spoken of#aj as the 8celebrated ,reusch02 nevertheless1 ,i have not been able to find a copy of his book in ,london1 even in the ,british ,museum ,library2 it is not mentioned in the printed catalogue of the ,bodleian ,library2 ,messrs4 ,asher have failed to obtain it for me by advertisement in ,germany2 and ,professor ,adamson has been equally unsuccessful4 ,from the way in which the principle of substitution is mentioned by ,reusch1 it would seem likely that other logicians of the early part of the eighteenth century were acquainted with it2 but1 if so1 it is still more curious that recent historians of logical science have overlooked the doctrine4 #i ,hamilton's ,lectures1 vol4 iv4 p4 #cai4 #aj ,ibid4 p4 #cbf4 ,it is a strange and discouraging fact1 that true views of logic should have been discovered and discussed from one to two centuries ago1 and yet should have remained1 like ,george ,bentham's work in this century1 without influence on the subsequent progress of the science4 ,it may be regarded as certain that none of the discoverers of the quantification of the predicate1 ,bentham1 ,hamilton1 ,thomson1 ,de ,morgan1 and ,boole1 were in any way assisted by the hints of the principle contained in previous writers4 ,as to my own views of logic1 they were originally moulded by a careful study of ,boole's works1 as fully stated in my first logical essay4#aa ,as to the process of substitution1 it was not learnt from any work on logic1 but is simply the process of substitution perfectly familiar to mathematicians1 and with which ,i necessarily became familiar in the course of my long-continued study of mathematics under the late ,professor ,de ,morgan4 #aa 99,pure ,logic1 or the ,logic of ,quality apart from ,quantity2 with ,remarks on ,boole's ,system1 and on the ,relation of ,logic and ,mathematics499 ,london1 #ahfd1 p4 #c4 ,i find that the ,theory of ,number1 which ,i explained in the eighth chapter of this work1 is also partially anticipated in a single scholium of ,leibnitz4 ,he first gives as an axiom the now well-known law of ,boole1 as follows3-- 8,axioma ,i4 ,si idem secum ipso sumatur1 nihil constituitur novum1 seu 99,a99 99,a99 99,a9940 ,then follows this remarkable scholium3 8,equidem in numeris #d #d facit #h1 seu bini nummi binis additi faciunt quatuor nummos1 sed tunc bini additi sunt alii a prioribus2 si iidem essent nihil novi prodiret et perinde esset ac si joco ex tribus ovis facere vellemus sex numerando1 primum #c ova1 deinde uno sublato residua #b1 ac denique uno rursus sublato residuum40 ,translated this would read as follows3-- 8,axiom ,i4 ,if the same thing is taken together with itself1 nothing new arises1 or 99,a99 99,a99 99,a994 8,scholium4 ,in numbers1 indeed1 #d #d makes #h1 or two coins added to two coins make four coins1 but then the two added are different from the former ones2 if they were the same nothing new would be produced1 and it would be just as if we tried in joke to make six eggs out of three1 by counting firstly the three eggs1 then1 one being removed1 counting the remaining two1 and lastly1 one being again removed1 counting the remaining egg40 ,compare the above with pp4 #aef to #afb of the present work4 ,m4 ,littr has quite recently pointed out#ab what he thinks is an analogy between the system of formal logic1 stated in the following pages1 and the logical devices of the celebrated ,raymond ,lully4 ,lully's method of invention was described in a great number of medival books1 but is best stated in his 99,ars ,compendiosa ,inveniendi ,veritatem1 seu ,ars ,magna et ,major994 ,this method consisted in placing various names of things in the sectors of concentric circles1 so that when the circles were turned1 every possible combination of the things was easily produced by mechanical means4 ,it might1 perhaps1 be possible to discover in this method a vague and rude anticipation of combinational logic2 but it is well known that the results of ,lully's method were usually of a fanciful1 if not absurd character4 #ab 99,la ,philosophie ,positive991 ,mai-,juin1 #ahgg1 tom4 xviii4 p4 #def4 ,a much closer analogue of the ,logical ,alphabet is probably to be found in the ,logical ,square1 invented by ,john ,christian ,lange1 and described in a rare and unnoticed work by him which ,i have recently found in the ,british ,museum4#ac ,this square involved the principle of bifurcate classification1 and was an improved form of the ,ramean and ,porphyrian tree 7see below1 p4 #gjb74 ,lange seems1 indeed1 to have worked out his ,logical ,square into a mechanical form1 and he suggests that it might be employed somewhat in the manner of ,napier's ,bones 7p4 #fe74 ,there is much analogy between his ,square and my ,abacus1 but ,lange had not arrived at a logical system enabling him to use his invention for logical inference in the manner of the ,logical ,abacus4 ,another work of ,lange is said to contain the first publication of the well known ,eulerian diagrams of proposition and syllogism4#ad #ac 99,inventum ,novum ,quadrati ,logici991 c41 ,giss ,hassorum1 #agad1 #hvo4 #ad ,see 99,ueberweg's ,system of ,logic991 c41 translated by ,lindsay1 p4 #cjb4 ,since the first edition was published1 an important work by ,mr4 ,george ,lewes has appeared1 namely1 his 99,problems of ,life and ,mind991 which to a great extent treats of scientific method1 and formulates the rules of philosophising4 ,i should have liked to discuss the bearing of ,mr4 ,lewes's views upon those here propounded1 but ,i have felt it to be impossible in a book already filling nearly #hjj pages1 to enter upon the discussion of a yet more extensive book4 ,for the same reason ,i have not been able to compare my own treatment of the subject of probability with the views expressed by ,mr4 ,venn in his 99,logic of ,chance994 ,with ,mr4 ,j4 ,j4 ,murphy's profound and remarkable works on 99,habit and ,intelligence991 and on 99,the ,scientific ,basis of ,faith991 ,i was unfortunately unacquainted when ,i wrote the following pages4 ,they cannot safely be overlooked by any one who wishes to comprehend the tendency of philosophy and scientific method in the present day4 ,it seems desirable that ,i should endeavour to answer some of the critics who have pointed out what they consider defects in the doctrines of this book1 especially in the first part1 which treats of deduction4 ,some of the notices of the work were indeed rather statements of its contents than critiques4 ,thus1 ,i am much indebted to ,m4 ,louis ,liard1 ,professor of ,philosophy at ,bordeaux1 for the very careful exposition#ae of the substitutional view of logic which he gave in the excellent 99,revue ,philosophique991 edited by ,m4 ,ribot4 7,mars1 #ahgg1 tom4 iii4 p4 #bgg47 ,an equally careful account of the system was given by ,m4 ,riehl1 ,professor of ,philosophy at ,graz1 in his article on 8,die ,englische ,logik der ,gegenwart10 published in the 99,vierteljahrsschrift fr wissenschaftliche ,philosophie994 7#a ,heft1 ,leipzig1 #ahgf47 ,i should like to acknowledge also the careful and able manner in which my book was reviewed by the 99,new ,york ,daily ,tribune99 and the 99,new ,york ,times994 #ae ,since the above was written ,m4 ,liard has republished this exposition as one chapter of an interesting and admirably lucid account of the progress of logical science in ,england4 ,after a brief but clear introduction1 treating of the views of ,herschel1 ,mill1 and others concerning ,inductive ,logic1 ,m4 ,liard describes in succession the logical systems of ,george ,bentham1 ,hamilton1 ,de ,morgan1 ,boole1 and that contained in the present work4 ,the title of the book is as follows3--99,les ,logiciens ,anglais ,contemporains994 ,par ,louis ,liard1 ,professeur de ,philosophie la ,facult des ,lettres de ,bordeaux4 ,paris3 ,librairie ,germer ,baillire4 #ahgh4 7,note added ,november1 #ahgh47 ,the most serious objections which have been brought against my treatment of logic have regard to my failure to enter into an analysis of the ultimate nature and origin of the ,laws of ,thought4 ,the 99,spectator991#af for instance1 in the course of a careful review1 says of the principle of substitution1 8,surely it is a great omission not to discuss whence we get this great principle itself2 whether it is a pure law of the mind1 or only an approximate lesson of experience2 and if a pure product of the mind1 whether there are any other products of the same kind1 furnished by our knowing faculty itself40 ,professor ,robertson1 in his very acute review1#ag likewise objects to the want of psychological and philosophical analysis4 8,if the book really corresponded to its title1 ,mr4 ,jevons could hardly have passed so lightly over the question1 which he does not omit to raise1 concerning those undoubted principles of knowledge commonly called the ,laws of ,thought4444 ,everywhere1 indeed1 he appears least at ease when he touches on questions properly philosophical2 nor is he satisfactory in his psychological references1 as on pp4 #d1 #e1 where he cannot commit himself to a statement without an accompaniment of #8probably1' #8almost1' or #8hardly4' ,reservations are often very much in place1 but there are fundamental questions on which it is proper to make up one's mind40 #af 99,spectator991 ,september #ai1 #ahgd1 p4 #aagh4 ,a second portion of the review appeared in the same journal for ,september #bf1 #ahgd1 p4 #abjd4 #ag 99,mind993 a ,quarterly ,review of ,psychology and ,philosophy4 ,no4 ,,ii4 ,april #ahgf4 ,vol4 ,i4 p4 #bjf4 ,these remarks appear to me to be well founded1 and ,i must state why it is that ,i have ventured to publish an extensive work on logic1 without properly making up my mind as to the fundamental nature of the reasoning process4 ,the fault after all is one of omission rather than of commission4 ,it is open to me on a future occasion to supply the deficiency if ,i should ever feel able to undertake the task4 ,but ,i do not conceive it to be an essential part of any treatise to enter into an ultimate analysis of its subject matter4 ,analyses must always end somewhere4 ,there were good treatises on light which described the laws of the phenomenon correctly before it was known whether light consisted of undulations or of projected particles4 ,now we have treatises on the ,undulatory ,theory which are very valuable and satisfactory1 although they leave us in almost complete doubt as to what the vibrating medium really is4 ,so ,i think that1 in the present day1 we need a correct and scientific exhibition of the formal laws of thought1 and of the forms of reasoning based on them1 although we may not be able to enter into any complete analysis of the nature of those laws4 ,what would the science of geometry be like now if the ,greek geometers had decided that it was improper to publish any propositions before they had decided on the nature of an axiom8 ,where would the science of arithmetic be now if an analysis of the nature of number itself were a necessary preliminary to a development of the results of its laws8 ,in recent times there have been enormous additions to the mathematical sciences1 but very few attempts at psychological analysis4 ,in the ,alexandrian and early medival schools of philosophy1 much attention was given to the nature of unity and plurality chiefly called forth by the question of the ,trinity4 ,in the last two centuries whole sciences have been created out of the notion of plurality1 and yet speculation on the nature of plurality has dwindled away4 ,this present treatise contains1 in the eighth chapter1 one of the few recent attempts to analyse the notion of number itself4 ,if further illustration is needed1 ,i may refer to the differential calculus4 ,nobody calls in question the formal truth of the results of that calculus4 ,all the more exact and successful parts of physical science depend upon its use1 and yet the mathematicians who have created so great a body of exact truths have never decided upon the basis of the calculus4 ,what is the nature of a limit or the nature of an infinitesimal8 ,start the question among a knot of mathematicians1 and it will be found that hardly two agree1 unless it is in regarding the question itself as a trifling one4 ,some hold that there are no such things as infinitesimals1 and that it is all a question of limits4 ,others would argue that the infinitesimal is the necessary outcome of the limit1 but various shades of intermediate opinion spring up4 ,now it is just the same with logic4 ,if the forms of deductive and inductive reasoning given in the earlier part of this treatise are correct1 they constitute a definite addition to logical science1 and it would have been absurd to decline to publish such results because ,i could not at the same time decide in my own mind about the psychology and philosophy of the subject4 ,it comes in short to this1 that my book is a book on ,formal ,logic and ,scientific ,method1 and not a book on psychology and philosophy4 ,it may be objected1 indeed1 as the 99,spectator99 objects1 that ,mill's ,system of ,logic is particularly strong in the discussion of the psychological foundations of reasoning1 so that ,mill would appear to have successfully treated that which ,i feel myself to be incapable of attempting at present4 ,if ,mill's analysis of knowledge is correct1 then ,i have nothing to say in excuse for my own deficiencies4 ,but it is well to do one thing at a time1 and therefore ,i have not occupied any considerable part of this book with controversy and refutation4 ,what ,i have to say of ,mill's logic will be said in a separate work1 in which his analysis of knowledge will be somewhat minutely analysed4 ,it will then be shown1 ,i believe1 that ,mill's psychological and philosophical treatment of logic has not yielded such satisfactory results as some writers seem to believe4#ah #ah ,portions of this work have already been published in my articles1 entitled 8,john ,stuart ,mill's ,philosophy ,tested10 printed in the 99,contemporary ,review99 for ,december1 #ahgg1 vol4 xxxi4 p4 #afg1 and for ,january and ,april1 #ahgh1 vol4 xxxi4 p4 #bef1 and vol4 xxxii4 p4 #hh4 7,note added in ,november1 #ahgh47 ,various minor but still important criticisms were made by ,professor ,robertson1 a few of which have been noticed in the text 7pp4 #bg1 #aja74 ,in other cases his objections hardly admit of any other answer than such as consists in asking the reader to judge between the work and the criticism4 ,thus ,mr4 ,robertson asserts#ai that the most complex logical problems solved in this book 7up to p4 #ajb of this edition7 might be more easily and shortly dealt with upon the principles and with the recognised methods of the traditional logic4 ,the burden of proof here lies upon ,mr4 ,robertson1 and his only proof consists in a single case1 where he is able1 as it seems to me accidentally1 to get a special conclusion by the old form of dilemma4 ,it would be a long labour to test the old logic upon every result obtained by my notation1 and ,i must leave such readers as are well acquainted with the syllogistic logic to pronounce upon the comparative simplicity and power of the new and old systems4 ,for other acute objections brought forward by ,mr4 ,robertson1 ,i must refer the reader to the article in question4 #ai 99,mind991 vol4 i4 p4 #bbb4 ,one point in my last chapter1 that on the ,results and ,limits of ,scientific ,method1 has been criticised by ,professor ,w4 ,k4 ,clifford in his lecture#bj on 8,the ,first and the ,last ,catastrophe40 ,in vol4 ii4 p4 #dch of the first edition 7p4 #gdd of this edition7 ,i referred to certain inferences drawn by eminent physicists as to a limit to the antiquity of the present order of things4 8,according to ,sir ,w4 ,thomson's deductions from ,fourier's 99theory of heat991 we can trace down the dissipation of heat by conduction and radiation to an infinitely distant time when all things will be uniformly cold4 ,but we cannot similarly trace the ,heat-history of the ,universe to an infinite distance in the past4 ,for a certain negative value of the time1 the formul give impossible values1 indicating that there was some initial distribution of heat which could not have resulted1 according to known laws of nature1 from any previous distribution40 #bj 99,fortnightly ,review991 ,new ,series1 ,april #ahge1 p4 #dhj4 ,lecture reprinted by the ,sunday ,lecture ,society1 p4 #bd4 ,now according to ,professor ,clifford ,i have here misstated ,thomson's results4 8,it is not according to the known laws of nature1 it is according to the known laws of conduction of heat1 that ,sir ,william ,thomson is speaking4444 ,all these physical writers1 knowing what they were writing about1 simply drew such conclusions from the facts which were before them as could be reasonably drawn4 ,they say1 here is a state of things which could not have been produced by the circumstances we are at present investigating4 ,then your speculator comes1 he reads a sentence and says1 #8,here is an opportunity for me to have my fling4' ,and he has his fling1 and makes a purely baseless theory about the necessary origin of the present order of nature at some definite point of time1 which might be calculated40 ,professor ,clifford proceeds to explain that ,thomson's formul only give a limit to the heat history of1 say1 the earth's crust in the solid state4 ,we are led back to the time when it became solidified from the fluid condition4 ,there is discontinuity in the history of the solid matter1 but still discontinuity which is within our comprehension4 ,still further back we should come to discontinuity again1 when the liquid was formed by the condensation of heated gaseous matter4 ,beyond that event1 however1 there is no need to suppose further discontinuity of law1 for the gaseous matter might consist of molecules which had been falling together from different parts of space through infinite past time4 ,as ,professor ,clifford says 7p4 #dha7 of the bodies of the universe1 8,what they have actually done is to fall together and get solid4 ,if we should reverse the process we should see them separating and getting cool1 and as a limit to that1 we should find that all these bodies would be resolved into molecules1 and all these would be flying away from each other4 ,there would be no limit to that process1 and we could trace it as far back as ever we liked to trace it40 ,assuming that ,i have erred1 ,i should like to point out that ,i have erred in the best company1 or more strictly1 being a speculator1 ,i have been led into error by the best physical writers4 ,professor ,tait1 in his 99,sketch of ,thermodynamics991 speaking of the laws discovered by ,fourier for the motion of heat in a solid1 says1 8,their mathematical expressions point also to the fact that a uniform distribution of heat1 or a distribution tending to become uniform1 must have arisen from some primitive distribution of heat of a kind not capable of being produced by known laws from any previous distribution40 ,in the latter words it will be seen that there is no limitation to the laws of conduction1 and1 although ,i had carefully referred to ,sir ,w4 ,thomson's original paper1 it is not unnatural that ,i should take ,professor ,tait's interpretation of its meaning4#ba #ba ,sir ,w4 ,thomson's words are as follows 799,cambridge ,mathematical ,journal991 ,nov4 #ahdb1 vol4 iii4 p4 #agd74 8,when 99x99 is negative1 the state represented cannot be the result of any 99possible99 distribution of temperature which has previously existed40 ,there is no limitation in the sentence to the laws of conduction1 but1 as the whole paper treats of the results of conduction in a solid1 it may no doubt be understood that there is a 99tacit99 limitation4 ,see also a second paper on the subject in the same journal for ,february1 #ahdd1 vol4 iv4 p4 #fg1 where again there is no expressed limitation4 ,in his new work 99,on some ,recent ,advances in ,physical ,science991 ,professor ,tait has recurred to the subject as follows3#bb 8,a profound lesson may be learned from one of the earliest little papers of ,sir ,w4 ,thomson1 published while he was an undergraduate at ,cambridge1 where he shows that ,fourier's magnificent treatment of the conduction of heat in a solid body leads to formul for its distribution which are intelligible 7and of course capable of being fully verified by experiment7 for all time future1 but which1 except in particular cases1 when extended to time past1 remain intelligible for a finite period only1 and 99then99 indicate a state of things which could not have resulted under known laws from any conceivable previous distribution of heat in the body4 ,so far as heat is concerned1 modern investigations have shown that a previous distribution of the 99matter99 involved may1 by its potential energy1 be capable of producing such a state of things at the moment of its aggregation2 but the example is now adduced not for its bearing on heat alone1 but as a simple illustration of the fact that all portions of our ,science1 especially that beautiful one1 the ,dissipation of ,energy1 point unanimously to a beginning1 to a state of things incapable of being derived by present laws of tangible matter and its energy from any conceivable previous arrangement40 ,as this was published nearly a year after ,professor ,clifford's lecture1 it may be inferred that ,professor ,tait adheres to his original opinion that the theory of heat does give evidence of 8a beginning40 #bb ,pp4 #be-bf4 ,the parentheses are in the original1 and show ,professor ,tait's corrections in the verbatim reports of his lectures4 ,the subject is treated again on pp4 #afh-i4 ,i may add that ,professor ,clerk ,maxwell's words seem to countenance the same view1 for he says1#bc 8,this is only one of the cases in which a consideration of the dissipation of energy leads to the determination of a superior limit to the antiquity of the observed order of things40 ,the expression 8observed order of things0 is open to much ambiguity1 but in the absence of qualification ,i should take it to include the aggregate of the laws of nature known to us4 ,i should interpret ,professor ,maxwell as meaning that the theory of heat indicates the occurrence of some event of which our science cannot give any further explanation4 ,the physical writers thus seem not to be so clear about the matter as ,professor ,clifford assumes4 #bc 99,theory of ,heat99 #ahga1 p4 #bde4 ,so far as ,i may venture to form an independent opinion on the subject1 it is to the effect that ,professor ,clifford is right1 and that the known laws of nature do not enable us to assign a 8beginning40 ,science leads us backwards into infinite past duration4 ,but that ,professor ,clifford is right on this point1 is no reason why we should suppose him to be right in his other opinions1 some of which ,i am sure are wrong4 ,nor is it a reason why other parts of my last chapter should be wrong4 ,the question only affects the single paragraph on pp4 #gdd-e of this book1 which might1 ,i believe1 be struck out without necessitating any alteration in the rest of the text4 ,it is always to be remembered that the failure of an argument in favour of a proposition does not1 generally speaking1 add much1 if any1 probability to the contradictory proposition4 ,i cannot conclude without expressing my acknowledgments to ,professor ,clifford for his kind expressions regarding my work as a whole4 #b1 ,the ,chestnuts1 ,west ,heath1 ,hampstead1 ,n4 ,w4 99,august #ae1 #ahgg499 ,,contents4 ,,book ,i4 ,,formal ,,logic1 ,,deductive ,,and ,,inductive4 ,,chapter ,i4 ,,introduction4 ,,section ,,page #a4 ,introduction #a #b4 ,the ,powers of ,mind concerned in the ,creation of ,science #d #c4 ,laws of ,identity and ,difference #e #d4 ,the ,nature of the ,laws of ,identity and ,difference #f #e4 ,the ,process of ,inference #i #f4 ,deduction and ,induction #aa #g4 ,symbolic ,expression of ,logical ,inference #ac #h4 ,expression of ,identity and ,difference #ad #i4 ,general ,formula of ,logical ,inference #ag #aj4 ,the ,propagating ,power of ,similarity #bj #aa4 ,anticipations of the ,principle of ,substitution #ba #ab4 ,the ,logic of ,relatives #bb ,,chapter ,,ii4 ,,terms4 #a4 ,terms #bd #b4 ,twofold meaning of ,general ,names #be #c4 ,abstract ,terms #bg #d4 ,substantial ,terms #bh #e4 ,collective ,terms #bi #f4 ,synthesis of ,terms #cj #g4 ,symbolic ,expression of the ,law of ,contradiction #ca #h4 ,certain ,special ,conditions of ,logical ,symbols #cb ,,chapter ,,iii4 ,,propositions4 #a4 ,propositions #cf #b4 ,simple ,identities #cg #c4 ,partial ,identities #dj #d4 ,limited ,identities #db #e4 ,negative ,propositions #dc #f4 ,conversion of ,propositions #df #g4 ,twofold ,interpretation of ,propositions #dg ,,chapter ,,iv4 ,,deductive ,,reasoning4 #a4 ,deductive ,reasoning #di #b4 ,immediate ,inference #ej #c4 ,inference with ,two ,simple ,identities #ea #d4 ,inference with a ,simple and a ,partial ,identity #ec #e4 ,inference of a ,partial from ,two ,partial ,identities #ee #f4 ,on the ,ellipsis of ,terms in ,partial ,identities #eg #g4 ,inference of a ,simple from ,two ,partial ,identities #eh #h4 ,inference of a ,limited from ,two ,partial ,identities #ei #i4 ,miscellaneous ,forms of ,deductive ,inference #fj #aj4 ,fallacies #fb ,,chapter ,v4 ,,disjunctive ,,propositions4 #a4 ,disjunctive ,propositions #ff #b4 ,expression of the ,alternative ,relation #fg #c4 ,nature of the ,alternative ,relation #fh #d4 ,laws of the ,disjunctive ,relation #ga #e4 ,symbolic ,expression of the ,law of ,duality #gc #f4 ,various ,forms of the ,disjunctive ,proposition #gd #g4 ,inference by ,disjunctive ,propositions #gf ,,chapter ,,vi4 ,,the ,,indirect ,,method ,,of ,,inference4 #a4 ,the ,indirect ,method of ,inference #ha #b4 ,simple ,illustrations #hc #c4 ,employment of the ,contrapositive ,proposition #hd #d4 ,contrapositive of a ,simple ,identity #hf #e4 ,miscellaneous ,examples of the ,method #hh #f4 ,mr4 ,venn's ,problem #ij #g4 ,abbreviation of the ,process #ia #h4 ,the ,logical ,alphabet #id #i4 ,the ,logical ,slate #ie #aj4 ,abstraction of ,indifferent ,circumstances #ig #aa4 ,illustrations of the ,indirect ,method #ih #ab4 ,second ,example #ii #ac4 ,third ,example #ajj #ad4 ,fourth ,example #aja #ae4 ,fifth ,example #aja #af4 ,fallacies ,analysed by the ,indirect ,method #ajb #ag4 ,the ,logical ,abacus #ajd #ah4 ,the ,logical ,machine #ajg #ai4 ,the ,order of ,premises #aad #bj4 ,the ,equivalence of ,propositions #aae #ba4 ,the ,nature of ,inference #aah ,,chapter ,,vii4 ,,induction4 #a4 ,induction #aba #b4 ,induction an ,inverse ,operation #abb #c4 ,inductive ,problems for ,solution by the ,reader #abf #d4 ,induction of ,simple ,identities #abg #e4 ,induction of ,partial ,identities #acj #f4 ,solution of the ,inverse or ,inductive ,problem1 involving ,two ,classes #acd #g4 ,the ,inverse ,logical ,problem1 involving ,three ,classes #acg #h4 ,professor ,clifford on the ,types of ,compound ,statement involving ,four ,classes #adc #i4 ,distinction between ,perfect and ,imperfect ,induction #adf #aj4 ,transition from ,perfect to ,imperfect ,induction #adi ,,book ,,ii4 ,,number1 ,,variety1 ,,and ,,probability4 ,,chapter ,,viii4 ,,principles ,,of ,,number4 #a4 ,principles of ,number #aec #b4 ,the ,nature of ,numbe #aef #c4 ,of ,numerical ,abstraction #aeh #d4 ,concrete and ,abstract ,number #aei #e4 ,analogy of ,logical and ,numerical ,terms #afj #f4 ,principle of ,mathematical ,inference #afb #g4 ,reasoning by ,inequalities #afe #h4 ,arithmetical ,reasoning #afg #i4 ,numerically ,definite ,reasoning #afh #aj4 ,numerical meaning of ,logical ,conditions #aga ,,chapter ,,ix4 ,,the ,,variety ,,of ,,nature1 ,,or ,,the ,,doctrine ,,of ,,combinations ,,and ,,permutations4 #a4 ,the ,variety of ,nature #agc #b4 ,distinction of ,combinations and ,permutations #agg #c4 ,calculation of ,number of ,combinations #ahj #d4 ,the ,arithmetical ,triangle #ahb #e4 ,connexion between the ,arithmetical ,triangle and the ,logical ,alphabet #ahi #f4 ,possible ,variety of ,nature and ,art #aij #g4 ,higher ,orders of ,variety #aib ,,chapter ,x4 ,,theory ,,of ,,probability4 #a4 ,theory of ,probability #aig #b4 ,fundamental ,principles of the ,theory #bjj #c4 ,rules for the ,calculation of ,probabilities #bjc #d4 ,the ,logical ,alphabet in questions of ,probability #bje #e4 ,comparison of the ,theory with ,experience #bjf #f4 ,probable ,deductive ,arguments #bji #g4 ,difficulties of the ,theory #bac ,,chapter ,,xi4 ,,philosophy ,,of ,,inductive ,,inference4 #a4 ,philosophy of ,inductive ,inference #bah #b4 ,various ,classes of ,inductive ,truths #bai #c4 ,the ,relation of ,cause and ,effect #bbj #d4 ,fallacious ,use of the ,term ,cause #bba #e4 ,confusion of ,two ,questions #bbb #f4 ,definition of the ,term ,cause #bbd #g4 ,distinction of ,inductive and ,deductive ,results #bbf #h4 ,the ,grounds of ,inductive ,inference #bbh #i4 ,illustrations of the ,inductive ,process #bbi #aj4 ,geometrical ,reasoning #bcc #aa4 ,discrimination of ,certainty and ,probability #bce ,,chapter ,,xii4 ,,the ,,inductive ,,or ,,inverse ,,application ,,of ,,the ,,theory ,,of ,,probability4 #a4 ,the ,inductive or ,inverse ,application of the ,theory #bdj #b4 ,principle of the ,inverse ,method #bdb #c4 ,simple ,applications of the ,inverse ,method #bdd #d4 ,the ,theory of ,probability in ,astronomy #bdg #e4 ,the ,general ,inverse ,problem #bej #f4 ,simple ,illustration of the ,inverse ,problem #bec #g4 ,general ,solution of the ,inverse ,problem #bee #h4 ,rules of the ,inverse ,method #beg #i4 ,fortuitous ,coincidences #bfa #aj4 ,summary of the ,theory of ,inductive ,inference #bfe ,,book ,,iii4 ,,methods ,,of ,,measurement4 ,,chapter ,,xiii4 ,,the ,,exact ,,measurement ,,of ,,phenomena4 #a4 ,the ,exact ,measurement of ,phenomena #bgj #b4 ,division of the ,subject #bgd #c4 ,continuous quantity #bgd #d4 ,the ,fallacious ,indications of the ,senses #bgf #e4 ,complexity of ,quantitative ,questions #bgh #f4 ,the ,methods of ,accurate ,measurement #bhb #g4 ,conditions of ,accurate ,measurement #bhb #h4 ,measuring ,instruments #bhd #i4 ,the ,method of ,repetition #bhh #aj4 ,measurements by ,natural ,coincidence #bib #aa4 ,modes of ,indirect ,measurement #bif #ab4 ,comparative ,use of ,measuring ,instruments #bii #ac4 ,systematic ,performance of ,measurements #cjj #ad4 ,the ,pendulum #cjb #ae4 ,attainable ,accuracy of ,measurement #cjc ,,chapter ,,xiv4 ,,units ,,and ,,standards ,,of ,,measurement4 #a4 ,units and ,standards of ,measurement #cje #b4 ,standard ,unit of ,time #cjg #c4 ,the ,unit of ,space and the ,bar ,standard #cab #d4 ,the ,terrestrial ,standard #cad #e4 ,the ,pendulum ,standard #cae #f4 ,unit of ,density #caf #g4 ,unit of ,mass #cag #h4 ,natural ,system of ,standards #cai #i4 ,subsidiary ,units #cbj #aj4 ,derived ,units #cba #aa4 ,provisional ,units #cbc #ab4 ,theory of ,dimensions #cbe #ac4 ,natural ,constants #cbh #ad4 ,mathematical ,constants #ccj #ae4 ,physical ,constants #cca #af4 ,astronomical ,constants #ccb #ag4 ,terrestrial ,numbers #ccc #ah4 ,organic ,numbers #ccc #ai4 ,social ,numbers #ccd ,,chapter ,,xv4 ,,analysis ,,of ,,quantitative ,,phenomena4 #a4 ,analysis of ,quantitative ,phenomena #cce #b4 ,illustrations of the ,complication of ,effects #ccf #c4 ,methods of ,eliminating ,error #cci #d4 ,method of ,avoidance of ,error #cdj #e4 ,differential ,method #cdd #f4 ,method of ,correction #cdf #g4 ,method of ,compensation #cej #h4 ,method of ,reversal #ced ,,chapter ,,xvi4 ,,the ,,method ,,of ,,means4 #a4 ,the ,method of ,means #ceg #b4 ,several ,uses of the ,mean ,result #cei #c4 ,the ,mean and the ,average #cfj #d4 ,on the ,average or ,fictitious ,mean #cfc #e4 ,the ,precise ,mean ,result #cfe #f4 ,determination of the ,zero ,point #cfh #g4 ,determination of ,maximum ,points #cga ,,chapter ,,xvii4 ,,the ,,law ,,of ,,error4 #a4 ,the ,law of ,error #cgd #b4 ,establishment of the ,law of ,error #cge #c4 ,herschel's ,geometrical ,proof #cgg #d4 ,laplace's and ,quetelet's ,proof of the ,law #cgh #e4 ,logical ,origin of the ,law of ,error #chc #f4 ,verification of the ,law of ,error #chc #g4 ,the ,probable ,mean ,result #che #h4 ,the ,probable ,error of ,results #chf #i4 ,rejection of the ,mean ,result #chi #aj4 ,method of ,least ,squares #cic #aa4 ,works upon the ,theory of ,probability #cid #ab4 ,detection of ,constant ,errors #cif ,,book ,,iv4 ,,inductive ,,investigation4 ,,chapter ,,xviii4 ,,observation4 #a4 ,observation #cii #b4 ,distinction of ,observation and ,experiment #djj #c4 ,mental ,conditions of ,correct ,observation #djb #d4 ,instrumental and ,sensual ,conditions of ,correct ,observation #djd #e4 ,external ,conditions of ,correct ,observation #djg #f4 ,apparent ,sequence of ,events #dji #g4 ,negative ,arguments from ,non-,observation #daa ,,chapter ,,xix4 ,,experiment4 #a4 ,experiment #daf #b4 ,exclusion of ,indifferent ,circumstances #dai #c4 ,simplification of ,experiments #dbb #d4 ,failure in the ,simplification of ,experiments #dbd #e4 ,removal of ,usual ,conditions #dbf #f4 ,interference of ,unsuspected ,conditions #dbh #g4 ,blind or ,test ,experiments #dcc #h4 ,negative ,results of ,experiment #dcd #i4 ,limits of ,experiment #dcg ,,chapter ,,xx4 ,,method ,,of ,,variations4 #a4 ,method of ,variations #dci #b4 ,the ,variable and the ,variant #ddj #c4 ,measurement of the ,variable #dda #d4 ,maintenance of ,similar ,conditions #ddc #e4 ,collective ,experiments #dde #f4 ,periodic ,variations #ddg #g4 ,combined ,periodic ,changes #dej #h4 ,principle of ,forced ,vibrations #dea #i4 ,integrated ,variations #deb ,,chapter ,,xxi4 ,,theory ,,of ,,approximation4 #a4 ,theory of ,approximation #def #b4 ,substitution of ,simple ,hypotheses #deh #c4 ,approximation to ,exact ,laws #dfb #d4 ,successive ,approximations to ,natural ,conditions #dfe #e4 ,discovery of ,hypothetically ,simple ,laws #dgj #f4 ,mathematical ,principles of ,approximation #dga #g4 ,approximate ,independence of ,small ,effects #dge #h4 ,four ,meanings of ,equality #dgi #i4 ,arithmetic of ,approximate ,quantities #dha ,,chapter ,,xxii4 ,,quantitative ,,induction4 #a4 ,quantitative ,induction #dhc #b4 ,probable ,connexion of ,varying ,quantities #dhd #c4 ,empirical ,mathematical ,laws #dhg #d4 ,discovery of ,rational ,formul #dhi #e4 ,the ,graphical ,method #dib #f4 ,interpolation and ,extrapolation #die #g4 ,illustrations of ,empirical ,quantitative ,laws #dii #h4 ,simple ,proportional ,variation #eja ,,chapter ,,xxiii4 ,,the ,,use ,,of ,,hypothesis4 #a4 ,the ,use of ,hypothesis #ejd #b4 ,requisites of a good ,hypothesis #eaj #c4 ,possibility of ,deductive ,reasoning #eaa #d4 ,consistency with the ,laws of ,nature #ead #e4 ,conformity with ,facts #eaf #f4 ,experimentum ,crucis #eah #g4 ,descriptive ,hypotheses #ebb ,,chapter ,,xxiv4 ,,empirical ,,knowledge1 ,,explanation ,,and ,,prediction4 #a4 ,empirical ,knowledge1 ,explanation and ,prediction #ebe #b4 ,empirical ,knowledge #ebf #c4 ,accidental ,discovery #ebi #d4 ,empirical ,observations subsequently ,explained #ecb #e4 ,overlooked ,results of ,theory #ecd #f4 ,predicted ,discoveries #ecf #g4 ,predictions in the ,science of ,light #ech #h4 ,predictions from the ,theory of ,undulations #edj #i4 ,prediction in other ,sciences #edb #aj4 ,prediction by ,inversion of ,cause and ,effect #ede #aa4 ,facts known only by ,theory #edg ,,chapter ,,xxv4 ,,accordance ,,of ,,quantitative ,,theories4 #a4 ,accordance of ,quantitative ,theories #eea #b4 ,empirical ,measurements #eeb #c4 ,quantities indicated by ,theory1 but ,empirically ,measured #eec #d4 ,explained ,results of ,measurement #eed #e4 ,quantities determined by ,theory and verified by ,measurement #eee #f4 ,quantities determined by ,theory and not verified #eef #g4 ,discordance of ,theory and ,experiment #eeh #h4 ,accordance of ,measurements of ,astronomical ,distances #efj #i4 ,selection of the best ,mode of ,measurement #efc #aj4 ,agreement of ,distinct ,modes of ,measurement #efd #aa4 ,residual ,phenomena #efi ,,chapter ,,xxvi4 ,,character ,,of ,,the ,,experimentalist4 #a4 ,character of the ,experimentalist #egd #b4 ,error of the ,baconian ,method #egf #c4 ,freedom of ,theorising #egg #d4 ,the ,newtonian ,method1 the ,true ,organum #eha #e4 ,candour and ,courage of the ,philosophic ,mind #ehf #f4 ,the ,philosophic ,character of ,faraday #ehg #g4 ,reservation of ,judgment #eib ,,book ,v4 ,,generalisation1 ,,analogy1 ,,and ,,classification4 ,,chapter ,,xxvii4 ,,generalisation4 #a4 ,generalisation #eid #b4 ,distinction of ,generalisation and ,analogy #eif #c4 ,two ,meanings of ,generalisation #eig #d4 ,value of ,generalisation #eii #e4 ,comparative ,generality of ,properties #fjj #f4 ,uniform ,properties of all ,matter #fjc #g4 ,variable ,properties of ,matter #fjf #h4 ,extreme ,instances of ,properties #fjg #i4 ,the ,detection of ,continuity #faj #aj4 ,the ,law of ,continuity #fae #aa4 ,failure of the ,law of ,continuity #fai #ab4 ,negative ,arguments on the ,principle of ,continuity #fba #ac4 ,tendency to ,hasty ,generalisation #fbc ,,chapter ,,xxviii4 ,,analogy4 #a4 ,analogy #fbg #b4 ,analogy as a ,guide in ,discovery #fbi #c4 ,analogy in the ,mathematical ,sciences #fca #d4 ,analogy in the ,theory of ,undulations #fce #e4 ,analogy in ,astronomy #fch #f4 ,failures of ,analogy #fda ,,chapter ,,xxix4 ,,exceptional ,,phenomena4 #a4 ,exceptional ,phenomena #fdd #b4 ,imaginary or ,false ,exceptions #fdg #c4 ,apparent but ,congruent ,exceptions #fdi #d4 ,singular ,exceptions #feb #e4 ,divergent ,exceptions #fee #f4 ,accidental ,exceptions #feh #g4 ,novel and ,unexplained ,exceptions #ffa #h4 ,limiting ,exceptions #ffc #i4 ,real ,exceptions to ,supposed ,laws #fff #aj4 ,unclassed ,exceptions #ffh ,,chapter ,,xxx4 ,,classification4 #a4 ,classification #fgc #b4 ,classification involving ,induction #fge #c4 ,multiplicity of ,modes of ,classification #fgg #d4 ,natural and ,artificial ,systems of ,classification #fgi #e4 ,correlation of ,properties #fha #f4 ,classification in ,crystallography #fhe #g4 ,classification an ,inverse and ,tentative ,operation #fhi #h4 ,symbolic ,statement of the ,theory of ,classification #fib #i4 ,bifurcate ,classification #fid #aj4 ,the ,five ,predicates #fih #aa4 ,summum ,genus and ,infima ,species #gja #ab4 ,the ,tree of ,porphyry #gjb #ac4 ,does ,abstraction imply ,generalisation8 #gjd #ad4 ,discovery of ,marks or ,characteristics #gjh #ae4 ,diagnostic ,systems of ,classification #gaj #af4 ,index ,classifications #gad #ag4 ,classification in the ,biological ,sciences #gah #ah4 ,classification by ,types #gbb #ai4 ,natural ,genera and ,species #gbd #bj4 ,unique or ,exceptional ,objects #gbh #ba4 ,limits of ,classification #gcj ,,book ,,vi4 ,,chapter ,,xxxi4 ,,reflections ,,on ,,the ,,results ,,and ,,limits ,,of ,,scientific ,,method4 #a4 ,reflections on the ,results and ,limits of ,scientific ,method #gce #b4 ,the ,meaning of ,natural ,law #gcg #c4 ,infiniteness of the ,universe #gch #d4 ,the ,indeterminate ,problem of ,creation #gdj #e4 ,hierarchy of ,natural ,laws #gdb #f4 ,the ,ambiguous ,expression--8,uniformity of ,nature0 #gde #g4 ,possible ,states of the ,universe #gdi #h4 ,speculations on the ,reconcentration of ,energy #gea #i4 ,the ,divergent ,scope for ,new ,discovery #geb #aj4 ,infinite ,incompleteness of the ,mathematical ,sciences #ged #aa4 ,the ,reign of ,law in ,mental and ,social ,phenomena #gei #ab4 ,the ,theory of ,evolution #gfa #ac4 ,possibility of ,divine ,interference #gfe #ad4 ,conclusion #gff ,,index #ggc ,,the ,,principles ,,of ,,science4 ,,chapter ,i4 ,,introduction4 ,science arises from the discovery of ,identity amidst ,diversity4 ,the process may be described in different words1 but our language must always imply the presence of one common and necessary element4 ,in every act of inference or scientific method we are engaged about a certain identity1 sameness1 similarity1 likeness1 resemblance1 analogy1 equivalence or equality apparent between two objects4 ,it is doubtful whether an entirely isolated phenomenon could present itself to our notice1 since there must always be some points of similarity between object and object4 ,but in any case an isolated phenomenon could be studied to no useful purpose4 ,the whole value of science consists in the power which it confers upon us of applying to one object the knowledge acquired from like objects2 and it is only so far1 therefore1 as we can discover and register resemblances that we can turn our observations to account4 ,nature is a spectacle continually exhibited to our senses1 in which phenomena are mingled in combinations of endless variety and novelty4 ,wonder fixes the mind's attention2 memory stores up a record of each distinct impression2 the powers of association bring forth the record when the like is felt again4 ,by the higher faculties of judgment and reasoning the mind compares the new with the old1 recognises essential identity1 even when disguised by diverse circumstances1 and expects to find again what was before experienced4 ,it must be the ground of all reasoning and inference that 99what is true of one thing will be true of its equivalent991 and that under carefully ascertained conditions 99,nature repeats herself994 ,were this indeed a ,chaotic ,universe1 the powers of mind employed in science would be useless to us4 ,did ,chance wholly take the place of order1 and did all phenomena come out of an 99,infinite ,lottery991 to use ,condorcet's expression1 there could be no reason to expect the like result in like circumstances4 ,it is possible to conceive a world in which no two things should be associated more often1 in the long run1 than any other two things4 ,the frequent conjunction of any two events would then be purely fortuitous1 and if we expected conjunctions to recur continually1 we should be disappointed4 ,in such a world we might recognise the same kind of phenomenon as it appeared from time to time1 just as we might recognise a marked ball as it was occasionally drawn and re-drawn from a ballot-box2 but the approach of any phenomenon would be in no way indicated by what had gone before1 nor would it be a sign of what was to come after4 ,in such a world knowledge would be no more than the memory of past coincidences1 and the reasoning powers1 if they existed at all1 would give no clue to the nature of the present1 and no presage of the future4 ,happily the ,universe in which we dwell is not the result of chance1 and where chance seems to work it is our own deficient faculties which prevent us from recognising the operation of ,law and of ,design4 ,in the material framework of this world1 substances and forces present themselves in definite and stable combinations4 ,things are not in perpetual flux1 as ancient philosophers held4 ,element remains element2 iron changes not into gold4 ,with suitable precautions we can calculate upon finding the same thing again endowed with the same properties4 ,the constituents of the globe1 indeed1 appear in almost endless combinations2 but each combination bears its fixed character1 and when resolved is found to be the compound of definite substances4 ,misapprehensions must continually occur1 owing to the limited extent of our experience4 ,we can never have examined and registered possible existences so thoroughly as to be sure that no new ones will occur and frustrate our calculations4 ,the same outward appearances may cover any amount of hidden differences which we have not yet suspected4 ,to the variety of substances and powers diffused through nature at its creation1 we should not suppose that our brief experience can assign a limit1 and the necessary imperfection of our knowledge must be ever borne in mind4 ,yet there is much to give us confidence in ,science4 ,the wider our experience1 the more minute our examination of the globe1 the greater the accumulation of well-reasoned knowledge1--the fewer in all probability will be the failures of inference compared with the successes4 ,exceptions to the prevalence of ,law are gradually reduced to ,law themselves4 ,certain deep similarities have been detected among the objects around us1 and have never yet been found wanting4 ,as the means of examining distant parts of the universe have been acquired1 those similarities have been traced there as here4 ,other worlds and stellar systems may be almost incomprehensively different from ours in magnitude1 condition and disposition of parts1 and yet we detect there the same elements of which our own limbs are composed4 ,the same natural laws can be detected in operation in every part of the universe within the scope of our instruments2 and doubtless these laws are obeyed irrespective of distance1 time1 and circumstance4 ,it is the prerogative of ,intellect to discover what is uniform and unchanging in the phenomena around us4 ,so far as object is different from object1 knowledge is useless and inference impossible4 ,but so far as object resembles object1 we can pass from one to the other4 ,in proportion as resemblance is deeper and more general1 the commanding powers of knowledge become more wonderful4 ,identity in one or other of its phases is thus always the bridge by which we pass in inference from case to case2 and it is my purpose in this treatise to trace out the various forms in which the one same process of reasoning presents itself in the ever-growing achievements of ,scientific ,method4 99,the ,powers of ,mind concerned in the ,creation of ,science499 ,it is no part of the purpose of this work to investigate the nature of mind4 ,people not uncommonly suppose that logic is a branch of psychology1 because reasoning is a mental operation4 ,on the same ground1 however1 we might argue that all the sciences are branches of psychology4 ,as will be further explained1 ,i adopt the opinion of ,mr4 ,herbert ,spencer1 that logic is really an objective science1 like mathematics or mechanics4 ,only in an incidental manner1 then1 need ,i point out that the mental powers employed in the acquisition of knowledge are probably three in number4 ,they are substantially as ,professor ,bain has stated them#bd3-- #bd 99,the ,senses and the ,intellect991 ,second ,ed41 pp4 #e1 #cbe1 c4 #a4 ,the ,power of ,discrimination4 #b4 ,the ,power of ,detecting ,identity4 #c4 ,the ,power of ,retention4 ,we exert the first power in every act of perception4 ,hardly can we have a sensation or feeling unless we discriminate it from something else which preceded4 ,consciousness would almost seem to consist in the break between one state of mind and the next1 just as an induced current of electricity arises from the beginning or the ending of the primary current4 ,we are always engaged in discrimination2 and the rudiment of thought which exists in the lower animals probably consists in their power of feeling difference and being agitated by it4 ,yet had we the power of discrimination only1 ,science could not be created4 ,to know that one feeling differs from another gives purely negative information4 ,it cannot teach us what will happen4 ,in such a state of intellect each sensation would stand out distinct from every other2 there would be no tie1 no bridge of affinity between them4 ,we want a unifying power by which the present and the future may be linked to the past2 and this seems to be accomplished by a different power of mind4 ,lord ,bacon has pointed out that different men possess in very different degrees the powers of discrimination and identification4 ,it may be said indeed that discrimination necessarily implies the action of the opposite process of identification2 and so it doubtless does in negative points4 ,but there is a rare property of mind which consists in penetrating the disguise of variety and seizing the common elements of sameness2 and it is this property which furnishes the true measure of intellect4 ,the name of 8intellect0 expresses the interlacing of the general and the single1 which is the peculiar province of mind4#be ,to 99cogitate99 is the ,latin 99coagitare991 resting on a like metaphor4 ,logic1 also1 is but another name for the same process1 the peculiar work of reason2 for is derived from 1 which like the ,latin 99legere99 meant originally to gather4 ,plato said of this unifying power1 that if he met the man who could detect 99the one in the many991 he would follow him as a god4 #be ,max ,mller1 99,lectures on the ,science of ,language991 ,second ,series1 vol4 ii4 p4 #fc2 or ,sixth ,edition1 vol4 ii4 p4 #fg4 ,the view of the etymological meaning of 8intellect0 is given above on the authority of ,professor ,max ,mller4 ,it seems to be opposed to the ordinary opinion1 according to which the ,latin 99intelligere99 means to choose between1 to see a difference between1 to discriminate1 instead of to unite4 99,laws of ,identity and ,difference499 ,at the base of all thought and science must lie the laws which express the very nature and conditions of the discriminating and identifying powers of mind4 ,these are the so-called ,fundamental ,laws of ,thought1 usually stated as follows3-- #a4 ,the ,law of ,identity4 99,whatever is1 is499 #b4 ,the ,law of ,contradiction4 99,a thing cannot both be and not be499 #c4 ,the ,law of ,duality4 99,a thing must either be or not be499 ,the first of these statements may perhaps be regarded as a description of identity itself1 if so fundamental a notion can admit of description4 ,a thing at any moment is perfectly identical with itself1 and1 if any person were unaware of the meaning of the word 8identity10 we could not better describe it than by such an example4 ,the second law points out that contradictory attributes can never be joined together4 ,the same object may vary in its different parts2 here it may be black1 and there white2 at one time it may be hard and at another time soft2 but at the same time and place an attribute cannot be both present and absent4 ,aristotle truly described this law as the first of all axioms--one of which we need not seek for any demonstration4 ,all truths cannot be proved1 otherwise there would be an endless chain of demonstration2 and it is in self-evident truths like this that we find the simplest foundations4 ,the third of these laws completes the other two4 ,it asserts that at every step there are two possible alternatives--presence or absence1 affirmation or negation4 ,hence ,i propose to name this law the ,law of ,duality1 for it gives to all the formul of reasoning a dual character4 ,it asserts also that between presence and absence1 existence and non-existence1 affirmation and negation1 there is no third alternative4 ,as ,aristotle said1 there can be no mean between opposite assertions3 we must either affirm or deny4 ,hence the inconvenient name by which it has been known--,the ,law of ,excluded ,middle4 ,it may be allowed that these laws are not three independent and distinct laws2 they rather express three different aspects of the same truth1 and each law doubtless presupposes and implies the other two4 ,but it has not hitherto been found possible to state these characters of identity and difference in less than the threefold formula4 ,the reader may perhaps desire some information as to the mode in which these laws have been stated1 or the way in which they have been regarded1 by philosophers in different ages of the world4 ,abundant information on this and many other points of logical history will be found in ,ueberweg's 99,system of ,logic991 of which an excellent translation has been published by ,professor ,t4 ,m4 ,lindsay 7see pp4 #bbh-bha74 99,the ,nature of the ,laws of ,identity and ,difference499 ,i must at least allude to the profoundly difficult question concerning the nature and authority of these ,laws of ,identity and ,difference4 ,are they ,laws of ,thought or ,laws of ,things8 ,do they belong to mind or to material nature8 ,on the one hand it may be said that science is a purely mental existence1 and must therefore conform to the laws of that which formed it4 ,science is in the mind and not in the things1 and the properties of mind are therefore all important4 ,it is true that these laws are verified in the observation of the exterior world2 and it would seem that they might have been gathered and proved by generalisation1 had they not already been in our possession4 ,but on the other hand1 it may well be urged that we cannot prove these laws by any process of reasoning or observation1 because the laws themselves are presupposed1 as ,leibnitz acutely remarked1 in the very notion of a proof4 ,they are the prior conditions of all thought and all knowledge1 and even to question their truth is to allow them true4 ,hartley ingeniously refined upon this argument1 remarking that if the fundamental laws of logic be not certain1 there must exist a logic of a second order whereby we may determine the degree of uncertainty3 if the second logic be not certain1 there must be a third2 and so on 99ad infinitum994 ,thus we must suppose either that absolutely certain laws of thought exist1 or that there is no such thing as certainty whatever4#bf #bf ,hartley on ,man1 vol4 i4 p4 #cei4 ,logicians1 indeed1 appear to me to have paid insufficient attention to the fact that mistakes in reasoning are always possible1 and of not unfrequent occurrence4 ,the ,laws of ,thought are often called necessary laws1 that is1 laws which cannot but be obeyed4 ,yet as a matter of fact1 who is there that does not often fail to obey them8 ,they are the laws which the mind ought to obey rather than what it always does obey4 ,our thoughts cannot be the criterion of truth1 for we often have to acknowledge mistakes in arguments of moderate complexity1 and we sometimes only discover our mistakes by collision between our expectations and the events of objective nature4 ,mr4 ,herbert ,spencer holds that the laws of logic are objective laws1#bg and he regards the mind as being in a state of constant education1 each act of false reasoning or miscalculation leading to results which are likely to prevent similar mistakes from being again committed4 ,i am quite inclined to accept such ingenious views2 but at the same time it is necessary to distinguish between the accumulation of knowledge1 and the constitution of the mind which allows of the acquisition of knowledge4 ,before the mind can perceive or reason at all it must have the conditions of thought impressed upon it4 ,before a mistake can be committed1 the mind must clearly distinguish the mistaken conclusion from all other assertions4 ,are not the ,laws of ,identity and ,difference the prior conditions of all consciousness and all existence8 ,must they not hold true1 alike of things material and immaterial8 and if so1 can we say that they are only subjectively true or objectively true8 ,i am inclined1 in short1 to regard them as true both 8in the nature of thought and things10 as ,i expressed it in my first logical essay2#bh and ,i hold that they belong to the common basis of all existence4 ,but this is one of the most difficult questions of psychology and metaphysics which can be raised1 and it is hardly one for the logician to decide4 ,as the mathematician does not inquire into the nature of unity and plurality1 but develops the formal laws of plurality1 so the logician1 as ,i conceive1 must assume the truth of the ,laws of ,identity and ,difference1 and occupy himself in developing the variety of forms of reasoning in which their truth may be manifested4 #bg 99,principles of ,psychology991 ,second ,ed41 vol4 ii4 p4 #hf4 #bh 99,pure ,logic1 or the ,logic of ,quality apart from ,quantity991 #ahfd1 pp4 #aj1 #af1 #bb1 #bi1 #cf1 c4 ,again1 ,i need hardly dwell upon the question whether logic treats of language1 notions1 or things4 ,as reasonably might we debate whether a mathematician treats of symbols1 quantities1 or things4 ,a mathematician certainly does treat of symbols1 but only as the instruments whereby to facilitate his reasoning concerning quantities2 and as the axioms and rules of mathematical science must be verified in concrete objects in order that the calculations founded upon them may have any validity or utility1 it follows that the ultimate objects of mathematical science are the things themselves4 ,in like manner ,i conceive that the logician treats of language so far as it is essential for the embodiment and exhibition of thought4 ,even if reasoning can take place in the inner consciousness of man without the use of any signs1 which is doubtful1 at any rate it cannot become the subject of discussion until by some system of material signs it is manifested to other persons4 ,the logician then uses words and symbols as instruments of reasoning1 and leaves the nature and peculiarities of language to the grammarian4 ,but signs again must correspond to the thoughts and things expressed1 in order that they shall serve their intended purpose4 ,we may therefore say that logic treats ultimately of thoughts and things1 and immediately of the signs which stand for them4 ,signs1 thoughts1 and exterior objects may be regarded as parallel and analogous series of phenomena1 and to treat any one of the three series is equivalent to treating either of the other series4 99,the ,process of ,inference499 ,the fundamental action of our reasoning faculties consists in inferring or carrying to a new instance of a phenomenon whatever we have previously known of its like1 analogue1 equivalent or equal4 ,sameness or identity presents itself in all degrees1 and is known under various names2 but the great rule of inference embraces all degrees1 and affirms that 99so far as there exists sameness1 identity or likeness1 what is true of one thing will be true of the other994 ,the great difficulty doubtless consists in ascertaining that there does exist a sufficient degree of likeness or sameness to warrant an intended inference2 and it will be our main task to investigate the conditions under which reasoning is valid4 ,in this place ,i wish to point out that there is something common to all acts of inference1 however different their apparent forms4 ,the one same rule lends itself to the most diverse applications4 ,the simplest possible case of inference1 perhaps1 occurs in the use of a 99pattern991 99example991 or1 as it is commonly called1 a 99sample994 ,to prove the exact similarity of two portions of commodity1 we need not bring one portion beside the other4 ,it is sufficient that we take a sample which exactly represents the texture1 appearance1 and general nature of one portion1 and according as this sample agrees or not with the other1 so will the two portions of commodity agree or differ4 ,whatever is true as regards the colour1 texture1 density1 material of the sample will be true of the goods themselves4 ,in such cases likeness of quality is the condition of inference4 ,exactly the same mode of reasoning holds true of magnitude and figure4 ,to compare the sizes of two objects1 we need not lay them beside each other4 ,a staff1 string1 or other kind of measure may be employed to represent the length of one object1 and according as it agrees or not with the other1 so must the two objects agree or differ4 ,in this case the proxy or sample represents length2 but the fact that lengths can be added and multiplied renders it unnecessary that the proxy should always be as large as the object4 ,any standard of convenient size1 such as a common foot-rule1 may be made the medium of comparison4 ,the height of a church in one town may be carried to that in another1 and objects existing immovably at opposite sides of the earth may be vicariously measured against each other4 ,we obviously employ the axiom that whatever is true of a thing as regards its length1 is true of its equal4 ,to every other simple phenomenon in nature the same principle of substitution is applicable4 ,we may compare weights1 densities1 degrees of hardness1 and degrees of all other qualities1 in like manner4 ,to ascertain whether two sounds are in unison we need not compare them directly1 but a third sound may be the go-between4 ,if a tuning-fork is in unison with the middle ,c of ,york ,minster organ1 and we afterwards find it to be in unison with the same note of the organ in ,westminster ,abbey1 then it follows that the two organs are tuned in unison4 ,the rule of inference now is1 that what is true of the tuning-fork as regards the tone or pitch of its sound1 is true of any sound in unison with it4 ,the skilful employment of this substitutive process enables us to make measurements beyond the powers of our senses4 ,no one can count the vibrations1 for instance1 of an organ-pipe4 ,but we can construct an instrument called the 99siren991 so that1 while producing a sound of any pitch1 it shall register the number of vibrations constituting the sound4 ,adjusting the sound of the siren in unison with an organ-pipe1 we measure indirectly the number of vibrations belonging to a sound of that pitch4 ,to measure a sound of the same pitch is as good as to measure the sound itself4 ,sir ,david ,brewster1 in a somewhat similar manner1 succeeded in measuring the refractive indices of irregular fragments of transparent minerals4 ,it was a troublesome1 and sometimes impracticable work to grind the minerals into prisms1 so that the power of refracting light could be directly observed2 but he fell upon the ingenious device of compounding a liquid possessing the same refractive power as the transparent fragment under examination4 ,the moment when this equality was attained could be known by the fragments ceasing to reflect or refract light when immersed in the liquid1 so that they became almost invisible in it4 ,the refractive power of the liquid being then measured gave that of the solid4 ,a more beautiful instance of representative measurement1 depending immediately upon the principle of inference1 could not be found4#bi #bi ,brewster1 99,treatise on ,new ,philosophical ,instruments991 p4 #bgc4 ,concerning this method see also ,whewell1 99,philosophy of the ,inductive ,sciences991 vol4 ii4 p4 #cee2 ,tomlinson1 99,philosophical ,magazine991 ,fourth ,series1 vol4 xl4 p4 #cbh2 ,tyndall1 in ,youmans' 99,modern ,culture991 p4 #af4 ,throughout the various logical processes which we are about to consider--,deduction1 ,induction1 ,generalisation1 ,analogy1 ,classification1 ,quantitative ,reasoning--we shall find the one same principle operating in a more or less disguised form4 99,deduction and ,induction499 ,the processes of inference always depend on the one same principle of substitution2 but they may nevertheless be distinguished according as the results are inductive or deductive4 ,as generally stated1 deduction consists in passing from more general to less general truths2 induction is the contrary process from less to more general truths4 ,we may however describe the difference in another manner4 ,in deduction we are engaged in developing the consequences of a law4 ,we learn the meaning1 contents1 results or inferences1 which attach to any given proposition4 ,induction is the exactly inverse process4 ,given certain results or consequences1 we are required to discover the general law from which they flow4 ,in a certain sense all knowledge is inductive4 ,we can only learn the laws and relations of things in nature by observing those things4 ,but the knowledge gained from the senses is knowledge only of particular facts1 and we require some process of reasoning by which we may collect out of the facts the laws obeyed by them4 ,experience gives us the materials of knowledge3 induction digests those materials1 and yields us general knowledge4 ,when we possess such knowledge1 in the form of general propositions and natural laws1 we can usefully apply the reverse process of deduction to ascertain the exact information required at any moment4 ,in its ultimate foundation1 then1 all knowledge is inductive--in the sense that it is derived by a certain inductive reasoning from the facts of experience4 ,it is nevertheless true1--and this is a point to which insufficient attention has been paid1 that all reasoning is founded on the principles of deduction4 ,i call in question the existence of any method of reasoning which can be carried on without a knowledge of deductive processes4 ,i shall endeavour to show that 99induction is really the inverse process of deduction994 ,there is no mode of ascertaining the laws which are obeyed in certain phenomena1 unless we have the power of determining what results would follow from a given law4 ,just as the process of division necessitates a prior knowledge of multiplication1 or the integral calculus rests upon the observation and remembrance of the results of the differential calculus1 so induction requires a prior knowledge of deduction4 ,an inverse process is the undoing of the direct process4 ,a person who enters a maze must either trust to chance to lead him out again1 or he must carefully notice the road by which he entered4 ,the facts furnished to us by experience are a maze of particular results2 we might by chance observe in them the fulfilment of a law1 but this is scarcely possible1 unless we thoroughly learn the effects which would attach to any particular law4 ,accordingly1 the importance of deductive reasoning is doubly supreme4 ,even when we gain the results of induction they would be of no use unless we could deductively apply them4 ,but before we can gain them at all we must understand deduction1 since it is the inversion of deduction which constitutes induction4 ,our first task in this work1 then1 must be to trace out fully the nature of identity in all its forms of occurrence4 ,having given any series of propositions we must be prepared to develop deductively the whole meaning embodied in them1 and the whole of the consequences which flow from them4 99,symbolic ,expression of ,logical ,inference499 ,in developing the results of the ,principle of ,inference we require to use an appropriate language of signs4 ,it would indeed be quite possible to explain the processes of reasoning by the use of words found in the dictionary4 ,special examples of reasoning1 too1 may seem to be more readily apprehended than general symbolic forms4 ,but it has been shown in the mathematical sciences that the attainment of truth depends greatly upon the invention of a clear1 brief1 and appropriate system of symbols4 ,not only is such a language convenient1 but it is almost essential to the expression of those general truths which are the very soul of science4 ,to apprehend the truth of special cases of inference does not constitute logic2 we must apprehend them as cases of more general truths4 ,the object of all science is the separation of what is common and general from what is accidental and different4 ,in a system of logic1 if anywhere1 we should esteem this generality1 and strive to exhibit clearly what is similar in very diverse cases4 ,hence the great value of 99general symbols99 by which we can represent the form of a reasoning process1 disentangled from any consideration of the special subject to which it is applied4 ,the signs required in logic are of a very simple kind4 ,as sameness or difference must exist between two things or notions1 we need signs to indicate the things or notions compared1 and other signs to denote the relations between them4 ,we need1 then1 7#a7 symbols for terms1 7#b7 a symbol for sameness1 7#c7 a symbol for difference1 and 7#d7 one or two symbols to take the place of conjunctions4 ,ordinary nouns substantive1 such as 99,iron991 99,metal991 99,electricity991 99,undulation991 might serve as terms1 but1 for the reasons explained above1 it is better to adopt blank letters1 devoid of special signification1 such as ,a1 ,b1 ,c1 c4 ,each letter must be understood to represent a noun1 and1 so far as the conditions of the argument allow1 99any noun994 ,just as in ,algebra1 99x991 99y991 99z991 99p991 99q991 c4 are used for 99any quantities991 undetermined or unknown1 except when the special conditions of the problem are taken into account1 so will our letters stand for undetermined or unknown things4 ,these letter-terms will be used indifferently for nouns substantive and adjective4 ,between these two kinds of nouns there may perhaps be differences in a metaphysical or grammatical point of view4 ,but grammatical usage sanctions the conversion of adjectives into substantives1 and 99vice vers992 we may avail ourselves of this latitude without in any way prejudging the metaphysical difficulties which may be involved4 ,here1 as throughout this work1 ,i shall devote my attention to truths which ,i can exhibit in a clear and formal manner1 believing that in the present condition of logical science1 this course will lead to greater advantage than discussion upon the metaphysical questions which may underlie any part of the subject4 ,every noun or term denotes an object1 and usually implies the possession by that object of certain qualities or circumstances common to all the objects denoted4 ,there are certain terms1 however1 which imply the absence of qualities or circumstances attaching to other objects4 ,it will be convenient to employ a special mode of indicating these 99negative terms991 as they are called4 ,if the general name ,a denotes an object or class of objects possessing certain defined qualities1 then the term ,not ,a will denote any object which does not possess the whole of those qualities2 in short1 ,not ,a is the sign for anything which differs from ,a in regard to any one or more of the assigned qualities4 ,if ,a denote 8transparent object10 ,not ,a will denote 8not transparent object40 ,brevity and facility of expression are of no slight importance in a system of notation1 and it will therefore be desirable to substitute for the negative term ,not ,a a briefer symbol4 ,de ,morgan represented negative terms by small ,roman letters1 or sometimes by small italic letters2#cj as the latter seem to be highly convenient1 ,i shall use 99a991 99b991 99c991 444 99p991 99q991 c41 as the negative terms corresponding to ,a1 ,b1 ,c1 444 ,p1 ,q1 c4 ,thus if ,a means 8fluid10 99a99 will mean 8not fluid40 #cj 99,formal ,logic991 p4 #ch4 99,expression of ,identity and ,difference499 ,to denote the relation of sameness or identity ,i unhesitatingly adopt the sign 1 so long used by mathematicians to denote equality4 ,this symbol was originally appropriated by ,robert ,recorde in his 99,whetstone of ,wit991 to avoid the tedious repetition of the words 8is equal to20 and he chose a pair of parallel lines1 because no two things can be more equal4#ca ,the meaning of the sign has however been gradually extended beyond that of equality of quantities2 mathematicians have themselves used it to indicate equivalence of operations4 ,the force of analogy has been so great that writers in most other branches of science have employed the same sign4 ,the philologist uses it to indicate the equivalence of meaning of words3 chemists adopt it to signify identity in kind and equality in weight of the elements which form two different compounds4 ,not a few logicians1 for instance ,lambert1 ,drobitsch1 ,george ,bentham1#cb ,boole1#cc have employed it as the copula of propositions4 ,de ,morgan declined to use it for this purpose1 but still further extended its meaning so as to include the equivalence of a proposition with the premises from which it can be inferred2#cd and ,herbert ,spencer has applied it in a like manner4#ce #ca ,hallam's 99,literature of ,europe991 ,first ,ed41 vol4 ii4 p4 #ddd4 #cb 99,outline of a ,new ,system of ,logic991 ,london1 #ahbg1 pp4 #acc1 c4 #cc 99,an ,investigation of the ,laws of ,thought991 pp4 #bg1 c4 #cd 99,formal ,logic991 pp4 #hb1 #ajf4 ,in his later work1 99,the ,syllabus of a ,new ,system of ,logic991 he discontinued the use of the sign4 #ce 99,principles of ,psychology991 ,second ,ed41 vol4 ii4 pp4 #ed1 #ee4 ,many persons may think that the choice of a symbol is a matter of slight importance or of mere convenience2 but ,i hold that the common use of this sign in so many different meanings is really founded upon a generalisation of the widest character and of the greatest importance--one indeed which it is a principal purpose of this work to explain4 ,the employment of the same sign in different cases would be unphilosophical unless there were some real analogy between its diverse meanings4 ,if such analogy exists1 it is not only allowable1 but highly desirable and even imperative1 to use the symbol of equivalence with a generality of meaning corresponding to the generality of the principles involved4 ,accordingly ,de ,morgan's refusal to use the symbol in logical propositions indicated his opinion that there was a want of analogy between logical propositions and mathematical equations4 ,i use the sign because ,i hold the contrary opinion4 ,i conceive that the sign as commonly employed1 always denotes some form or degree of sameness1 and the particular form is usually indicated by the nature of the terms joined by it4 ,thus 8#f1gbj pounds #c tons0 is evidently an equation of quantities4 ,the formula - - expresses the equivalence of operations4 8,exogens ,dicotyledons0 is a logical identity expressing a profound truth concerning the character and origin of a most important group of plants4 ,we have great need in logic of a distinct sign for the copula1 because the little verb 99is99 7or 99are9971 hitherto used both in logic and ordinary discourse1 is thoroughly ambiguous4 ,it sometimes denotes identity1 as in 8,st4 ,paul's is the 99chef-d'uvre99 of ,sir ,christopher ,wren20 but it more commonly indicates inclusion of class within class1 or partial identity1 as in 8,bishops are members of the ,house of ,lords40 ,this latter relation involves identity1 but requires careful discrimination from simple identity1 as will be shown further on4 ,when with this sign of equality we join two nouns or logical terms1 as in ,hydrogen ,the least dense element1 we signify that the object or group of objects denoted by one term is identical with that denoted by the other1 in everything except the names4 ,the general formula ,a ,b must be taken to mean that ,a and ,b are symbols for the same object or group of objects4 ,this identity may sometimes arise from the mere imposition of names1 but it may also arise from the deepest laws of the constitution of nature2 as when we say ,gravitating matter ,matter possessing inertia1 ,exogenous plants ,dicotyledonous plants1 ,plagihedral quartz crystals ,quartz crystals causing the plane of polarisation of light to rotate4 ,we shall need carefully to distinguish between relations of terms which can be modified at our own will and those which are fixed as expressing the laws of nature2 but at present we are considering only the mode of expression which may be the same in either case4 ,sometimes1 but much less frequently1 we require a symbol to indicate difference or the absence of complete sameness4 ,for this purpose we may generalise in like manner the symbol 1 which was introduced by ,wallis to signify difference between quantities4 ,the general formula ,b ,c denotes that ,b and ,c are the names of two objects or groups which are not identical with each other4 ,thus we may say ,acrogens ,flowering plants4 ,snowdon ,the highest mountain in ,great ,britain4 ,i shall also occasionally use the sign to signify in the most general manner the existence of any relation between the two terms connected by it4 ,thus might mean not only the relations of equality or inequality1 sameness or difference1 but any special relation of time1 place1 size1 causation1 c4 in which one thing may stand to another4 ,by ,a ,b ,i mean1 then1 any two objects of thought related to each other in any conceivable manner4 99,general ,formula of ,logical ,inference499 ,the one supreme rule of inference consists1 as ,i have said1 in the direction to affirm of anything whatever is known of its like1 equal or equivalent4 ,the 99,substitution of ,similars99 is a phrase which seems aptly to express the capacity of mutual replacement existing in any two objects which are like or equivalent to a sufficient degree4 ,it is matter for further investigation to ascertain when and for what purposes a degree of similarity less than complete identity is sufficient to warrant substitution4 ,for the present we think only of the exact sameness expressed in the form ,a ,b4 ,now if we take the letter ,c to denote any third conceivable object1 and use the sign in its stated meaning of 99indefinite relation991 then the general formula of all inference may be thus exhibited3-- ,from ,a ,b ,c we may infer ,a ,c or1 in words--99,in whatever relation a thing stands to a second thing1 in the same relation it stands to the like or equivalent of that second thing499 ,the identity between ,a and ,b allows us indifferently to place ,a where ,b was1 or ,b where ,a was2 and there is no limit to the variety of special meanings which we can bestow upon the signs used in this formula consistently with its truth4 ,thus if we first specify only the meaning of the sign 1 we may say that if 99,c is the weight of ,b991 then 99,c is also the weight of ,a994 ,similarly ,if ,c is the father of ,b1 ,c is the father of ,a2 ,if ,c is a fragment of ,b1 ,c is a fragment of ,a2 ,if ,c is a quality of ,b1 ,c is a quality of ,a2 ,if ,c is a species of ,b1 ,c is a species of ,a2 ,if ,c is the equal of ,b1 ,c is the equal of ,a2 and so on 99ad infinitum994 ,we may also endow with special meanings the letter-terms ,a1 ,b1 and ,c1 and the process of inference will never be false4 ,thus let the sign mean 8is height of10 and let ,a ,snowdon1 ,b ,highest mountain in ,england or ,wales1 ,c #c1eij feet2 then it obviously follows since 8#c1eij feet is the height of ,snowdon10 and 8,snowdon the highest mountain in ,england or ,wales10 that1 8#c1eij feet is the height of the highest mountain in ,england or ,wales40 ,one result of this general process of inference is that we may in any aggregate or complex whole replace any part by its equivalent without altering the whole4 ,to alter is to make a difference2 but if in replacing a part ,i make no difference1 there is no alteration of the whole4 ,many inferences which have been very imperfectly included in logical formulas at once follow4 ,i remember the late ,prof4 ,de ,morgan remarking that all ,aristotle's logic could not prove that 8,because a horse is an animal1 the head of a horse is the head of an animal40 ,i conceive that this amounts merely to replacing in the complete notion 99head of a horse991 the term 8horse10 by its equivalent 99some animal99 or 99an animal994 ,similarly1 since ,the ,lord ,chancellor ,the ,speaker of the ,house of ,lords1 it follows that ,the death of the ,lord ,chancellor ,the death of the ,speaker of the ,house of ,lords2 and any event1 circumstance or thing1 which stands in a certain relation to the one will stand in like relation to the other4 ,milton reasons in this way when he says1 in his ,areopagitica1 8,who kills a man1 kills a reasonable creature1 ,god's image40 ,if we may suppose him to mean ,god's image man some reasonable creature1 it follows that 8,the killer of a man is the killer of some reasonable creature10 and also 8,the killer of ,god's image40 ,this replacement of equivalents may be repeated over and over again to any extent4 ,thus if 99person99 is identical in meaning with 99individual991 it follows that ,meeting of persons meeting of individuals2 and if 99assemblage99 99meeting991 we may make a new replacement and show that ,meeting of persons assemblage of individuals4 ,we may in fact found upon this principle of substitution a most general axiom in the following terms#cf3-- #cf 99,pure ,logic1 or the ,logic of ,quality991 p4 #ad4 99,same parts samely related make same wholes499 ,if1 for instance1 exactly similar bricks and other materials be used to build two houses1 and they be similarly placed in each house1 the two houses must be similar4 ,there are millions of cells in a human body1 but if each cell of one person were represented by an exactly similar cell similarly placed in another body1 the two persons would be undistinguishable1 and would be only 99numerically99 different4 ,it is upon this principle1 as we shall see1 that all accurate processes of measurement depend4 ,if for a weight in a scale of a balance we substitute another weight1 and the equilibrium remains entirely unchanged1 then the weights must be exactly equal4 ,the general test of equality is substitution4 ,objects are equally bright when on replacing one by the other the eye perceives no difference4 ,objects are equal in dimensions when tested by the same gauge they fit in the same manner4 ,generally speaking1 two objects are alike so far as when substituted one for another no alteration is produced1 and 99vice vers99 when alike no alteration is produced by the substitution4 99,the ,propagating ,power of ,similarity499 ,the relation of similarity in all its degrees is reciprocal4 ,so far as things are alike1 either may be substituted for the other2 and this may perhaps be considered the very meaning of the relation4 ,but it is well worth notice that there is in similarity a peculiar power of extending itself among all the things which are similar4 ,to render a number of things similar to each other we need only render them similar to one standard object4 ,each coin struck from a pair of dies not only resembles the matrix or original pattern from which the dies were struck1 but resembles every other coin manufactured from the same original pattern4 ,among a million such coins there are not less than #dii1iii1ejj1jjj 99pairs of coins99 resembling each other4 ,similars to the same are similars to all4 ,it is one great advantage of printing that all copies of a document struck from the same type are necessarily identical each with each1 and whatever is true of one copy will be true of every copy4 ,similarly1 if fifty rows of pipes in an organ be tuned in perfect unison with one row1 usually the ,principal1 they must be in unison with each other4 ,similarity can also reproduce or propagate itself 99ad infinitum993 for if a number of tuning-forks be adjusted in perfect unison with one standard fork1 all instruments tuned to any one fork will agree with any instrument tuned to any other fork4 ,standard measures of length1 capacity1 weight1 or any other measurable quality1 are propagated in the same manner4 ,so far as copies of the original standard1 or copies of copies1 or copies again of those copies1 are accurately executed1 they must all agree each with every other4 ,it is the capability of mutual substitution which gives such great value to the modern methods of mechanical construction1 according to which all the parts of a machine are exact facsimiles of a fixed pattern4 ,the rifles used in the ,british army are constructed on the ,american interchangeable system1 so that any part of any rifle can be substituted for the same part of another4 ,a bullet fitting one rifle will fit all others of the same bore4 ,sir ,j4 ,whitworth has extended the same system to the screws and screw-bolts used in connecting together the parts of machines1 by establishing a series of standard screws4 99,anticipations of the ,principle of ,substitution499 ,in such a subject as logic it is hardly possible to put forth any opinions which have not been in some degree previously entertained4 ,the germ at least of every doctrine will be found in earlier writers1 and novelty must arise chiefly in the mode of harmonising and developing ideas4 ,when ,i first employed the process and name of 99substitution99 in logic1#cg ,i was led to do so from analogy with the familiar mathematical process of substituting for a symbol its value as given in an equation4 ,in writing my first logical essay ,i had a most imperfect conception of the importance and generality of the process1 and ,i described1 as if they were of equal importance1 a number of other laws which now seem to be but particular cases of the one general rule of substitution4 #cg 99,pure ,logic991 pp4 #ah1 #ai4 ,my second essay1 8,the ,substitution of ,similars10 was written shortly after ,i had become aware of the great simplification which may be effected by a proper application of the principle of substitution4 ,i was not then acquainted with the fact that the ,german logician ,beneke had employed the principle of substitution1 and had used the word itself in forming a theory of the syllogism4 ,my imperfect acquaintance with the ,german language had prevented me from acquiring a complete knowledge of ,beneke's views2 but there is no doubt that ,professor ,lindsay is right in saying that he1 and probably other logicians1 were in some degree familiar with the principle4#ch ,even ,aristotle's dictum may be regarded as an imperfect statement of the principle of substitution2 and1 as ,i have pointed out1 we have only to modify that dictum in accordance with the quantification of the predicate in order to arrive at the complete process of substitution4#ci ,the ,port-,royal logicians appear to have entertained nearly equivalent views1 for they considered that all moods of the syllogism might be reduced under one general principle4#dj ,of two premises they regard one as the 99containing proposition99 7propositio continens71 and the other as the 99applicative proposition994 ,the latter proposition must always be affirmative1 and represents that by which a substitution is made2 the former may or may not be negative1 and is that in which a substitution is effected4 ,they also show that this method will embrace certain cases of complex reasoning which had no place in the ,aristotelian syllogism4 ,their views probably constitute the greatest improvement in logical doctrine made up to that time since the days of ,aristotle4 ,but a true reform in logic must consist1 not in explaining the syllogism in one way or another1 but in doing away with all the narrow restrictions of the ,aristotelian system1 and in showing that there exists an infinite variety of logical arguments immediately deducible from the principle of substitution of which the ancient syllogism forms but a small and not even the most important part4 #ch ,ueberweg's 99,system of ,logic991 transl4 by ,lindsay1 pp4 #ddb-ddf1 #ega1 #egb4 ,the anticipations of the principle of substitution to be found in the works of ,leibnitz1 ,reusch1 and perhaps other ,german logicians1 will be noticed in the preface to this second edition4 #ci 99,substitution of ,similars99 7#ahfi71 p4 #i4 #dj 99,port-,royal ,logic991 transl4 by ,spencer ,baynes1 pp4 #bab-bai4 ,part ,,iii4 chap4 x4 and xi4 99,the ,logic of ,relatives499 ,there is a difficult and important branch of logic which may be called the ,logic of ,relatives4 ,if ,i argue1 for instance1 that because ,daniel ,bernoulli was the son of ,john1 and ,john the brother of ,james1 therefore ,daniel was the nephew of ,james1 it is not possible to prove this conclusion by any simple logical process4 ,we require at any rate to assume that the son of a brother is a nephew4 ,a simple logical relation is that which exists between properties and circumstances of the same object or class4 ,but objects and classes of objects may also be related according to all the properties of time and space4 ,i believe it may be shown1 indeed1 that where an inference concerning such relations is drawn1 a process of substitution is really employed and an identity must exist2 but ,i will not undertake to prove the assertion in this work4 ,the relations of time and space are logical relations of a complicated character demanding much abstract and difficult investigation4 ,the subject has been treated with such great ability by ,peirce1#da ,de ,morgan1#db ,ellis1#dc and ,harley1 that ,i will not in the present work attempt any review of their writings1 but merely refer the reader to the publications in which they are to be found4 #da 99,description of a ,notation for the ,logic of ,relatives1 resulting from an ,amplification of the ,conceptions of ,boole's ,calculus of ,logic499 ,by ,c4 ,s4 ,peirce4 99,memoirs of the ,american ,academy991 vol4 ix4 ,cambridge1 ,u4,s41 #ahgj4 #db 99,on the ,syllogism ,no ,,iv41 and on the ,logic of ,relations499 ,by ,augustus ,de ,morgan4 99,transactions of the ,cambridge ,philosophical ,society991 vol4 x4 part ii41 #ahfj4 #dc 99,observations on ,boole's ,laws of ,thought499 ,by the late ,r4 ,leslie ,ellis2 communicated by the ,rev4 ,robert ,harley1 ,f4,r4,s4 99,report of the ,british ,association991 #ahgj4 99,report of ,sections991 p4 #ab4 ,also1 99,on ,boole's ,laws of ,thought994 ,by the ,rev4 ,robert ,harley1 ,f4,r4,s41 99ibid499 p4 #ad4 ,,chapter ,,ii4 ,,terms4 ,every proposition expresses the resemblance or difference of the things denoted by its terms4 ,as inference treats of the relation between two or more propositions1 so a proposition expresses a relation between two or more terms4 ,in the portion of this work which treats of deduction it will be convenient to follow the usual order of exposition4 ,we will consider in succession the various kinds of terms1 propositions1 and arguments1 and we commence in this chapter with terms4 ,the simplest and most palpable meaning which can belong to a term consists of some single material object1 such as ,westminster ,abbey1 ,stonehenge1 the ,sun1 ,sirius1 c4 ,it is probable that in early stages of intellect only concrete and palpable things are the objects of thought4 ,the youngest child knows the difference between a hot and a cold body4 ,the dog can recognise his master among a hundred other persons1 and animals of much lower intelligence know and discriminate their haunts4 ,in all such acts there is judgment concerning the likeness of physical objects1 but there is little or no power of analysing each object and regarding it as a group of qualities4 ,the dignity of intellect begins with the power of separating points of agreement from those of difference4 ,comparison of two objects may lead us to perceive that they are at once like and unlike4 ,two fragments of rock may differ entirely in outward form1 yet they may have the same colour1 hardness1 and texture4 ,flowers which agree in colour may differ in odour4 ,the mind learns to regard each object as an aggregate of qualities1 and acquires the power of dwelling at will upon one or other of those qualities to the exclusion of the rest4 ,logical abstraction1 in short1 comes into play1 and the mind becomes capable of reasoning1 not merely about objects which are physically complete and concrete1 but about things which may be thought of separately in the mind though they exist not separately in nature4 ,we can think of the hardness of a rock1 or the colour of a flower1 and thus produce abstract notions1 denoted by abstract terms1 which will form a subject for further consideration4 ,at the same time arise general notions and classes of objects4 ,we cannot fail to observe that the quality 99hardness99 exists in many objects1 for instance in many fragments of rock2 mentally joining these together1 we create the class 99hard object991 which will include1 not only the actual objects examined1 but all others which may happen to agree with them1 as they agree with each other4 ,as our senses cannot possibly report to us all the contents of space1 we cannot usually set any limits to the number of objects which may fall into any such class4 ,at this point we begin to perceive the power and generality of thought1 which enables us in a single act to treat of indefinitely or even infinitely numerous objects4 ,we can safely assert that whatever is true of any one object coming under a class is true of any of the other objects so far as they possess the common qualities implied in their belonging to the class4 ,we must not place a thing in a class unless we are prepared to believe of it all that is believed of the class in general2 but it remains a matter of important consideration to decide how far and in what manner we can safely undertake thus to assign the place of objects in that general system of classification which constitutes the body of science4 99,twofold ,meaning of ,general ,names499 ,etymologically the 99meaning99 of a name is that which we are caused to think of when the name is used4 ,now every general name causes us to think of some one or more of the objects belonging to a class2 it may also cause us to think of the common qualities possessed by those objects4 ,a name is said to 99denote99 the object of thought to which it may be applied2 it 99implies99 at the same time the possession of certain qualities or circumstances4 ,the objects denoted form the 99extent99 of meaning of the term2 the qualities implied form the 99intent99 of meaning4 ,crystal is the name of any substance of which the molecules are arranged in a regular geometrical manner4 ,the substances or objects in question form the extent of meaning2 the circumstance of having the molecules so arranged forms the intent of meaning4 ,when we compare general terms together1 it may often be found that the meaning of one is included in the meaning of another4 ,thus all 99crystals99 are included among 99material substances991 and all 99opaque crystals99 are included among 99crystals992 here the inclusion is in extension4 ,we may also have inclusion of meaning in regard to intension4 ,for1 as all crystals are material substances1 the qualities implied by the term material substance must be among those implied by crystal4 ,again1 it is obvious that while in extension of meaning opaque crystals are but a part of crystals1 in intension of meaning crystal is but part of opaque crystal4 ,we increase the intent of meaning of a term by joining to it adjectives1 or phrases equivalent to adjectives1 and the removal of such adjectives of course decreases the intensive meaning4 ,now1 concerning such changes of meaning1 the following all-important law holds universally true3--99,when the intent of meaning of a term is increased the extent is decreased2 and99 vice vers1 99when the extent is increased the intent is decreased994 ,in short1 as one is increased the other is decreased4 ,this law refers only to logical changes4 ,the number of steam-engines in the world may be undergoing a rapid increase without the intensive meaning of the name being altered4 ,the law will only be verified1 again1 when there is a real change in the intensive meaning1 and an adjective may often be joined to a noun without making a change4 99,elementary metal99 is identical with 99metal992 99mortal man99 with 99man992 it being a 99property99 of all metals to be elements1 and of all men to be mortals4 ,there is no limit to the amount of meaning which a term may have4 ,a term may denote one object1 or many1 or an infinite number2 it may imply a single quality1 if such there be1 or a group of any number of qualities1 and yet the law connecting the extension and intension will infallibly apply4 ,taking the general name 99planet991 we increase its intension and decrease its extension by prefixing the adjective 99exterior992 and if we further add 99nearest to the earth991 there remains but one planet1 99,mars991 to which the name can then be applied4 ,singular terms1 which denote a single individual only1 come under the same law of meaning as general names4 ,they may be regarded as general names of which the meaning in extension is reduced to a minimum4 ,logicians have erroneously asserted1 as it seems to me1 that singular terms are devoid of meaning in intension1 the fact being that they exceed all other terms in that kind of meaning1 as ,i have elsewhere tried to show4#dd #dd ,jevons' 99,elementary ,lessons in ,logic991 pp4 #da-dc2 99,pure ,logic991 p4 #f4 ,see also ,j4 ,s4 ,mill1 99,system of ,logic991 ,book ,i4 chap4 ii4 section #e1 and ,shedden's 99,elements of ,logic991 ,london1 #ahfd1 pp4 #ad1 c4 ,professor ,robertson objects 799,mind991 vol4 i4 p4 #baj7 that ,i confuse 99singular99 and 99proper99 names2 if so1 it is because ,i hold that the same remarks apply to proper names1 which do not seem to me to differ logically from singular names4 99,abstract ,terms499 ,comparison of objects1 and analysis of the complex resemblances and differences which they present1 lead us to the conception of 99abstract qualities994 ,we learn to think of one object as not only different from another1 but as differing in some particular point1 such as colour1 or weight1 or size4 ,we may then convert points of agreement or difference into separate objects of thought which we call qualities and denote by 99abstract terms994 ,thus the term 99redness99 means something in which a number of objects agree as to colour1 and in virtue of which they are called red4 ,redness forms1 in fact1 the intensive meaning of the term red4 ,abstract terms are strongly distinguished from general terms by possessing only one kind of meaning2 for as they denote qualities there is nothing which they cannot in addition imply4 ,the adjective 8red0 is the name of red objects1 but it implies the possession by them of the quality 99redness992 but this latter term has one single meaning--the quality alone4 ,thus it arises that abstract terms are incapable of plurality4 ,red objects are numerically distinct each from each1 and there are multitudes of such objects2 but redness is a single quality which runs through all those objects1 and is the same in one as it is in another4 ,it is true that we may speak of 99rednesses991 meaning different kinds or tints of redness1 just as we may speak of 99colours991 meaning different kinds of colours4 ,but in distinguishing kinds1 degrees1 or other differences1 we render the terms so far concrete4 ,in that they are merely red there is but a single nature in red objects1 and so far as things are merely coloured1 colour is a single indivisible quality4 ,redness1 so far as it is redness merely1 is one and the same everywhere1 and possesses absolute oneness4 ,in virtue of this unity we acquire the power of treating all instances of such quality as we may treat any one4 ,we possess1 in short1 general knowledge4 99,substantial ,terms499 ,logicians appear to have taken little notice of a class of terms which partake in certain respects of the character of abstract terms and yet are undoubtedly the names of concrete existing things4 ,these terms are the names of substances1 such as gold1 carbonate of lime1 nitrogen1 c4 ,we cannot speak of two golds1 twenty carbonates of lime1 or a hundred nitrogens4 ,there is no such distinction between the parts of a uniform substance as will allow of a discrimination of numerous individuals4 ,the qualities of colour1 lustre1 malleability1 density1 c41 by which we recognise gold1 extend through its substance irrespective of particular size or shape4 ,so far as a substance is gold1 it is one and the same everywhere2 so that terms of this kind1 which ,i propose to call 99substantial terms991 possess the peculiar unity of abstract terms4 ,yet they are not abstract2 for gold is of course a tangible visible body1 entirely concrete1 and existing independently of other bodies4 ,it is only when1 by actual mechanical division1 we break up the uniform whole which forms the meaning of a substantial term1 that we introduce number4 99,piece of gold99 is a term capable of plurality2 for there may be a great many pieces discriminated either by their various shapes and sizes1 or1 in the absence of such marks1 by simultaneously occupying different parts of space4 ,in substance they are one2 as regards the properties of space they are many4#de ,we need not further pursue this question1 which involves the distinction between unity and plurality1 until we consider the principles of number in a subsequent chapter4 #de ,professor ,robertson has criticised my introduction of 8,substantial ,terms0 799,mind991 vol4 i4 p4 #baj71 and objects1 perhaps correctly1 that the distinction if valid is extra-logical4 ,i am inclined to think1 however1 that the doctrine of terms is1 strictly speaking1 for the most part extra-logical4 99,collective ,terms499 ,we must clearly distinguish between the 99collective99 and the 99general meanings99 of terms4 ,the same name may be used to denote the whole body of existing objects of a certain kind1 or any one of those objects taken separately4 8,man0 may mean the aggregate of existing men1 which we sometimes describe as 99mankind992 it is also the general name applying to any man4 ,the vegetable kingdom is the name of the whole aggregate of 99plants991 but 8plant0 itself is a general name applying to any one or other plant4 ,every material object may be conceived as divisible into parts1 and is therefore collective as regards those parts4 ,the animal body is made up of cells and fibres1 a crystal of molecules2 wherever physical division1 or as it has been called 99partition991 is possible1 there we deal in reality with a collective whole4 ,thus the greater number of general terms are at the same time collective as regards each individual whole which they denote4 ,it need hardly be pointed out that we must not infer of a collective whole what we know only of the parts1 nor of the parts what we know only of the whole4 ,the relation of whole and part is not one of identity1 and does not allow of substitution4 ,there may nevertheless be qualities which are true alike of the whole and of its parts4 ,a number of organ-pipes tuned in unison produce an aggregate of sound which is of exactly the same pitch as each separate sound4 ,in the case of substantial terms1 certain qualities may be present equally in each minutest part as in the whole4 ,the chemical nature of the largest mass of pure carbonate of lime is the same as the nature of the smallest particle4 ,in the case of abstract terms1 again1 we cannot draw a distinction between whole and part2 what is true of redness in any case is always true of redness1 so far as it is merely red4 99,synthesis of ,terms499 ,we continually combine simple terms together so as to form new terms of more complex meaning4 ,thus1 to increase the intension of meaning of a term we write it with an adjective or a phrase of adjectival nature4 ,by joining 8brittle0 to 8metal10 we obtain a combined term1 8brittle metal10 which denotes a certain portion of the metals1 namely1 such as are selected on account of possessing the quality of 99brittleness994 ,as we have already seen1 8brittle metal0 possesses less extension and greater intension than metal4 ,nouns1 prepositional phrases1 participial phrases and subordinate propositions may also be added to terms so as to increase their intension and decrease their extension4 ,in our symbolic language we need some mode of indicating this junction of terms1 and the most convenient device will be the juxtaposition of the letter-terms4 ,thus if ,a mean brittle1 and ,b mean metal1 then ,,ab will mean brittle metal4 ,nor need there be any limit to the number of letters thus joined together1 or the complexity of the notions which they may represent4 ,thus if we take the letters ,p metal1 ,q white1 ,r monovalent1 ,s of specific gravity #aje1 ,t melting above #ajjj ,c41 ,v good conductor of heat and electricity1 then we can form a combined term ,,pqrstv1 which will denote 8a white monovalent metal1 of specific gravity #aje1 melting above #ajjj ,c41 and a good conductor of heat and electricity40 ,there are many grammatical usages concerning the junction of words and phrases to which we need pay no attention in logic4 ,we can never say in ordinary language 8of wood table10 meaning 8table of wood20 but we may consider 8of wood0 as logically an exact equivalent of 8wooden02 so that if ,x of wood1 ,y table1 there is no reason why1 in our symbols1 ,,xy should not be just as correct an expression for 8table of wood 0 as ,,yx4 ,in this case indeed we might substitute for 8of wood 0 the corresponding adjective 8wooden10 but we should often fail to find any adjective answering exactly to a phrase4 ,there is no single word by which we could express the notion 8of specific gravity #aje30 but logically we may consider these words as forming an adjective2 and denoting this by ,s and metal by ,p1 we may say that ,,sp means 8metal of specific gravity #aje40 ,it is one of many advantages in these blank letter-symbols that they enable us completely to neglect all grammatical peculiarities and to fix our attention solely on the purely logical relations involved4 ,investigation will probably show that the rules of grammar are mainly founded upon traditional usage and have little logical signification4 ,this indeed is sufficiently proved by the wide grammatical differences which exist between languages1 though the logical foundation must be the same4 99,symbolic ,expression of the ,law of ,contradiction499 ,the synthesis of terms is subject to the all-important ,law of ,thought1 described in a previous section 7p4 #e7 and called the ,law of ,contradiction1 ,it is self-evident that no quality can be both present and absent at the same time and place4 ,this fundamental condition of all thought and of all existence is expressed symbolically by a rule that a term and its negative shall never be allowed to come into combination4 ,such combined terms as ,a99a991 ,b99b991 ,c99c991 c41 are self-contradictory and devoid of all intelligible meaning4 ,if they could represent anything1 it would be what cannot exist1 and cannot even be imagined in the mind4 ,they can therefore only enter into our consideration to suffer immediate exclusion4 ,the criterion of false reasoning1 as we shall find1 is that it involves self-contradiction1 the affirming and denying of the same statement4 ,we might represent the object of all reasoning as the separation of the consistent and possible from the inconsistent and impossible2 and we cannot make any statement except a truism without implying that certain combinations of terms are contradictory and excluded from thought4 ,to assert that 8all ,a's are ,b's0 is equivalent to the assertion that 8,a's which are not ,b's cannot exist40 ,it will be convenient to have the means of indicating the exclusion of the self-contradictory1 and we may use the familiar sign for 99nothing991 the cipher #j4 ,thus the second law of thought may be symbolised in the forms ,a99a99 #j ,,ab99b99 #j ,,abc99a99 #j ,we may variously describe the meaning of #j in logic as the 99non-existent991 the 99impossible991 the 99self-inconsistent991 the 99inconceivable994 ,close analogy exists between this meaning and its mathematical signification4 99,certain ,special ,conditions of ,logical ,symbols499 ,in order that we may argue and infer truly we must treat our logical symbols according to the fundamental laws of ,identity and ,difference4 ,but in thus using our symbols we shall frequently meet with combinations of which the meaning will not at first sight be apparent4 ,if in one case we learn that an object is 8yellow and round10 and in another case that it is 8round and yellow10 there arises the question whether these two descriptions are identical in meaning or not4 ,again1 if we proved that an object was 8round round10 the meaning of such an expression would be open to doubt4 ,accordingly we must take notice1 before proceeding further1 of certain special laws which govern the combination of logical terms4 ,in the first place the combination of a logical term with itself is without effect1 just as the repetition of a statement does not alter the meaning of the statement2 8a round round object0 is simply 8a round object40 ,what is yellow yellow is merely yellow2 metallic metals cannot differ from metals1 nor circular circles from circles4 ,in our symbolic language we may similarly hold that ,,aa is identical with ,a1 or ,a ,,aa ,,aaa c4 ,the late ,professor ,boole is the only logician in modern times who has drawn attention to this remarkable property of logical terms2#df but in place of the name which he gave to the law1 ,i have proposed to call it ,the ,law of ,simplicity4#dg ,its high importance will only become apparent when we attempt to determine the relations of logical and mathematical science4 ,two symbols of quantity1 and only two1 seem to obey this law2 we may say that #a #a #a1 and #j #j #j 7taking #j to mean absolute zero or #a - #a72 there is apparently no other number which combined with itself gives an unchanged result4 ,i shall point out1 however1 in the chapter upon ,number1 that in reality all numerical symbols obey this logical principle4 #df 99,mathematical ,analysis of ,logic991 ,cambridge1 #ahdg1 p4 #ag4 99,an ,investigation of the ,laws of ,thought991 ,london1 #ahed1 p4 #ca4 #dg 99,pure ,logic991 p4 #ae4 ,it is curious that this ,law of ,simplicity1 though almost unnoticed in modern times1 was known to ,bothius1 who makes a singular remark in his treatise 99,de ,trinitate et ,unitate ,dei99 7p4 #iei74 ,he says3 8,if ,i should say sun1 sun1 sun1 ,i should not have made three suns1 but ,i should have named one sun so many times40#dh ,ancient discussions about the doctrine of the ,trinity drew more attention to subtle questions concerning the nature of unity and plurality than has ever since been given to them4 #dh 8,velut si dicam1 ,sol1 ,sol1 ,sol1 non tres soles effecerim1 sed uno toties prdicaverim40 ,it is a second law of logical symbols that order of combination is a matter of indifference4 8,rich and rare gems0 are the same as 8rare and rich gems10 or even as 8gems1 rich and rare40 ,grammatical1 rhetorical1 or poetic usage may give considerable significance to order of expression4 ,the limited power of our minds prevents our grasping many ideas at once1 and thus the order of statement may produce some effect1 but not in a simply logical manner4 ,all life proceeds in the succession of time1 and we are obliged to write1 speak1 or even think of things and their qualities one after the other2 but between the things and their qualities there need be no such relation of order in time or space4 ,the sweetness of sugar is neither before nor after its weight and solubility4 ,the hardness of a metal1 its colour1 weight1 opacity1 malleability1 electric and chemical properties1 are all coexistent and coextensive1 pervading the metal and every part of it in perfect community1 none before nor after the others4 ,in our words and symbols we cannot observe this natural condition2 we must name one quality first and another second1 just as some one must be the first to sign a petition1 or to walk foremost in a procession4 ,in nature there is no such precedence4 ,i find that the opinion here stated1 to the effect that relations of space and time do not apply to many of our ideas1 is clearly adopted by ,hume in his celebrated 99,treatise on ,human ,nature99 7vol4 i4 p4 #daj74 ,he says3#di--8;,an object may be said to be no where1 when its parts are not so situated with respect to each other1 as to form any figure or quantity2 nor the whole with respect to other bodies so as to answer to our notions of contiguity or distance4 ,now this is evidently the case with all our perceptions and objects1 except those of sight and feeling4 ,a moral reflection cannot be placed on the right hand or on the left hand of a passion1 nor can a smell or sound be either of a circular or a square figure4 ,these objects and perceptions1 so far from requiring any particular place1 are absolutely incompatible with it1 and even the imagination cannot attribute it to them40 #di ,book i41 ,part iv41 ,section #e4 ,a little reflection will show that knowledge in the highest perfection would consist in the 99simultaneous99 possession of a multitude of facts4 ,to comprehend a science perfectly we should have every fact present with every other fact4 ,we must write a book and we must read it successively word by word1 but how infinitely higher would be our powers of thought if we could grasp the whole in one collective act of consciousness6 ,compared with the brutes we do possess some slight approximation to such power1 and it is conceivable that in the indefinite future mind may acquire an increase of capacity1 and be less restricted to the piecemeal examination of a subject4 ,but ,i wish here to make plain that there is no logical foundation for the successive character of thought and reasoning unavoidable under our present mental conditions4 99,we are logically weak and imperfect in respect of the fact that we are obliged to think of one thing after another499 ,we must describe metal as 8hard and opaque10 or 8opaque and hard10 but in the metal itself there is no such difference of order2 the properties are simultaneous and coextensive in existence4 ,setting aside all grammatical peculiarities which render a substantive less moveable than an adjective1 and disregarding any meaning indicated by emphasis or marked order of words1 we may state1 as a general law of logic1 that ,,ab is identical with ,,ba1 or ,,ab ,,ba4 ,similarly1 ,,abc ,,acb ,,bca c4 ,boole first drew attention in recent years to this property of logical terms1 and he called it the property of ,commutativeness4#ej ,he not only stated the law with the utmost clearness1 but pointed out that it is a ,law of ,thought rather than a ,law of ,things4 ,i shall have in various parts of this work to show how the necessary imperfection of our symbols expressed in this law clings to our modes of expression1 and introduces complication into the whole body of mathematical formul1 which are really founded on a logical basis4 #ej 99,laws of ,thought991 p4 #bi4 ,it is pointed out in the preface to this ,second ,edition that ,leibnitz was acquainted with the ,laws of ,simplicity and of ,commutativeness4 ,it is of course apparent that the power of commutation belongs only to terms related in the simple logical mode of synthesis4 ,no one can confuse 8a house of bricks0 with 8bricks of a house10 8twelve square feet0 with 8twelve feet square10 8the water of crystallization0 with 8the crystallization of water40 ,all relations which involve differences of time and space are inconvertible2 the higher must not be made to change places with the lower1 nor the first with the last4 ,for the parties concerned there is all the difference in the world between ,a killing ,b and ,b killing ,a4 ,the law of commutativeness simply asserts that difference of order does not attach to the connection between the properties and circumstances of a thing--to what ,i call 99simple logical relation994 ,,chapter ,,iii4 ,,propositions4 ,we now proceed to consider the variety of forms of propositions in which the truths of science must be expressed4 ,i shall endeavour to show that1 however diverse these forms may be1 they all admit the application of the one same principle of inference that what is true of a thing is true of the like or same4 ,this principle holds true whatever be the kind or manner of the likeness1 provided proper regard be had to its nature4 ,propositions may assert an identity of time1 space1 manner1 quantity1 degree1 or any other circumstance in which things may agree or differ4 ,we find an instance of a proposition concerning time in the following3--8,the year in which ,newton was born1 was the year in which ,galileo died40 ,this proposition expresses an approximate identity of time between two events2 hence whatever is true of the year in which ,galileo died is true of that in which ,newton was born1 and 99vice vers994 8,tower ,hill is the place where ,raleigh was executed0 expresses an identity of place2 and whatever is true of the one spot is true of the spot otherwise defined1 but in reality the same4 ,in ordinary language we have many propositions obscurely expressing identities of number1 quantity1 or degree4 8,so many men1 so many minds10 is a proposition concerning number1 that is to say1 an equation2 whatever is true of the number of men is true of the number of minds1 and 99vice vers994 8,the density of ,mars is 7nearly7 the same as that of the ,earth10 8,the force of gravity is directly as the product of the masses1 and inversely as the square of the distance10 are propositions concerning magnitude or degree4 ,logicians have not paid adequate attention to the great variety of propositions which can be stated by the use of the little conjunction 99as991 together with 99so994 8,as the home so the people10 is a proposition expressing identity of manner2 and a great number of similar propositions all indicating some kind of resemblance might be quoted4 ,whatever be the special kind of identity1 all such expressions are subject to the great principle of inference2 but as we shall in later parts of this work treat more particularly of inference in cases of number and magnitude1 we will here confine our attention to logical propositions which involve only notions of quality4 99,simple ,identities499 ,the most important class of propositions consists of those which fall under the formula ,a ,b1 and may be called 99simple identities994 ,i may instance1 in the first place1 those most elementary propositions which express the exact similarity of a quality encountered in two or more objects4 ,i may compare the colour of the ,pacific ,ocean with that of the ,atlantic1 and declare them identical4 ,i may assert that 8the smell of a rotten egg is like that of hydrogen sulphide20 8the taste of silver hyposulphite is like that of cane sugar20 8the sound of an earthquake resembles that of distant artillery40 ,such are propositions stating1 accurately or otherwise1 the identity of simple physical sensations4 ,judgments of this kind are necessarily pre-supposed in more complex judgments4 ,if ,i declare that 8this coin is made of gold10 ,i must base the judgment upon the exact likeness of the substance in several qualities to other pieces of substance which are undoubtedly gold4 ,i must make judgments of the colour1 the specific gravity1 the hardness1 and of other mechanical and chemical properties2 each of these judgments is expressed in an elementary proposition1 8the colour of this coin is the colour of gold10 and so on4 ,even when we establish the identity of a thing with itself under a different name or aspect1 it is by distinct judgments concerning single circumstances4 ,to prove that the ,homeric is copper we must show the identity of each quality recorded of with a quality of copper4 ,to establish ,deal as the landing-place of ,csar all material circumstances must be shown to agree4 ,if the modern ,wroxeter is the ancient ,uriconium1 there must be the like agreement of all features of the country not subject to alteration by time4 ,such identities must be expressed in the form ,a ,b4 ,we may say ,colour of ,pacific ,ocean ,colour of ,atlantic ,ocean4 ,smell of rotten egg ,smell of hydrogen sulphide4 ,in these and similar propositions we assert identity of single qualities or causes of sensation4 ,in the same form we may also express identity of any group of qualities1 as in ,copper4 ,deal ,landing-place of ,csar4 ,a multitude of propositions involving singular terms fall into the same form1 as in ,the ,pole star ,the slowest-moving star4 ,jupiter ,the greatest of the planets4 ,the ringed planet ,the planet having seven satellites4 ,the ,queen of ,england ,the ,empress of ,india4 ,the number two ,the even prime number4 ,honesty ,the best policy4 ,in mathematical and scientific theories we often meet with simple identities capable of expression in the same form4 ,thus in mechanical science 8,the process for finding the resultant of forces the process for finding the resultant of simultaneous velocities40 ,theorems in geometry often give results in this form1 as ,equilateral triangles ,equiangular triangles4 ,circle ,finite plane curve of constant curvature4 ,circle ,curve of least perimeter4 ,the more profound and important laws of nature are often expressible in the form of simple identities2 in addition to some instances which have already been given1 ,i may suggest1 ,crystals of cubical system ,crystals not possessing the power of double refraction4 ,all definitions are necessarily of this form1 whether the objects defined be many1 few1 or singular4 ,thus we may say1 ,common salt ,sodium chloride4 ,chlorophyl ,green colouring matter of leaves4 ,square ,equal-sided rectangle4 ,it is an extraordinary fact that propositions of this elementary form1 all-important and very numerous as they are1 had no recognised place in ,aristotle's system of ,logic4 ,accordingly their importance was overlooked until very recent times1 and logic was the most deformed of sciences4 ,but it is impossible that ,aristotle or any other person should avoid constantly using them2 not a term could be defined without their use4 ,in one place at least ,aristotle actually notices a proposition of the kind4 ,he observes3 8,we sometimes say that that white thing is ,socrates1 or that the object approaching is ,callias40#ea ,here we certainly have simple identity of terms2 but he considered such propositions purely accidental1 and came to the unfortunate conclusion1 that 8,singulars cannot be predicated of other terms40 #ea 99,prior ,analytics991 i4 cap4 xxvii4 #c4 ,propositions may also express the identity of extensive groups of objects taken collectively or in one connected whole2 as when we say1 ,the ,queen1 ,lords1 and ,commons ,the ,legislature of the ,united ,kingdom4 ,when ,blackstone asserts that 8,the only true and natural foundation of society are the wants and fears of individuals10 we must interpret him as meaning that the whole of the wants and fears of individuals in the aggregate form the foundation of society4 ,but many propositions which might seem to be collective are but groups of singular propositions or identities4 ,when we say 8,potassium and sodium are the metallic bases of potash and soda10 we obviously mean1 ,potassium ,metallic base of potash2 ,sodium ,metallic base of soda4 ,it is the work of grammatical analysis to separate the various propositions often combined into a single sentence4 ,logic cannot be properly required to interpret the forms and devices of language1 but only to treat the meaning when clearly exhibited4 99,partial ,identities499 ,a second highly important kind of proposition is that which ,i propose to call 99a partial identity994 ,when we say that 8,all mammalia are vertebrata10 we do not mean that mammalian animals are identical with vertebrate animals1 but only that the mammalia form a 99part of the class vertebrata994 ,such a proposition was regarded in the old logic as asserting the inclusion of one class in another1 or of an object in a class4 ,it was called a universal affirmative proposition1 because the attribute 99vertebrate99 was affirmed of the whole subject 99mammalia992 but the attribute was said to be 99undistributed991 because not all vertebrata were of necessity involved in the proposition4 ,aristotle1 overlooking the importance of simple identities1 and indeed almost denying their existence1 unfortunately founded his system upon the notion of inclusion in a class1 instead of adopting the basis of identity4 ,he regarded inference as resting upon the rule that what is true of the containing class is true of the contained1 in place of the vastly more general rule that what is true of a class or thing is true of the like4 ,thus he not only reduced logic to a fragment of its proper self1 but destroyed the deep analogies which bind together logical and mathematical reasoning4 ,hence a crowd of defects1 difficulties and errors which will long disfigure the first and simplest of the sciences4 ,it is surely evident that the relation of inclusion rests upon the relation of identity4 ,mammalian animals cannot be included among vertebrates unless they be identical with part of the vertebrates4 ,cabinet ,ministers are included almost always in the class ,members of ,parliament1 because they are identical with some who sit in ,parliament4 ,we may indicate this identity with a part of the larger class in various ways2 as for instance1 ,mammalia part of the vertebrata4 ,diatomace a class of plants4 ,cabinet ,ministers some members of ,parliament4 ,iron a metal4 ,in ordinary language the verbs 99is99 and 99are99 express mere inclusion more often than not4 99,men are mortals991 means that 99men99 form a part of the class 99mortal992 but great confusion exists between this sense of the verb and that in which it expresses identity1 as in 8,the sun is the centre of the planetary system40 ,the introduction of the indefinite article 99a99 often expresses partiality2 when we say 8,iron is a metal0 we clearly mean that iron is 99one only99 of several metals4 ,certain recent logicians have proposed to avoid the indefiniteness in question by what is called the ,quantification of the ,predicate1 and they have generally used the little word 99some99 to show that only a part of the predicate is identical with the subject4 99,some99 is an 99indeterminate adjective992 it implies unknown qualities by which we might select the part in question if the qualities were known1 but it gives no hint as to their nature4 ,i might make use of such an indeterminate sign to express partial identities in this work4 ,thus1 taking the special symbol ,v ,some1 the general form of a partial identity would be ,a ,,vb1 and in ,boole's ,logic expressions of the kind were much used4 ,but ,i believe that indeterminate symbols only introduce complexity1 and destroy the beauty and simple universality of the system which may be created without their use4 ,a vague word like 99some99 is only used in ordinary language by 99ellipsis991 and to avoid the trouble of attaining accuracy4 ,we can always employ more definite expressions if we like2 but when once the indefinite 99some99 is introduced we cannot replace it by the special description4 ,we do not know whether 99some99 colour is red1 yellow1 blue1 or what it is2 but on the other hand 99red99 colour is certainly 99some99 colour4 ,throughout this system of logic ,i shall dispense with such indefinite expressions2 and this can readily be done by substituting one of the other terms4 ,to express the proposition 8,all ,a's are some ,b's0 ,i shall not use the form ,a ,,vb1 but ,a ,,ab4 ,this formula states that the class ,a is identical with the class ,,ab2 and as the latter must be a part at least of the class ,b1 it implies the inclusion of the class ,a in that of ,b4 ,we might represent our former example thus1 ,mammalia ,mammalian vertebrata4 ,this proposition asserts identity between a part 7or it may be the whole7 of the vertebrata and the mammalia4 ,if it is asked ,what part8 the proposition affords no answer1 except that it is the part which is mammalian2 but the assertion 8mammalia some vertebrata0 tells us no more4 ,it is quite likely that some readers will think this mode of representing the universal affirmative proposition artificial and complicated4 ,i will not undertake to convince them of the opposite at this point of my exposition4 ,justification for it will be found1 not so much in the immediate treatment of this proposition1 as in the general harmony which it will enable us to disclose between all parts of reasoning4 ,i have no doubt that this is the critical difficulty in the relation of logical to other forms of reasoning4 ,grant this mode of denoting that 8all ,a's are ,b's10 and ,i fear no further difficulties2 refuse it1 and we find want of analogy and endless anomaly in every direction4 ,it is on general grounds that ,i hope to show overwhelming reasons for seeking to reduce every kind of proposition to the form of an identity4 ,i may add that not a few logicians have accepted this view of the universal affirmative proposition4 ,leibnitz1 in his 99,difficultates ,qudam ,logic991 adopts it1 saying1 8,omne ,a est ,b2 id est quivalent ,,ab et ,a1 seu ,a non ,b est nonens40 ,boole employed the logical equation 99x99 99xy99 concurrently with 99x99 99vy992 and ,spalding#eb distinctly says that the proposition 8all metals are minerals0 might be described as an assertion of 99partial identity99 between the two classes4 ,hence the name which ,i have adopted for the proposition4 #eb 99,encyclopdia ,britannica991 ,eighth ,ed4 art4 ,logic1 sect4 #cg1 note4 #hvo4 reprint1 p4 #gi4 99,limited ,identities499 ,an important class of propositions have the form ,,ab ,,ac1 expressing the identity of the class ,,ab with the class ,,ac4 ,in other words1 8,within the sphere of the class ,a1 all the ,b's are all the ,c's20 or again1 8,the ,b's and ,c's1 which are ,a's1 are identical40 ,but it will be observed that nothing is asserted concerning things which are outside of the class ,a2 and thus the identity is of limited extent4 ,it is the proposition ,b ,c limited to the sphere of things called ,a4 ,thus we may say1 with some approximation to truth1 that 8,large plants are plants devoid of locomotive power40 ,a barrister may make numbers of most general statements concerning the relations of persons and things in the course of an argument1 but it is of course to be understood that he speaks only of persons and things under the ,english ,law4 ,even mathematicians make statements which are not true with absolute generality4 ,they say that imaginary roots enter into equations by pairs2 but this is only true under the tacit condition that the equations in question shall not have imaginary coefficients4#ec ,the universe1 in short1 within which they habitually discourse is that of equations with real coefficients4 ,these implied limitations form part of that great mass of tacit knowledge which accompanies all special arguments4 #ec ,de ,morgan1 99,on the ,root of any ,function994 ,cambridge ,philosophical ,transactions1 #ahfg1 vol4 xi4 p4 #be4 ,to ,de ,morgan is due the remark1 that we do usually think and argue in a limited universe or sphere of notions1 even when it is not expressly stated4#ed #ed 99,syllabus of a proposed ,system of ,logic991 #abb1 #abc4 ,it is worthy of inquiry whether all identities are not really limited to an implied sphere of meaning4 ,when we make such a plain statement as 8,gold is malleable0 we obviously speak of gold only in its solid state2 when we say that 8,mercury is a liquid metal0 we must be understood to exclude the frozen condition to which it may be reduced in the ,arctic regions4 ,even when we take such a fundamental law of nature as 8,all substances gravitate10 we must mean by substance1 material substance1 not including that basis of heat1 light1 and electrical undulations which occupies space and possesses many wonderful mechanical properties1 but not gravity4 ,the proposition then is really of the form ,material substance ,material gravitating substance4 99,negative ,propositions499 ,in every act of intellect we are engaged with a certain identity or difference between things or sensations compared together4 ,hitherto ,i have treated only of identities2 and yet it might seem that the relation of difference must be infinitely more common than that of likeness4 ,one thing may resemble a great many other things1 but then it differs from all remaining things in the world4 ,diversity may almost be said to constitute life1 being to thought what motion is to a river4 ,the perception of an object involves its discrimination from all other objects4 ,but we may nevertheless be said to detect resemblance as often as we detect difference4 ,we cannot1 in fact1 assert the existence of a difference1 without at the same time implying the existence of an agreement4 ,if ,i compare mercury1 for instance1 with other metals1 and decide that it is 99not solid991 here is a difference between mercury and solid things1 expressed in a negative proposition2 but there must be implied1 at the same time1 an agreement between mercury and the other substances which are not solid4 ,as it is impossible to separate the vowels of the alphabet from the consonants without at the same time separating the consonants from the vowels1 so ,i cannot select as the object of thought 99solid things991 without thereby throwing together into another class all things which are 99not solid994 ,the very fact of not possessing a quality1 constitutes a new quality which may be the ground of judgment and classification4 ,in this point of view1 agreement and difference are ever the two sides of the same act of intellect1 and it becomes equally possible to express the same judgment in the one or other aspect4 ,between affirmation and negation there is accordingly a perfect equilibrium4 ,every affirmative proposition implies a negative one1 and 99vice vers994 ,it is even a matter of indifference1 in a logical point of view1 whether a positive or negative term be used to denote a given quality and the class of things possessing it4 ,if the ordinary state of a man's body be called 99good health991 then in other circumstances he is said 99not to be in good health992 but we might equally describe him in the latter state as 99sickly991 and in his normal condition he would be 99not sickly994 ,animal and vegetable substances are now called 99organic991 so that the other substances1 forming an immensely greater part of the globe1 are described negatively as 99inorganic994 ,but we might1 with at least equal logical correctness1 have described the preponderating class of substances as 99mineral991 and then vegetable and animal substances would have been 99non-mineral994 ,it is plain that any positive term and its corresponding negative divide between them the whole universe of thought3 whatever does not fall into one must fall into the other1 by the third fundamental ,law of ,thought1 the ,law of ,duality4 ,it follows at once that there are two modes of representing a difference4 ,supposing that the things represented by ,a and ,b are found to differ1 we may indicate 7see p4 #ag7 the result of the judgment by the notation ,a ,b4 ,we may now represent the same judgment by the assertion that ,a agrees with those things which differ from ,b1 or that ,a agrees with the not-,b's4 ,using our notation for negative terms 7see p4 #ad71 we obtain ,a ,a99b99 as the expression of the ordinary negative proposition4 ,thus if we take ,a to mean quicksilver1 and ,b solid1 then we have the following proposition3-- ,quicksilver ,quicksilver not-solid4 ,there may also be several other classes of negative propositions1 of which no notice was taken in the old logic4 ,we may have cases where all ,a's are not-,b's1 and at the same time all not-,b's are ,a's2 there may1 in short1 be a simple identity between ,a and not-,b1 which may be expressed in the form ,a 99b994 ,an example of this form would be ,conductors of electricity non-electrics4 ,we shall also frequently have to deal as results of deduction1 with simple1 partial1 or limited identities between negative terms1 as in the forms 99a99 99b991 99a99 99a9999b991 99a99,c 99b99,c1 etc4 ,it would be possible to represent affirmative propositions in the negative form4 ,thus 8,iron is solid10 might be expressed as 8,iron is not not-solid10 or 8,iron is not fluid20 or1 taking ,a and 99b99 for the terms 8iron10 and 8not-solid10 the form would be ,a 99b994 ,but there are very strong reasons why we should employ all propositions in their affirmative form4 ,all inference proceeds by the substitution of equivalents1 and a proposition expressed in the form of an identity is ready to yield all its consequences in the most direct manner4 ,as will be more fully shown1 we can infer 99in99 a negative proposition1 but not 99by99 it4 ,difference is incapable of becoming the ground of inference2 it is only the implied agreement with other differing objects which admits of deductive reasoning2 and it will always be found advantageous to employ propositions in the form which exhibits clearly the implied agreements4 99,conversion of ,propositions499 ,the old books of logic contain many rules concerning the conversion of propositions1 that is1 the transposition of the subject and predicate in such a way as to obtain a new proposition which will be true when the original proposition is true4 ,the reduction of every proposition to the form of an identity renders all such rules and processes needless4 ,identity is essentially reciprocal4 ,if the colour of the ,atlantic ,ocean is the same as that of the ,pacific ,ocean1 that of the ,pacific must be the same as that of the ,atlantic4 ,sodium chloride being identical with common salt1 common salt must be identical with sodium chloride4 ,if the number of windows in ,salisbury ,cathedral equals the number of days in the year1 the number of days in the year must equal the number of the windows4 ,lord ,chesterfield was not wrong when he said1 8,i will give anybody their choice of these two truths1 which amount to the same thing2 ,he who loves himself best is the honestest man2 or1 ,the honestest man loves himself best40 ,scotus ,erigena exactly expresses this reciprocal character of identity in saying1 8,there are not two studies1 one of philosophy and the other of religion2 true philosophy is true religion1 and true religion is true philosophy40 ,a mathematician would not think it worth while to mention that if 99x99 99y99 then also 99y99 99x994 ,he would not consider these to be two equations at all1 but one equation accidentally written in two different manners4 ,in written symbols one of two names must come first1 and the other second1 and a like succession must perhaps be observed in our thoughts3 but in the relation of identity there is no need for succession in order 7see p4 #cc71 each is simultaneously equal and identical to the other4 ,these remarks will hold true both of logical and mathematical identity2 so that ,i shall consider the two forms ,a ,b and ,b ,a to express exactly the same identity differently written4 ,all need for rules of conversion disappears1 and there will be no single proposition in the system which may not be written with either end foremost4 ,thus ,a ,,ab is the same as ,,ab ,a1 99a99,c 99b99,c is the same as 99b99,c 99a99,c1 and so forth4 ,the same remarks are partially true of differences and inequalities1 which are also reciprocal to the extent that one thing cannot differ from a second without the second differing from the first4 ,mars differs in colour from ,venus1 and ,venus must differ from ,mars4 ,the ,earth differs from ,jupiter in density2 therefore ,jupiter must differ from the ,earth4 ,speaking generally1 if ,a ,b we shall also have ,b ,a1 and these two forms may be considered expressions of the same difference4 ,but the relation of differing things is not wholly reciprocal4 ,the density of ,jupiter does not differ from that of the ,earth in the same way that that of the ,earth differs from that of ,jupiter4 ,the change of sensation which we experience in passing from ,venus to ,mars is not the same as what we experience in passing back to ,venus1 but just the opposite in nature4 ,the colour of the sky is lighter than that of the ocean2 therefore that of the ocean cannot be lighter than that of the sky1 but darker4 ,in these and all similar cases we gain a notion of 99direction99 or character of change1 and results of immense importance may be shown to rest on this notion4 ,for the present we shall be concerned with the mere fact of identity existing or not existing4 99,twofold ,interpretation of ,propositions499 ,terms1 as we have seen 7p4 #be71 may have a meaning either in extension or intension2 and according as one or the other meaning is attributed to the terms of a proposition1 so may a different interpretation be assigned to the proposition itself4 ,when the terms are abstract we must read them in intension1 and a proposition connecting such terms must denote the identity or non-identity of the qualities respectively denoted by the terms4 ,thus if we say ,equality ,identity of magnitude1 the assertion means that the circumstance of being equal exactly corresponds with the circumstance of being identical in magnitude4 ,similarly in ,opacity ,incapability of transmitting light1 the quality of being incapable of transmitting light is declared to be the same as the intended meaning of the word opacity4 ,when general names form the terms of a proposition we may apply a double interpretation4 ,thus ,exogens ,dicotyledons means either that the qualities which belong to all exogens are the same as those which belong to all dicotyledons1 or else that every individual falling under one name falls equally under the other4 ,hence it may be said that there are two distinct fields of logical thought4 ,we may argue either by the qualitative meaning of names or by the quantitative1 that is1 the extensive meaning4 ,every argument involving concrete plural terms might be converted into one involving only abstract singular terms1 and 99vice vers994 ,but there are reasons for believing that the intensive or qualitative form of reasoning is the primary and fundamental one4 ,it is sufficient to point out that the extensive meaning of a name is a changeable and fleeting thing1 while the intensive meaning may nevertheless remain fixed4 ,very numerous additions have been lately made to the extensive meanings both of planet and element4 ,every iron steam-ship which is made or destroyed adds to or subtracts from the extensive meaning of the name steam-ship1 without necessarily affecting the intensive meaning4 ,stage coach means as much as ever in one way1 but in extension the class is nearly extinct4 ,chinese railway1 on the other hand1 is a term represented only by a single instance2 in twenty years it may be the name of a large class4 ,,chapter ,,iv4 ,,deductive ,,reasoning4 ,the general principle of inference having been explained in the previous chapters1 and a suitable system of symbols provided1 we have now before us the comparatively easy task of tracing out the most common and important forms of deductive reasoning4 ,the general problem of deduction is as follows3--99,from one or more propositions called premises to draw such other propositions as will necessarily be true when the premises are true499 ,by deduction we investigate and unfold the information contained in the premises2 and this we can do by one single rule--99,for any term occurring in any proposition substitute the term which is asserted in any premise to be identical with it499 ,to obtain certain deductions1 especially those involving negative conclusions1 we shall require to bring into use the second and third ,laws of ,thought1 and the process of reasoning will then be called 99,indirect ,deduction994 ,in the present chapter1 however1 ,i shall confine my attention to those results which can be obtained by the process of 99,direct ,deduction991 that is1 by applying to the premises themselves the rule of substitution4 ,it will be found that we can combine into one harmonious system1 not only the various moods of the ancient syllogism but a great number of equally important forms of reasoning1 which had no recognised place in the old logic4 ,we can at the same time dispense entirely with the elaborate apparatus of logical rules and mnemonic lines1 which were requisite so long as the vital principle of reasoning was not clearly expressed4 99,immediate ,inference499 ,probably the simplest of all forms of inference is that which has been called 99,immediate ,inference991 because it can be performed upon a single proposition4 ,it consists in joining an adjective1 or other qualifying clause of the same nature1 to both sides of an identity1 and asserting the equivalence of the terms thus produced4 ,for instance1 since ,conductors of electricity ,non-electrics1 it follows that ,liquid conductors of electricity ,liquid non-electrics4 ,if we suppose that ,plants ,bodies decomposing carbonic acid1 it follows that ,microscopic plants ,microscopic bodies decomposing carbonic acid4 ,in general terms1 from the identity ,a ,b we can infer the identity ,,ac ,,bc4 ,this is but a case of plain substitution2 for by the first ,law of ,thought it must be admitted that ,,ac ,,ac1 and if1 in the second side of this identity1 we substitute for ,a its equivalent ,b1 we obtain ,,ac ,,bc4 ,in like manner from the partial identity ,a ,,ab we may obtain ,,ac ,,abc by an exactly similar act of substitution2 and in every other case the rule will be found capable of verification by the principle of inference4 ,the process when performed as here described will be quite free from the liability to error which ,i have shown#ee to exist in 8,immediate ,inference by added ,determinants10 as described by ,dr4 ,thomson4#ef #ee 99,elementary ,lessons in ,logic991 p4 #hf4 #ef 99,outline of the ,laws of ,thought991 #hg4 99,inference with ,two ,simple ,identities499 ,one of the most common forms of inference1 and one to which ,i shall especially direct attention1 is practised with two simple identities4 ,from the two statements that 8,london is the capital of ,england0 and 8,london is the most populous city in the world10 we instantaneously draw the conclusion that 8,the capital of ,england is the most populous city in the world40 ,similarly1 from the identities ,hydrogen ,substance of least density1 ,hydrogen ,substance of least atomic weight1 we infer ,substance of least density ,substance of least atomic weight4 ,the general form of the argument is exhibited in the symbols ,b ,a 7#a7 ,b ,c 7#b7 hence ,a ,c4 7#c7 ,we may describe the result by saying that terms identical with the same term are identical with each other2 and it is impossible to overlook the analogy to the first axiom of ,euclid that 8things equal to the same thing are equal to each other40 ,it has been very commonly supposed that this is a fundamental principle of thought1 incapable of reduction to anything simpler4 ,but ,i entertain no doubt that this form of reasoning is only one case of the general rule of inference4 ,we have two propositions1 ,a ,b and ,b ,c1 and we may for a moment consider the second one as affirming a truth concerning ,b1 while the former one informs us that ,b is identical with ,a2 hence by substitution we may affirm the same truth of ,a4 ,it happens in this particular case that the truth affirmed is identity to ,c1 and we might1 if we preferred it1 have considered the substitution as made by means of the second identity in the first4 ,having two identities we have a choice of the mode in which we will make the substitution1 though the result is exactly the same in either case4 ,now compare the three following formul1 7#a7 ,a ,b ,c1 hence ,a ,c 7#b7 ,a ,b ,c1 hence ,a ,c 7#c7 ,a ,b ,c1 no inference4 ,in the second formula we have an identity and a difference1 and we are able to infer a difference2 in the third we have two differences and are unable to make any inference at all4 ,because ,a and ,c both differ from ,b1 we cannot tell whether they will or will not differ from each other4 ,the flowers and leaves of a plant may both differ in colour from the earth in which the plant grows1 and yet they may differ from each other2 in other cases the leaves and stem may both differ from the soil and yet agree with each other4 ,where we have difference only we can make no inference2 where we have identity we can infer4 ,this fact gives great countenance to my assertion that inference proceeds always through identity1 but may be equally well effected in propositions asserting difference or identity4 ,deferring a more complete discussion of this point1 ,i will only mention now that arguments from double identity occur very frequently1 and are usually taken for granted1 owing to their extreme simplicity4 ,in regard to the equivalence of words this form of inference must be constantly employed4 ,if the ancient ,greek is our 99copper991 then it must be the ,french 99cuivre991 the ,german 99kupfer991 the ,latin 99cuprum991 because these are words1 in one sense at least1 equivalent to copper4 ,whenever we can give two definitions or expressions for the same term1 the formula applies2 thus ,senior defined wealth as 8,all those things1 and those things only1 which are transferable1 are limited in supply1 and are directly or indirectly productive of pleasure or preventive of pain40 ,wealth is also equivalent to 8things which have value in exchange20 hence obviously1 8things which have value in exchange all those things1 and those things only1 which are transferable1 c40 ,two expressions for the same term are often given in the same sentence1 and their equivalence implied4 ,thus ,thomson and ,tait say1#eg 8,the naturalist may be content to know matter as that which can be perceived by the senses1 or as that which can be acted upon by or can exert force40 ,i take this to mean-- ,matter what can be perceived by the senses2 ,matter what can be acted upon by or can exert force4 #eg 99,treatise on ,natural ,philosophy991 vol4 i4 p4 #afa4 ,for the term 8matter0 in either of these identities we may substitute its equivalent given in the other definition4 ,elsewhere they often employ sentences of the form exemplified in the following3#eh 8,the integral curvature1 or whole change of direction of an arc of a plane curve1 is the angle through which the tangent has turned as we pass from one extremity to the other40 ,this sentence is certainly of the form-- ,the integral curvature the whole change of direction1 c4 the angle through which the tangent has turned1 c4 #eh 99,treatise on ,natural ,philosophy991 vol4 i4 p4 #f4 ,disguised cases of the same kind of inference occur throughout all sciences1 and a remarkable instance is found in algebraic geometry4 ,mathematicians readily show that every equation of the form 99y99 99mx99 99c99 corresponds to or represents a straight line2 it is also easily proved that the same equation is equivalent to one of the general form ,a99x99 ,b99y99 ,c #j1 and 99vice vers994 ,hence it follows that every equation of the form in question1 that is to say1 every equation of the first degree1 corresponds to or represents a straight line4#ei #ei ,todhunter's 99,plane ,co-ordinate ,geometry991 chap4 ii4 pp4 #aa-ad4 99,inference with a ,simple and a ,partial ,identity499 ,a form of reasoning somewhat different from that last considered consists in inference-between a simple and a partial identity4 ,if we have two propositions of the forms ,a ,b1 ,b ,,bc1 we may then substitute for ,b in either proposition its equivalent in the other1 getting in both cases ,a ,,bc2 in this we may if we like make a second substitution for ,b1 getting ,a ,,ac4 ,thus1 since 8,the ,mont ,blanc is the highest mountain in ,europe1 and the ,mont ,blanc is deeply covered with snow10 we infer by an obvious substitution that 8,the highest mountain in ,europe is deeply covered with snow40 ,these propositions when rigorously stated fall into the forms above exhibited4 ,this mode of inference is constantly employed when for a term we substitute its definition1 or 99vice vers994 ,the very purpose of a definition is to allow a single noun to be employed in place of a long descriptive phrase4 ,thus1 when we say 8,a circle is a curve of the second degree10 we may substitute a definition of the circle1 getting 8,a curve1 all points of which are at equal distances from one point1 is a curve of the second degree40 ,the real forms of the propositions here given are exactly those shown in the symbolic statement1 but in this and many other cases it will be sufficient to state them in ordinary elliptical language for sake of brevity4 ,in scientific treatises a term and its definition are often both given in the same sentence1 as in 8,the weight of a body in any given locality1 or the force with which the earth attracts it1 is proportional to its mass40 ,the conjunction 99or99 in this statement gives the force of equivalence to the parenthetic phrase1 so that the propositions really are ,weight of a body force with which the earth attracts it4 ,weight of a body weight1 c4 proportional to its mass4 ,a slightly different case of inference consists in substituting in a proposition of the form ,a ,,ab1 a definition of the term ,b4 ,thus from ,a ,,ab and ,b ,c we get ,a ,,ac4 ,for instance1 we may say that 8,metals are elements0 and 8,elements are incapable of decomposition40 ,metal metal element4 ,element what is incapable of decomposition4 ,hence ,metal metal incapable of decomposition4 ,it is almost needless to point out that the form of these arguments does not suffer any real modification if some of the terms happen to be negative2 indeed in the last example 8incapable of decomposition0 may be treated as a negative term4 ,taking ,a metal ,b element ,c capable of decomposition 99c99 incapable of decomposition2 the propositions are of the forms ,a ,,ab ,b 99c99 whence1 by substitution1 ,a ,a99c994 99,inference of a ,partial from ,two ,partial ,identities499 ,however common be the cases of inference already noticed1 there is a form occurring almost more frequently1 and which deserves much attention1 because it occupied a prominent place in the ancient syllogistic system4 ,that system strangely overlooked all the kinds of argument we have as yet considered1 and selected1 as the type of all reasoning1 one which employs two partial identities as premises4 ,thus from the propositions ,sodium is a metal 7#a7 ,metals conduct electricity1 7#b7 we may conclude that ,sodium conducts electricity4 7#c7 ,taking ,a1 ,b1 ,c to represent the three terms respectively1 the premises are of the forms ,a ,,ab 7#a7 ,b ,,bc4 7#b7 ,now for ,b in 7#a7 we can substitute its expression as given in 7#b71 obtaining ,a ,,abc1 7#c7 or1 in words1 from ,sodium sodium metal1 7#a7 ,metal metal conducting electricity1 7#b7 we infer ,sodium sodium metal conducting electricity1 7#c7 which1 in the elliptical language of common life1 becomes 8,sodium conducts electricity40 ,the above is a syllogism in the mood called ,barbara#fj in the truly barbarous language of ancient logicians2 and the first figure of the syllogism contained ,barbara and three other moods which were esteemed distinct forms of argument4 ,but it is worthy of notice that1 without any real change in our form of inference1 we readily include these three other moods under ,barbara4 ,the negative mood ,celarent will be represented by the example #fj ,an explanation of this and other technical terms of the old logic will be found in my 99,elementary ,lessons in ,logic991 ,sixth ,edition1 #ahgf2 ,macmillan4 ,neptune is a planet1 7#a7 ,no planet has retrograde motion2 7#b7 ,hence ,neptune has not retrograde motion4 7#c7 ,if we put ,a for ,neptune1 ,b for planet1 and ,c for 8having retrograde motion10 then by the corresponding negative term c1 we denote 8not having retrograde motion40 ,the premises now fall into the forms ,a ,,ab 7#a7 ,b ,b99c991 7#b7 and by substitution for ,b1 exactly as before1 we obtain ,a ,,ab99c994 7#c7 ,what is called in the old logic a particular conclusion may be deduced without any real variation in the symbols4 ,particular quantity is indicated1 as before mentioned 7p4 #da71 by joining to the term an indefinite adjective of quantity1 such as 99some991 99a part of991 99certain991 c41 meaning that an unknown part of the term enters into the proposition as subject4 ,considerable doubt and ambiguity arise out of the question whether the part may not in some cases be the whole1 and in the syllogism at least it must be understood in this sense4#fa ,now1 if we take a letter to represent this indefinite part1 we need make no change in our formul to express the syllogisms ,darii and ,ferio4 ,consider the example-- #fa 99,elementary ,lessons in ,logic991 pp4 #fg1 #gi4 ,some metals are of less density than water1 7#a7 ,all bodies of less density than water will float upon the surface of water2 hence 7#b7 ,some metals will float upon the surface of water4 7#c7 ,let ,a some metals1 ,b body of less density than water1 ,c floating on the surface of water then the propositions are evidently as before1 ,a ,,ab1 7#a7 ,b ,,bc2 7#b7 hence ,a ,,abc1 7#c7 ,thus the syllogism ,darii does not really differ from ,barbara4 ,if the reader prefer it1 we can readily employ a distinct symbol for the indefinite sign of quantity4 ,let ,p some1 ,q metal1 ,b and ,c having the same meanings as before4 ,then the premises become ,,pq ,,pqb1 7#a7 ,b ,,bc2 7#b7 hence1 by substitution1 as before1 ,,pq ,,pqbc4 7#c7 ,except that the formul look a little more complicated there is no difference whatever4 ,the mood ,ferio is of exactly the same character as ,darii or ,barbara1 except that it involves the use of a negative term4 ,take the example1 ,bodies which are equally elastic in all directions do not doubly refract light2 ,some crystals are bodies equally elastic in all directions2 therefore1 some crystals do not doubly refract light4 ,assigning the letters as follows3-- ,a some crystals1 ,b bodies equally elastic in all directions1 ,c doubly refracting light1 99c99 not doubly refracting light4 ,our argument is of the same form as before1 and may be concisely stated in one line1 ,a ,,ab ,,ab99c994 ,if it is preferred to put ,,pq for the indefinite 99some crystals991 we have ,,pq ,,pqb ,,pqb99c994 ,the only difference is that the negative term c takes the place of ,c in the mood ,darii4 99,ellipsis of ,terms in ,partial ,identities499 ,the reader will probably have noticed that the conclusion which we obtain from premises is often more full than that drawn by the old ,aristotelian processes4 ,thus from 8,sodium is a metal10 and 8,metals conduct electricity10 we inferred 7p4 #ee7 that 8,sodium sodium1 metal1 conducting electricity10 whereas the old logic simply concludes that 8,sodium conducts electricity40 ,symbolically1 from ,a ,,ab1 and ,b ,,bc1 we get ,a ,,abc1 whereas the old logic gets at the most ,a ,,ac4 ,it is therefore well to show that without employing any other principles of inference than those already described1 we may infer ,a ,,ac from ,a ,,abc1 though we cannot infer the latter more full and accurate result from the former4 ,we may show this most simply as follows3-- ,by the first ,law of ,thought it is evident that ,,aa ,,aa2 and if we have given the proposition ,a ,,abc1 we may substitute for both the ,a's in the second side of the above1 obtaining ,,aa ,,abc 4 ,,abc4 ,but from the property of logical symbols expressed in the ,law of ,simplicity 7p4 #cc7 some of the repeated letters may be made to coalesce1 and we have ,a ,,abc 4 ,c4 ,substituting again for ,,abc its equivalent ,a1 we obtain ,a ,,ac1 the desired result4 ,by a similar process of reasoning it may be shown that we can always drop out any term appearing in one member of a proposition1 provided that we substitute for it the whole of the other member4 ,this process was described in my first logical ,essay1#fb as 99,intrinsic ,elimination991 but it might perhaps be better entitled the 99,ellipsis of ,terms994 ,it enables us to get rid of needless terms by strict substitutive reasoning4 #fb 99,pure ,logic991 p4 #ai4 99,inference of a ,simple from ,two ,partial ,identities499 ,two terms may be connected together by two partial identities in yet another manner1 and a case of inference then arises which is of the highest importance4 ,in the two premises ,a ,,ab 7#a7 ,b ,,ab 7#b7 the second member of each is the same2 so that we can by obvious substitution obtain ,a ,b4 ,thus1 in plain geometry we readily prove that 8,every equilateral triangle is also an equiangular triangle10 and we can with equal ease prove that 8,every equiangular triangle is an equilateral triangle40 ,thence by substitution1 as explained above1 we pass to the simple identity1 ,equilateral triangle equiangular triangle4 ,we thus prove that one class of triangles is entirely identical with another class2 that is to say1 they differ only in our way of naming and regarding them4 ,the great importance of this process of inference arises from the fact that the conclusion is more simple and general than either of the premises1 and contains as much information as both of them put together4 ,it is on this account constantly employed in inductive investigation1 as will afterwards be more fully explained1 and it is the natural mode by which we arrive at a conviction of the truth of simple identities as existing between classes of numerous objects4 99,inference of a ,limited from ,two ,partial ,identities499 ,we have considered some arguments which are of the type treated by ,aristotle in the first figure of the syllogism4 ,but there exist two other types of argument which employ a pair of partial identities4 ,if our premises are as shown in these symbols1 ,b ,,ab 7#a7 ,b ,,cb1 7#b7 we may substitute for ,b either by 7#a7 in 7#b7 or by 7#b7 in 7#a71 and by both modes we obtain the conclusion ,,ab ,,cb1 7#c7 a proposition of the kind which we have called a limited identity 7p4 #db74 ,thus1 for example1 ,potassium potassium metal 7#a7 ,potassium potassium capable of floating on water2 7#b7 hence ,potassium metal potassium capable of floating on water4 7#c7 ,this is really a syllogism of the mood ,darapti in the third figure1 except that we obtain a conclusion of a more exact character than the old syllogism gives4 ,from the premises 8,potassium is a metal0 and 8,potassium floats on water10 ,aristotle would have inferred that 8,some metals float on water40 ,but if inquiry were made what the 8some metals0 are1 the answer would certainly be 8,metal which is potassium40 ,hence ,aristotle's conclusion simply leaves out some of the information afforded in the premises4 ,it even leaves us open to interpret the 99some metals99 in a wider sense than we are warranted in doing4 ,from these distinct defects of the old syllogism the process of substitution is free1 and the new process only incurs the possible objection of being tediously minute and accurate4 99,miscellaneous ,forms of ,deductive ,inference499 ,the more common forms of deductive reasoning having been exhibited and demonstrated on the principle of substitution1 there still remain many1 in fact an indefinite number1 which may be explained with nearly equal ease4 ,such as involve the use of disjunctive propositions will be described in a later chapter1 and several of the syllogistic moods which include negative terms will be more conveniently treated after we have introduced the symbolic use of the second and third laws of thought4 ,we sometimes meet with a chain of propositions which allow of repeated substitution1 and form an argument called in the old logic a ,sorites4 ,take1 for instance1 the premises ,iron is a metal1 7#a7 ,metals are good conductors of electricity1 7#b7 ,good conductors of electricity are useful for telegraphic purposes4 7#c7 ,it obviously follows that ,iron is useful for telegraphic purposes4 7#d7 ,now if we take our letters thus1 ,a ,iron1 ,b metal1 ,c good conductor of electricity1 ,d useful for telegraphic purposes1 the premises will assume the forms ,a ,,ab1 7#a7 ,b ,,bc1 7#b7 ,c ,,cd4 7#c7 ,for ,b in 7#a7 we can substitute its equivalent in 7#b7 obtaining1 as before1 ,a ,,abc4 ,substituting for ,c in this intermediate result its equivalent as given in 7#c71 we obtain the complete conclusion ,a ,,abcd4 7#d7 ,the full interpretation is that 99,iron is iron1 metal1 good conductor of electricity1 useful for telegraphic purposes991 which is abridged in common language by the ellipsis of the circumstances which are not of immediate importance4 ,instead of all the propositions being exactly of the same kind as in the last example1 we may have a series of premises of various character2 for instance1 ,common salt is sodium chloride1 7#a7 ,sodium chloride crystallizes in a cubical form1 7#b7 ,what crystallizes in a cubical form does not possess the power of double refraction2 7#c7 it will follow that ,common salt does not possess the power of double refraction4 7#d7 ,taking our letter-terms thus1 ,a ,common salt1 ,b ,sodium chloride1 ,c ,crystallizing in a cubical form1 ,d ,possessing the power of double refraction1 we may state the premises in the forms ,a ,b1 7#a7 ,b ,,bc1 7#b7 ,c ,c99d994 7#c7 ,substituting by 7#c7 in 7#b7 and then by 7#b7 as thus altered in 7#a7 we obtain ,a ,,bc99d991 7#d7 which is a more precise version of the common conclusion4 ,we often meet with a series of propositions describing the qualities or circumstances of the one same thing1 and we may combine them all into one proposition by the process of substitution4 ,this case is1 in fact1 that which ,dr4 ,thomson has called 8,immediate ,inference by the sum of several predicates10 and his example will serve my purpose well4#fc ,he describes copper as 8,a metal--of a red colour--and disagreeable smell--and taste--all the preparations of which are poisonous--which is highly malleable--ductile--and tenacious--with a specific gravity of about #h4hc40 ,if we assign the letter ,a to copper1 and the succeeding letters of the alphabet in succession to the series of predicates1 we have nine distinct statements1 of the form ,a ,,ab 7#a7 ,a ,,ac 7#b7 ,a ,,ad 7#c7 444 ,a ,,ak 7#i74 ,we can readily combine these propositions into one by substituting for ,a in the second side of 7#a7 its expression in 7#b74 ,we thus get #fc 99,an ,outline of the ,necessary ,laws of ,thought991 ,fifth ,ed4 p4 #afa4 ,a ,,abc1 and by repeating the process over and over again we obviously get the single proposition ,a ,,abcd 444 ,,jk4 ,but ,dr4 ,thomson is mistaken in supposing that we can obtain in this manner a 99definition99 of copper4 ,strictly speaking1 the above proposition is only a 99description99 of copper1 and all the ordinary descriptions of substances in scientific works may be summed up in this form4 ,thus we may assert of the organic substances called ,paraffins that they are all saturated hydrocarbons1 incapable of uniting with other substances1 produced by heating the alcoholic iodides with zinc1 and so on4 ,it may be shown that no amount of ordinary description can be equivalent to a definition of any substance4 99,fallacies499 ,i have hitherto been engaged in showing that all the forms of reasoning of the old syllogistic logic1 and an indefinite number of other forms in addition1 may be readily and clearly explained on the single principle of substitution4 ,it is now desirable to show that the same principle will prevent us falling into fallacies4 ,so long as we exactly observe the one rule of substitution of equivalents it will be impossible to commit a 99paralogism991 that is to break any one of the elaborate rules of the ancient system4 ,the one new rule is thus proved to be as powerful as the six1 eight1 or more rules by which the correctness of syllogistic reasoning was guarded4 ,it was a fundamental rule1 for instance1 that two negative premises could give no conclusion4 ,if we take the propositions ,granite is not a sedimentary rock1 7#a7 ,basalt is not a sedimentary rock1 7#b7 we ought not to be able to draw any inference concerning the relation between granite and basalt4 ,taking our letter-terms thus3 ,a granite1 ,b sedimentary rock1 ,c basalt1 the premises may be expressed in the forms ,a ,b1 7#a7 ,c ,b4 7#b7 ,we have in this form two statements of difference2 but the principle of inference can only work with a statement of agreement or identity 7p4 #fc74 ,thus our rule gives us no power whatever of drawing any inference2 this is exactly in accordance with the fifth rule of the syllogism4 ,it is to be remembered1 indeed1 that we claim the power of always turning a negative proposition into an affirmative one 7p4 #de72 and it might seem that the old rule against negative premises would thus be circumvented4 ,let us try4 ,the premises 7#a7 and 7#b7 when affirmatively stated take the forms ,a ,a99b99 7#a7 ,c ,c99b994 7#b7 ,the reader will find it impossible by the rule of substitution to discover a relation between ,a and ,c4 ,three terms occur in the above premises1 namely ,a1 99b991 and ,c2 but they are so combined that no term occurring in one has its exact equivalent stated in the other4 ,no substitution can therefore be made1 and the principle of the fifth rule of the syllogism holds true4 ,fallacy is impossible4 ,it would be a mistake1 however1 to suppose that the mere occurrence of negative terms in both premises of a syllogism renders them incapable of yielding a conclusion4 ,the old rule informed us that from two negative premises no conclusion could be drawn1 but it is a fact that the rule in this bare form does not hold universally true2 and ,i am not aware that any precise explanation has been given of the conditions under which it is or is not imperative4 ,consider the following example3 ,whatever is not metallic is not capable of powerful magnetic influence1 7#a7 ,carbon is not metallic1 7#b7 ,therefore1 carbon is not capable of powerful magnetic influence4 7#c7 ,here we have two distinctly negative premises 7#a7 and 7#b71 and yet they yield a perfectly valid negative conclusion 7#c74 ,the syllogistic rule is actually falsified in its bare and general statement4 ,in this and many other cases we can convert the propositions into affirmative ones which will yield a conclusion by substitution without any difficulty4 ,to show this let ,a carbon1 ,b metallic1 ,c capable of powerful magnetic influence4 ,the premises readily take the forms 99b99 99bc991 7#a7 ,a ,a99b991 7#b7 and substitution for 99b99 in 7#b7 by means of 7#a7 gives the conclusion ,a ,a99bc994 7#c7 ,our principle of inference then includes the rule of negative premises whenever it is true1 and discriminates correctly between the cases where it does and does not hold true4 ,the paralogism1 anciently called 99the ,fallacy of ,undistributed ,middle991 is also easily exhibited and infallibly avoided by our system4 ,let the premises be ,hydrogen is an element1 7#a7 ,all metals are elements4 7#b7 ,according to the syllogistic rules the middle term 8element0 is here undistributed1 and no conclusion can be obtained2 we cannot tell then whether hydrogen is or is not a metal4 ,represent the terms as follows ,a hydrogen1 ,b element1 ,c metal4 ,the premises then become ,a ,,ab1 7#a7 ,c ,,cb4 7#b7 ,the reader will here1 as in a former page 7p4 #fb71 find it impossible to make any substitution4 ,the only term which occurs in both premises is ,b1 but it is differently combined in the two premises4 ,for ,b we must not substitute ,a1 which is equivalent to ,,ab1 not to ,b4 ,nor must we confuse together ,,cb and ,,ab1 which1 though they contain one common letter1 are different aggregate terms4 ,the rule of substitution gives us no right to decompose combinations2 and if we adhere rigidly to the rule1 that if two terms are stated to be equivalent we may substitute one for the other1 we cannot commit the fallacy4 ,it is apparent that the form of premises stated above is the same as that which we obtained by translating two negative premises into the affirmative form4 ,the old fallacy1 technically called the 99,illicit ,process of the ,major ,term991 is more easy to commit and more difficult to detect than any other breach of the syllogistic rules4 ,in our system it could hardly occur4 ,from the premises ,all planets are subject to gravity1 7#a7 ,fixed stars are not planets1 7#b7 we might inadvertently but fallaciously infer that1 8,fixed stars are not subject to gravity40 ,to reduce the premises to symbolic form1 let ,a planet ,b fixed star ,c subject to gravity2 then we have the propositions ,a ,,ac 7#a7 ,b ,b99a994 7#b7 ,the reader will try in vain to produce from these premises by legitimate substitution any relation between ,b and ,c2 he could not then commit the fallacy of asserting that ,b is not ,c4 ,there remain two other kinds of paralogism1 commonly known as the fallacy of ,four ,terms and the ,illicit ,process of the ,minor ,term4 ,they are so evidently impossible while we obey the rule of the substitution of equivalents1 that it is not necessary to give any illustrations4 ,when there are four distinct terms in two propositions as in ,a ,b and ,c ,d1 there could evidently be no opening for substitution4 ,as to the ,illicit ,process of the ,minor ,term it consists in a flagrant substitution for a term of another wider term which is not known to be equivalent to it1 and which is therefore not allowed by our rule to be substituted for it4 ,,chapter ,v4 ,,disjunctive ,,propositions4 ,in the previous chapter ,i have exhibited various cases of deductive reasoning by the process of substitution1 avoiding the introduction of disjunctive propositions2 but we cannot long defer the consideration of this more complex class of identities4 ,general terms arise1 as we have seen 7p4 #bd71 from classifying or mentally uniting together all objects which agree in certain qualities1 the value of this union consisting in the fact that the power of knowledge is multiplied thereby4 ,in forming such classes or general notions1 we overlook or abstract the points of difference which exist between the objects joined together1 and fix our attention only on the points of agreement4 ,but every process of thought may be said to have its inverse process1 which consists in undoing the effects of the direct process4 ,just as division undoes multiplication1 and evolution undoes involution1 so we must have a process which undoes generalization1 or the operation of forming general notions4 ,this inverse process will consist in distinguishing the separate objects or minor classes which are the constituent parts of any wider class4 ,if we mentally unite together certain objects visible in the sky and call them planets1 we shall afterwards need to distinguish the contents of this general notion1 which we do in the disjunctive proposition-- ,a planet is either ,mercury or ,venus or the ,earth or 444 or ,neptune4 ,having formed the very wide class 8vertebrate animal10 we may specify its subordinate classes thus3--8,a vertebrate animal is either a mammal1 bird1 reptile1 or fish40 ,nor is there any limit to the number of possible alternatives4 8,an exogenous plant is either a ranunculus1 a poppy1 a crucifer1 a rose1 or it belongs to some one of the other seventy natural orders of exogens at present recognized by botanists40 ,a cathedral church in ,england must be either that of ,london1 ,canterbury1 ,winchester1 ,salisbury1 ,manchester1 or of one of about twenty-four cities possessing such churches4 ,and if we were to attempt to specify the meaning of the term 8star10 we should require to enumerate as alternatives1 not only the many thousands of stars recorded in catalogues1 but the many millions unnamed4 ,whenever we thus distinguish the parts of a general notion we employ a disjunctive proposition1 in at least one side of which are several alternatives joined by the so-called disjunctive conjunction or1 a contracted form of 99other994 ,there must be some relation between the parts thus connected in one proposition2 we may call it the 99disjunctive99 or 99alternative99 relation1 and we must carefully inquire into its nature4 ,this relation is that of ignorance and doubt1 giving rise to choice4 ,whenever we classify and abstract we must open the way to such uncertainty4 ,by fixing our attention on certain attributes to the exclusion of others1 we necessarily leave it doubtful what those other attributes are4 ,the term 8molar tooth0 bears upon the face of it that it is a part of the wider term 8tooth40 ,but if we meet with the simple term 8tooth0 there is nothing to indicate whether it is an incisor1 a canine1 or a molar tooth4 ,this doubt1 however1 may be resolved by further information1 and we have to consider what are the appropriate logical processes for treating disjunctive propositions in connection with other propositions disjunctive or otherwise4 99,expression of the ,alternative ,relation499 ,in order to represent disjunctive propositions with convenience we require a sign of the alternative relation1 equivalent to one meaning at least of the little conjunction 99or99 so frequently used in common language4 ,i propose to use for this purpose the symbol 4 ,in my first logical essay ,i followed the practice of ,boole and adopted the sign 2 but this sign should not be employed unless there exists exact analogy between mathematical addition and logical alternation4 ,we shall find that the analogy is imperfect1 and that there is such profound difference between logical and mathematical terms as should prevent our uniting them by the same symbol4 ,accordingly ,i have chosen a sign 1 which seems aptly to suggest whatever degree of analogy may exist without implying more4 ,the exact meaning of the symbol we will now proceed to investigate4 99,nature of the ,alternative ,relation499 ,before treating disjunctive propositions it is indispensable to decide whether the alternatives must be considered exclusive or unexclusive4 ,by 99exclusive alternatives99 we mean those which cannot contain the same things4 ,if we say 8,arches are circular or pointed10 it is certainly to be understood that the same arch cannot be described as both circular and pointed4 ,many examples1 on the other hand1 can readily be suggested in which two or more alternatives may hold true of the same object4 ,thus ,luminous bodies are self-luminous or luminous by reflection4 ,it is undoubtedly possible1 by the laws of optics1 that the same surface may at one and the same moment give off light of its own and reflect light from other bodies4 ,we speak familiarly of 99deaf or dumb99 persons1 knowing that the majority of those who are deaf from birth are also dumb4 ,there can be no doubt that in a great many cases1 perhaps the greater number of cases1 alternatives are exclusive as a matter of fact4 ,any one number is incompatible with any other2 one point of time or place is exclusive of all others4 ,roger ,bacon died either in #abhd or #abib2 it is certain that he could not die in both years4 ,henry ,fielding was born either in ,dublin or ,somersetshire2 he could not be born in both places4 ,there is so much more precision and clearness in the use of exclusive alternatives that we ought doubtless to select them when possible4 ,old works on logic accordingly contained a rule directing that the 99,membra dividentia991 the parts of a division or the constituent species of a genus1 should be exclusive of each other4 ,it is no doubt owing to the great prevalence and convenience of exclusive divisions that the majority of logicians have held it necessary to make every alternative in a disjunctive proposition exclusive of every other one4 ,aquinas considered that when this was not the case the proposition was actually 99false991 and ,kant adopted the same opinion4#fd ,a multitude of statements to the same effect might readily be quoted1 and if the question were to be determined by the weight of historical evidence1 it would certainly go against my view4 ,among recent logicians ,hamilton1 as well as ,boole1 took the exclusive side4 ,but there are authorities to the opposite effect4 ,whately1 ,mansel1 and ,j4 ,s4 ,mill have all pointed out that we may often treat alternatives as 99,compossible991 or true at the same time4 ,whately gives us an example1#fe 8,virtue tends to procure us either the esteem of mankind1 or the favour of ,god10 and he adds--8,here both members are true1 and consequently from one being affirmed we are not authorized to deny the other4 ,of course we are left to conjecture in each case1 from the context1 whether it is meant to be implied that the members are or are not exclusive40 ,mansel says1#ff 899,we may happen to know99 that two alternatives cannot be true together1 so that the affirmation of the second necessitates the denial of the first2 but this1 as ,boethius observes1 is a 99material991 not a 99formal99 consequence40 ,mill has also pointed out the absurdities which would arise from always interpreting alternatives as exclusive4 8,if we assert10 he says1#fg 8that a man who has acted in some particular way must be either a knave or a fool1 we by no means assert1 or intend to assert1 that he cannot be both40 ,again1 8to make an entirely unselfish use of despotic power1 a man must be either a saint or a philosopher4444 ,does the disjunctive premise necessarily imply1 or must it be construed as supposing1 that the same person cannot be both a saint and a philosopher8 ,such a construction would be ridiculous40 #fd ,mansel's 99,aldrich991 p4 #ajc1 and 99,prolegomena ,logica991 p4 #bba4 #fe 99,elements of ,logic991 ,book ,,ii4 chap4 iv4 sect4 #d4 #ff ,aldrich1 99,artis ,logic ,rudimenta991 p4 #ajd4 #fg 99,examination of ,sir ,w4 ,hamilton's ,philosophy991 pp4 #deb-ded4 ,i discuss this subject fully because it is really the point which separates my logical system from that of ,boole4 ,in his 99,laws of ,thought99 7p4 #cb7 he expressly says1 8,in strictness1 the words #8and1' #8or1' interposed between the terms descriptive of two or more classes of objects1 imply that those classes are quite distinct1 so that no member of one is found in another40 ,this ,i altogether dispute4 ,in the ordinary use of these conjunctions we do not join distinct terms only2 and when terms so joined do prove to be logically distinct1 it is by virtue of a 99tacit premise991 something in the meaning of the names and our knowledge of them1 which teaches us that they are distinct4 ,if our knowledge of the meanings of the words joined is defective it will often be impossible to decide whether terms joined by conjunctions are exclusive or not4 ,in the sentence 8,repentance is not a single act1 but a habit or virtue10 it cannot be implied that a virtue is not a habit2 by ,aristotle's definition it is4 ,milton has the expression in one of his sonnets1 8,unstain'd by gold or fee10 where it is obvious that if the fee is not always gold1 the gold is meant to be a fee or bribe4 ,tennyson has the expression 8wreath or anadem40 ,most readers would be quite uncertain whether a wreath may be an anadem1 or an anadem a wreath1 or whether they are quite distinct or quite the same4 ,from ,darwin's 99,origin of ,species991 ,i take the expression1 8,when we see any 99part or organ99 developed in a remarkable 99degree or manner9940 ,in this1 99or99 is used twice1 and neither time exclusively4 ,for if 99part99 and 99organ99 are not synonymous1 at any rate an organ is a part4 ,and it is obvious that a part may be developed at the same time both in an extraordinary degree and an extraordinary manner1 although such cases may be comparatively rare4 ,from a careful examination of ordinary writings1 it will thus be found that the meanings of terms joined by 8and10 8or0 vary from absolute identity up to absolute contrariety4 ,there is no logical condition of distinctness at all1 and when we do choose exclusive alternatives1 it is because our subject demands it4 ,the matter1 not the form of an expression1 points out whether terms are exclusive or not4#fh ,in bills1 policies1 and other kinds of legal documents1 it is sometimes necessary to express very distinctly that alternatives are not exclusive4 ,the form and_/or is then used1 and1 as ,mr4 ,j4 ,j4 ,murphy has remarked1 this form coincides exactly in meaning with the symbol 4 #fh 99,pure ,logic991 pp #gf1 #gg4 ,in the first edition of this work 7vol4 i41 p4 #ha71 ,i took the disjunctive proposition 8,matter is solid1 or liquid1 or gaseous10 and treated it as an instance of exclusive alternatives1 remarking that the same portion of matter cannot be at once solid and liquid1 properly speaking1 and that still less can we suppose it to be solid and gaseous1 or solid1 liquid1 and gaseous all at the same time4 ,but the experiments of ,professor ,andrews show that1 under certain conditions of temperature and pressure1 there is no abrupt change from the liquid to the gaseous state4 ,the same substance may be in such a state as to be indifferently described as liquid and gaseous4 ,in many cases1 too1 the transition from solid to liquid is gradual1 so that the properties of solidity are at least partially joined with those of liquidity4 ,the proposition then1 instead of being an instance of exclusive alternatives1 seems to afford an excellent instance to the opposite effect4 ,when such doubts can arise1 it is evidently impossible to treat alternatives as absolutely exclusive by the logical nature of the relation4 ,it becomes purely a question of the matter of the proposition4 ,the question1 as we shall afterwards see more fully1 is one of the greatest theoretical importance1 because it concerns the true distinction between the sciences of ,logic and ,mathematics4 ,it is the foundation of number that every unit shall be distinct from every other unit2 but ,boole imported the conditions of number into the science of ,logic1 and produced a system which1 though wonderful in its results1 was not a system of logic at all4 99,laws of the ,disjunctive ,relation499 ,in considering the combination or synthesis of terms 7p4 #cj71 we found that certain laws1 those of ,simplicity and ,commutativeness1 must be observed4 ,in uniting terms by the disjunctive symbol we shall find that the same or closely similar laws hold true4 ,the alternatives of either member of a disjunctive proposition are certainly commutative4 ,just as we cannot properly distinguish between 99rich and rare gems99 and 99rare and rich gems991 so we must consider as identical the expression 99rich or rare gems991 and 99rare or rich gems994 ,in our symbolic language we may say ,a ,b ,b ,a4 ,the order of statement1 in short1 has no effect upon the meaning of an aggregate of alternatives1 so that the ,law of ,commutativeness holds true of the disjunctive symbol4 ,as we have admitted the possibility of joining as alternatives terms which are not really different1 the question arises1 ,how shall we treat two or more alternatives when they are clearly shown to be the same8 ,if we have it asserted that ,p is ,q or ,r1 and it is afterwards proved that ,q is but another name for ,r1 the result is that ,p is either ,r or ,r4 ,how shall we interpret such a statement8 ,what would be the meaning1 for instance1 of 8wreath or anadem0 if1 on referring to a dictionary1 we found 99anadem99 described as a wreath8 ,i take it to be self-evident that the meaning would then become simply 8wreath40 ,accordingly we may affirm the general law ,a ,a ,a4 ,any number of identical alternatives may always be reduced to1 and are logically equivalent to1 any one of those alternatives4 ,this is a law which distinguishes mathematical terms from logical terms1 because it obviously does not apply to the former4 ,i propose to call it the 99,law of ,unity991 because it must really be involved in any definition of a mathematical unit4 ,this law is closely analogous to the ,law of ,simplicity1 ,,aa ,a2 and the nature of the connection is worthy of attention4 ,few or no logicians except ,de ,morgan have adequately noticed the close relation between combined and disjunctive terms1 namely1 that every disjunctive term is the negative of a corresponding combined term1 and 99vice vers994 ,consider the term ,malleable dense metal4 ,how shall we describe the class of things which are not malleable-dense-metals8 ,whatever is included under that term must have all the qualities of malleability1 denseness1 and metallicity4 ,wherever any one or more of the qualities is wanting1 the combined term will not apply4 ,hence the negative of the whole term is ,not-malleable or not-dense or not-metallic4 ,in the above the conjunction 99or99 must clearly be interpreted as unexclusive2 for there may readily be objects which are both not-malleable1 and not-dense1 and perhaps not-metallic at the same time4 ,if in fact we were required to use 99or99 in a strictly exclusive manner1 it would be requisite to specify seven distinct alternatives in order to describe the negative of a combination of three terms4 ,the negatives of four or five terms would consist of fifteen or thirty-one alternatives4 ,this consideration alone is sufficient to prove that the meaning of 99or99 cannot be always exclusive in common language4 ,expressed symbolically1 we may say that the negative of ,,abc is not-,a or not-,b or not-,c2 that is1 99a99 99b99 99c994 ,reciprocally the negative of ,p ,q ,r is 99pqr994 ,every disjunctive term1 then1 is the negative of a combined term1 and 99vice vers994 ,apply this result to the combined term ,,aaa1 and its negative is 99a99 99a99 99a994 ,since ,,aaa is by the ,law of ,simplicity equivalent to ,a1 so 99a99 99a99 99a99 must be equivalent to 99a991 and the ,law of ,unity holds true4 ,each law thus necessarily presupposes the other4 99,symbolic expression of the ,law of ,duality499 ,we may now employ our symbol of alternation to express in a clear and formal manner the third ,fundamental ,law of ,thought1 which ,i have called the ,law of ,duality 7p4 #f74 ,taking ,a to represent any class or object or quality1 and ,b any other class1 object or quality1 we may always assert that ,a either agrees with ,b1 or does not agree4 ,thus we may say ,a ,,ab ,a99b994 ,this is a formula which will henceforth be constantly employed1 and it lies at the basis of reasoning4 ,the reader may perhaps wish to know why ,a is inserted in both alternatives of the second member of the identity1 and why the law is not stated in the form ,a ,b 99b994 ,but if he will consider the contents of the last section 7p4 #gc71 he will see that the latter expression cannot be correct1 otherwise no term could have a corresponding negative term4 ,for the negative of ,b 99b99 is 99b99,b1 or a self-contradictory term2 thus if ,a were identical with ,b 99b991 its negative 99a99 would be non-existent4 ,to say the least1 this result would in most cases be an absurd one1 and ,i see much reason to think that in a strictly logical point of view it would always be absurd4 ,in all probability we ought to assume as a fundamental logical axiom that 99every term has its negative in thought994 ,we cannot think at all without separating what we think about from other things1 and these things necessarily form the negative notion4#fi ,it follows that any proposition of the form ,a ,b 99b99 is just as self-contradictory as one of the form ,a ,b99b994 #fi 99,pure ,logic991 p4 #fe4 ,see also the criticism of this point by ,de ,morgan in the 99,athenum991 ,no4 #ahib1 #cjth ,january1 #ahfd2 p4 #aee4 ,it is convenient to recapitulate in this place the three ,laws of ,thought in their symbolic form1 thus ,law of ,identity ,a ,a4 ,law of ,contradiction ,a99a99 #j4 ,law of ,duality ,a ,,ab ,a99b994 99,various ,forms of the ,disjunctive ,proposition499 ,disjunctive propositions may occur in a great variety of forms1 of which the old logicians took insufficient notice4 ,there may be any number of alternatives1 each of which may be a combination of any number of simple terms4 ,a proposition1 again1 may be disjunctive in one or both members4 ,the proposition ,solids or liquids or gases are electrics or conductors of electricity is an example of the doubly disjunctive form4 ,the meaning of such a proposition is that whatever falls under any one or more alternatives on one side must fall under one or more alternatives on the other side4 ,from what has been said before1 it is apparent that the proposition ,a ,b ,c ,d will correspond to 99ab99 99cd991 each member of the latter being the negative of a member of the former proposition4 ,as an instance of a complex disjunctive proposition ,i may give ,senior's definition of wealth1 which1 briefly stated1 amounts to the proposition 8,wealth is what is transferable1 limited in supply1 and either productive of pleasure or preventive of pain40#gj #gj ,boole's 99,laws of ,thought991 p4 #ajf4 ,jevons' 99,pure ,logic991 p4 #fi4 ,let ,a wealth ,b transferable ,c limited in supply ,d productive of pleasure ,e preventive of pain4 ,the definition takes the form ,a ,,bc7,d ,e72 but if we develop the alternatives by a method to be afterwards more fully considered1 it becomes ,a ,,bcde ,,bcd99e99 ,,bc99d99,e4 ,an example of a still more complex proposition is found in ,de ,morgan's writings1#ga as follows3--8,he must have been rich1 and if not absolutely mad was weakness itself1 subjected either to bad advice or to most unfavourable circumstances40 #ga 99,on the ,syllogism991 ,no4 iii4 p4 #ab4 ,camb4 ,phil4 ,trans4 vol4 x1 part i4 ,if we assign the letters of the alphabet in succession1 thus1 ,a he ,b rich ,c absolutely mad ,d weakness itself ,e subjected to bad advice ,f subjected to most unfavourable circumstances1 the proposition will take the form ,a ,,ab,c ,d 7,e ,f71 and if we develop the alternatives1 expressing some of the different cases which may happen1 we obtain ,a ,,abc ,,ab99c99,,def ,,ab99c99,,de99f99 ,,ab99c99,d99e99,f4 ,the above gives the strict logical interpretation of the sentence1 and the first alternative ,,abc is capable of development into eight cases1 according as ,d1 ,e and ,f are or are not present4 ,although from our knowledge of the matter1 we may infer that weakness of character cannot be asserted of a person absolutely mad1 there is no explicit statement to this effect4 99,inference by ,disjunctive ,propositions499 ,before we can make a free use of disjunctive propositions in the processes of inference we must consider how disjunctive terms can be combined together or with simple terms4 ,in the first place1 to combine a simple term with a disjunctive one1 we must combine it with every alternative of the disjunctive term4 ,a vegetable1 for instance1 is either a herb1 a shrub1 or a tree4 ,hence an exogenous vegetable is either an exogenous herb1 or an exogenous shrub1 or an exogenous tree4 ,symbolically stated1 this process of combination is as follows1 ,a7,b ,c7 ,,ab ,,ac4 ,secondly1 to combine two disjunctive terms with each other1 combine each alternative of one with each alternative of the other4 ,since flowering plants are either exogens or endogens1 and are at the same time either herbs1 shrubs or trees1 it follows that there are altogether six alternatives--namely1 exogenous herbs1 exogenous shrubs1 exogenous trees1 endogenous herbs1 endogenous shrubs1 endogenous trees4 ,this process of combination is shown in the general form 7,a ,b7 7,c ,d ,e7 ,,ac ,,ad ,,ae ,,bc ,,bd ,,be4 ,it is hardly necessary to point out that1 however numerous the terms combined1 or the alternatives in those terms1 we may effect the combination1 provided each alternative is combined with each alternative of the other terms1 as in the algebraic process of multiplication4 ,some processes of deduction may be at once exhibited4 ,we may always1 for instance1 unite the same qualifying term to each side of an identity even though one or both members of the identity be disjunctive4 ,thus let ,a ,b ,c4 ,now it is self-evident that ,,ad ,,ad1 and in one side of this identity we may for ,a substitute its equivalent ,b ,c1 obtaining ,,ad ,,bd ,,cd4 ,since 8a gaseous element is either hydrogen1 or oxygen1 or nitrogen1 or chlorine1 or fluorine10 it follows that 8a free gaseous element is either free hydrogen1 or free oxygen1 or free nitrogen1 or free chlorine1 or free fluorine40 ,this process of combination will lead to most useful inferences when the qualifying adjective combined with both sides of the proposition is a negative of one or more alternatives4 ,since chlorine is a coloured gas1 we may infer that 8a colourless gaseous element is either 7colourless7 hydrogen1 oxygen1 nitrogen1 or fluorine40 ,the alternative chlorine disappears because colourless chlorine does not exist4 ,again1 since 8a tooth is either an incisor1 canine1 bicuspid1 or molar10 it follows that 8a not-incisor tooth is either canine1 bicuspid1 or molar40 ,the general rule is that from the denial of any of the alternatives the affirmation of the remainder can be inferred4 ,now this result clearly follows from our process of substitution2 for if we have the proposition ,a ,b ,c ,d1 and we insert this expression for ,a on one side of the self-evident identity ,a99b99 ,a99b991 we obtain ,a99b99 ,,ab99b99 ,a99b99,c ,a99b99,d2 and1 as the first of the three alternatives is self-contradictory1 we strike it out according to the law of contradiction3 there remains ,a99b99 ,a99b99,c ,a99b99,d4 ,thus our system fully includes and explains that mood of the ,disjunctive ,syllogism technically called the 99modus tollendo ponens994 ,but the reader must carefully observe that the ,disjunctive ,syllogism of the mood 99ponendo tollens991 which affirms one alternative1 and thence infers the denial of the rest1 cannot be held true in this system4 ,if ,i say1 indeed1 that ,water is either salt or fresh water1 it seems evident that 8water which is salt is not fresh40 ,but this inference really proceeds from our knowledge that water cannot be at once salt and fresh4 ,this inconsistency of the alternatives1 as ,i have fully shown1 will not always hold4 ,thus1 if ,i say ,gems are either rare stones or beautiful stones1 7#a7 it will obviously not follow that ,a rare gem is not a beautiful stone1 7#b7 nor that ,a beautiful gem is not a rare stone4 7#c7 ,our symbolic method gives only true conclusions2 for if we take ,a gem ,b rare stone ,c beautiful stone1 the proposition 7#a7 is of the form ,a ,b ,c hence ,,ab ,b ,,bc and ,,ac ,,bc ,c2 but these inferences are not equivalent to the false ones 7#b7 and 7#c74 ,we can readily represent disjunctive reasoning by the 99modus ponendo tollens991 when it is valid1 by expressing the inconsistency of the alternatives explicitly4 ,thus if we resort to our instance of ,water is either salt or fresh1 and take ,a ,water ,b salt ,c fresh1 then the premise is apparently of the form ,a ,,ab ,,ac2 but in reality there is an unexpressed condition that 8what is salt is not fresh10 from which follows1 by a process of inference to be afterwards described1 that 8what is fresh is not salt40 ,we have then1 in letter-terms1 the two propositions ,b ,b99c99 ,c 99b99,c4 ,if we substitute these descriptions in the original proposition1 we obtain _/99 ,a ,,ab99c99 ,a99b99,c2 99_/ uniting ,b to each side we infer ,,ab ,,ab99c99 ,,ab99b99,c or ,,ab ,,ab99c992 that is1 ,water which is salt is water salt and not fresh4 ,i should weary the reader if ,i attempted to illustrate the multitude of forms which disjunctive reasoning may take2 and as in the next chapter we shall be constantly treating the subject1 ,i must here restrict myself to a single instance4 ,a very common process of reasoning consists in the determination of the name of a thing by the successive exclusion of alternatives1 a process called by the old name 99abscissio infiniti994 ,take the case3 ,red-coloured metal is either copper or gold 7#a7 ,copper is dissolved by nitric acid 7#b7 ,this specimen is red-coloured metal 7#c7 ,this specimen is not dissolved by nitric acid 7#d7 ,therefore1 this specimen consists of gold 7#e7 ,let us assign the letter-symbols thus-- ,a this specimen ,b red-coloured metal ,c copper ,d gold ,e dissolved by nitric acid4 ,assuming that the alternatives copper or gold are intended to be exclusive1 as just explained in the case of fresh and salt water1 the premises may be stated in the forms ,b ,,bc99d99 ,b99c99,d 7#a7 ,c ,,ce 7#b7 ,a ,,ab 7#c7 ,a ,a99e99 7#d7 ,substituting for ,c in 7#a7 by means of 7#b7 we get ,b ,,bc99d99,e ,b99c99,d ,from 7#c7 and 7#d7 we may infer likewise ,a ,,ab99e99 and if in this we substitute for ,b its equivalent just stated1 it follows that ,a ,,abc99d99,e99e99 ,,ab99c99,d99e99 ,the first of the alternatives being contradictory the result is ,a ,,ab99c99,d99e99 which contains a full description of 8this specimen10 as furnished in the premises1 but by ellipsis asserts that it is gold4 ,it will be observed that in the symbolic expression 7#a7 ,i have explicitly stated what is certainly implied1 that copper is not gold1 and gold not copper1 without which condition the inference would not hold good4 ,,chapter ,,vi4 ,,the ,,indirect ,,method ,,of ,,inference4 ,the forms of deductive reasoning as yet considered1 are mostly cases of ,direct ,deduction as distinguished from those which we are now about to treat4 ,the method of ,indirect ,deduction may be described as that which points out what a thing is1 by showing that it cannot be anything else4 ,we can define a certain space upon a map1 either by colouring that space1 or by colouring all except the space2 the first mode is positive1 the second negative4 ,the difference1 it will be readily seen1 is exactly analogous to that between the direct and indirect modes of proof in geometry4 ,euclid often shows that two lines are equal1 by showing that they cannot be unequal1 and the proof rests upon the known number of alternatives1 greater1 equal or less1 which are alone conceivable4 ,in other cases1 as for instance in the seventh proposition of the first book1 he shows that two lines must meet in a particular point1 by showing that they cannot meet elsewhere4 ,in logic we can always define with certainty the utmost number of alternatives which are conceivable4 ,the ,law of ,duality 7pp4 #f1 #gd7 enables us always to assert that any quality or circumstance whatsoever is either present or absent4 ,whatever may be the meaning of the terms ,a and ,b it is certainly true that ,a ,,ab ,a99b99 ,b ,,ab 99a99,b4 ,these are universal tacit premises which may be employed in the solution of every problem1 and which are such invariable and necessary conditions of all thought1 that they need not be specially laid down4 ,the ,law of ,contradiction is a further condition of all thought and of all logical symbols2 it enables1 and in fact obliges1 us to reject from further consideration all terms which imply the presence and absence of the same quality4 ,now1 whenever we bring both these ,laws of ,thought into explicit action by the method of substitution1 we employ the ,indirect ,method of ,inference4 ,it will be found that we can treat not only those arguments already exhibited according to the direct method1 but we can include an infinite multitude of other arguments which are incapable of solution by any other means4 ,some philosophers1 especially those of ,france1 have held that the ,indirect ,method of ,proof has a certain inferiority to the direct method1 which should prevent our using it except when obliged4 ,but there are many truths which we can prove only indirectly4 ,we can prove that a number is a prime only by the purely indirect method of showing that it is not any of the numbers which have divisors1 and the remarkable process known as ,eratosthenes' ,sieve is the only mode by which we can select the prime numbers4#gb ,it bears a strong analogy to the indirect method here to be described4 ,we can prove that the side and diameter of a square are incommensurable1 but only in the negative or indirect manner1 by showing that the contrary supposition inevitably leads to contradiction4#gc ,many other demonstrations in various branches of the mathematical sciences proceed upon a like method4 ,now1 if there is only one important truth which must be1 and can only be1 proved indirectly1 we may say that the process is a necessary and sufficient one1 and the question of its comparative excellence or usefulness is not worth discussion4 ,as a matter of fact ,i believe that nearly half our logical conclusions rest upon its employment4 #gb ,see ,horsley1 99,philosophical ,transactions991 #aggb2 vol4 lxii4 p4 #cbg4 ,montucla1 99,histoire des ,mathematiques991 vol4 i4 p4 #bci4 99,penny ,cyclopdia991 article 8,eratosthenes40 #gc ,euclid1 ,book x4 ,prop4 #aag4 99,simple ,illustrations499 ,in tracing out the powers and results of this method1 we will begin with the simplest possible instance4 ,let us take a proposition of the common form1 ,a ,,ab1 say1 99,a ,metal is an ,element199 and let us investigate its full meaning4 ,any person who has had the least logical training1 is aware that we can draw from the above proposition an apparently different one1 namely1 99,a ,not-element is a ,not-metal499 ,while some logicians1 as for instance ,de ,morgan1#gd have considered the relation of these two propositions to be purely self-evident1 and neither needing nor allowing analysis1 a great many more persons1 as ,i have observed while teaching logic1 are at first unable to perceive the close connection between them4 ,i believe that a true and complete system of logic will furnish a clear analysis of this process1 which has been called 99,contrapositive ,conversion992 the full process is as follows3-- #gd 99,philosophical ,magazine991 ,december #aheb2 ,fourth ,series1 vol4 iv4 p4 #dce1 8,on ,indirect ,demonstration40 ,firstly1 by the ,law of ,duality we know that 99,not-element is either ,metal or ,not-metal499 ,if it be metal1 we know that it is by the premise 99an element992 we should thus be supposing that the same thing is an element and a not-element1 which is in opposition to the ,law of ,contradiction4 ,according to the only other alternative1 then1 the not-element must be a not-metal4 ,to represent this process of inference symbolically we take the premise in the form ,a ,,ab4 7#a7 ,we observe that by the ,law of ,duality the term not-,b is thus described 99b99 ,a99b99 99ab994 7#b7 ,for ,a in this proposition we substitute its description as given in 7#a71 obtaining 99b99 ,,ab99b99 99ab994 ,but according to the ,law of ,contradiction the term ,,ab99b99 must be excluded from thought1 or ,,ab99b99 #j4 ,hence it results that 99b99 is either nothing at all1 or it is 99ab992 and the conclusion is 99b99 99ab994 ,as it will often be necessary to refer to a conclusion of this kind ,i shall call it1 as is usual1 the 99,contrapositive ,proposition99 of the original4 ,the reader need hardly be cautioned to observe that from all ,a's are ,b's it does not follow that all not-,a's are not-,b's4 ,for by the ,law of ,duality we have 99a99 99a99,b 99ab991 and it will not be found possible to make any substitution in this by our original premise ,a ,,ab4 ,it still remains doubtful1 therefore1 whether not-metal is element or not-element4 ,the proof of the ,contrapositive ,proposition given above is exactly the same as that which ,euclid applies in the case of geometrical notions4 ,de ,morgan describes ,euclid's process as follows#ge3--8;,from every not-,b is not-,a he produces ,every ,a is ,b1 thus3 ,if it be possible1 let this ,a be not-,b1 but every not-,b is not-,a1 therefore this ,a is not-,a1 which is absurd3 whence every ,a is ,b40 ,now ,de ,morgan thinks that this proof is entirely needless1 because common logic gives the inference without the use of any geometrical reasoning4 ,i conceive however that logic gives the inference only by an indirect process4 ,de ,morgan claims 8to see identity in ,every ,a is ,b and every not-,b is not-,a1 by a process of thought prior to syllogism40 ,whether prior to syllogism or not1 ,i claim that it is not prior to the laws of thought and the process of substitutive inference1 by which it may be undoubtedly demonstrated4 #ge 99,philosophical ,magazine991 ,dec4 #aheb2 p4 #dcg4 99,employment of the ,contrapositive ,proposition499 ,we can frequently employ the contrapositive form of a proposition by the method of substitution2 and certain moods of the ancient syllogism1 which we have hitherto passed over1 may thus be satisfactorily comprehended in our system4 ,take for instance the following syllogism in the mood ,camestres3-- 8,whales are not true fish2 for they do not respire water1 whereas true fish do respire water40 ,let us take ,a whale ,b true fish ,c respiring water ,the premises are of the forms ,a ,a99c99 7#a7 ,b ,,bc 7#b7 ,now1 by the process of contraposition we obtain from the second premise 99c99 99bc99 and we can substitute this expression for 99c99 in 7#a71 obtaining ,a ,a99bc99 or 8,whales are not true fish1 not respiring water40 ,the mood ,cesare does not really differ from ,camestres except in the order of the premises1 and it could be exhibited in an exactly similar manner4 ,the mood ,baroko gave much trouble to the old logicians1 who could not 99reduce99 it to the first figure in the same manner as the other moods1 and were obliged to invent1 specially for it and for ,bokardo1 a method of ,indirect ,reduction closely analogous to the indirect proof of ,euclid4 ,now these moods require no exceptional treatment in this system4 ,let us take as an instance of ,baroko1 the argument ,all heated solids give continuous spectra 7#a7 ,some nebul do not give continuous spectra 7#b7 ,therefore1 some nebul are not heated solids 7#c7 ,treating the little word some as an indeterminate adjective of selection1 to which we assign a symbol like any other adjective1 let ,a some ,b nebul ,c giving continuous spectra ,d heated solids ,the premises then become ,d ,,dc 7#a7 ,,ab ,,ab99c99 7#b7 ,now from 7#a7 we obtain by the indirect method the contrapositive proposition 99c99 99cd99 and if we substitute this expression for 99c99 in 7#b7 we have ,,ab ,,ab99cd99 the full meaning of which is that 8some nebul do not give continuous spectra and are not heated solids40 ,we might similarly apply the contrapositive in many other instances4 ,take the argument1 8,all fixed stars are self-luminous2 but some of the heavenly bodies are not self-luminous1 and are therefore not fixed stars40 ,taking our terms ,a fixed stars ,b self-luminous ,c some ,d heavenly bodies we have the premises ,a ,,ab1 7#a7 ,,cd 99b99,,cd 7#b7 ,now from 7#a7 we can draw the contrapositive 99b99 99ab99 and substituting this expression for 99b99 in 7#b7 we obtain ,,cd 99ab99,,cd which expresses the conclusion of the argument that some heavenly bodies are not fixed stars4 99,contrapositive of a ,simple ,identity499 ,the reader should carefully note that when we apply the process of ,indirect ,inference to a simple identity of the form ,a ,b we may obtain further results4 ,if we wish to know what is the term not-,b1 we have as before1 by the ,law of ,duality1 99b99 ,a99b99 99ab99 and substituting for ,a we obtain 99b99 ,b99b99 99ab99 99ab994 ,but we may now also draw a second contrapositive2 for we have 99a99 99a99,b 99ab991 and substituting for ,b its equivalent ,a we have 99a99 99a99,a 99ab99 99ab994 ,hence from the single identity ,a ,b we can draw the two propositions 99a99 99ab99 99b99 99ab991 and observing that these propositions have a common term 99ab99 we can make a new substitution1 getting 99a99 99b994 ,this result is in strict accordance with the fundamental principles of inference1 and it may be a question whether it is not a self-evident result1 independent of the steps of deduction by which we have reached it4 ,for where two classes are coincident like ,a and ,b1 whatever is true of the one is true of the other2 what is excluded from the one must be excluded from the other similarly4 ,now as 99a99 bears to ,a exactly the same relation that 99b99 bears to ,b1 the identity of either pair follows from the identity of the other pair4 ,in every identity1 equality1 or similarity1 we may argue from the negative of the one side to the negative of the other4 ,thus at ordinary temperatures ,mercury liquid-metal1 hence obviously ,not-mercury not liquid-metal2 or since ,sirius brightest fixed star1 it follows that whatever star is not the brightest is not ,sirius1 and 99vice vers994 ,every correct definition is of the form ,a ,b1 and may often require to be applied in the equivalent negative form4 ,let us take as an illustration of the mode of using this result the argument following3 ,vowels are letters which can be sounded alone1 7#a7 ,the letter 99w99 cannot be sounded alone2 7#b7 ,therefore the letter 99w99 is not a vowel4 7#c7 ,here we have a definition 7#a71 and a comparison of a thing with that definition 7#b71 leading to exclusion of the thing from the class defined4 ,taking the terms ,a vowel1 ,b letter which can be sounded alone1 ,c letter 99w991 the premises are plainly of the forms ,a ,b1 7#a7 ,c 99b99,c4 7#b7 ,now by the ,indirect method we obtain from 7#a7 the ,contrapositive 99b99 99a991 and inserting in 7#b7 the equivalent for 99b99 we have ,c 99a99,c1 7#c7 or 8the letter 99w99 is not a vowel40 99,miscellaneous ,examples of the ,method499 ,we can apply the ,indirect ,method of ,inference however many may be the terms involved or the premises containing those terms4 ,as the working of the method is best learnt from examples1 ,i will take a case of two premises forming the syllogism ,barbara3 thus ,iron is metal 7#a7 ,metal is element4 7#b7 ,if we want to ascertain what inference is possible concerning the term 99,iron991 we develop the term by the ,law of ,duality4 ,iron must be either metal or not-metal2 iron which is metal must be either element or not-element2 and similarly iron which is not-metal must be either element or not-element4 ,there are then altogether four alternatives among which the description of iron must be contained2 thus ,iron1 metal1 element1 77 ,iron1 metal1 not-element1 77 ,iron1 not-metal1 element1 77 ,iron1 not-metal1 not-element4 77 ,our first premise informs us that iron is a metal1 and if we substitute this description in 77 and 77 we shall have self-contradictory combinations4 ,our second premise likewise informs us that metal is element1 and applying this description to 77 we again have self-contradiction1 so that there remains only 77 as a description of iron--our inference is ,iron iron1 metal1 element4 ,to represent this process of reasoning in general symbols1 let ,a iron ,b metal ,c element1 ,the premises of the problem take the forms ,a ,,ab 7#a7 ,b ,,bc4 7#b7 ,by the ,law of ,duality we have ,a ,,ab ,a99b99 7#c7 ,a ,,ac ,a99c994 7#d7 ,now1 if we insert for ,a in the second side of 7#c7 its description in 7#d71 we obtain what ,i shall call the 99development of ,a with respect to ,b and ,c991 namely ,a ,,abc ,,ab99c99 ,a99b99,c ,a99bc994 7#e7 ,wherever the letters ,a or ,b appear in the second side of 7#e7 substitute their equivalents given in 7#a7 and 7#b71 and the results stated at full length are ,a ,,abc ,,abc99c99 ,,ab99b99,c ,,ab99b99,c99c994 ,the last three alternatives break the ,law of ,contradiction1 so that ,a ,,abc #j #j #j ,,abc4 ,this conclusion is1 indeed1 no more than we could obtain by the direct process of substitution1 that is by substituting for ,b in 7#a71 its description in 7#b7 as in p4 #ee2 it is the characteristic of the ,indirect process that it gives all possible logical conclusions1 both those which we have previously obtained1 and an immense number of others or which the ancient logic took little or no account4 ,from the same premises1 for instance1 we can obtain a description of the class 99not-element99 or 99c994 ,by the ,law of ,duality we can develop 99c99 into four alternatives1 thus 99c99 ,,ab99c99 ,a99bc99 99a99,b99c99 99abc994 ,if we substitute for ,a and ,b as before1 we get 99c99 ,,abc99c99 ,,ab99bc99 99a99,,bc99c99 99abc991 and1 striking out the terms which break the ,law of ,contradiction1 there remains 99c99 99abc991 or what is not element is also not iron and not metal4 ,this ,indirect ,method of ,inference thus furnishes a complete solution of the following problem--99,given any number of logical premises or conditions1 required the description of any class of objects1 or of any term1 as governed by those conditions499 ,the steps of the process of inference may thus be concisely stated-- #a4 ,by the ,law of ,duality develop the utmost number of alternatives which may exist in the description of the required class or term as regards the terms involved in the premises4 #b4 ,for each term in these alternatives substitute its description as given in the premises4 #c4 ,strike out every alternative which is then found to break the ,law of ,contradiction4 #d4 ,the remaining terms may be equated to the term in question as the desired description4 99,mr4 ,venn's ,problem499 ,the need of some logical method more powerful and comprehensive than the old logic of ,aristotle is strikingly illustrated by ,mr4 ,venn in his most interesting and able article on ,boole's logic4#gf ,an easy example1 originally got1 as he says1 by the aid of my method as simply described in the 99,elementary ,lessons in ,logic991 was proposed in examination and lecture-rooms to some hundred and fifty students as a problem in ordinary logic4 ,it was answered by1 at most1 five or six of them4 ,it was afterwards set1 as an example on ,boole's method1 to a small class who had attended a few lectures on the nature of these symbolic methods4 ,it was readily answered by half or more of their number4 #gf 99,mind992 a ,quarterly ,review of ,psychology and ,philosophy2 ,october1 #ahgf1 vol4 i4 p4 #dhg4 ,the problem was as follows3--8,the members of a board were all of them either bondholders1 or shareholders1 but not both2 and the bondholders as it happened1 were all on the board4 ,what conclusion can be drawn80 ,the conclusion wanted is1 8,no shareholders are bondholders40 ,now1 as ,mr4 ,venn says1 nothing can look simpler than the following reasoning1 99when stated993--8,there can be no bondholders who are shareholders2 for if there were they must be either on the board1 or off it4 ,but they are not on it1 by the first of the given statements2 nor off it1 by the second40 ,yet from the want of any systematic mode of treating such a question only five or six of some hundred and fifty students could succeed in so simple a problem4 ,by symbolic statement the problem is instantly solved4 ,taking ,a member of board ,b bondholder ,c shareholder the premises are evidently ,a ,,ab99c99 ,a99b99,c ,b ,,ab4 ,the class ,c or shareholders may in respect of ,a and ,b be developed into four alternatives1 ,c ,,abc ,a99b99,c 99a99,,bc 99ab99,c4 ,but substituting for ,a in the first and for ,b in the third alternative we get ,c ,,abc99c99 ,,ab99b99,c ,a99b99,c 99a99,,abc 99ab99,c4 ,the first1 second1 and fourth alternatives in the above are self-contradictory combinations1 and only these2 striking them out there remain ,c ,a99b99,c 99ab99,c 99b99,c1 the required answer4 ,this symbolic reasoning is1 ,i believe1 the exact equivalent of ,mr4 ,venn's reasoning1 and ,i do not believe that the result can be attained in a simpler manner4 ,mr4 ,venn adds that he could adduce other similar instances1 that is1 instances showing the necessity of a better logical method4 99,abbreviation of the ,process499 ,before proceeding to further illustrations of the use of this method1 ,i must point out how much its practical employment can be simplified1 and how much more easy it is than would appear from the description4 ,when we want to effect at all a thorough solution of a logical problem it is best to form1 in the first place1 a complete series of all the combinations of terms involved in it4 ,if there be two terms ,a and ,b1 the utmost variety of combinations in which they can appear are ,,ab 99a99,b ,a99b99 99ab994 ,the term ,a appears in the first and second2 ,b in the first and third2 99a99 in the third and fourth2 and 99b99 in the second and fourth4 ,now if we have any premise1 say ,a ,b1 we must ascertain which of these combinations will be rendered self-contradictory by substitution2 the second and third will have to be struck out1 and there will remain only ,,ab 99ba994 ,hence we draw the following inferences ,a ,,ab1 ,b ,,ab1 99a99 99ab991 99b99 99ab994 ,exactly the same method must be followed when a question involves a greater number of terms4 ,thus by the ,law of ,duality the three terms ,a1 ,b1 ,c1 give rise to eight conceivable combinations1 namely ,,abc 77 99a99,,bc 77 ,,ab99c99 77 99a99,b99c99 77 ,a99b99,c 77 99ab99,c 77 ,a99bc99 77 99abc994 77 ,the development of the term ,a is formed by the first four of these2 for ,b we must select 771 771 771 772 ,c consists of 771 771 771 772 99b99 of 771 771 771 771 and so on4 ,now if we want to investigate completely the meaning of the premises ,a ,,ab 7#a7 ,b ,,bc 7#b7 we examine each of the eight combinations as regards each premise2 77 and 77 are contradicted by 7#a71 and 77 and 77 by 7#b71 so that there remain only ,,abc 77 99a99,,bc 77 99ab99,c 77 99abc994 77 ,to describe any term under the conditions of the premises 7#a7 and 7#b71 we have simply to draw out the proper combinations from this list2 thus1 ,a is represented only by ,,abc1 that is to say ,a ,,abc1 similarly 99c99 99abc994 ,for ,b we have two alternatives thus stated1 ,b ,,abc 99a99,,bc2 and for 99b99 we have 99b99 99ab99,c 99abc994 ,when we have a problem involving four distinct terms we need to double the number of combinations1 and as we add each new term the combinations become twice as numerous4 ,thus ,a1 ,b produce four combinations ,a1 ,b1 ,c1 eight ,a1 ,b1 ,c1 ,d sixteen ,a1 ,b1 ,c1 ,d1 ,e thirty-two ,a1 ,b1 ,c1 ,d1 ,e1 ,f sixty-four and so on4 ,i propose to call any such series of combinations the 99,logical ,alphabet994 ,it holds in logical science a position the importance of which cannot be exaggerated1 and as we proceed from logical to mathematical considerations1 it will become apparent that there is a close connection between these combinations and the fundamental theorems of mathematical science4 ,for the convenience of the reader who may wish to employ the 99,alphabet99 in logical questions1 ,i have had printed on the next page a complete series of the combinations up to those of six terms4 ,at the very commencement1 in the first column1 is placed a single letter ,x1 which might seem to be superfluous4 ,this letter serves to denote that it is always some higher class which is divided up4 ,thus the combination ,,ab really means ,,abx1 or that part of some larger class1 say ,x1 which has the qualities of ,a and ,b present4 ,the letter ,x is omitted in the greater part of the table merely for the sake of brevity and clearness4 ,in a later chapter on ,combinations it will become apparent that the introduction of this unit class is requisite in order to complete the analogy with the ,arithmetical ,triangle there described4 ,the reader ought to bear in mind that though the ,logical ,alphabet seems to give mere lists of combinations1 these combinations are intended in every case to constitute the development of a term of a proposition4 ,thus the four combinations ,,ab1 ,a99b991 99a99,b1 99ab99 really mean that any class ,x is described by the following proposition1 ,x ,,xab ,,xa99b99 ,x99a99,b ,x99ab994 ,if we select the ,a's1 we obtain the following proposition ,,ax ,,xab ,,xa99b994 ,thus whatever group of combinations we treat must be conceived as part of a higher class1 99summum genus99 or universe symbolised in the term ,x2 but1 bearing this in mind1 it is needless to complicate our formul by always introducing the letter4 ,all inference consists in passing from propositions to propositions1 and combinations 99per se99 have no meaning4 ,they are consequently to be regarded in all cases as forming parts of propositions4 ,,the ,,logical ,,alphabet4 ,i4 ,,ii4 ,,iii4 ,,iv4 ,v4 ,,vi4 ,,vii4 ,x ,,ax ,,ab ,,abc ,,abcd ,,abcde ,,abcdef 99a99,x ,a99b99 ,,ab99c99 ,,abc99d99 ,,abcd99e99 ,,abcde99f99 99a99,b ,a99b99,c ,,ab99c99,d ,,abc99d99,e ,,abcd99e99,f 99ab99 ,a99bc99 ,,ab99cd99 ,,abc99de99 ,,abcd99ef99 99a99,,bc ,a99b99,,cd ,,ab99c99,,de ,,abc99d99,,ef 99a99,b99c99 ,a99b99,c99d99 ,,ab99c99,d99e99 ,,abc99d99,e99f99 99ab99,c ,a99bc99,d ,,ab99cd99,e ,,abc99de99,f 99abc99 ,ab99cd99 ,,ab99cde99 ,,abc99def99 99a99,,bcd ,a99b99,,cde ,,ab99c99,,def 99a99,,bc99d99 ,a99b99,,cd99e99 ,,ab99c99,,de99f99 99a99,b99c99,d ,a99b99,c99d99,e ,,ab99c99,d99e99,f 99a99,b99cd99 ,a99b99,c99de99 ,,ab99c99,d99ef99 99ab99,,cd ,a99bc99,,de ,,ab99cd99,,ef 99ab99,c99d99 ,a99bc99,d99e99 ,,ab99cd99,e99f99 99abc99,d ,a99bcd99,e ,,ab99cde99,f 99abcd99 ,a99bcde99 ,,ab99cdef99 99a99,,bcde ,a99b99,,cdef 99a99,,bcd99e99 ,a99b99,,cde99f99 99a99,,bc99d99,e ,a99b99,,cd99e99,f 99a99,,bc99de99 ,a99b99,,cd99ef99 99a99,b99c99,,de ,a99b99,c99d99,,ef 99a99,b99c99,d99e99 ,a99b99,c99d99,e99f99 99a99,b99cd99,e ,a99b99,c99de99,f 99a99,b99cde99 ,a99b99,c99def99 99ab99,,cde ,a99bc99,,def 99ab99,,cd99e99 ,a99bc99,,de99f99 99ab99,c99d99,e ,a99bc99,d99e99,f 99ab99,cd99e99 ,a99bc99,d99ef99 99abc99,,de ,a99bcd99,,ef 99abc99,d99e99 ,a99bcd99,e99f99 99abcd99,e ,a99bcde99,f 99abcde99 ,a99bcdef99 99a99,,bcdef 99a99,,bcde99f99 99a99,,bcd99e99,f 99a99,,bcd99ef99 99a99,,bc99d99,,ef 99a99,,bc99d99,e99f99 99a99,,bc99de99,f 99a99,,bc99def99 99a99,b99c99,,def 99a99,b99c99,,de99f99 99a99,b99c99,d99e99,f 99a99,b99c99,d99ef99 99a99,b99cd99,,ef 99a99,b99cd99,e99f99 99a99,b99cde99,f 99a99,b99cdef99 99ab99,,cdef 99ab99,,cde99f99 99ab99,,cd99e99,f 99ab99,,cd99ef99 99ab99,c99d99,,ef 99ab99,c99d99,e99f99 99ab99,c99de99,f 99ab99,c99def99 99abc99,,def 99abc99,,de99f99 99abc99,d99e99,f 99abc99,d99ef99 99abcd99,,ef 99abcd99,e99f99 99abcde99,f 99abcdef99 ,in a theoretical point of view we may conceive that the ,logical ,alphabet is infinitely extended4 ,every new quality or circumstance which can belong to an object1 subdivides each combination or class1 so that the number of such combinations1 when unrestricted by logical conditions1 is represented by an infinitely high power of two4 ,the extremely rapid increase in the number of subdivisions obliges us to confine our attention to a few qualities at a time4 ,when contemplating the properties of this ,alphabet ,i am often inclined to think that ,pythagoras perceived the deep logical importance of duality2 for while unity was the symbol of identity and harmony1 he described the number two as the origin of contrasts1 or the symbol of diversity1 division and separation4 ,the number four1 or the 99,tetractys991 was also regarded by him as one of the chief elements of existence1 for it represented the generating virtue whence come all combinations4 ,in one of the golden verses ascribed to ,pythagoras1 he conjures his pupil to be virtuous3#gg 8,by him who stampt 99,the ,four99 upon the ,mind1 99,the ,four991 the fount of ,nature's endless stream40 #gg ,whewell1 99,history of the ,inductive ,sciences991 vol4 i4 p4 #bbb4 ,now four and the higher powers of duality do represent in this logical system the numbers of combinations which can be generated in the absence of logical restrictions4 ,the followers of ,pythagoras may have shrouded their master's doctrines in mysterious and superstitious notions1 but in many points these doctrines seem to have some basis in logical philosophy4 99,the ,logical ,slate499 ,to a person who has once comprehended the extreme significance and utility of the ,logical ,alphabet the indirect process of inference becomes reduced to the repetition of a few uniform operations of classification1 selection1 and elimination of contradictories4 ,logical deduction1 even in the most complicated questions1 becomes a matter of mere routine1 and the amount of labour required is the only impediment1 when once the meaning of the premises is rendered clear4 ,but the amount of labour is often found to be considerable4 ,the mere writing down of sixty-four combinations of six letters each is no small task1 and1 if we had a problem of five premises1 each of the sixty-four combinations would have to be examined in connection with each premise4 ,the requisite comparison is often of a very tedious character1 and considerable chance of error intervenes4 ,i have given much attention1 therefore1 to lessening both the manual and mental labour of the process1 and ,i shall describe several devices which may be adopted for saving trouble and risk of mistake4 ,in the first place1 as the same sets of combinations occur over and over again in different problems1 we may avoid the labour of writing them out by having the sets of letters ready printed upon small sheets of writing-paper4 ,it has also been suggested by a correspondent that1 if any one series of combinations were marked upon the margin of a sheet of paper1 and a slit cut between each pair of combinations1 it would be easy to fold down any particular combination1 and thus strike it out of view4 ,the combinations consistent with the premises would then remain in a broken series4 ,this method answers sufficiently well for occasional use4 ,a more convenient mode1 however1 is to have the series of letters shown on p4 #id1 engraved upon a common school writing slate1 of such a size1 that the letters may occupy only about a third of the space on the left hand side of the slate4 ,the conditions of the problem can then be written down on the unoccupied part of the slate1 and the proper series of combinations being chosen1 the contradictory combinations can be struck out with the pencil4 ,i have used a slate of this kind1 which ,i call a 99,logical ,slate991 for more than twelve years1 and it has saved me much trouble4 ,it is hardly possible to apply this process to problems of more than six terms1 owing to the large number of combinations which would require examination4 99,abstraction of ,indifferent ,circumstances499 ,there is a simple but highly important process of inference which enables us to abstract1 eliminate or disregard all circumstances indifferently present and absent4 ,thus if ,i were to state that 8a triangle is a three-sided rectilinear figure1 either large or not large10 these two alternatives would be superfluous1 because1 by the ,law of ,duality1 ,i know that everything must be either large or not large4 ,to add the qualification gives no new knowledge1 since the existence of the two alternatives will be understood in the absence of any information to the contrary4 ,accordingly1 when two alternatives differ only as regards a single component term which is positive in one and negative in the other1 we may reduce them to one term by striking out their indifferent part4 ,it is really a process of substitution which enables us to do this2 for having any proposition of the form ,a ,,abc ,,ab99c991 7#a7 we know by the ,law of ,duality that ,,ab ,,abc ,,ab99c994 7#b7 ,as the second member of this is identical with the second member of 7#a7 we may substitute1 obtaining ,a ,,ab4 ,this process of reducing useless alternatives may be applied again and again2 for it is plain that ,a ,,ab 7,,cd ,c99d99 99c99,d 99cd997 communicates no more information than that ,a is ,b4 ,abstraction of indifferent terms is in fact the converse process to that of development described in p4 #hi2 and it is one of the most important operations in the whole sphere of reasoning4 ,the reader should observe that in the proposition ,,ac ,,bc we cannot abstract ,c and infer ,a ,b2 but from ,,ac ,a99c99 ,,bc ,b99c99 we may abstract all reference to the term ,c4 ,it ought to be carefully remarked1 however1 that alternatives which seem to be without meaning often imply important knowledge4 ,thus if ,i say that 8a triangle is a three-sided rectilinear figure1 with or without three equal angles10 the last alternatives really express a property of triangles1 namely1 that some triangles have three equal angles1 and some do not have them4 ,if we put ,p 8,some10 meaning by the indefinite adjective 8,some10 one or more of the undefined properties of triangles with three equal angles1 and take ,a triangle ,b three-sided rectilinear figure ,c with three equal angles1 then the knowledge implied is expressed in the two propositions ,,pa ,,pbc 99p99,a 99p99,b99c994 ,these may also be thrown into the form of one proposition1 namely1 ,a ,,pbc 99p99,b99c992 but these alternatives cannot be reduced1 and the proposition is quite different from ,a ,,bc ,b99c994 99,illustrations of the ,indirect ,method499 ,a great variety of arguments and logical problems might be introduced here to show the comprehensive character and powers of the ,indirect ,method4 ,we can treat either a single premise or a series of premises4 ,take in the first place a simple definition1 such as 8a triangle is a three-sided rectilinear figure40 ,let ,a triangle ,b three-sided ,c rectilinear figure1 then the definition is of the form ,a ,,bc4 ,if we take the series of eight combinations of three letters in the ,logical ,alphabet 7p4 #id7 and strike out those which are inconsistent with the definition1 we have the following result3-- ,,abc 99a99,b99c99 99ab99,c 99abc499 ,for the description of the class ,c we have ,c ,,abc 99ab99,c1 that is1 8a rectilinear figure is either a triangle and three-sided1 or not a triangle and not three-sided40 ,for the class 99b99 we have 99b99 99ab99,c 99abc994 ,to the second side of this we may apply the process of simplification by abstraction described in the last section2 for by the ,law of ,duality 99ab99 99ab99,c 99abc992 and as we have two propositions identical in the second side of each we may substitute1 getting 99b99 99ab991 or what is not three-sided is not a triangle 7whether it be rectilinear or not74 99,second ,example499 ,let us treat by this method the following argument3-- 8,blende is not an elementary substance2 elementary substances are those which are undecomposable2 blende1 therefore1 is decomposable40 ,taking our letters thus-- ,a blende1 ,b elementary substance1 ,c undecomposable1 the premises are of the forms ,a ,a99b991 7#a7 ,b ,c4 7#b7 ,no immediate substitution can be made2 but if we take the contrapositive of 7#b7 7see p4 #hf71 namely 99b99 99c991 7#c7 we can substitute in 7#a7 obtaining the conclusion ,a ,a99c994 ,but the same result may be obtained by taking the eight combinations of ,a1 ,b1 ,c1 of the ,logical ,alphabet2 it will be found that only three combinations1 namely1 ,a99bc99 99a99,,bc 99abc991 are consistent with the premises1 whence it results that ,a ,a99bc991 or by the process of ,ellipsis before described 7p4 #eg7 ,a ,a99c994 99,third ,example499 ,as a somewhat more complex example ,i take the argument thus stated1 one which could not be thrown into the syllogistic form3-- 8,all metals except gold and silver are opaque2 therefore what is not opaque is either gold or silver or is not-metal40 ,there is more implied in this statement than is distinctly asserted1 the full meaning being as follows3 ,all metals not gold or silver are opaque1 7#a7 ,gold is not opaque but is a metal1 7#b7 ,silver is not opaque but is a metal1 7#c7 ,gold is not silver4 7#d7 ,taking our letters thus-- ,a metal ,c silver ,b gold ,d opaque1 we may state the premises in the forms ,a99bc99 ,a99bc99,d 7#a7 ,b ,,ab99d99 7#b7 ,c ,,ac99d99 7#c7 ,b ,b99c994 7#d7 ,to obtain a complete solution of the question we take the sixteen combinations of ,a1 ,b1 ,c1 ,d1 and striking out those which are inconsistent with the premises1 there remain only ,,ab99cd99 ,a99b99,c99d99 ,a99bc99,d 99abc99,d 99abcd994 ,the expression for not-opaque things consists of the three combinations containing 99d991 thus 99d99 ,,ab99cd99 ,a99b99,c99d99 99abcd991 or 99d99 ,a99d99 7,b99c99 99b99,c7 99abcd994 ,in ordinary language1 what is not-opaque is either metal which is gold1 and then not-silver1 or silver and then not-gold1 or else it is not-metal and neither gold nor silver4 99,fourth ,example499 ,a good example for the illustration of the ,indirect ,method is to be found in ,de ,morgan's 99,formal ,logic99 7p4 #abc71 the premises being substantially as follows3-- ,from ,a follows ,b1 and from ,c follows ,d2 but ,b and ,d are inconsistent with each other2 therefore ,a and ,c are inconsistent4 ,the meaning no doubt is that where ,a is1 ,b will be found1 or that every ,a is a ,b1 and similarly every ,c is a ,d2 but ,b and ,d cannot occur together4 ,the premises therefore appear to be of the forms ,a ,,ab1 7#a7 ,c ,,cd1 7#b7 ,b ,b99d994 7#c7 ,on examining the series of sixteen combinations1 only five are found to be consistent with the above conditions1 namely1 ,,ab99cd99 99a99,b99cd99 99ab99,,cd 99abc99,d 99abcd994 ,in these combinations the only ,a which appears is joined to 99c991 and similarly ,c is joined to 99a991 or ,a is inconsistent with ,c4 99,fifth ,example499 ,a more complex argument1 also given by ,de ,morgan1#gh contains five terms1 and is as stated below1 except that the letters are altered4 ,every ,a is one only of the two ,b or ,c2 ,d is both ,b and ,c1 except when ,b is ,e1 and then it is neither2 therefore no ,a is ,d4 #gh 99,formal ,logic991 p4 #abd4 ,as ,professor ,croom ,robertson has pointed out to me1 the second and third premises may be thrown into a single proposition1 ,d ,d99e99,,bc ,,de99bc994 ,the meaning of the above premises is difficult to interpret1 but seems to be capable of expression in the following symbolic forms-- ,a ,,ab99c99 ,a99b99,c1 7#a7 ,de ,d99e99,,bc1 7#b7 ,,de ,,de99bc994 7#c7 ,as five terms enter into these premises it is requisite to treat their thirty-two combinations1 and it will be found that fourteen of them remain consistent with the premises1 namely ,,ab99cd99,e 99a99,,bcd99e99 99ab99,c99d99,e ,,ab99cde99 99a99,,bc99d99,e 99ab99,c99de99 ,a99b99,c99d99,e 99a99,,bc99de99 99abc99,,de ,a99b99,c99de99 99a99,b99cd99,e 99abcd99,e 99a99,b99cde99 99abcde994 ,if we examine the first four combinations1 all of which contain ,a1 we find that they none of them contain ,d2 or again1 if we select those which contain ,d1 we have only two1 thus-- ,d 99a99,,bcd99e99 99abc99,,de4 ,hence it is clear that no ,a is ,d1 and 99vice vers99 no ,d is ,a4 ,we might draw many other conclusions from the same premises2 for instance-- ,,de 99abc99,,de1 or ,d and ,e never meet but in the absence of ,a1 ,b1 and ,c4 99,fallacies analysed by the ,indirect ,method499 ,it has been sufficiently shown1 perhaps1 that we can by the ,indirect ,method of ,inference extract the whole truth from a series of propositions1 and exhibit it anew in any required form of conclusion4 ,but it may also need to be shown by examples that so long as we follow correctly the almost mechanical rules of the method1 we cannot fall into any of the fallacies or paralogisms which are often committed in ordinary discussion4 ,let us take the example of a fallacious argument1 previously treated by the ,method of ,direct ,inference 7p4 #fb71 ,granite is not a sedimentary rock1 7#a7 ,basalt is not a sedimentary rock1 7#b7 and let us ascertain whether any precise conclusion can be drawn concerning the relation of granite and basalt4 ,taking as before ,a granite1 ,b sedimentary rock1 ,c basalt1 the premises become ,a ,a99b991 7#a7 ,c ,c99b994 7#b7 ,of the eight conceivable combinations of ,a1 ,b1 ,c1 five agree with these conditions1 namely ,a99b99,c 99a99,b99c99 ,a99bc99 99ab99,c 99abc994 ,selecting the combinations which contain ,a1 we find the description of granite to be ,a ,a99b99,c ,a99bc99 ,a99b997,c 99c9971 that is1 granite is not a sedimentary rock1 and is either basalt or not-basalt4 ,if we want a description of basalt the answer is of like form ,c ,a99b99,c 99ab99,c 99b99,c7,a 99a9971 that is basalt is not a sedimentary rock1 and is either granite or not-granite4 ,as it is already perfectly evident that basalt must be either granite or not1 and 99vice vers991 the premises fail to give us any information on the point1 that is to say the ,method of ,indirect ,inference saves us from falling into any fallacious conclusions4 ,this example sufficiently illustrates both the fallacy of ,negative premises and that of ,undistributed ,middle of the old logic4 ,the fallacy called the ,illicit ,process of the ,major ,term is also incapable of commission in following the rules of the method4 ,our example was 7p4 #fe7 ,all planets are subject to gravity1 7#a7 ,fixed stars are not planets4 7#b7 ,the false conclusion is that 8fixed stars are not subject to gravity40 ,the terms are ,a planet ,b fixed star ,c subject to gravity4 ,and the premises are ,a ,,ac1 7#a7 ,b 99a99,b4 7#b7 ,the combinations which remain uncontradicted on comparison with these premises are ,a99b99,c 99a99,b99c99 99a99,,bc 99ab99,c 99abc994 ,for fixed star we have the description ,b 99a99,,bc 99a99,b99c991 that is1 8a fixed star is not a planet1 but is either subject or not1 as the case may be1 to gravity40 ,here we have no conclusion concerning the connection of fixed stars and gravity4 99,the ,logical ,abacus499 ,the ,indirect ,method of ,inference has now been sufficiently described1 and a careful examination of its powers will show that it is capable of giving a full analysis and solution of every question involving only logical relations4 ,the chief difficulty of the method consists in the great number of combinations which may have to be examined2 not only may the requisite labour become formidable1 but a considerable chance of mistake arises4 ,i have therefore given much attention to modes of facilitating the work1 and have succeeded in reducing the method to an almost mechanical form4 ,it soon appeared obvious that if the conceivable combinations of the ,logical ,alphabet1 for any number of letters1 instead of being printed in fixed order on a piece of paper or slate1 were marked upon light movable pieces of wood1 mechanical arrangements could readily be devised for selecting any required class of the combinations4 ,the labour of comparison and rejection might thus be immensely reduced4 ,this idea was first carried out in the ,logical ,abacus1 which ,i have found useful in the lecture-room for exhibiting the complete solution of logical problems4 ,a minute description of the construction and use of the ,abacus1 together with figures of the parts1 has already been given in my essay called 99,the ,substitution of ,similars991#gi and ,i will here give only a general description4 #gi ,pp4 #ee-ei1 #ha-hf4 ,the ,logical ,abacus consists of a common school black-board placed in a sloping position and furnished with four horizontal and equi-distant ledges4 ,the combinations of the letters shown in the first four columns of the ,logical ,alphabet are printed in somewhat large type1 so that each letter is about an inch from the neighbouring one1 but the letters are placed one above the other instead of being in horizontal lines as in p4 #id4 ,each combination of letters is separately fixed to the surface of a thin slip of wood one inch broad and about one-eighth inch thick4 ,short steel pins are then driven in an inclined position into the wood4 ,when a letter is a large capital representing a positive term1 the pin is fixed in the upper part of its space2 when the letter is a small italic representing a negative term1 the pin is fixed in the lower part of the space4 ,now1 if one of the series of combinations be ranged upon a ledge of the black-board1 the sharp edge of a flat rule can be inserted beneath the pins belonging to any one letter--say ,a1 so that all the combinations marked ,a can be lifted out and placed upon a separate ledge4 ,thus we have represented the act of thought which separates the class ,a from what is not-,a4 ,the operation can be repeated2 out of the ,a's we can in like manner select those which are ,b's1 obtaining the ,,ab's2 and in like manner we may select any other classes such as the 99a99,b's1 the 99ab99's1 or the 99abc99's4 ,if now we take the series of eight combinations of the letters ,a1 ,b1 ,c1 99a991 99b991 99c991 and wish to analyse the argument anciently called ,barbara1 having the premises ,a ,,ab 7#a7 ,b ,,bc1 7#b7 we proceed as follows--,we raise the combinations marked 99a991 leaving the ,a's behind2 out of these ,a's we move to a lower ledge such as are 99b99's1 and to the remaining ,,ab's we join the 99a99's which have been raised4 ,the result is that we have divided all the combinations into two classes1 namely1 the ,a99b99's which are incapable of existing consistently with premise 7#a71 and the combinations which are consistent with the premise4 ,turning now to the second premise1 we raise out of those which agree with 7#a7 the 99b99's1 then we lower the ,b99c99's2 lastly we join the 99b99's to the ,,bc's4 ,we now find our combinations arranged as below4 -------------------------------------- ,a 99a99 99a99 99a99 ,b ,b 99b99 99b99 ,c ,c ,c 99c99 -------------------------------------- ,a ,a ,a 99a99 ,b 99b99 99b99 ,b 99c99 ,c 99c99 99c99 -------------------------------------- ,the lower line contains all the combinations which are inconsistent with either premise2 we have carried out in a mechanical manner that exclusion of self-contradictories which was formerly done upon the slate or upon paper4 ,accordingly1 from the combinations remaining in the upper line we can draw any inference which the premises yield4 ,if we raise the ,a's we find only one1 and that is ,c1 so that ,a must be ,c4 ,if we select the 99c99's we again find only one1 which is 99a99 and also 99b992 thus we prove that not-,c is not-,a and not-,b4 ,when a disjunctive proposition occurs among the premises the requisite movements become rather more complicated4 ,take the disjunctive argument ,a is either ,b or ,c or ,d1 ,a is not ,c and not ,d1 ,therefore ,a is ,b4 ,the premises are represented accurately as follows3-- ,a ,,ab ,,ac ,,ad 7#a7 ,a ,a99c99 7#b7 ,a ,a99d994 7#c7 ,as there are four terms1 we choose the series of sixteen combinations and place them on the highest ledge of the board but one4 ,we raise the 99a99's and out of the ,a's1 which remain1 we lower the 99b99's4 ,but we are not to reject all the ,a99b99's as contradictory1 because by the first premise ,a's may be either ,b's or ,c's or ,d's4 ,accordingly out of the ,a99b99's we must select the 99c99's1 and out of these again the 99d99's1 so that only ,a99bcd99 will remain to be rejected finally4 ,joining all the other fifteen combinations together again1 and proceeding to premise 7#b71 we raise the 99a99's and lower the ,,ac's1 and thus reject the combinations inconsistent with 7#b72 similarly we reject the ,,ad's which are inconsistent with 7#c74 ,it will be found that there remain1 in addition to all the eight combinations containing 99a991 only one containing ,a1 namely ,,ab99cd991 whence it is apparent that ,a must be ,b1 the ordinary conclusion of the argument4 ,in my 8,substitution of ,similars0 7pp4 #ef-ei7 ,i have described the working upon the ,abacus of two other logical problems1 which it would be tedious to repeat in this place4 99,the ,logical ,machine499 ,although the ,logical ,abacus considerably reduced the labour of using the ,indirect ,method1 it was not free from the possibility of error4 ,i thought moreover that it would afford a conspicuous proof of the generality and power of the method if ,i could reduce it to a purely mechanical form4 ,logicians had long been accustomed to speak of ,logic as an ,organon or ,instrument1 and even ,lord ,bacon1 while he rejected the old syllogistic logic1 had insisted1 in the second aphorism of his 8,new ,instrument10 that the mind required some kind of systematic aid4 ,in the kindred science of mathematics mechanical assistance of one kind or another had long been employed4 ,orreries1 globes1 mechanical clocks1 and such like instruments1 are really aids to calculation and are of considerable antiquity4 ,the ,arithmetical ,abacus is still in common use in ,russia and ,china4 ,the calculating machine of ,pascal is more than two centuries old1 having been constructed in #afdb-de4 ,m4 ,thomas of ,colmar manufactures an arithmetical machine on ,pascal's principles which is employed by engineers and others who need frequently to multiply or divide4 ,to ,babbage and ,scheutz is due the merit of embodying the ,calculus of ,differences in a machine1 which thus became capable of calculating the most complicated tables of figures4 ,it seemed strange that in the more intricate science of quantity mechanism should be applicable1 whereas in the simple science of qualitative reasoning1 the syllogism was only called an instrument by a figure of speech4 ,it is true that ,swift satirically described the ,professors of ,laputa as in possession of a thinking machine1 and in #ahea ,mr4 ,alfred ,smee actually proposed the construction of a ,relational machine and a ,differential machine1 the first of which would be a mechanical dictionary and the second a mode of comparing ideas2 but with these exceptions ,i have not yet met with so much as a suggestion of a reasoning machine4 ,it may be added that ,mr4 ,smee's designs1 though highly ingenious1 appear to be impracticable1 and in any case they do not attempt the performance of logical inference4#hj #hj ,see his work called 99,the ,process of ,thought adapted to ,words and ,language1 together with a ,description of the ,relational and ,differential ,machines994 ,also 99,philosophical ,transactions991 #ahgj vol4 #afj1 p4 #eah4 ,the ,logical ,abacus soon suggested the notion of a ,logical ,machine1 which1 after two unsuccessful attempts1 ,i succeeded in constructing in a comparatively simple and effective form4 ,the details of the ,logical ,machine have been fully described by the aid of plates in the ,philosophical ,transactions1#ha and it would be needless to repeat the account of the somewhat intricate movements of the machine in this place4 #ha 99,philosophical ,transactions99 #ahgj1 vol4 #afj1 p4 #dig4 99,proceedings of the ,royal ,society991 vol4 xviii4 p4 #aff1 ,jan4 #bj1 #ahgj4 99,nature991 vol1 i4 p4 #cdc4 ,the general appearance of the machine is shown in a plate facing the title-page of this volume4 ,it somewhat resembles a very small upright piano or organ1 and has a keyboard containing twenty-one keys4 ,these keys are of two kinds1 sixteen of them representing the terms or letters ,a1 99a991 ,b1 99b991 ,c1 99c991 ,d1 99d991 which have so often been employed in our logical notation4 ,when letters occur on the left-hand side of a proposition1 formerly called the subject1 each is represented by a key on the left-hand half of the keyboard2 but when they occur on the right-hand side1 or as it used to be called the predicate of the proposition1 the letter-keys on the right-hand side of the keyboard are the proper representatives4 ,the five other keys may be called operation keys1 to distinguish them from the letter or term keys4 ,they stand for the stops1 copula1 and disjunctive conjunctions of a proposition4 ,the middle key of all is the copula1 to be pressed when the verb 99is99 or the sign is met4 ,the key to the extreme right-hand is called the ,full ,stop1 because it should be pressed when a proposition is completed1 in fact in the proper place of the full stop4 ,the key to the extreme left-hand is used to terminate an argument or to restore the machine to its initial condition2 it is called the ,finis key4 ,the last keys but one on the right and left complete the whole series1 and represent the conjunction 99or99 in its unexclusive meaning1 or the sign which ,i have employed1 according as it occurs in the right or left hand side of the proposition4 ,the whole keyboard is arranged as shown on the next page-- ---------------------------------------- ----------------------------------- ,c ,f ,left-hand side of ,proposition4 o ,right-hand side of ,proposition4 ,f ,s i p u t n---------------------------u----------- ----------------l o i l l p s99d99 ,d 99c99 ,c 99b99 ,b 99a99 ,a a ,a 99a99 ,b 99b99 ,c 99c99 ,d 99d99 4 4,or 4 ,or ---------------------------------------- ------------------- ,to work the machine it is only requisite to press the keys in succession as indicated by the letters and signs of a symbolical proposition4 ,all the premises of an argument are supposed to be reduced to the simple notation which has been employed in the previous pages4 ,taking then such a simple proposition as ,a ,,ab1 we press the keys ,a 7left71 copula1 ,a 7right71 ,b 7right71 and full stop4 ,if there be a second premise1 for instance ,b ,,bc1 we press in like manner the keys-- ,b 7left71 copula1 ,b 7right71 ,c 7right71 full stop4 ,the process is exactly the same however numerous the premises may be4 ,when they are completed the operator will see indicated on the face of the machine the exact combinations of letters which are consistent with the premises according to the principles of thought4 ,as shown in the figure opposite the title-page1 the machine exhibits in front a ,logical ,alphabet of sixteen combinations1 exactly like that of the ,abacus1 except that the letters of each combination are separated by a certain interval4 ,after the above problem has been worked upon the machine the ,logical ,alphabet will have been modified so as to present the following appearance-- ---------------------------------------- --------------- ---------------------------------------- ,a ,a 99a9999a99 99a9999a9999a9999a99 ---------------------------------------- ---------------------------------------- ,b ,b ,b ,b 99b9999b9999b9999b99 ---------------------------------------- ---------------------------------------- ,c ,c ,c ,c ,c ,c 99c9999c99 ---------------------------------------- ---------------------------------------- ,d 99d99 ,d 99d99 ,d 99d99 ,d 99d99 ---------------------------------------- ---------------------------------------- --------------- ,the operator will readily collect the various conclusions in the manner described in previous pages1 as1 for instance that ,a is always ,c1 that not-,c is not-,b and not-,a2 and not-,b is not-,a but either ,c or not-,c4 ,the results are thus to be read off exactly as in the case of the ,logical ,slate1 or the ,logical ,abacus4 ,disjunctive propositions are to be treated in an exactly similar manner4 ,thus1 to work the premises ,a ,,ab ,,ac ,b ,c ,,bd ,,cd1 it is only necessary to press in succession the keys ,a 7left71 copula1 ,a 7right71 ,b1 1 ,a1 ,c1 full stop4 ,b 7left71 1 ,c1 copula1 ,b 7right71 ,d1 1 ,c1 ,d1 full stop4 ,the combinations then remaining will be as follows ,,abcd 99a99,,bcd 99abc99,d ,,ab99c99,d 99a99,b99c99,d 99abcd499 ,a99c99,,cd 99ab99,,cd ,on pressing the left-hand key ,a1 all the possible combinations which do not contain ,a will disappear1 and the description of ,a may be gathered from what remain1 namely that it is always ,d4 ,the full-stop key restores all combinations consistent with the premises and any other selection may be made1 as say not-,d1 which will be found to be always not-,a1 not-,b1 and not-,c4 ,at the end of every problem1 when no further questions need be addressed to the machine1 we press the ,finis key1 which has the effect of bringing into view the whole of the conceivable combinations of the alphabet4 ,this key in fact obliterates the conditions impressed upon the machine by moving back into their ordinary places those combinations which had been rejected as inconsistent with the premises4 ,before beginning any new problem it is requisite to observe that the whole sixteen combinations are visible4 ,after the ,finis key has been used the machine represents a mind endowed with powers of thought1 but wholly devoid of knowledge4 ,it would not in that condition give any answer but such as would consist in the primary laws of thought themselves4 ,but when any proposition is worked upon the keys1 the machine analyses and digests the meaning of it and becomes charged with the knowledge embodied in that proposition4 ,accordingly it is able to return as an answer any description of a term or class so far as furnished by that proposition in accordance with the ,laws of ,thought4 ,the machine is thus the embodiment of a true logical system4 ,the combinations are classified1 selected or rejected1 just as they should be by a reasoning mind1 so that at each step in a problem1 the ,logical ,alphabet represents the proper condition of a mind exempt from mistake4 ,it cannot be asserted indeed that the machine entirely supersedes the agency of conscious thought2 mental labour is required in interpreting the meaning of grammatical expressions1 and in correctly impressing that meaning on the machine2 it is further required in gathering the conclusion from the remaining combinations4 ,nevertheless the true process of logical inference is really accomplished in a purely mechanical manner4 ,it is worthy of remark that the machine can detect any self-contradiction existing between the premises presented to it2 should the premises be self-contradictory it will be found that one or more of the letter-terms disappears entirely from the ,logical ,alphabet4 ,thus if we work the two propositions1 ,a is ,b1 and ,a is not-,b1 and then inquire for a description of ,a1 the machine will refuse to give it by exhibiting no combination at all containing ,a4 ,this result is in agreement with the law1 which ,i have explained1 that every term must have its negative 7p4 #gd74 ,accordingly1 whenever any one of the letters ,a1 ,b1 ,c1 ,d1 99a991 99b991 99c991 99d991 wholly disappears from the alphabet1 it may be safely inferred that some act of self-contradiction has been committed4 ,it ought to be carefully observed that the logical machine cannot receive a simple identity of the form ,a ,b except in the double form of ,a ,b and ,b ,a4 ,to work the proposition ,a ,b1 it is therefore necessary to press the keys-- ,a 7left71 copula1 ,b 7right71 full stop2 ,b 7left71 copula1 ,a 7right71 full stop4 ,the same double operation will be necessary whenever the proposition is not of the kind called a partial identity 7p4 #dj74 ,thus ,,ab ,,cd1 ,,ab ,,ac1 ,a ,b ,c1 ,a ,b ,c ,d1 all require to be read from both ends separately4 ,the proper rule for using the machine may in fact be given in the following way3--7#a7 99,read each proposition as it stands1 and play the corresponding keys993 7#b7 99,convert the proposition and read and play the keys again in the transposed order of the terms499 ,so long as this rule is observed the true result must always be obtained4 ,there can be no mistake4 ,but it will be found that in the case of partial identities1 and some other similar forms of propositions1 the transposed reading has no effect upon the combinations of the ,logical ,alphabet4 ,one reading is in such cases all that is practically needful4 ,after some experience has been gained in the use of the machine1 the worker naturally saves himself the trouble of the second reading when possible4 ,it is no doubt a remarkable fact that a simple identity cannot be impressed upon the machine except in the form of two partial identities1 and this may be thought by some logicians to militate against the equational mode of representing propositions4 ,before leaving the subject ,i may remark that these mechanical devices are not likely to possess much practical utility4 ,we do not require in common life to be constantly solving complex logical questions4 ,even in mathematical calculation the ordinary rules of arithmetic are generally sufficient1 and a calculating machine can only be used with advantage in peculiar cases4 ,but the machine and abacus have nevertheless two important uses4 ,in the first place ,i hope that the time is not very far distant when the predominance of the ancient ,aristotelian ,logic will be a matter of history only1 and when the teaching of logic will be placed on a footing more worthy of its supreme importance4 ,it will then be found that the solution of logical questions is an exercise of mind at least as valuable and necessary as mathematical calculation4 ,i believe that these mechanical devices1 or something of the same kind1 will then become useful for exhibiting to a class of students a clear and visible analysis of logical problems of any degree of complexity1 the nature of each step being rendered plain to the eyes of the students4 ,i often used the machine or abacus for this purpose in my class lectures while ,i was ,professor of ,logic at ,owens ,college4 ,secondly1 the more immediate importance of the machine seems to consist in the unquestionable proof which it affords that correct views of the fundamental principles of reasoning have now been attained1 although they were unknown to ,aristotle and his followers4 ,the time must come when the inevitable results of the admirable investigations of the late ,dr4 ,boole must be recognised at their true value1 and the plain and palpable form in which the machine presents those results will1 ,i hope1 hasten the time4 ,undoubtedly ,boole's life marks an era in the science of human reason4 ,it may seem strange that it had remained for him first to set forth in its full extent the problem of logic1 but ,i am not aware that anyone before him had treated logic as a symbolic method for evolving from any premises the description of any class whatsoever as defined by those premises4 ,in spite of several serious errors into which he fell1 it will probably be allowed that ,boole discovered the true and general form of logic1 and put the science substantially into the form which it must hold for evermore4 ,he thus effected a reform with which there is hardly anything comparable in the history of logic between his time and the remote age of ,aristotle4 ,nevertheless1 ,boole's quasi-mathematical system could hardly be regarded as a final and unexceptionable solution of the problem4 ,not only did it require the manipulation of mathematical symbols in a very intricate and perplexing manner1 but the results when obtained were devoid of demonstrative force1 because they turned upon the employment of unintelligible symbols1 acquiring meaning only by analogy4 ,i have also pointed out that he imported into his system a condition concerning the exclusive nature of alternatives 7p4 #gj71 which is not necessarily true of logical terms4 ,i shall have to show in the next chapter that logic is really the basis of the whole science of mathematical reasoning1 so that ,boole inverted the true order of proof when he proposed to infer logical truths by algebraic processes4 ,it is wonderful evidence of his mental power that by methods fundamentally false he should have succeeded in reaching true conclusions and widening the sphere of reason4 ,the mechanical performance of logical inference affords a demonstration both of the truth of ,boole's results and of the mistaken nature of his mode of deducing them4 ,conclusions which he could obtain only by pages of intricate calculation1 are exhibited by the machine after one or two minutes of manipulation4 ,and not only are those conclusions easily reached1 but they are demonstratively true1 because every step of the process involves nothing more obscure than the three fundamental ,laws of ,thought4 99,the ,order of ,premises499 ,before quitting the subject of deductive reasoning1 ,i may remark that the order in which the premises of an argument are placed is a matter of logical indifference4 ,much discussion has taken place at various times concerning the arrangement of the premises of a syllogism2 and it has been generally held1 in accordance with the opinion of ,aristotle1 that the so-called major premise1 containing the major term1 or the predicate of the conclusion1 should stand first4 ,this distinction however falls to the ground in our system1 since the proposition is reduced to an identical form1 in which there is no distinction of subject and predicate4 ,in a strictly logical point of view the order of statement is wholly devoid of significance4 ,the premises are simultaneously coexistent1 and are not related to each other according to the properties of space and time4 ,just as the qualities of the same object are neither before nor after each other in nature 7p4 #cc71 and are only thought of in some one order owing to the limited capacity of mind1 so the premises of an argument are neither before nor after each other1 and are only thought of in succession because the mind cannot grasp many ideas at once4 ,the combinations of the logical alphabet are exactly the same in whatever order the premises be treated on the logical slate or machine4 ,some difference may doubtless exist as regards convenience to human memory4 ,the mind may take in the results of an argument more easily in one mode of statement than another1 although there is no real difference in the logical results4 ,but in this point of view ,i think that ,aristotle and the old logicians were clearly wrong4 ,it is more easy to gather the conclusion that 8all ,a's are ,c's0 from 8all ,a's are ,b's and all ,b's are ,c's10 than from the same propositions in inverted order1 8all ,b's are ,c's and all ,a's are ,b's40 99,the ,equivalence of ,propositions994 ,one great advantage which arises from the study of this ,indirect ,method of ,inference consists in the clear notion which we gain of the ,equivalence of ,propositions4 ,the older logicians showed how from certain simple premises we might draw an inference1 but they failed to point out whether that inference contained the whole1 or only a part1 of the information embodied in the premises4 ,any one proposition or group of propositions may be classed with respect to another proposition or group of propositions1 as #a4 ,equivalent1 #b4 ,inferrible1 #c4 ,consistent1 #d4 ,contradictory4 ,taking the proposition 8,all men are mortals0 as the original1 then 8,all immortals are not men0 is its equivalent2 8,some mortals are men0 is inferrible1 or capable of inference1 but is not equivalent2 8,all not-men are not mortals0 cannot be inferred1 but is consistent1 that is1 may be true at the same time2 8,all men are immortals0 is of course contradictory4 ,one sufficient test of equivalence is capability of mutual inference4 ,thus from ,all electrics all non-conductors1 ,i can infer ,all non-electrics all conductors1 and 99vice vers99 from the latter ,i can pass back to the former4 ,in short1 ,a ,b is equivalent to 99a99 99b994 ,again1 from the union of the two propositions1 ,a ,,ab and ,b ,,ab1 ,i get ,a ,b1 and from this ,i might as easily deduce the two with which ,i started4 ,in this case one proposition is equivalent to two other propositions4 ,there are in fact no less than four modes in which we may express the identity of two classes ,a and ,b1 namely1 ,,first ,,mode4 ,,second ,,mode4 ,,third ,,mode4 ,,fourth ,,mode4 ,a ,b 99a99 99b99 ,a ,,ab 99a99 99ab99 ,b ,,ab 99b99 99ab99 ,the ,indirect ,method of ,inference furnishes a universal and clear criterion as to the relationship of propositions4 ,the import of a statement is always to be measured by the combinations of terms which it destroys4 ,hence two propositions are equivalent when they remove the same combinations from the ,logical ,alphabet1 and neither more nor less4 ,a proposition is inferrible but not equivalent to another when it removes some but not all the combinations which the other removes1 and none except what this other removes4 ,again1 propositions are consistent provided that they jointly allow each term and the negative of each term to remain somewhere in the ,logical ,alphabet4 ,if after all the combinations inconsistent with two propositions are struck out1 there still appears each of the letters ,a1 99a991 ,b1 99b991 ,c1 99c991 ,d1 99d991 which were there before1 then no inconsistency between the propositions exists1 although they may not be equivalent or even inferrible4 ,finally1 contradictory propositions are those which taken together remove any one or more letter-terms from the ,logical ,alphabet4 ,what is true of single propositions applies also to groups of propositions1 however large or complicated2 that is to say1 one group may be equivalent1 inferrible1 consistent1 or contradictory as regards another1 and we may similarly compare one proposition with a group of propositions4 ,to give in this place illustrations of all the four kinds of relation would require much space3 as the examples given in previous sections or chapters may serve more or less to explain the relations of inference1 consistency1 and contradiction1 ,i will only add a few instances of equivalent propositions or groups4 ,in the following list each proposition or group of propositions is exactly equivalent in meaning to the corresponding one in the other column1 and the truth of this statement may be tested by working out the combinations of the alphabet1 which ought to be found exactly the same in the case of each pair of equivalents4 ,a ,a99b99 4 4 4 4 4 4 4 ,b 99a99,b ,a 99b99 4 4 4 4 4 4 4 4 99a99 ,b ,a ,,bc 4 4 4 4 4 4 4 4 99a99 99b99 99c99 ,a ,,ab ,,ac 4 4 4 4 4 4 99b99 99ab99 ,a99b99,c ,a ,b ,c ,d 4 4 4 4 4 4 4 99ab99 99cd99 ,a 99c99 ,b 99d99 4 4 4 4 4 4 99a99,c 99b99,d ,a ,,ab99c99 ,a99b99,c 4 4 4 ,a ,,ab ,,ac ,,ab ,,ab99c99 ,a ,b ,a ,b ,b ,c 4 4 4 4 4 4 4 4 4 ,a ,c ,a ,,ab ,a ,,ac ,b ,,bc 4 4 4 4 4 4 4 4 4 ,b ,a 99a99,,bc ,although in these and many other cases the equivalents of certain propositions can readily be given1 yet ,i believe that no uniform and infallible process can be pointed out by which the exact equivalents of premises can be ascertained4 ,ordinary deductive inference usually gives us only a portion of the contained information4 ,it is true that the combinations consistent with a set of premises may always be thrown into the form of a proposition which must be logically equivalent to those premises2 but the difficulty consists in detecting the other forms of propositions which will be equivalent to the premises4 ,the task is here of a different character from any which we have yet attempted4 ,it is in reality an inverse process1 and is just as much more troublesome and uncertain than the direct process1 as seeking is compared with hiding4 ,not only may several different answers equally apply1 but there is no method of discovering any of those answers except by repeated trial4 ,the problem which we have here met is really that of induction1 the inverse of deduction2 and1 as ,i shall soon show1 induction is always tentative1 and1 unless conducted with peculiar skill and insight1 must be exceedingly laborious in cases of complexity4 ,de ,morgan was unfortunately led by this equivalence of propositions into the most serious error of his ingenious system of ,logic4 ,he held that because the proposition 8,all ,a's are all ,b's10 is but another expression for the two propositions 8,all ,a's are ,b's0 and 8,all ,b's are ,a's10 it must be a composite and not really an elementary form of proposition4#hb ,but on taking a general view of the equivalence of propositions such an objection seems to have no weight4 ,logicians have1 with few exceptions1 persistently upheld the original error of ,aristotle in rejecting from their science the one simple relation of identity on which all more complex logical relations must really rest4 #hb 99,syllabus of a proposed system of ,logic991 #eg1 #aba1 c4 99,formal ,logic991 p4 #ff4 99,the ,nature of ,inference499 ,the question1 ,what is ,inference8 is involved1 even to the present day1 in as much uncertainty as that ancient question1 ,what is ,truth8 ,i shall in more than one part of this work endeavour to show that inference never does more than explicate1 unfold1 or develop the information contained in certain premises or facts4 ,neither in deductive nor inductive reasoning can we add a tittle to our implicit knowledge1 which is like that contained in an unread book or a sealed letter4 ,sir ,w4 ,hamilton has well said1 8,reasoning is the showing out explicitly that a proposition not granted or supposed1 is implicitly contained in something different1 which is granted or supposed40#hc #hc ,lectures on ,metaphysics1 vol4 iv4 p4 #cfi4 ,professor ,bowen has explained#hd with much clearness that the conclusion of an argument states explicitly what is virtually or implicitly thought4 8,the process of reasoning is not so much a mode of evolving a new truth1 as it is of establishing or proving an old one1 by showing how much was admitted in the concession of the two premises taken together40 ,it is true that the whole meaning of these statements rests upon that of such words as 8explicit10 8implicit10 8virtual40 ,that is implicit which is wrapped up1 and we render it explicit when we unfold it4 ,just as the conception of a circle involves a hundred important geometrical properties1 all following from what we know1 if we have acuteness to unfold the results1 so every fact and statement involves more meaning than seems at first sight4 ,reasoning explicates or brings to conscious possession what was before unconscious4 ,it does not create1 nor does it destroy1 but it transmutes and throws the same matter into a new form4 #hd ,bowen1 99,treatise on ,logic991 ,cambridge1 ,u4,s41 #ahff2 p4 #cfb4 ,the difficult question still remains1 ,where does novelty of form begin8 ,is it a case of inference when we pass from 8,sincerity is the parent of truth0 to 8,the parent of truth is sincerity80 ,the old logicians would have called this change 99conversion991 one case of immediate inference4 ,but as all identity is necessarily reciprocal1 and the very meaning of such a proposition is that the two terms are identical in their signification1 ,i fail to see any difference between the statements whatever4 ,as well might we say that 99x99 99y99 and 99y99 99x99 are different equations4 ,another point of difficulty is to decide when a change is merely grammatical and when it involves a real logical transformation4 ,between a 99table of wood99 and a 99wooden table99 there is no logical difference 7p4 #ca71 the adjective being merely a convenient substitute for the prepositional phrase4 ,but it is uncertain to my mind whether the change from 8,all men are mortal0 to 8,no men are not mortal0 is purely grammatical4 ,logical change may perhaps be best described as consisting in the determination of a relation between certain classes of objects from a relation between certain other classes4 ,thus ,i consider it a truly logical inference when we pass from 8,all men are mortal0 to 8,all immortals are not-men10 because the classes 99immortals99 and 99not-men99 are different from 99mortals99 and 99men991 and yet the propositions contain at the bottom the very same truth1 as shown in the combinations of the ,logical ,alphabet4 ,the passage from the qualitative to the quantitative mode of expressing a proposition is another kind of change which we must discriminate from true logical inference4 ,we state the same truth when we say that 8mortality belongs to all men10 as when we assert that 8all men are mortals40 ,here we do not pass from class to class1 but from one kind of term1 the abstract1 to another kind1 the concrete4 ,but inference probably enters when we pass from either of the above propositions to the assertion that the class of immortal men is zero1 or contains no objects4 ,it is of course a question of words to what processes we shall or shall not apply the name 8inference10 and ,i have no wish to continue the trifling discussions which have already taken place upon the subject4 ,what we need to do is to define accurately the sense in which we use the word 8inference10 and to distinguish the relation of inferrible propositions from other possible relations4 ,it seems to be sufficient to recognise four modes in which two apparently different propositions may be related4 ,thus two propositions may be-- #a4 99,tautologous99 or 99identical991 involving the same relation between the same terms and classes1 and only differing in the order of statement2 thus 8,victoria is the ,queen of ,england0 is tautologous with 8,the ,queen of ,england is ,victoria40 #b4 99,grammatically related991 when the classes or objects are the same and similarly related1 and the only difference is in the words2 thus 8,victoria is the ,queen of ,england0 is grammatically equivalent to 8,victoria is ,england's ,queen40 #c4 99,equivalents99 in qualitative and quantitative form1 the classes being the same1 but viewed in a different manner4 #d4 99,logically inferrible991 one from the other1 or it may be 99equivalent991 when the classes and relations are different1 but involve the same knowledge of the possible combinations4 ,,chapter ,,vii4 ,,induction4 ,we enter in this chapter upon the second great department of logical method1 that of ,induction or the ,inference of general from particular truths4 ,it cannot be said that the ,inductive process is of greater importance than the ,deductive process already considered1 because the latter process is absolutely essential to the existence of the former4 ,each is the complement and counterpart of the other4 ,the principles of thought and existence which underlie them are at the bottom the same1 just as subtraction of numbers necessarily rests upon the same principles as addition4 ,induction is1 in fact1 the inverse operation of deduction1 and cannot be conceived to exist without the corresponding operation1 so that the question of relative importance cannot arise4 ,who thinks of asking whether addition or subtraction is the more important process in arithmetic8 ,but at the same time much difference in difficulty may exist between a direct and inverse operation2 the integral calculus1 for instance1 is infinitely more difficult than the differential calculus of which it is the inverse4 ,similarly1 it must be allowed that inductive investigations are of a far higher degree of difficulty and complexity than any questions of deduction2 and it is this fact no doubt which led some logicians1 such as ,francis ,bacon1 ,locke1 and ,j4 ,s4 ,mill1 to erroneous opinions concerning the exclusive importance of induction4 ,hitherto we have been engaged in considering how from certain conditions1 laws1 or identities governing the combinations of qualities1 we may deduce the nature of the combinations agreeing with those conditions4 ,our work has been to unfold the results of what is contained in any statements1 and the process has been one of 99,synthesis994 ,the terms or combinations of which the character has been determined have usually1 though by no means always1 involved more qualities1 and therefore1 by the relation of extension and intension1 fewer objects than the terms in which they were described4 ,the truths inferred were thus usually less general than the truths from which they were inferred4 ,in induction all is inverted4 ,the truths to be ascertained are more general than the data from which they are drawn4 ,the process by which they are reached is 99analytical991 and consists in separating the complex combinations in which natural phenomena are presented to us1 and determining the relations of separate qualities4 ,given events obeying certain unknown laws1 we have to discover the laws obeyed4 ,instead of the comparatively easy task of finding what effects will follow from a given law1 the effects are now given and the law is required4 ,we have to interpret the will by which the conditions of creation were laid down4 99,induction an ,inverse ,operation99 ,i have already asserted that induction is the inverse operation of deduction1 but the difference is one of such great importance that ,i must dwell upon it4 ,there are many cases in which we can easily and infallibly do a certain thing but may have much trouble in undoing it4 ,a person may walk into the most complicated labyrinth or the most extensive catacombs1 and turn hither and thither at his will2 it is when he wishes to return that doubt and difficulty commence4 ,in entering1 any path served him2 in leaving1 he must select certain definite paths1 and in this selection he must either trust to memory of the way he entered or else make an exhaustive trial of all possible ways4 ,the explorer entering a new country makes sure his line of return by barking the trees4 ,the same difficulty arises in many scientific processes4 ,given any two numbers1 we may by a simple and infallible process obtain their product2 but when a large number is given it is quite another matter to determine its factors4 ,can the reader say what two numbers multiplied together will produce the number #h1faf1dfj1gii8 ,i think it unlikely that anyone but myself will ever know2 for they are two large prime numbers1 and can only be rediscovered by trying in succession a long series of prime divisors until the right one be fallen upon4 ,the work would probably occupy a good computer for many weeks1 but it did not occupy me many minutes to multiply the two factors together4 ,similarly there is no direct process for discovering whether any number is a prime or not2 it is only by exhaustively trying all inferior numbers which could be divisors1 that we can show there is none1 and the labour of the process would be intolerable were it not performed systematically once for all in the process known as the ,sieve of ,eratosthenes1 the results being registered in tables of prime numbers4 ,the immense difficulties which are encountered in the solution of algebraic equations afford another illustration4 ,given any algebraic factors1 we can easily and infallibly arrive at the product2 but given a product it is a matter of infinite difficulty to resolve it into factors4 ,given any series of quantities however numerous1 there is very little trouble in making an equation which shall have those quantities as roots4 ,let 99a991 99b991 99c991 99d991 c41 be the quantities2 then 799x99 - 99a997799x99 - 99b997799x99 - 99c997799x99 - d7 444 #j is the equation required1 and we only need to multiply out the expression on the left hand by ordinary rules4 ,but having given a complex algebraic expression equated to zero1 it is a matter of exceeding difficulty to discover all the roots4 ,mathematicians have exhausted their highest powers in carrying the complete solution up to the fourth degree4 ,in every other mathematical operation the inverse process is far more difficult than the direct process1 subtraction than addition1 division than multiplication1 evolution than involution2 but the difficulty increases vastly as the process becomes more complex4 ,differentiation1 the direct process1 is always capable of performance by fixed rules1 but as these rules produce considerable variety of results1 the inverse process of integration presents immense difficulties1 and in an infinite majority of cases surpasses the present resources of mathematicians4 ,there are no infallible and general rules for its accomplishment2 it must be done by trial1 by guesswork1 or by remembering the results of differentiation1 and using them as a guide4 ,coming more nearly to our own immediate subject1 exactly the same difficulty exists in determining the law which certain things obey4 ,given a general mathematical expression1 we can infallibly ascertain its value for any required value of the variable4 ,but ,i am not aware that mathematicians have ever attempted to lay down the rules of a process by which1 having given certain numbers1 one might discover a rational or precise formula from which they proceed4 ,the reader may test his power of detecting a law1 by contemplation of its results1 if he1 not being a mathematician1 will attempt to point out the law obeyed by the following numbers3 #a_/f1 #a_/cj1 #a_/db1 #a_/cj1 #e_/ff1 #fia_/bgcj1 #g_/f1 #cfag_/eaj1 #dchfg_/gih1 etc4 ,these numbers are sometimes in low terms1 but unexpectedly spring up to high terms2 in absolute magnitude they are very variable4 ,they seem to set all regularity and method at defiance1 and it is hardly to be supposed that anyone could1 from contemplation of the numbers1 have detected the relations between them4 ,yet they are derived from the most regular and symmetrical laws of relation1 and are of the highest importance in mathematical analysis1 being known as the numbers of ,bernoulli4 ,compare again the difficulty of decyphering with that of cyphering4 ,anyone can invent a secret language1 and with a little steady labour can translate the longest letter into the character4 ,but to decypher the letter1 having no key to the signs adopted1 is a wholly different matter4 ,as the possible modes of secret writing are infinite in number and exceedingly various in kind1 there is no direct mode of discovery whatever4 ,repeated trial1 guided more or less by knowledge of the customary form of cypher1 and resting entirely on the principles of probability and logical induction1 is the only resource4 ,a peculiar tact or skill is requisite for the process1 and a few men1 such as ,wallis or ,wheatstone1 have attained great success4 ,induction is the decyphering of the hidden meaning of natural phenomena4 ,given events which happen in certain definite combinations1 we are required to point out the laws which govern those combinations4 ,any laws being supposed1 we can1 with ease and certainty1 decide whether the phenomena obey those laws4 ,but the laws which may exist are infinite in variety1 so that the chances are immensely against mere random guessing4 ,the difficulty is much increased by the fact that several laws will usually be in operation at the same time1 the effects of which are complicated together4 ,the only modes of discovery consist either in exhaustively trying a great number of supposed laws1 a process which is exhaustive in more senses than one1 or else in carefully contemplating the effects1 endeavouring to remember cases in which like effects followed from known laws4 ,in whatever manner we accomplish the discovery1 it must be done by the more or less conscious application of the direct process of deduction4 ,the ,logical ,alphabet illustrates induction as well as deduction4 ,in considering the ,indirect ,process of ,inference we found that from certain propositions we could infallibly determine the combinations of terms agreeing with those premises4 ,the inductive problem is just the inverse4 ,having given certain combinations of terms1 we need to ascertain the propositions with which the combinations are consistent1 and from which they may have proceeded4 ,now1 if the reader contemplates the following combinations1 ,,abc 99ab99,c 99a99,,bc 99abc991 he will probably remember at once that they belong to the premises ,a ,,ab1 ,b ,,bc 7p4 #ib74 ,if not1 he will require a few trials before he meets with the right answer1 and every trial will consist in assuming certain laws and observing whether the deduced results agree with the data4 ,to test the facility with which he can solve this inductive problem1 let him casually strike out any of the combinations of the fourth column of the ,logical ,alphabet1 7p4 #id71 and say what laws the remaining combinations obey1 observing that every one of the letter-terms and their negatives ought to appear in order to avoid self-contradiction in the premises 7pp4 #gd1 #aaa74 ,let him say1 for instance1 what laws are embodied in the combinations ,,abc 99a99,,bc ,a99bc99 99ab99,c4 ,the difficulty becomes much greater when more terms enter into the combinations4 ,it would require some little examination to ascertain the complete conditions fulfilled in the combinations ,,ac99e99 99ab99,c99e99 99a99,,bc99e99 99abc99,e4 99a99,b99cd99,e ,the reader may discover easily enough that the principal laws are ,c 99e991 and ,a ,a99e992 but he would hardly discover without some trouble the remaining law1 namely1 that ,,bd ,,bd99e994 ,the difficulties encountered in the inductive investigations of nature1 are of an exactly similar kind4 ,we seldom observe any law in uninterrupted and undisguised operation4 ,the acuteness of ,aristotle and the ancient ,greeks did not enable them to detect that all terrestrial bodies tend to fall towards the centre of the earth4 ,a few nights of observation might have convinced an astronomer viewing the solar system from its centre1 that the planets travelled round the sun2 but the fact that our place of observation is one of the travelling planets1 so complicates the apparent motions of the other bodies1 that it required all the sagacity of ,copernicus to prove the real simplicity of the planetary system4 ,it is the same throughout nature2 the laws may be simple1 but their combined effects are not simple1 and we have no clue to guide us through their intricacies4 8,it is the glory of ,god10 said ,solomon1 8to conceal a thing1 but the glory of a king to search it out40 ,the laws of nature are the invaluable secrets which ,god has hidden1 and it is the kingly prerogative of the philosopher to search them out by industry and sagacity4 99,inductive ,problems for ,solution by the ,reader499 ,in the first edition 7vol4 ii4 p4 #cgj7 ,i gave a logical problem involving six terms1 and requested readers to discover the laws governing the combinations given4 ,i received satisfactory replies from readers both in the ,united ,states and in ,england4 ,i formed the combinations deductively from four laws of correction1 but my correspondents found that three simpler laws1 equivalent to the four more complex ones1 were the best answer2 these laws are as follows3 99a99 99ac991 99b99 99cd991 99d99 ,e99f994 ,in case other readers should like to test their skill in the inductive or inverse problem1 ,i give below several series of combinations forming problems of graduated difficulty4 ,,problem ,i4 ,a ,b 99c99 ,a 99b99 ,c 99a99 ,b ,c ,,problem ,,ii4 ,a ,b ,c ,a 99b99 ,c 99a99 ,b ,c 99a99 ,b 99c99 ,,problem ,,iii4 ,a ,b ,c ,a 99b99 ,c 99a99 ,b ,c 99a99 ,b 99c99 99a99 99b99 99c99 ,,problem ,,iv4 ,a ,b ,c ,d ,a 99b99 99c99 ,d 99a99 ,b 99c99 99d99 99a99 99b99 ,c 99d99 ,,problem ,v4 ,a ,b ,c ,d ,a ,b ,c 99d99 ,a ,b 99c99 99d99 ,a 99b99 ,c ,d ,a 99b99 99c99 ,d 99a99 ,b ,c ,d 99a99 ,b 99c99 ,d 99a99 ,b 99c99 99d99 99a99 99b99 ,c 99d99 ,,problem ,,vi4 ,a ,b ,c ,d ,e ,a ,b ,c 99d99 99e99 ,a ,b 99c99 ,d ,e ,a ,b 99c99 99d99 99e99 ,a 99b99 ,c ,d ,e 99a99 ,b ,c ,d ,e 99a99 ,b ,c 99d99 99e99 99a99 99b99 ,c ,d ,e 99a99 99b99 99c99 99d99 99e99 ,,problem ,,vii4 ,a 99b99 99c99 ,d 99e99 99a99 ,b ,c 99d99 ,e 99a99 99b99 ,c 99d99 ,e ,,problem ,,viii4 ,a ,b ,c ,d ,e ,a ,b ,c ,d 99e99 ,a ,b ,c 99d99 99e99 ,a ,b 99c99 99d99 99e99 ,a 99b99 ,c ,d ,e ,a 99b99 99c99 99d99 ,e ,a 99b99 99c99 99d99 99e99 99a99 ,b ,c ,d 99e99 99a99 ,b ,c 99d99 99e99 99a99 ,b 99c99 ,d 99e99 99a99 99b99 ,c ,d 99e99 99a99 99b99 ,c 99d99 ,e 99a99 99b99 99c99 ,d 99e99 99a99 99b99 99c99 99d99 ,e ,,problem ,,ix4 ,a ,b 99c99 ,d ,e ,f ,a ,b 99c99 ,d 99e99 ,f ,a 99b99 ,c ,d 99e99 99f99 ,a 99b99 99c99 ,d ,e 99f99 ,a 99b99 99c99 ,d 99e99 99f99 ,a 99b99 99c99 99d99 ,e ,f ,a 99b99 99c99 99d99 99e99 ,f 99a99 ,b 99c99 ,d ,e ,f 99a99 ,b 99c99 ,d 99e99 ,f 99a99 ,b 99c99 99d99 ,e ,f 99a99 99b99 ,c ,d ,e ,f 99a99 99b99 ,c ,d 99e99 ,f 99a99 99b99 ,c ,d 99e99 99f99 99a99 99b99 99c99 ,d 99e99 99f99 99a99 99b99 99c99 ,d ,e 99f99 99a99 99b99 99c99 99d99 99e99 ,f ,,problem ,x4 ,a ,b ,c ,d 99e99 ,f ,a ,b 99c99 ,d ,e 99f99 ,a 99b99 ,c ,d ,e ,f ,a 99b99 ,c ,d 99e99 ,f ,a 99b99 99c99 ,d 99e99 ,f 99a99 ,b ,c ,d ,e 99f99 99a99 ,b 99c99 ,d ,e 99f99 99a99 99b99 ,c ,d 99e99 ,f 99a99 99b99 ,c 99d99 99e99 ,f 99a99 99b99 99c99 ,d 99e99 99f99 99a99 99b99 99c99 99d99 99e99 99f99 99,induction of ,simple ,identities994 ,many important laws of nature are expressible in the form of simple identities1 and ,i can at once adduce them as examples to illustrate what ,i have said of the difficulty of the inverse process of induction4 ,two phenomena are conjoined4 ,thus all gravitating matter is exactly coincident with all matter possessing inertia2 where one property appears1 the other likewise appears4 ,all crystals of the cubical system1 are all the crystals which do not doubly refract light4 ,all exogenous plants are1 with some exceptions1 those which have two cotyledons or seed-leaves4 ,a little reflection will show that there is no direct and infallible process by which such complete coincidences may be discovered4 ,natural objects are aggregates of many qualities1 and any one of those qualities may prove to be in close connection with some others4 ,if each of a numerous group of objects is endowed with a hundred distinct physical or chemical qualities1 there will be no less than 7#a_/b77ajj #ii7 or #diej pairs of qualities1 which may be connected1 and it will evidently be a matter of great intricacy and labour to ascertain exactly which qualities are connected by any simple law4 ,one principal source of difficulty is that the finite powers of the human mind are not sufficient to compare by a single act any large group of objects with another large group4 ,we cannot hold in the conscious possession of the mind at any one moment more than five or six different ideas4 ,hence we must treat any more complex group by successive acts of attention4 ,the reader will perceive by an almost individual act of comparison that the words 99,roma99 and 99,mora99 contain the same letters4 ,he may perhaps see at a glance whether the same is true of 99,causal99 and 99,casual991 and of 99,logica99 and 99,caligo994 ,to assure himself that the letters in 99,astronomers99 make 99,no more stars991 that 99,serpens in akuleo99 is an anagram of 99,joannes ,keplerus991 or 99,great gun do us a sum99 an anagram of 99,augustus de ,morgan991 it will certainly be necessary to break up the act of comparison into several successive acts4 ,the process will acquire a double character1 and will consist in ascertaining that each letter of the first group is among the letters of the second group1 and 99vice vers991 that each letter of the second is among those of the first group4 ,in the same way we can only prove that two long lists of names are identical1 by showing that each name in one list occurs in the other1 and 99vice vers994 ,this process of comparison really consists in establishing two partial identities1 which are1 as already shown 7p4 #eh71 equivalent in conjunction to one simple identity4 ,we first ascertain the truth of the two propositions ,a ,,ab1 ,b ,,ab1 and we then rise by substitution to the single law ,a ,b4 ,there is another process1 it is true1 by which we may get to exactly the same result2 for the two propositions ,a ,,ab1 99a99 99ab99 are also equivalent to the simple identity ,a ,b4 ,if then we can show that all objects included under ,a are included under ,b1 and also that all objects not included under ,a are not included under ,b1 our purpose is effected4 ,by this process we should usually compare two lists if we are allowed to mark them4 ,for each name in the first list we should strike off one in the second1 and if1 when the first list is exhausted1 the second list is also exhausted1 it follows that all names absent from the first must be absent from the second1 and the coincidence must be complete4 ,these two modes of proving an identity are so closely allied that it is doubtful how far we can detect any difference in their powers and instances of application4 ,the first method is perhaps more convenient when the phenomena to be compared are rare4 ,thus we prove that all the musical concords coincide with all the more simple numerical ratios1 by showing that each concord arises from a simple ratio of undulations1 and then showing that each simple ratio gives rise to one of the concords4 ,to examine all the possible cases of discord or complex ratio of undulation would be impossible4 ,by a happy stroke of induction ,sir ,john ,herschel discovered that all crystals of quartz which cause the plane of polarization of light to rotate are precisely those crystals which have plagihedral faces1 that is1 oblique faces on the corners of the prism unsymmetrical with the ordinary faces4 ,this singular relation would be proved by observing that all plagihedral crystals possessed the power of rotation1 and 99vice vers99 all crystals possessing this power were plagihedral4 ,but it might at the same time be noticed that all ordinary crystals were devoid of the power4 ,there is no reason why we should not detect any of the four propositions ,a ,,ab1 ,b ,,ab1 99a99 99ab991 99b99 99ab991 all of which follow from ,a ,b 7p4 #aae74 ,sometimes the terms of the identity may be singular objects2 thus we observe that diamond is a combustible gem1 and being unable to discover any other that is1 we affirm-- ,diamond combustible gem4 ,in a similar manner we ascertain that ,mercury metal liquid at ordinary temperatures1 ,substance of least density substance of least atomic weight4 ,two or three objects may occasionally enter into the induction1 as when we learn that ,sodium potassium metal of less density than water1 ,venus ,mercury ,mars major planet devoid of satellites4 99,induction of ,partial ,identities994 ,we found in the last section that the complete identity of two classes is almost always discovered not by direct observation of the fact1 but by first establishing two partial identities4 ,there are also a multitude of cases in which the partial identity of one class with another is the only relation to be discovered4 ,thus the most common of all inductive inferences consists in establishing the fact that all objects having the properties of ,a have also those of ,b1 or that ,a ,,ab4 ,to ascertain the truth of a proposition of this kind it is merely necessary to assemble together1 mentally or physically1 all the objects included under ,a1 and then observe whether ,b is present in each of them1 or1 which is the same1 whether it would be impossible to select from among them any not-,b4 ,thus1 if we mentally assemble together all the heavenly bodies which move with apparent rapidity1 that is to say1 the planets1 we find that they all possess the property of not scintillating4 ,we cannot analyse any vegetable substance without discovering that it contains carbon and hydrogen1 but it is not true that all substances containing carbon and hydrogen are vegetable substances4 ,the great mass of scientific truths consists of propositions of this form ,a ,,ab4 ,thus in astronomy we learn that all the planets are spheroidal bodies2 that they all revolve in one direction round the sun2 that they all shine by reflected light2 that they all obey the law of gravitation4 ,but of course it is not to be asserted that all bodies obeying the law of gravitation1 or shining by reflected light1 or revolving in a particular direction1 or being spheroidal in form1 are planets4 ,in other sciences we have immense numbers of propositions of the same form1 as1 for instance1 all substances in becoming gaseous absorb heat2 all metals are elements2 they are all good conductors of heat and electricity2 all the alkaline metals are monad elements2 all foraminifera are marine organisms2 all parasitic animals are non-mammalian2 lightning never issues from stratous clouds2 pumice never occurs where only ,labrador felspar is present2 milkmaids do not suffer from small-pox2 and1 in the works of ,darwin1 scientific importance may attach even to such an apparently trifling observation as that 8white tom-cats having blue eyes are deaf40 ,the process of inference by which all such truths are obtained may readily be exhibited in a precise symbolic form4 ,we must have one premise specifying in a disjunctive form all the possible individuals which belong to a class2 we resolve the class1 in short1 into its constituents4 ,we then need a number of propositions1 each of which affirms that one of the individuals possesses a certain property4 ,thus the premises must be of the forms ,a ,b ,c ,d 4444 ,p ,q ,b ,,bx ,c ,,cx 444 444 444 444 ,q ,,qx4 ,now1 if we substitute for each alternative of the first premise its description as found among the succeeding premises1 we obtain ,a ,,bx ,,cx 4444 ,,px ,,qx or ,a 7,b ,c 4444 ,q7,x ,but for the aggregate of alternatives we may now substitute their equivalent as given in the first premise1 namely ,a1 so that we get the required result3 ,a ,,ax4 ,we should have reached the same result if the first premise had been of the form ,a ,,ab ,,ac 4444 ,,aq4 ,we can always prove a proposition1 if we find it more convenient1 by proving its equivalent4 ,to assert that all not-,b's are not-,a's1 is exactly the same as to assert that all ,a's are ,b's4 ,accordingly we may ascertain that ,a ,,ab by first ascertaining that 99b99 99ab994 ,if we observe1 for instance1 that all substances which are not solids are also not capable of double refraction1 it follows necessarily that all double refracting substances are solids4 ,we may convince ourselves that all electric substances are nonconductors of electricity1 by reflecting that all good conductors do not1 and in fact cannot1 retain electric excitation4 ,when we come to questions of probability it will be found desirable to prove1 as far as possible1 both the original proposition and its equivalent1 as there is then an increased area of observation4 ,the number of alternatives which may arise in the division of a class varies greatly1 and may be any number from two upwards4 ,thus it is probable that every substance is either magnetic or diamagnetic1 and no substance can be both at the same time4 ,the division then must be made in the form ,a ,,ab99c99 ,a99b99,c4 ,if now we can prove that all magnetic substances are capable of polarity1 say ,b ,,bd1 and also that all diamagnetic substances are capable of polarity1 ,c ,,cd1 it follows by substitution that all substances are capable of polarity1 or ,a ,,ad4 ,we commonly divide the class substance into the three subclasses1 solid1 liquid1 and gas2 and if we can show that in each of these forms it obeys ,carnot's thermodynamic law1 it follows that all substances obey that law4 ,similarly we may show that all vertebrate animals possess red blood1 if we can show separately that fish1 reptiles1 birds1 marsupials1 and mammals possess red blood1 there being1 as far as is known1 only five principal subclasses of vertebrata4 ,our inductions will often be embarrassed by exceptions1 real or apparent4 ,we might affirm that all gems are incombustible were not diamonds undoubtedly combustible4 ,nothing seems more evident than that all the metals are opaque until we examine them in fine films1 when gold and silver are found to be transparent4 ,all plants absorb carbonic acid except certain fungi2 all the bodies of the planetary system have a progressive motion from west to east1 except the satellites of ,uranus and ,neptune4 ,even some of the profoundest laws of matter are not quite universal2 all solids expand by heat except india-rubber1 and possibly a few other substances2 all liquids which have been tested expand by heat except water below #d ,c4 and fused bismuth2 all gases have a coefficient of expansion increasing with the temperature1 except hydrogen4 ,in a later chapter ,i shall consider how such anomalous cases may be regarded and classified2 here we have only to express them in a consistent manner by our notation4 ,let us take the case of the transparency of metals1 and assign the terms thus3-- ,a metal ,d iron ,b gold ,e1 ,f1 c4 copper1 lead1 c4 ,c silver ,x opaque4 ,our premises will be ,a ,b ,c ,d ,e1 c4 ,b ,b99x99 ,c ,c99x99 ,d ,,dx ,e ,,ex1 and so on for the rest of the metals4 ,now evidently ,a99bc99 7,d ,e ,f 444799bc991 and by substitution as before we shall obtain ,a99bc99 ,a99bc99,x1 or in words1 8,all metals not gold nor silver are opaque20 at the same time we have ,a7,b ,c7 ,,ab ,,ac ,,ab99x99 ,,ac99x99 ,a7,b ,c799x991 or 8,metals which are either gold or silver are not opaque40 ,in some cases the problem of induction assumes a much higher degree of complexity4 ,if we examine the properties of crystallized substances we may find some properties which are common to all1 as cleavage or fracture in definite planes2 but it would soon become requisite to break up the class into several minor ones4 ,we should divide crystals according to the seven accepted systems--and we should then find that crystals of each system possess many common properties4 ,thus crystals of the ,regular or ,cubical system expand equally by heat1 conduct heat and electricity with uniform rapidity1 and are of like elasticity in all directions2 they have but one index of refraction for light2 and every facet is repeated in like relation to each of the three axes4 ,crystals of the system having one principal axis will be found to possess the various physical powers of conduction1 refraction1 elasticity1 c41 uniformly in directions perpendicular to the principal axis2 in other directions their properties vary according to complicated laws4 ,the remaining systems in which the crystals possess three unequal axes1 or have inclined axes1 exhibit still more complicated results1 the effects of the crystal upon light1 heat1 electricity1 c41 varying in all directions4 ,but when we pursue induction into the intricacies of its application to nature we really enter upon the subject of classification1 which we must take up again in a later part of this work4 99,solution of the ,inverse or ,inductive ,problem1 involving ,two ,classes994 ,it is now plain that ,induction consists in passing back from a series of combinations to the laws by which such combinations are governed4 ,the natural law that all metals are conductors of electricity really means that in nature we find three classes of objects1 namely-- #a4 ,metals1 conductors2 #b4 ,not-metals1 conductors2 #c4 ,not-metals1 not-conductors4 ,it comes to the same thing if we say that it excludes the existence of the class1 8metals not-conductors40 ,in the same way every other law or group of laws will really mean the exclusion from existence of certain combinations of the things1 circumstances or phenomena governed by those laws4 ,now in logic1 strictly speaking1 we treat not the phenomena1 nor the laws1 but the general forms of the laws2 and a little consideration will show that for a finite number of things the possible number of forms or kinds of law governing them must also be finite4 ,using general terms1 we know that ,a and ,b can be present or absent in four ways and no more--thus3 ,,ab1 ,a99b991 99a99,b1 99ab992 therefore every possible law which can exist concerning the relation of ,a and ,b must be marked by the exclusion of one or more of the above combinations4 ,the number of possible laws then cannot exceed the number of selections which we can make from these four combinations4 ,since each combination may be present or absent1 the number of cases to be considered is #b #b #b #b1 or sixteen2 and these cases are all shown in the following table1 in which the sign #j indicates absence or non-existence of the combination shown at the left-hand column in the same line1 and the mark #a its presence3-- #a #b #c #d #e #f #g #h #i #aj #aa #ab #ac #ad #ae #af 99 99 99 99 99 99 99 ,,ab #j #j #j #j #j #j #j #j #a #a #a #a #a #a #a #a ,a99b99 #j #j #j #j #a #a #a #a #j #j #j #j #a #a #a #a 99a99,b #j #j #a #a #j #j #a #a #j #j #a #a #j #j #a #a 99ab99 #j #a #j #a #j #a #j #a #j #a #j #a #j #a #j #a ,thus in column sixteen we find that all the conceivable combinations are present1 which means that there are no special laws in existence in such a case1 and that the combinations are governed only by the universal ,laws of ,identity and ,difference4 ,the example of metals and conductors of electricity would be represented by the twelfth column2 and every other mode in which two things or qualities might present themselves is shown in one or other of the columns4 ,more than half the cases may indeed be at once rejected1 because they involve the entire absence of a term or its negative4 ,it has been shown to be a logical principle that every term must have its negative 7p4 #aaa71 and when this is not the case1 inconsistency between the conditions of combination must exist4 ,thus if we laid down the two following propositions1 8,graphite conducts electricity10 and 8,graphite does not conduct electricity10 it would amount to asserting the impossibility of graphite existing at all2 or in general terms1 ,a is ,b and ,a is not ,b result in destroying altogether the combinations containing ,a1 a case shown in the fourth column of the above table4 ,we therefore restrict our attention to those cases which may be represented in natural phenomena when at least two combinations are present1 and which correspond to those columns of the table in which each of ,a1 99a991 ,b1 99b99 appears4 ,these cases are shown in the columns marked with an asterisk4 ,we find that seven cases remain for examination1 thus characterised-- ,four cases exhibiting three combinations1 ,two cases exhibiting two combinations1 ,one case exhibiting four combinations4 ,it has already been pointed out that a proposition of the form ,a ,,ab destroys one combination1 ,a99b991 so that this is the form of law applying to the twelfth column4 ,but by changing one or more of the terms in ,a ,,ab into its negative1 or by interchanging ,a and ,b1 99a99 and 99b991 we obtain no less than eight different varieties of the one form2 thus-- #abth case4 #hth case4 #aeth case4 #adth case4 ,a ,,ab ,a ,a99b99 99a99 99a99,b 99a99 99ab99 99b99 99ab99 ,b 99a99,b 99b99 ,a99b99 ,b ,,ab ,the reader of the preceding sections will see that each proposition in the lower line is logically equivalent to1 and is in fact the contrapositive of1 that above it 7p4 #hc74 ,thus the propositions ,a ,a99b99 and ,b 99a99,b both give the same combinations1 shown in the eighth column of the table1 and trial shows that the twelfth1 eighth1 fifteenth and fourteenth columns are thus accounted for4 ,we come to this conclusion then--99,the general form of proposition99 ,a ,,ab 99admits of four logically distinct varieties1 each capable of expression in two modes994 ,in two columns of the table1 namely the seventh and tenth1 we observe that two combinations are missing4 ,now a simple identity ,a ,b renders impossible both ,a99b99 and 99a99,b1 accounting for the tenth case2 and if we change ,b into 99b99 the identity ,a 99b99 accounts for the seventh case4 ,there may indeed be two other varieties of the simple identity1 namely 99a99 99b99 and 99a99 ,b2 but it has already been shown repeatedly that these are equivalent respectively to ,a ,b and ,a 99b99 7p4 #aae74 ,as the sixteenth column has already been accounted for as governed by no special conditions1 we come to the following general conclusion3--,the laws governing the combinations of two terms must be capable of expression either in a partial identity or a simple identity2 the partial identity is capable of only four logically distinct varieties1 and the simple identity of two4 ,every logical relation between two terms must be expressed in one of these six forms of law1 or must be logically equivalent to one of them4 ,in short1 we may conclude that in treating of partial and complete identity1 we have exhaustively treated the modes in which two terms or classes of objects can be related4 ,of any two classes it can be said that one must either be included in the other1 or must be identical with it1 or a like relation must exist between one class and the negative of the other4 ,we have thus completely solved the inverse logical problem concerning two terms4#he #he ,the contents of this and the following section nearly correspond with those of a paper read before the ,manchester ,literary and ,philosophical ,society on ,december #bfth1 #ahga4 ,see ,proceedings of the ,society1 vol4 xi4 pp4 #fe-fh1 and ,memoirs1 ,third ,series1 vol4 v4 pp4 #aai-acj4 99,the ,inverse ,logical ,problem involving ,three ,classes499 ,no sooner do we introduce into the problem a third term ,c1 than the investigation assumes a far more complex character1 so that some readers may prefer to pass over this section4 ,three terms and their negatives may be combined1 as we have frequently seen1 in eight different combinations1 and the effect of laws or logical conditions is to destroy any one or more of these combinations4 ,now we may make selections from eight things in #bh or #bef ways2 so that we have no less than #bef different cases to treat1 and the complete solution is at least fifty times as troublesome as with two terms4 ,many series of combinations1 indeed1 are contradictory1 as in the simpler problem1 and may be passed over1 the test of consistency being that each of the letters ,a1 ,b1 ,c1 99a991 99b991 99c991 shall appear somewhere in the series of combinations4 ,my mode of solving the problem was as follows3--,having written out the whole of the #bef series of combinations1 ,i examined them separately and struck out such as did not fulfil the test of consistency4 ,i then chose some form of proposition involving two or three terms1 and varied it in every possible manner1 both by the circular interchange of letters 7,a1 ,b1 ,c into ,b1 ,c1 ,a and then into ,c1 ,a1 ,b71 and by the substitution for any one or more of the terms of the corresponding negative terms4 ,for instance1 the proposition ,,ab ,,abc can be first varied by circular interchange so as to give ,,bc ,,bca and then ,,ca ,,cab4 ,each of these three can then be thrown into eight varieties by negative change4 ,thus ,,ab ,,abc gives 99a99,b 99a99,,bc1 ,a99b99 ,a99b99,c1 ,,ab ,,ab99c991 99ab99 99ab99,c1 and so on4 ,thus there may possibly exist no less than twenty-four varieties of the law having the general form ,,ab ,,abc1 meaning that whatever has the properties of ,a and ,b has those also of ,c4 ,it by no means follows that some of the varieties may not be equivalent to others2 and trial shows1 in fact1 that ,,ab ,,abc is exactly the same in meaning as ,a99c99 ,a99bc99 or ,b99c99 ,b99ca994 ,thus the law in question has but eight varieties of distinct logical meaning4 ,i now ascertain by actual deductive reasoning which of the #bef series of combinations result from each of these distinct laws1 and mark them off as soon as found4 ,i then proceed to some other form of law1 for instance ,a ,,abc1 meaning that whatever has the qualities of ,a has those also of ,b and ,c4 ,i find that it admits of twenty-four variations1 all of which are found to be logically distinct2 the combinations being worked out1 ,i am able to mark off twenty-four more of the list of #bef series4 ,i proceed in this way to work out the results of every form of law which ,i can find or invent4 ,if in the course of this work ,i obtain any series of combinations which had been previously marked off1 ,i learn at once that the law giving these combinations is logically equivalent to some law previously treated4 ,it may be safely inferred that every variety of the apparently new law will coincide in meaning with some variety of the former expression of the same law4 ,i have sufficiently verified this assumption in some cases1 and have never found it lead to error4 ,thus as ,,ab ,,abc is equivalent to ,a99c99 ,a99bc991 so we find that 99ab99 99ab99,c is equivalent to 99ac99 99ac99,b4 ,among the laws treated were the two ,a ,,ab and ,a ,b which involve only two terms1 because it may of course happen that among three things two only are in special logical relation1 and the third independent2 and the series of combinations representing such cases of relation are sure to occur in the complete enumeration4 ,all single propositions which ,i could invent having been treated1 pairs of propositions were next investigated4 ,thus we have the relations1 8,all ,a's are ,b's1 and all ,b's are ,c's10 of which the old logical syllogism is the development4 ,we may also have 8all ,a's are all ,b's1 and all ,b's are ,c's10 or even 8all ,a's are all ,b's1 and all ,b's are all ,c's40 ,all such premises admit of variations1 greater or less in number1 the logical distinctness of which can only be determined by trial in detail4 ,disjunctive propositions either singly or in pairs were also treated1 but were often found to be equivalent to other propositions of a simpler form2 thus ,a ,,abc ,a99bc99 is exactly the same in meaning as ,,ab ,,ac4 ,this mode of exhaustive trial bears some analogy to that ancient mathematical process called the ,sieve of ,eratosthenes4 ,having taken a long series of the natural numbers1 ,eratosthenes is said to have calculated out in succession all the multiples of every number1 and to have marked them off1 so that at last the prime numbers alone remained1 and the factors of every number were exhaustively discovered4 ,my problem of #bef series of combinations is the logical analogue1 the chief points of difference being that there is a limit to the number of cases1 and that prime numbers have no analogue in logic1 since every series of combinations corresponds to a law or group of conditions4 ,but the analogy is perfect in the point that they are both inverse processes4 ,there is no mode of ascertaining that a number is prime but by showing that it is not the product of any assignable factors4 ,so there is no mode of ascertaining what laws are embodied in any series of combinations but trying exhaustively the laws which would give them4 ,just as the results of ,eratosthenes' method have been worked out to a great extent and registered in tables for the convenience of other mathematicians1 ,i have endeavoured to work out the inverse logical problem to the utmost extent which is at present practicable or useful4 ,i have thus found that there are altogether fifteen conditions or series of conditions which may govern the combinations of three terms1 forming the premises of fifteen essentially different kinds of arguments4 ,the following table contains a statement of these conditions1 together with the numbers of combinations which are contradicted or destroyed by each1 and the numbers of logically distinct variations of which the law is capable4 ,there might be also added1 as a sixteenth case1 that case where no special logical condition exists1 so that all the eight combinations remain4 ---------------------------------------- ----------------------- ,number of ,number of ,reference ,propositions expressing the distinct combinations ,number4 general type of the logical logical contradicted conditions4 variations4 by each4 ---------------------------------------- ----------------------- ,i4 ,a ,b #f #d ,,ii4 ,a ,,ab #ab #b ,,iii4 ,a ,b1 ,b ,c #d #f ,,iv4 ,a ,b1 ,b ,,bc #bd #e ,v4 ,a ,,ab1 ,b ,,bc #bd #d ,,vi4 ,a ,,bc #bd #d ,,vii4 ,a ,,abc #bd #c ,,viii4 ,,ab ,,abc #h #a ,,ix4 ,a ,,ab1 99a99,b 99a99,b99c99 #bd #c ,x4 ,a ,,abc1 99ab99 99ab99,c #h #d ,,xi4 ,,ab ,,abc1 99ab99 99abc99 #d #b ,,xii4 ,,ab ,,ac #ab #b ,,xiii4 ,a ,,bc ,a99bc99 #h #c ,,xiv4 ,a ,,bc 99bc99 #b #d ,,xv4 ,a ,,abc1 99a99 ,b99c99 99b99,c #h #e ---------------------------------------- ----------------------- ,there are sixty-three series of combinations derived from self-contradictory premises1 which with #aib1 the sum of the numbers of distinct logical variations stated in the third column of the table1 and with the one case where there are no conditions or laws at all1 make up the whole conceivable number of #bef series4 ,we learn from this table1 for instance1 that two propositions of the form ,a ,,ab1 ,b ,,bc1 which are such as constitute the premises of the old syllogism ,barbara1 exclude as impossible four of the eight combinations in which three terms may be united1 and that these propositions are capable of taking twenty-four variations by transpositions of the terms or the introduction of negatives4 ,this table then presents the results of a complete analysis of all the possible logical relations arising in the case of three terms1 and the old syllogism forms but one out of fifteen typical forms4 ,generally speaking1 every form can be converted into apparently different propositions2 thus the fourth type ,a ,b1 ,b ,,bc may appear in the form ,a ,,abc1 99a99 99ab991 or again in the form of three propositions ,a ,,ab1 ,b ,,bc1 99a99,b 99a99,b99c992 but all these sets of premises yield identically the same series of combinations1 and are therefore of equivalent logical meaning4 ,the fifth type1 or ,barbara1 can also be thrown into the equivalent forms ,a ,,abc1 99a99,b 99a99,,bc and ,a ,,ac1 ,b ,a 99a99,,bc4 ,in other cases ,i have obtained the very same logical conditions in four modes of statements4 ,as regards mere appearance and form of statement1 the number of possible premises would be very great1 and difficult to exhibit exhaustively4 ,the most remarkable of all the types of logical condition is the fourteenth1 namely1 ,a ,,bc 99bc994 ,it is that which expresses the division of a genus into two doubly marked species1 and might be illustrated by the example--8,component of the physical universe matter1 gravitating1 or not-matter 7ether71 not-gravitating40 ,it is capable of only two distinct logical variations1 namely1 ,a ,,bc 99bc99 and ,a ,b99c99 99b99,c4 ,by transposition or negative change of the letters we can indeed obtain six different expressions of each of these propositions2 but when their meanings are analysed1 by working out the combinations1 they are found to be logically equivalent to one or other of the above two4 ,thus the proposition ,a ,,bc 99bc99 can be written in any of the following five other modes1 99a99 99b99,c ,b99c991 ,b ,,ca 99ca991 99b99 99c99,a ,c99a991 ,c ,,ab 99ab991 99c99 99a99,b ,a99b994 ,i do not think it needful to publish at present the complete table of #aic series of combinations and the premises corresponding to each4 ,such a table enables us by mere inspection to learn the laws obeyed by any set of combinations of three things1 and is to logic what a table of factors and prime numbers is to the theory of numbers1 or a table of integrals to the higher mathematics4 ,the table already given 7p4 #adj7 would enable a person with but little labour to discover the law of any combinations4 ,if there be seven combinations 7one contradicted7 the law must be of the eighth type1 and the proper variety will be apparent4 ,if there be six combinations 7two contradicted71 either the second1 eleventh1 or twelfth type applies1 and a certain number of trials will disclose the proper type and variety4 ,if there be but two combinations the law must be of the third type1 and so on4 ,the above investigations are complete as regards the possible logical relations of two or three terms4 ,but when we attempt to apply the same kind of method to the relations of four or more terms1 the labour becomes impracticably great4 ,four terms give sixteen combinations compatible with the laws of thought1 and the number of possible selections of combinations is no less than #baf or #fe1ecf4 ,the following table shows the extraordinary manner in which the number of possible logical relations increases with the number of terms involved4 ---------------------------------------- --------------------- ,number of ,number of ,number of possible selections of combi- terms4 possible nations corresponding to consistent combinations4 or inconsistent logical relations4 ---------------------------------------- ---------------------- #b #d #af #c #h #bef #d #af #fe1ecf #e #cb #d1bid1ifg1bif #f #fd #ah1ddf1gdd1jgc1gji1eea1faf ---------------------------------------- --------------------- ,some years of continuous labour would be required to ascertain the types of laws which may govern the combinations of only four things1 and but a small part of such laws would be exemplified or capable of practical application in science4 ,the purely logical inverse problem1 whereby we pass from combinations to their laws1 is solved in the preceding pages1 as far as it is likely to be for a long time to come2 and it is almost impossible that it should ever be carried more than a single step further4 ,in the first edition1 vol i4 p4 #aeh1 ,i stated that ,i had not been able to discover any mode of calculating the number of cases in which inconsistency would be implied in the selection of combinations from the ,logical ,alphabet4 ,the logical complexity of the problem appeared to be so great that the ordinary modes of calculating numbers of combinations failed1 in my opinion1 to give any aid1 and exhaustive examination of the combinations in detail seemed to be the only method applicable4 ,this opinion1 however1 was mistaken1 for both ,mr4 ,r4 ,b4 ,hayward1 of ,harrow1 and ,mr4 ,w4 ,h4 ,brewer have calculated the numbers of inconsistent cases both for three and for four terms1 without much difficulty4 ,in the case of four terms they find that there are #agfa inconsistent selections and #fc1ggd consistent1 which with one case where no condition exists1 make up the total of #fe1ecf possible selections4 ,the inconsistent cases are distributed in the manner shown in the following table3-- ---------------------------------------- ------------------------- ,number of ,combinations #j #a #b #c #d #e #f #g #h #i #aj1 c4 remaining4 ---------------------------------------- ------------------------- ,number of ,inconsistent #a #af #aab #ceb #ecf #ddh #bbd #fd #h #j #j1 c4 ,cases4 ---------------------------------------- ------------------------- ,when more than eight combinations of the ,logical ,alphabet 7p4 #id1 column ,v47 remain unexcluded1 there cannot be inconsistency4 ,the whole numbers of ways of selecting #j1 #a1 #b1 c41 combinations out of #af are given in the #agth line of the ,arithmetical ,triangle given further on in the ,chapter on ,combinations and ,permutations1 the sum of the numbers in that line being #fe1ecf4 99,professor ,clifford on the ,types of ,compound ,statement involving ,four ,classes499 ,in the first edition 7vol4 i4 p4 #afc71 ,i asserted that some years of labour would be required to ascertain even the precise number of types of law governing the combinations of four classes of things4 ,though ,i still believe that some years' labour would be required to work out the types themselves1 it is clearly a mistake to suppose that the 99numbers99 of such types cannot be calculated with a reasonable amount of labour1 ,professor ,w4 ,k4 ,clifford having actually accomplished the task4 ,his solution of the numerical problem involves the use of a complete new system of nomenclature and is far too intricate to be fully described here4 ,i can only give a brief abstract of the results1 and refer readers1 who wish to follow out the reasoning1 to the ,proceedings of the ,literary and ,philosophical ,society of ,manchester1 for the #ith ,january1 #ahgg1 vol4 xvi41 p4 #hh1 where ,professor ,clifford's paper is printed in full4 ,by a 99simple statement99 ,professor ,clifford means the denial of the existence of any single combination or 99cross-division991 of the classes1 as in ,,abcd #j1 or ,a99b99,c99d99 #j4 ,the denial of two or more such combinations is called a 99compound statement991 and is further said to be 99twofold991 99threefold991 c41 according to the number denied4 ,thus ,,abc #j is a twofold compound statement in regard to four classes1 because it involves both ,,abcd #j and ,,abc99d99 #j4 ,when two compound statements can be converted into one another by interchange of the classes1 ,a1 ,b1 ,c1 ,d1 with each other or with their complementary classes1 99a991 99b991 99c991 99d991 they are called 99similar991 and all similar statements are said to belong to the same 99type994 ,two statements are called 99complementary99 when they deny between them all the sixteen combinations without both denying any one2 or1 which is the same thing1 when each denies just those combinations which the other permits to exist4 ,it is obvious that when two statements are similar1 the complementary statements will also be similar1 and consequently for every type of 99n99-fold statement1 there is a complementary type of 7#af--99n997-fold statement4 ,it follows that we need only enumerate the types as far as the eighth order2 for the types of more-than-eight-fold statement will already have been given as complementary to types of lower orders4 ,one combination1 ,,abcd1 may be converted into another ,a99b99,c99d99 by interchanging one or more of the classes with the complementary classes4 ,the number of such changes is called the 99distance991 which in the above case is #b4 ,in two similar compound statements the distances of the combinations denied must be the same2 but it does not follow that when all the distances are the same1 the statements are similar4 ,there is1 however1 only one example of two dissimilar statements having the same distances4 ,when the distance is #d1 the two combinations are said to be 99obverse99 to one another1 and the statements denying them are called 99obverse statements991 as in ,,abcd #j and 99abcd99 #j or again ,a99b99,c99d99 #j and 99a99,b99c99,d #j4 ,when any one combination is given1 called the 99origin991 all the others may be grouped in respect of their relations to it as follows3--,four are at distance 99one99 from it1 and may be called 99proximates992 six are at distance 99two991 and may be called 99mediates992 four are at distance 99three991 and may be called 99ultimates992 finally the obverse is at distance 99four994 ,origin and ,six ,obverse and four proximates4 mediates4 four ultimates4 99ab99,,cd 99a99,,bcd ,a99bc99,d ,a99b99,c99d99 ,a99bcd99 _/ _/ _/ ,,abc99d99--,,abcd--,a99b99,,cd 99abc99,d--99abcd99--99a99,b99cd99 _/ _/ _/ ,,ab99c99,d 99a99,b99c99,d 99a99,,bc99d99 99ab99,c99d994 ,,ab99cd99 ,it will be seen that the four proximates are respectively obverse to the four ultimates1 and that the mediates form three pairs of obverses4 ,every proximate or ultimate is distant #a and #c respectively from such a pair of mediates4 ,aided by this system of nomenclature ,professor ,clifford proceeds to an exhaustive enumeration of types1 in which it is impossible to follow him4 ,the results are as follows3-- #a-;fold statements #a type #b #d types #c #f #d #ai #aei #e #bg #f #dg #g #ee #h-;fold statements #gh ,now as each seven-fold or less-than-seven-fold statement is complementary to a nine-fold or more-than-nine-fold statement1 it follows that the complete number of types will be #aei #b #gh #cif4 ,it appears then that the types of statement concerning four classes are only about #bf times as numerous as those concerning three classes1 fifteen in number1 although the number of possible combinations is #bef times as great4 ,professor ,clifford informs me that the knowledge of the possible groupings of subdivisions of classes which he obtained by this inquiry has been of service to him in some applications of hyper-elliptic functions to which he has subsequently been led4 ,professor ,cayley has since expressed his opinion that this line of investigation should be followed out1 owing to the bearing of the theory of compound combinations upon the higher geometry4#hf ,it seems likely that many unexpected points of connection will in time be disclosed between the sciences of logic and mathematics4 #hf 99,proceedings of the ,manchester ,literary and ,philosophical ,society991 #fth ,february1 #ahgg1 vol4 xvi41 p4 #aac4 99,distinction between ,perfect and ,imperfect ,induction499 ,we cannot proceed with advantage before noticing the extreme difference which exists between cases of perfect and those of imperfect induction4 ,we call an induction 99perfect99 when all the objects or events which can possibly come under the class treated have been examined4 ,but in the majority of cases it is impossible to collect together1 or in any way to investigate1 the properties of all portions of a substance or of all the individuals of a race4 ,the number of objects would often be practically infinite1 and the greater part of them might be beyond our reach1 in the interior of the earth1 or in the most distant parts of the ,universe4 ,in all such cases induction is 99imperfect991 and is affected by more or less uncertainty4 ,as some writers have fallen into much error concerning the functions and relative importance of these two branches of reasoning1 ,i shall have to point out that-- #a4 ,perfect ,induction is a process absolutely requisite1 both in the performance of imperfect induction and in the treatment of large bodies of facts of which our knowledge is complete4 #b4 ,imperfect ,induction is founded on ,perfect ,induction1 but involves another process of inference of a widely different character4 ,it is certain that if ,i can draw any inference at all concerning objects not examined1 it must be done on the data afforded by the objects which have been examined4 ,if ,i judge that a distant star obeys the law of gravity1 it must be because all other material objects sufficiently known to me obey that law4 ,if ,i venture to assert that all ruminant animals have cloven hoofs1 it is because all ruminant animals which have come under my notice have cloven hoofs4 ,on the other hand1 ,i cannot safely say that all cryptogamous plants possess a purely cellular structure1 because some cryptogamous plants1 which have been examined by botanists1 have a partially vascular structure4 ,the probability that a new cryptogam will be cellular only can be estimated1 if at all1 on the ground of the comparative numbers of known cryptogams which are and are not cellular4 ,thus the first step in every induction will consist in accurately summing up the number of instances of a particular phenomenon which have fallen under our observation4 ,adams and ,leverrier1 for instance1 must have inferred that the undiscovered planet ,neptune would obey ,bode's law1 because 99all the planets known at that time obeyed it994 ,on what principles the passage from the known to the apparently unknown is warranted1 must be carefully discussed in the next section1 and in various parts of this work4 ,it would be a great mistake1 however1 to suppose that ,perfect ,induction is in itself useless4 ,even when the enumeration of objects belonging to any class is complete1 and admits of no inference to unexamined objects1 the statement of our knowledge in a general proposition is a process of so much importance that we may consider it necessary4 ,in many cases we may render our investigations exhaustive2 all the teeth or bones of an animal2 all the cells in a minute vegetable organ2 all the caves in a mountain side2 all the strata in a geological section2 all the coins in a newly found hoard1 may be so completely scrutinized that we may make some general assertion concerning them without fear of mistake4 ,every bone might be proved to contain phosphate of lime2 every cell to enclose a nucleus2 every cave to hide remains of extinct animals2 every stratum to exhibit signs of marine origin2 every coin to be of ,roman manufacture4 ,these are cases where our investigation is limited to a definite portion of matter1 or a definite area on the earth's surface4 ,there is another class of cases where induction is naturally and necessarily limited to a definite number of alternatives4 ,of the regular solids we can say without the least doubt that no one has more than twenty faces1 thirty edges1 and twenty corners2 for by the principles of geometry we learn that there cannot exist more than five regular solids1 of each of which we easily observe that the above statements are true4 ,in the theory of numbers1 an endless variety of perfect inductions might be made2 we can show that no number less than sixty possesses so many divisors1 and the like is true of #cfj2 for it does not require a great amount of labour to ascertain and count all the divisors of numbers up to sixty or #cfj4 ,i can assert that between #fj1jda and #fj1jgg no prime number occurs1 because the exhaustive examination of those who have constructed tables of prime numbers proves it to be so4 ,in matters of human appointment or history1 we can frequently have a complete limitation of the number of instances to be included in an induction4 ,we might show that the propositions of the third book of ,euclid treat only of circles2 that no part of the works of ,galen mentions the fourth figure of the syllogism2 that none of the other kings of ,england reigned so long as ,george ,,iii42 that ,magna ,charta has not been repealed by any subsequent statute2 that the price of corn in ,england has never been so high since #ahdg as it was in that year2 that the price of the ,english funds has never been lower than it was on the #bcrd of ,january1 #agih1 when it fell to #dg-a_/d4 ,it has been urged against this process of ,perfect ,induction that it gives no new information1 and is merely a summing up in a brief form of a multitude of particulars4 ,but mere abbreviation of mental labour is one of the most important aids we can enjoy in the acquisition of knowledge4 ,the powers of the human mind are so limited that multiplicity of detail is alone sufficient to prevent its progress in many directions4 ,thought would be practically impossible if every separate fact had to be separately thought and treated4 ,economy of mental power may be considered one of the main conditions on which our elevated intellectual position depends4 ,mathematical processes are for the most part but abbreviations of the simpler acts of addition and subtraction4 ,the invention of logarithms was one of the most striking additions ever made to human power3 yet it was a mere abbreviation of operations which could have been done before had a sufficient amount of labour been available4 ,similar additions to our power will1 it is hoped1 be made from time to time2 for the number of mathematical problems hitherto solved is but an indefinitely small fraction of those which await solution1 because the labour they have hitherto demanded renders them impracticable4 ,so it is throughout all regions of thought4 ,the amount of our knowledge depends upon our power of bringing it within practicable compass4 ,unless we arrange and classify facts and condense them into general truths1 they soon surpass our powers of memory1 and serve but to confuse4 ,hence ,perfect ,induction1 even as a process of abbreviation1 is absolutely essential to any high degree of mental achievement4 99,transition from ,perfect to ,imperfect ,induction499 ,it is a question of profound difficulty on what grounds we are warranted in inferring the future from the present1 or the nature of undiscovered objects from those which we have examined with our senses4 ,we pass from ,perfect to ,imperfect ,induction when once we allow our conclusion to apply1 at all events apparently1 beyond the data on which it was founded4 ,in making such a step we seem to gain a net addition to our knowledge2 for we learn the nature of what was unknown4 ,we reap where we have never sown4 ,we appear to possess the divine power of creating knowledge1 and reaching with our mental arms far beyond the sphere of our own observation4 ,i shall have1 indeed1 to point out certain methods of reasoning in which we do pass altogether beyond the sphere of the senses1 and acquire accurate knowledge which observation could never have given2 but it is not imperfect induction that accomplishes such a task4 ,of imperfect induction itself1 ,i venture to assert that it never makes any real addition to our knowledge1 in the meaning of the expression sometimes accepted4 ,as in other cases of inference1 it merely unfolds the information contained in past observations2 it merely renders explicit what was implicit in previous experience4 ,it transmutes1 but certainly does not create knowledge4 ,there is no fact which ,i shall more constantly keep before the reader's mind in the following pages than that the results of imperfect induction1 however well authenticated and verified1 are never more than probable4 ,we never can be sure that the future will be as the present4 ,we hang ever upon the will of the ,creator3 and it is only so far as ,he has created two things alike1 or maintains the framework of the world unchanged from moment to moment1 that our most careful inferences can be fulfilled4 ,all predictions1 all inferences which reach beyond their data1 are purely hypothetical1 and proceed on the assumption that new events will conform to the conditions detected in our observation of past events4 ,no experience of finite duration can give an exhaustive knowledge of the forces which are in operation4 ,there is thus a double uncertainty2 even supposing the ,universe as a whole to proceed unchanged1 we do not really know the ,universe as a whole4 ,we know only a point in its infinite extent1 and a moment in its infinite duration4 ,we cannot be sure1 then1 that our observations have not escaped some fact1 which will cause the future to be apparently different from the past2 nor can we be sure that the future really will be the outcome of the past4 ,we proceed then in all our inferences to unexamined objects and times on the assumptions-- #a4 ,that our past observation gives us a complete knowledge of what exists4 #b4 ,that the conditions of things which did exist will continue to be the conditions which will exist4 ,we shall often need to illustrate the character of our knowledge of nature by the simile of a ballot-box1 so often employed by mathematical writers in the theory of probability4 ,nature is to us like an infinite ballot-box1 the contents of which are being continually drawn1 ball after ball1 and exhibited to us4 ,science is but the careful observation of the succession in which balls of various character present themselves2 we register the combinations1 notice those which seem to be excluded from occurrence1 and from the proportional frequency of those which appear we infer the probable character of future drawings4 ,but under such circumstances certainty of prediction depends on two conditions3-- #a4 ,that we acquire a perfect knowledge of the comparative numbers of balls of each kind within the box4 #b4 ,that the contents of the ballot-box remain unchanged4 ,of the latter assumption1 or rather that concerning the constitution of the world which it illustrates1 the logician or physicist can have nothing to say4 ,as the ,creation of the ,universe is necessarily an act passing all experience and all conception1 so any change in that ,universe1 or1 it may be1 a termination of it1 must likewise be infinitely beyond the bounds of our mental faculties4 ,no science no reasoning upon the subject1 can have any validity2 for without experience we are without the basis and materials of knowledge4 ,it is the fundamental postulate accordingly of all inference concerning the future1 that there shall be no arbitrary change in the subject of inference2 of the probability or improbability of such a change ,i conceive that our faculties can give no estimate4 ,the other condition of inductive inference--that we acquire an approximately complete knowledge of the combinations in which events do occur1 is in some degree within our power4 ,there are branches of science in which phenomena seem to be governed by conditions of a most fixed and general character4 ,we have ground in such cases for believing that the future occurrence of such phenomena can be calculated and predicted4 ,but the whole question now becomes one of probability and improbability4 ,we seem to leave the region of logic to enter one in which the number of events is the ground of inference4 ,we do not really leave the region of logic2 we only leave that where certainty1 affirmative or negative1 is the result1 and the agreement or disagreement of qualities the means of inference4 ,for the future1 number and quantity will commonly enter into our processes of reasoning2 but then ,i hold that number and quantity are but portions of the great logical domain4 ,i venture to assert that number is wholly logical1 both in its fundamental nature and in its developments4 ,quantity in all its forms is but a development of number4 ,that which is mathematical is not the less logical2 if anything it is more logical1 in the sense that it presents logical results in a higher degree of complexity and variety4 ,before proceeding then from ,perfect to ,imperfect ,induction ,i must devote a portion of this work to treating the logical conditions of number4 ,i shall then employ number to estimate the variety of combinations in which natural phenomena may present themselves1 and the probability or improbability of their occurrence under definite circumstances4 ,it is in later parts of the work that ,i must endeavour to establish the notions which ,i have set forth upon the subject of ,imperfect ,induction1 as applied in the investigation of ,nature1 which notions maybe thus briefly stated3-- #a4 ,imperfect ,induction entirely rests upon ,perfect ,induction for its materials4 #b4 ,the logical process by which we seem to pass directly from examined to unexamined cases consists in an inverse application of deductive inference1 so that all reasoning may be said to be either directly or inversely deductive4 #c4 ,the result is always of a hypothetical character1 and is never more than probable4 #d4 ,no net addition is ever made to our knowledge by reasoning2 what we know of future events or unexamined objects is only the unfolded contents of our previous knowledge1 and it becomes less probable as it is more boldly extended to remote cases4 ,,book ,,ii4 ,,number1 ,,variety1 ,,and ,,probability4 ,,chapter ,,viii4 ,,principles ,,of ,,number4 ,not without reason did ,pythagoras represent the world as ruled by number4 ,into almost all our acts of thought number enters1 and in proportion as we can define numerically we enjoy exact and useful knowledge of the ,universe4 ,the science of numbers1 too1 has hitherto presented the widest and most practicable training in logic4 ,so free and energetic has been the study of mathematical forms1 compared with the forms of logic1 that mathematicians have passed far in advance of pure logicians4 ,occasionally1 in recent times1 they have condescended to apply their algebraic instrument to a reflex treatment of the primary logical science4 ,it is thus that we owe to profound mathematicians1 such as ,john ,herschel1 ,whewell1 ,de ,morgan1 or ,boole1 the regeneration of logic in the present century4 ,i entertain no doubt that it is in maintaining a close alliance with quantitative reasoning that we must look for further progress in our comprehension of qualitative inference4 ,i cannot assent1 indeed1 to the common notion that certainty begins and ends with numerical determination4 ,nothing is more certain than logical truth4 ,the laws of identity and difference are the tests of all that is certain throughout the range of thought1 and mathematical reasoning is cogent only when it conforms to these conditions1 of which logic is the first development4 ,and if it be erroneous to suppose that all certainty is mathematical1 it is equally an error to imagine that all which is mathematical is certain4 ,many processes of mathematical reasoning are of most doubtful validity4 ,there are points of mathematical doctrine which must long remain matter of opinion2 for instance1 the best form of the definition and axiom concerning parallel lines1 or the true nature of a limit4 ,in the use of symbolic reasoning questions occur on which the best mathematicians may differ1 as ,bernoulli and ,leibnitz differed irreconcileably concerning the existence of the logarithms of negative quantities4#hg ,in fact we no sooner leave the simple logical conditions of number1 than we find ourselves involved in a mazy and mysterious science of symbols4 #hg ,montucla4 99,histoire des ,mathmatiques991 vol4 iii4 p4 #cgc4 ,mathematical science enjoys no monopoly1 and not even a supremacy1 in certainty of results4 ,it is the boundless extent and variety of quantitative questions that delights the mathematical student4 ,when simple logic can give but a bare answer ,yes or ,no1 the algebraist raises a score of subtle questions1 and brings out a crowd of curious results4 ,the flower and the fruit1 all that is attractive and delightful1 fall to the share of the mathematician1 who too often despises the plain but necessary stem from which all has arisen4 ,in no region of thought can a reasoner cast himself free from the prior conditions of logical correctness4 ,the mathematician is only strong and true as long as he is logical1 and if number rules the world1 it is logic which rules number4 ,nearly all writers have hitherto been strangely content to look upon numerical reasoning as something apart from logical inference4 ,a long divorce has existed between quality and quantity1 and it has not been uncommon to treat them as contrasted in nature and restricted to independent branches of thought4 ,for my own part1 ,i believe that all the sciences meet somewhere4 ,no part of knowledge can stand wholly disconnected from other parts of the universe of thought2 it is incredible1 above all1 that the two great branches of abstract science1 interlacing and co-operating in every discourse1 should rest upon totally distinct foundations4 ,i assume that a connection exists1 and care only to inquire1 ,what is its nature8 ,does the science of quantity rest upon that of quality2 or1 99vice vers991 does the science of quality rest upon that of quantity8 ,there might conceivably be a third view1 that they both rest upon some still deeper set of principles4 ,it is generally supposed that ,boole adopted the second view1 and treated logic as an application of algebra1 a special case of analytical reasoning which admits only two quantities1 unity and zero4 ,it is not easy to ascertain clearly which of these views really was accepted by ,boole4 ,in his interesting biographical sketch of ,boole1#hh the ,rev4 ,r4 ,harley protests against the statement that ,boole's logical calculus imported the conditions of number and quantity into logic4 ,he says3 8,logic is never identified or confounded with mathematics2 the two systems of thought are kept perfectly distinct1 each being subject to its own laws and conditions4 ,the symbols are the same for both systems1 but they have not the same interpretation40 ,the ,rev4 ,j4 ,venn1 again1 in his review of ,boole's logical system1#hi holds that ,boole's processes are at bottom logical1 not mathematical1 though stated in a highly generalized form and with a mathematical dress4 ,but it is quite likely that readers of ,boole should be misled4 ,not only have his logical works an entirely mathematical appearance1 but ,i find on p4 #ab of his 99,laws of ,thought99 the following unequivocal statement3 8,that logic1 as a science1 is susceptible of very wide applications is admitted2 but it is equally certain that its ultimate forms and processes are mathematical40 ,a few lines below he adds1 8,it is not of the essence of mathematics to be conversant with the ideas of number and quantity40 #hh 99,british ,quarterly ,review991 ,no4 lxxxvii1 ,july #ahff4 #hi 99,mind991 ,october #ahgf1 vol4 i4 p4 #dhd4 ,the solution of the difficulty is that ,boole used the term mathematics in a wider sense than that usually attributed to it4 ,he probably adopted the third view1 so that his mathematical 99,laws of ,thought99 are the common basis both of logic and of quantitative mathematics4 ,but ,i do not care to pursue the subject because ,i think that1 in either case ,boole was wrong4 ,in my opinion logic is the superior science1 the general basis of mathematics as well as of all other sciences4 ,number is but logical discrimination1 and algebra a highly developed logic4 ,thus it is easy to understand the deep analogy which ,boole pointed out between the forms of algebraic and logical deduction4 ,logic resembles algebra as the mould resembles that which is cast in it4 ,boole mistook the cast for the mould4 ,considering that logic imposes its own laws upon every branch of mathematical science1 it is no wonder that we constantly meet with the traces of logical laws in mathematical processes4 99,the ,nature of ,number499 ,number is but another name for 99diversity994 ,exact identity is unity1 and with difference arises plurality4 ,an abstract notion1 as was pointed out 7p4 #bh71 possesses a certain 99oneness994 ,the quality of 99justice991 for instance1 is one and the same in whatever just acts it is manifested4 ,in justice itself there are no marks of difference by which to discriminate justice from justice4 ,but one just act can be discriminated from another just act by circumstances of time and place1 and we can count many acts thus discriminated each from each4 ,in like manner pure gold is simply pure gold1 and is so far one and the same throughout4 ,but besides its intrinsic qualities1 gold occupies space and must have shape and size4 ,portions of gold are always mutually exclusive and capable of discrimination1 in respect that they must be each without the other4 ,hence they may be numbered4 ,plurality arises when and only when we detect difference4 ,for instance1 in counting a number of gold coins ,i must count each coin once1 and not more than once4 ,let ,c denote a coin1 and the mark above it the order of counting4 ,then ,i must count the coins ,c ,c ,c ,c 4444 ,if ,i were to count them as follows ,c ,c ,c ,c ,c 4441 ,i should make the third coin into two1 and should imply the existence of difference where there is no difference4#ij ,c and ,c are but the names of one coin named twice over4 ,but according to one of the conditions of logical symbols1 which ,i have called the ,law of ,unity 7p4 #gb71 the same name repeated has no effect1 and ,a ,a ,a4 #ij 99,pure ,logic991 ,appendix1 p4 #hb1 #aib4 ,we must apply the ,law of ,unity1 and must reduce all identical alternatives before we can count with certainty and use the processes of numerical calculation4 ,identical alternatives are harmless in logic1 but are wholly inadmissible in number4 ,thus logical science ascertains the nature of the mathematical unit1 and the definition may be given in these terms--99,a unit is any object of thought which can be discriminated from every other object treated as a unit in the same problem499 ,it has often been said that units are units in respect of being perfectly similar to each other2 but though they may be perfectly similar in some respects1 they must be different in at least one point1 otherwise they would be incapable of plurality4 ,if three coins were so similar that they occupied the same space at the same time1 they would not be three coins1 but one coin4 ,it is a property of space that every point is discriminable from every other point1 and in time every moment is necessarily distinct from any other moment before or after4 ,hence we frequently count in space or time1 and ,locke1 with some other philosophers1 has held that number arises from repetition in time4 ,beats of a pendulum may be so perfectly similar that we can discover no difference except that one beat is before and another after4 ,time alone is here the ground of difference and is a sufficient foundation for the discrimination of plurality2 but it is by no means the only foundation4 ,three coins are three coins1 whether we count them successively or regard them all simultaneously4 ,in many cases neither time nor space is the ground of difference1 but pure quality alone enters4 ,we can discriminate the weight1 inertia1 and hardness of gold as three qualities1 though none of these is before nor after the other1 neither in space nor time4 ,every means of discrimination may be a source of plurality4 ,our logical notation may be used to express the rise of number4 ,the symbol ,a stands for one thing or one class1 and in itself must be regarded as a unit1 because no difference is specified4 ,but the combinations ,,ab and ,a99b99 are necessarily 99two991 because they cannot logically coalesce1 and there is a mark ,b which distinguishes one from the other4 ,a logical definition of the number 99four99 is given in the combinations ,,abc1 ,,ab99c991 ,a99b99,c1 ,a99bc991 where there is a double difference4 ,as ,puck says-- 8,yet but three8 ,come one more2 ,two of both kinds makes up four40 ,i conceive that all numbers might be represented as arising out of the combinations of the ,logical ,alphabet1 more or less of each series being struck out by various logical conditions4 ,the number three1 for instance1 arises from the condition that ,a must be either ,b or ,c1 so that the combinations are ,,abc1 ,,ab99c991 ,a99b99,c4 99,of ,numerical ,abstraction499 ,there will now be little difficulty in forming a clear notion of the nature of numerical abstraction4 ,it consists in abstracting the character of the difference from which plurality arises1 retaining merely the fact4 ,when ,i speak of 99three men99 ,i need not at once specify the marks by which each may be known from each4 ,those marks must exist if they are really three men and not one and the same1 and in speaking of them as many ,i imply the existence of the requisite differences4 ,abstract number1 then1 is 99the empty form of difference992 the abstract number 99three99 asserts the existence of marks without specifying their kind4 ,numerical abstraction is thus seen to be a different process from logical abstraction 7p4 #bg71 for in the latter process we drop out of notice the very existence of difference and plurality4 ,in forming the abstract notion 99hardness991 we ignore entirely the diverse circumstances in which the quality may appear4 ,it is the concrete notion 99three hard objects991 which asserts the existence of hardness along with sufficient other undefined qualities1 to mark out 99three99 such objects4 ,numerical thought is indeed closely interwoven with logical thought4 ,we cannot use a concrete term in the plural1 as 99men991 without implying that there are marks of difference4 ,but when we use an abstract term1 we deal with unity4 ,the origin of the great generality of number is now apparent4 ,three sounds differ from three colours1 or three riders from three horses2 but they agree in respect of the variety of marks by which they can be discriminated4 ,the symbols #a #a #a are thus the empty marks asserting the existence of discrimination4 ,but in dropping out of sight the character of the differences we give rise to new agreements on which mathematical reasoning is founded4 ,numerical abstraction is so far from being incompatible with logical abstraction that it is the origin of our widest acts of generalization4 99,concrete and ,abstract ,number499 ,the common distinction between concrete and abstract number can now be easily stated4 ,in proportion as we specify the logical characters of the things numbered1 we render them concrete4 ,in the abstract number three there is no statement of the points in which the 99three99 objects agree2 but in 99three coins991 99three men991 or 99three horses991 not only are the objects numbered but their nature is restricted4 ,concrete number thus implies the same consciousness of difference as abstract number1 but it is mingled with a groundwork of similarity expressed in the logical terms4 ,there is identity so far as logical terms enter2 difference so far as the terms are merely numerical4 ,the reason of the important ,law of ,homogeneity will now be apparent4 ,this law asserts that in every arithmetical calculation the logical nature of the things numbered must remain unaltered4 ,the specified logical agreement of the things must not be affected by the unspecified numerical differences4 ,a calculation would be palpably absurd which1 after commencing with length1 gave a result in hours4 ,it is equally absurd1 in a purely arithmetical point of view1 to deduce areas from the calculation of lengths1 masses from the combination of volume and density1 or momenta from mass and velocity4 ,it must remain for subsequent consideration to decide in what sense we may truly say that two linear feet multiplied by two linear feet give four superficial feet2 arithmetically it is absurd1 because there is a change of unit4 ,as a general rule we treat in each calculation only objects of one nature4 ,we do not1 and cannot properly add1 in the same sum yards of cloth and pounds of sugar4 ,we cannot even conceive the result of adding area to velocity1 or length to density1 or weight to value4 ,the units added must have a basis of homogeneity1 or must be reducible to some common denominator4 ,nevertheless it is possible1 and in fact common1 to treat in one complex calculation the most heterogeneous quantities1 on the condition that each kind of object is kept distinct1 and treated numerically only in conjunction with its own kind4 ,different units1 so far as their logical differences are specified1 must never be substituted one for the other4 ,chemists continually use equations which assert the equivalence of groups of atoms4 ,ordinary fermentation is represented by the formula ,c#f ,h#ab ,o#f #b;,c#b ,h#f ,o #b;,,co#b4 ,three kinds of units1 the atoms respectively of carbon1 hydrogen1 and oxygen1 are here intermingled1 but there is really a separate equation in regard to each kind4 ,mathematicians also employ compound equations of the same kind2 for in1 99a99 99b99 - #a 99c99 99d99 - #a1 it is impossible by ordinary addition to add 99a99 to 99b99 - #a4 ,hence we really have the separate equations 99a99 99b991 and 99c99 - #a 99d99 - #a4 ,similarly an equation between two quaternions is equivalent to four equations between ordinary quantities1 whence indeed the name 99quaternion994 99,analogy of ,logical and ,numerical ,terms499 ,if my assertion is correct that number arises out of logical conditions1 we ought to find number obeying all the laws of logic4 ,it is almost superfluous to point out that this is the case with the fundamental laws of identity and difference1 and it only remains to show that mathematical symbols do really obey the special conditions of logical symbols which were formerly pointed out 7p4 #cb74 ,thus the ,law of ,commutativeness1 is equally true of quality and quantity4 ,as in logic we have ,,ab ,,ba1 so in mathematics it is familiarly known that #b #c #c #b1 or 99x99 99y99 99y99 99x994 ,the properties of space are as indifferent in multiplication as we found them in pure logical thought4 ,similarly1 as in logic triangle or square square or triangle1 or generally ,a ,b ,b ,a1 so in quantity #b #c #c #b1 or generally 99x99 99y99 99y99 99x994 ,the symbol is not identical with 1 but it is thus far analogous4 ,how far1 now1 is it true that mathematical symbols obey the ,law of ,simplicity expressed in the form ,,aa ,a1 or the example ,round round round8 ,apparently there are but two numbers which obey this law2 for it is certain that 99x99 99x99 99x99 is true only in the two cases when 99x99 #a1 or 99x99 #j4 ,in reality all numbers obey the law1 for #b #b #b is not really analogous to ,,aa ,a4 ,according to the definition of a unit already given1 each unit is discriminated from each other in the same problem1 so that in #b #b1 the first 99two99 involves a different discrimination from the second 99two994 ,i get four kinds of things1 for instance1 if ,i first discriminate 8heavy and light0 and then 8cubical and spherical10 for we now have the following classes-- heavy1 cubical4 light1 cubical4 heavy1 spherical4 light1 spherical4 ,but suppose that my two classes are in both cases discriminated by the same difference of light and heavy1 then we have heavy heavy heavy1 heavy light #j1 light heavy #j1 light light light4 ,thus1 7heavy or light7 7heavy or light7 7heavy or light74 ,in short1 99twice two is two99 unless we take care that the second two has a different meaning from the first4 ,but under similar circumstances logical terms give the like result1 and it is not true that ,a,a ,a1 when ,a is different in meaning from ,a4 ,in a similar manner it may be shown that the ,law of ,unity ,a ,a ,a4 holds true alike of logical and mathematical terms4 ,it is absurd indeed to say that 99x99 99x99 99x99 except in the one case when 99x99 absolute zero4 ,but this contradiction 99x99 99x99 99x99 arises from the fact that we have already defined the units in one x as differing from those in the other4 ,under such circumstances the ,law of ,unity does not apply4 ,for if in ,a ,a ,a we mean that ,a is in any way different from ,a the assertion of identity is evidently false4 ,the contrast then which seems to exist between logical and mathematical symbols is only apparent4 ,it is because the ,laws of ,simplicity and ,unity must always be observed in the operation of counting that those laws seem no further to apply4 ,this is the understood condition under which we use all numerical symbols4 ,whenever ,i write the symbol #e ,i really mean #a #a #a #a #a1 and it is perfectly understood that each of these units is distinct from each other4 ,if requisite ,i might mark them thus #a #a #a #a #a4 ,were this not the case and were the units really #a #a #a #a #a1 the ,law of ,unity would1 as before remarked1 apply1 and #a #a #a4 ,mathematical symbols then obey all the laws of logical symbols1 but two of these laws seem to be inapplicable simply because they are presupposed in the definition of the mathematical unit4 ,logic thus lays down the conditions of number1 and the science of arithmetic developed as it is into all the wondrous branches of mathematical calculus is but an outgrowth of logical discrimination4 99,principle of ,mathematical ,inference499 ,the universal principle of all reasoning1 as ,i have asserted1 is that which allows us to substitute like for like4 ,i have now to point out how in the mathematical sciences this principle is involved in each step of reasoning4 ,it is in these sciences indeed that we meet with the clearest cases of substitution1 and it is the simplicity with which the principle can be applied which probably led to the comparatively early perfection of the sciences of geometry and arithmetic4 ,euclid1 and the ,greek mathematicians from the first1 recognised 99equality99 as the fundamental relation of quantitative thought1 but ,aristotle rejected the exactly analogous1 but far more general relation of identity1 and thus crippled the formal science of logic as it has descended to the present day4 ,geometrical reasoning starts from the axiom that 8things equal to the same thing are equal to each other40 ,two equalities enable us to infer a third equality2 and this is true not only of lines and angles1 but of areas1 volumes1 numbers1 intervals of time1 forces1 velocities1 degrees of intensity1 or1 in short1 anything which is capable of being equal or unequal4 ,two stars equally bright with the same star must be equally bright with each other1 and two forces equally intense with a third force are equally intense with each other4 ,it is remarkable that ,euclid has not explicitly stated two other axioms1 the truth of which is necessarily implied4 ,the second axiom should be that 8,two things of which one is equal and the other unequal to a third common thing1 are unequal to each other40 ,an equality and inequality1 in short1 give an inequality1 and this is equally true with the first axiom of all kinds of quantity4 ,if ,venus1 for instance1 agrees with ,mars in density1 but ,mars differs from ,jupiter1 then ,venus differs from ,jupiter4 ,a third axiom must exist to the effect that 8,things unequal to the same thing may or may not be equal to each other40 99,two inequalities give no ground of inference whatever499 ,if we only know1 for instance1 that ,mercury and ,jupiter differ in density from ,mars1 we cannot say whether or not they agree between themselves4 ,as a fact they do not agree2 but ,venus and ,mars on the other hand both differ from ,jupiter and yet closely agree with each other4 ,the force of the axioms can be most clearly illustrated by drawing equal and unequal lines4#ia #ia 99,elementary ,lessons in ,logic99 7,macmillan71 p4 #abc4 ,it is pointed out in the preface to this ,second ,edition1 that the views here given were partially stated by ,leibnitz4 ,the general conclusion then must be that where there is equality there may be inference1 but where there is not equality there cannot be inference4 ,a plain induction will lead us to believe that 99equality is the condition of inference concerning quantity994 ,all the three axioms may in fact be summed up in one1 to the effect1 that 899in whatever relation one quantity stands to another1 it stands in the same relation to the equal of that other9940 ,the active power is always the substitution of equals1 and it is an accident that in a pair of equalities we can make the substitution in two ways4 ,from 99a99 99b99 99c99 we can infer 99a99 99c991 either by substituting in 99a99 99b99 the value of 99b99 as given in 99b99 99c991 or else by substituting in 99b99 99c99 the value of 99b99 as given in 99a99 99b994 ,in 99a99 99b99 99d99 we can make but the one substitution of 99a99 for 99b994 ,in 99e99 99f99 99g99 we can make no substitution and get no inference4 ,in mathematics the relations in which terms may stand to each other are far more varied than in pure logic1 yet our principle of substitution always holds true4 ,we may say in the most general manner that 99,in whatever relation one quantity stands to another1 it stands in the same relation to the equal of that other499 ,in this axiom we sum up a number of axioms which have been stated in more or less detail by algebraists4 ,thus1 8,if equal quantities be added to equal quantities1 the sums will be equal40 ,to explain this1 let 99a99 99b991 99c99 99d994 ,now 99a99 99c991 whatever it means1 must be identical with itself1 so that 99a99 99c99 99a99 99c994 ,in one side of this equation substitute for the quantities their equivalents1 and we have the axiom proved 99a99 99c99 99b99 99d994 ,the similar axiom concerning subtraction is equally evident1 for whatever 99a99 - 99c99 may mean it is equal to 99a99 - 99c991 and therefore by substitution to 99b99 - 99d994 ,again1 8if equal quantities be multiplied by the same or equal quantities1 the products will be equal10 ,for evidently 99ac99 99ac991 and if for 99c99 in one side we substitute its equal 99d991 we have 99ac99 99ad991 and a second similar substitution gives us 99ac99 99bd994 ,we might prove a like axiom concerning division in an exactly similar manner4 ,i might even extend the list of axioms and say that 8,equal powers of equal numbers are equal40 ,for certainly1 whatever 99a99 99a99 99a99 may mean1 it is equal to 99a99 99a99 99a992 hence by our usual substitution it is equal to 99b99 99b99 99b994 ,the same will be true of roots of numbers and c99a99 d99b99 provided that the roots are so taken that the root of 99a99 shall really be related to 99a99 as the root of 99b99 is to 99b994 ,the ambiguity of meaning of an operation thus fails in any way to shake the universality of the principle4 ,we may go further and assert that1 not only the above common relations1 but all other known or conceivable mathematical relations obey the same principle4 ,let ,q99a99 denote in the most general manner that we do something with the quantity 99a992 then if 99a99 99b99 it follows that ,q99a99 ,q99b994 ,the reader will also remember that one of the most frequent operations in mathematical reasoning is to substitute for a quantity its equal1 as known either by assumed1 natural1 or self-evident conditions4 ,whenever a quantity appears twice over in a problem1 we may apply what we learn of its relations in one place to its relations in the other4 ,all reasoning in mathematics1 as in other branches of science1 thus involves the principle of treating equals equally1 or similars similarly4 ,in whatever way we employ quantitative reasoning in the remaining parts of this work1 we never can desert the simple principle on which we first set out4 99,reasoning by ,inequalities499 ,i have stated that all the processes of mathematical reasoning may be deduced from the principle of substitution4 ,exceptions to this assertion may seem to exist in the use of inequalities4 ,the greater of a greater is undoubtedly a greater1 and what is less than a less is certainly less4 ,snowdon is higher than the ,wrekin1 and ,ben ,nevis than ,snowdon2 therefore ,ben ,nevis is higher than the ,wrekin4 ,but a little consideration discloses sufficient reason for believing that even in such cases1 where equality does not apparently enter1 the force of the reasoning entirely depends upon underlying and implied equalities4 ,in the first place1 two statements of mere difference do not give any ground of inference4 ,we learn nothing concerning the comparative heights of ,st4 ,paul's and ,westminster ,abbey from the assertions that they both differ in height from ,st4 ,peter's at ,rome4 ,we need something more than inequality2 we require one identity in addition1 namely the identity in direction of the two differences4 ,thus we cannot employ inequalities in the simple way in which we do equalities1 and1 when we try to express what other conditions are requisite1 we find ourselves lapsing into the use of equalities or identities4 ,in the second place1 every argument by inequalities may be represented in the form of equalities4 ,we express that 99a99 is greater than 99b99 by the equation 99a99 99b99 99p991 7#a7 where 99p99 is an intrinsically positive quantity1 denoting the difference of 99a99 and 99b994 ,similarly we express that 99b99 is greater than 99c99 by the equation 99b99 99c99 99q991 7#b7 and substituting for 99b99 in 7#a7 its value in 7#b7 we have 99a99 99c99 99q99 99p994 7#c7 ,now as 99p99 and 99q99 are both positive1 it follows that 99a99 is greater than 99c991 and we have the exact amount of excess specified4 ,it will be easily seen that the reasoning concerning that which is less than a less will result in an equation of the form 99c99 99a99 - 99r99 - 99s994 ,every argument by inequalities may then be thrown into the form of an equality2 but the converse is not true4 ,we cannot possibly prove that two quantities are equal by merely asserting that they are both greater or both less than another quantity4 ,from 99e99 99f99 and 99g99 99f991 or 99e99 99f99 and 99g99 99f991 we can infer no relation between 99e99 and 99g994 ,and if the reader take the equations 99x99 99y99 #c and attempt to prove that therefore 99x99 #c1 by throwing them into inequalities1 he will find it impossible to do so4 ,from these considerations ,i gather that reasoning in arithmetic or algebra by so-called inequalities1 is only an imperfectly expressed reasoning by equalities1 and when we want to exhibit exactly and clearly the conditions of reasoning1 we are obliged to use equalities explicitly4 ,just as in pure logic a negative proposition1 as expressing mere difference1 cannot be the means of inference1 so inequality can never really be the true ground of inference4 ,i do not deny that affirmation and negation1 agreement and difference1 equality and inequality1 are pairs of equally fundamental relations1 but ,i assert that inference is possible only where affirmation1 agreement1 or equality1 some species of identity in fact1 is present1 explicitly or implicitly4 99,arithmetical ,reasoning499 ,it may seem somewhat inconsistent that ,i assert number to arise out of difference or discrimination1 and yet hold that no reasoning can be grounded on difference4 ,number1 of course1 opens a most wide sphere for inference1 and a little consideration shows that this is due to the unlimited series of identities which spring up out of numerical abstraction4 ,if six people are sitting on six chairs1 there is no resemblance between the chairs and the people in logical character4 ,but if we overlook all the qualities both of a chair and a person and merely remember that there are marks by which each of six chairs may be discriminated from the others1 and similarly with the people1 then there arises a resemblance between the chairs and the people1 and this resemblance in number may be the ground of inference4 ,if on another occasion the chairs are filled by people again1 we may infer that these people resemble the others in number though they need not resemble them in any other points4 ,groups of units are what we really treat in arithmetic4 ,the number 99five99 is really #a #a #a #a #a1 but for the sake of conciseness we substitute the more compact sign #e1 or the name 99five994 ,these names being arbitrarily imposed in any one manner1 an infinite variety of relations spring up between them which are not in the least arbitrary4 ,if we define 99four99 as #a #a #a #a1 and 99five99 as #a #a #a #a #a1 then of course it follows that 99five99 99four99 #a2 but it would be equally possible to take this latter equality as a definition1 in which case one of the former equalities would become an inference4 ,it is hardly requisite to decide how we define the names of numbers1 provided we remember that out of the infinitely numerous relations of one number to others1 some one relation expressed in an equality must be a definition of the number in question and the other relations immediately become necessary inferences4 ,in the science of number the variety of classes which can be formed is altogether infinite1 and statements of perfect generality may be made subject only to difficulty or exception at the lower end of the scale4 ,every existing number for instance belongs to the class 99m99 #g2 that is1 every number must be the sum of another number and seven1 except of course the first six or seven numbers1 negative quantities not being here taken into account4 ,every number is the half of some other1 and so on4 ,the subject of generalization1 as exhibited in mathematical truths1 is an infinitely wide one4 ,in number we are only at the first step of an extensive series of generalizations4 ,as number is general compared with the particular things numbered1 so we have general symbols for numbers1 and general symbols for relations between undetermined numbers4 ,there is an unlimited hierarchy of successive generalizations4 99,numerically ,definite ,reasoning499 ,it was first discovered by ,de ,morgan that many arguments are valid which combine logical and numerical reasoning1 although they cannot be included in the ancient logical formulas4 ,he developed the doctrine of the 8,numerically ,definite ,syllogism10 fully explained in his 99,formal ,logic99 7pp4 #ada-agj74 ,boole also devoted considerable attention to the determination of what he called 8,statistical ,conditions10 meaning the numerical conditions of logical classes4 ,in a paper published among the ,memoirs of the ,manchester ,literary and ,philosophical ,society1 ,third ,series1 vol4 ,,iv4 p4 #ccj 7,session #ahfi-gj71 ,i have pointed out that we can apply arithmetical calculation to the ,logical ,alphabet4 ,having given certain logical conditions and the numbers of objects in certain classes1 we can either determine the numbers of objects in other classes governed by those conditions1 or can show what further data are required to determine them4 ,as an example of the kind of questions treated in numerical logic1 and the mode of treatment1 ,i give the following problem suggested by ,de ,morgan1 with my mode of representing its solution4 8,for every man in the house there is a person who is aged2 some of the men are not aged4 ,it follows that some persons in the house are not men40#ib #ib 99,syllabus of a ,proposed ,system of ,logic991 p4 #bi4 ,now let ,a person in house1 ,b male1 ,c aged4 ,by enclosing a logical symbol in brackets1 let us denote the number of objects belonging to the class indicated by the symbol4 ,thus let 7,a7 number of persons in house1 7,,ab7 number of male persons in house1 7,,abc7 number of aged male persons in house1 and so on4 ,now if we use 99w99 and 99w99 to denote unknown numbers1 the conditions of the problem may be thus stated according to my interpretation of the words-- 7,,ab7 7,,ac7 - 99w991 7#a7 that is to say1 the number of persons in the house who are aged is at least equal to1 and may exceed1 the number of male persons in the house2 7,,ab99c997 99w991 7#b7 that is to say1 the number of male persons in the house who are not aged is some unknown positive quantity4 ,if we develop the terms in 7#a7 by the ,law of ,duality 7pp4 #gd1 #ha1 #hi71 we obtain 7,,abc7 7,,ab99c997 7,,abc7 7,a99b99,c7 - 99w994 ,subtracting the common term 7,,abc7 from each side and substituting for 7,,ab99c997 its value as given in 7#b71 we get at once 7,a99b99,c7 99w99 99w991 and adding 7,a99bc997 to each side1 we have 7,a99b997 7,a99bc997 99w99 99w994 ,the meaning of this result is that the number of persons in the house who are not men is at least equal to 99w99 99w991 and exceeds it by the number of persons in the house who are neither men nor aged 7,a99bc9974 ,it should be understood that this solution applies only to the terms of the example quoted above1 and not to the general problem for which ,de ,morgan intended it to serve as an illustration4 ,as a second instance1 let us take the following question3--,the whole number of voters in a borough is 99a992 the number against whom objections have been lodged by liberals is 99b992 and the number against whom objections have been lodged by conservatives is 99c992 required the number1 if any1 who have been objected to on both sides4 ,taking ,a voter1 ,b objected to by liberals1 ,c objected to by conservatives1 then we require the value of 7,,abc74 ,now the following equation is identically true-- 7,,abc7 7,,ab7 7,,ac7 7,a99bc997 - 7,a74 7#a7 ,for if we develop all the terms on the second side we obtain 7,,abc7 7,,abc7 7,,ab99c997 7,,abc7 7,a99b99,c7 7,a99bc997 - 7,,abc7 - 7,,ab99c997 - 7,a99b99,c7 - 7,a99bc9972 and striking out the corresponding positive and negative terms1 we have left only 7,,abc7 7,,abc74 ,since then 7#a7 is necessarily true1 we have only to insert the known values1 and we have 7,,abc7 99b99 99c99 - 99a99 7,a99bc9974 ,hence the number who have received objections from both sides is equal to the excess1 if any1 of the whole number of objections over the number of voters together with the number of voters who have received no objection 7,a99bc9974 ,the following problem illustrates the expression for the common part of any three classes3--,the number of paupers who are blind males1 is equal to the excess1 if any1 of the sum of the whole number of blind persons1 added to the whole number of male persons1 added to the number of those who being paupers are neither blind nor males1 above the sum of the whole number of paupers added to the number of those who1 not being paupers1 are blind1 and to the number of those who1 not being paupers1 are male4 ,the reader is requested to prove the truth of the above statement1 7#a7 by his own unaided common sense2 7#b7 by the ,aristotelian ,logic2 7#c7 by the method of numerical logic just expounded2 and then to decide which method is most satisfactory4 99,numerical meaning of ,logical ,conditions499 ,in many cases classes of objects may exist under special logical conditions1 and we must consider how these conditions can be interpreted numerically4 ,every logical proposition gives rise to a corresponding numerical equation4 ,sameness of qualities occasions sameness of numbers4 ,hence if ,a ,b denotes the identity of the qualities of ,a and ,b1 we may conclude that 7,a7 7,b74 ,it is evident that exactly those objects1 and those objects only1 which are comprehended under ,a must be comprehended under ,b4 ,it follows that wherever we can draw an equation of qualities1 we can draw a similar equation of numbers4 ,thus1 from ,a ,b ,c we infer ,a ,c2 and similarly from 7,a7 7,b7 7,c71 meaning that the numbers of ,a's and ,c's are equal to the number of ,b's1 we can infer 7,a7 7,c74 ,but1 curiously enough1 this does not apply to negative propositions and inequalities4 ,for if ,a ,b ,d means that ,a is identical with ,b1 which differs from ,d1 it does not follow that 7,a7 7,b7 7,d74 ,two classes of objects may differ in qualities1 and yet they may agree in number4 ,this point strongly confirms me in the opinion which ,i have already expressed1 that all inference really depends upon equations1 not differences4 ,the ,logical ,alphabet thus enables us to make a complete analysis of any numerical problem1 and though the symbolical statement may sometimes seem prolix1 ,i conceive that it really represents the course which the mind must follow in solving the question4 ,although thought may outstrip the rapidity with which the symbols can be written down1 yet the mind does not really follow a different course from that indicated by the symbols4 ,for a fuller explanation of this natural system of ,numerically ,definite ,reasoning1 with more abundant illustrations and an analysis of ,de ,morgan's ,numerically ,definite ,syllogism1 ,i must refer the reader to the paper#ic in the ,memoirs of the ,manchester ,literary and ,philosophical ,society1 already mentioned1 portions of which1 however1 have been embodied in the present section4 #ic ,it has been pointed out to me by ,mr4 ,c4 ,j4 ,monroe1 that section #ad 7p4 #cci7 of this paper is erroneous1 and ought to be cancelled4 ,the problem concerning the number of paupers illustrates the answer which should have been obtained4 ,mr4 ,a4 ,j4 ,ellis1 ,f4,r4,s41 had previously observed that my solution in the paper of ,de ,morgan's problem about 8men in the house0 did not answer the conditions intended by ,de ,morgan1 and ,i therefore give in the text a more satisfactory solution4 ,the reader may be referred1 also1 to ,boole's writings upon the subject in the 99,laws of ,thought991 chap4 xix4 p4 #bie1 and in a paper on 8,propositions ,numerically ,definite10 communicated by ,de ,morgan1 in #ahfh1 to the ,cambridge ,philosophical ,society1 and printed in their 99,transactions991 vol4 xi4 part ii4 ,,chapter ,,ix4 ,,the ,,variety ,,of ,,nature1 ,,or ,,the ,,doctrine ,,of ,,combinations ,,and ,,permutations4 ,nature may be said to be evolved from the monotony of non-existence by the creation of diversity4 ,it is plausibly asserted that we are conscious only so far as we experience difference4 ,life is change1 and perfectly uniform existence would be no better than non-existence4 ,certain it is that life demands incessant novelty1 and that nature1 though it probably never fails to obey the same fixed laws1 yet presents to us an apparently unlimited series of varied combinations of events4 ,it is the work of science to observe and record the kinds and comparative numbers of such combinations of phenomena1 occurring spontaneously or produced by our interference4 ,patient and skilful examination of the records may then disclose the laws imposed on matter at its creation1 and enable us more or less successfully to predict1 or even to regulate1 the future occurrence of any particular combination4 ,the ,laws of ,thought are the first and most important of all the laws which govern the combinations of phenomena1 and1 though they be binding on the mind1 they may also be regarded as verified in the external world4 ,the ,logical ,alphabet develops the utmost variety of things and events which may occur1 and it is evident that as each new quality is introduced1 the number of combinations is doubled4 ,thus four qualities may occur in #af combinations2 five qualities in #cb2 six qualities in #fd2 and so on4 ,in general language1 if n be the number of qualities1 #bn is the number of varieties of things which may be formed from them1 if there be no conditions but those of logic4 ,this number1 it need hardly be said1 increases after the first few terms1 in an extraordinary manner1 so that it would require #cjb figures to express the number of combinations in which #a1jjj qualities might conceivably present themselves4 ,if all the combinations allowed by the ,laws of ,thought occurred indifferently in nature1 then science would begin and end with those laws4 ,to observe nature would give us no additional knowledge1 because no two qualities would in the long run be oftener associated than any other two4 ,we could never predict events with more certainty than we now predict the throws of dice1 and experience would be without use4 ,but the universe1 as actually created1 presents a far different and much more interesting problem4 ,the most superficial observation shows that some things are constantly associated with other things4 ,the more mature our examination1 the more we become convinced that each event depends upon the prior occurrence of some other series of events4 ,action and reaction are gradually discovered to underlie the whole scene1 and an independent or casual occurrence does not exist except in appearance4 ,even dice as they fall are surely determined in their course by prior conditions and fixed laws4 ,thus the combinations of events which can really occur are found to be comparatively restricted1 and it is the work of science to detect these restricting conditions4 ,in the ,english alphabet1 for instance1 we have twenty-six letters4 ,were the combinations of such letters perfectly free1 so that any letter could be indifferently sounded with any other1 the number of words which could be formed without any repetition would be #bbf - #a1 or #fg1ajh1hfc1 equal in number to the combinations of the twenty-seventh column of the ,logical ,alphabet1 excluding one for the case in which all the letters would be absent4 ,but the formation of our vocal organs prevents us from using the far greater part of these conjunctions of letters4 ,at least one vowel must be present in each word2 more than two consonants cannot usually be brought together2 and to produce words capable of smooth utterance a number of other rules must be observed4 ,to determine exactly how many words might exist in the ,english language under these circumstances1 would be an exceedingly complex problem1 the solution of which has never been attempted4 ,the number of existing ,english words may perhaps be said not to exceed one hundred thousand1 and it is only by investigating the combinations presented in the dictionary1 that we can learn the ,laws of ,euphony or calculate the possible number of words4 ,in this example we have an epitome of the work and method of science4 ,the combinations of natural phenomena are limited by a great number of conditions which are in no way brought to our knowledge except so far as they are disclosed in the examination of nature4 ,it is often a very difficult matter to determine the numbers of permutations or combinations which may exist under various restrictions4 ,many learned men puzzled themselves in former centuries over what were called ,protean verses1 or verses admitting many variations in accordance with the ,laws of ,metre4 ,the most celebrated of these verses was that invented by ,bernard ,bauhusius1 as follows3#id-- 8,tot tibi sunt dotes1 ,virgo1 quot sidera clo40 #id ,montucla1 99,histoire991 c41 vol4 iii4 p4 #chh4 ,one author1 ,ericius ,puteanus1 filled forty-eight pages of a work in reckoning up its possible transpositions1 making them only #ajbb4 ,other calculators gave #baif1 #cbgf1 #behj as their results4 ,wallis assigned #cjif1 but without much confidence in the accuracy of his result4#ie ,it required the skill of ,james ,bernoulli to decide that the number of transpositions was #ccab1 under the condition that the sense and metre of the verse shall be perfectly preserved4 #ie ,wallis1 99,of ,combinations991 c41 p4 #aai4 ,in approaching the consideration of the great ,inductive problem1 it is very necessary that we should acquire correct notions as to the comparative numbers of combinations which may exist under different circumstances4 ,the doctrine of combinations is that part of mathematical science which applies numerical calculation to determine the numbers of combinations under various conditions4 ,it is a part of the science which really lies at the base not only of other sciences1 but of other branches of mathematics4 ,the forms of algebraical expressions are determined by the principles of combination1 and ,hindenburg recognised this fact in his ,combinatorial ,analysis4 ,the greatest mathematicians have1 during the last three centuries1 given their best powers to the treatment of this subject2 it was the favourite study of ,pascal2 it early attracted the attention of ,leibnitz1 who wrote his curious essay1 99,de ,arte ,combinatoria991 at twenty years of age2 ,james ,bernoulli1 one of the very profoundest mathematicians1 devoted no small part of his life to the investigation of the subject1 as connected with that of ,probability2 and in his celebrated work1 99,de ,arte ,conjectandi991 he has so finely described the importance of the doctrine of combinations1 that ,i need offer no excuse for quoting his remarks at full length4 8,it is easy to perceive that the prodigious variety which appears both in the works of nature and in the actions of men1 and which constitutes the greatest part of the beauty of the universe1 is owing to the multitude of different ways in which its several parts are mixed with1 or placed near1 each other4 ,but1 because the number of causes that concur in producing a given event1 or effect1 is oftentimes so immensely great1 and the causes themselves are so different one from another1 that it is extremely difficult to reckon up all the different ways in which they may be arranged or combined together1 it often happens that men1 even of the best understandings and greatest circumspection1 are guilty of that fault in reasoning which the writers on logic call 99the insufficient or imperfect enumeration of parts or cases993 insomuch that ,i will venture to assert1 that this is the chief1 and almost the only1 source of the vast number of erroneous opinions1 and those too very often in matters of great importance1 which we are apt to form on all the subjects we reflect upon1 whether they relate to the knowledge of nature1 or the merits and motives of human actions4 8,it must therefore be acknowledged1 that that art which affords a cure to this weakness1 or defect1 of our understandings1 and teaches us so to enumerate all the possible ways in which a given number of things may be mixed and combined together1 that we may be certain that we have not omitted any one arrangement of them that can lead to the object of our inquiry1 deserves to be considered as most eminently useful and worthy of our highest esteem and attention4 ,and this is the business of 99the art or doctrine of combinations994 ,nor is this art or doctrine to be considered merely as a branch of the mathematical sciences4 ,for it has a relation to almost every species of useful knowledge that the mind of man can be employed upon4 ,it proceeds indeed upon mathematical principles1 in calculating the number of the combinations of the things proposed3 but by the conclusions that are obtained by it1 the sagacity of the natural philosopher1 the exactness of the historian1 the skill and judgment of the physician1 and the prudence and foresight of the politician may be assisted2 because the business of all these important professions is but 99to form reasonable conjectures99 concerning the several objects which engage their attention1 and all wise conjectures are the results of a just and careful examination of the several different effects that may possibly arise from the causes that are capable of producing them40#if #if ,james ,bernoulli1 99,de ,arte ,conjectandi991 translated by ,baron ,maseres4 ,london1 #agie1 pp4 #ce1 #cf4 99,distinction of ,combinations and ,permutations499 ,we must first consider the deep difference which exists between ,combinations and ,permutations1 a difference involving important logical principles1 and influencing the form of mathematical expressions4 ,in 99permutation99 we recognise varieties of order1 treating ,,ab as a different group from ,,ba4 ,in 99combination99 we take notice only of the presence or absence of a certain thing1 and pay no regard to its place in order of time or space4 ,thus the four letters 99a991 99e991 99m991 99n99 can form but one combination1 but they occur in language in several permutations1 as 99name991 99amen991 99mean991 99mane994 ,we have hitherto been dealing with purely logical questions1 involving only combination of qualities4 ,i have fully pointed out in more than one place that1 though our symbols could not but be written in order of place and read in order of time1 the relations expressed had no regard to place or time 7pp4 #cc1 #aad74 ,the ,law of ,commutativeness1 in fact1 expresses the condition that in logic we deal with combinations1 and the same law is true of all the processes of algebra4 ,in some cases1 order may be a matter of indifference2 it makes no difference1 for instance1 whether gunpowder is a mixture of sulphur1 carbon1 and nitre1 or carbon1 nitre1 and sulphur1 or nitre1 sulphur1 and carbon1 provided that the substances are present in proper proportions and well mixed4 ,but this indifference of order does not usually extend to the events of physical science or the operations of art4 ,the change of mechanical energy into heat is not exactly the same as the change from heat into mechanical energy2 thunder does not indifferently precede and follow lightning2 it is a matter of some importance that we load1 cap1 present1 and fire a rifle in this precise order4 ,time is the condition of all our thoughts1 space of all our actions1 and therefore both in art and science we are to a great extent concerned with permutations4 ,language1 for instance1 treats different permutations of letters as having different meanings4 ,permutations of things are far more numerous than combinations of those things1 for the obvious reason that each distinct thing is regarded differently according to its place4 ,thus the letters ,a1 ,b1 ,c1 will make different permutations according as ,a stands first1 second1 or third2 having decided the place of ,a1 there are two places between which we may choose for ,b2 and then there remains but one place for ,c4 ,accordingly the permutations of these letters will be altogether #c #b #a or #f in number4 ,with four things or letters1 ,a1 ,b1 ,c1 ,d1 we shall have four choices of place for the first letter1 three for the second1 two for the third1 and one for the fourth1 so that there will be altogether1 #d #c #b #a1 or #bd permutations4 ,the same simple rule applies in all cases2 beginning with the whole number of things we multiply at each step by a number decreased by a unit4 ,in general language1 if 99n99 be the number of things in a combination1 the number of permutations is 99n99 799n99 - #a7799n99 - #b7 4444 #d 4 #c 4 #b 4 #a4 ,if we were to re-arrange the names of the days of the week1 the possible arrangements out of which we should have to choose the new order1 would be no less than #g 4 #f 4 #e 4 #d 4 #c 4 #b 4 #a1 or #ejdj1 or1 excluding the existing order1 #ejci4 ,the reader will see that the numbers which we reach in questions of permutation1 increase in a more extraordinary manner even than in combination4 ,each new object or term doubles the number of combinations1 but increases the permutations by a factor continually growing4 ,instead of #b #b #b #b 4444 we have #b #c #d #e 4444 and the products of the latter expression immensely exceed those of the former4 ,these products of increasing factors are frequently employed1 as we shall see1 in questions both of permutation and combination4 ,they are technically called 99factorials991 that is to say1 the product of all integer numbers1 from unity up to any number 99n99 is the 99factorial99 of 99n991 and is often indicated symbolically by 99n9964 ,i give below the factorials up to that of twelve3-- #bd #a 4 #b 4 #c 4 #d #abj #a 4 #b 444 #e #gbj #a 4 #b 444 #f #e1jdj #g6 #dj1cbj #h6 #cfb1hhj #i6 #c1fbh1hjj #aj6 #ci1iaf1hjj #aa6 #dgi1jja1fjj #ab6 ,the factorials up to #cf6 are given in ,rees's #8,cyclopdia1' art4 99,cipher991 and the logarithms of factorials up to #bfe6 are to be found at the end of the table of logarithms published under the superintendence of the ,society for the ,diffusion of ,useful ,knowledge 7p4 #bae74 ,to express the factorial #bfe6 would require #ebi places of figures4 ,many writers have from time to time remarked upon the extraordinary magnitude of the numbers with which we deal in this subject4 ,tacquet calculated#ig that the twenty-four sic letters of the alphabet may be arranged in more than #fbj thousand trillions of orders2 and ,schott estimated#ih that if a thousand millions of men were employed for the same number of years in writing out these arrangements1 and each man filled each day forty pages with forty arrangements in each1 they would not have accomplished the task1 as they would have written only #ehd thousand trillions instead of #fbj thousand trillions4 #ig 99,arithmetic ,theoria499 ,ed4 ,amsterd4 #agjd4 p4 #eag4 #ih ,rees's 99,cyclopdia991 art4 99,cipher994 ,in some questions the number of permutations may be restricted and reduced by various conditions4 ,some things in a group may be undistinguishable from others1 so that change of order will produce no difference4 ,thus if we were to permutate the letters of the name 99,ann991 according to our previous rule1 we should obtain #c #b #a1 or #f orders2 but half of these arrangements would be identical with the other half1 because the interchange of the two 99n99's has no effect4 ,the really different orders will therefore be 7#c 4 #b 4 #a7_/7a 4 #b7 or #c1 namely 99,ann991 99,nan991 99,nna994 ,in the word 99utility99 there are two 99i99's and two 99t99's1 in respect of both of which pairs the numbers of permutations must be halved4 ,thus we obtain 7#g 4 #f 4 #e 4 #d 4 #c 4 #b 4 #a7_/7a 4 #b 4 #a 4 #b7 or #abfj1 as the number of permutations4 ,the simple rule evidently is--when some things or letters are undistinguished1 proceed in the first place to calculate all the possible permutations as if all were different1 and then divide by the numbers of possible permutations of those series of things which are not distinguished1 and of which the permutations have therefore been counted in excess4 ,thus since the word 99,utilitarianism99 contains fourteen letters1 of which four are 99i99's1 two 99a99's1 and two 99t99's1 the number of distinct arrangements will be found by dividing the factorial of #ad1 by the factorials of #d1 #b1 and #b1 the result being #ijh1ajg1bjj4 ,from the letters of the word 99,mississippi99 we can get in like manner #aa6_/7d6 #d6 #b67 or #cd1fej permutations1 which is not the one-thousandth part of what we should obtain were all the letters different4 99,calculation of ,number of ,combinations499 ,although in many questions both of art and science we need to calculate the number of permutations on account of their own interest1 it far more frequently happens in scientific subjects that they possess but an indirect interest4 ,as ,i have already pointed out1 we almost always deal in the logical and mathematical sciences with 99combinations991 and variety of order enters only through the inherent imperfections of our symbols and modes of calculation4 ,signs must be used in some order1 and we must withdraw our attention from this order before the signs correctly represent the relations of things which exist neither before nor after each other4 ,now1 it often happens that we cannot choose all the combinations of things1 without first choosing them subject to the accidental variety of order1 and we must then divide by the number of possible variations of order1 that we may get to the true number of pure combinations4 ,suppose that we wish to determine the number of ways in which we can select a group of three letters out of the alphabet1 without allowing the same letter to be repeated4 ,at the first choice we can take any one of #bf letters2 at the next step there remain #be letters1 any one of which may be joined with that already taken2 at the third step there will be #bd choices1 so that apparently the whole number of ways of choosing is #bf #be #bd4 ,but the fact that one choice succeeded another has caused us to obtain the same combinations of letters in different orders2 we should get1 for instance1 99a991 99p991 99r99 at one time1 and 99p991 99r991 99a99 at another1 and every three distinct letters will appear six times over1 because three things can be arranged in six permutations4 ,to get the number of combinations1 then1 we must divide the whole number of ways of choosing1 by six1 the number of permutations of three things1 obtaining 7#bf #be #bd7_/7a #b #c7 or #b1fjj4 ,it is apparent that we need the doctrine of combinations in order that we may in many questions counteract the exaggerating effect of successive selection4 ,if out of a senate of #cj persons we have to choose a committee of #e1 we may choose any of #cj first1 any of #bi next1 and so on1 in fact there will be #cj #bi #bh #bg #bf selections2 but as the actual character of the members of the committee will not be affected by the accidental order of their selection1 we divide by #a #b #c #d #e1 and the possible number of different committees will be #adb1ejf4 ,similarly if we want to calculate the number of ways in which the eight major planets may come into conjunction1 it is evident that they may meet either two at a time or three at a time1 or four or more at a time1 and as nothing is said as to the relative order or place in the conjunction1 we require the number of combinations4 ,now a selection of #b out of #h is possible in 7#h 4 #g7_/7a 4 #b7 or #bh ways2 of #c out of #h in 7#h 4 #g 4 #f7_/7a 4 #b 4 #c7 or #ef ways2 of #d out of #h in 7#h 4 #g 4 #f 4 #e7_/7a 4 #b 4 #c 4 #d7 or #gj ways2 and it may be similarly shown that for #e1 #f1 #g1 and #h planets1 meeting at one time1 the numbers of ways are #ef1 #bh1 #h1 and #a4 ,thus we have solved the whole question of the variety of conjunctions of eight planets2 and adding all the numbers together1 we find that #bdg is the utmost possible number of modes of meeting4 ,in general algebraic language1 we may say that a group of 99m99 things may be chosen out of a total number of 99n99 things1 in a number of combinations denoted by the formula 799n99 4 799n99-#a7799n99-#b7799n99-#c7 4444 799n99 - 99m99 #a77_/7a 4 #b 4 #c 4 #d 4444 99m997 ,the extreme importance and significance of this formula seems to have been first adequately recognised by ,pascal1 although its discovery is attributed by him to a friend1 ,m4 de ,ganires4#ii ,we shall find it perpetually recurring in questions both of combinations and probability1 and throughout the formul of mathematical analysis traces of its influence may be noticed4 #ii 99uvres ,compltes de ,pascal99 7#ahfe71 vol4 iii4 p4 #cjb4 ,montucla states the name as ,de ,gruires1 99,histoire des ,mathmatiques991 vol4 iii4 p4 #chi4 99,the ,arithmetical ,triangle499 ,the ,arithmetical ,triangle is a name long since given to a series of remarkable numbers connected with the subject we are treating4 ,according to ,montucla#ajj 8this triangle is in the theory of combinations and changes of order1 almost what the table of ,pythagoras is in ordinary arithmetic1 that is to say1 it places at once under the eyes the numbers required in a multitude of cases of this theory40 ,as early as #aedd ,stifels had noticed the remarkable properties of these numbers and the mode of their evolution4 ,briggs1 the inventor of the common system of logarithms1 was so struck with their importance that he called them the ,abacus ,panchrestus4 ,pascal1 however1 was the first who wrote a distinct treatise on these numbers1 and gave them the name by which they are still known4 ,but ,pascal did not by any means exhaust the subject1 and it remained for ,james ,bernoulli to demonstrate fully the importance of the 99figurate numbers991 as they are also called4 ,in his treatise 99,de ,arte ,conjectandi991 he points out their application in the theory of combinations and probabilities1 and remarks of the ,arithmetical ,triangle1 8,it not only contains the clue to the mysterious doctrine of combinations1 but it is also the ground or foundation of most of the important and abstruse discoveries that have been made in the other branches of the mathematics40#aja #ajj 99,histoire des ,mathmatiques991 vol4 iii4 p4 #cgh4 #aja ,bernoulli1 99,de ,arte ,conjectandi991 translated by ,francis ,maseres4 ,london1 #agie1 p4 #ge4 ,the numbers of the triangle can be calculated in a very easy manner by successive additions4 ,we commence with unity at the apex2 in the next line we place a second unit to the right of this2 to obtain the third line of figures we move the previous line one place to the right1 and add them to the same figures as they were before removal2 we can then repeat the same process 99ad infinitum994 ,the fourth line of figures1 for instance1 contains #a1 #c1 #c1 #a2 moving them one place and adding as directed we obtain3-- ,fourth line 444 #a #c #c #a #a #c #c #a -------------- ,fifth line 4444 #a #d #f #d #a #a #d #f #d #a ---------------- ,sixth line 4444 #a #e #aj #aj #e #a ,carrying out this simple process through ten more steps we obtain the first seventeen lines of the ,arithmetical ,triangle as printed on the next page4 ,theoretically speaking the ,triangle must be regarded as infinite in extent1 but the numbers increase so rapidly that it soon becomes impracticable to continue the table4 ,the longest table of the numbers which ,i have found is in ,fortia's 8,trait des ,progressions0 7p4 #hj71 where they are given up to the fortieth line and the ninth column4 ,,the ,,arithmetical ,,triangle4 ,line4 ,first ,column4 #a #a ,second ,column4 #b #a #a ,third ,column4 #c #a #b #a ,fourth ,column4 #d #a #c #c #a ,fifth ,column4 #e #a #d #f #d #a ,sixth ,column4 #f #a #e #aj #aj #e #a ,seventh ,column4 #g #a #f #ae #bj #ae #f #a ,eighth ,column4 #h #a #g #ba #ce #ce #ba #g #a ,ninth ,column4 #i #a #h #bh #ef #gj #ef #bh #h #a ,tenth ,column4 #aj #a #i #cf #hd #abf #abf #hd #cf #i #a ,eleventh ,column4 #aa #a #aj #de #abj #baj #beb #baj #abj #de #aj #a ,twelfth ,column4 #ab #a #aa #ee #afe #ccj #dfb #dfb #ccj #afe #ee #aa #a ,thirteenth ,column4 #ac #a #ab #ff #bbj #die #gib #ibd #gib #die #bbj #ff #ab #a ,fourteenth ,column4 #ad #a #ac #gh #bhf #gae #abhg #agaf #agaf #abhg #gae #bhf #gh #ac #a ,fifteenth ,column4 #ae #a #ad #ia #cfd #ajja #bjjb #cjjc #cdcb #cjjc #bjjb #ajja #cfd #ia #ad #a ,sixteenth ,column4 #af #a #ae #aje #dee #acfe #cjjc #ejje #fdce #fdce #ejje #cjjc #acfe #dee #aje #ae #a ,seventeenth ,col4 #ag #a #af #abj #efj #ahbj #dcfh #hjjh #aaddj #abhgj #aaddj #hjjh #dcfh #ahbj #efj #abj #af #a ,examining these numbers1 we find that they are connected by an unlimited series of relations1 a few of the more simple of which may be noticed4 ,each vertical column of numbers exactly corresponds with an oblique series descending from left to right1 so that the triangle is perfectly symmetrical in its contents4 ,the first column contains only 99units992 the second column contains the 99natural numbers991 #a1 #b1 #c1 c42 the third column contains a remarkable series of numbers1 #a1 #c1 #f1 #aj1 #ae1 c41 which have long been called 99the triangular numbers991 because they correspond with the numbers of balls which may be arranged in a triangular form1 thus-- ,illustration ,the fourth column contains the 99pyramidal numbers991 so called because they correspond to the numbers of equal balls which can be piled in regular triangular pyramids4 ,their differences are the triangular numbers4 ,the numbers of the fifth column have the pyramidal numbers for their differences1 but as there is no regular figure of which they express the contents1 they have been arbitrarily called the 99trianguli-triangular numbers994 ,the succeeding columns have1 in a similar manner1 been said to contain the 99trianguli-pyramidal991 the 99pyramidi-pyramidal99 numbers1 and so on4#ajb #ajb ,wallis's 99,algebra991 ,discourse of ,combinations1 c41 p4 #aji4 ,from the mode of formation of the table1 it follows that the differences of the numbers in each column will be found in the preceding column to the left4 ,hence the 99second differences991 or the 99differences of differences991 will be in the second column to the left of any given column1 the third differences in the third column1 and so on4 ,thus we may say that unity which appears in the first column is the 99first difference99 of the numbers in the second column2 the 99second difference99 of those in the third column2 the 99third difference99 of those in the fourth1 and so on4 ,the triangle is seen to be a complete classification of all numbers according as they have unity for any of their differences4 ,since each line is formed by adding the previous line to itself1 it is evident that the sum of the numbers in each horizontal line must be double the sum of the numbers in the line next above4 ,hence we know1 without making the additions1 that the successive sums must be #a1 #b1 #d1 #h1 #af1 #cb1 #fd1 c41 the same as the numbers of combinations in the ,logical ,alphabet4 ,speaking generally1 the sum of the numbers in the 99n99th line will be #b99n99 - #a4 ,again1 if the whole of the numbers down to any line be added together1 we shall obtain a number less by unity than some power of #b2 thus1 the first line gives #a or #ba - #a2 the first two lines give #c or #bb - #a2 the first three lines #g or #bc - #a2 the first six lines give #fc or #bf - #a2 or1 speaking in general language1 the sum of the first 99n99 lines is #b99n99 - #a4 ,it follows that the sum of the numbers in any one line is equal to the sum of those in all the preceding lines increased by a unit4 ,for the sum of the 99n99th line is1 as already shown1 #b99n99 - #a1 and the sum of the first 99n99 - #a lines is #b99n99 - #a - #a1 or less by a unit4 ,this account of the properties of the figurate numbers does not approach completeness2 a considerable1 probably an unlimited1 number of less simple and obvious relations might be traced out4 ,pascal1 after giving many of the properties1 exclaims#ajc3 8,mais j'en laisse bien plus que je n'en donne2 c'est une chose trange combien il est fertile en proprits6 ,chacun peut s'y exercer40 ,the arithmetical triangle may be considered a natural classification of numbers1 exhibiting1 in the most complete manner1 their evolution and relations in a certain point of view4 ,it is obvious that in an unlimited extension of the triangle1 each number1 with the single exception of the number 99two991 has at least two places4 #ajc 99uvres ,compltes991 vol4 iii4 p4 #bea4 ,though the properties above explained are highly curious1 the greatest value of the triangle arises from the fact that it contains a complete statement of the values of the formula 7p4 #ahb71 for the numbers of combinations of 99m99 things out of 99n991 for all possible values of 99m99 and 99n994 ,out of seven things one may be chosen in seven ways1 and seven occurs in the eighth line of the second column4 ,the combinations of two things chosen out of seven are 7#g #f7_/7a #b7 or #ba1 which is the third number in the eighth line4 ,the combinations of three things out of seven are 7#g #f #e7_/7a #b #c7 or #ce1 which appears fourth in the eighth line4 ,in a similar manner1 in the fifth1 sixth1 seventh1 and eighth columns of the eighth line ,i find it stated in how many ways ,i can select combinations of #d1 #e1 #f1 and #g things out of #g4 ,proceeding to the ninth line1 ,i find in succession the number of ways in which ,i can select #a1 #b1 #c1 #d1 #e1 #f1 #g1 and #h things1 out of #h things4 ,in general language1 if ,i wish to know in how many ways 99m99 things can be selected in combinations out of 99n99 things1 ,i must look in the 99n99 #ath line1 and take the 99m99 #ath number1 as the answer4 ,in how many ways1 for instance1 can a subcommittee of five be chosen out of a committee of nine4 ,the answer is #abf1 and is the sixth number in the tenth line2 it will be found equal to 7#i 4 #h 4 #g 4 #f 4 #e7_/7a 4 #b 4 #c 4 #d 4 #e71 which our formula 7p4 #ahb7 gives4 ,the full utility of the figurate numbers will be more apparent when we reach the subject of probabilities1 but ,i may give an illustration or two in this place4 ,in how many ways can we arrange four pennies as regards head and tail8 ,the question amounts to asking in how many ways we can select #j1 #a1 #b1 #c1 or #d heads1 out of #d heads1 and the 99fifth99 line of the triangle gives us the complete answer1 thus-- ,we can select ,no head and #d tails in #a way4 #a head and #c tails in #d ways4 #b heads and #b tails in #f ways4 #c heads and #a tail in #d ways4 #d heads and #j tail in #a way4 ,the total number of different cases is #af1 or #bd1 and when we come to the next chapter1 it will be found that these numbers give us the respective probabilities of all throws with four pennies4 ,i gave in p4 #aha a calculation of the number of ways in which eight planets can meet in conjunction2 the reader will find all the numbers detailed in the ninth line of the arithmetical triangle4 ,the sum of the whole line is #bh or #bef2 but we must subtract a unit for the case where no planet appears1 and #h for the #h cases in which only one planet appears2 so that the total number of conjunctions is #bh -#a - #h or #bdg4 ,if an organ has eleven stops we find in the twelfth line the numbers of ways in which we can draw them1 #a1 #b1 #c1 or more at a time4 ,thus there are #dfb ways of drawing five stops at once1 and as many of drawing six stops4 ,the total number of ways of varying the sound is #bjdh1 including the single case in which no stop at all is drawn4 ,one of the most important scientific uses of the arithmetical triangle consists in the information which it gives concerning the comparative frequency of divergencies from an average4 ,suppose1 for the sake of argument1 that all persons were naturally of the equal stature of five feet1 but enjoyed during youth seven independent chances of growing one inch in addition4 ,of these seven chances1 one1 two1 three1 or more1 may happen favourably to any individual2 but1 as it does not matter what the chances are1 so that the inch is gained1 the question really turns upon the number of combinations of #j1 #a1 #b1 #c1 c41 things out of seven4 ,hence the eighth line of the triangle gives us a complete answer to the question1 as follows3-- ,out of every #abh people-- ,feet ,inches4 ,one person would have the stature of #e #j #g persons #e #a #ba persons #e #b #ce persons #e #c #ce persons #e #d #ba persons #e #e #g persons #e #f #a person #e #g ,by taking a proper line of the triangle1 an answer may be had under any more natural supposition4 ,this theory of comparative frequency of divergence from an average1 was first adequately noticed by ,quetelet1 and has lately been employed in a very interesting and bold manner by ,mr4 ,francis ,galton1#ajd in his remarkable work on 8,hereditary ,genius40 ,we shall afterwards find that the theory of error1 to which is made the ultimate appeal in cases of quantitative investigation1 is founded upon the comparative numbers of combinations as displayed in the triangle4 #ajd ,see also ,galton's ,lecture at the ,royal ,institution1 #bgth ,february1 #ahgd2 ,catalogue of the ,special ,loan ,collection of ,scientific ,instruments1 ,south ,kensington1 ,nos4 #dh1 #di2 and ,galton1 99,philosophical ,magazine991 ,january #ahge4 99,connection between the ,arithmetical ,triangle and the ,logical ,alphabet499 ,there exists a close connection between the arithmetical triangle described in the last section1 and the series of combinations of letters called the ,logical ,alphabet4 ,the one is to mathematical science what the other is to logical science4 ,in fact the figurate numbers1 or those exhibited in the triangle1 are obtained by summing up the logical combinations4 ,accordingly1 just as the total of the numbers in each line of the triangle is twice as great as that for the preceding line 7p4 #ahf71 so each column of the ,alphabet 7p4 #id7 contains twice as many combinations as the preceding one4 ,the like correspondence also exists between the sums of all the lines of figures down to any particular line1 and of the combinations down to any particular column4 ,by examining any column of the ,logical ,alphabet we find that the combinations naturally group themselves according to the figurate numbers4 ,take the combinations of the letters ,a1 ,b1 ,c1 ,d2 they consist of all the ways in which ,i can choose four1 three1 two1 one1 or none of the four letters1 filling up the vacant spaces with negative terms4 ,there is one combination1 ,,abcd1 in which all the positive letters are present2 there are four combinations in each of which three positive letters are present2 six in which two are present2 four in which only one is present2 and1 finally1 there is the single case1 99abcd991 in which all positive letters are absent4 ,these numbers1 #a1 #d1 #f1 #d1 #a1 are those of the fifth line of the arithmetical triangle1 and a like correspondence will be found to exist in each column of the ,logical ,alphabet4 ,numerical abstraction1 it has been asserted1 consists in overlooking the kind of difference1 and retaining only a consciousness of its existence 7p4 #aeh74 ,while in logic1 then1 we have to deal with each combination as a separate kind of thing1 in arithmetic we distinguish only the classes which depend upon more or less positive terms being present1 and the numbers of these classes immediately produce the numbers of the arithmetical triangle4 ,it may here be pointed out that there are two modes in which we can calculate the whole number of combinations of certain things4 ,either we may take the whole number at once as shown in the ,logical ,alphabet1 in which case the number will be some power of two1 or else we may calculate successively1 by aid of permutations1 the number of combinations of none1 one1 two1 three things1 and so on4 ,hence we arrive at a necessary identity between two series of numbers4 ,in the case of four things we shall have #b #a #d_/a 7#d 4 #c7_/7a 4 #b7 7#d 4 #c 4 #b7_/7a 4 #b 4 #c7 7#d 4 #c 4 #b 4 #a7_/7a 4 #b 4 #c 4 #d74 ,in a general form of expression we shall have #b #a 99n99_/#a 799n99 4 799n99 - #a77_/7a 4 #b7 799n99 799n99 - #a7799n99 - #b77_/7a 4 #b 4 #c7 c41 the terms being continued until they cease to have any value4 ,thus we arrive at a proof of simple cases of the ,binomial ,theorem1 of which each column of the ,logical ,alphabet is an exemplification4 ,it may be shown that all other mathematical expansions likewise arise out of simple processes of combination1 but the more complete consideration of this subject must be deferred to another work4 99,possible ,variety of ,nature and ,art499 ,we cannot adequately understand the difficulties which beset us in certain branches of science1 unless we have some clear idea of the vast numbers of combinations or permutations which may be possible under certain conditions4 ,thus only can we learn how hopeless it would be to attempt to treat nature in detail1 and exhaust the whole number of events which might arise4 ,it is instructive to consider1 in the first place1 how immensely great are the numbers of combinations with which we deal in many arts and amusements4 ,in dealing a pack of cards1 the number of hands1 of thirteen cards each1 which can be produced is evidently #eb #ea #ej 444 #dj divided by #a #b #c 444 #ac4 or #fce1jac1eei1fjj4 ,but in whist four hands are simultaneously held1 and the number of distinct deals becomes so vast that it would require twenty-eight figures to express it4 ,if the whole population of the world1 say one thousand millions of persons1 were to deal cards day and night1 for a hundred million of years1 they would not in that time have exhausted one hundred-thousandth part of the possible deals4 ,even with the same hands of cards the play may be almost infinitely varied1 so that the complete variety of games at whist which may exist is almost incalculably great4 ,it is in the highest degree improbable that any one game of whist was ever exactly like another1 except it were intentionally so4 ,the end of novelty in art might well be dreaded1 did we not find that nature at least has placed no attainable limit1 and that the deficiency will lie in our inventive faculties4 ,it would be a cheerless time indeed when all possible varieties of melody were exhausted1 but it is readily shown that if a peal of twenty-four bells had been rung continuously from the so-called beginning of the world to the present day1 no approach could have been made to the completion of the possible changes4 ,nay1 had every single minute been prolonged to #aj1jjj years1 still the task would have been unaccomplished4#aje ,as regards ordinary melodies1 the eight notes of a single octave give more than #dj1jjj permutations1 and two octaves more than a million millions4 ,if we were to take into account the semitones1 it would become apparent that it is impossible to exhaust the variety of music4 ,when the late ,mr4 ,j4 ,s4 ,mill1 in a depressed state of mind1 feared the approaching exhaustion of musical melodies1 he had certainly not bestowed sufficient study on the subject of permutations4 #aje ,wallis1 99,of ,combinations991 p4 #aaf1 quoting ,vossius4 ,similar considerations apply to the possible number of natural substances1 though we cannot always give precise numerical results4 ,it was recommended by ,hatchett#ajf that a systematic examination of all alloys of metals should be carried out1 proceeding from the binary ones to more complicated ternary or quaternary ones4 ,he can hardly have been aware of the extent of his proposed inquiry4 ,if we operate only upon thirty of the known metals1 the number of binary alloys would be #dce1 of ternary alloys #djfj1 of quaternary #bg1dje1 without paying regard to the varying proportions of the metals1 and only regarding the kind of metal4 ,if we varied all the ternary alloys by quantities not less than one per cent41 the number of these alloys would be #aa1dde1jfj4 ,an exhaustive investigation of the subject is therefore out of the question1 and unless some laws connecting the properties of the alloy and its components can be discovered1 it is not apparent how our knowledge of them can ever be more than fragmentary4 #ajf 99,philosophical ,transactions99 7#ahjc71 vol4 xciii4 p4 #aic4 ,the possible variety of definite chemical compounds1 again1 is enormously great4 ,chemists have already examined many thousands of inorganic substances1 and a still greater number of organic compounds2#ajg they have nevertheless made no appreciable impression on the number which may exist4 ,taking the number of elements at sixty-one1 the number of compounds containing different selections of four elements each would be more than half a million 7#eba1hee74 ,as the same elements often combine in many different proportions1 and some of them1 especially carbon1 have the power of forming an almost endless number of compounds1 it would hardly be possible to assign any limit to the number of chemical compounds which may be formed4 ,there are branches of physical science1 therefore1 of which it is unlikely that scientific men1 with all their industry1 can ever obtain a knowledge in any appreciable degree approaching to completeness4 #ajg ,hofmann's 99,introduction to ,chemistry991 p4 #cf4 99,higher ,orders of ,variety499 ,the consideration of the facts already given in this chapter will not produce an adequate notion of the possible variety of existence1 unless we consider the comparative numbers of combinations of different orders4 ,by a combination of a higher order1 ,i mean a combination of groups1 which are themselves groups4 ,the immense numbers of compounds of carbon1 hydrogen1 and oxygen1 described in organic chemistry1 are combinations of a second order1 for the atoms are groups of groups4 ,the wave of sound produced by a musical instrument may be regarded as a combination of motions2 the body of sound proceeding from a large orchestra is therefore a complex aggregate of sounds1 each in itself a complex combination of movements4 ,all literature may be said to be developed out of the difference of white paper and black ink4 ,from the unlimited number of marks which might be chosen we select twenty-six conventional letters4 ,the pronounceable combinations of letters are probably some trillions in number4 ,now1 as a sentence is a selection of words1 the possible sentences must be inconceivably more numerous than the words of which it may be composed4 ,a book is a combination of sentences1 and a library is a combination of books4 ,a library1 therefore1 may be regarded as a combination of the fifth order1 and the powers of numerical expression would be severely tasked in attempting to express the number of distinct libraries which might be constructed4 ,the calculation1 of course1 would not be possible1 because the union of letters in words1 of words in sentences1 and of sentences in books1 is governed by conditions so complex as to defy analysis4 ,i wish only to point out that the infinite variety of literature1 existing or possible1 is all developed out of one fundamental difference4 ,galileo remarked that all truth is contained in the compass of the alphabet4 ,he ought to have said that it is all contained in the difference of ink and paper4 ,one consequence of successive combination is that the simplest marks will suffice to express any information4 ,francis ,bacon proposed for secret writing a biliteral cipher1 which resolves all letters of the alphabet into permutations of the two letters 99a99 and 99b994 ,thus ,a was 99aaaaa991 ,b 99aaaab991 ,x 99babab991 and so on4#ajh ,in a similar way1 as ,bacon clearly saw1 any one difference can be made the ground of a code of signals2 we can express1 as he says1 99omnia per omnia994 ,the ,morse alphabet uses only a succession of long and short marks1 and other systems of telegraphic language employ right and left strokes4 ,a single lamp obscured at various intervals1 long or short1 may be made to spell out any words1 and with two lamps1 distinguished by colour1 position1 or any other circumstance1 we could at once represent ,bacon's biliteral alphabet4 ,babbage ingeniously suggested that every lighthouse in the world should be made to spell out its own name or number perpetually1 by flashes or obscurations of various duration and succession4 ,a system like that of ,babbage is now being applied to lighthouses in the ,united ,kingdom by ,sir ,w4 ,thomson and ,dr4 ,john ,hopkinson4 #ajh 99,works991 edited by ,shaw1 vol4 i4 pp4 #ada-ade1 quoted in ,rees's 99,encyclopdia991 art4 99,cipher994 ,let us calculate the numbers of combinations of different orders which may arise out of the presence or absence of a single mark1 say ,a4 ,in these figures ------ ------ ------ ------ ,a ,a ,a ,a ------ ------ ------ ------ we have four distinct varieties4 ,form them into a group of a higher order1 and consider in how many ways we may vary that group by omitting one or more of the component parts4 ,now1 as there are four parts1 and any one may be present or absent1 the possible varieties will be #b #b #b #b1 or #af in number4 ,form these into a new whole1 and proceed again to create variety by omitting any one or more of the sixteen4 ,the number of possible changes will now be #b 4 #b 4 #b 4 #b 4 #b 4 #b 4 #b 4 #b 4 #b 4 #b 4 #b 4 #b 4 #b 4 #b 4 #b 4 #b1 or #baf1 and we can repeat the process again and again4 ,we are imagining the creation of objects1 whose numbers are represented by the successive orders of the powers of 99two994 ,at the first step we have #b2 at the next #bb1 or #d2 at the third 7#bb7b1 or #af1 numbers of very moderate amount4 ,let the reader calculate the next term1 77#bb7b7b1 and he will be surprised to find it leap up to #fe1ecf4 ,but at the next step he has to calculate the value of #fe1ecf 99two99's multiplied together1 and it is so great that we could not possibly compute it1 the mere expression of the result requiring #ai1gbi places of figures4 ,but go one step more and we pass the bounds of all reason4 ,the sixth order of the powers of 99two99 becomes so great1 that we could not even express the number of figures required in writing it down1 without using about #ai1gbi figures for the purpose4 ,the successive orders of the powers of two have then the following values so far as we can succeed in describing them3-- ,first order #b ,second order #d ,third order #af ,fourth order #fe1ecf ,fifth order1 number expressed by #ai1gbi figures4 ,sixth order1 number expressed by figures1 to express the number of which figures would require about #ai1gbi figures4 ,it may give us some notion of infinity to remember that at this sixth step1 having long surpassed all bounds of intuitive conception1 we make no approach to a limit4 ,nay1 were we to make a hundred such steps1 we should be as far away as ever from actual infinity4 ,it is well worth observing that our powers of expression rapidly overcome the possible multitude of finite objects which may exist in any assignable space4 ,archimedes showed long ago1 in one of the most remarkable writings of antiquity1 the 99,liber de ,arcn ,numero991 that the grains of sand in the world could be numbered1 or rather1 that if numbered1 the result could readily be expressed in arithmetical notation4 ,let us extend his problem1 and ascertain whether we could express the number of atoms which could exist in the visible universe4 ,the most distant stars which can now be seen by telescopes--those of the sixteenth magnitude--are supposed to have a distance of about #cc1ijj1jjj1jjj1jjj1jjj miles4 ,sir ,w4 ,thomson has shown reasons for supposing that there do not exist more than from #c #ajbd to #ajbf molecules in a cubic centimetre of a solid or liquid substance4#aji ,assuming these data to be true1 for the sake of argument1 a simple calculation enables us to show that the almost inconceivably vast sphere of our stellar system if entirely filled with solid matter1 would not contain more than about #fh #ajij atoms1 that is to say1 a number requiring for its expression #ib places of figures4 ,now1 this number would be immensely less than the fifth order of the powers of two4 #aji 99,nature991 vol4 i4 p4 #eec4 ,in the variety of logical relations1 which may exist between a certain number of logical terms1 we also meet a case of higher combinations4 ,we have seen 7p4 #adb7 that with only six terms the number of possible selections of combinations is #ah1ddf1gdd1jgc1gji1eea1faf4 ,considering that it is the most common thing in the world to use an argument involving six objects or terms1 it may excite some surprise that the complete investigation of the relations in which six such terms may stand to each other1 should involve an almost inconceivable number of cases4 ,yet these numbers of possible logical relations belong only to the second order of combinations4 ,,chapter ,x4 ,,the ,,theory ,,of ,,probability4 ,the subject upon which we now enter must not be regarded as an isolated and curious branch of speculation4 ,it is the necessary basis of the judgments we make in the prosecution of science1 or the decisions we come to in the conduct of ordinary affairs4 ,as ,butler truly said1 8,probability is the very guide of life40 ,had the science of numbers been studied for no other purpose1 it must have been developed for the calculation of probabilities4 ,all our inferences concerning the future are merely probable1 and a due appreciation of the degree of probability depends upon a comprehension of the principles of the subject4 ,i am convinced that it is impossible to expound the methods of induction in a sound manner1 without resting them upon the theory of probability4 ,perfect knowledge alone can give certainty1 and in nature perfect knowledge would be infinite knowledge1 which is clearly beyond our capacities4 ,we have1 therefore1 to content ourselves with partial knowledge--knowledge mingled with ignorance1 producing doubt4 ,a great difficulty in this subject consists in acquiring a precise notion of the matter treated4 ,what is it that we number1 and measure1 and calculate in the theory of probabilities8 ,is it belief1 or opinion1 or doubt1 or knowledge1 or chance1 or necessity1 or want of art8 ,does probability exist in the things which are probable1 or in the mind which regards them as such8 ,the etymology of the name lends us no assistance3 for1 curiously enough1 99probable99 is ultimately the same word as 99provable991 a good instance of one word becoming differentiated to two opposite meanings4 ,chance cannot be the subject of the theory1 because there is really no such thing as chance1 regarded as producing and governing events4 ,the word chance signifies 99falling991 and the notion of falling is continually used as a simile to express uncertainty1 because we can seldom predict how a die1 a coin1 or a leaf will fall1 or when a bullet will hit the mark4 ,but everyone sees1 after a little reflection1 that it is in our knowledge the deficiency lies1 not in the certainty of nature's laws4 ,there is no doubt in lightning as to the point it shall strike2 in the greatest storm there is nothing capricious2 not a grain of sand lies upon the beach1 but infinite knowledge would account for its lying there2 and the course of every falling leaf is guided by the principles of mechanics which rule the motions of the heavenly bodies4 ,chance then exists not in nature1 and cannot coexist with knowledge2 it is merely an expression1 as ,laplace remarked1 for our ignorance of the causes in action1 and our consequent inability to predict the result1 or to bring it about infallibly4 ,in nature the happening of an event has been pre-determined from the first fashioning of the universe4 99,probability belongs wholly to the mind499 ,this is proved by the fact that different minds may regard the very same event at the same time with widely different degrees of probability4 ,a steam-vessel1 for instance1 is missing and some persons believe that she has sunk in mid-ocean2 others think differently4 ,in the event itself there can be no such uncertainty2 the steam-vessel either has sunk or has not sunk1 and no subsequent discussion of the probable nature of the event can alter the fact4 ,yet the probability of the event will really vary from day to day1 and from mind to mind1 according as the slightest information is gained regarding the vessels met at sea1 the weather prevailing there1 the signs of wreck picked up1 or the previous condition of the vessel4 ,probability thus belongs to our mental condition1 to the light in which we regard events1 the occurrence or non-occurrence of which is certain in themselves4 ,many writers accordingly have asserted that probability is concerned with degree or quantity of belief4 ,de ,morgan says1#aaj 8,by degree of probability we really mean or ought to mean degree of belief40 ,the late ,professor ,donkin expressed the meaning of probability as 8quantity of belief20 but ,i have never felt satisfied with such definitions of probability4 ,the nature of 99belief99 is not more clear to my mind than the notion which it is used to define4 ,but an all-sufficient objection is1 that 99the theory does not measure what the belief is1 but what it ought to be994 ,few minds think in close accordance with the theory1 and there are many cases of evidence in which the belief existing is habitually different from what it ought to be4 ,even if the state of belief in any mind could be measured and expressed in figures1 the results would be worthless4 ,the value of the theory consists in correcting and guiding our belief1 and rendering our states of mind and consequent actions harmonious with our knowledge of exterior conditions4 #aaj 99,formal ,logic991 p4 #agb4 ,this objection has been clearly perceived by some of those who still used quantity of belief as a definition of probability4 ,thus ,de ,morgan adds--8,belief is but another name for imperfect knowledge40 ,donkin has well said that the quantity of belief is 8always relative to a particular state of knowledge or ignorance2 but it must be observed that it is absolute in the sense of not being relative to any individual mind2 since1 the same information being presupposed1 all minds 99ought99 to distribute their belief in the same way40#aaa ,boole seemed to entertain a like view1 when he described the theory as engaged with 8the equal distribution of ignorance20#aab but we may just as well say that it is engaged with the equal distribution of knowledge4 #aaa 99,philosophical ,magazine991 #dth ,series1 vol4 i4 p4 #cee4 #aab 99,transactions of the ,royal ,society of ,edinburgh991 vol4 xxi4 part #d4 ,i prefer to dispense altogether with this obscure word belief1 and to say that the theory of probability deals with 99quantity of knowledge991 an expression of which a precise explanation and measure can presently be given4 ,an event is only probable when our knowledge of it is diluted with ignorance1 and exact calculation is needed to discriminate how much we do and do not know4 ,the theory has been described by some writers as professing 99to evolve knowledge out of ignorance992 but as ,donkin admirably remarked1 it is really 8a method of avoiding the erection of belief upon ignorance40 ,it defines rational expectation by measuring the comparative amounts of knowledge and ignorance1 and teaches us to regulate our actions with regard to future events in a way which will1 in the long run1 lead to the least disappointment4 ,it is1 as ,laplace happily said1 99good sense reduced to calculation994 ,this theory appears to me the noblest creation of intellect1 and it passes my conception how two such men as ,auguste ,comte and ,j4 ,s4 ,mill could be found depreciating it and vainly questioning its validity4 ,to eulogise the theory ought to be as needless as to eulogise reason itself4 99,fundamental ,principles of the ,theory499 ,the calculation of probabilities is really founded1 as ,i conceive1 upon the principle of reasoning set forth in preceding chapters4 ,we must treat equals equally1 and what we know of one case may be affirmed of every case resembling it in the necessary circumstances4 ,the theory consists in putting similar cases on a par1 and distributing equally among them whatever knowledge we possess4 ,throw a penny into the air1 and consider what we know with regard to its way of falling4 ,we know that it will certainly fall upon a side1 so that either head or tail will be uppermost2 but as to whether it will be head or tail1 our knowledge is equally divided4 ,whatever we know concerning head1 we know also concerning tail1 so that we have no reason for expecting one more than the other4 ,the least predominance of belief to either side would be irrational2 it would consist in treating unequally things of which our knowledge is equal4 ,the theory does not require1 as some writers have erroneously supposed1 that we should first ascertain by experiment the equal facility of the events we are considering4 ,so far as we can examine and measure the causes in operation1 events are removed out of the sphere of probability4 ,the theory comes into play where ignorance begins1 and the knowledge we possess requires to be distributed over many cases4 ,nor does the theory show that the coin will fall as often on the one side as the other4 ,it is almost impossible that this should happen1 because some inequality in the form of the coin1 or some uniform manner in throwing it up1 is almost sure to occasion a slight preponderance in one direction4 ,but as we do not previously know in which way a preponderance will exist1 we have no reason for expecting head more than tail4 ,our state of knowledge will be changed should we throw up the coin many times and register the results4 ,every throw gives us some slight information as to the probable tendency of the coin1 and in subsequent calculations we must take this into account4 ,in other cases experience might show that we had been entirely mistaken2 we might expect that a die would fall as often on each of the six sides as on each other side in the long run2 trial might show that the die was a loaded one1 and falls most often on a particular face4 ,the theory would not have misled us3 it treated correctly the information we had1 which is all that any theory can do4 ,it may be asked1 as ,mill asks1 ,why spend so much trouble in calculating from imperfect data1 when a little trouble would enable us to render a conclusion certain by actual trial8 ,why calculate the probability of a measurement being correct1 when we can try whether it is correct8 ,but ,i shall fully point out in later parts of this work that in measurement we never can attain perfect coincidence4 ,two measurements of the same base line in a survey may show a difference of some inches1 and there may be no means of knowing which is the better result4 ,a third measurement would probably agree with neither4 ,to select any one of the measurements1 would imply that we knew it to be the most nearly correct one1 which we do not4 ,in this state of ignorance1 the only guide is the theory of probability1 which proves that in the long run the mean of divergent results will come most nearly to the truth4 ,in all other scientific operations whatsoever1 perfect knowledge is impossible1 and when we have exhausted all our instrumental means in the attainment of truth1 there is a margin of error which can only be safely treated by the principles of probability4 ,the method which we employ in the theory consists in calculating the number of all the cases or events concerning which our knowledge is equal4 ,if we have the slightest reason for suspecting that one event is more likely to occur than another1 we should take this knowledge into account4 ,this being done1 we must determine the whole number of events which are1 so far as we know1 equally likely4 ,thus1 if we have no reason for supposing that a penny will fall more often one way than another1 there are two cases1 head and tail1 equally likely4 ,but if from trial or otherwise we know1 or think we know1 that of #ajj throws #ee will give tail1 then the probability is measured by the ratio of #ee to #ajj4 ,the mathematical formul of the theory are exactly the same as those of the theory of combinations4 ,in this latter theory we determine in how many ways events may be joined together1 and we now proceed to use this knowledge in calculating the number of ways in which a certain event may come about4 ,it is the comparative numbers of ways in which events can happen which measure their comparative probabilities4 ,if we throw three pennies into the air1 what is the probability that two of them will fall tail uppermost8 ,this amounts to asking in how many possible ways can we select two tails out of three1 compared with the whole number of ways in which the coins can be placed4 ,now1 the fourth line of the ,arithmetical ,triangle 7p4 #ahd7 gives us the answer4 ,the whole number of ways in which we can select or leave three things is eight1 and the possible combinations of two things at a time is three2 hence the probability of two tails is the ratio of three to eight4 ,from the numbers in the triangle we may similarly draw all the following probabilities3-- ,one combination gives #j tail4 ,probability #a_/h4 ,three combinations gives #a tail4 ,probability #c_/h4 ,three combinations give #b tails4 ,probability #c_/h4 ,one combination gives #c tails4 ,probability #a_/h4 ,we can apply the same considerations to the imaginary causes of the difference of stature1 the combinations of which were shown in p4 #ahh4 ,there are altogether #abh ways in which seven causes can be present or absent4 ,now1 twenty-one of these combinations give an addition of two inches1 so that the probability of a person under the circumstances being five feet two inches is #ba_/abh4 ,the probability of five feet three inches is #ce_/abh2 of five feet one inch #g_/abh2 of five feet #a_/abh1 and so on4 ,thus the eighth line of the ,arithmetical ,triangle gives all the probabilities arising out of the combinations of seven causes4 99,rules for the ,calculation of ,probabilities499 ,i will now explain as simply as possible the rules for calculating probabilities4 ,the principal rule is as follows3-- ,calculate the number of events which may happen independently of each other1 and which1 as far as is known1 are equally probable4 ,make this number the denominator of a fraction1 and take for the numerator the number of such events as imply or constitute the happening of the event1 whose probability is required4 ,thus1 if the letters of the word 99,roma99 be thrown down casually in a row1 what is the probability that they will form a significant ,latin word8 ,the possible arrangements of four letters are #d #c #b #a1 or #bd in number 7p4 #agh71 and if all the arrangements be examined1 seven of these will be found to have meaning1 namely 99,roma991 99ramo991 99oram991 99mora991 99maro991 99armo991 and 99amor994 ,hence the probability of a significant result is #g_/bd4 ,we must distinguish comparative from absolute probabilities4 ,in drawing a card casually from a pack1 there is no reason to expect any one card more than any other4 ,now1 there are four kings and four queens in a pack1 so that there are just as many ways of drawing one as the other1 and the probabilities are equal4 ,but there are thirteen diamonds1 so that the probability of a king is to that of a diamond as four to thirteen4 ,thus the probabilities of each are proportional to their respective numbers of ways of happening4 ,again1 ,i can draw a king in four ways1 and not draw one in forty-eight1 so that the probabilities are in this proportion1 or1 as is commonly said1 the 99odds99 against drawing a king are forty-eight to four4 ,the odds are seven to seventeen in favour1 or seventeen to seven against the letters ,r1o1m1a1 accidentally forming a significant word4 ,the odds are five to three against two tails appearing in three throws of a penny4 ,conversely1 when the odds of an event are given1 and the probability is required1 99take the odds in favour of the event for numerator1 and the sum of the odds for denominator994 ,it is obvious that an event is certain when all the combinations of causes which can take place produce that event4 ,if we represent the probability of such event according to our rule1 it gives the ratio of some number to itself1 or unity4 ,an event is certain not to happen when no possible combination of causes gives the event1 and the ratio by the same rule becomes that of #j to some number4 ,hence it follows that in the theory of probability certainty is expressed by #a1 and impossibility by #j2 but no mystical meaning should be attached to these symbols1 as they merely express the fact that 99all99 or 99no99 possible combinations give the event4 ,by a 99compound event991 we mean an event which may be decomposed into two or more simpler events4 ,thus the firing of a gun may be decomposed into pulling the trigger1 the fall of the hammer1 the explosion of the cap1 c4 ,in this example the simple events are not 99independent991 because if the trigger is pulled1 the other events will under proper conditions necessarily follow1 and their probabilities are therefore the same as that of the first event4 ,events are 99independent99 when the happening of one does not render the other either more or less probable than before4 ,thus the death of a person is neither more nor less probable because the planet ,mars happens to be visible4 ,when the component events are independent1 a simple rule can be given for calculating the probability of the compound event1 thus--99,multiply together the fractions expressing the probabilities of the independent component events499 ,the probability of throwing tail twice with a penny is #a_/b #a_/b1 or #a_/d2 the probability of throwing it three times running is #a_/b #a_/b #a_/b1 or #a_/h2 a result agreeing with that obtained in an apparently different manner 7p4 #bjb74 ,in fact1 when we multiply together the denominators1 we get the whole number of ways of happening of the compound event1 and when we multiply the numerators1 we get the number of ways favourable to the required event4 ,probabilities may be added to or subtracted from each other under the important condition that the events in question are exclusive of each other1 so that not more than one of them can happen4 ,it might be argued that1 since the probability of throwing head at the first trial is #a_/b1 and at the second trial also #a_/b1 the probability of throwing it in the first two throws is #a_/b #a_/b1 or certainty4 ,not only is this result evidently absurd1 but a repetition of the process would lead us to a probability of #a-a_/b or of any greater number1 results which could have no meaning whatever4 ,the probability we wish to calculate is that of one head in two throws1 but in our addition we have included the case in which two heads appear4 ,the true result is #a_/b #a_/b #a_/b or #c_/d1 or the probability of head at the first throw1 added to the exclusive probability that if it does not come at the first1 it will come at the second4 ,the greatest difficulties of the theory arise from the confusion of exclusive and unexclusive alternatives4 ,i may remind the reader that the possibility of unexclusive alternatives was a point previously discussed 7p4 #fh71 and to the reasons then given for considering alternation as logically unexclusive1 may be added the existence of these difficulties in the theory of probability4 ,the erroneous result explained above really arose from overlooking the fact that the expression 8head first throw or head second throw0 might include the case of head at both throws4 99,the ,logical ,alphabet in questions of ,probability499 ,when the probabilities of certain simple events are given1 and it is required to deduce the probabilities of compound events1 the ,logical ,alphabet may give assistance1 provided that there are no special logical conditions so that all the combinations are possible4 ,thus1 if there be three events1 ,a1 ,b1 ,c1 of which the probabilities are1 1 1 1 then the negatives of those events1 expressing the absence of the events1 will have the probabilities #a - 1 #a - 1 #a - 4 ,we have only to insert these values for the letters of the combinations and multiply1 and we obtain the probability of each combination4 ,thus the probability of ,,abc is 2 of ,a99bc991 7#a - 77#a - 74 ,we can now clearly distinguish between the probabilities of exclusive and unexclusive events4 ,thus1 if ,a and ,b are events which may happen together like rain and high tide1 or an earthquake and a storm1 the probability of ,a or ,b happening is not the sum of their separate probabilities4 ,for by the ,laws of ,thought we develop ,a ,b into ,,ab ,a99b99 99a99,b1 and substituting and 1 the probabilities of ,a and ,b respectively1 we obtain 4 4 7#a - 7 7#a - 7 4 or - 4 4 ,but if events are 99incompossible99 or incapable of happening together1 like a clear sky and rain1 or a new moon and a full moon1 then the events are not really ,a or ,b1 but ,a not-,b1 or ,b not-,a1 or in symbols ,a99b99 99a99,b4 ,now if we take probability of ,a99b99 and probability of 99a99,b1 then we may add simply1 and the probability of ,a99b99 99a99,b is 4 ,let the reader carefully observe that if the combination ,,ab cannot exist1 the probability of ,a99b99 is not the product of the probabilities of ,a and 99b994 ,when certain combinations are logically impossible1 it is no longer allowable to substitute the probability of each term for the term1 because the multiplication of probabilities presupposes the independence of the events4 ,a large part of ,boole's ,laws of ,thought is devoted to an attempt to overcome this difficulty and to produce a ,general ,method in ,probabilities by which from certain logical conditions and certain given probabilities it would be possible to deduce the probability of any other combinations of events under those conditions4 ,boole pursued his task with wonderful ingenuity and power1 but after spending much study on his work1 ,i am compelled to adopt the conclusion that his method is fundamentally erroneous4 ,as pointed out by ,mr4 ,wilbraham1#aac ,boole obtained his results by an arbitrary assumption1 which is only the most probable1 and not the only possible assumption4 ,the answer obtained is therefore not the real probability1 which is usually indeterminate1 but only1 as it were1 the most probable probability4 ,certain problems solved by ,boole are free from logical conditions and therefore may admit of valid answers4 ,these1 as ,i have shown1#aad may be solved by the combinations of the ,logical ,alphabet1 but the rest of the problems do not admit of a determinate answer1 at least by ,boole's method4 #aac 99,philosophical ,magazine991 #dth ,series1 vol4 vii4 p4 #dfe2 vol4 viii4 p4 #ia4 #aad 99,memoirs of the ,manchester ,literary and ,philosophical ,society991 #crd ,series1 vol4 iv4 p4 #cdg4 99,comparison of the ,theory with ,experience499 ,the ,laws of ,probability rest upon the fundamental principles of reasoning1 and cannot be really negatived by any possible experience4 ,it might happen that a person should always throw a coin head uppermost1 and appear incapable of getting tail by chance4 ,the theory would not be falsified1 because it contemplates the possibility of the most extreme runs of luck4 ,our actual experience might be counter to all that is probable2 the whole course of events might seem to be in complete contradiction to what we should expect1 and yet a casual conjunction of events might be the real explanation4 ,it is just possible that some regular coincidences1 which we attribute to fixed laws of nature1 are due to the accidental conjunction of phenomena in the cases to which our attention is directed4 ,all that we can learn from finite experience is capable1 according to the theory of probabilities1 of misleading us1 and it is only infinite experience that could assure us of any inductive truths4 ,at the same time1 the probability that any extreme runs of luck will occur is so excessively slight1 that it would be absurd seriously to expect their occurrence4 ,it is almost impossible1 for instance1 that any whist player should have played in any two games where the distribution of the cards was exactly the same1 by pure accident 7p4 #aia74 ,such a thing as a person always losing at a game of pure chance1 is wholly unknown4 ,coincidences of this kind are not impossible1 as ,i have said1 but they are so unlikely that the lifetime of any person1 or indeed the whole duration of history1 does not give any appreciable probability of their being encountered4 ,whenever we make any extensive series of trials of chance results1 as in throwing a die or coin1 the probability is great that the results will agree nearly with the predictions yielded by theory4 ,precise agreement must not be expected1 for that1 as the theory shows1 is highly improbable4 ,several attempts have been made to test1 in this way1 the accordance of theory and experience4 ,buffon caused the first trial to be made by a young child who threw a coin many times in succession1 and he obtained #aiib tails to #bjdh heads4 ,a pupil of ,de ,morgan repeated the trial for his own satisfaction1 and obtained #bjdd tails to #bjdh heads4 ,in both cases the coincidence with theory is as close as could be expected1 and the details may be found in ,de ,morgan's 8,formal ,logic10 p4 #ahe4 ,quetelet also tested the theory in a rather more complete manner1 by placing #bj black and #bj white balls in an urn and drawing a ball out time after time in an indifferent manner1 each ball being replaced before a new drawing was made4 ,he found1 as might be expected1 that the greater the number of drawings made1 the more nearly were the white and black balls equal in number4 ,at the termination of the experiment he had registered #bjff white and #bjcj black balls1 the ratio being #ajb4aae #aae 99,letters on the ,theory of ,probabilities991 translated by ,downes1 #ahdi1 pp4 #cf1 #cg4 ,i have made a series of experiments in a third manner1 which seemed to me even more interesting1 and capable of more extensive trial4 ,taking a handful of ten coins1 usually shillings1 ,i threw them up time after time1 and registered the numbers of heads which appeared each time4 ,now the probability of obtaining #aj1 #i1 #h1 #g1 c41 heads is proportional to the number of combinations of #aj1 #i1 #h1 #g1 c41 things out of #aj things4 ,consequently the results ought to approximate to the numbers in the eleventh line of the ,arithmetical ,triangle4 ,i made altogether #bjdh throws1 in two sets of #ajbd throws each1 and the numbers obtained are given in the following table3-- ---------------------------------------- ----------------------------- ,character of ,throw4,theoretical ,first ,second ,average4 ,divergence4 ,numbers4 ,series4 ,series4 ---------------------------------------- ----------------------------- #aj ,heads #j ,tail #a #c #a #b #a #i #a #aj #ab #bc #ag-a_/b #g-a_/b #h #b #de #eg #gc #fe #bj #g #c #abj #abi #abc #abf #f #f #d #baj #aha #aij #ahe-a_/b - #be #e #e #beb #beg #bcb #bdd-a_/b - #g-a_/b #d #f #baj #bja #aig #aii - #aa #c #g #abj #aaa #aai #aae - #e #b #h #de #eb #ej #ea #f #a #i #aj #ba #ae #ah #h #j #aj #a #j #a #a_/b - #a_/b ---------------------------------------- ----------------------------- ,totals #ajbd #ajbd #ajbd #ajbd - #a ---------------------------------------- ----------------------------- ,the whole number of single throws of coins amounted to #aj #bjdh1 or #bj1dhj in all1 one half of which or #aj1bdj should theoretically give head4 ,the total number of heads obtained was actually #aj1cec1 or #ebbb in the first series1 and #eaca in the second4 ,the coincidence with theory is pretty close1 but considering the large number of throws there is some reason to suspect a tendency in favour of heads4 ,the special interest of this trial consists in the exhibition1 in a practical form1 of the results of ,bernoulli's theorem1 and the law of error or divergence from the mean to be afterwards more fully considered4 ,it illustrates the connection between combinations and permutations1 which is exhibited in the ,arithmetical ,triangle1 and which underlies many important theorems of science4 99,probable ,deductive ,arguments994 ,with the aid of the theory of probabilities1 we may extend the sphere of deductive argument4 ,hitherto we have treated propositions as certain1 and on the hypothesis of certainty have deduced conclusions equally certain4 ,but the information on which we reason in ordinary life is seldom or never certain1 and almost all reasoning is really a question of probability4 ,we ought therefore to be fully aware of the mode and degree in which deductive reasoning is affected by the theory of probability1 and many persons may be surprised at the results which must be admitted4 ,some controversial writers appear to consider1 as ,de ,morgan remarked1#aaf that an inference from several equally probable premises is itself as probable as any of them1 but the true result is very different4 ,if an argument involves many propositions1 and each of them is uncertain1 the conclusion will be of very little force4 #aaf 99,encyclopdia ,metropolitana991 art4 99,probabilities991 p4 #cif4 ,the validity of a conclusion may be regarded as a compound event1 depending upon the premises happening to be true2 thus1 to obtain the probability of the conclusion1 we must multiply together the fractions expressing the probabilities of the premises4 ,if the probability is #a_/b that ,a is ,b1 and also #a_/b that ,b is ,c1 the conclusion that ,a is ,c1 on the ground of these premises1 is #a_/b #a_/b or #a_/d4 ,similarly if there be any number of premises requisite to the establishment of a conclusion and their probabilities be 99p991 99q991 99r991 c41 the probability of the conclusion on the ground of these premises is 99p99 99q99 99r99 444 ,this product has but a small value1 unless each of the quantities 99p991 99q991 c41 be nearly unity4 ,but it is particularly to be noticed that the probability thus calculated is not the whole probability of the conclusion1 but that only which it derives from the premises in question4 ,whately's#aag remarks on this subject might mislead the reader into supposing that the calculation is completed by multiplying together the probabilities of the premises4 ,but it has been fully explained by ,de ,morgan#aah that we must take into account the antecedent probability of the conclusion2 ,a may be ,c for other reasons besides its being ,b1 and as he remarks1 8,it is difficult1 if not impossible1 to produce a chain of argument of which the reasoner can rest the result on those arguments only40 ,the failure of one argument does not1 except under special circumstances1 disprove the truth of the conclusion it is intended to uphold1 otherwise there are few truths which could survive the ill-considered arguments adduced in their favour4 ,as a rope does not necessarily break because one or two strands in it fail1 so a conclusion may depend upon an endless number of considerations besides those immediately in view4 ,even when we have no other information we must not consider a statement as devoid of all probability4 ,the true expression of complete doubt is a ratio of equality between the chances in favour of and against it1 and this ratio is expressed in the probability #a_/b4 #aag 99,elements of ,logic991 ,book ,,iii4 sections #aa and #ah4 #aah 99,encyclopdia ,metropolitana991 art4 99,probabilities991 p4 #djj4 ,now if ,a and ,c are wholly unknown things1 we have no reason to believe that ,a is ,c rather than ,a is not ,c4 ,the antecedent probability is then #a_/b4 ,if we also have the probabilities that ,a is ,b1 #a_/b and that ,b is ,c1 #a_/b we have no right to suppose that the probability of ,a being ,c is reduced by the argument in its favour4 ,if the conclusion is true on its own grounds1 the failure of the argument does not affect it2 thus its total probability is its antecedent probability1 added to the probability that this failing1 the new argument in question establishes it4 ,there is a probability #a_/b that we shall not require the special argument2 a probability #a_/b that we shall1 and a probability #a_/d that the argument does in that case establish it4 ,thus the complete result is #a_/b #a_/b #a_/d1 or #e_/h4 ,in general language1 if 99a99 be the probability founded on a particular argument1 and 99c99 the antecedent probability of the event1 the general result is #a - 7#a - 99a9977#a - 99c9971 or 99a99 99c99 - 99ac994 ,we may put it still more generally in this way3--,let 99a991 99b991 99c991 c4 be the probabilities of a conclusion grounded on various arguments4 ,it is only when all the arguments fail that our conclusion proves finally untrue2 the probabilities of each failing are respectively1 #a - 99a991 #a - 99b991 #a - 99c991 c42 the probability that they will all fail is 7#a - 99a9977#a - 99b9977#a - 99c997 4442 therefore the probability that the conclusion will not fail is #a - 7#a - 99a9977#a - 99b9977#a - 99c997 444 c4 ,it follows that every argument in favour of a conclusion1 however flimsy and slight1 adds probability to it4 ,when it is unknown whether an overdue vessel has foundered or not1 every slight indication of a lost vessel will add some probability to the belief of its loss1 and the disproof of any particular evidence will not disprove the event4 ,we must apply these principles of evidence with great care1 and observe that in a great proportion of cases the adducing of a weak argument does tend to the disproof of its conclusion4 ,the assertion may have in itself great inherent improbability as being opposed to other evidence or to the supposed law of nature1 and every reasoner may be assumed to be dealing plainly1 and putting forward the whole force of evidence which he possesses in its favour4 ,if he brings but one argument1 and its probability 99a99 is small1 then in the formula #a - 7#a - 99a9977#a - 99c997 both 99a99 and 99c99 are small1 and the whole expression has but little value4 ,the whole effect of an argument thus turns upon the question whether other arguments remain1 so that we can introduce other factors 7#a - 99b9971 7#a - 99d9971 c41 into the above expression4 ,in a court of justice1 in a publication having an express purpose1 and in many other cases1 it is doubtless right to assume that the whole evidence considered to have any value as regards the conclusion asserted1 is put forward4 ,to assign the antecedent probability of any proposition1 may be a matter of difficulty or impossibility1 and one with which logic and the theory of probability have little concern4 ,from the general body of science in our possession1 we must in each case make the best judgment we can4 ,but in the absence of all knowledge the probability should be considered #a_/b1 for if we make it less than this we incline to believe it false rather than true4 ,thus1 before we possessed any means of estimating the magnitudes of the fixed stars1 the statement that ,sirius was greater than the sun had a probability of exactly #a_/b2 it was as likely that it would be greater as that it would be smaller2 and so of any other star4 ,this was the assumption which ,michell made in his admirable speculations4#aai ,it might seem1 indeed1 that as every proposition expresses an agreement1 and the agreements or resemblances between phenomena are infinitely fewer than the differences 7p4 #dd71 every proposition should in the absence of other information be infinitely improbable4 ,but in our logical system every term may be indifferently positive or negative1 so that we express under the same form as many differences as agreements4 ,it is impossible therefore that we should have any reason to disbelieve rather than to believe a statement about things of which we know nothing4 ,we can hardly indeed invent a proposition concerning the truth of which we are absolutely ignorant1 except when we are entirely ignorant of the terms used4 ,if ,i ask the reader to assign the odds that a 8,platythliptic ,coefficient is positive0 he will hardly see his way to doing so1 unless he regard them as even4 #aai 99,philosophical ,transactions99 7#agfg74 ,abridg4 vol4 xii4 p4 #dce4 ,the assumption that complete doubt is properly expressed by #a_/b has been called in question by ,bishop ,terrot1#abj who proposes instead the indefinite symbol #j_/j2 and he considers that 8the 99 priori99 probability derived from absolute ignorance has no effect upon the force of a subsequently admitted probability40 ,but if we grant that the probability may have any value between #j and #a1 and that every separate value is equally likely1 then 99n99 and #a - 99n99 are equally likely1 and the average is always #a_/b4 ,or we may take 99p99 4 99dp99 to express the probability that our estimate concerning any proposition should lie between 99p99 and 99p99 99dp994 ,the complete probability of the proposition is then the integral taken between the limits #a and #j1 or again #a_/b4 #abj 99,transactions of the ,edinburgh ,philosophical ,society991 vol4 xxi4 p4 #cge4 99,difficulties of the ,theory499 ,the theory of probability1 though undoubtedly true1 requires very careful application4 ,not only is it a branch of mathematics in which oversights are frequently committed1 but it is a matter of great difficulty in many cases1 to be sure that the formula correctly represents the data of the problem4 ,these difficulties often arise from the logical complexity of the conditions1 which might be1 perhaps1 to some extent cleared up by constantly bearing in mind the system of combinations as developed in the ,indirect ,logical ,method4 ,in the study of probabilities1 mathematicians had unconsciously employed logical processes far in advance of those in possession of logicians1 and the ,indirect ,method is but the full statement of these processes4 ,it is very curious how often the most acute and powerful intellects have gone astray in the calculation of probabilities4 ,seldom was ,pascal mistaken1 yet he inaugurated the science with a mistaken solution4#aba ,leibnitz fell into the extraordinary blunder of thinking that the number twelve was as probable a result in the throwing of two dice as the number eleven4#abb ,in not a few cases the false solution first obtained seems more plausible to the present day than the correct one since demonstrated4 ,james ,bernoulli candidly records two false solutions of a problem which he at first thought self-evident2 and he adds a warning against the risk of error1 especially when we attempt to reason on this subject without a rigid adherence to methodical rules and symbols4 ,montmort was not free from similar mistakes4 ,d',alembert constantly fell into blunders1 and could not perceive1 for instance1 that the probabilities would be the same when coins are thrown successively as when thrown simultaneously4 ,some men of great reputation1 such as ,ancillon1 ,moses ,mendelssohn1 ,garve1 ,auguste ,comte1#abc ,poinsot1 and ,j4 ,s4 ,mill1#abd have so far misapprehended the theory1 as to question its value or even to dispute its validity4 ,the erroneous statements about the theory given in the earlier editions of ,mill's 99,system of ,logic99 were partially withdrawn in the later editions4 #aba ,montucla1 99,histoire des ,mathmatiques991 vol4 iii4 p4 #chf4 #abb ,leibnitz 99,opera991 ,dutens' ,edition1 vol4 vi4 part i4 p4 #bag4 ,todhunter's 99,history of the ,theory of ,probability991 p4 #dh4 ,to the latter work ,i am indebted for many of the statements in the text4 #abc 99,positive ,philosophy991 translated by ,martineau1 vol4 ii4 p4 #abj4 #abd 99,system of ,logic991 bk4 iii4 chap4 #ah1 #eth ,ed4 vol4 ii4 p4 #fa4 ,many persons have a fallacious tendency to believe that when a chance event has happened several times together in an unusual conjunction1 it is less likely to happen again4 ,d',alembert seriously held that if head was thrown three times running with a coin1 tail would more probably appear at the next trial4#abe ,bequelin adopted the same opinion1 and yet there is no reason for it whatever4 ,if the event be really casual1 what has gone before cannot in the slightest degree influence it4 ,as a matter of fact1 the more often a casual event takes place the more likely it is to happen again2 because there is some slight empirical evidence of a tendency4 ,the source of the fallacy is to be found entirely in the feelings of surprise with which we witness an event happening by chance1 in a manner which seems to proceed from design4 #abe ,montucla1 99,histoire991 vol4 iii4 p4 #dje2 ,todhunter1 p4 #bfc4 ,misapprehension may also arise from overlooking the difference between permutations and combinations4 ,to throw ten heads in succession with a coin is no more unlikely than to throw any other particular succession of heads and tails1 but it is much less likely than five heads and five tails without regard to their order1 because there are no less than #beb different particular throws which will give this result1 when we abstract the difference of order4 ,difficulties arise in the application of the theory from our habitual disregard of slight probabilities4 ,we are obliged practically to accept truths as certain which are nearly so1 because it ceases to be worth while to calculate the difference4 ,no punishment could be inflicted if absolutely certain evidence of guilt were required1 and as ,locke remarks1 8,he that will not stir till he infallibly knows the business he goes about will succeed1 will have but little else to do but to sit still and perish40#abf ,there is not a moment of our lives when we do not lie under a slight danger of death1 or some most terrible fate4 ,there is not a single action of eating1 drinking1 sitting down1 or standing up1 which has not proved fatal to some person4 ,several philosophers have tried to assign the limit of the probabilities which we regard as zero2 ,buffon named #a_/aj1jjj1 because it is the probability1 practically disregarded1 that a man of #ef years of age will die the next day4 ,pascal remarked that a man would be esteemed a fool for hesitating to accept death when three dice gave sixes twenty times running1 if his reward in case of a different result was to be a crown2 but as the chance of death in question is only #a #ffj1 or unity divided by a number of #dg places of figures1 we may be said to incur greater risks every day for less motives4 ,there is far greater risk of death1 for instance1 in a game of cricket or a visit to the rink4 #abf 99,essay concerning ,human ,understanding991 bk4 iv4 ch4 #ad4 #a4 ,nothing is more requisite than to distinguish carefully between the truth of a theory and the truthful application of the theory to actual circumstances4 ,as a general rule1 events in nature and art will present a complexity of relations exceeding our powers of treatment4 ,the intricate action of the mind often intervenes and renders complete analysis hopeless4 ,if1 for instance1 the probability that a marksman shall hit the target in a single shot be #a in #aj1 we might seem to have no difficulty in calculating the probability of any succession of hits2 thus the probability of three successive hits would be one in a thousand4 ,but1 in reality1 the confidence and experience derived from the first successful shot would render a second success more probable4 ,the events are not really independent1 and there would generally be a far greater preponderance of runs of apparent luck1 than a simple calculation of probabilities could account for4 ,in some persons1 however1 a remarkable series of successes will produce a degree of excitement rendering continued success almost impossible4 ,attempts to apply the theory of probability to the results of judicial proceedings have proved of little value1 simply because the conditions are far too intricate4 ,as ,laplace said1 8,tant de passions1 d'intrts divers et de circonstances compliquent les questions relatives ces objets1 qu'elles sont presque toujours insolubles40 ,men acting on a jury1 or giving evidence before a court1 are subject to so many complex influences that no mathematical formulas can be framed to express the real conditions4 ,jurymen or even judges on the bench cannot be regarded as acting independently1 with a definite probability in favour of each delivering a correct judgment4 ,each man of the jury is more or less influenced by the opinion of the others1 and there are subtle effects of character and manner and strength of mind which defy analysis4 ,even in physical science we can in comparatively few cases apply the theory in a definite manner1 because the data required are too complicated and difficult to obtain4 ,but such failures in no way diminish the truth and beauty of the theory itself2 in reality there is no branch of science in which our symbols can cope with the complexity of ,nature4 ,as ,donkin said1-- 8,i do not see on what ground it can be doubted that every definite state of belief concerning a proposed hypothesis1 is in itself capable of being represented by a numerical expression1 however difficult or impracticable it may be to ascertain its actual value4 ,it would be very difficult to estimate in numbers the 99vis viva99 of all the particles of a human body at any instant2 but no one doubts that it is capable of numerical expression40#abg #abg 99,philosophical ,magazine991 #dth ,series1 vol4 i4 p4 #ced4 ,the difficulty1 in short1 is merely relative to our knowledge and skill1 and is not absolute or inherent in the subject4 ,we must distinguish between what is theoretically conceivable and what is practicable with our present mental resources4 ,provided that our aspirations are pointed in a right direction1 we must not allow them to be damped by the consideration that they pass beyond what can now be turned to immediate use4 ,in spite of its immense difficulties of application1 and the aspersions which have been mistakenly cast upon it1 the theory of probabilities1 ,i repeat1 is the noblest1 as it will in course of time prove1 perhaps the most fruitful branch of mathematical science4 ,it is the very guide of life1 and hardly can we take a step or make a decision of any kind without correctly or incorrectly making an estimation of probabilities4 ,in the next chapter we proceed to consider how the whole cogency of inductive reasoning rests upon probabilities4 ,the truth or untruth of a natural law1 when carefully investigated1 resolves itself into a high or low degree of probability1 and this is the case whether or not we are capable of producing precise numerical data4 ,,chapter ,,xi4 ,,philosophy ,,of ,,inductive ,,inference4 ,we have inquired into the nature of perfect induction1 whereby we pass backwards from certain observed combinations of events1 to the logical conditions governing such combinations4 ,we have also investigated the grounds of that theory of probability1 which must be our guide when we leave certainty behind1 and dilute knowledge with ignorance4 ,there is now before us the difficult task of endeavouring to decide how1 by the aid of that theory1 we can ascend from the facts to the laws of nature2 and may then with more or less success anticipate the future course of events4 ,all our knowledge of natural objects must be ultimately derived from observation1 and the difficult question arises--,how can we ever know anything which we have not directly observed through one of our senses1 the apertures of the mind8 ,the utility of reasoning is to assure ourselves that1 at a determinate time and place1 or under specified conditions1 a certain phenomenon will be observed4 ,when we can use our senses and perceive that the phenomenon does occur1 reasoning is superfluous4 ,if the senses cannot be used1 because the event is in the future1 or out of reach1 how can reasoning take their place8 ,apparently1 at least1 we must infer the unknown from the known1 and the mind must itself create an addition to the sum of knowledge4 ,but ,i hold that it is quite impossible to make any real additions to the contents of our knowledge1 except through new impressions upon the senses1 or upon some seat of feeling4 ,i shall attempt to show that inference1 whether inductive or deductive1 is never more than an unfolding of the contents of our experience1 and that it always proceeds upon the assumption that the future and the unperceived will be governed by the same conditions as the past and the perceived1 an assumption which will often prove to be mistaken4 ,in inductive as in deductive reasoning the conclusion never passes beyond the premises4 ,reasoning adds no more to the implicit contents of our knowledge1 than the arrangement of the specimens in a museum adds to the number of those specimens4 ,arrangement adds to our knowledge in a certain sense3 it allows us to perceive the similarities and peculiarities of the specimens1 and on the assumption that the museum is an adequate representation of nature1 it enables us to judge of the prevailing forms of natural objects4 ,bacon's first aphorism holds perfectly true1 that man knows nothing but what he has observed1 provided that we include his whole sources of experience1 and the whole implicit contents of his knowledge4 ,inference but unfolds the hidden meaning of our observations1 and 99the theory of probability shows how far we go beyond our data in assuming that new specimens will resemble the old ones991 or that the future may be regarded as proceeding uniformly with the past4 99,various ,classes of ,inductive ,truths499 ,it will be desirable1 in the first place1 to distinguish between the several kinds of truths which we endeavour to establish by induction4 ,although there is a certain common and universal element in all our processes of reasoning1 yet diversity arises in their application4 ,similarity of condition between the events from which we argue1 and those to which we argue1 must always be the ground of inference2 but this similarity may have regard either to time or place1 or the simple logical combination of events1 or to any conceivable junction of circumstances involving quality1 time1 and place4 ,having met with many pieces of substance possessing ductility and a bright yellow colour1 and having discovered1 by perfect induction1 that they all possess a high specific gravity1 and a freedom from the corrosive action of acids1 we are led to expect that every piece of substance1 possessing like ductility and a similar yellow colour1 will have an equally high specific gravity1 and a like freedom from corrosion by acids4 ,this is a case of the coexistence of qualities2 for the character of the specimens examined alters not with time nor place4 ,in a second class of cases1 time will enter as a principal ground of similarity4 ,when we hear a clock pendulum beat time after time1 at equal intervals1 and with a uniform sound1 we confidently expect that the stroke will continue to be repeated uniformly4 ,a comet having appeared several times at nearly equal intervals1 we infer that it will probably appear again at the end of another like interval4 ,a man who has returned home evening after evening for many years1 and found his house standing1 may1 on like grounds1 expect that it will be standing the next evening1 and on many succeeding evenings4 ,even the continuous existence of an object in an unaltered state1 or the finding again of that which we have hidden1 is but a matter of inference depending on experience4 ,a still larger and more complex class of cases involves the relations of space1 in addition to those of time and quality4 ,having observed that every triangle drawn upon the diameter of a circle1 with its apex upon the circumference1 apparently contains a right angle1 we may ascertain that all triangles in similar circumstances will contain right angles4 ,this is a case of pure space reasoning1 apart from circumstances of time or quality1 and it seems to be governed by different principles of reasoning4 ,i shall endeavour to show1 however1 that geometrical reasoning differs but in degree from that which applies to other natural relations4 99,the ,relation of ,cause and ,effect499 ,in a very large part of the scientific investigations which must be considered1 we deal with events which follow from previous events1 or with existences which succeed existences4 ,science1 indeed1 might arise even were material nature a fixed and changeless whole4 ,endow mind with the power to travel about1 and compare part with part1 and it could certainly draw inferences concerning the similarity of forms1 the coexistence of qualities1 or the preponderance of a particular kind of matter in a changeless world4 ,a solid universe1 in at least approximate equilibrium1 is not inconceivable1 and then the relation of cause and effect would evidently be no more than the relation of before and after4 ,as nature exists1 however1 it is a progressive existence1 ever moving and changing as time1 the great independent variable1 proceeds4 ,hence it arises that we must continually compare what is happening now with what happened a moment before1 and a moment before that moment1 and so on1 until we reach indefinite periods of past time4 ,a comet is seen moving in the sky1 or its constituent particles illumine the heavens with their tails of fire4 ,we cannot explain the present movements of such a body without supposing its prior existence1 with a definite amount of energy and a definite direction of motion2 nor can we validly suppose that our task is concluded when we find that it came wandering to our solar system through the unmeasured vastness of surrounding space4 ,every event must have a cause1 and that cause again a cause1 until we are lost in the obscurity of the past1 and are driven to the belief in one ,first ,cause1 by whom the course of nature was determined4 99,fallacious ,use of the ,term ,cause499 ,the words ,cause and ,causation have given rise to infinite trouble and obscurity1 and have in no slight degree retarded the progress of science4 ,from the time of ,aristotle1 the work of philosophy has been described as the discovery of the causes of things1 and ,francis ,bacon adopted the notion when he said 899vere scire esse per causas scire9940 ,even now it is not uncommonly supposed that the knowledge of causes is something different from other knowledge1 and consists1 as it were1 in getting possession of the keys of nature4 ,a single word may thus act as a spell1 and throw the clearest intellect into confusion1 as ,i have often thought that ,locke was thrown into confusion when endeavouring to find a meaning for the word 99power994#abh ,in ,mill's 99,system of ,logic99 the term 99cause99 seems to have re-asserted its old noxious power4 ,not only does ,mill treat the ,laws of ,causation as almost coextensive with science1 but he so uses the expression as to imply that when once we pass within the circle of causation we deal with certainties4 #abh 99,essay concerning ,human ,understanding991 bk4 ii4 chap4 xxi4 ,the philosophical danger which attaches to the use of this word may be thus described4 ,a cause is defined as the necessary or invariable antecedent of an event1 so that when the cause exists the effect will also exist or soon follow4 ,if then we know the cause of an event1 we know what will certainly happen2 and as it is implied that science1 by a proper experimental method1 may attain to a knowledge of causes1 it follows that experience may give us a certain knowledge of future events4 ,but nothing is more unquestionable than that finite experience can never give us certain knowledge of the future1 so that either a cause is not an invariable antecedent1 or else we can never gain certain knowledge of causes4 ,the first horn of this dilemma is hardly to be accepted4 ,doubtless there is in nature some invariably acting mechanism1 such that from certain fixed conditions an invariable result always emerges4 ,but we1 with our finite minds and short experience1 can never penetrate the mystery of those existences which embody the ,will of the ,creator1 and evolve it throughout time4 ,we are in the position of spectators who witness the productions of a complicated machine1 but are not allowed to examine its intimate structure4 ,we learn what does happen and what does appear1 but if we ask for the reason1 the answer would involve an infinite depth of mystery4 ,the simplest bit of matter1 or the most trivial incident1 such as the stroke of two billiard balls1 offers infinitely more to learn than ever the human intellect can fathom4 ,the word cause covers just as much untold meaning as any of the words 99substance991 99matter991 99thought991 99existence994 99,confusion of ,two ,questions499 ,the subject is much complicated1 too1 by the confusion of two distinct questions4 ,an event having happened1 we may ask-- 7#a7 ,is there any cause for the event8 7#b7 ,of what kind is that cause8 ,no one would assert that the mind possesses any faculty capable of inferring1 prior to experience1 that the occurrence of a sudden noise with flame and smoke indicates the combustion of a black powder1 formed by the mixture of black1 white1 and yellow powders4 ,the greatest upholder of 99 priori99 doctrines will allow that the particular aspect1 shape1 size1 colour1 texture1 and other qualities of a cause must be gathered through the senses4 ,the question whether there is any cause at all for an event1 is of a totally different kind4 ,if an explosion could happen without any prior existing conditions1 it must be a new creation--a distinct addition to the universe4 ,it may be plausibly held that we can imagine neither the creation nor annihilation of anything4 ,as regards matter1 this has long been held true2 as regards force1 it is now almost universally assumed as an axiom that energy can neither come into nor go out of existence without distinct acts of ,creative ,will4 ,that there exists any instinctive belief to this effect1 indeed1 seems doubtful4 ,we find ,lucretius1 a philosopher of the utmost intellectual power and cultivation1 gravely assuming that his raining atoms could turn aside from their straight paths in a self-determining manner1 and by this spontaneous origination of energy determine the form of the universe4#abi ,sir ,george ,airy1 too1 seriously discussed the mathematical conditions under which a perpetual motion1 that is1 a perpetual source of self-created energy1 might exist4#acj ,the larger part of the philosophic world has long held that in mental acts there is free will--in short1 self-causation4 ,it is in vain to attempt to reconcile this doctrine with that of an intuitive belief in causation1 as ,sir ,w4 ,hamilton candidly allowed4 #abi 99,de ,rerum ,natura991 bk4 ii4 ll4 #baf-bic4 #acj 99,cambridge ,philosophical ,transactions99 7#ahcj71 vol4 iii4 pp4 #cfi-cgb4 ,it is obvious1 moreover1 that to assert the existence of a cause for every event cannot do more than remove into the indefinite past the inconceivable fact and mystery of creation4 ,at any given moment matter and energy were equal to what they are at present1 or they were not2 if equal1 we may make the same inquiry concerning any other moment1 however long prior1 and we are thus obliged to accept one horn of the dilemma--existence from infinity1 or creation at some moment4 ,this is but one of the many cases in which we are compelled to believe in one or other of two alternatives1 both inconceivable4 ,my present purpose1 however1 is to point out that we must not confuse this supremely difficult question with that into which inductive science inquires on the foundation of facts4 ,by induction we gain no certain knowledge2 but by observation1 and the inverse use of deductive reasoning1 we estimate the probability that an event which has occurred was preceded by conditions of specified character1 or that such conditions will be followed by the event4 99,definition of the ,term ,cause499 ,clear definitions of the word cause have been given by several philosophers4 ,hobbes has said1 8,a cause is the sum or aggregate of all such accidents1 both in the agents and the patients1 as concur in the producing of the effect propounded2 all which existing together1 it cannot be understood but that the effect existeth with them2 or that it can possibly exist if any of them be absent40 ,brown1 in his 99,essay on ,causation991 gave a nearly corresponding statement4 8,a cause10 he says1#aca 8may be defined to be the object or event which immediately precedes any change1 and which existing again in similar circumstances will be always immediately followed by a similar change40 ,of the kindred word 99power991 he likewise says3#acb 8,power is nothing more than that invariableness of antecedence which is implied in the belief of causation40 #aca 99,observations on the ,nature and ,tendency of the ,doctrine of ,mr4 ,hume1 concerning the ,relation of ,cause and ,effect499 ,second ed4 p4 #dd4 #acb ,ibid4 p4 #ig4 ,these definitions may be accepted with the qualification that our knowledge of causes in such a sense can be probable only4 ,the work of science consists in ascertaining the combinations in which phenomena present themselves4 ,concerning every event we shall have to determine its probable conditions1 or the group of antecedents from which it probably follows4 ,an antecedent is anything which exists prior to an event2 a consequent is anything which exists subsequently to an antecedent4 ,it will not usually happen that there is any probable connection between an antecedent and consequent4 ,thus nitrogen is an antecedent to the lighting of a common fire2 but it is so far from being a cause of the lighting1 that it renders the combustion less active4 ,daylight is an antecedent to all fires lighted during the day1 but it probably has no appreciable effect upon their burning4 ,but in the case of any given event it is usually possible to discover a certain number of antecedents which seem to be always present1 and with more or less probability we conclude that when they exist the event will follow4 ,let it be observed that the utmost latitude is at present enjoyed in the use of the term 99cause994 ,not only may a cause be an existent thing endowed with powers1 as oxygen is the cause of combustion1 gunpowder the cause of explosion1 but the very absence or removal of a thing may also be a cause4 ,it is quite correct to speak of the dryness of the ,egyptian atmosphere1 or the absence of moisture1 as being the cause of the preservation of mummies1 and other remains of antiquity4 ,the cause of a mountain elevation1 ,ingleborough for instance1 is the excavation of the surrounding valleys by denudation4 ,it is not so usual to speak of the existence of a thing at one moment as the cause of its existence at the next1 but to me it seems the commonest case of causation which can occur4 ,the cause of motion of a billiard ball may be the stroke of another ball2 and recent philosophy leads us to look upon all motions and changes1 as but so many manifestations of prior existing energy4 ,in all probability there is no creation of energy and no destruction1 so that as regards both mechanical and molecular changes1 the cause is really the manifestation of existing energy4 ,in the same way ,i see not why the prior existence of matter is not also a cause as regards its subsequent existence4 ,all science tends to show us that the existence of the universe in a particular state at one moment1 is the condition of its existence at the next moment1 in an apparently different state4 ,when we analyse the meaning which we can attribute to the word 99cause991 it amounts to the existence of suitable portions of matter endowed with suitable quantities of energy4 ,if we may accept ,horne ,tooke's assertion1 99cause99 has etymologically the meaning of 99thing before994 ,though1 indeed1 the origin of the word is very obscure1 its derivatives1 the ,italian 99cosa991 and the ,french 99chose991 mean simply 99thing994 ,in the ,german equivalent 99ursache991 we have plainly the original meaning of 99thing before991 the 99sache99 denoting 8interesting or important object10 the ,english 99sake991 and 99ur99 being the equivalent of the ,english 99ere991 99before994 ,we abandon1 then1 both etymology and philosophy1 when we attribute to the 99laws of causation99 any meaning beyond that of the 99conditions99 under which an event may be expected to happen1 according to our observation of the previous course of nature4 ,i have no objection to use the words cause and causation1 provided they are never allowed to lead us to imagine that our knowledge of nature can attain to certainty4 ,i repeat that if a cause is an invariable and necessary condition of an event1 we can never know certainly whether the cause exists or not4 ,to us1 then1 a cause is not to be distinguished from the group of positive or negative conditions which1 with more or less probability1 precede an event4 ,in this sense1 there is no particular difference between knowledge of causes and our general knowledge of the succession of combinations1 in which the phenomena of nature are presented to us1 or found to occur in experimental inquiry4 99,distinction of ,inductive and ,deductive ,results499 ,we must carefully avoid confusing together inductive investigations which terminate in the establishment of general laws1 and those which seem to lead directly to the knowledge of future particular events4 ,that process only can be called induction which gives general laws1 and it is by the subsequent employment of deduction that we anticipate particular events4 ,if the observation of a number of cases shows that alloys of metals fuse at lower temperatures than their constituent metals1 ,i may with more or less probability draw a general inference to that effect1 and may thence deductively ascertain the probability that the next alloy examined will fuse at a lower temperature than its constituents4 ,it has been asserted1 indeed1 by ,mill1#acc and partially admitted by ,mr4 ,fowler1#acd that we can argue directly from case to case1 so that what is true of some alloys will be true of the next4 ,professor ,bain has adopted the same view of reasoning4 ,he thinks that ,mill has extricated us from the dead lock of the syllogism and effected a total revolution in logic4 ,he holds that reasoning from particulars to particulars is not only the usual1 the most obvious and the most ready method1 but that it is the type of reasoning which best discloses the real process4#ace ,doubtless1 this is the usual result of our reasoning1 regard being had to degrees of probability2 but these logicians fail entirely to give any explanation of the process by which we get from case to case4 #acc 99,system of ,logic991 bk4 ,,ii4 chap1 iii4 #acd 99,inductive ,logic991 pp4 #ac1 #ad4 #ace ,bain1 99,deductive ,logic991 pp4 #bjh1 #bji4 ,it may be allowed that the knowledge of future particular events is the main purpose of our investigations1 and if there were any process of thought by which we could pass directly from event to event without ascending into general truths1 this method would be sufficient1 and certainly the briefest4 ,it is true1 also1 that the laws of mental association lead the mind always to expect the like again in apparently like circumstances1 and even animals of very low intelligence must have some trace of such powers of association1 serving to guide them more or less correctly1 in the absence of true reasoning faculties4 ,but it is the purpose of logic1 according to ,mill1 to ascertain whether inferences have been correctly drawn1 rather than to discover them4#acf ,even if we can1 then1 by habit1 association1 or any rude process of inference1 infer the future directly from the past1 it is the work of logic to analyse the conditions on which the correctness of this inference depends4 ,even ,mill would admit that such analysis involves the consideration of general truths1#acg and in this1 as in several other important points1 we might controvert ,mill's own views by his own statements4 ,it seems to me undesirable in a systematic work like this to enter into controversy at any length1 or to attempt to refute the views of other logicians4 ,but ,i shall feel bound to state1 in a separate publication1 my very deliberate opinion that many of ,mill's innovations in logical science1 and especially his doctrine of reasoning from particulars to particulars1 are entirely groundless and false4 #acf 99,system of ,logic499 ,introduction1 #d4 ,fifth ed4 pp4 #h1 #i4 #acg ,ibid4 bk4 ,,ii4 chap4 iii4 #e1 pp4 #bbe1 c4 99,the ,grounds of ,inductive ,inference499 ,i hold that in all cases of inductive inference we must invent hypotheses1 until we fall upon some hypothesis which yields deductive results in accordance with experience4 ,such accordance renders the chosen hypothesis more or less probable1 and we may then deduce1 with some degree of likelihood1 the nature of our future experience1 on the assumption that no arbitrary change takes place in the conditions of nature4 ,we can only argue from the past to the future1 on the general principle set forth in this work1 that what is true of a thing will be true of the like4 ,so far then as one object or event differs from another1 all inference is impossible1 particulars as particulars can no more make an inference than grains of sand can make a rope4 ,we must always rise to something which is general or same in the cases1 and assuming that sameness to be extended to new cases we learn their nature4 ,hearing a clock tick five thousand times without exception or variation1 we adopt the very probable hypothesis that there is some invariably acting machine which produces those uniform sounds1 and which will1 in the absence of change1 go on producing them4 ,meeting twenty times with a bright yellow ductile substance1 and finding it always to be very heavy and incorrodible1 ,i infer that there was some natural condition which tended in the creation of things to associate these properties together1 and ,i expect to find them associated in the next instance4 ,but there always is the possibility that some unknown change may take place between past and future cases4 ,the clock may run down1 or be subject to a hundred accidents altering its condition4 ,there is no reason in the nature of things1 so far as known to us1 why yellow colour1 ductility1 high specific gravity1 and incorrodibility1 should always be associated together1 and in other cases1 if not in this1 men's expectations have been deceived4 ,our inferences1 therefore1 always retain more or less of a hypothetical character1 and are so far open to doubt4 ,only in proportion as our induction approximates to the character of perfect induction1 does it approximate to certainty4 ,the amount of uncertainty corresponds to the probability that other objects than those examined may exist and falsity our inferences2 the amount of probability corresponds to the amount of information yielded by our examination2 and the theory of probability will be needed to prevent us from over-estimating or under-estimating the knowledge we possess4 99,illustrations of the ,inductive ,process499 ,to illustrate the passage from the known to the apparently unknown1 let us suppose that the phenomena under investigation consist of numbers1 and that the following six numbers being exhibited to us1 we are required to infer the character of the next in the series3-- #e1 #ae1 #ce1 #de1 #fe1 #ie4 ,the question first of all arises1 ,how may we describe this series of numbers8 ,what is uniformly true of them8 ,the reader cannot fail to perceive at the first glance that they all end in five1 and the problem is1 from the properties of these six numbers1 to infer the properties of the next number ending in five4 ,if we test their properties by the process of perfect induction1 we soon perceive that they have another common property1 namely that of being 99divisible by five without remainder994 ,may we then assert that the next number ending in five is also divisible by five1 and1 if so1 upon what grounds8 ,or extending the question1 ,is every number ending in five divisible by five8 ,does it follow that because six numbers obey a supposed law1 therefore #cgf1fhe1ige or any other number1 however large1 obeys the law8 ,i answer 99certainly not994 ,the law in question is undoubtedly true2 but its truth is not proved by any finite number of examples4 ,all that these six numbers can do is to suggest to my mind the possible existence of such a law2 and ,i then ascertain its truth1 by proving deductively from the rules of decimal numeration1 that any number ending in five must be made up of multiples of five1 and must therefore be itself a multiple4 ,to make this more plain1 let the reader now examine the numbers-- #g1 #ag1 #cg1 #dg1 #fg1 #ig4 ,they all end in #g instead of #e1 and though not at equal intervals1 the intervals are the same as in the previous case4 ,after consideration1 the reader will perceive that these numbers all agree in being 99prime numbers991 or multiples of unity only4 ,may we then infer that the next1 or any other number ending in #g1 is a prime number8 ,clearly not1 for on trial we find that #bg1 #eg1 #aag are not primes4 ,six instances1 then1 treated empirically1 lead us to a true and universal law in one case1 and mislead us in another case4 ,we ought1 in fact1 to have no confidence in any law until we have treated it deductively1 and have shown that from the conditions supposed the results expected must ensue4 ,no one can show from the principles of number1 that numbers ending in #g should be primes4 ,from the history of the theory of numbers some good examples of false induction can be adduced4 ,taking the following series of prime numbers1 #da1 #dc1 #dg1 #ec1 #fa1 #ga1 #hc1 #ig1 #aac1 #aca1 #aea1 c41 it will be found that they all agree in being values of the general expression 99x99#b 99x99 #da1 putting for 99x99 in succession the values1 #j1 #a1 #b1 #c1 #d1 c4 ,we seem always to obtain a prime number1 and the induction is apparently strong1 to the effect that this expression always will give primes4 ,yet a few more trials disprove this false conclusion4 ,put 99x99 #dj1 and we obtain #dj #dj #dj #da1 or #da #da4 ,such a failure could never have happened1 had we shown any deductive reason why 99x99#b 99x99 #da should give primes4 ,there can be no doubt that what here happens with forty instances1 might happen with forty thousand or forty million instances4 ,an apparent law never once failing up to a certain point may then suddenly break down1 so that inductive reasoning1 as it has been described by some writers1 can give no sure knowledge of what is to come4 ,babbage pointed out1 in his ,ninth ,bridgewater ,treatise1 that a machine could be constructed to give a perfectly regular series of numbers through a vast series of steps1 and yet to break the law of progression suddenly at any required point4 ,no number of particular cases as particulars enables us to pass by inference to any new case4 ,it is hardly needful to inquire here what can be inferred from an infinite series of facts1 because they are never practically within our power2 but we may unhesitatingly accept the conclusion1 that no finite number of instances can ever prove a general law1 or can give us certain knowledge of even one other instance4 ,general mathematical theorems have indeed been discovered by the observation of particular cases1 and may again be so discovered4 ,we have ,newton's own statement1 to the effect that he was thus led to the all-important ,binomial ,theorem1 the basis of the whole structure of mathematical analysis4 ,speaking of a certain series of terms1 expressing the area of a circle or hyperbola1 he says3 8,i reflected that the denominators were in arithmetical progression2 so that only the numerical co-efficients of the numerators remained to be investigated4 ,but these1 in the alternate areas1 were the figures of the powers of the number eleven1 namely #aaj1 #aaa1 #aab1 #aac1 #aad2 that is1 in the first #a2 in the second #a1 #a2 in the third #a1 #b1 #a2 in the fourth #a1 #c1 #c1 #a2 in the fifth #a1 #d1 #f1 #d1 #a4ach ,i inquired1 therefore1 in what manner all the remaining figures could be found from the first two2 and ,i found that if the first figure be called 99m991 all the rest could be found by the continual multiplication of the terms of the formula 7799m99 - #j7_/a7 7799m99 - #a7_/b7 7799m99 - #b7_/c7 7799m99 - #c7_/d7 c40#aci #ach ,these are the figurate numbers considered in pages #ahc1 #ahg1 c4 #aci 99,commercium ,epistolicum499 99,epistola ad ,oldenburgum991 ,oct4 #bd1 #afgf4 ,horsley's 99,works of ,newton991 vol4 iv4 p4 #eda4 ,see ,de ,morgan in 99,penny ,cyclopdia991 art4 8,binomial ,theorem10 p4 #dab4 ,it is pretty evident1 from this most interesting statement1 that ,newton1 having simply observed the succession of the numbers1 tried various formul until he found one which agreed with them all4 ,he was so little satisfied with this process1 however1 that he verified particular results of his new theorem by comparison with the results of common multiplication1 and the rule for the extraction of the square root4 ,newton1 in fact1 gave no demonstration of his theorem2 and the greatest mathematicians of the last century1 ,james ,bernoulli1 ,maclaurin1 ,landen1 ,euler1 ,lagrange1 c41 occupied themselves with discovering a conclusive method of deductive proof4 ,there can be no doubt that in geometry also discoveries have been suggested by direct observation4 ,many of the now trivial propositions of ,euclid's ,elements were probably thus discovered1 by the ancient ,greek geometers2 and we have pretty clear evidence of this in the ,commentaries of ,proclus4#adj ,galileo was the first to examine the remarkable properties of the cycloid1 the curve described by a point in the circumference of a wheel rolling on a plane4 ,by direct observation he ascertained that the area of the curve is apparently three times that of the generating circle or wheel1 but he was unable to prove this exactly1 or to verify it by strict geometrical reasoning4 ,sir ,george ,airy has recorded a curious case1 in which he fell accidentally by trial on a new geometrical property of the sphere4#ada ,but discovery in such cases means nothing more than suggestion1 and it is always by pure deduction that the general law is really established4 ,as ,proclus puts it1 99we must pass from sense to consideration994 #adj ,bk4 ii4 chap4 iv4 #ada 99,philosophical ,transactions99 7#ahff71 vol4 #adf1 p4 #ccd4 ,illustration ,given1 for instance1 the series of figures in the accompanying diagram1 measurement will show that the curved lines approximate to semicircles1 and the rectilinear figures to right-angled triangles4 ,these figures may seem to suggest to the mind the general law that angles inscribed in semicircles are right angles2 but no number of instances1 and no possible accuracy of measurement would really establish the truth of that general law4 ,availing ourselves of the suggestion furnished by the figures1 we can only investigate deductively the consequences which flow from the definition of a circle1 until we discover among them the property of containing right angles4 ,persons have thought that they had discovered a method of trisecting angles by plane geometrical construction1 because a certain complex arrangement of lines and circles had appeared to trisect an angle in every case tried by them1 and they inferred1 by a supposed act of induction1 that it would succeed in all other cases4 ,de ,morgan has recorded a proposed mode of trisecting the angle which could not be discriminated by the senses from a true general solution1 except when it was applied to very obtuse angles4#adb ,in all such cases1 it has always turned out either that the angle was not trisected at all1 or that only certain particular angles could be thus trisected4 ,the trisectors were misled by some apparent or special coincidence1 and only deductive proof could establish the truth and generality of the result4 ,in this particular case1 deductive proof shows that the problem attempted is impossible1 and that angles generally cannot be trisected by common geometrical methods4 #adb 99,budget of ,paradoxes991 p4 #beg4 99,geometrical ,reasoning499 ,this view of the matter is strongly supported by the further consideration of geometrical reasoning4 ,no skill and care could ever enable us to verify absolutely any one geometrical proposition4 ,rousseau1 in his 99,emile991 tells us that we should teach a child geometry by causing him to measure and compare figures by superposition4 ,while a child was yet incapable of general reasoning1 this would doubtless be an instructive exercise2 but it never could teach geometry1 nor prove the truth of any one proposition4 ,all our figures are rude approximations1 and they may happen to seem unequal when they should be equal1 and equal when they should be unequal4 ,moreover figures may from chance be equal in case after case1 and yet there may be no general reason why they should be so4 ,the results of deductive geometrical reasoning are absolutely certain1 and are either exactly true or capable of being carried to any required degree of approximation4 ,in a perfect triangle1 the angles must be equal to one half-revolution precisely2 even an infinitesimal divergence would be impossible2 and ,i believe with equal confidence1 that however many are the angles of a figure1 provided there are no re-entrant angles1 the sum of the angles will be precisely and absolutely equal to twice as many right-angles as the figure has sides1 less by four right-angles4 ,in such cases1 the deductive proof is absolute and complete2 empirical verification can at the most guard against accidental oversights4 ,there is a second class of geometrical truths which can only be proved by approximation2 but1 as the mind sees no reason why that approximation should not always go on1 we arrive at complete conviction4 ,we thus learn that the surface of a sphere is equal exactly to two-thirds of the whole surface of the circumscribing cylinder1 or to four times the area of the generating circle4 ,the area of a parabola is exactly two-thirds of that of the circumscribing parallelogram4 ,the area of the cycloid is exactly three times that of the generating circle4 ,these are truths that we could never ascertain1 nor even verify by observation2 for any finite amount of difference1 less than what the senses can discern1 would falsify them4 ,there are geometrical relations again which we cannot assign exactly1 but can carry to any desirable degree of approximation4 ,the ratio of the circumference to the diameter of a circle is that of #cadaeibfecehigicbchdf4444 to #a1 and the approximation may be carried to any extent by the expenditure of sufficient labour4 ,mr4 ,w4 ,shanks has given the value of this natural constant1 known as 1 to the extent of #gjg places of decimals4#adc ,some years since1 ,i amused myself by trying how near ,i could get to this ratio1 by the careful use of compasses1 and ,i did not come nearer than #a part in #edj4 ,we might imagine measurements so accurately executed as to give us eight or ten places correctly4 ,but the power of the hands and senses must soon stop1 whereas the mental powers of deductive reasoning can proceed to an unlimited degree of approximation4 ,geometrical truths1 then1 are incapable of verification2 and1 if so1 they cannot even be learnt by observation4 ,how can ,i have learnt by observation a proposition of which ,i cannot even prove the truth by observation1 when ,i am in possession of it8 ,all that observation or empirical trial can do is to suggest propositions1 of which the truth may afterwards be proved deductively4 #adc 99,proceedings of the ,royal ,society99 7#ahgb-c71 vol4 xxi4 p4 #cai4 ,if ,viviani's story is to be believed1 ,galileo endeavoured to satisfy himself about the area of the cycloid by cutting out several large cycloids in pasteboard1 and then comparing the areas of the curve and the generating circle by weighing them4 ,in every trial the curve seemed to be rather less than three times the circle1 so that ,galileo1 we are told1 began to suspect that the ratio was not precisely #c to #a4 ,it is quite clear1 however1 that no process of weighing or measuring could ever prove truths like these1 and it remained for ,torricelli to show what his master ,galileo had only guessed at4#add #add 99,life of ,galileo991 ,society for the ,diffusion of ,useful ,knowledge1 p4 #ajb4 ,much has been said about the peculiar certainty of mathematical reasoning1 but it is only certainty of deductive reasoning1 and equal certainty attaches to all correct logical deduction4 ,if a triangle be right-angled1 the square on the hypothenuse will undoubtedly equal the sum of the two squares on the other sides2 but ,i can never be sure that a triangle is right-angled3 so ,i can be certain that nitric acid will not dissolve gold1 provided ,i know that the substances employed really correspond to those on which ,i tried the experiment previously4 ,here is like certainty of inference1 and like doubt as to the facts4 99,discrimination of ,certainty and ,probability499 ,we can never recur too often to the truth that our knowledge of the laws and future events of the external world is only probable4 ,the mind itself is quite capable of possessing certain knowledge1 and it is well to discriminate carefully between what we can and cannot know with certainty4 ,in the first place1 whatever feeling is actually present to the mind is certainly known to that mind4 ,if ,i see blue sky1 ,i may be quite sure that ,i do experience the sensation of blueness4 ,whatever ,i do feel1 ,i do feel beyond all doubt4 ,we are indeed very likely to confuse what we really feel with what we are inclined to associate with it1 or infer inductively from it2 but the whole of our consciousness1 as far as it is the result of pure intuition and free from inference1 is certain knowledge beyond all doubt4 ,in the second place1 we may have certainty of inference2 the fundamental laws of thought1 and the rule of substitution 7p4 #i71 are certainly true2 and if my senses could inform me that ,a was indistinguishable in colour from ,b1 and ,b from ,c1 then ,i should be equally certain that ,a was indistinguishable from ,c4 ,in short1 whatever truth there is in the premises1 ,i can certainly embody in their correct logical result4 ,but the certainty generally assumes a hypothetical character4 ,i never can be quite sure that two colours are exactly alike1 that two magnitudes are exactly equal1 or that two bodies whatsoever are identical even in their apparent qualities4 ,almost all our judgments involve quantitative relations1 and1 as will be shown in succeeding chapters1 we can never attain exactness and certainty where continuous quantity enters4 ,judgments concerning discontinuous quantity or numbers1 however1 allow of certainty2 ,i may establish beyond doubt1 for instance1 that the difference of the squares of #ag and #ac is the product of #ag #ac and #ag - #ac1 and is therefore #cj #d1 or #abj4 ,inferences which we draw concerning natural objects are never certain except in a hypothetical point of view4 ,it might seem to be certain that iron is magnetic1 or that gold is incapable of solution in nitric acid2 but1 if we carefully investigate the meanings of these statements1 they will be found to involve no certainty but that of consciousness and that of hypothetical inference4 ,for what do ,i mean by iron or gold8 ,if ,i choose a remarkable piece of yellow substance1 call it gold1 and then immerse it in a liquid which ,i call nitric acid1 and find that there is no change called solution1 then consciousness has certainly informed me that1 with my meaning of the terms1 8,gold is insoluble in nitric acid40 ,i may further be certain of something else2 for if this gold and nitric acid remain what they were1 ,i may be sure there will be no solution on again trying the experiment4 ,if ,i take other portions of gold and nitric acid1 and am sure that they really are identical in properties with the former portions1 ,i can be certain that there will be no solution4 ,but at this point my knowledge becomes purely hypothetical2 for how can ,i be sure without trial that the gold and acid are really identical in nature with what ,i formerly called gold and nitric acid4 ,how do ,i know gold when ,i see it8 ,if ,i judge by the apparent qualities--colour1 ductility1 specific gravity1 c41 ,i may be misled1 because there may always exist a substance which to the colour1 ductility1 specific gravity1 and other specified qualities1 joins others which we do not expect4 ,similarly1 if iron is magnetic1 as shown by an experiment with objects answering to those names1 then all iron is magnetic1 meaning all pieces of matter identical with my assumed piece4 ,but in trying to identify iron1 ,i am always open to mistake4 ,nor is this liability to mistake a matter of speculation only4#ade #ade ,professor ,bowen has excellently stated this view4 99,treatise on ,logic499 ,cambridge1 ,u4,s4,a41 #ahff1 p4 #ced4 ,the history of chemistry shows that the most confident inferences may have been falsified by the confusion of one substance with another4 ,thus strontia was never discriminated from baryta until ,klaproth and ,hay detected differences between some of their properties4 ,accordingly chemists must often have inferred concerning strontia what was only true of baryta1 and 99vice vers994 ,there is now no doubt that the recently discovered substances1 csium and rubidium1 were long mistaken for potassium4#adf ,other elements have often been confused together--for instance1 tantalum and niobium2 sulphur and selenium2 cerium1 lanthanum1 and didymium2 yttrium and erbium4 #adf ,roscoe's 99,spectrum ,analysis991 #ast edit41 p4 #ih4 ,even the best known laws of physical science do not exclude false inference4 ,no law of nature has been better established than that of universal gravitation1 and we believe with the utmost confidence that any body capable of affecting the senses will attract other bodies1 and fall to the earth if not prevented4 ,euler remarks that1 although he had never made trial of the stones which compose the church of ,magdeburg1 yet he had not the least doubt that all of them were heavy1 and would fall if unsupported4 ,but he adds1 that it would be extremely difficult to give any satisfactory explanation of this confident belief4#adg ,the fact is1 that the belief ought not to amount to certainty until the experiment has been tried1 and in the meantime a slight amount of uncertainty enters1 because we cannot be sure that the stones of the ,magdeburg ,church resemble other stones in all their properties4 #adg ,euler's 99,letters to a ,german ,princess991 translated by ,hunter4 #bnd ed41 vol4 ii4 pp4 #ag1 #ah4 ,in like manner1 not one of the inductive truths which men have established1 or think they have established1 is really safe from exception or reversal4 ,lavoisier1 when laying the foundations of chemistry1 met with so many instances tending to show the existence of oxygen in all acids1 that he adopted a general conclusion to that effect1 and devised the name oxygen accordingly4 ,he entertained no appreciable doubt that the acid existing in sea salt also contained oxygen2#adh yet subsequent experience falsified his expectations4 ,this instance refers to a science in its infancy1 speaking relatively to the possible achievements of men4 ,but all sciences are and ever will remain in their infancy1 relatively to the extent and complexity of the universe which they undertake to investigate4 ,euler expresses no more than the truth when he says that it would be impossible to fix on any one thing really existing1 of which we could have so perfect a knowledge as to put us beyond the reach of mistake4#adi ,we may be quite certain that a comet will go on moving in a similar path 99if99 all circumstances remain the same as before2 but if we leave out this extensive qualification1 our predictions will always be subject to the chance of falsification by some unexpected event1 such as the division of ,biela's comet or the interference of an unknown gravitating body4 #adh ,lavoisier's 99,chemistry991 translated by ,kerr4 #crd ed41 pp4 #aad1 #aba1 #abc4 #adi ,euler's 99,letters991 vol4 ii4 p4 #ba4 ,inductive inference might attain to certainty if our knowledge of the agents existing throughout the universe were complete1 and if we were at the same time certain that the same ,power which created the universe would allow it to proceed without arbitrary change4 ,there is always a possibility of causes being in existence without our knowledge1 and these may at any moment produce an unexpected effect4 ,even when by the theory of probabilities we succeed in forming some notion of the comparative confidence with which we should receive inductive results1 it yet appears to me that we must make an assumption4 ,events come out like balls from the vast ballot-box of nature1 and close observation will enable us to form some notion1 as we shall see in the next chapter1 of the contents of that ballot-box4 ,but we must still assume that1 between the time of an observation and that to which our inferences relate1 no change in the ballot-box has been made4 ,,chapter ,,xii4 ,,the ,,inductive ,,or ,,inverse ,,application ,,of ,,the ,,theory ,,of ,,probability4 ,we have hitherto considered the theory of probability only in its simple deductive employment1 in which it enables us to determine from given conditions the probable character of events happening under those conditions4 ,but as deductive reasoning when inversely applied constitutes the process of induction1 so the calculation of probabilities may be inversely applied2 from the known character of certain events we may argue backwards to the probability of a certain law or condition governing those events4 ,having satisfactorily accomplished this work1 we may indeed calculate forwards to the probable character of future events happening under the same conditions2 but this part of the process is a direct use of deductive reasoning 7p4 #bbf74 ,now it is highly instructive to find that whether the theory of probability be deductively or inductively applied1 the calculation is always performed according to the principles and rules of deduction4 ,the probability that an event has a particular condition entirely depends upon the probability that if the condition existed the event would follow4 ,if we take up a pack of common playing cards1 and observe that they are arranged in perfect numerical order1 we conclude beyond all reasonable doubt that they have been thus intentionally arranged by some person acquainted with the usual order of sequence4 ,this conclusion is quite irresistible1 and rightly so2 for there are but two suppositions which we can make as to the reason of the cards being in that particular order3-- #a4 ,they may have been intentionally arranged by some one who would probably prefer the numerical order4 #b4 ,they may have fallen into that order by chance1 that is1 by some series of conditions which1 being unknown to us1 cannot be known to lead by preference to the particular order in question4 ,the latter supposition is by no means absurd1 for any one order is as likely as any other when there is no preponderating tendency4 ,but we can readily calculate by the doctrine of permutations the probability that fifty-two objects would fall by chance into any one particular order4 ,fifty-two objects can be arranged in #eb #ea 444 #c #b #a or about #hjff 7#aj7fd possible orders1 the number obtained requiring #fh places of figures for its full expression4 ,hence it is excessively unlikely that anyone should ever meet with a pack of cards arranged in perfect order by accident4 ,if we do meet with a pack so arranged1 we inevitably adopt the other supposition1 that some person1 having reasons for preferring that special order1 has thus put them together4 ,we know that of the immense number of possible orders the numerical order is the most remarkable2 it is useful as proving the perfect constitution of the pack1 and it is the intentional result of certain games4 ,at any rate1 the probability that intention should produce that order is incomparably greater than the probability that chance should produce it2 and as a certain pack exists in that order1 we rightly prefer the supposition which most probably leads to the observed result4 ,by a similar mode of reasoning we every day arrive1 and validly arrive1 at conclusions approximating to certainty4 ,whenever we observe a perfect resemblance between two objects1 as1 for instance1 two printed pages1 two engravings1 two coins1 two foot-prints1 we are warranted in asserting that they proceed from the same type1 the same plate1 the same pair of dies1 or the same boot4 ,and why8 ,because it is almost impossible that with different types1 plates1 dies1 or boots some apparent distinction of form should not be produced4 ,it is impossible for the hand of the most skilful artist to make two objects alike1 so that mechanical repetition is the only probable explanation of exact similarity4 ,we can often establish with extreme probability that one document is copied from another4 ,suppose that each document contains #aj1jjj words1 and that the same word is incorrectly spelt in each4 ,there is then a probability of less than #a in #aj1jjj that the same mistake should be made in each4 ,if we meet with a second error occurring in each document1 the probability is less than #a in #aj1jjj #iiii1 that two such coincidences should occur by chance1 and the numbers grow with extreme rapidity for more numerous coincidences4 ,we cannot make any precise calculations without taking into account the character of the errors committed1 concerning the conditions of which we have no accurate means of estimating probabilities4 ,nevertheless1 abundant evidence may thus be obtained as to the derivation of documents from each other4 ,in the examination of many sets of logarithmic tables1 six remarkable errors were found to be present in all but two1 and it was proved that tables printed at ,paris1 ,berlin1 ,florence1 ,avignon1 and even in ,china1 besides thirteen sets printed in ,england between the years #afcc and #ahbb1 were derived directly or indirectly from some common source4#aej ,with a certain amount of labour1 it is possible to establish beyond reasonable doubt the relationship or genealogy of any number of copies of one document1 proceeding possibly from parent copies now lost4 ,the relations between the manuscripts of the ,new ,testament have been elaborately investigated in this manner1 and the same work has been performed for many classical writings1 especially by ,german scholars4 #aej ,lardner1 99,edinburgh ,review991 ,july #ahcd1 p4 #bgg4 99,principle of the ,inverse ,method499 ,the inverse application of the rules of probability entirely depends upon a proposition which may be thus stated1 nearly in the words of ,laplace4#aea 99,if an event can be produced by any one of a certain number of different causes1 all equally probable priori1 the probabilities of the existence of these causes as inferred from the event1 are proportional to the probabilities of the event as derived from these causes499 ,in other words1 the most probable cause of an event which has happened is that which would most probably lead to the event supposing the cause to exist2 but all other possible causes are also to be taken into account with probabilities proportional to the probability that the event would happen if the cause existed4 ,suppose1 to fix our ideas clearly1 that ,e is the event1 and ,c#a ,c#b ,c#c are the three only conceivable causes4 ,if ,c#a exist1 the probability is 99p99#a that ,e would follow2 if ,c#b or ,c#c exist1 the like probabilities are respectively 99p99#b and 99p99#c4 ,then as 99p99#a is to 99p99#b1 so is the probability of ,c#a being the actual cause to the probability of ,c#b being it2 and1 similarly1 as 99p99#b is to 99p99#c1 so is the probability of ,c#b being the actual cause to the probability of ,c#c being it4 ,by a simple mathematical process we arrive at the conclusion that the actual probability of ,c#a being the cause is 99p99#a_/799p99#a 99p99#b 99p99#c72 #aea 99,mmoires par divers ,savans991 tom4 vi42 quoted by ,todhunter in his 99,history of the ,theory of ,probability991 p4 #deh4 and the similar probabilities of the existence of ,c#b and ,c#c are1 99p99#b_/799p99#a 99p99#b 99p99#c7 and 99p99#c_/799p99#a 99p99#b 99p99#c74 ,the sum of these three fractions amounts to unity1 which correctly expresses the certainty that one cause or other must be in operation4 ,we may thus state the result in general language4 99,if it is certain that one or other of the supposed causes exists1 the probability that any one does exist is the probability that if it exists the event happens1 divided by the sum of all the similar probabilities499 ,there may seem to be an intricacy in this subject which may prove distasteful to some readers2 but this intricacy is essential to the subject in hand4 ,no one can possibly understand the principles of inductive reasoning1 unless he will take the trouble to master the meaning of this rule1 by which we recede from an event to the probability of each of its possible causes4 ,this rule or principle of the indirect method is that which common sense leads us to adopt almost instinctively1 before we have any comprehension of the principle in its general form4 ,it is easy to see1 too1 that it is the rule which will1 out of a great multitude of cases1 lead us most often to the truth1 since the most probable cause of an event really means that cause which in the greatest number of cases produces the event4 ,donkin and ,boole have given demonstrations of this principle1 but the one most easy to comprehend is that of ,poisson4 ,he imagines each possible cause of an event to be represented by a distinct ballot-box1 containing black and white balls1 in such a ratio that the probability of a white ball being drawn is equal to that of the event happening4 ,he further supposes that each box1 as is possible1 contains the same total number of balls1 black and white2 then1 mixing all the contents of the boxes together1 he shows that if a white ball be drawn from the aggregate ballot-box thus formed1 the probability that it proceeded from any particular ballot-box is represented by the number of white balls in that particular box1 divided by the total number of white balls in all the boxes4 ,this result corresponds to that given by the principle in question4#aeb #aeb ,poisson1 99,recherches sur la ,probabilit des ,jugements991 ,paris1 #ahcg1 pp4 #hb1 #hc4 ,thus1 if there be three boxes1 each containing ten balls in all1 and respectively containing seven1 four1 and three white balls1 then on mixing all the balls together we have fourteen white ones2 and if we draw a white ball1 that is if the event happens1 the probability that it came out of the first box is #g_/ad2 which is exactly equal to 7#g_/aj7_/7g_/aj #d_/aj #c_/aj71 the fraction given by the rule of the ,inverse ,method4 99,simple ,applications of the ,inverse ,method499 ,in many cases of scientific induction we may apply the principle of the inverse method in a simple manner4 ,if only two1 or at the most a few hypotheses1 may be made as to the origin of certain phenomena1 we may sometimes easily calculate the respective probabilities4 ,it was thus that ,bunsen and ,kirchhoff established1 with a probability little short of certainty1 that iron exists in the sun4 ,on comparing the spectra of sunlight and of the light proceeding from the incandescent vapour of iron1 it became apparent that at least sixty bright lines in the spectrum of iron coincided with dark lines in the sun's spectrum4 ,such coincidences could never be observed with certainty1 because1 even if the lines only closely approached1 the instrumental imperfections of the spectroscope would make them apparently coincident1 and if one line came within half a millimetre of another1 on the map of the spectra1 they could not be pronounced distinct4 ,now the average distance of the solar lines on ,kirchhoff's map is #b mm41 and if we throw down a line1 as it were1 by pure chance on such a map1 the probability is about one-half that the new line will fall within #a_/b mm4 on one side or the other of some one of the solar lines4 ,to put it in another way1 we may suppose that each solar line1 either on account of its real breadth1 or the defects of the instrument1 possesses a breadth of #a_/b mm41 and that each line in the iron spectrum has a like breadth4 ,the probability then is just one-half that the centre of each iron line will come by chance within #a mm4 of the centre of a solar line1 so as to appear to coincide with it4 ,the probability of casual coincidence of each iron line with a solar line is in like manner #a_/b4 ,coincidence in the case of each of the sixty iron lines is a very unlikely event if it arises casually1 for it would have a probability of only 7#a_/b7fj or less than #a in a trillion4 ,the odds1 in short1 are more than a million million millions to unity against such casual coincidence4#aec ,but on the other hypothesis1 that iron exists in the sun1 it is highly probable that such coincidences would be observed2 it is immensely more probable that sixty coincidences would be observed if iron existed in the sun1 than that they should arise from chance4 ,hence by our principle it is immensely probable that iron does exist in the sun4 #aec ,kirchhoff's 99,researches on the ,solar ,spectrum994 ,first part1 translated by ,roscoe1 pp4 #ah1 #ai4 ,all the other interesting results1 given by the comparison of spectra1 rest upon the same principle of probability4 ,the almost complete coincidence between the spectra of solar1 lunar1 and planetary light renders it practically certain that the light is all of solar origin1 and is reflected from the surfaces of the moon and planets1 suffering only slight alteration from the atmospheres of some of the planets4 ,a fresh confirmation of the truth of the ,copernican theory is thus furnished4 ,herschel proved in this way the connection between the direction of the oblique faces of quartz crystals1 and the direction in which the same crystals rotate the plane of polarisation of light4 ,for if it is found in a second crystal that the relation is the same as in the first1 the probability of this happening by chance is #a_/b2 the probability that in another crystal also the direction will be the same is #a_/d1 and so on4 ,the probability that in 99n99 #a crystals there would be casual agreement of direction is the nth power of #a_/b4 ,thus1 if in examining fourteen crystals the same relation of the two phenomena is discovered in each1 the odds that it proceeds from uniform conditions are more than #hjjj to #a4aed ,since the first observations on this subject were made in #ahbj1 no exceptions have been observed1 so that the probability of invariable connection is incalculably great4 #aed 99,edinburgh ,review991 ,no4 #ahe1 vol4 xcii4 ,july #ahej1 p4 #cb2 ,herschel's 99,essays991 p4 #dba2 99,transactions of the ,cambridge ,philosophical ,society991 vol4 i4 p4 #dc4 ,it is exceedingly probable that the ancient ,egyptians had exactly recorded the eclipses occurring during long periods of time1 for ,diogenes ,laertius mentions that #cgc solar and #hcb lunar eclipses had been observed1 and the ratio between these numbers exactly expresses that which would hold true of the eclipses of any long period1 of say #abjj or #acjj years1 as estimated on astronomical grounds4 ,it is evident that an agreement between small numbers1 or customary numbers1 such as seven1 one hundred1 a myriad1 c41 is much more likely to happen from chance1 and therefore gives much less presumption of dependence4 ,if two ancient writers spoke of the sacrifice of oxen1 they would in all probability describe it as a hecatomb1 and there would be nothing remarkable in the coincidence4 ,but it is impossible to point out any special reason why an old writer should select such numbers as #cgc and #hcb1 unless they had been the results of observation4 ,on similar grounds1 we must inevitably believe in the human origin of the flint flakes so copiously discovered of late years4 ,for though the accidental stroke of one stone against another may often produce flakes1 such as are occasionally found on the sea-shore1 yet when several flakes are found in close company1 and each one bears evidence1 not of a single blow only1 but of several successive blows1 all conducing to form a symmetrical knife-like form1 the probability of a natural and accidental origin becomes incredibly small1 and the contrary supposition1 that they are the work of intelligent beings1 approximately certain4#aee #aee ,evans' 99,ancient ,stone ,implements of ,great ,britain994 ,london1 #ahgb 7,longmans74 99,the ,theory of ,probability in ,astronomy499 ,the science of astronomy1 occupied with the simple relations of distance1 magnitude1 and motion of the heavenly bodies1 admits more easily than almost any other science of interesting conclusions founded on the theory of probability4 ,more than a century ago1 in #agfg1 ,michell showed the extreme probability of bonds connecting together systems of stars4 ,he was struck by the unexpected number of fixed stars which have companions close to them4 ,such a conjunction might happen casually by one star1 although possibly at a great distance from the other1 happening to lie on a straight line passing near the earth4 ,but the probabilities are so greatly against such an optical union happening often in the expanse of the heavens1 that ,michell asserted the existence of some connection between most of the double stars4 ,it has since been estimated by ,struve1 that the odds are #iegj to #a against any two stars of not less than the seventh magnitude falling within the apparent distance of four seconds of each other by chance1 and yet ninety-one such cases were known when the estimation was made1 and many more cases have since been discovered4 ,there were also four known triple stars1 and yet the odds against the appearance of any one such conjunction are #agc1ebd to #a4aef ,the conclusions of ,michell have been entirely verified by the discovery that many double stars are connected by gravitation4 #aef ,herschel1 99,outlines of ,astronomy991 #ahdi1 p4 #efe2 but ,todhunter1 in his 99,history of the ,theory of ,probability991 p4 #cce1 states that the calculations do not agree with those published by ,struve4 ,michell also investigated the probability that the six brightest stars in the ,pleiades should have come by accidents into such striking proximity4 ,estimating the number of stars of equal or greater brightness at #aejj1 be found the odds to be nearly #ejj1jjj to #a against casual conjunction4 ,extending the same kind of argument to other clusters1 such as that of ,prsepe1 the nebula in the hilt of ,perseus' sword1 he says3#aeg 8,we may with the highest probability conclude1 the odds against the contrary opinion being many million millions to one1 that the stars are really collected together in clusters in some places1 where they form a kind of system1 while in others there are either few or none of them1 to whatever cause this may be owing1 whether to their mutual gravitation1 or to some other law or appointment of the ,creator40 #aeg 99,philosophical ,transactions991 #agfg1 vol4 lvii4 p4 #dca4 ,the calculations of ,michell have been called in question by the late ,james ,d4 ,forbes1#aeh and ,mr4 ,todhunter vaguely countenances his objections1#aei otherwise ,i should not have thought them of much weight4 ,certainly ,laplace accepts ,michell's views1#afj and if ,michell be in error it is in the methods of calculation1 not in the general validity of his reasoning and conclusions4 #aeh 99,philosophical ,magazine991 #crd ,series1 vol4 xxxvii4 p4 #dja1 ,december #ahej2 also ,august #ahdi4 #aei 99,history991 c41 p4 #ccd4 #afj 99,essai ,philosophique991 p4 #eg4 ,similar calculations might no doubt be applied to the peculiar drifting motions which have been detected by ,mr4 ,r ,a4 ,proctor in some of the constellations4#afa ,the odds are very greatly against any numerous group of stars moving together in any one direction by chance4 ,on like grounds1 there can be no doubt that the sun has a considerable proper motion because on the average the fixed stars show a tendency to move apparently from one point of the heavens towards that diametrically opposite4 ,the sun's motion in the contrary direction would explain this tendency1 otherwise we must believe that thousands of stars accidentally agree in their direction of motion1 or are urged by some common force from which the sun is exempt4 ,it may be said that the rotation of the earth is proved in like manner1 because it is immensely more probable that one body would revolve than that the sun1 moon1 planets1 comets1 and the whole of the stars of the heavens should be whirled round the earth daily1 with a uniform motion superadded to their own peculiar motions4 ,this appears to be mainly the reason which led ,gilbert1 one of the earliest ,english ,copernicans1 and in every way an admirable physicist1 to admit the rotation of the earth1 while ,francis ,bacon denied it4 #afa 99,proceedings of the ,royal ,society992 #bj ,january1 #ahgj2 99,philosophical ,magazine991 #dth ,series1 vol4 xxxix4 p4 #cha4 ,in contemplating the planetary system1 we are struck with the similarity in direction of nearly all its movements4 ,newton remarked upon the regularity and uniformity of these motions1 and contrasted them with the eccentricity and irregularity of the cometary orbits4#afb ,could we1 in fact1 look down upon the system from the northern side1 we should see all the planets moving round from west to east1 the satellites moving round their primaries1 and the sun1 planets1 and satellites rotating in the same direction1 with some exceptions on the verge of the system4 ,in the time of ,laplace eleven planets were known1 and the directions of rotation were known for the sun1 six planets1 the satellites of ,jupiter1 ,saturn's ring1 and one of his satellites4 ,thus there were altogether #dc motions all concurring1 namely3-- ,orbital motions of eleven planets #aa ,orbital motions of eighteen satellites #ah ,axial rotations #ad -- #dc #afb 99,principia991 bk4 ii4 ,general scholium4 ,the probability that #dc motions independent of each other would coincide by chance is the #dbnd power of #a_/b1 so that the odds are about #d1djj1jjj1jjj1jjj to #a in favour of some common cause for the uniformity of direction4 ,this probability1 as ,laplace observes1#afc is higher than that of many historical events which we undoubtingly believe4 ,in the present day1 the probability is much increased by the discovery of additional planets1 and the rotation of other satellites1 and it is only slightly weakened by the fact that some of the outlying satellites are exceptional in direction1 there being considerable evidence of an accidental disturbance in the more distant parts of the system4 #afc 99,essai ,philosophique991 p4 #ee4 ,laplace appears to count the rings of ,saturn as giving two independent movements4 ,hardly less remarkable than the uniform direction of motion is the near approximation of the orbits of the planets to a common plane4 ,daniel ,bernoulli roughly estimated the probability of such an agreement arising from accident as #a 7#ab7f the greatest inclination of any orbit to the sun's equator being #a-abth part of a quadrant4 ,laplace devoted to this subject some of his most ingenious investigations4 ,he found the probability that the sum of the inclinations of the planetary orbits would not exceed by accident the actual amount 7#iadahg of a right angle for the ten planets known in #ahja7 to be 7#a_/aj76 7#iadahg7aj or about #jjjjjjaabce4 ,this probability may be combined with that derived from the direction of motion1 and it then becomes immensely probable that the constitution of the planetary system arose out of uniform conditions1 or1 as we say1 from some common cause4#afd #afd ,lubbock1 99,essay on ,probability991 p4 #ad4 ,de ,morgan1 99,encyc4 ,metrop499 art4 99,probability991 p4 #dab4 ,todhunter's 99,history of the ,theory of ,probability991 p4 #edc4 ,concerning the objections raised to these conclusions by ,boole1 see the 99,philosophical ,magazine991 #dth ,series1 vol4 ii4 p4 #ih4 ,boole's 99,laws of ,thought991 pp4 #cfd-cge4 ,if the same kind of calculation be applied to the orbits of comets1 the result is very different4#afe ,of the orbits which have been determined #dhi per cent4 only are direct or in the same direction as the planetary motions4#aff ,hence it becomes apparent that comets do not properly belong to the solar system1 and it is probable that they are stray portions of nebulous matter which have accidentally become attached to the system by the attractive powers of the sun or ,jupiter4 #afe ,laplace1 99,essai ,philosophique991 pp4 #ee1 #ef4 #aff ,chambers' 99,astronomy991 #bnd ed4 pp4 #cdf-di4 99,the ,general ,inverse ,problem499 ,in the instances described in the preceding sections1 we have been occupied in receding from the occurrence of certain similar events to the probability that there must have been a condition or cause for such events4 ,we have found that the theory of probability1 although never yielding a certain result1 often enables us to establish an hypothesis beyond the reach of reasonable doubt4 ,there is1 however1 another method of applying the theory1 which possesses for us even greater interest1 because it illustrates1 in the most complete manner1 the theory of inference adopted in this work1 which theory indeed it suggested4 ,the problem to be solved is as follows3-- 99,an event having happened a certain number of times1 and failed a certain number of times1 required the probability that it will happen any given number of times in the future under the same circumstances499 ,all the 99larger99 planets hitherto discovered move in one direction round the sun2 what is the probability that1 if a new planet exterior to ,neptune be discovered1 it will move in the same direction8 ,all known permanent gases1 except chlorine1 are colourless2 what is the probability that1 if some new permanent gas should be discovered1 it will be colourless8 ,in the general solution of this problem1 we wish to infer the future happening of any event from the number of times that it has already been observed to happen4 ,now1 it is very instructive to find that there is no known process by which we can pass directly from the data to the conclusion4 ,it is always requisite to recede from the data to the probability of some hypothesis1 and to make that hypothesis the ground of our inference concerning future events4 ,mathematicians1 in fact1 make every hypothesis which is applicable to the question in hand2 they then calculate1 by the inverse method1 the probability of every such hypothesis according to the data1 and the probability that if each hypothesis be true1 the required future event will happen4 ,the total probability that the event will happen is the sum of the separate probabilities contributed by each distinct hypothesis4 ,to illustrate more precisely the method of solving the problem1 it is desirable to adopt some concrete mode of representation1 and the ballot-box1 so often employed by mathematicians1 will best serve our purpose4 ,let the happening of any event be represented by the drawing of a white ball from a ballot-box1 while the failure of an event is represented by the drawing of a black ball4 ,now1 in the inductive problem we are supposed to be ignorant of the contents of the ballot-box1 and are required to ground all our inferences on our experience of those contents as shown in successive drawings4 ,rude common sense would guide us nearly to a true conclusion4 ,thus1 if we had drawn twenty balls one after another1 replacing the ball after each drawing1 and the ball had in each case proved to be white1 we should believe that there was a considerable preponderance of white balls in the urn1 and a probability in favour of drawing a white ball on the next occasion4 ,though we had drawn white balls for thousands of times without fail1 it would still be possible that some black balls lurked in the urn and would at last appear1 so that our inferences could never be certain4 ,on the other hand1 if black balls came at intervals1 we should expect that after a certain number of trials the black balls would appear again from time to time with somewhat the same frequency4 ,the mathematical solution of the question consists in little more than a close analysis of the mode in which our common sense proceeds4 ,if twenty white balls have been drawn and no black ball1 my common sense tells me that any hypothesis which makes the black balls in the urn considerable compared with the white ones is improbable2 a preponderance of white balls is a more probable hypothesis1 and as a deduction from this more probable hypothesis1 ,i expect a recurrence of white balls4 ,the mathematician merely reduces this process of thought to exact numbers4 ,taking1 for instance1 the hypothesis that there are #ii white and one black ball in the urn1 he can calculate the probability that #bj white balls would be drawn in succession in those circumstances2 he thus forms a definite estimate of the probability of this hypothesis1 and knowing at the same time the probability of a white ball reappearing if such be the contents of the urn1 he combines these probabilities1 and obtains an exact estimate that a white ball will recur in consequence of this hypothesis4 ,but as this hypothesis is only one out of many possible ones1 since the ratio of white and black balls may be #ih to #b1 or #ig to #c1 or #if to #d1 and so on1 he has to repeat the estimate for every such possible hypothesis4 ,to make the method of solving the problem perfectly evident1 ,i will describe in the next section a very simple case of the problem1 originally devised for the purpose by ,condorcet1 which was also adopted by ,lacroix1#afg and has passed into the works of ,de ,morgan1 ,lubbock1 and others4 #afg 99,trait lmentaire du ,calcul des ,probabilits991 #crd ed4 7#ahcc71 p4 #adh4 99,simple ,illustration of the ,inverse ,problem499 ,suppose it to be known that a ballot-box contains only four black or white balls1 the ratio of black and white balls being unknown4 ,four drawings having been made with replacement1 and a white ball having appeared on each occasion but one1 it is required to determine the probability that a white ball will appear next time4 ,now the hypotheses which can be made as to the contents of the urn are very limited in number1 and are at most the following five3-- #d white and #j black balls #c #a #b #b #a #c #j #d ,the actual occurrence of black and white balls in the drawings puts the first and last hypothesis out of the question1 so that we have only three left to consider4 ,if the box contains three white and one black1 the probability of drawing a white each time is #c_/d1 and a black #a_/d2 so that the compound event observed1 namely1 three white and one black1 has the probability #c_/d #c_/d #c_/d #a_/d1 by the rule already given 7p4 #bjd74 ,but as it is indifferent in what order the balls are drawn1 and the black ball might come first1 second1 third1 or fourth1 we must multiply by four1 to obtain the probability of three white and one black in any order1 thus getting #bg_/fd4 ,taking the next hypothesis of two white and two black balls in the urn1 we obtain for the same probability the quantity #a_/b #a_/b #a_/b #a_/b #d1 or #af_/fd1 and from the third hypothesis of one white and three black we deduce likewise #a_/d #a_/d #a_/d #c_/d #d1 or #c_/fd4 ,according1 then1 as we adopt the first1 second1 or third hypothesis1 the probability that the result actually noticed would follow is #bg_/fd1 #af_/fd1 and #c_/fd4 ,now it is certain that one or other of these hypotheses must be the true one1 and their absolute probabilities are proportional to the probabilities that the observed events would follow from them 7pp4 #bdb1 #bdc74 ,all we have to do1 then1 in order to obtain the absolute probability of each hypothesis1 is to alter these fractions in a uniform ratio1 so that their sum shall be unity1 the expression of certainty4 ,now1 since #bg #af #c #df1 this will be effected by dividing each fraction by #df1 and multiplying by #fd4 ,thus the probabilities of the first1 second1 and third hypotheses are respectively-- #bg_/df1 #af_/df1 #c_/df4 ,the inductive part of the problem is completed1 since we have found that the urn most likely contains three white and one black ball1 and have assigned the exact probability of each possible supposition4 ,but we are now in a position to resume deductive reasoning1 and infer the probability that the next drawing will yield1 say a white ball4 ,for if the box contains three white and one black ball1 the probability of drawing a white one is certainly #c_/d2 and as the probability of the box being so constituted is #bg_/df1 the compound probability that the box will be so filled and will give a white ball at the next trial1 is #bg_/df #c_/d or #ha_/ahd4 ,again1 the probability is #af_/df that the box contains two white and two black1 and under those conditions the probability is #a_/b that a white ball will appear2 hence the probability that a white ball will appear in consequence of that condition1 is #af_/df #a_/b or #cb_/ahd4 ,from the third supposition we get in like manner the probability #c_/df #a_/d or #c_/ahd4 ,since one and not more than one hypothesis can be true1 we may add together these separate probabilities1 and we find that #ha_/ahd #cb_/ahd #c_/ahd or #aaf_/ahd is the complete probability that a white ball will be next drawn under the conditions and data supposed4 99,general ,solution of the ,inverse ,problem499 ,in the instance of the inverse method described in the last section1 the balls supposed to be in the ballot-box were few1 for the purpose of simplifying the calculation4 ,in order that our solution may apply to natural phenomena1 we must render our hypotheses as little arbitrary as possible4 ,having no 99 priori99 knowledge of the conditions of the phenomena in question1 there is no limit to the variety of hypotheses which might be suggested4 ,mathematicians have therefore had recourse to the most extensive suppositions which can be made1 namely1 that the ballot-box contains an infinite number of balls2 they have then varied the proportion of white to black balls continuously1 from the smallest to the greatest possible proportion1 and estimated the aggregate probability which results from this comprehensive supposition4 ,to explain their procedure1 let us imagine that1 instead of an infinite number1 the ballot-box contains a large finite number of balls1 say #ajjj4 ,then the number of white balls might be #a or #b or #c or #d1 and so on1 up to #iii4 ,supposing that three white and one black ball have been drawn from the urn as before1 there is a certain very small probability that this would have occurred in the case of a box containing one white and #iii black balls2 there is also a small probability that from such a box the next ball would be white4 ,compound these probabilities1 and we have the probability that the next ball really will be white1 in consequence of the existence of that proportion of balls4 ,if there be two white and #iih black balls in the box1 the probability is greater and will increase until the balls are supposed to be in the proportion of those drawn4 ,now #iii different hypotheses are possible1 and the calculation is to be made for each of these1 and their aggregate taken as the final result4 ,it is apparent that as the number of balls in the box is increased1 the absolute probability of any one hypothesis concerning the exact proportion of balls is decreased1 but the aggregate results of all the hypotheses will assume the character of a wider average4 ,when we take the step of supposing the balls within the urn to be infinite in number1 the possible proportions of white and black balls also become infinite1 and the probability of any one proportion actually existing is infinitely small4 ,hence the final result that the next ball drawn will be white is really the sum of an infinite number of infinitely small quantities4 ,it might seem impossible to calculate out a problem having an infinite number of hypotheses1 but the wonderful resources of the integral calculus enable this to be done with far greater facility than if we supposed any large finite number of balls1 and then actually computed the results4 ,i will not attempt to describe the processes by which ,laplace finally accomplished the complete solution of the problem4 ,they are to be found described in several ,english works1 especially ,de ,morgan's 99,treatise on ,probabilities991 in the 99,encyclopdia ,metropolitana991 and ,mr4 ,todhunter's 99,history of the ,theory of ,probability994 ,the abbreviating power of mathematical analysis was never more strikingly shown4 ,but ,i may add that though the integral calculus is employed as a means of summing infinitely numerous results1 we in no way abandon the principles of combinations already treated4 ,we calculate the values of infinitely numerous factorials1 not1 however1 obtaining their actual products1 which would lead to an infinite number of figures1 but obtaining the final answer to the problem by devices which can only be comprehended after study of the integral calculus4 ,it must be allowed that the hypothesis adopted by ,laplace is in some degree arbitrary1 so that there was some opening for the doubt which ,boole has cast upon it4#afh ,but it may be replied1 7#a7 that the supposition of an infinite number of balls treated in the manner of ,laplace is less arbitrary and more comprehensive than any other that can be suggested4 7#b7 ,the result does not differ much from that which would be obtained on the hypothesis of any large finite number of balls4 7#c7 ,the supposition leads to a series of simple formulas which can be applied with ease in many cases1 and which bear all the appearance of truth so far as it can be independently judged by a sound and practiced understanding4 #afh 99,laws of ,thought991 pp4 #cfh-cge4 99,rules of the ,inverse ,method499 ,by the solution of the problem1 as described in the last section1 we obtain the following series of simple rules4 #a4 99,to find the probability that an event which has not hitherto been observed to fail will happen once more1 divide the number of times the event has been observed increased by one1 by the same number increased by two499 ,if there have been 99m99 occasions on which a certain event might have been observed to happen1 and it has happened on all those occasions1 then the probability that it will happen on the next occasion of the same kind 799m99 #a7_/799m99 #b74 ,for instance1 we may say that there are nine places in the planetary system where planets might exist obeying ,bode's law of distance1 and in every place there is a planet obeying the law more or less exactly1 although no reason is known for the coincidence4 ,hence the probability that the next planet beyond ,neptune will conform to the law is #aj_/aa4 #b4 99,to find the1 probability that an event which has not hitherto failed will not fail for a certain number of new occasions1 divide the number of times the event has happened increased by one1 by the same number increased by one and the number of times it is to happen499 ,an event having happened 99m99 times without fail1 the probability that it will happen 99n99 more times is 799m99 #a7_/799m99 99n99 #a74 ,thus the probability that three new planets would obey ,bode's law is #aj_/ac2 but it must be allowed that this1 as well as the previous result1 would be much weakened by the fact that ,neptune can barely be said to obey the law4 99#c4 ,an event having happened and failed a certain number of times1 to find the probability that it will happen the next time1 divide the number of times the event has happened increased by one1 by the whole number of times the event has happened or failed increased by two499 ,if an event has happened 99m99 times and failed 99n99 times1 the probability that it will happen on the next occasion is 799m99 #a7_/799m99 99n99 #b74 ,thus1 if we assume that of the elements discovered up to the year #ahgc1 #ej are metallic and #ad non-metallic1 then the probability that the next element discovered will be metallic is #ea_/ff4 ,again1 since of #cg metals which have been sufficiently examined only four1 namely1 sodium1 potassium1 lanthanum1 and lithium1 are of less density than water1 the probability that the next metal examined or discovered will be less dense than water is 7#d #a7_/7cg #b7 or #e_/ci4 ,we may state the results of the method in a more general manner thus1#afi--;,if under given circumstances certain events ,a1 ,b1 ,c1 c41 have happened respectively 99m991 99n991 99p991 c41 times1 and one or other of these events must happen1 then the probabilities of these events are proportional to 99m99 #a1 99n99 #a1 99p99 #a1 c41 so that the probability of ,a will be 799m99 #a7_/799m99 #a 99n99 #a 99p99 #a c47 ,but if new events may happen in addition to those which have been observed1 we must assign unity for the probability of such new event4 ,the odds then become #a for a new event1 99m99 #a for ,a1 99n99 #a for ,b1 and so on1 and the absolute probability of ,a is 799m99 #a7_/7a 99m99 #a 99n99 #a c47 #afi ,de ,morgan's 99,essay on ,probabilities991 ,cabinet ,cyclopdia1 p4 #fg4 ,it is interesting to trace out the variations of probability according to these rules4 ,the first time a casual event happens it is #b to #a that it will happen again2 if it does happen it is #c to #a that it will happen a third time2 and on successive occasions of the like kind the odds become #d1 #e1 #f1 c41 to #a4 ,the odds of course will be discriminated from the probabilities which are successively #b_/c1 #c_/d1 #d_/e1 c4 ,thus on the first occasion on which a person sees a shark1 and notices that it is accompanied by a little pilot fish1 the odds are #b to #a1 or the probability #b_/c1 that the next shark will be so accompanied4 ,when an event has happened a very great number of times1 its happening once again approaches nearly to certainty4 ,if we suppose the sun to have risen one thousand million times1 the probability that it will rise again1 on the ground of this knowledge merely1 is 7#a1jjj1jjj1jjj #a7_/7a1jjj1jjj1jjj #a #a74 ,but then the probability that it will continue to rise for as long a period in the future is only 7#a1jjj1jjj1jjj #a7_/7b1jjj1jjj1jjj #a71 or almost exactly #a_/b4 ,the probability that it will continue so rising a thousand times as long is only about #a_/ajja4 ,the lesson which we may draw from these figures is quite that which we should adopt on other grounds1 namely1 that experience never affords certain knowledge1 and that it is exceedingly improbable that events will always happen as we observe them4 ,inferences pushed far beyond their data soon lose any considerable probability4 ,de ,morgan has said1#agj 8,no finite experience whatsoever can justify us in saying that the future shall coincide with the past in all time to come1 or that there is any probability for such a conclusion40 ,on the other hand1 we gain the assurance that experience sufficiently extended and prolonged will give us the knowledge of future events with an unlimited degree of probability1 provided indeed that those events are not subject to arbitrary interference4 #agj 99,essay on ,probabilities991 p4 #abh4 ,it must be clearly understood that these probabilities are only such as arise from the mere happening of the events1 irrespective of any knowledge derived from other sources concerning those events or the general laws of nature4 ,all our knowledge of nature is indeed founded in like manner upon observation1 and is therefore only probable4 ,the law of gravitation itself is only probably true4 ,but when a number of different facts1 observed under the most diverse circumstances1 are found to be harmonized under a supposed law of nature1 the probability of the law approximates closely to certainty4 ,each science rests upon so many observed facts1 and derives so much support from analogies or connections with other sciences1 that there are comparatively few cases where our judgment of the probability of an event depends entirely upon a few antecedent events1 disconnected from the general body of physical science4 ,events1 again1 may often exhibit a regularity of succession or preponderance of character1 which the simple formula will not take into account4 ,for instance1 the majority of the elements recently discovered are metals1 so that the probability of the next discovery being that of a metal1 is doubtless greater than we calculated 7p4 #beh74 ,at the more distant parts of the planetary system1 there are symptoms of disturbance which would prevent our placing much reliance on any inference from the prevailing order of the known planets to those undiscovered ones which may possibly exist at great distances4 ,these and all like complications in no way invalidate the theoretic truth of the formulas1 but render their sound application much more difficult4 ,erroneous objections have been raised to the theory of probability1 on the ground that we ought not to trust to our 99 priori99 conceptions of what is likely to happen1 but should always endeavour to obtain precise experimental data to guide us4#aga ,this course1 however1 is perfectly in accordance with the theory1 which is our best and only guide1 whatever data we possess4 ,we ought to be always applying the inverse method of probabilities so as to take into account all additional information4 ,when we throw up a coin for the first time1 we are probably quite ignorant whether it tends more to fall head or tail upwards1 and we must therefore assume the probability of each event as #a_/b4 ,but if it shows head in the first throw1 we now have very slight experimental evidence in favour of a tendency to show head4 ,the chance of two heads is now slightly greater than #a_/d1 which it appeared to be at first1#agb and as we go on throwing the coin time after time1 the probability of head appearing next time constantly varies in a slight degree according to the character of our previous experience4 ,as ,laplace remarks1 we ought always to have regard to such considerations in common life4 ,events when closely scrutinized will hardly ever prove to be quite independent1 and the slightest preponderance one way or the other is some evidence of connection1 and in the absence of better evidence should be taken into account4 #aga ,j4 ,s4 ,mill1 99,system of ,logic991 #eth edition1 bk4 iii4 chap4 xviii4 #c4 #agb ,todhunter's 99,history991 pp4 #dgb1 #eih4 ,the grand object of seeking to estimate the probability of future events from past experience1 seems to have been entertained by ,james ,bernoulli and ,de ,moivre1 at least such was the opinion of ,condorcet2 and ,bernoulli may be said to have solved one case of the problem4#agc ,the ,english writers ,bayes and ,price are1 however1 undoubtedly the first who put forward any distinct rules on the subject4#agd ,condorcet and several other eminent mathematicians advanced the mathematical theory of the subject2 but it was reserved to the immortal ,laplace to bring to the subject the full power of his genius1 and carry the solution of the problem almost to perfection4 ,it is instructive to observe that a theory which arose from petty games of chance1 the rules and the very names of which are forgotten1 gradually advanced1 until it embraced the most sublime problems of science1 and finally undertook to measure the value and certainty of all our inductions4 #agc ,todhunter's 99,history991 pp4 #cgh1 #cgi4 #agd 99,philosophical ,transactions991 #agfc1 vol4 liii4 p4 #cgj1 and #agfd1 vol4 liv4 p4 #bif4 ,todhunter1 pp4 #bid-cjj4 99,fortuitous ,coincidences499 ,we should have studied the theory of probability to very little purpose1 if we thought that it would furnish us with an infallible guide4 ,the theory itself points out the approximate certainty1 that we shall sometimes be deceived by extraordinary fortuitous coincidences4 ,there is no run of luck so extreme that it may not happen1 and it may happen to us1 or in our time1 as well as to other persons or in other times4 ,we may be forced by correct calculation to refer such coincidences to a necessary cause1 and yet we may be deceived4 ,all that the calculus of probability pretends to give1 is 99the result in the long run991 as it is called1 and this really means in 99an infinity of cases994 ,during any finite experience1 however long1 chances may be against us4 ,nevertheless the theory is the best guide we can have4 ,if we always think and act according to its well-interpreted indications1 we shall have the best chance of escaping error2 and if all persons1 throughout all time to come1 obey the theory in like manner1 they will undoubtedly thereby reap the greatest advantage4 ,no rule can be given for discriminating between coincidences which are casual and those which are the effects of law4 ,by a fortuitous or casual coincidence1 we mean an agreement between events1 which nevertheless arise from wholly independent and different causes or conditions1 and which will not always so agree4 ,it is a fortuitous coincidence1 if a penny thrown up repeatedly in various ways always falls on the same side2 but it would not be fortuitous if there were any similarity in the motions of the hand1 and the height of the throw1 so as to cause or tend to cause a uniform result4 ,now among the infinitely numerous events1 objects1 or relations in the universe1 it is quite likely that we shall occasionally notice casual coincidences4 ,there are seven intervals in the octave1 and there is nothing very improbable in the colours of the spectrum happening to be apparently divisible into the same or similar series of seven intervals4 ,it is hardly yet decided whether this apparent coincidence1 with which ,newton was much struck1 is well founded or not1#age but the question will probably be decided in the negative4 #age ,newton's 99,opticks991 ,bk4 ,i41 ,part ii4 ,prop4 #c2 99,nature991 vol4 i4 p4 #bhf4 ,it is certainly a casual coincidence which the ancients noticed between the seven vowels1 the seven strings of the lyre1 the seven ,pleiades1 and the seven chiefs at ,thebes4#agf ,the accidents connected with the number seven have misled the human intellect throughout the historical period4 ,pythagoras imagined a connection between the seven planets and the seven intervals of the monochord4 ,the alchemists were never tired of drawing inferences from the coincidence in numbers of the seven planets and the seven metals1 not to speak of the seven days of the week4 #agf ,aristotle's 99,metaphysics991 xiii4 #f4 #c4 ,a singular circumstance was pointed out concerning the dimensions of the earth1 sun1 and moon2 the sun's diameter was almost exactly #aaj times as great as the earth's diameter1 while in almost exactly the same ratio the mean distance of the earth was greater than the sun's diameter1 and the mean distance of the moon from the earth was greater than the moon's diameter4 ,the agreement was so close that it might have proved more than casual1 but its fortuitous character is now sufficiently shown by the fact1 that the coincidence ceases to be remarkable when we adopt the amended dimensions of the planetary system4 ,a considerable number of the elements have atomic weights1 which are apparently exact multiples of that of hydrogen4 ,if this be not a law to be ultimately extended to all the elements1 as supposed by ,prout1 it is a most remarkable coincidence4 ,but1 as ,i have observed1 we have no means of absolutely discriminating accidental coincidences from those which imply a deep producing cause4 ,a coincidence must either be very strong in itself1 or it must be corroborated by some explanation or connection with other laws of nature4 ,little attention was ever given to the coincidence concerning the dimensions of the sun1 earth1 and moon1 because it was not very strong in itself1 and had no apparent connection with the principles of physical astronomy4 ,prout's ,law bears more probability because it would bring the constitution of the elements themselves in close connection with the atomic theory1 representing them as built up out of a simpler substance4 ,in historical and social matters1 coincidences are frequently pointed out which are due to chance1 although there is always a strong popular tendency to regard them as the work of design1 or as having some hidden meaning4 ,if to #agid1 the number of the year in which ,robespierre fell1 we add the sum of its digits1 the result is #ahae1 the year in which ,napoleon fell2 the repetition of the process gives #ahcj the year in which ,charles the ,tenth abdicated4 ,again1 the ,french ,chamber of ,deputies1 in #ahcj1 consisted of #djb members1 of whom #bba formed the party called 8,la queue de ,robespierre10 while the remainder1 #aha in number1 were named 8,les honntes gens40 ,if we give to each letter a numerical value corresponding to its place in the alphabet1 it will be found that the sum of the values of the letters in each name exactly indicates the number of the party4 ,a number of such coincidences1 often of a very curious character1 might be adduced1 and the probability against the occurrence of each is enormously great4 ,they must be attributed to chance1 because they cannot be shown to have the slightest connection with the general laws of nature2 but persons are often found to be greatly influenced by such coincidences1 regarding them as evidence of fatality1 that is of a system of causation governing human affairs independently of the ordinary laws of nature4 ,let it be remembered that there are an infinite number of opportunities in life for some strange coincidence to present itself1 so that it is quite to be expected that remarkable conjunctions will sometimes happen4 ,in all matters of judicial evidence1 we must bear in mind the probable occurrence from time to time of unaccountable coincidences4 ,the ,roman jurists refused for this reason to invalidate a testamentary deed1 the witnesses of which had sealed it with the same seal4 ,for witnesses independently using their own seals might be found to possess identical ones by accident4#agg ,it is well known that circumstantial evidence of apparently overwhelming completeness will sometimes lead to a mistaken judgment1 and as absolute certainty is never really attainable1 every court must act upon probabilities of a high amount1 and in a certain small proportion of cases they must almost of necessity condemn the innocent victims of a remarkable conjuncture of circumstances4#agh ,popular judgments usually turn upon probabilities of far less amount1 as when the palace of ,nicomedia1 and even the bedchamber of ,diocletian1 having been on fire twice within fifteen days1 the people entirely refused to believe that it could be the result of accident4 ,the ,romans believed that there was fatality connected with the name of ,sextus4 8,semper sub ,sextis perdita ,roma fuit40 #agg ,possunt autem omnes testes et uno annulo signare testamentum ,quid enim si septem annuli una sculptura fuerint1 secundum quod ,pomponio visum est8--99,justinian991 ii4 tit4 x4 #e4 #agh ,see ,wills on 99,circumstantial ,evidence991 p4 #adh4 ,the utmost precautions will not provide against all contingencies4 ,to avoid errors in important calculations1 it is usual to have them repeated by different computers2 but a case is on record in which three computers made exactly the same calculations of the place of a star1 and yet all did it wrong in precisely the same manner1 for no apparent reason4#agi #agi 99,memoirs of the ,royal ,astronomical ,society991 vol4 iv4 p4 #bij1 quoted by ,lardner1 99,edinburgh ,review991 ,july #ahcd1 p4 #bgh4 99,summary of the ,theory of ,inductive ,inference499 ,the theory of inductive inference stated in this and the previous chapters1 was suggested by the study of the ,inverse ,method of ,probability1 but it also bears much resemblance to the so-called ,deductive ,method described by ,mill1 in his celebrated 99,system of ,logic994 ,mill's views concerning the ,deductive ,method1 probably form the most original and valuable part of his treatise1 and ,i should have ascribed the doctrine entirely to him1 had ,i not found that the opinions put forward in other parts of his work are entirely inconsistent with the theory here upheld4 ,as this subject is the most important and difficult one with which we have to deal1 ,i will try to remedy the imperfect manner in which ,i have treated it1 by giving a recapitulation of the views adopted4 ,all inductive reasoning is but the inverse application of deductive reasoning4 ,being in possession of certain particular facts or events expressed in propositions1 we imagine some more general proposition expressing the existence of a law or cause2 and1 deducing the particular results of that supposed general proposition1 we observe whether they agree with the facts in question4 ,hypothesis is thus always employed1 consciously or unconsciously4 ,the sole conditions to which we need conform in framing any hypothesis is1 that we both have and exercise the power of inferring deductively from the hypothesis to the particular results1 which are to be compared with the known facts4 ,thus there are but three steps in the process of induction3-- 7#a7 ,framing some hypothesis as to the character of the general law4 7#b7 ,deducing consequences from that law4 7#c7 ,observing whether the consequences agree with the particular facts under consideration4 ,in very simple cases of inverse reasoning1 hypothesis may seem altogether needless4 ,to take numbers again as a convenient illustration1 ,i have only to look at the series1 #a1 #b1 #d1 #h1 #af1 #cb1 c41 to know at once that the general law is that of geometrical progression2 ,i need no successive trial of various hypotheses1 because ,i am familiar with the series1 and have long since learnt from what general formula it proceeds4 ,in the same way a mathematician becomes acquainted with the integrals of a number of common formulas1 so that he need not go through any process of discovery4 ,but it is none the less true that whenever previous reasoning does not furnish the knowledge1 hypotheses must be framed and tried 7p4 #abd74 ,there naturally arise two cases1 according as the nature of the subject admits of certain or only probable deductive reasoning4 ,certainty1 indeed1 is but a singular case of probability1 and the general principles of procedure are always the same4 ,nevertheless1 when certainty of inference is possible1 the process is simplified4 ,of several mutually inconsistent hypotheses1 the results of which can be certainly compared with fact1 but one hypothesis can ultimately be entertained4 ,thus in the inverse logical problem1 two logically distinct conditions could not yield the same series of possible combinations4 ,accordingly1 in the case of two terms we had to choose one of six different kinds of propositions 7p4 #acf71 and in the case of three terms1 our choice lay among #aib possible distinct hypotheses 7p4 #adj74 ,natural laws1 however1 are often quantitative in character1 and the possible hypotheses are then infinite in variety4 ,when deduction is certain1 comparison with fact is needed only to assure ourselves that we have rightly selected the hypothetical conditions4 ,the law establishes itself1 and no number of particular verifications can add to its probability4 ,having once deduced from the principles of algebra that the difference of the squares of two numbers is equal to the product of their sum and difference1 no number of particular trials of its truth will render it more certain4 ,on the other hand1 no finite number of particular verifications of a supposed law will render that law certain4 ,in short1 certainty belongs only to the deductive process1 and to the teachings of direct intuition2 and as the conditions of nature are not given by intuition1 we can only be certain that we have got a correct hypothesis when1 out of a limited number conceivably possible1 we select that one which alone agrees with the facts to be explained4 ,in geometry and kindred branches of mathematics1 deductive reasoning is conspicuously certain1 and it would often seem as if the consideration of a single diagram yields us certain knowledge of a general proposition4 ,but in reality all this certainty is of a purely hypothetical character4 ,doubtless if we could ascertain that a supposed circle was a true and perfect circle1 we could be certain concerning a multitude of its geometrical properties4 ,but geometrical figures are physical objects1 and the senses can never assure us as to their exact forms4 ,the figures really treated in ,euclid's 99,elements99 are imaginary1 and we never can verify in practice the conclusions which we draw with certainty in inference2 questions of degree and probability enter4 ,passing now to subjects in which deduction is only probable1 it ceases to be possible to adopt one hypothesis to the exclusion of the others4 ,we must entertain at the same time all conceivable hypotheses1 and regard each with the degree of esteem proportionate to its probability4 ,we go through the same steps as before4 7#a7 ,we frame an hypothesis4 7#b7 ,we deduce the probability of various series of possible consequences4 7#c7 ,we compare the consequences with the particular facts1 and observe the probability that such facts would happen under the hypothesis4 ,the above processes must be performed for every conceivable hypothesis1 and then the absolute probability of each will be yielded by the principle of the inverse method 7p4 #bdb74 ,as in the case of certainty we accept that hypothesis which certainly gives the required results1 so now we accept as most probable that hypothesis which most probably gives the results2 but we are obliged to entertain at the same time all other hypotheses with degrees of probability proportionate to the probabilities that they would give the same results4 ,so far we have treated only of the process by which we pass from special facts to general laws1 that inverse application of deduction which constitutes induction4 ,but the direct employment of deduction is often combined with the inverse4 ,no sooner have we established a general law1 than the mind rapidly draws particular consequences from it4 ,in geometry we may almost seem to infer that 99because99 one equilateral triangle is equiangular1 therefore another is so4 ,in reality it is not because one is that another is1 but because all are4 ,the geometrical conditions are perfectly general1 and by what is sometimes called 99parity of reasoning99 whatever is true of one equilateral triangle1 so far as it is equilateral1 is true of all equilateral triangles4 ,similarly1 in all other cases of inductive inference1 where we seem to pass from some particular instances to a new instance1 we go through the same process4 ,we form an hypothesis as to the logical conditions under which the given instances might occur2 we calculate inversely the probability of that hypothesis1 and compounding this with the probability that a new instance would proceed from the same conditions1 we gain the absolute probability of occurrence of the new instance in virtue of this hypothesis4 ,but as several1 or many1 or even an infinite number of mutually inconsistent hypotheses may be possible1 we must repeat the calculation for each such conceivable hypothesis1 and then the complete probability of the future instance will be the sum of the separate probabilities4 ,the complication of this process is often very much reduced in practice1 owing to the fact that one hypothesis may be almost certainly true1 and other hypotheses1 though conceivable1 may be so improbable as to be neglected without appreciable error4 ,when we possess no knowledge whatever of the conditions from which the events proceed1 we may be unable to form any probable hypotheses as to their mode of origin4 ,we have now to fall back upon the general solution of the problem effected by ,laplace1 which consists in admitting on an equal footing every conceivable ratio of favourable and unfavourable chances for the production of the event1 and then accepting the aggregate result as the best which can be obtained4 ,this solution is only to be accepted in the absence of all better means1 but like other results of the calculus of probability1 it comes to our aid where knowledge is at an end and ignorance begins1 and it prevents us from over-estimating the knowledge we possess4 ,the general results of the solution are in accordance with common sense1 namely1 that the more often an event has happened the more probable1 as a general rule1 is its subsequent recurrence4 ,with the extension of experience this probability increases1 but at the same time the probability is slight that events will long continue to happen as they have previously happened4 ,we have now pursued the theory of inductive inference1 as far as can be done with regard to simple logical or numerical relations4 ,the laws of nature deal with time and space1 which are infinitely divisible4 ,as we passed from pure logic to numerical logic1 so we must now pass from questions of discontinuous1 to questions of continuous quantity1 encountering fresh considerations of much difficulty4 ,before1 therefore1 we consider how the great inductions and generalisations of physical science illustrate the views of inductive reasoning just explained1 we must break off for a time1 and review the means which we possess of measuring and comparing magnitudes of time1 space1 mass1 force1 momentum1 energy1 and the various manifestations of energy in motion1 heat1 electricity1 chemical change1 and the other phenomena of nature4 ,,book ,,iii4 ,,methods ,,of ,,measurement4 ,,chapter ,,xiii4 ,,the ,,exact ,,measurement ,,of ,,phenomena4 ,as physical science advances1 it becomes more and more accurately quantitative4 ,questions of simple logical fact after a time resolve themselves into questions of degree1 time1 distance1 or weight4 ,forces hardly suspected to exist by one generation1 are clearly recognised by the next1 and precisely measured by the third generation4 ,but one condition of this rapid advance is the invention of suitable instruments of measurement4 ,we need what ,francis ,bacon called 99,instanti citantes991 or 99evocantes991 methods of rendering minute phenomena perceptible to the senses2 and we also require 99,instanti radii99 or 99curriculi991 that is measuring instruments4 ,accordingly1 the introduction of a new instrument often forms an epoch in the history of science4 ,as ,davy said1 8,nothing tends so much to the advancement of knowledge as the application of a new instrument4 ,the native intellectual powers of men in different times are not so much the causes of the different success of their labours1 as the peculiar nature of the means and artificial resources in their possession40 ,in the absence indeed of advanced theory and analytical power1 a very precise instrument would be useless4 ,measuring apparatus and mathematical theory should advance 99pari passu991 and with just such precision as the theorist can anticipate results1 the experimentalist should be able to compare them with experience4 ,the scrupulously accurate observations of ,flamsteed were the proper complement to the intense mathematical powers of ,newton4 ,every branch of knowledge commences with quantitative notions of a very rude character4 ,after we have far progressed1 it is often amusing to look back into the infancy of the science1 and contrast present with past methods4 ,at ,greenwich ,observatory in the present day1 the hundredth part of a second is not thought an inconsiderable portion of time4 ,the ancient ,chaldans recorded an eclipse to the nearest hour1 and the early ,alexandrian astronomers thought it superfluous to distinguish between the edge and centre of the sun4 ,by the introduction of the astrolabe1 ,ptolemy and the later ,alexandrian astronomers could determine the places of the heavenly bodies within about ten minutes of arc4 ,little progress then ensued for thirteen centuries1 until ,tycho ,brahe made the first great step towards accuracy1 not only by employing better instruments1 but even more by ceasing to regard an instrument as correct4 ,tycho1 in fact1 determined the errors of his instruments1 and corrected his observations4 ,he also took notice of the effects of atmospheric refraction1 and succeeded in attaining an accuracy often sixty times as great as that of ,ptolemy4 ,yet ,tycho and ,hevelius often erred several minutes in the determination of a star's place1 and it was a great achievement of ,rmer and ,flamsteed to reduce this error to seconds4 ,bradley1 the modern ,hipparchus1 carried on the improvement1 his errors in right ascension1 according to ,bessel1 being under one second of time1 and those of declination under four seconds of arc4 ,in the present day the average error of a single observation is probably reduced to the half or quarter of what it was in ,bradley's time2 and further extreme accuracy is attained by the multiplication of observations1 and their skilful combination according to the theory of error4 ,some of the more important constants1 for instance that of nutation1 have been determined within the tenth part of a second of space4#ahj #ahj ,baily1 99,british ,association ,catalogue of ,stars991 pp4 #g1 #bc4 ,it would be a matter of great interest to trace out the dependence of this progress upon the introduction of new instruments4 ,the astrolabe of ,ptolemy1 the telescope of ,galileo1 the pendulum of ,galileo and ,huyghens1 the micrometer of ,horrocks1 and the telescopic sights and micrometer of ,gascoygne and ,picard1 ,rmer's transit instrument1 ,newton's and ,hadley's quadrant1 ,dollond's achromatic lenses1 ,harrison's chronometer1 and ,ramsden's dividing engine--such were some of the principal additions to astronomical apparatus4 ,the result is1 that we now take note of quantities1 #cjj1jjj or #djj1jjj times as small as in the time of the ,chaldans4 ,it would be interesting again to compare the scrupulous accuracy of a modern trigonometrical survey with ,eratosthenes' rude but ingenious guess at the difference of latitude between ,alexandria and ,syene--or with ,norwood's measurement of a degree of latitude in #afce4 8,sometimes ,i measured1 sometimes ,i paced10 said ,norwood2 8and ,i believe ,i am within a scantling of the truth40 ,such was the germ of those elaborate geodesical measurements which have made the dimensions of the globe known to us within a few hundred yards4 ,in other branches of science1 the invention of an instrument has usually marked1 if it has not made1 an epoch4 ,the science of heat might be said to commence with the construction of the thermometer1 and it has recently been advanced by the introduction of the thermo-electric pile4 ,chemistry has been created chiefly by the careful use of the balance1 which forms a unique instance of an instrument remaining substantially in the form in which it was first applied to scientific purposes by ,archimedes4 ,the balance never has been and probably never can be improved1 except in details of construction4 ,the torsion balance1 introduced by ,coulomb towards the end of last century1 has rapidly become essential in many branches of investigation4 ,in the hands of ,cavendish and ,baily1 it gave a determination of the earth's density2 applied in the galvanometer1 it gave a delicate measure of electrical forces1 and is indispensable in the thermo-electric pile4 ,this balance is made by simply suspending any light rod by a thin wire or thread attached to the middle point4 ,and we owe to it almost all the more delicate investigations in the theories of heat1 electricity1 and magnetism4 ,though we can now take note of the millionth of an inch in space1 and the millionth of a second in time1 we must not overlook the fact that in other operations of science we are yet in the position of the ,chaldans4 ,not many years have elapsed since the magnitudes of the stars1 meaning the amounts of light they send to the observer's eye1 were guessed at in the rudest manner1 and the astronomer adjudged a star to this or that order of magnitude by a rough comparison with other stars of the same order4 ,to ,sir ,john ,herschel we owe an attempt to introduce a uniform method of measurement and expression1 bearing some relation to the real photometric magnitudes of the stars4#aha ,previous to the researches of ,bunsen and ,roscoe on the chemical action of light1 we were devoid of any mode of measuring the energy of light2 even now the methods are tedious1 and it is not clear that they give the energy of light so much as one of its special effects4 ,many natural phenomena have hardly yet been made the subject of measurement at all1 such as the intensity of sound1 the phenomena of taste and smell1 the magnitude of atoms1 the temperature of the electric spark or of the sun's photosphere4 #aha 99,outlines of ,astronomy991 #dth ed4 sect4 #gha1 p4 #ebb4 99,results of ,observations at the ,cape of ,good ,hope991 c41 p4 #cg4 ,to suppose1 then1 that quantitative science treats only of exactly measurable quantities1 is a gross if it be a common mistake4 ,whenever we are treating of an event which either happens altogether or does not happen at all1 we are engaged with a non-quantitative phenomenon1 a matter of fact1 not of degree2 but whenever a thing may be greater or less1 or twice or thrice as great as another1 whenever1 in short1 ratio enters even in the rudest manner1 there science will have a quantitative character4 ,there can be little doubt1 indeed1 that every science as it progresses will become gradually more and more quantitative4 ,numerical precision is the soul of science1 as ,herschel said1 and as all natural objects exist in space1 and involve molecular movements1 measurable in velocity and extent1 there is no apparent limit to the ultimate extension of quantitative science4 ,but the reader must not for a moment suppose that1 because we depend more and more upon mathematical methods1 we leave logical methods behind us4 ,number1 as ,i have endeavoured to show1 is logical in its origin1 and quantity is but a development of number1 or analogous thereto4 99,division of the ,subject499 ,the general subject of quantitative investigation will have to be divided into several parts4 ,we shall firstly consider the means at our disposal for measuring phenomena1 and thus rendering them more or less amenable to mathematical treatment4 ,this task will involve an analysis of the principles on which accurate methods of measurement are founded1 forming the subject of the remainder of the present chapter4 ,as measurement1 however1 only yields ratios1 we have in the next chapter to consider the establishment of unit magnitudes1 in terms of which our results may be expressed4 ,as every phenomenon is usually the sum of several distinct quantities depending upon different causes1 we have next to investigate in ,chapter ,,xv4 the methods by which we may disentangle complicated effects1 and refer each part of the joint effect to its separate cause4 ,it yet remains for us in subsequent chapters to treat of quantitative induction1 properly so called4 ,we must follow out the inverse logical method1 as it presents itself in problems of a far higher degree of difficulty than those which treat of objects related in a simple logical manner1 and incapable of merging into each other by addition and subtraction4 99,continuous ,quantity499 ,the phenomena of nature are for the most part manifested in quantities which increase or decrease continuously4 ,when we inquire into the precise meaning of continuous quantity1 we find that it can only be described as that which is divisible without limit4 ,we can divide a millimetre into ten1 or a hundred1 or a thousand1 or ten thousand parts1 and mentally at any rate we can carry on the division 99ad infinitum994 ,any finite space1 then1 must be conceived as made up of an infinite number of parts each infinitely small4 ,we cannot entertain the simplest geometrical notions without allowing this4 ,the conception of a square involves the conception of a side and diagonal1 which1 as ,euclid beautifully proves in the #aagth proposition of his tenth book1 have no common measure1#ahb meaning no finite common measure4 ,incommensurable quantities are1 in fact1 those which have for their only common measure an infinitely small quantity4 ,it is somewhat startling to find1 too1 that in theory incommensurable quantities will be infinitely more frequent than commensurable4 ,let any two lines be drawn haphazard2 it is infinitely unlikely that they will be commensurable1 so that the commensurable quantities1 which we are supposed to deal with in practice1 are but singular cases among an infinitely greater number of incommensurable cases4 #ahb ,see ,de ,morgan1 99,study of ,mathematics991 in ,u4,k4,s4 ,library1 p4 #ha4 ,practically1 however1 we treat all quantities as made up of the least quantities which our senses1 assisted by the best measuring instruments1 can perceive4 ,so long as microscopes were uninvented1 it was sufficient to regard an inch as made up of a thousand thousandths of an inch2 now we must treat it as composed of a million millionths4 ,we might apparently avoid all mention of infinitely small quantities1 by never carrying our approximations beyond quantities which the senses can appreciate4 ,in geometry1 as thus treated1 we should never assert two quantities to be equal1 but only to be 99apparently99 equal4 ,legendre really adopts this mode of treatment in the twentieth proposition of the first book of his ,geometry2 and it is practically adopted throughout the physical sciences1 as we shall afterwards see4 ,but though our fingers1 and senses1 and instruments must stop somewhere1 there is no reason why the mind should not go on4 ,we can see that a proof which is only carried through a few steps in fact1 might be carried on without limit1 and it is this consciousness of no stopping-place1 which renders ,euclid's proof of his #aagth proposition so impressive4 ,try how we will to circumvent the matter1 we cannot really avoid the consideration of the infinitely small and the infinitely great4 ,the same methods of approximation which seem confined to the finite1 mentally extend themselves to the infinite4 ,one result of these considerations is1 that we cannot possibly adjust two quantities in absolute equality4 ,the suspension of ,mahomet's coffin between two precisely equal magnets is theoretically conceivable but practically impossible4 ,the story of the 99,merchant of ,venice99 turns upon the infinite improbability that an exact quantity of flesh could be cut4 ,unstable equilibrium cannot exist in nature1 for it is that which is destroyed by an infinitely small displacement4 ,it might be possible to balance an egg on its end practically1 because no egg has a surface of perfect curvature4 ,suppose the egg shell to be perfectly smooth1 and the feat would become impossible4 99,the ,fallacious ,indications of the ,senses499 ,i may briefly remind the reader how little we can trust to our unassisted senses in estimating the degree or magnitude of any phenomenon4 ,the eye cannot correctly estimate the comparative brightness of two luminous bodies which differ much in brilliancy2 for we know that the iris is constantly adjusting itself to the intensity of the light received1 and thus admits more or less light according to circumstances4 ,the moon which shines with almost dazzling brightness by night1 is pale and nearly imperceptible while the eye is yet affected by the vastly more powerful light of day4 ,much has been recorded concerning the comparative brightness of the zodiacal light at different times1 but it would be difficult to prove that these changes are not due to the varying darkness at the time1 or the different acuteness of the observer's eye4 ,for a like reason it is exceedingly difficult to establish the existence of any change in the form or comparative brightness of nebul2 the appearance of a nebula greatly depends upon the keenness of sight of the observer1 or the accidental condition of freshness or fatigue of his eye4 ,the same is true of lunar observations2 and even the use of the best telescope fails to remove this difficulty4 ,in judging of colours1 again1 we must remember that light of any given colour tends to dull the sensibility of the eye for light of the same colour4 ,nor is the eye when unassisted by instruments a much better judge of magnitude4 ,our estimates of the size of minute bright points1 such as the fixed stars1 are completely falsified by the effects of irradiation4 ,tycho calculated from the apparent size of the star-discs1 that no one of the principal fixed stars could be contained within the area of the earth's orbit4 ,apart1 however1 from irradiation or other distinct causes of error our visual estimates of sizes and shapes are often astonishingly incorrect4 ,artists almost invariably draw distant mountains in ludicrous disproportion to nearer objects1 as a comparison of a sketch with a photograph at once shows4 ,the extraordinary apparent difference of size of the sun or moon1 according as it is high in the heavens or near the horizon1 should be sufficient to make us cautious in accepting the plainest indications of our senses1 unassisted by instrumental measurement4 ,as to statements concerning the height of the aurora and the distance of meteors1 they are to be utterly distrusted4 ,when ,captain ,parry says that a ray of the aurora shot suddenly downwards between him and the land which was only #c1jjj yards distant1 we must consider him subject to an illusion of sense4#ahc #ahc ,loomis1 99,on the ,aurora ,borealis994 ,smithsonian ,transactions1 quoting ,parry's ,third ,voyage1 p4 #fa4 ,it is true that errors of observation are more often errors of judgment than of sense4 ,that which is actually seen must be so far truly seen2 and if we correctly interpret the meaning of the phenomenon1 there would be no error at all4 ,but the weakness of the bare senses as measuring instruments1 arises from the fact that they import varying conditions of unknown amount1 and we cannot make the requisite corrections and allowances as in the case of a solid and invariable instrument4 ,bacon has excellently stated the insufficiency of the senses for estimating the magnitudes of objects1 or detecting the degrees in which phenomena present themselves4 8,things escape the senses10 he says1 8because the object is not sufficient in quantity to strike the sense3 as all minute bodies2 because the percussion of the object is too great to be endured by the senses3 as the form of the sun when looking directly at it in mid-day2 because the time is not proportionate to actuate the sense3 as the motion of a bullet in the air1 or the quick circular motion of a firebrand1 which are too fast1 or the hour-hand of a common clock1 which is too slow2 from the distance of the object as to place3 as the size of the celestial bodies1 and the size and nature of all distant bodies2 from prepossession by another object3 as one powerful smell renders other smells in the same room imperceptible2 from the interruption of interposing bodies3 as the internal parts of animals2 and because the object is unfit to make an impression upon the sense3 as the air or the invisible and untangible spirit which is included in every living body40 99,complexity of ,quantitative ,questions499 ,one remark which we may well make in entering upon quantitative questions1 has regard to the great variety and extent of phenomena presented to our notice4 ,so long as we deal only with a simply logical question1 that question is merely1 ,does a certain event happen8 or1 ,does a certain object exist8 ,no sooner do we regard the event or object as capable of more and less1 than the question branches out into many4 ,we must now ask1 ,how much is it compared with its cause8 ,does it change when the amount of the cause changes8 ,if so1 does it change in the same or opposite direction8 ,is the change in simple proportion to that of the cause8 ,if not1 what more complex law of connection holds true8 ,this law determined satisfactorily in one series of circumstances may be varied under new conditions1 and the most complex relations of several quantities may ultimately be established4 ,in every question of physical science there is thus a series of steps the first one or two of which are usually made with ease while the succeeding ones demand more and more careful measurement4 ,we cannot lay down any invariable series of questions which must be asked from nature4 ,the exact character of the questions will vary according to the nature of the case1 but they will usually be of an evident kind1 and we may readily illustrate them by examples4 ,suppose that we are investigating the solution of some salt in water4 ,the first is a purely logical question3 ,is there solution1 or is there not8 ,assuming the answer to be in the affirmative1 we next inquire1 ,does the solubility vary with the temperature1 or not8 ,in all probability some variation will exist1 and we must have an answer to the further question1 ,does the quantity dissolved increase1 or does it diminish with the temperature8 ,in by far the greatest number of cases salts and substances of all kinds dissolve more freely the higher the temperature of the water2 but there are a few salts1 such as calcium sulphate1 which follow the opposite rule4 ,a considerable number of salts resemble sodium sulphate in becoming more soluble up to a certain temperature1 and then varying in the opposite direction4 ,we next require to assign the amount of variation as compared with that of the temperature1 assuming at first that the increase of solubility is proportional to the increase of temperature4 ,common salt is an instance of very slight variation1 and potassium nitrate of very considerable increase with temperature4 ,accurate observations will probably show1 however1 that the simple law of proportionate variation is only approximately true1 and some more complicated law involving the second1 third1 or higher powers of the temperature may ultimately be established4 ,all these investigations have to be carried out for each salt separately1 since no distinct principles by which we may infer from one substance to another have yet been detected4 ,there is still an indefinite field for further research open2 for the solubility of salts will probably vary with the pressure under which the medium is placed2 the presence of other salts already dissolved may have effects yet unknown4 ,the researches already effected as regards the solvent power of water must be repeated with alcohol1 ether1 carbon bisulphide1 and other media1 so that unless general laws can be detected1 this one phenomenon of solution can never be exhaustively treated4 ,the same kind of questions recur as regards the solution or absorption of gases in liquids1 the pressure as well as the temperature having then a most decided effect1 and ,professor ,roscoe's researches on the subject present an excellent example of the successive determination of various complicated laws4#ahd #ahd ,watts' 99,dictionary of ,chemistry991 vol4 ii4 p4 #gij4 ,there is hardly a branch of physical science in which similar complications are not ultimately encountered4 ,in the case of gravity1 indeed1 we arrive at the final law1 that the force is the same for all kinds of matter1 and varies only with the distance of action4 ,but in other subjects the laws1 if simple in their ultimate nature1 are disguised and complicated in their apparent results4 ,thus the effect of heat in expanding solids1 and the reverse effect of forcible extension or compression upon the temperature of a body1 will vary from one substance to another1 will vary as the temperature is already higher or lower1 and1 will probably follow a highly complex law1 which in some cases gives negative or exceptional results4 ,in crystalline substances the same researches have to be repeated in each distinct axial direction4 ,in the sciences of pure observation1 such as those of astronomy1 meteorology1 and terrestrial magnetism1 we meet with many interesting series of quantitative determinations4 ,the so-called fixed stars1 as ,giordano ,bruno divined1 are not really fixed1 and may be more truly described as vast wandering orbs1 each pursuing its own path through space4 ,we must then determine separately for each star the following questions3-- #a4 ,does it move8 #b4 ,in what direction8 #c4 ,at what velocity8 #d4 ,is this velocity variable or uniform8 #e4 ,if variable1 according to what law8 #f4 ,is the direction uniform8 #g4 ,if not1 what is the form of the apparent path8 #h4 ,does it approach or recede8 #i4 ,what is the form of the real path8 ,the successive answers to such questions in the case of certain binary stars1 have afforded a proof that the motions are due to a central force coinciding in law with gravity1 and doubtless identical with it4 ,in other cases the motions are usually so small that it is exceedingly difficult to distinguish them with certainty4 ,and the time is yet far off when any general results as regards stellar motions can be established4 ,the variation in the brightness of stars opens an unlimited field for curious observation4 ,there is not a star in the heavens concerning which we might not have to determine3-- #a4 ,does it vary in brightness8 #b4 ,is the brightness increasing or decreasing8 #c4 ,is the variation uniform8 #d4 ,if not1 according to what law does it vary8 ,in a majority of cases the change will probably be found to have a periodic character1 in which case several other questions will arise1 such as-- #e4 ,what is the length of the period8 #f4 ,are there minor periods8 #g4 ,what is the law of variation within the period8 #h4 ,is there any change in the amount of variation8 #i4 ,if so1 is it a secular1 99i4e499 a continually growing change1 or does it give evidence of a greater period8 ,already the periodic changes of a certain number of stars have been determined with accuracy1 and the lengths of the periods vary from less than three days up to intervals of time at least #bej times as great4 ,periods within periods have also been detected4 ,there is1 perhaps1 no subject in which more complicated quantitative conditions have to be determined than terrestrial magnetism4 ,since the time when the declination of the compass was first noticed1 as some suppose by ,columbus1 we have had successive discoveries from time to time of the progressive change of declination from century to century2 of the periodic character of this change2 of the difference of the declination in various parts of the earth's surface2 of the varying laws of the change of declination2 of the dip or inclination of the needle1 and the corresponding laws of its periodic changes2 the horizontal and perpendicular intensities have also been the subject of exact measurement1 and have been found to vary with place and time1 like the directions of the needle2 daily and yearly periodic changes have also been detected1 and all the elements are found to be subject to occasional storms or abnormal perturbations1 in which the eleven year period1 now known to be common to many planetary relations1 is apparent4 ,the complete solution of these motions of the compass needle involves nothing less than a determination of its position and oscillations in every part of the world at any epoch1 the like determination for another epoch1 and so on1 time after time1 until the periods of all changes are ascertained4 ,this one subject offers to men of science an almost inexhaustible field for interesting quantitative research1 in which we shall doubtless at some future time discover the operation of causes now most mysterious and unaccountable4 99,the ,methods of ,accurate ,measurement499 ,in studying the modes by which physicists have accomplished very exact measurements1 we find that they are very various1 but that they may perhaps be reduced under the following three classes3-- #a4 ,the increase or decrease1 in some determinate ratio1 of the quantity to be measured1 so as to bring it within the scope of our senses1 and to equate it with the standard unit1 or some determinate multiple or sub-multiple of this unit4 #b4 ,the discovery of some natural conjunction of events which will enable us to compare directly the multiples of the quantity with those of the unit1 or a quantity related in a definite ratio to that unit4 #c4 ,indirect measurement1 which gives us not the quantity itself1 but some other quantity connected with it by known mathematical relations4 99,conditions of ,accurate ,measurement499 ,several conditions are requisite in order that a measurement may be made with great accuracy1 and that the results may be closely accordant when several independent measurements are made4 ,in the first place the magnitude must be exactly defined by sharp terminations1 or precise marks of inconsiderable thickness4 ,when a boundary is vague and graduated1 like the penumbra in a lunar eclipse1 it is impossible to say where the end really is1 and different people will come to different results4 ,we may sometimes overcome this difficulty to a certain extent1 by observations repeated in a special manner1 as we shall afterwards see2 but when possible1 we should choose opportunities for measurement when precise definition is easy4 ,the moment of occultation of a star by the moon can be observed with great accuracy1 because the star disappears with perfect suddenness2 but there are other astronomical conjunctions1 eclipses1 transits1 c41 which occupy a certain length of time in happening1 and thus open the way to differences of opinion4 ,it would be impossible to observe with precision the movements of a body possessing no definite points of reference4 ,the colours of the complete spectrum shade into each other so continuously that exact determinations of refractive indices would have been impossible1 had we not the dark lines of the solar spectrum as precise points for measurement1 or various kinds of homogeneous light1 such as that of sodium1 possessing a nearly uniform length of vibration4 ,in the second place1 we cannot measure accurately unless we have the means of multiplying or dividing a quantity without considerable error1 so that we may correctly equate one magnitude with the multiple or submultiple of the other4 ,in some cases we operate upon the quantity to be measured1 and bring it into accurate coincidence with the actual standard1 as when in photometry we vary the distance of our luminous body1 until its illuminating power at a certain point is equal to that of a standard lamp4 ,in other cases we repeat the unit until it equals the object1 as in surveying land1 or determining a weight by the balance4 ,the requisites of accuracy now are3--7#a7 ,that we can repeat unit after unit of exactly equal magnitude2 7#b7 ,that these can be joined together so that the aggregate shall really be the sum of the parts4 ,the same conditions apply to subdivision1 which may be regarded as a multiplication of subordinate units4 ,in order to measure to the thousandth of an inch1 we must be able to add thousandth after thousandth without error in the magnitude of these spaces1 or in their conjunction4 99,measuring ,instruments499 ,to consider the mechanical construction of scientific instruments1 is no part of my purpose in this book4 ,i wish to point out merely the general purpose of such instruments1 and the methods adopted to carry out that purpose with great precision4 ,in the first place we must distinguish between the instrument which effects a comparison between two quantities1 and the standard magnitude which often forms one of the quantities compared4 ,the astronomer's clock1 for instance1 is no standard of the efflux of time2 it serves but to subdivide1 with approximate accuracy1 the interval of successive passages of a star across the meridian1 which it may effect perhaps to the tenth part of a second1 or #a_/hfdjjj part of the whole4 ,the moving globe itself is the real standard clock1 and the transit instrument the finger of the clock1 while the stars are the hour1 minute1 and second marks1 none the less accurate because they are disposed at unequal intervals4 ,the photometer is a simple instrument1 by which we compare the relative intensity of rays of light falling upon a given spot4 ,the galvanometer shows the comparative intensity of electric currents passing through a wire4 ,the calorimeter gauges the quantity of heat passing from a given object4 ,but no such instruments furnish the standard unit in terms of which our results are to be expressed4 ,in one peculiar case alone does the same instrument combine the unit of measurement and the means of comparison4 ,a theodolite1 mural circle1 sextant1 or other instrument for the measurement of angular magnitudes has no need of an additional physical unit2 for the circle itself1 or complete revolution1 is the natural unit to which all greater or lesser amounts of angular magnitude are referred4 ,the result of every measurement is to make known the purely numerical ratio existing between the magnitude to be measured1 and a certain other magnitude1 which should1 when possible1 be a fixed unit or standard magnitude1 or at least an intermediate unit of which the value can be ascertained in terms of the ultimate standard4 ,but though a ratio is the required result1 an equation is the mode in which the ratio is determined and expressed4 ,in every measurement we equate some multiple or submultiple of one quantity1 with some multiple or submultiple of another1 and equality is always the fact which we ascertain by the senses4 ,by the eye1 the ear1 or the touch1 we judge whether there is a discrepancy or not between two lights1 two sounds1 two intervals of time1 two bars of metal4 ,often indeed we substitute one sense for the other1 as when the efflux of time is judged by the marks upon a moving slip of paper1 so that equal intervals of time are represented by equal lengths4 ,there is a tendency to reduce all comparisons to the comparison of space magnitudes1 but in every case one of the senses must be the ultimate judge of coincidence or non-coincidence4 ,since the equation to be established may exist between any multiples or submultiples of the quantities compared1 there naturally arise several different modes of comparison adapted to different cases4 ,let 99p99 be the magnitude to be measured1 and 99q99 that in terms of which it is to be expressed4 ,then we wish to find such numbers 99x99 and 99y991 that the equation 99p 7x_/y7q99 may be true4 ,this equation may be presented in four forms1 namely3-- ,first ,form4 ,second ,form4 ,third ,form4 ,fourth ,form4 99p 7x_/y7q99 99p7y_/x7 q99 99py qx99 99p_/x q_/y99 ,each of these modes of expressing the same equation corresponds to one mode of effecting a measurement4 ,when the standard quantity is greater than that to be measured1 we often adopt the first mode1 and subdivide the unit until we get a magnitude equal to that measured4 ,the angles observed in surveying1 in astronomy1 or in goniometry are usually smaller than a whole revolution1 and the measuring circle is divided by the use of the screw and microscope1 until we obtain an angle undistinguishable from that observed4 ,the dimensions of minute objects are determined by subdividing the inch or centimetre1 the screw micrometer being the most accurate means of subdivision4 ,ordinary temperatures are estimated by division of the standard interval between the freezing and boiling points of water1 as marked on a thermometer tube4 ,in a still greater number of cases1 perhaps1 we multiply the standard unit until we get a magnitude equal to that to be measured4 ,ordinary measurement by a foot rule1 a surveyor's chain1 or the excessively careful measurements of the base line of a trigonometrical survey by standard bars1 are sufficient instances of this procedure4 ,in the second case1 where 99p7y_/x7 q991 we multiply or divide a magnitude until we get what is equal to the unit1 or to some magnitude easily comparable with it4 ,as a general rule the quantities which we desire to measure in physical science are too small rather than too great for easy determination1 and the problem consists in multiplying them without introducing error4 ,thus the expansion of a metallic bar when heated from #j;,c to #ajj may be multiplied by a train of levers or cog wheels4 ,in the common thermometer the expansion of the mercury1 though slight1 is rendered very apparent1 and easily measurable by the fineness of the tube1 and many other cases might be quoted4 ,there are some phenomena1 on the contrary1 which are too great or rapid to come within the easy range of our senses1 and our task is then the opposite one of diminution4 ,galileo found it difficult to measure the velocity of a falling body1 owing to the considerable velocity acquired in a single second4 ,he adopted the elegant device1 therefore1 of lessening the rapidity by letting the body roll down an inclined plane1 which enables us to reduce the accelerating force in any required ratio4 ,the same purpose is effected in the well-known experiments performed on ,attwood's machine1 and the measurement of gravity by the pendulum really depends on the same principle applied in a far more advantageous manner4 ,wheatstone invented a beautiful method of galvanometry for strong currents1 which consists in drawing off from the main current a certain determinate portion1 which is equated by the galvanometer to a standard current4 ,in short1 he measures not the current itself but a known fraction of it4 ,in many electrical and other experiments1 we wish to measure the movements of a needle or other body1 which are not only very slight in themselves1 but the manifestations of exceedingly small forces4 ,we cannot even approach a delicately balanced needle without disturbing it4 ,under these circumstances the only mode of proceeding with accuracy1 is to attach a very small mirror to the moving body1 and employ a ray of light reflected from the mirror as an index of its movements4 ,the ray may be considered quite incapable of affecting the body1 and yet by allowing the ray to pass to a sufficient distance1 the motions of the mirror may be increased to almost any extent4 ,a ray of light is in fact a perfectly weightless finger or index of indefinite length1 with the additional advantage that the angular deviation is by the law of reflection double that of the mirror4 ,this method was introduced by ,gauss1 and is now of great importance2 but in ,wollaston's reflecting goniometer a ray of light had previously been employed as an index4 ,lavoisier and ,laplace had also used a telescope in connection with the pyrometer4 ,it is a great advantage in some instruments that they can be readily made to manifest a phenomenon in a greater or less degree1 by a very slight change in the construction4 ,thus either by enlarging the bulb or contracting the tube of the thermometer1 we can make it give more conspicuous indications of change of temperature4 ,the ordinary barometer1 on the other hand1 always gives the variations of pressure on one scale4 ,the torsion balance is remarkable for the extreme delicacy which may be attained by increasing the length and lightness of the rod1 and the length and thinness of the supporting thread4 ,forces so minute as the attraction of gravitation between two balls1 or the magnetic and diamagnetic attraction of common liquids and gases1 may thus be made apparent1 and even measured4 ,the common chemical balance1 too1 is capable theoretically of unlimited sensibility4 ,the third mode of measurement1 which may be called the ,method of ,repetition1 is of such great importance and interest that we must consider it in a separate section4 ,it consists in multiplying both magnitudes to be compared until some multiple of the first is found to coincide very nearly with some multiple of the second4 ,if the multiplication can be effected to an unlimited extent1 without the introduction of countervailing errors1 the accuracy with which the required ratio can be determined is unlimited1 and we thus account for the extraordinary precision with which intervals of time in astronomy are compared together4 ,the fourth mode of measurement1 in which we equate submultiples of two magnitudes1 is comparatively seldom employed1 because it does not conduce to accuracy4 ,in the photometer1 perhaps1 we may be said to use it2 we compare the intensity of two sources of light1 by placing them both at such distances from a given surface1 that the light falling on the surface is tolerable to the eye1 and equally intense from each source4 ,since the intensity of light varies inversely as the square of the distance1 the relative intensities of the luminous bodies are proportional to the squares of their distances4 ,the equal intensity of two rays of similarly coloured light may be most accurately ascertained in the mode suggested by ,arago1 namely1 by causing the rays to pass in opposite directions through two nearly flat lenses pressed together4 ,there is an exact equation between the intensities of the beams when ,newton's rings disappear1 the ring created by one ray being exactly the complement of that created by the other4 99,the ,method of ,repetition499 ,the ratio of two quantities can be determined with unlimited accuracy1 if we can multiply both the object of measurement and the standard unit without error1 and then observe what multiple of the one coincides or nearly coincides with some multiple of the other4 ,although perfect coincidence can never be really attained1 the error thus arising may be indefinitely reduced4 ,for if the equation 99py99 99qx99 be uncertain to the amount 99e991 so that 99py99 99qx99 99e991 then we have 99p99 99q7x_/y799 99e_/y99 1 and as we are supposed to be able to make 99y99 as great as we like without increasing the error 99e991 it follows that we can make 99e99 99y99 as small as we like1 and thus approximate within an inconsiderable quantity to the required ratio 99x99 99y994 ,this method of repetition is naturally employed whenever quantities can be repeated1 or repeat themselves without error of juxtaposition1 which is especially the case with the motions of the earth and heavenly bodies4 ,in determining the length of the sidereal day1 we determine the ratio between the earth's revolution round the sun1 and its rotation on its own axis4 ,we might ascertain the ratio by observing the successive passages of a star across the zenith1 and comparing the interval by a good clock with that between two passages of the sun1 the difference being due to the angular movement of the earth round the sun4 ,in such observations we should have an error of a considerable part of a second at each observation1 in addition to the irregularities of the clock4 ,but the revolutions of the earth repeat themselves day after day1 and year after year1 without the slightest interval between the end of one period and the beginning of another4 ,the operation of multiplication is perfectly performed for us by nature4 ,if1 then1 we can find an observation of the passage of a star across the meridian a hundred years ago1 that is of the interval of time between the passage of the sun and the star1 the instrumental errors in measuring this interval by a clock and telescope may be greater than in the present day1 but will be divided by about #cf1ebd days1 and rendered excessively small4 ,it is thus that astronomers have been able to ascertain the ratio of the mean solar to the sidereal day to the #hth place of decimals 7#ajjbgcgia to #a71 or to the hundred millionth part1 probably the most accurate result of measurement in the whole range of science4 ,the antiquity of this mode of comparison is almost as great as that of astronomy itself4 ,hipparchus made the first clear application of it1 when he compared his own observations with those of ,aristarchus1 made #ade years previously1 and thus ascertained the length of the year4 ,this calculation may in fact be regarded as the earliest attempt at an exact determination of the constants of nature4 ,the method is the main resource of astronomers2 ,tycho1 for instance1 detected the slow diminution of the obliquity of the earth's axis1 by the comparison of observations at long intervals4 ,living astronomers use the method as much as earlier ones2 but so superior in accuracy are all observations taken during the last hundred years to all previous ones1 that it is often found preferable to take a shorter interval1 rather than incur the risk of greater instrumental errors in the earlier observations4 ,it is obvious that many of the slower changes of the heavenly bodies must require the lapse of large intervals of time to render their amount perceptible4 ,hipparchus could not possibly have discovered the smaller inequalities of the heavenly motions1 because there were no previous observations of sufficient age or exactness to exhibit them4 ,and just as the observations of ,hipparchus formed the starting-point for subsequent comparisons1 so a large part of the labour of present astronomers is directed to recording the present state of the heavens so exactly1 that future generations of astronomers may detect changes1 which cannot possibly become known in the present age4 ,the principle of repetition was very ingeniously employed in an instrument first proposed by ,mayer in #agfg1 and carried into practice in the ,repeating ,circle of ,borda4 ,the exact measurement of angles is indispensable1 not only in astronomy but also in trigonometrical surveys1 and the highest skill in the mechanical execution of the graduated circle and telescope will not prevent terminal errors of considerable amount4 ,if instead of one telescope1 the circle be provided with two similar telescopes1 these may be alternately directed to two distant points1 say the marks in a trigonometrical survey1 so that the circle shall be turned through any multiple of the angle subtended by those marks1 before the amount of the angular revolution is read off upon the graduated circle4 ,theoretically speaking1 all error arising from imperfect graduation might thus be indefinitely reduced1 being divided by the number of repetitions4 ,in practice1 the advantage of the invention is not found to be very great1 probably because a certain error is introduced at each observation in the changing and fixing of the telescopes4 ,it is moreover inapplicable to moving objects like the heavenly bodies1 so that its use is confined to important trigonometrical surveys4 ,the pendulum is the most perfect of all instruments1 chiefly because it admits of almost endless repetition4 ,since the force of gravity never ceases1 one swing of the pendulum is no sooner ended than the other is begun1 so that the juxtaposition of successive units is absolutely perfect4 ,provided that the oscillations be equal1 one thousand oscillations will occupy exactly one thousand times as great an interval of time as one oscillation4 ,not only is the subdivision of time entirely dependent on this fact1 but in the accurate measurement of gravity1 and many other important determinations1 it is of the greatest service4 ,in the deepest mine1 we could not observe the rapidity of fall of a body for more than a quarter of a minute1 and the measurement of its velocity would be difficult1 and subject to uncertain errors from resistance of air1 c4 ,in the pendulum1 we have a body which can be kept rising and falling for many hours1 in a medium entirely under our command or if desirable in a vacuum4 ,moreover1 the comparative force of gravity at different points1 at the top and bottom of a mine for instance1 can be determined with wonderful precision1 by comparing the oscillations of two exactly similar pendulums1 with the aid of electric clock signals4 ,to ascertain the comparative times of vibration of two pendulums1 it is only requisite to swing them one in front of the other1 to record by a clock the moment when they coincide in swing1 so that one hides the other1 and then count the number of vibrations until they again come to coincidence4 ,if one pendulum makes 99m99 vibrations and the other 99n991 we at once have our equation 99pn99 99qm992 which gives the length of vibration of either pendulum in terms of the other4 ,this method of coincidence1 embodying the principle of repetition in perfection1 was employed with wonderful skill by ,sir ,george ,airy1 in his experiments on the ,density of the ,earth at the ,harton ,colliery1 the pendulums above and below being compared with clocks1 which again were compared with each other by electric signals4 ,so exceedingly accurate was this method of observation1 as carried out by ,sir ,george ,airy1 that he was able to measure a total difference in the vibrations at the top and bottom of the shaft1 amounting to only #bbd seconds in the twenty-four hours1 with an error of less than one hundredth part of a second1 or one part in #h1fdj1jjj of the whole day4#ahe #ahe 99,philosophical ,transactions991 7#ahef7 vol4 #adf1 ,part i4 p4 #big4 ,the principle of repetition has been elegantly applied in observing the motion of waves in water4 ,if the canal in which the experiments are made be short1 say twenty feet long1 the waves will pass through it so rapidly that an observation of one length1 as practised by ,walker1 will be subject to much terminal error1 even when the observer is very skilful4 ,but it is a result of the undulatory theory that a wave is unaltered1 and loses no time by complete reflection1 so that it may be allowed to travel backwards and forwards in the same canal1 and its motion1 say through sixty lengths1 or #abjj feet1 may be observed with the same accuracy as in a canal #abjj feet long1 with the advantage of greater uniformity in the condition of the canal and water4#ahf ,it is always desirable1 if possible1 to bring an experiment into a small compass1 so that it may be well under command1 and yet we may often by repetition enjoy at the same time the advantage of extensive trial4 #ahf ,airy1 99,on ,tides and ,waves991 ,encyclopdia ,metropolitana1 p4 #cde4 ,scott ,russell1 99,british ,association ,report991 #ahcg1 p4 #dcb4 ,one reason of the great accuracy of weighing with a good balance is the fact1 that weights placed in the same scale are naturally added together without the slightest error4 ,there is no difficulty in the precise juxtaposition of two grams1 but the juxtaposition of two metre measures can only be effected with tolerable accuracy1 by the use of microscopes and many precautions4 ,hence1 the extreme trouble and cost attaching to the exact measurement of a base line for a survey1 the risk of error entering at every juxtaposition of the measuring bars1 and indefatigable attention to all the requisite precautions being necessary throughout the operation4 99,measurements by ,natural ,coincidence499 ,in certain cases a peculiar conjunction of circumstances enables us to dispense more or less with instrumental aids1 and to obtain very exact numerical results in the simplest manner4 ,the mere fact1 for instance1 that no human being has ever seen a different face of the moon from that familiar to us1 conclusively proves that the period of rotation of the moon on its own axis is equal to that of its revolution round the earth4 ,not only have we the repetition of these movements during #ajjj or #bjjj years at least1 but we have observations made for us at very remote periods1 free from instrumental error1 no instrument being needed4 ,we learn that the seventh satellite of ,saturn is subject to a similar law1 because its light undergoes a variation in each revolution1 owing to the existence of some dark tract of land2 now this failure of light always occurs while it is in the same position relative to ,saturn1 clearly proving the equality of the axial and revolutional periods1 as ,huygens perceived4#ahg ,a like peculiarity in the motions of ,jupiter's fourth satellite was similarly detected by ,maraldi in #agac4 #ahg 99,hugenii ,cosmotheoros991 pp4 #aag1 #aah4 ,laplace's 99,systme991 translated1 vol4 i4 p4 #fg4 ,remarkable conjunctions of the planets may sometimes allow us to compare their periods of revolution1 through great intervals of time1 with much accuracy4 ,laplace in explaining the long inequality in the motions of ,jupiter and ,saturn1 was assisted by a conjunction of these planets1 observed at ,cairo1 towards the close of the eleventh century4 ,laplace calculated that such a conjunction must have happened on the #cast of ,october1 ,a4,d4 #ajhg2 and the discordance between the distances of the planets as recorded1 and as assigned by theory1 was less than one-fifth part of the apparent diameter of the sun4 ,this difference being less than the probable error of the early record1 the theory was confirmed as far as facts were available4#ahh #ahh ,grant's 99,history of ,physical ,astronomy991 p4 #abi4 ,ancient astronomers often showed the highest ingenuity in turning any opportunities of measurement which occurred to good account4 ,eratosthenes1 as early as #bej ,b4,c41 happening to hear that the sun at ,syene1 in ,upper ,egypt1 was visible at the summer solstice at the bottom of a well1 proving that it was in the zenith1 proposed to determine the dimensions of the earth1 by measuring the length of the shadow of a rod at ,alexandria on the same day of the year4 ,he thus learnt in a rude manner the difference of latitude between ,alexandria and ,syene and finding it to be about one fiftieth part of the whole circumference1 he ascertained the dimensions of the earth within about one sixth part of the truth4 ,the use of wells in astronomical observation appears to have been occasionally practised in comparatively recent times as by ,flamsteed in #afgi4ahi ,the ,alexandrian astronomers employed the moon as an instrument of measurement in several sagacious modes4 ,when the moon is exactly half full1 the moon1 sun1 and earth1 are at the angles of a right-angled triangle4 ,aristarchus measured at such a time the moon's elongation from the sun1 which gave him the two other angles of the triangle1 and enabled him to judge of the comparative distances of the moon and sun from the earth4 ,his result1 though very rude1 was far more accurate than any notions previously entertained1 and enabled him to form some estimate of the comparative magnitudes of the bodies4 ,eclipses of the moon were very useful to ,hipparchus in ascertaining the longitude of the stars1 which are invisible when the sun is above the horizon4 ,for the moon when eclipsed must be #ahj distant from the sun2 hence it is only requisite to measure the distance of a fixed star in longitude from the eclipsed moon to obtain with ease its angular distance from the sun4 #ahi ,baily's 99,account of ,flamsteed991 p4 lix4 ,in later times the eclipses of ,jupiter have served to measure an angle2 for at the middle moment of the eclipse the satellite must be in the same straight line with the planet and sun1 so that we can learn from the known laws of movement of the satellite the longitude of ,jupiter as seen from the sun4 ,if at the same time we measure the elongation or apparent angular distance of ,jupiter from the sun1 as seen from the earth1 we have all the angles of the triangle between ,jupiter1 the sun1 and the earth1 and can calculate the comparative magnitudes of the sides of the triangle by trigonometry4 ,the transits of ,venus over the sun's face are other natural events which give most accurate measurements of the sun's parallax1 or apparent difference of position as seen from distant points of the earth's surface4 ,the sun forms a kind of background on which the place of the planet is marked1 and serves as a measuring instrument free from all the errors of construction which affect human instruments4 ,the rotation of the earth1 too1 by variously affecting the apparent velocity of ingress or egress of ,venus1 as seen from different places1 discloses the amount of the parallax4 ,it has been sufficiently shown that by rightly choosing the moments of observation1 the planetary bodies may often be made to reveal their relative distance1 to measure their own position1 to record their own movements with a high degree of accuracy4 ,with the improvement of astronomical instruments1 such conjunctions become less necessary to the progress of the science1 but it will always remain advantageous to choose those moments for observation when instrumental errors enter with the least effect4 ,in other sciences1 exact quantitative laws can occasionally be obtained without instrumental measurement1 as when we learn the exactly equal velocity of sounds of different pitch1 by observing that a peal of bells or a musical performance is heard harmoniously at any distance to which the sound penetrates2 this could not be the case1 as ,newton remarked1 if one sound overtook the other4 ,one of the most important principles of the atomic theory1 was proved by implication before the use of the balance was introduced into chemistry4 ,wenzel observed1 before #aggg1 that when two neutral substances decompose each other1 the resulting salts are also neutral4 ,in mixing sodium sulphate and barium nitrate1 we obtain insoluble barium sulphate and neutral sodium nitrate4 ,this result could not follow unless the nitric acid1 requisite to saturate one atom of sodium1 were exactly equal to that required by one atom of barium1 so that an exchange could take place without leaving either acid or base in excess4 ,an important principle of mechanics may also be established by a simple acoustical observation4 ,when a rod or tongue of metal fixed at one end is set in vibration1 the pitch of the sound may be observed to be exactly the same1 whether the vibrations be small or great2 hence the oscillations are isochronous1 or equally rapid1 independently of their magnitude4 ,on the ground of theory1 it can be shown that such a result only happens when the flexure is proportional to the deflecting force4 ,thus the simple observation that the pitch of the sound of a harmonium1 for instance1 does not change with its loudness establishes an exact law of nature4#aij #aij ,jamin1 99,cours de ,physique991 vol4 i4 p4 #aeb4 ,a closely similar instance is found in the proof that the intensity of light or heat rays varies inversely as the square of the distance increases4 ,for the apparent magnitude certainly varies according to this law2 hence1 if the intensity of light varied according to any other law1 the brightness of an object would be different at different distances1 which is not observed to be the case4 ,melloni applied the same kind of reasoning1 in a somewhat different form1 to the radiation of heat-rays4 99,modes of ,indirect ,measurement499 ,some of the most conspicuously beautiful experiments in the whole range of science1 have been devised for the purpose of indirectly measuring quantities1 which in their extreme greatness or smallness surpass the powers of sense4 ,all that we need to do1 is to discover some other conveniently measurable phenomenon1 which is related in a known ratio or according to a known law1 however complicated1 with that to be measured4 ,having once obtained experimental data1 there is no further difficulty beyond that of arithmetic or algebraic calculation4 ,gold is reduced by the gold-beater to leaves so thin1 that the most powerful microscope would not detect any measurable thickness4 ,if we laid several hundred leaves upon each other to multiply the thickness1 we should still have no more than #a_/ajjth of an inch at the most to measure1 and the errors arising in the superposition and measurement would be considerable4 ,but we can readily obtain an exact result through the connected amount of weight4 ,faraday weighed #bjjj leaves of gold1 each #c-c_/h inch square1 and found them equal to #chd grains4 ,from the known specific gravity of gold it was easy to calculate that the average thickness of the leaves was #a_/bhb1jjj of an inch4#aia #aia ,faraday1 99,chemical ,researches991 p4 #cic4 ,we must ascribe to ,newton the honour of leading the way in methods of minute measurement4 ,he did not call waves of light by their right name1 and did not understand their nature2 yet he measured their length1 though it did not exceed the #b1jjj1jjjth part of a metre or the one fifty-thousandth part of an inch4 ,he pressed together two lenses of large but known radii4 ,it was easy to calculate the interval between the lenses at any point1 by measuring the distance from the central point of contact4 ,now1 with homogeneous rays the successive rings of light and darkness mark the points at which the interval between the lenses is equal to one half1 or any multiple of half a vibration of the light1 so that the length of the vibration became known4 ,in a similar manner many phenomena of interference of rays of light admit of the measurement of the wave lengths4 ,fringes of interference arise from rays of light which cross each other at a small angle1 and an excessively minute difference in the lengths of the waves makes a very perceptible difference in the position of the point at which two rays will interfere and produce darkness4 ,fizeau has recently employed ,newton's rings to measure small amounts of motion4 ,by merely counting the number of rings of sodium monochromatic light passing a certain point where two glass plates are in close proximity1 he is able to ascertain with the greatest accuracy and ease the change of distance between these glasses1 produced1 for instance1 by the expansion of a metallic bar1 connected with one of the glass plates4#aib #aib 99,proceedings of the ,royal ,society991 #cjth ,november1 #ahff4 ,nothing excites more admiration than the mode in which scientific observers can occasionally measure quantities1 which seem beyond the bounds of human observation4 ,we know the 99average99 depth of the ,pacific ,ocean to be #ad1aij feet1 not by actual sounding1 which would be impracticable in sufficient detail1 but by noticing the rate of transmission of earthquake waves from the ,south ,american to the opposite coasts1 the rate of movement being connected by theory with the depth of the water4#aic ,in the same way the average depth of the ,atlantic ,ocean is inferred to be no less than #bb1aeg feet1 from the velocity of the ordinary tidal waves4 ,a tidal wave again gives beautiful evidence of an effect of the law of gravity1 which we could never in any other way detect4 ,newton estimated that the moon's force in moving the ocean is only one part in #b1hga1djj of the whole force of gravity1 so that even the pendulum1 used with the utmost skill1 would fail to render it apparent4 ,yet1 the immense extent of the ocean allows the accumulation of the effect into a very palpable amount2 and from the comparative heights of the lunar and solar tides1 ,newton roughly estimated the comparative forces of the moon's and sun's gravity at the earth4#aid #aic ,herschel1 99,physical ,geography991 #dj4 #aid 99,principia991 bk4 iii4 ,prop4 #cg1 99,corollaries991 #b and #c4 ,motte's translation1 vol4 ii4 p4 #caj4 ,a few years ago it might have seemed impossible that we should ever measure the velocity with which a star approaches or recedes from the earth1 since the apparent position of the star is thereby unaltered4 ,but the spectroscope now enables us to detect and even measure such motions with considerable accuracy1 by the alteration which it causes in the apparent rapidity of vibration1 and consequently in the refrangibility of rays of light of definite colour4 ,and while our estimates of the lateral movements of stars depend upon our very uncertain knowledge of their distances1 the spectroscope gives the motions of approach and recess irrespective of other motions excepting that of the earth4 ,it gives in short the motions of approach and recess of the stars relatively to the earth4#aie #aie ,roscoe's 99,spectrum ,analysis991 #ast ed4 p4 #bif4 ,the rapidity of vibration for each musical tone1 having been accurately determined by comparison with the ,syren 7p4 #aj71 we can use sounds as indirect indications of rapid vibrations4 ,it is now known that the contraction of a muscle arises from the periodical contractions of each separate fibre1 and from a faint sound or susurrus which accompanies the action of a muscle1 it is inferred that each contraction lasts for about one #cjjth part of a second4 ,minute quantities of radiant heat are now always measured indirectly by the electricity which they produce when falling upon a thermopile4 ,the extreme delicacy of the method seems to be due to the power of multiplication at several points in the apparatus4 ,the number of elements or junctions of different metals in the thermopile can be increased so that the tension of the electric current derived from the same intensity of radiation is multiplied2 the effect of the current upon the magnetic needle can be multiplied within certain bounds1 by passing the current many times round it in a coil2 the excursions of the needle can be increased by rendering it astatic and increasing the delicacy of its suspension2 lastly1 the angular divergence can be observed1 with any required accuracy1 by the use of an attached mirror and distant scale viewed through a telescope 7p4 #bhg74 ,such is the delicacy of this method of measuring heat1 that ,dr4 ,joule succeeded in making a thermopile which would indicate a difference of #jjjjaad ,cent4#aif #aif 99,philosophical ,transactions99 7#ahei71 vol4 cxlix4 p4 #id4 ,a striking case of indirect measurement is furnished by the revolving mirror of ,wheatstone and ,foucault1 whereby a minute interval of time is estimated in the form of an angular deviation4 ,wheatstone viewed an electric spark in a mirror rotating so rapidly1 that if the duration of the spark had been more than one #gb1jjjth part of a second1 the point of light would have appeared elongated to an angular extent of one-half degree4 ,in the spark1 as drawn directly from a ,leyden jar1 no elongation was apparent1 so that the duration of the spark was immeasurably small2 but when the discharge took place through a bad conductor1 the elongation of the spark denoted a sensible duration4#aig ,in the hands of ,foucault the rotating mirror gave a measure of the time occupied by light in passing through a few metres of space4 #aig ,watts' 99,dictionary of ,chemistry991 vol4 ii4 p4 #cic4 99,comparative ,use of ,measuring ,instruments499 ,in almost every case a measuring instrument serves1 and should serve only as a means of comparison between two or more magnitudes4 ,as a general rule1 we should not attempt to make the divisions of the measuring scale exact multiples or submultiples of the unit1 but1 regarding them as arbitrary marks1 should determine their values by comparison with the standard itself4 ,the perpendicular wires in the field of a transit telescope1 are fixed at nearly equal but arbitrary distances1 and those distances are afterwards determined1 as first suggested by ,malvasia1 by watching the passage of star after star across them1 and noting the intervals of time by the clock4 ,owing to the perfectly regular motion of the earth1 these time intervals give exact determinations of the angular intervals4 ,in the same way1 the angular value of each turn of the screw micrometer attached to a telescope1 can be easily and accurately ascertained4 ,when a thermopile is used to observe radiant heat1 it would be almost impossible to calculate on 99 priori99 grounds what is the value of each division of the galvanometer circle1 and still more difficult to construct a galvanometer1 so that each division should have a given value4 ,but this is quite unnecessary1 because by placing the thermopile before a body of known dimensions1 at a known distance1 with a known temperature and radiating power1 we measure a known amount of radiant heat1 and inversely measure the value of the indications of the thermopile4 ,in a similar way ,dr4 ,joule ascertained the actual temperature produced by the compression of bars of metal4 ,for having inserted a small thermopile composed of a single junction of copper and iron wire1 and noted the deflections of the galvanometer1 he had only to dip the bars into water of different temperatures1 until he produced a like deflection1 in order to ascertain the temperature developed by pressure4#aih #aih 99,philosophical ,transactions99 7#ahei71 vol4 cxlix4 p4 #aai1 c4 ,in some cases we are obliged to accept a very carefully constructed instrument as a standard1 as in the case of a standard barometer or thermometer4 ,but it is then best to treat all inferior instruments comparatively only1 and determine the values of their scales by comparison with the assumed standard4 99,systematic ,performance of ,measurements499 ,when a large number of accurate measurements have to be effected1 it is usually desirable to make a certain number of determinations with scrupulous care1 and afterwards use them as points of reference for the remaining determinations4 ,in the trigonometrical survey of a country1 the principal triangulation fixes the relative positions and distances of a few points with rigid accuracy4 ,a minor triangulation refers every prominent hill or village to one of the principal points1 and then the details are filled in by reference to the secondary points4 ,the survey of the heavens is effected in a like manner4 ,the ancient astronomers compared the right ascensions of a few principal stars with the moon1 and thus ascertained their positions with regard to the sun2 the minor stars were afterwards referred to the principal stars4 ,tycho followed the same method1 except that he used the more slowly moving planet ,venus instead of the moon4 ,flamsteed was in the habit of using about seven stars1 favourably situated at points all round the heavens4 ,in his early observations the distances of the other stars from these standard points were determined by the use of the quadrant4#aii ,even since the introduction of the transit telescope and the mural circle1 tables of standard stars are formed at ,greenwich1 the positions being determined with all possible accuracy1 so that they can be employed for purposes of reference by astronomers4 #aii ,baily's 99,account of ,flamsteed991 pp4 #cgh-chj4 ,in ascertaining the specific gravities of substances1 all gases are referred to atmospheric air at a given temperature and pressure2 all liquids and solids are referred to water4 ,we require to compare the densities of water and air with great care1 and the comparative densities of any two substances whatever can then be ascertained4 ,in comparing a very great with a very small magnitude1 it is usually desirable to break up the process into several steps1 using intermediate terms of comparison4 ,we should never think of measuring the distance from ,london to ,edinburgh by laying down measuring rods1 throughout the whole length4 ,a base of several miles is selected on level ground1 and compared on the one hand with the standard yard1 and on the other with the distance of ,london and ,edinburgh1 or any other two points1 by trigonometrical survey4 ,again1 it would be exceedingly difficult to compare the light of a star with that of the sun1 which would be about thirty thousand million times greater2 but ,herschel#bjj effected the comparison by using the full moon as an intermediate unit4 ,wollaston ascertained that the sun gave #hja1jgb times as much light as the full moon1 and ,herschel determined that the light of the latter exceeded that of ,centauri #bg1djh times1 so that we find the ratio between the light of the sun and star to be that of about #bb1jjj1jjj1jjj to #a4 #bjj ,herschel's 99,astronomy991 #hag1 #dth4 ed4 p4 #eec4 99,the ,pendulum499 ,by far the most perfect and beautiful of all instruments of measurement is the pendulum4 ,consisting merely of a heavy body suspended freely at an invariable distance from a fixed point1 it is most simple in construction2 yet all the highest problems of physical measurement depend upon its careful use4 ,its excessive value arises from two circumstances4 7#a7 ,the method of repetition is eminently applicable to it1 as already described 7p4 #bij74 7#b7 ,unlike other instruments1 it connects together three different quantities1 those of space1 time1 and force4 ,in most works on natural philosophy it is shown1 that when the oscillations of the pendulum are infinitely small1 the square of the time occupied by an oscillation is directly proportional to the length of the pendulum1 and indirectly proportional to the force affecting it1 of whatever kind4 ,the whole theory of the pendulum is contained in the formula1 first given by ,huygens in his 99,horologium ,oscillatorium994 ,time of oscillation #cadaei 7length of pendulum_/force74 ,the quantity #cadaei is the constant ratio of the circumference and radius of a circle1 and is of course known with accuracy4 ,hence1 any two of the three quantities concerned being given1 the third may be found2 or any two being maintained invariable1 the third will be invariable4 ,thus a pendulum of invariable length suspended at the same place1 where the force of gravity may be considered constant1 furnishes a measure of time4 ,the same invariable pendulum being made to vibrate at different points of the earth's surface1 and the times of vibration being astronomically determined1 the force of gravity becomes accurately known4 ,finally1 with a known force of gravity1 and time of vibration ascertained by reference to the stars1 the length is determinate4 ,all astronomical observations depend upon the first manner of using the pendulum1 namely1 in the astronomical clock4 ,in the second employment it has been almost equally indispensable4 ,the primary principle that gravity is equal in all matter was proved by ,newton's and ,gauss' pendulum experiments4 ,the torsion pendulum of ,michell1 ,cavendish1 and ,baily1 depending upon exactly the same principles as the ordinary pendulum1 gave the density of the earth1 one of the foremost natural constants4 ,kater and ,sabine1 by pendulum observations in different parts of the earth1 ascertained the variation of gravity1 whence comes a determination of the earth's ellipticity4 ,the laws of electric and magnetic attraction have also been determined by the method of vibrations1 which is in constant use in the measurement of the horizontal force of terrestrial magnetism4 ,we must not confuse with the ordinary use of the pendulum its application by ,newton1 to show the absence of internal friction against space1#bja or to ascertain the laws of motion and elasticity4#bjb ,in these cases the extent of vibration is the quantity measured1 and the principles of the instrument are different4 #bja 99,principia991 bk4 ii4 ,sect4 #f4 ,prop4 #ca4 ,motte's ,translation1 vol4 ii4 p4 #ajg4 #bjb ,ibid4 bk4 i4 ,law iii4 ,corollary #f4 ,motte's ,translation1 vol4 i4 p4 #cc4 99,attainable ,accuracy of ,measurement499 ,it is a matter of some interest to compare the degrees of accuracy which can be attained in the measurement of different kinds of magnitude4 ,few measurements of any kind are exact to more than six significant figures1#bjc but it is seldom that such accuracy can be hoped for4 ,time is the magnitude which seems to be capable of the most exact estimation1 owing to the properties of the pendulum1 and the principle of repetition described in previous sections4 ,as regards short intervals of time1 it has already been stated that ,sir ,george ,airy was able to estimate one part in #h1fdj1jjj1 an exactness1 as he truly remarks1 8almost beyond conception40#bjd ,the ratio between the mean solar and the sidereal day is known to be about one part in one hundred millions1 or to the eighth place of decimals1 7p4 #bhi74 #bjc ,thomson and ,tait's 99,natural ,philosophy991 vol4 i4 p4 #ccc4 #bjd 99,philosophical ,transactions991 7#ahef71 vol4 cxlvi4 pp4 #ccj1 #cca4 ,determinations of weight seem to come next in exactness1 owing to the fact that repetition without error is applicable to them4 ,an ordinary good balance should show about one part in #ejj1jjj of the load4 ,the finest balance employed by ,m4 ,stas1 turned with one part in #hbe1jjj of the load4#bje ,but balances have certainly been constructed to show one part in a million1#bjf and ,ramsden is said to have constructed a balance for the ,royal ,society1 to indicate one part in seven millions1 though this is hardly credible4 ,professor ,clerk ,maxwell takes it for granted that one part in five millions can be detected1 but we ought to discriminate between what a balance can do when first constructed1 and when in continuous use4 #bje 99,first ,annual ,report of the ,mint991 p4 #ajf4 #bjf ,jevons1 in ,watts' 99,dictionary of ,chemistry991 vol4 i4 p4 #dhc4 ,determinations of length1 unless performed with extraordinary care1 are open to much error in the junction of the measuring bars4 ,even in measuring the base line of a trigonometrical survey1 the accuracy generally attained is only that of about one part in #fj1jjj1 or an inch in the mile2 but it is said that in four measurements of a base line carried out very recently at ,cape ,comorin1 the greatest error was #jjgg inch in #afh mile1 or one part in #a1chb1djj1 an almost incredible degree of accuracy4 ,sir ,j4 ,whitworth has shown that touch is even a more delicate mode of measuring lengths than sight1 and by means of a splendidly executed screw1 and a small cube of iron placed between two flat-ended iron bars1 so as to be suspended when touching them1 he can detect a change of dimension in a bar1 amounting to no more than one-millionth of an inch4#bjg #bjg ,british ,association1 ,glasgow1 #ahef4 99,address of the ,president of the ,mechanical ,section994 ,,chapter ,,xiv4 ,,units ,,and ,,standards ,,of ,,measurement4 ,as we have seen1 instruments of measurement are only means of comparison between one magnitude and another1 and as a general rule we must assume some one arbitrary magnitude1 in terms of which all results of measurement are to be expressed4 ,mere ratios between any series of objects will never tell us their absolute magnitudes2 we must have at least one ratio for each1 and we must have one absolute magnitude4 ,the number of ratios 99n99 are expressible in 99n99 equations1 which will contain at least 99n99 #a quantities1 so that if we employ them to make known 99n99 magnitudes1 we must have one magnitude known4 ,hence1 whether we are measuring time1 space1 density1 mass1 weight1 energy1 or any other physical quantity1 we must refer to some concrete standard1 some actual object1 which if once lost and irrecoverable1 all our measures lose their absolute meaning4 ,this concrete standard is in all cases arbitrary in point of theory1 and its selection a question of practical convenience4 ,there are two kinds of magnitude1 indeed1 which do not need to be expressed in terms of arbitrary concrete units1 since they pre-suppose the existence of natural standard units4 ,one case is that of abstract number itself1 which needs no special unit1 because any object which exists or is thought of as separate from other objects 7p4 #aeg7 furnishes us with a unit1 and is the only standard required4 ,angular magnitude is the second case in which we have a natural unit of reference1 namely the whole revolution or 99perigon991 as it has been called by ,mr4 ,sandeman4#bjh ,it is a necessary result of the uniform properties of space1 that all complete revolutions are equal to each other1 so that we need not select any one revolution1 but can always refer anew to space itself4 ,whether we take the whole perigon1 its half1 or its quarter1 is really immaterial2 ,euclid took the right angle1 because the ,greek geometers had never generalised their notions of angular magnitude sufficiently to treat angles of all magnitudes1 or of unlimited 99quantity of revolution994 ,euclid defines a right angle as half that made by a line with its own continuation1 which is of course equal to half a revolution1 but which was not treated as an angle by him4 ,in mathematical analysis a different fraction of the perigon is taken1 namely1 such a fraction that the arc or portion of the circumference included within it is equal to the radius of the circle4 ,in this point of view angular magnitude is an abstract ratio1 namely1 the ratio between the length of arc subtended and the length of the radius4 ,the geometrical unit is then necessarily the angle corresponding to the ratio unity4 ,this angle is equal to about #eg1 #ag1 #ddh1 or decimally #egbieggieac444 4#bji ,it was called by ,de ,morgan the 99arcual unit991 but a more convenient name for common use would be 99radian991 as suggested by ,professor ,everett4 ,though this standard angle is naturally employed in mathematical analysis1 and any other unit would introduce great complexity1 we must not look upon it as a distinct unit1 since its amount is connected with that of the half perigon1 by the natural constant #cadaei444 usually denoted by the letter 4 #bjh 99,pelicotetics1 or the ,science of ,quantity2 an ,elementary ,treatise on ,algebra1 and its groundwork ,arithmetic499 ,by ,archibald ,sandeman1 ,m4 ,a4 ,cambridge 7,deighton1 ,bell1 and ,co471 #ahfh1 p4 #cjd4 #bji ,de ,morgan's 99,trigonometry and ,double ,algebra991 p4 #e4 ,when we pass to other species of quantity1 the choice of unit is found to be entirely arbitrary4 ,there is absolutely no mode of defining a length1 but by selecting some physical object exhibiting that length between certain obvious points--as1 for instance1 the extremities of a bar1 or marks made upon its surface4 99,standard ,unit of ,time499 ,time is the great independent variable of all change--that which itself flows on uninterruptedly1 and brings the variety which we call motion and life4 ,when we reflect upon its intimate nature1 ,time1 like every other element of existence1 proves to be an inscrutable mystery4 ,we can only say with ,st4 ,augustin1 to one who asks us what is time1 8,i know when you do not ask me40 ,the mind of man will ask what can never be answered1 but one result of a true and rigorous logical philosophy must be to convince us that scientific explanation can only take place between phenomena which have something in common1 and that when we get down to primary notions1 like those of time and space1 the mind must meet a point of mystery beyond which it cannot penetrate4 ,a definition of time must not be looked for2 if we say with ,hobbes1#baj that it is 8the phantasm of before and after in motion10 or with ,aristotle that it is 8the number of motion according to former and latter10 we obviously gain nothing1 because the notion of time is involved in the expressions 99before and after991 99former and latter994 ,time is undoubtedly one of those primary notions which can only be defined physically1 or by observation of phenomena which proceed in time4 #baj 99,english ,works of ,thos4 ,hobbes991 ,edit4 by ,molesworth1 vol4 i4 p4 #ie4 ,if we have not advanced a step beyond ,augustin's acute reflections on this subject1#baa it is curious to observe the wonderful advances which have been made in the practical measurement of its efflux4 ,in earlier centuries the rude sun-dial or the rising of a conspicuous star gave points of reference1 while the flow of water from the clepsydra1 the burning of a candle1 or1 in the monastic ages1 even the continuous chanting of psalms1 were the means of roughly subdividing periods1 and marking the hours of the day and night4#bab ,the sun and stars still furnish the standard of time1 but means of accurate subdivision have become requisite1 and this has been furnished by the pendulum and the chronograph4 ,by the pendulum we can accurately divide the day into seconds of time4 ,by the chronograph we can subdivide the second into a hundred1 a thousand1 or even a million parts4 ,wheatstone measured the duration of an electric spark1 and found it to be no more than one #aae1bjjth part of a second1 while more recently ,captain ,noble has been able to appreciate intervals of time not exceeding the millionth part of a second4 #baa 99,confessions991 bk4 xi4 chapters #bj-bh4 #bab ,sir ,g4 ,c4 ,lewis gives many curious particulars concerning the measurement of time in his 99,astronomy of the ,ancients991 pp4 #bda1 c4 ,when we come to inquire precisely what phenomenon it is that we thus so minutely measure1 we meet insurmountable difficulties4 ,newton distinguished time according as it was 99absolute99 or 99apparent99 time1 in the following words3--8,absolute1 true1 and mathematical time1 of itself and from its own nature1 flows equably without regard to anything external1 and by another name is called 99duration992 relative1 apparent and common time1 is some sensible and external measure of duration by the means of motion40#bac ,though we are perhaps obliged to assume the existence of a uniformly increasing quantity which we call time1 yet we cannot feel or know abstract and absolute time4 ,duration must be made manifest to us by the recurrence of some phenomenon4 ,the succession of our own thoughts is no doubt the first and simplest measure of time1 but a very rude one1 because in some persons and circumstances the thoughts evidently flow with much greater rapidity than in other persons and circumstances4 ,in the absence of all other phenomena1 the interval between one thought and another would necessarily become the unit of time1 but the most cursory observations show that there are changes in the outward world much better fitted by their constancy to measure time than the change of thoughts within us4 #bac 99,principia991 bk4 i4 99,scholium to ,definitions994 ,translated by ,motte1 vol4 i4 p4 #i4 ,see also p4 #aa4 ,the earth1 as ,i have already said1 is the real clock of the astronomer1 and is practically assumed as invariable in its movements4 ,but on what ground is it so assumed8 ,according to the first law of motion1 every body perseveres in its state of rest or of uniform motion in a right line1 unless it is compelled to change that state by forces impressed thereon4 ,rotatory motion is subject to a like condition1 namely1 that it perseveres uniformly unless disturbed by extrinsic forces4 ,now uniform motion means motion through equal spaces in equal times1 so that if we have a body entirely free from all resistance or perturbation1 and can measure equal spaces of its path1 we have a perfect measure of time4 ,but let it be remembered that this law has never been absolutely proved by experience2 for we cannot point to any body1 and say that it is wholly unresisted or undisturbed2 and even if we had such a body1 we should need some independent standard of time to ascertain whether its motion was really uniform4 ,as it is in moving bodies that we find the best standard of time1 we cannot use them to prove the uniformity of their own movements1 which would amount to a 99petitio principii994 ,our experience comes to this1 that when we examine and compare the movements of bodies which seem to us nearly free from disturbance1 we find them giving nearly harmonious measures of time4 ,if any one body which seems to us to move uniformly is not doing so1 but is subject to fits and starts unknown to us1 because we have no absolute standard of time1 then all other bodies must be subject to the same arbitrary fits and starts1 otherwise there would be discrepancy disclosing the irregularities4 ,just as in comparing together a number of chronometers1 we should soon detect bad ones by their going irregularly1 as compared with the others1 so in nature we detect disturbed movement by its discrepancy from that of other bodies which we believe to be undisturbed1 and which agree nearly among themselves4 ,but inasmuch as the measure of motion involves time1 and the measure of time involves motion1 there must be ultimately an assumption4 ,we may define equal times1 as times during which a moving body under the influence of no force describes equal spaces2#bad but all we can say in support of this definition is1 that it leads us into no known difficulties1 and that to the best of our experience one freely moving body gives the same results as any other4 #bad ,rankine1 99,philosophical ,magazine991 ,feb4 #ahfg1 vol4 xxxiii4 p4 #ia4 ,when we inquire where the freely moving body is1 no perfectly satisfactory answer can be given4 ,practically the rotating globe is sufficiently accurate1 and ,thomson and ,tait say3 8,equal times are times during which the earth turns through equal angles40#bae ,no long time has passed since astronomers thought it impossible to detect any inequality in its movement4 ,poisson was supposed to have proved that a change in the length of the sidereal day amounting to one ten-millionth part in #b1ejj years was incompatible with an ancient eclipse recorded by the ,chaldans1 and similar calculations were made by ,laplace4 ,but it is now known that these calculations were somewhat in error1 and that the dissipation of energy arising out of the friction of tidal waves1 and the radiation of the heat into space1 has slightly decreased the rapidity of the earth's rotatory motion4 ,the sidereal day is now longer by one part in #b1gjj1jjj1 than it was in #gbj ,b4,c4 ,even before this discovery1 it was known that invariability of rotation depended upon the perfect maintenance of the earth's internal heat1 which is requisite in order that the earth's dimensions shall be unaltered4 ,now the earth being superior in temperature to empty space1 must cool more or less rapidly1 so that it cannot furnish an absolute measure of time4 ,similar objections could be raised to all other rotating bodies within our cognisance4 #bae 99,treatise on ,natural ,philosophy991 vol4 i4 p4 #agi4 ,the moon's motion round the earth1 and the earth's motion round the sun1 form the next best measure of time4 ,they are subject1 indeed1 to disturbance from other planets1 but it is believed that these perturbations must in the course of time run through their rhythmical courses1 leaving the mean distances unaffected1 and consequently1 by the third ,law of ,kepler1 the periodic times unchanged4 ,but there is more reason than not to believe that the earth encounters a slight resistance in passing through space1 like that which is so apparent in ,encke's comet4 ,there may also be dissipation of energy in the electrical relations of the earth to the sun1 possibly identical with that which is manifested in the retardation of comets4#baf ,it is probably an untrue assumption then1 that the earth's orbit remains quite invariable4 ,it is just possible that some other body may be found in the course of time to furnish a better standard of time than the earth in its annual motion4 ,the greatly superior mass of ,jupiter and its satellites1 and their greater distance from the sun1 may render the electrical dissipation of energy less considerable than in the case of the earth4 ,but the choice of the best measure will always be an open one1 and whatever moving body we choose may ultimately be shown to be subject to disturbing forces4 #baf 99,proceedings of the ,manchester ,philosophical ,society991 #bhth ,nov4 #ahga1 vol4 xi4 p4 #cc4 ,the pendulum1 although so admirable an instrument for subdivision of time1 fails as a standard2 for though the same pendulum affected by the same force of gravity performs equal vibrations in equal times1 yet the slightest change in the form or weight of the pendulum1 the least corrosion of any part1 or the most minute displacement of the point of suspension1 falsifies the results1 and there enter many other difficult questions of temperature1 friction1 resistance1 length of vibration1 c4 ,thomson and ,tait are of opinion#bag that the ultimate standard of chronometry must be founded on the physical properties of some body of more constant character than the earth2 for instance1 a carefully arranged metallic spring1 hermetically sealed in an exhausted glass vessel4 ,but it is hard to see how we can be sure that the dimensions and elasticity of a piece of wrought metal will remain perfectly unchanged for the few millions of years contemplated by them4 ,a nearly perfect gas1 like hydrogen1 is perhaps the only kind of substance in the unchanged elasticity of which we could have confidence4 ,moreover1 it is difficult to perceive how the undulations of such a spring could be observed with the requisite accuracy4 ,more recently ,professor ,clerk ,maxwell has made the novel suggestion1 discussed in a subsequent section1 that undulations of light 99in vacuo99 would form the most universal standard of reference1 both as regards time and space4 ,according to this system the unit of time would be the time occupied by one vibration of the particular kind of light whose wave length is taken as the unit of length4 #bag 99,the ,elements of ,natural ,philosophy991 part i4 p4 #aai4 99,the ,unit of ,space and the ,bar ,standard499 ,next in importance after the measurement of time is that of space4 ,time comes first in theory1 because phenomena1 our internal thoughts for instance1 may change in time without regard to space4 ,as to the phenomena of outward nature1 they tend more and more to resolve themselves into motions of molecules1 and motion cannot be conceived or measured without reference both to time and space4 ,turning now to space measurement1 we find it almost equally difficult to fix and define once and for ever1 a unit magnitude4 ,there are three different modes in which it has been proposed to attempt the perpetuation of a standard length4 7#a7 ,by constructing an actual specimen of the standard yard or metre1 in the form of a bar4 7#b7 ,by assuming the globe itself to be the ultimate standard of magnitude1 the practical unit being a submultiple of some dimension of the globe4 7#c7 ,by adopting the length of the simple seconds pendulum1 as a standard of reference4 ,at first sight it might seem that there was no great difficulty in this matter1 and that any one of these methods might serve well enough2 but the more minutely we inquire into the details1 the more hopeless appears to be the attempt to establish an invariable standard4 ,we must in the first place point out a principle not of an obvious character1 namely1 that 99the standard length must be defined by one single object994#bah ,to make two bars of exactly the same length1 or even two bars bearing a perfectly defined ratio to each other1 is beyond the power of human art4 ,if two copies of the standard metre be made and declared equally correct1 future investigators will certainly discover some discrepancy between them1 proving of course that they cannot both be the standard1 and giving cause for dispute as to what magnitude should then be taken as correct4 #bah ,see ,harris' 99,essay upon ,money and ,coins991 part4 ii4 #ageh p4 #abg4 ,if one invariable bar could be constructed and maintained as the absolute standard1 no such inconvenience could arise4 ,each successive generation as it acquired higher powers of measurement1 would detect errors in the copies of the standard1 but the standard itself would be unimpeached1 and would1 as it were1 become by degrees more and more accurately known4 ,unfortunately to construct and preserve a metre or yard is also a task which is either impossible1 or what comes nearly to the same thing1 cannot be shown to be possible4 ,passing over the practical difficulty of defining the ends of the standard length with complete accuracy1 whether by dots or lines on the surface1 or by the terminal points of the bar1 we have no means of proving that substances remain of invariable dimensions4 ,just as we cannot tell whether the rotation of the earth is uniform1 except by comparing it with other moving bodies1 believed to be more uniform in motion1 so we cannot detect the change of length in a bar1 except by comparing it with some other bar supposed to be invariable4 ,but how are we to know which is the invariable bar8 ,it is certain that many rigid and apparently invariable substances do change in dimensions4 ,the bulb of a thermometer certainly contracts by age1 besides undergoing rapid changes of dimensions when warmed or cooled through #ajj ,cent4 ,can we be sure that even the most solid metallic bars do not slightly contract by age1 or undergo variations in their structure by change of temperature4 ,fizeau was induced to try whether a quartz crystal1 subjected to several hundred alternations of temperature1 would be modified in its physical properties1 and he was unable to detect any change in the coefficient of expansion4#bai ,it does not follow1 however1 that1 because no apparent change was discovered in a quartz crystal1 newly-constructed bars of metal would undergo no change4 #bai 99,philosophical ,magazine991 7#ahfh71 #dth ,series1 vol4 xxxvi4 p4 #cb4 ,the best principle1 as it seems to me1 upon which the perpetuation of a standard of length can be rested1 is that1 if a variation of length occurs1 it will in all probability be of different amount in different substances4 ,if then a great number of standard metres were constructed of all kinds of different metals and alloys2 hard rocks1 such as granite1 serpentine1 slate1 quartz1 limestone2 artificial substances1 such as porcelain1 glass1 c41 c41 careful comparison would show from time to time the comparative variations of length of these different substances4 ,the most variable substances would be the most divergent1 and the standard would be furnished by the mean length of those which agreed most closely with each other just as uniform motion is that of those bodies which agree most closely in indicating the efflux of time4 99,the ,terrestrial ,standard499 ,the second method assumes that the globe itself is a body of invariable dimensions and the founders of the metrical system selected the ten-millionth part of the distance from the equator to the pole as the definition of the metre4 ,the first imperfection in such a method is that the earth is certainly not invariable in size2 for we know that it is superior in temperature to surrounding space1 and must be slowly cooling and contracting4 ,there is much reason to believe that all earthquakes1 volcanoes1 mountain elevations1 and changes of sea level are evidences of this contraction as asserted by ,mr4 ,mallet4#bbj ,but such is the vast bulk of the earth and the duration of its past existence1 that this contraction is perhaps less rapid in proportion than that of any bar or other material standard which we can construct4 #bbj 99,proceedings of the ,royal ,society991 #bjth ,june1 #ahgb1 vol4 xx4 p4 #dch4 ,the second and chief difficulty of this method arises from the vast size of the earth1 which prevents us from making any comparison with the ultimate standard1 except by a trigonometrical survey of a most elaborate and costly kind4 ,the ,french physicists1 who first proposed the method1 attempted to obviate this inconvenience by carrying out the survey once for all1 and then constructing a standard metre1 which should be exactly the one ten millionth part of the distance from the pole to the equator4 ,but since all measuring operations are merely approximate1 it was impossible that this operation could be perfectly achieved4 ,accordingly1 it was shown in #ahch that the supposed ,french metre was erroneous to the considerable extent of one part in #eebg4 ,it then became necessary either to alter the length of the assumed metre1 or to abandon its supposed relation to the earth's dimensions4 ,the ,french ,government and the ,international ,metrical ,commission have for obvious reasons decided in favour of the latter course1 and have thus reverted to the first method of defining the metre by a given bar4 ,as from time to time the ratio between this assumed standard metre and the quadrant of the earth becomes more accurately known1 we have better means of restoring that metre by reference to the globe if required4 ,but until lost1 destroyed1 or for some clear reason discredited1 the bar metre and not the globe is the standard4 ,thomson and ,tait remark that any of the more accurate measurements of the ,english trigonometrical survey might in like manner be employed to restore our standard yard1 in terms of which the results are recorded4 99,the ,pendulum ,standard499 ,the third method of defining a standard length1 by reference to the seconds pendulum1 was first proposed by ,huyghens1 and was at one time adopted by the ,english ,government4 ,from the principle of the pendulum 7p4 #cjb7 it clearly appears that if the time of oscillation and the force actuating the pendulum be the same1 the length of the pendulum must be the same4 ,we do not get rid of theoretical difficulties1 for we must assume the attraction of gravity at some point of the earth's surface1 say ,london1 to be unchanged from time to time1 and the sidereal day to be invariable1 neither assumption being absolutely correct so far as we can judge4 ,the pendulum1 in short1 is only an indirect means of making one physical quantity of space depend upon two other physical quantities of time and force4 ,the practical difficulties are1 however1 of a far more serious character than the theoretical ones4 ,the length of a pendulum is not the ordinary length of the instrument1 which might be greatly varied without affecting the duration of a vibration1 but the distance from the centre of suspension to the centre of oscillation4 ,there are no direct means of determining this latter centre1 which depends upon the average momentum of all the particles of the pendulum as regards the centre of suspension4 ,huyghens discovered that the centres of suspension and oscillation are interchangeable1 and ,kater pointed out that if a pendulum vibrates with exactly the same rapidity when suspended from two different points1 the distance between these points is the true length of the equivalent simple pendulum4#bba ,but the practical difficulties in employing ,kater's reversible pendulum are considerable1 and questions regarding the disturbance of the air1 the force of gravity1 or even the interference of electrical attractions have to be entertained4 ,it has been shown that all the experiments made under the authority of ,government for determining the ratio between the standard yard and the seconds pendulum1 were vitiated by an error in the corrections for the resisting1 adherent1 or buoyant power of the air in which the pendulums were swung4 ,even if such corrections were rendered unnecessary by operating in a vacuum1 other difficult questions remain4#bbb ,gauss' mode of comparing the vibrations of a wire pendulum when suspended at two different lengths is open to equal or greater practical difficulties4 ,thus it is found that the pendulum standard cannot compete in accuracy and certainty with the simple bar standard1 and the method would only be useful as an accessory mode of restoring the bar standard if at any time again destroyed4 #bba ,kater's 99,treatise on ,mechanics991 ,cabinet ,cyclopdia1 p4 #aed4 #bbb ,grant's 99,history of ,physical ,astronomy991 p4 #aef4 99,unit of ,density499 ,before we can measure the phenomena of nature1 we require a third independent unit1 which shall enable us to define the quantity of matter occupying any given space4 ,all the changes of nature1 as we shall see1 are probably so many manifestations of energy2 but energy requires some substratum or material machinery of molecules1 in and by which it may be manifested4 ,observation shows that1 as regards force1 there may be two modes of variation of matter4 ,as ,newton says in the first definition of the ,principia1 8the quantity of matter is the measure of the same1 arising from its density and bulk conjunctly40 ,thus the force required to set a body in motion varies both according to the bulk of the matter1 and also according to its quality4 ,two cubic inches of iron of uniform quality1 will require twice as much force as one cubic inch to produce a certain velocity in a given time2 but one cubic inch of gold will require more force than one cubic inch of iron4 ,there is then some new measurable quality in matter apart from its bulk1 which we may call 99density991 and which is1 strictly speaking1 indicated by its capacity to resist and absorb the action of force4 ,for the unit of density we may assume that of any substance which is uniform in quality1 and can readily be referred to from time to time4 ,pure water at any definite temperature1 for instance that of snow melting under inappreciable pressure1 furnishes an invariable standard of density1 and by comparing equal bulks of various substances with a like bulk of ice-cold water1 as regards the velocity produced in a unit of time by the same force1 we should ascertain the densities of those substances as expressed in that of water4 ,practically the force of gravity is used to measure density2 for a beautiful experiment with the pendulum1 performed by ,newton and repeated by ,gauss1 shows that all kinds of matter gravitate equally4 ,two portions of matter then which are in equilibrium in the balance1 may be assumed to possess equal inertia1 and their densities will therefore be inversely as their cubic dimensions4 99,unit of ,mass499 ,multiplying the number of units of density of a portion of matter1 by the number of units of space occupied by it1 we arrive at the quantity of matter1 or1 as it is usually called1 the 99unit of mass991 as indicated by the inertia and gravity it possesses4 ,to proceed in the most simple manner1 the unit of mass ought to be that of a cubic unit of matter of the standard density2 but the founders of the metrical system took as their unit of mass1 the cubic centimetre of water1 at the temperature of maximum density 7about #d ,cent474 ,they called this unit of mass the 99gramme991 and constructed standard specimens of the kilogram1 which might be readily referred to by all who required to employ accurate weights4 ,unfortunately the determination of the bulk of a given weight of water at a certain temperature is an operation involving many difficulties1 and it cannot be performed in the present day with a greater exactness than that of about one part in #ejjj1 the results of careful observers being sometimes found to differ as much as one part in #ajjj4bbc #bbc ,clerk ,maxwell's 99,theory of ,heat991 p4 #gi4 ,weights1 on the other hand1 can be compared with each other to at least one part in a million4 ,hence if different specimens of the kilogram be prepared by direct weighing against water1 they will not agree closely with each other2 the two principal standard kilograms agree neither with each other1 nor with their definition4 ,according to ,professor ,miller the so-called ,kilogramme des ,archives weighs #aedcbcdhgd grains1 while the kilogram deposited at the ,ministry of the ,interior in ,paris1 as the standard for commercial purposes1 weighs #aedcbcdd grains4 ,since a standard weight constructed of platinum1 or platinum and iridium1 can be preserved free from any appreciable alteration1 and since it can be very accurately compared with other weights1 we shall ultimately attain the greatest exactness in our measurements of mass1 by assuming some single kilogram as a 99provisional standard991 leaving the determination of its actual mass in units of space and density for future investigation4 ,this is what is practically done at the present day1 and thus a unit of mass takes the place of the unit of density1 both in the ,french and ,english systems4 ,the ,english pound is defined by a certain lump of platinum1 preserved at ,westminster1 and is an arbitrary mass1 chosen merely that it may agree as nearly as possible with old ,english pounds4 ,the gallon1 the old ,english unit of cubic measurement1 is defined by the condition that it shall contain exactly ten pounds weight of water at #fb ,fahr42 and although it is stated that it has the capacity of about #bggbgd cubic inches1 this ratio between the cubic and linear systems of measurement is not legally enacted1 but left open to investigation4 ,while the ,french metric system as originally designed was theoretically perfect1 it does not differ practically in this point from the ,english system4 99,natural ,system of ,standards499 ,quite recently ,professor ,clerk ,maxwell has suggested that the vibrations of light and the atoms of matter might conceivably be employed as the ultimate standards of length1 time1 and mass4 ,we should thus arrive at a 99natural system of standards991 which1 though possessing no present practical importance1 has considerable theoretical interest4 8,in the present state of science10 he says1 8the most universal standard of length which we could assume would be the wave-length in vacuum of a particular kind of light1 emitted by some widely diffused substance such as sodium1 which has well-defined lines in its spectrum4 ,such a standard would be independent of any changes in the dimensions of the earth1 and should be adopted by those who expect their writings to be more permanent than that body40#bbd ,in the same way we should get a universal standard unit of time1 independent of all questions about the motion of material bodies1 by taking as the unit the periodic time of vibration of that particular kind of light whose wave-length is the unit of length4 ,it would follow that with these units of length and time the unit of velocity would coincide with the velocity of light in empty space4 ,as regards the unit of mass1 ,professor ,maxwell1 humorously as ,i should think1 remarks that if we expect soon to be able to determine the mass of a single molecule of some standard substance1 we may wait for this determination before fixing a universal standard of mass4 #bbd 99,treatise on ,electricity and ,magnetism991 vol4 i4 p4 #c4 ,in a theoretical point of view there can be no reasonable doubt that vibrations of light are1 as far as we can tell1 the most fixed in magnitude of all phenomena4 ,there is as usual no certainty in the matter1 for the properties of the basis of light may vary to some extent in different parts of space4 ,but no differences could ever be established in the velocity of light in different parts of the solar system1 and the spectra of the stars show that the times of vibration there do not differ perceptibly from those in this part of the universe4 ,thus all presumption is in favour of the absolute constancy of the vibrations of light--absolute1 that is1 so far as regards any means of investigation we are likely to possess4 ,nearly the same considerations apply to the atomic weight as the standard of mass4 ,it is impossible to prove that all atoms of the same substance are of equal mass1 and some physicists think that they differ1 so that the fixity of combining proportions may be due only to the approximate constancy of the mean of countless millions of discrepant weights4 ,but in any case the detection of difference is probably beyond our powers4 ,in a theoretical point of view1 then1 the magnitudes suggested by ,professor ,maxwell seem to be the most fixed ones of which we have any knowledge1 so that they necessarily become the natural units4 ,in a practical point of view1 as ,professor ,maxwell would be the first to point out1 they are of little or no value1 because in the present state of science we cannot measure a vibration or weigh an atom with any approach to the accuracy which is attainable in the comparison of standard metres and kilograms4 ,the velocity of light is not known probably within a thousandth part1 and as we progress in the knowledge of light1 so we shall progress in the accurate fixation of other standards4 ,all that can be said then1 is that it is very desirable to determine the wave-lengths and periods of the principal lines of the solar spectrum1 and the absolute atomic weights of the elements1 with all attainable accuracy1 in terms of our existing standards4 ,the numbers thus obtained would admit of the reproduction of our standards in some future age of the world to a corresponding degree of accuracy1 were there need of such reference2 but so far as we can see at present1 there is no considerable probability that this mode of reproduction would ever be the best mode4 99,subsidiary ,units499 ,having once established the standard units of time1 space1 and density or mass1 we might employ them for the expression of all quantities of such nature4 ,but it is often convenient in particular branches of science to use multiples or submultiples of the original units1 for the expression of quantities in a simple manner4 ,we use the mile rather than the yard when treating of the magnitude of the globe1 and the mean distance of the earth and sun is not too large a unit when we have to describe the distances of the stars4 ,on the other hand1 when we are occupied with microscopic objects1 the inch1 the line or the millimetre1 become the most convenient terms of expression4 ,it is allowable for a scientific man to introduce a new unit in any branch of knowledge1 provided that it assists precise expression1 and is carefully brought into relation with the primary units4 ,thus ,professor ,a4 ,w4 ,williamson has proposed as a convenient unit of volume in chemical science1 an absolute volume equal to about #aab litres representing the bulk of one gram of hydrogen gas at standard temperature and pressure1 or the 99equivalent99 weight of any other gas1 such as #af grams of oxygen1 #ad grams of nitrogen1 c42 in short1 the bulk of that quantity of any one of those gases which weighs as many grams as there are units in the number expressing its atomic weight4#bbe ,hofmann has proposed a new unit of weight for chemists1 called a 99crith991 to be defined by the weight of one litre of hydrogen gas at #j ,c4 and #jgf mm41 weighing about #jjhif gram4#bbf ,both of these units must be regarded as purely subordinate units1 ultimately defined by reference to the primary units1 and not involving any new assumption4 #bbe 99,chemistry for ,students991 by ,a4 ,w4 ,williamson4 ,clarendon ,press ,series1 #bnd ed4 ,preface p4 vi4 #bbf 99,introduction to ,chemistry991 p4 #aca4 99,derived ,units499 ,the standard units of time1 space1 and mass having been once fixed1 many kinds of magnitude are naturally measured by units derived from them4 ,from the metre1 the unit of linear magnitude follows in the most obvious manner the centiare or square metre1 the unit of superficial magnitude1 and the litre that is the cube of the tenth part of a metre1 the unit of capacity or volume4 ,velocity of motion is expressed by the ratio of the space passed over1 when the motion is uniform1 to the time occupied2 hence the unit of velocity is that of a body which passes over a unit of space in a unit of time4 ,in physical science the unit of velocity might be taken as one metre per second4 ,momentum is measured by the mass moving1 regard being paid both to the amount of matter and the velocity at which it is moving4 ,hence the unit of momentum will be that of a unit volume of matter of the unit density moving with the unit velocity1 or in the ,french system1 a cubic centimetre of water of the maximum density moving one metre per second4 ,an accelerating force is measured by the ratio of the momentum generated to the time occupied1 the force being supposed to act uniformly4 ,the unit of force will therefore be that which generates a unit of momentum in a unit of time1 or which causes1 in the ,french system1 one cubic centimetre of water at maximum density to acquire in one second a velocity of one metre per second4 ,the force of gravity is the most familiar kind of force1 and as1 when acting unimpeded upon any substance1 it produces in a second a velocity of #ihjhfh 4 4 metres per second in ,paris1 it follows that the absolute unit of force is about the tenth part of the force of gravity4 ,if we employ ,british weights and measures1 the absolute unit of force is represented by the gravity of about half an ounce1 since the force of gravity of any portion of matter acting upon that matter during one second1 produces a final velocity of #cbahhi feet per second or about #cb units of velocity4 ,although from its perpetual action and approximate uniformity we find in gravity the most convenient force for reference1 and thus habitually employ it to estimate quantities of matter1 we must remember that it is only one of many instances of force4 ,strictly speaking1 we should express weight in terms of force1 but practically we express other forces in terms of weight4 ,we still require the unit of energy1 a more complex notion4 ,the momentum of a body expresses the quantity of motion which belongs or would belong to the aggregate of the particles2 but when we consider how this motion is related to the action of a force producing or removing it1 we find that the effect of a force is proportional to the mass multiplied by the square of the velocity and it is convenient to take half this product as the expression required4 ,but it is shown in books upon dynamics that it will be exactly the same thing if we define energy by a force acting through a space4 ,the natural unit of energy will then be that which overcomes a unit of force acting through a unit of space2 when we lift one kilogram through one metre1 against gravity1 we therefore accomplish #ihjhfh 4 4 units of work1 that is1 we turn so many units of potential energy existing in the muscles1 into potential energy of gravitation4 ,in lifting one pound through one foot there is in like manner a conversion of #cbahhi units of energy4 ,accordingly the unit of energy will be in the ,english system1 that required to lift one pound through about the thirty-second part of a foot2 in terms of metric units1 it will be that required to lift a kilogram through about one tenth part of a metre4 ,every person is at liberty to measure and record quantities in terms of any unit which he likes4 ,he may use the yard for linear measurement and the litre for cubic measurement1 only there will then be a complicated relation between his different results4 ,the system of derived units which we have been briefly considering1 is that which gives the most simple and natural relations between quantitative expressions of different kinds1 and therefore conduces to ease of comprehension and saving of laborious calculation4 ,it would evidently be a source of great convenience if scientific men could agree upon some single system of units1 original and derived1 in terms of which all quantities could be expressed4 ,statements would thus be rendered easily comparable1 a large part of scientific literature would be made intelligible to all1 and the saving of mental labour would be immense4 ,it seems to be generally allowed1 too1 that the metric system of weights and measures presents the best basis for the ultimate system2 it is thoroughly established in ,western ,europe2 it is legalised in ,england2 it is already commonly employed by scientific men2 it is in itself the most simple and scientific of systems4 ,there is every reason then why the metric system should be accepted at least in its main features4 99,provisional ,units499 ,ultimately1 as we can hardly doubt1 all phenomena will be recognised as so many manifestations of energy2 and1 being expressed in terms of the unit of energy1 will be referable to the primary units of space1 time1 and density4 ,to effect this reduction1 however1 in any particular case1 we must not only be able to compare different quantities of the phenomenon1 but to trace the whole series of steps by which it is connected with the primary notions4 ,we can readily observe that the intensity of one source of light is greater than that of another2 and1 knowing that the intensity of light decreases as the square of the distance increases1 we can easily determine their comparative brilliance4 ,hence we can express the intensity of light falling upon any surface1 if we have a unit in which to make the expression4 ,light is undoubtedly one form of energy1 and the unit ought therefore to be the unit of energy4 ,but at present it is quite impossible to say how much energy there is in any particular amount of light4 ,the question then arises1--,are we to defer the measurement of light until we can assign its relation to other forms of energy8 ,if we answer ,yes1 it is equivalent to saying that the science of light must stand still perhaps for a generation2 and not only this science but many others4 ,the true course evidently is to select1 as the provisional unit of light1 some light of convenient intensity1 which can be reproduced from time to time in the same intensity1 and which is defined by physical circumstances4 ,all the phenomena of light may be experimentally investigated relatively to this unit1 for instance that obtained after much labour by ,bunsen and ,roscoe4#bbg ,in after years it will become a matter of inquiry what is the energy exerted in such unit of light2 but it may be long before the relation is exactly determined4 #bbg 99,philosophical ,transactions99 7#ahei71 vol4 cxlix4 p4 #hhd1 c4 ,a provisional unit1 then1 means one which is assumed and physically defined in a safe and reproducible manner1 in order that particular quantities may be compared 99inter se99 more accurately than they can yet be referred to the primary units4 ,in reality the great majority of our measurements are expressed in terms of such provisionally independent units1 and even the unit of mass1 as we have seen1 ought to be considered as provisional4 ,the unit of heat ought to be simply the unit of energy1 already described4 ,but a weight can be measured to the one-millionth part1 and temperature to less than the thousandth part of a degree ,fahrenheit1 and to less therefore than the five-hundred thousandth part of the absolute temperature1 whereas the mechanical equivalent of heat is probably not known to the thousandth part4 ,hence the need of a provisional unit of heat1 which is often taken as that requisite to raise one gram of water through one degree ,centigrade1 that is from #j to #a4 ,this quantity of heat is capable of approximate expression in terms of time1 space1 and mass2 for by the natural constant1 determined by ,dr4 ,joule1 and called the mechanical equivalent of heat1 we know that the assumed unit of heat is equal to the energy of #dbcee gram-metres1 or that energy which will raise the mass of #dbcee grams through one metre against #ih444 absolute units of force4 ,heat may also be expressed in terms of the quantity of ice at #j ,cent41 which it is capable of converting into water under inappreciable pressure4 99,theory of ,dimensions499 ,in order to understand the relations between the quantities dealt with in physical science1 it is necessary to pay attention to the ,theory of ,dimensions1 first clearly stated by ,joseph ,fourier1#bbh but in later years developed by several physicists4 ,this theory investigates the manner in which each derived unit depends upon or involves one or more of the fundamental units4 ,the number of units in a rectangular area is found by multiplying together the numbers of units in the sides2 thus the unit of length enters twice into the unit of area1 which is therefore said to have two dimensions with respect to length4 ,denoting length by 99,l991 we may say that the dimensions of area are 99,l99 99,l99 or 99,l99#b4 ,it is obvious in the same way that the dimensions of volume or bulk will be 99,l99#c4 #bbh 99,thorie ,analytique de la ,chaleur991 ,paris2 #ahbb1 #aeg-afb4 ,the number of units of mass in a body is found by multiplying the number of units of volume1 by those of density4 ,hence mass is of three dimensions as regards length1 and one as regards density4 ,calling density 99,d991 the dimensions of mass are 99,l99#c99;,d994 ,as already explained1 however1 it is usual to substitute an arbitrary provisional unit of mass1 symbolised by 99,m992 according to the view here taken we may say that the dimensions of 99,m99 are 99,l99#c99;,d994 ,introducing time1 denoted by 99,t991 it is easy to see that the dimensions of velocity will be 99,l_/,t99 or 99,,lt99-#a1 because the number of units in the velocity of a body is found by 99dividing99 the units of length passed over by the units of time occupied in passing4 ,the acceleration of a body is measured by the increase of velocity in relation to the time1 that is1 we must divide the units of velocity gained by the units of time occupied in gaining it2 hence its dimensions will be 99,,lt99-#b4 ,momentum is the product of mass and velocity1 so that its dimensions are 99,,mlt99-#a4 ,the effect of a force is measured by the acceleration produced in a unit of mass in a unit of time2 hence the dimensions of force are 99,,mlt99-#b4 ,work done is proportional to the force acting and to the space through which it acts2 so that it has the dimensions of force with that of length added1 giving 99,,ml99#b99,t99-#b4 ,it should be particularly noticed that angular magnitude has no dimensions at all1 being measured by the ratio of the arc to the radius 7p4 #cje74 ,thus we have the dimensions 99,,ll99-#a or 99,l99#j4 ,this agrees with the statement previously made1 that no arbitrary unit of angular magnitude is needed4 ,similarly1 all pure numbers expressing ratios only1 such as sines and other trigonometrical functions1 logarithms1 exponents1 c41 are devoid of dimensions4 ,they are absolute numbers necessarily expressed in terms of unity itself1 and are quite unaffected by the selection of the arbitrary physical units4 ,angular magnitude1 however1 enters into other quantities1 such as angular velocity1 which has the dimensions #a_/99,t99 or 99,t99-#a1 the units of angle being divided by the units of time occupied4 ,the dimensions of angular acceleration are denoted by 99,t99-#b4 ,the quantities treated in the theories of heat and electricity are numerous and complicated as regards their dimensions4 ,thermal capacity has the dimensions 99,,ml99-#c1 thermal conductivity1 99,,ml99-#a99,t99-#a4 ,in ,magnetism the dimensions of the strength of pole are 99,m99#a_/b99,l99#c_/b99,t99-#a1 the dimensions of field-intensity are 99,m99#a_/b99,l99-#a_/b99,t99-#a1 and the intensity of magnetisation has the same dimensions4 ,in the science of electricity physicists have to deal with numerous kinds of quantity1 and their dimensions are different too in the electro-static and the electro-magnetic systems4 ,thus electro-motive force has the dimensions 99,m99#a_/b99,l99#a_/b99,t99-#a1 in the former1 and 99,m99#a_/b99,l99#c_/b99,t99-#b in the latter system4 ,capacity simply depends upon length in electro-statics1 but upon 99,l99-#a99,t99#b in electro-magnetics4 ,it is worthy of particular notice that electrical quantities have simple dimensions when expressed in terms of density instead of mass4 ,the instances now given are sufficient to show the difficulty of conceiving and following out the relations of the quantities treated in physical science without a systematic method of calculating and exhibiting their dimensions4 ,it is only in quite recent years that clear ideas about these quantities have been attained4 ,half a century ago probably no one but ,fourier could have explained what he meant by temperature or capacity for heat4 ,the notion of measuring electricity had hardly been entertained4 ,besides affording us a clear view of the complex relations of physical quantities1 this theory is specially useful in two ways4 ,firstly1 it affords a test of the correctness of mathematical reasoning4 ,according to the 99,principle of ,homogeneity991 all the quantities 99added99 together1 and equated in any equation1 must have the same dimensions4 ,hence if1 on estimating the dimensions of the terms in any equation1 they be not homogeneous1 some blunder must have been committed4 ,it is impossible to add a force to a velocity1 or a mass to a momentum4 ,even if the numerical values of the two members of a non-homogeneous equation were equal1 this would be accidental1 and any alteration in the physical units would produce inequality and disclose the falsity of the law expressed in the equation4 ,secondly1 the theory of units enables us readily and infallibly to deduce the change in the numerical expression of any physical quantity1 produced by a change in the fundamental units4 ,it is of course obvious that in order to represent the same absolute quantity1 a number must vary inversely as the magnitude of the units which are numbered4 ,the yard expressed in feet is #c2 taking the inch as the unit instead of the foot it becomes #cf4 ,every quantity into which the dimension length enters positively must be altered in like manner4 ,changing the unit from the foot to the inch1 numerical expressions of volume must be multiplied by #ab #ab #ab4 ,when a dimension enters negatively the opposite rule will hold4 ,if for the minute we substitute the second as unit of time1 then we must divide all numbers expressing angular velocities by #fj1 and numbers expressing angular acceleration by #fj #fj4 ,the rule is that a numerical expression varies inversely as the magnitude of the unit as regards each whole dimension entering positively1 and it varies directly as the magnitude of the unit for each whole dimension entering negatively4 ,in the case of fractional exponents1 the proper root of the ratio of change has to be taken4 ,the study of this subject may be continued in ,professor ,j4 ,d4 ,everett's 8,illustrations of the ,centimetre-gramme-second ,system of ,units10 published by ,taylor and ,francis1 #ahge2 in ,professor ,maxwell's 8,theory of ,heat20 or ,professor ,fleeming ,jenkin's 8,text ,book of ,electricity40 99,natural ,constants499 ,having acquired accurate measuring instruments1 and decided upon the units in which the results shall be expressed1 there remains the question1 ,what use shall be made of our powers of measurement8 ,our principal object must be to discover general quantitative laws of nature2 but a very large amount of preliminary labour is employed in the accurate determination of the dimensions of existing objects1 and the numerical relations between diverse forces and phenomena4 ,step by step every part of the material universe is surveyed and brought into known relations with other parts4 ,each manifestation of energy is correlated with each other kind of manifestation4 ,professor ,tyndall has described the care with which such operations are conducted4#bbi #bbi ,tyndall's 99,sound991 #ast ed4 p4 #bf4 8,those who are unacquainted with the details of scientific investigation1 have no idea of the amount of labour expended on the determination of those numbers on which important calculations or inferences depend4 ,they have no idea of the patience shown by a ,berzelius in determining atomic weights2 by a ,regnault in determining coefficients of expansion2 or by a ,joule in determining the mechanical equivalent of heat4 ,there is a morality brought to bear upon such matters which1 in point of severity1 is probably without a parallel in any other domain of intellectual action40 ,every new natural constant which is recorded brings many fresh inferences within our power4 ,for if 99n99 be the number of such constants known1 then #a_/b 799n99#b--99n997 is the number of ratios which are within our powers of calculation1 and this increases with the square of 99n994 ,we thus gradually piece together a map of nature1 in which the lines of inference from one phenomenon to another rapidly grow in complexity1 and the powers of scientific prediction are correspondingly augmented4 ,babbage#bcj proposed the formation of a collection of the constant numbers of nature1 a work which has at last been taken in hand by the ,smithsonian ,institution4#bca ,it is true that a complete collection of such numbers would be almost co-extensive with scientific literature1 since almost all the numbers occurring in works on chemistry1 mineralogy1 physics1 astronomy1 c41 would have to be included4 ,still a handy volume giving all the more important numbers and their logarithms1 referred when requisite to the different units in common use1 would be very useful4 ,a small collection of constant numbers will be found at the end of ,babbage's1 ,hutton's1 and many other tables of logarithms1 and a somewhat larger collection is given in ,templeton's 99,millwright and ,engineer's ,pocket ,companion994 #bcj ,british ,association1 ,cambridge1 #ahcc4 ,report1 pp4 #dhd-dij4 #bca 99,smithsonian ,miscellaneous ,collections991 vol4 xii41 the ,constants of ,nature1 part4 i4 ,specific gravities compiled by ,f4 ,w4 ,clarke1 #hvo4 ,washington1 #ahgc4 ,our present object will be to classify these constant numbers roughly1 according to their comparative generality and importance1 under the following heads3-- 7#a7 ,mathematical constants4 7#b7 ,physical constants4 7#c7 ,astronomical constants4 7#d7 ,terrestrial numbers4 7#e7 ,organic numbers4 7#f7 ,social numbers4 99,mathematical ,constants499 ,at the head of the list of natural constants must come those which express the necessary relations of numbers to each other4 ,the ordinary ,multiplication ,table is the most familiar and the most important of such series of constants1 and is1 theoretically speaking1 infinite in extent4 ,next we must place the ,arithmetical ,triangle1 the significance of which has already been pointed out 7p4 #ahb74 ,tables of logarithms also contain vast series of natural constants1 arising out of the relations of pure numbers4 ,at the base of all logarithmic theory is the mysterious natural constant commonly denoted by 99e991 or 1 being equal to the infinite series #a #a_/a #a_/a4b #a_/a4b4c #a_/a4b4c4d 44441 and thus consisting of the sum of the ratios between the numbers of permutations and combinations of #j1 #a1 #b1 #c1 #d1 c4 things4 ,tables of prime numbers and of the factors of composite numbers must not be forgotten4 ,another vast and in fact infinite series of numerical constants contains those connected with the measurement of angles1 and embodied in trigonometrical tables1 whether as natural or logarithmic sines1 cosines1 and tangents4 ,it should never be forgotten that though these numbers find their chief employment in connection with trigonometry1 or the measurement of the sides of a right-angled triangle1 yet the numbers themselves arise out of numerical relations bearing no special relation to space4 ,foremost among trigonometrical constants is the well known number 1 usually employed as expressing the ratio of the circumference and the diameter of a circle2 from follows the value of the arcual or natural unit of angular value as expressed in ordinary degrees 7p4 #cjf74 ,among other mathematical constants not uncommonly used may be mentioned tables of factorials 7p4 #agi71 tables of ,bernoulli's numbers1 tables of the error function1#bcb which latter are indispensable not only in the theory of probability but also in several other branches of science4 #bcb ,j4 ,w4 ,l4 ,glaisher1 99,philosophical ,magazine991 #dth ,series1 vol4 xlii4 p4 #dba4 ,it should be clearly understood that the mathematical constants and tables of reference already in our possession1 although very extensive1 are only an infinitely small part of what might be formed4 ,with the progress of science the tabulation of new functions will be continually demanded1 and it is worthy of consideration whether public money should not be available to reward the severe1 long continued1 and generally thankless labour which must be gone through in calculating tables4 ,such labours are a benefit to the whole human race as long as it shall exist1 though there are few who can appreciate the extent of this benefit4 ,a most interesting and excellent description of many mathematical tables will be found in ,de ,morgan's article on 99,tables991 in the 99,english ,cyclopdia991 ,division of ,arts and ,sciences1 vol4 vii4 p4 #igf4 ,an almost exhaustive critical catalogue of extant tables is being published by a ,committee of the ,british ,association1 two portions1 drawn up chiefly by ,mr4 ,j4 ,w4 ,l4 ,glaisher and ,professor ,cayley1 having appeared in the ,reports of the ,association for #ahgc and #ahge4 99,physical ,constants499 ,the second class of constants contains those which refer to the actual constitution of matter4 ,for the most part they depend upon the peculiarities of the chemical substance in question1 but we may begin with those which are of the most general character4 ,in a first sub-class we may place the velocity of light or heat undulations1 the numbers expressing the relation between the lengths of the undulations1 and the rapidity of the undulations1 these numbers depending only on the properties of the ethereal medium1 and being probably the same in all parts of the universe4 ,the theory of heat gives rise to several numbers of the highest importance1 especially ,joule's mechanical equivalent of heat1 the absolute zero of temperature1 the mean temperature of empty space1 c4 ,taking into account the diverse properties of the elements we must have tables of the atomic weights1 the specific heats1 the specific gravities1 the refractive powers1 not only of the elements1 but their almost infinitely numerous compounds4 ,the properties of hardness1 elasticity1 viscosity1 expansion by heat1 conducting powers for heat and electricity1 must also be determined in immense detail4 ,there are1 however1 certain of these numbers which stand out prominently because they serve as intermediate units or terms of comparison4 ,such are1 for instance1 the absolute coefficients of expansion of air1 water and mercury1 the temperature of the maximum density of water1 the latent heats of water and steam1 the boiling-point of water under standard pressure1 the melting and boiling-points of mercury1 and so forth4 99,astronomical ,constants499 ,the third great class consists of numbers possessing far less generality because they refer not to the properties of matter1 but to the special forms and distances in which matter has been disposed in the part of the universe open to our examination4 ,we have1 first of all1 to define the magnitude and form of the earth1 its mean density1 the constant of aberration of light expressing the relation between the earth's mean velocity in space and the velocity of light4 ,from the earth1 as our observatory1 we then proceed to lay down the mean distances of the sun1 and of the planets from the same centre2 all the elements of the planetary orbits1 the magnitudes1 densities1 masses1 periods of axial rotation of the several planets are by degrees determined with growing accuracy4 ,the same labours must be gone through for the satellites4 ,catalogues of comets with the elements of their orbits1 as far as ascertainable1 must not be omitted4 ,from the earth's orbit as a new base of observations1 we next proceed to survey the heavens and lay down the apparent positions1 magnitudes1 motions1 distances1 periods of variation1 c4 of the stars4 ,all catalogues of stars from those of ,hipparchus and ,tycho1 are full of numbers expressing rudely the conformation of the visible universe4 ,but there is obviously no limit to the labours of astronomers2 not only are millions of distant stars awaiting their first measurements1 but those already registered require endless scrutiny as regards their movements in the three dimensions of space1 their periods of revolution1 their changes of brilliance and colour4 ,it is obvious that though astronomical numbers are conventionally called 99constant991 they are probably in all cases subject to more or less rapid variation4 99,terrestrial ,numbers499 ,our knowledge of the globe we inhabit involves many numerical determinations1 which have little or no connection with astronomical theory4 ,the extreme heights of the principal mountains1 the mean elevations of continents1 the mean or extreme depths of the oceans1 the specific gravities of rocks1 the temperature of mines1 the host of numbers expressing the meteorological or magnetic conditions of every part of the surface1 must fall into this class4 ,many such numbers are not to be called constant1 being subject to periodic or secular changes1 but they are hardly more variable in fact than some which in astronomical science are set down as constant4 ,in many cases quantities which seem most variable may go through rhythmical changes resulting in a nearly uniform average1 and it is only in the long progress of physical investigation that we can hope to discriminate successfully between those elemental numbers which are fixed and those which vary4 ,in the latter case the law of variation becomes the constant relation which is the object of our search4 99,organic ,numbers499 ,the forms and properties of brute nature having been sufficiently defined by the previous classes of numbers1 the organic world1 both vegetable and animal1 remains outstanding1 and offers a higher series of phenomena for our investigation4 ,all exact knowledge relating to the forms and sizes of living things1 their numbers1 the quantities of various compounds which they consume1 contain1 or excrete1 their muscular or nervous energy1 c4 must be placed apart in a class by themselves4 ,all such numbers are doubtless more or less subject to variation1 and but in a minor degree capable of exact determination4 ,man1 so far as he is an animal1 and as regards his physical form1 must also be treated in this class4 99,social ,numbers499 ,little allusion need be made in this work to the fact that man in his economic1 sanitary1 intellectual1 sthetic1 or moral relations may become the subject of sciences1 the highest and most useful of all sciences4 ,every one who is engaged in statistical inquiry must acknowledge the possibility of natural laws governing such statistical facts4 ,hence we must allot a distinct place to numerical information relating to the numbers1 ages1 physical and sanitary condition1 mortality1 c41 of different peoples1 in short1 to vital statistics4 ,economic statistics1 comprehending the quantities of commodities produced1 existing1 exchanged and consumed1 constitute another extensive body of science4 ,in the progress of time exact investigation may possibly subdue regions of phenomena which at present defy all scientific treatment4 ,that scientific method can ever exhaust the phenomena of the human mind is incredible4 ,,chapter ,,xv4 ,,analysis ,,of ,,quantitative ,,phenomena4 ,in the two preceding chapters we have been engaged in considering how a phenomenon may be accurately measured and expressed4 ,so delicate and complex an operation is a measurement which pretends to any considerable degree of exactness1 that no small part of the skill and patience of physicists is usually spent upon this work4 ,much of this difficulty arises from the fact that it is scarcely ever possible to measure a single effect at a time4 ,the ultimate object must be to discover the mathematical equation or law connecting a quantitative cause with its quantitative effect2 this purpose usually involves1 as we shall see1 the varying of one condition at a time1 the other conditions being maintained constant4 ,the labours of the experimentalist would be comparatively light if he could carry out this rule of varying one circumstance at a time4 ,he would then obtain a series of corresponding values of the variable quantities concerned1 from which he might by proper hypothetical treatment obtain the required law of connection4 ,but in reality it is seldom possible to carry out this direction except in an approximate manner4 ,before then we proceed to the consideration of the actual process of quantitative induction1 it is necessary to review the several devices by which a complicated series of effects can be disentangled4 ,every phenomenon measured will usually be the sum1 difference1 or it may be the product or quotient1 of two or more different effects1 and these must be in some way analysed and separately measured before we possess the materials for inductive treatment4 99,illustrations of the ,complication of ,effects499 ,it is easy to bring forward a multitude of instances to show that a phenomenon is seldom to be observed simple and alone4 ,a more or less elaborate process of analysis is almost always necessary4 ,thus if an experimentalist wishes to observe and measure the expansion of a liquid by heat1 he places it in a thermometer tube and registers the rise of the column of liquid in the narrow tube4 ,but he cannot heat the liquid without also heating the glass1 so that the change observed is really the difference between the expansions of the liquid and the glass4 ,more minute investigation will show the necessity perhaps of allowing for further minute effects1 namely the compression of the liquid and the expansion of the bulb due to the increased pressure of the column as it becomes lengthened4 ,in a great many cases an observed effect will be apparently at least the simple sum of two separate and independent effects4 ,the heat evolved in the combustion of oil is partly due to the carbon and partly to the hydrogen4 ,a measurement of the heat yielded by the two jointly1 cannot inform us how much proceeds from the one and how much from the other4 ,if by some separate determination we can ascertain how much the hydrogen yields1 then by mere subtraction we learn what is due to the carbon2 and 99vice vers994 ,the heat conveyed by a liquid1 may be partly conveyed by true conduction1 partly by convection4 ,the light dispersed in the interior of a liquid consists both of what is reflected by floating particles and what is due to true fluorescence2#bcc and we must find some mode of determining one portion before we can learn the other4 ,the apparent motion of the spots on the sun1 is the algebraic sum of the sun's axial rotation1 and of the proper motion of the spots upon the sun's surface2 hence the difficulty of ascertaining by direct observations the period of the sun's rotation4 #bcc ,stokes1 99,philosophical ,transactions99 7#aheb71 vol4 cxlii4 p4 #ebi4 ,we cannot obtain the weight of a portion of liquid in a chemical balance without weighing it with the containing vessel4 ,hence to have the real weight of the liquid operated upon in an experiment1 we must make a separate weighing of the vessel1 with or without the adhering film of liquid according to circumstances4 ,this is likewise the mode in which a cart and its load are weighed together1 the 99tare99 of the cart previously ascertained being deducted4 ,the variation in the height of the barometer is a joint effect1 partly due to the real variation of the atmospheric pressure1 partly to the expansion of the mercurial column by heat4 ,the effects may be discriminated1 if1 instead of one barometer tube we have two tubes containing mercury placed closely side by side1 so as to have the same temperature4 ,if one of them be closed at the bottom so as to be unaffected by the atmospheric pressure1 it will show the changes due to temperature only1 and1 by subtracting these changes from those shown in the other tube1 employed as a barometer1 we get the real oscillations of atmospheric pressure4 ,but this correction1 as it is called1 of the barometric reading1 is better effected by calculation from the readings of an ordinary thermometer4 ,in other cases a quantitative effect will be the difference of two causes acting in opposite directions4 ,sir ,john ,herschel invented an instrument like a large thermometer1 which he called the ,actinometer1#bcd and ,pouillet constructed a somewhat similar instrument called the ,pyrheliometer1 for ascertaining the heating power of the sun's rays4 ,in both instruments the heat of the sun was absorbed by a reservoir containing water1 and the rise of temperature of the water was exactly observed1 either by its own expansion1 or by the readings of a delicate thermometer immersed in it4 ,but in exposing the actinometer to the sun1 we do not obtain the full effect of the heat absorbed1 because the receiving surface is at the same time radiating heat into empty space4 ,the observed increment of temperature is in short the difference between what is received from the sun and lost by radiation4 ,the latter quantity is capable of ready determination2 we have only to shade the instrument from the direct rays of the sun1 leaving it exposed to the sky1 and we can observe how much it cools in a certain time4 ,the total effect of the sun's rays will obviously be the apparent effect 99plus99 the cooling effect in an equal time4 ,by alternate exposure in sun and shade during equal intervals the desired result may be obtained with considerable accuracy4#bce #bcd 99,admiralty ,manual of ,scientific ,enquiry991 #bnd ed4 p4 #bii4 #bce ,pouillet1 99,taylor's ,scientific ,memoirs991 vol4 iv4 p4 #de4 ,two quantitative effects were beautifully distinguished in an experiment of ,john ,canton1 devised in #agfa for the purpose of demonstrating the compressibility of water4 ,he constructed a thermometer with a large bulb full of water and a short capillary tube1 the part of which above the water was freed from air4 ,under these circumstances the water was relieved from the pressure of the atmosphere1 but the glass bulb in bearing that pressure was somewhat contracted4 ,he next placed the instrument under the receiver of an air-pump1 and on exhausting the air1 the water sank in the tube4 ,having thus obtained a measure of the effect of atmospheric pressure on the bulb1 he opened the top of the thermometer tube and admitted the air4 ,the level of the water now sank still more1 partly from the pressure on the bulb being now compensated1 and partly from the compression of the water by the atmospheric pressure4 ,it is obvious that the amount of the latter effect was approximately the difference of the two observed depressions4 ,not uncommonly the actual phenomenon which we wish to measure is considerably less than various disturbing effects which enter into the question4 ,thus the compressibility of mercury is considerably less than the expansion of the vessels in which it is measured under pressure1 so that the attention of the experimentalist has chiefly to be concentrated on the change of magnitude of the vessels4 ,many astronomical phenomena1 such as the parallax or the proper motions of the fixed stars1 are far less than the errors caused by instrumental imperfections1 or motions arising from precession1 nutation1 and aberration4 ,we need not be surprised that astronomers have from time to time mistaken one phenomenon for another1 as when ,flamsteed imagined that he had discovered the parallax of the ,pole star4#bcf #bcf ,baily's 99,account of the ,rev4 ,john ,flamsteed991 p4 #eh4 99,methods of ,eliminating ,error499 ,in any particular experiment it is the object of the experimentalist to measure a single effect only1 and he endeavours to obtain that effect free from interfering effects4 ,if this cannot be1 as it seldom or never can really be1 he makes the effect as considerable as possible compared with the other effects1 which he reduces to a minimum1 and treats as noxious errors4 ,those quantities1 which are called 99errors99 in one case1 may really be most important and interesting phenomena in another investigation4 ,when we speak of eliminating error we really mean disentangling the complicated phenomena of nature4 ,the physicist rightly wishes to treat one thing at a time1 but as this object can seldom be rigorously carried into practice1 he has to seek some mode of counteracting the irrelevant and interfering causes4 ,the general principle is that a single observation can render known only a single quantity4 ,hence1 if several different quantitative effects are known to enter into any investigation1 we must have at least as many distinct observations as there are quantities to be determined4 ,every complete experiment will therefore consist in general of several operations4 ,guided if possible by previous knowledge of the causes in action1 we must arrange the determinations1 so that by a simple mathematical process we may distinguish the separate quantities4 ,there appear to be five principal methods by which we may accomplish this object2 these methods are specified below and illustrated in the succeeding sections4 7#a7 99,the ,method of ,avoidance499 ,the physicist may seek for some special mode of experiment or opportunity of observation1 in which the error is non-existent or inappreciable4 7#b7 99,the ,differential ,method499 ,he may find opportunities of observation when all interfering phenomena remain constant1 and only the subject of observation is at one time present and another time absent2 the difference between two observations then gives its amount4 7#c7 99,the ,method of ,correction499 ,he may endeavour to estimate the amount of the interfering effect by the best available mode1 and then make a corresponding correction in the results of observation4 7#d7 99,the ,method of ,compensation499 ,he may invent some mode of neutralising the interfering cause by balancing against it an exactly equal and opposite cause of unknown amount4 7#e7 99,the ,method of ,reversal499 ,he may so conduct the experiment that the interfering cause may act in opposite directions1 in alternate observations1 the mean result being free from interference4 ,i4 99,method of ,avoidance of ,error499 ,astronomers seek opportunities of observation when errors will be as small as possible4 ,in spite of elaborate observations and long-continued theoretical investigation1 it is not practicable to assign any satisfactory law to the refractive power of the atmosphere4 ,although the apparent change of place of a heavenly body produced by refraction may be more or less accurately calculated yet the error depends upon the temperature and pressure of the atmosphere1 and1 when a ray is highly inclined to the perpendicular1 the uncertainty in the refraction becomes very considerable4 ,hence astronomers always make their observations1 if possible1 when the object is at the highest point of its daily course1 99i4e499 on the meridian4 ,in some kinds of investigation1 as1 for instance1 in the determination of the latitude of an observatory1 the astronomer is at liberty to select one or more stars out of the countless number visible4 ,there is an evident advantage in such a case1 in selecting a star which passes close to the zenith1 so that it may be observed almost entirely free from atmospheric refraction1 as was done by ,hooke4 ,astronomers endeavour to render their clocks as accurate as possible1 by removing the source of variation4 ,the pendulum is perfectly isochronous so long as its length remains invariable1 and the vibrations are exactly of equal length4 ,they render it nearly invariable in length1 that is in the distance between the centres of suspension and oscillation1 by a compensatory arrangement for the change of temperature4 ,but as this compensation may not be perfectly accomplished1 some astronomers place their chief controlling clock in a cellar1 or other apartment1 where the changes of temperature may be as slight as possible4 ,at the ,paris ,observatory a clock has been placed in the caves beneath the building1 where there is no appreciable difference between the summer and winter temperature4 ,to avoid the effect of unequal oscillations ,huyghens made his beautiful investigations1 which resulted in the discovery that a pendulum1 of which the centre of oscillation moved upon a cycloidal path1 would be perfectly isochronous1 whatever the variation in the length of oscillations4 ,but though a pendulum may be easily rendered in some degree cycloidal by the use of a steel suspension spring1 it is found that the mechanical arrangements requisite to produce a truly cycloidal motion introduce more error than they remove4 ,hence astronomers seek to reduce the error to the smallest amount by maintaining their clock pendulums in uniform movement2 in fact1 while a clock is in good order and has the same weights1 there need be little change in the length of oscillation4 ,when a pendulum cannot be made to swing uniformly1 as in experiments upon the force of gravity1 it becomes requisite to resort to the third method1 and a correction is introduced1 calculated on theoretical grounds from the amount of the observed change in the length of vibration4 ,it has been mentioned that the apparent expansion of a liquid by heat1 when contained in a thermometer tube or other vessel1 is the difference between the real expansion of the liquid and that of the containing vessel4 ,the effects can be accurately distinguished provided that we can learn the real expansion by heat of any one convenient liquid2 for by observing the apparent expansion of the same liquid in any required vessel we can by difference learn the amount of expansion of the vessel due to any given change of temperature4 ,when we once know the change of dimensions of the vessel1 we can of course determine the absolute expansion of any other liquid tested in it4 ,thus it became an all-important object in scientific research to measure with accuracy the absolute dilatation by heat of some one liquid1 and mercury owing to several circumstances was by far the most suitable4 ,dulong and ,petit devised a beautiful mode of effecting this by simply avoiding altogether the effect of the change of size of the vessel4 ,two upright tubes full of mercury were connected by a fine tube at the bottom1 and were maintained at two different temperatures4 ,as mercury was free to flow from one tube to the other by the connecting tube1 the two columns necessarily exerted equal pressures by the principles of hydrostatics4 ,hence it was only necessary to measure very accurately by a cathetometer the difference of level of the surfaces of the two columns of mercury1 to learn the difference of length of columns of equal hydrostatic pressure1 which at once gives the difference of density of the mercury1 and the dilatation by heat4 ,the changes of dimension in the containing tubes became a matter of entire indifference1 and the length of a column of mercury at different temperatures was measured as easily as if it had formed a solid bar4 ,the experiment was carried out by ,regnault with many improvements of detail1 and the absolute dilatation of mercury1 at temperatures between #j ,cent4 and #cej1 was determined almost as accurately as was needful4#bcg #bcg ,jamin1 99,cours de ,physique991 vol4 ii4 pp4 #ae-bh4 ,the presence of a large and uncertain amount of error may render a method of experiment valueless4 ,foucault devised a beautiful experiment with the pendulum for demonstrating popularly the rotation of the earth1 but it could be of no use for measuring the rotation exactly4 ,it is impossible to make the pendulum swing in a perfect plane1 and the slightest lateral motion gives it an elliptic path with a progressive motion of the axis of the ellipse1 which disguises and often entirely overpowers that due to the rotation of the earth4#bch #bch 99,philosophical ,magazine991 #ahea1 #dth ,series1 vol4 ii4 99passim994 ,faraday's laborious experiments on the relation of gravity and electricity were much obstructed by the fact that it is impossible to move a large weight of metal without generating currents of electricity1 either by friction or induction4 ,to distinguish the electricity1 if any1 directly due to the action of gravity from the greater quantities indirectly produced was a problem of excessive difficulty4 ,baily in his experiments on the density of the earth was aware of the existence of inexplicable disturbances which have since been referred with much probability to the action of electricity4#bci ,the skill and ingenuity of the experimentalist are often exhausted in trying to devise a form of apparatus in which such causes of error shall be reduced to a minimum4 #bci ,hearn1 99,philosophical ,transactions991 #ahdg1 vol4 cxxxvii4 pp4 #bag-bba4 ,in some rudimentary experiments we wish merely to establish the existence of a quantitative effect without precisely measuring its amount2 if there exist causes of error of which we can neither render the amount known or inappreciable1 the best way is to make them all negative so that the quantitative effects will be less than the truth rather than greater4 ,grove1 for instance1 in proving that the magnetisation or demagnetisation of a piece of iron raises its temperature1 took care to maintain the electro-magnet by which the iron was magnetised at a lower temperature than the iron1 so that it would cool rather than warm the iron by radiation or conduction4#bdj #bdj 99,the ,correlation of ,physical ,forces991 #crd ed4 p4 #aei4 ,rumford's celebrated experiment to prove that heat was generated out of mechanical force in the boring of a cannon was subject to the difficulty that heat might be brought to the cannon by conduction from neighbouring bodies4 ,it was an ingenious device of ,davy to produce friction by a piece of clock-work resting upon a block of ice in an exhausted receiver2 as the machine rose in temperature above #cb1 it was certain that no heat was received by conduction from the support4#bda ,in many other experiments ice may be employed to prevent the access of heat by conduction1 and this device1 first put in practice by ,murray1#bdb is beautifully employed in ,bunsen's calorimeter4 #bda 99,collected ,works of ,sir ,h4 ,davy991 vol4 ii4 pp4 #ab-ad4 99,elements of ,chemical ,philosophy991 p4 #id4 #bdb 99,nicholson's ,journal991 vol4 i4 p4 #bda2 quoted in 99,treatise on ,heat991 ,useful ,knowledge ,society1 p4 #bd4 ,to observe the true temperature of the air1 though apparently so easy1 is really a very difficult matter1 because the thermometer is sure to be affected either by the sun's rays1 the radiation from neighbouring objects1 or the escape of heat into space4 ,these sources of error are too fluctuating to allow of correction1 so that the only accurate mode of procedure is that devised by ,dr4 ,joule1 of surrounding the thermometer with a copper cylinder ingeniously adjusted to the temperature of the air1 as described by him1 so that the effect of radiation shall be nullified4#bdc #bdc ,clerk ,maxwell1 99,theory of ,heat991 p4 #bbh4 99,proceedings of the ,manchester ,philosophical ,society991 ,nov4 #bf1 #ahfg1 vol4 vii4 p4 #ce4 ,when the avoidance of error is not practicable1 it will yet be desirable to reduce the absolute amount of the interfering error as much as possible before employing the succeeding methods to correct the result4 ,as a general rule we can determine a quantity with less inaccuracy as it is smaller1 so that if the error itself be small the error in determining that error will be of a still lower order of magnitude4 ,but in some cases the absolute amount of an error is of no consequence1 as in the index error of a divided circle1 or the difference between a chronometer and astronomical time4 ,even the rate at which a clock gains or loses is a matter of little importance provided it remain constant1 so that a sure calculation of its amount can be made4 #b4 99,differential ,method499 ,when we cannot avoid the existence of error1 we can often resort with success to the second mode by measuring phenomena under such circumstances that the error shall remain very nearly the same in all the observations1 and neutralise itself as regards the purposes in view4 ,this mode is available whenever we want a difference between quantities and not the absolute quantity of either4 ,the determination of the parallax of the fixed stars is exceedingly difficult1 because the amount of parallax is far less than most of the corrections for atmospheric refraction1 nutation1 aberration1 precession1 instrumental irregularities1 c41 and can with difficulty be detected among these phenomena of various magnitude4 ,but1 as ,galileo long ago suggested1 all such difficulties would be avoided by the differential observation of stars1 which1 though apparently close together1 are really far separated on the line of sight4 ,two such stars in close apparent proximity will be subject to almost exactly equal errors1 so that all we need do is to observe the apparent change of place of the nearer star as referred to the more distant one4 ,a good telescope furnished with an accurate micrometer is alone needed for the application of the method4 ,huyghens appears to have been the first observer who actually tried to employ the method practically1 but it was not until #ahce that the improvement of telescopes and micrometers enabled ,struve to detect in this way the parallax of the star ,lyr4 ,it is one of the many advantages of the observation of transits of ,venus for the determination of the solar parallax that the refraction of the atmosphere affects in an exactly equal degree the planet and the portion of the sun's face over which it is passing4 ,thus the observations are strictly of a differential nature4 ,by the process of substitutive weighing it is possible to ascertain the equality or inequality of two weights with almost perfect freedom from error4 ,if two weights ,a and ,b be placed in the scales of the best balance we cannot be sure that the equilibrium of the beam indicates exact equality1 because the arms of the beam may be unequal or unbalanced4 ,but if we take ,b out and put another weight ,c in1 and equilibrium still exists1 it is apparent that the same causes of erroneous weighing exist in both cases1 supposing that the balance has not been disarranged2 ,b then must be exactly equal to ,c1 since it has exactly the same effect under the same circumstances4 ,in like manner it is a general rule that1 if by any uniform mechanical process we get a copy of an object1 it is unlikely that this copy will be precisely the same as the original in magnitude and form1 but two copies will equally diverge from the original1 and will therefore almost exactly resemble each other4 ,leslie's ,differential ,thermometer#bdd was well adapted to the experiments for which it was invented4 ,having two equal bulbs any alteration in the temperature of the air will act equally by conduction on each and produce no change in the indications of the instrument4 ,only that radiant heat which is purposely thrown upon one of the bulbs will produce any effect4 ,this thermometer in short carries out the principle of the differential method in a mechanical manner4 #bdd ,leslie1 99,inquiry into the ,nature of ,heat991 p4 #aj4 #c4 99,method of ,correction499 ,whenever the result of an experiment is affected by an interfering cause to a calculable amount1 it is sufficient to add or subtract this amount4 ,we are said to correct observations when we thus eliminate what is due to extraneous causes1 although of course we are only separating the correct effects of several agents4 ,the variation in the height of the barometer is partly due to the change of temperature1 but since the coefficient of absolute dilatation of mercury has been exactly determined1 as already described 7p4 #cda71 we have only to make calculations of a simple character1 or1 what is better still1 tabulate a series of such calculations for general use1 and the correction for temperature can be made with all desired accuracy4 ,the height of the mercury in the barometer is also affected by capillary attraction1 which depresses it by a constant amount depending mainly on the diameter of the tube4 ,the requisite corrections can be estimated with accuracy sufficient for most purposes1 more especially as we can check the correctness of the reading of a barometer by comparison with a standard barometer1 and introduce if need be an index error including both the error in the affixing of the scale and the effect due to capillarity4 ,but in constructing the standard barometer itself we must take greater precautions2 the capillary depression depends somewhat upon the quality of the glass1 the absence of air1 and the perfect cleanliness of the mercury1 so that we cannot assign the exact amount of the effect4 ,hence a standard barometer is constructed with a wide tube1 sometimes even an inch in diameter1 so that the capillary effect may be rendered almost zero4#bde ,gay-,lussac made barometers in the form of a uniform siphon tube1 so that the capillary forces acting at the upper and lower surfaces should balance and destroy each other2 but the method fails in practice because the lower surface1 being open to the air1 becomes sullied and subject to a different force of capillarity4 #bde ,jevons1 ,watts' 99,dictionary of ,chemistry991 vol4 i4 pp4 #eac-eae4 ,in mechanical experiments friction is an interfering condition1 and drains away a portion of the energy intended to be operated upon in a definite manner4 ,we should of course reduce the friction in the first place to the lowest possible amount1 but as it cannot be altogether prevented1 and is not calculable with certainty from any general laws1 we must determine it separately for each apparatus by suitable experiments4 ,thus ,smeaton1 in his admirable but almost forgotten researches concerning water-wheels1 eliminated friction in the most simple manner by determining by trial what weight1 acting by a cord and roller upon his model water-wheel1 would make it turn without water as rapidly as the water made it turn4 ,in short1 he ascertained what weight concurring with the water would exactly compensate for the friction4#bdf ,in ,dr4 ,joule's experiments to determine the mechanical equivalent of heat by the condensation of air1 a considerable amount of heat was produced by friction of the condensing pump1 and a small portion by stirring the water employed to absorb the heat4 ,this heat of friction was measured by simply repeating the experiment in an exactly similar manner except that no condensation was effected1 and observing the change of temperature then produced4#bdg #bdf 99,philosophical ,transactions991 vol4 li4 p4 #ajj4 #bdg 99,philosophical ,magazine991 #crd ,series1 vol4 xxvi4 p4 #cgb4 ,we may describe as 99test experiments99 any in which we perform operations not intended to give the quantity of the principal phenomenon1 but some quantity which would otherwise remain as an error in the result4 ,thus in astronomical observations almost every instrumental error may be avoided by increasing the number of observations and distributing them in such a manner as to produce in the final mean as much error in one way as in the other4 ,but there is one source of error1 first discovered by ,maskelyne1 which cannot be thus avoided1 because it affects all observations in the same direction and to the same average amount1 namely the ,personal ,error of the observer or the inclination to record the passage of a star across the wires of the telescope a little too soon or a little too late4 ,this personal error was first carefully described in the 99,edinburgh ,journal of ,science991 vol4 i4 p4 #agh4 ,the difference between the judgment of observers at the ,greenwich ,observatory usually varies from #a_/ajj to #a_/c of a second1 and remains pretty constant for the same observers4#bdh ,one practised observer in ,sir ,george ,airy's pendulum experiments recorded all his time observations half a second too early on the average as compared with the chief observer4#bdi ,in some observers it has amounted to seven or eight-tenths of a second4#bej ,de ,morgan appears to have entertained the opinion that this source of error was essentially incapable of elimination or correction4#bea ,but it seems clear1 as ,i suggested without knowing what had been done1#beb that this personal error might be determined absolutely with any desirable degree of accuracy by test experiments1 consisting in making an artificial star move at a considerable distance and recording by electricity the exact moment of its passage over the wire4 ,this method has in fact been successfully employed in ,leyden1 ,paris1 and ,neuchatel4#bec ,more recently1 observers were trained for the ,transit of ,venus ,expeditions by means of a mechanical model representing the motion of ,venus over the sun1 this model being placed at a little distance and viewed through a telescope1 so that differences in the judgments of different observers would become apparent4 ,it seems likely that tests of this nature might be employed with advantage in other cases4 #bdh 99,greenwich ,observations for99 #ahff1 p4 xlix4 #bdi 99,philosophical ,transactions991 #ahef1 p4 #cji4 #bej ,penny 99,cyclopdia991 art4 99,transit991 vol4 xxv4 pp4 #abi1 #acj4 #bea ,ibid4 art4 99,observation991 p4 #cij4 #beb 99,nature991 vol4 i4 p4 #he4 #bec 99,nature991 vol4 i4 p #ccg4 ,see references to the ,memoirs describing the method4 ,newton employed the pendulum for making experiments on the impact of balls4 ,two balls were hung in contact1 and one of them1 being drawn aside through a measured arc1 was then allowed to strike the other1 the arcs of vibration giving sufficient data for calculating the distribution of energy at the moment of impact4 ,the resistance of the air was an interfering cause which he estimated very simply by causing one of the balls to make several complete vibrations without impact and then marking the reduction in the lengths of the arcs1 a proper fraction of which reduction was added to each of the other arcs of vibration when impact took place4#bed #bed 99,principia991 ,book ,i4 ,law ,,iii4 ,corollary ,,vi4 ,scholium4 ,motte's translation1 vol4 i4 p4 #cc4 ,the exact definition of the standard of length is one of the most important1 as it is one of the most difficult questions in physical science1 and the different practice of different nations introduces needless confusion4 ,were all standards constructed so as to give the true length at a fixed uniform temperature1 for instance the freezing-point1 then any two standards could be compared without the interference of temperature by bringing them both to exactly the same fixed temperature4 ,unfortunately the ,french metre was defined by a bar of platinum at #j;,c1 while our yard was defined by a bronze bar at #fb;,f4 ,it is quite impossible1 then1 to make a comparison of the yard and metre without the introduction of a correction1 either for the expansion of platinum or bronze1 or both4 ,bars of metal differ too so much in their rates of expansion according to their molecular condition that it is dangerous to infer from one bar to another4 ,when we come to use instruments with great accuracy there are many minute sources of error which must be guarded against4 ,if a thermometer has been graduated when perpendicular1 it will read somewhat differently when laid flat1 as the pressure of a column of mercury is removed from the bulb4 ,the reading may also be somewhat altered if it has recently been raised to a higher temperature than usual1 if it be placed under a vacuous receiver1 or if the tube be unequally heated as compared with the bulb4 ,for these minute causes of error we may have to introduce troublesome corrections1 unless we adopt the simple precaution of using the thermometer in circumstances of position1 c41 exactly similar to those in which it was graduated4 ,there is no end to the number of minute corrections which may ultimately be required4 ,a large number of experiments on gases1 standard weights and measures1 c41 depend upon the height of the barometer2 but when experiments in different parts of the world are compared together we ought as a further refinement to take into account the varying force of gravity1 which even between ,london and ,paris makes a difference of #jjh inch of mercury4 ,the measurement of quantities of heat is a matter of great difficulty1 because there is no known substance impervious to heat1 and the problem is therefore as difficult as to measure liquids in porous vessels4 ,to determine the latent heat of steam we must condense a certain amount of the steam in a known weight of water1 and then observe the rise of temperature of the water4 ,but while we are carrying out the experiment1 part of the heat will escape by radiation and conduction from the condensing vessel or calorimeter4 ,we may indeed reduce the loss of heat by using vessels with double sides and bright surfaces1 surrounded with swans-down wool or other non-conducting materials2 and we may also avoid raising the temperature of the water much above that of the surrounding air4 ,yet we cannot by any such means render the loss of heat inconsiderable4 ,rumford ingeniously proposed to reduce the loss to zero by commencing the experiment when the temperature of the calorimeter is as much below that of the air as it is at the end of the experiment above it4 ,thus the vessel will first gain and then lose by radiation and conduction1 and these opposite errors will approximately balance each other4 ,but ,regnault has shown that the loss and gain do not proceed by exactly the same laws1 so that in very accurate investigations ,rumford's method is not sufficient4 ,there remains the method of correction which was beautifully carried out by ,regnault in his determination of the latent heat of steam4 ,he employed two calorimeters1 made in exactly the same way and alternately used to condense a certain amount of steam1 so that while one was measuring the latent heat1 the other calorimeter was engaged in determining the corrections to be applied1 whether on account of radiation and conduction from the vessel or on account of heat reaching the vessel by means of the connecting pipes4#bee #bee ,graham's 99,chemical ,reports and ,memoirs991 ,cavendish ,society1 pp4 #bdg1 #bfh1 c4 #d4 99,method of ,compensation499 ,there are many cases in which a cause of error cannot conveniently be rendered null1 and is yet beyond the reach of the third method1 that of calculating the requisite correction from independent observations4 ,the magnitude of an error may be subject to continual variations1 on account of change of weather1 or other fickle circumstances beyond our control4 ,it may either be impracticable to observe the variation of those circumstances in sufficient detail1 or1 if observed1 the calculation of the amount of error may be subject to doubt4 ,in these cases1 and only in these cases1 it will be desirable to invent some artificial mode of counterpoising the variable error against an equal error subject to exactly the same variation4 ,we cannot weigh an object with great accuracy unless we make a correction for the weight of the air displaced by the object1 and add this to the apparent weight4 ,in very accurate investigations relating to standard weights1 it is usual to note the barometer and thermometer at the time of making a weighing1 and1 from the measured bulks of the objects compared1 to calculate the weight of air displaced2 the third method in fact is adopted4 ,to make these calculations in the frequent weighings requisite in chemical analysis would be exceedingly laborious1 hence the correction is usually neglected4 ,but when the chemist wishes to weigh gas contained in a large glass globe for the purpose of determining its specific gravity1 the correction becomes of much importance4 ,hence chemists avoid at once the error1 and the labour of correcting it1 by attaching to the opposite scale of the balance a dummy sealed glass globe of equal capacity to that containing the gas to be weighed1 noting only the difference of weight when the operating globe is full and empty4 ,the correction1 being the same for both globes1 may be entirely neglected4#bef #bef ,regnault's 99,cours ,elmentaire de ,chimie991 #ahea1 vol i4 p4 #ada4 ,a device of nearly the same kind is employed in the construction of galvanometers which measure the force of an electric current by the deflection of a suspended magnetic needle4 ,the resistance of the needle is partly due to the directive influence of the earth's magnetism1 and partly to the torsion of the thread4 ,but the former force may often be inconveniently great as well as troublesome to determine for different inclinations4 ,hence it is customary to connect together two equally magnetised needles1 with their poles pointing in opposite directions1 one needle being within and another without the coil of wire4 ,as regards the earth's magnetism1 the needles are now 99astatic99 or indifferent1 the tendency of one needle towards the pole being balanced by that of the other4 ,an elegant instance of the elimination of a disturbing force by compensation is found in ,faraday's researches upon the magnetism of gases4 ,to observe the magnetic attraction or repulsion of a gas seems impossible unless we enclose the gas in an envelope1 probably best made of glass4 ,but any such envelope is sure to be more or less affected by the magnet1 so that it becomes difficult to distinguish between three forces which enter into the problem1 namely1 the magnetism of the gas in question1 that of the envelope1 and that of the surrounding atmospheric air4 ,faraday avoided all difficulties by employing two equal and similar glass tubes connected together1 and so suspended from the arm of a torsion balance that the tubes were in similar parts of the magnetic field4 ,one tube being filled with nitrogen and the other with oxygen1 it was found that the oxygen seemed to be attracted and the nitrogen repelled4 ,the suspending thread of the balance was then turned until the force of torsion restored the tubes to their original places1 where the magnetism of the tubes as well as that of the surrounding air1 being the same and in the opposite directions upon the two tubes1 could not produce any interference4 ,the force required to restore the tubes was measured by the amount of torsion of the thread1 and it indicated correctly the difference between the attractive powers of oxygen and nitrogen4 ,the oxygen was then withdrawn from one of the tubes1 and a second experiment made1 so as to compare a vacuum with nitrogen4 ,no force was now required to maintain the tubes in their places1 so that nitrogen was found to be1 approximately speaking1 indifferent to the magnet1 that is1 neither magnetic nor diamagnetic1 while oxygen was proved to be positively magnetic4#beg ,it required the highest experimental skill on the part of ,faraday and ,tyndall1 to distinguish between what is apparent and real in magnetic attraction and repulsion4 #beg ,tyndall's 99,faraday991 pp4 #aad1 #aae4 ,experience alone can finally decide when a compensating arrangement is conducive to accuracy4 ,as a general rule mechanical compensation is the last resource1 and in the more accurate observations it is likely to introduce more uncertainty than it removes4 ,a multitude of instruments involving mechanical compensation have been devised1 but they are usually of an unscientific character1#beh because the errors compensated can be more accurately determined and allowed for4 ,but there are exceptions to this rule1 and it seems to be proved that in the delicate and tiresome operation of measuring a base line1 invariable bars1 compensated for expansion by heat1 give the most accurate results4 ,this arises from the fact that it is very difficult to determine accurately the temperature of the measuring bars under varying conditions of weather and manipulation4#bei ,again1 the last refinement in the measurement of time at ,greenwich ,observatory depends upon mechanical compensation4 ,sir ,george ,airy1 observing that the standard clock increased its losing rate #jcj second for an increase of one inch in atmospheric pressure1 placed a magnet moved by a barometer in such a position below the pendulum1 as almost entirely to neutralise this cause of irregularity4 ,the thorough remedy1 however1 would be to remove the cause of error altogether by placing the clock in a vacuous case4 #beh ,see1 for instance1 the ,compensated ,sympiesometer1 99,philosophical ,magazine991 #dth ,series1 vol4 xxxix4 p4 #cga4 #bei ,grant1 99,history of ,physical ,astronomy991 pp4 #adf1 #adg4 ,we thus see that the choice of one or other mode of eliminating an error depends entirely upon circumstances and the object in view2 but we may safely lay down the following conclusions4 ,first of all1 seek to avoid the source of error altogether if it can be conveniently done2 if not1 make the experiment so that the error may be as small1 but more especially as constant1 as possible4 ,if the means are at hand for determining its amount by calculation from other experiments and principles of science1 allow the error to exist and make a correction in the result4 ,if this cannot be accurately done or involves too much labour for the purposes in view1 then throw in a counteracting error which shall as nearly as possible be of equal amount in all circumstances with that to be eliminated4 ,there yet remains1 however1 one important method1 that of ,reversal1 which will form an appropriate transition to the succeeding chapters on the ,method of ,mean ,results and the ,law of ,error4 #e4 99,method of ,reversal499 ,the fifth method of eliminating error is most potent and satisfactory when it can be applied1 but it requires that we shall be able to reverse the apparatus and mode of procedure1 so as to make the interfering cause act alternately in opposite directions4 ,if we can get two experimental results1 one of which is as much too great as the other is too small1 the error is equal to half the difference1 and the true result is the mean of the two apparent results4 ,it is an unavoidable defect of the chemical balance1 for instance1 that the points of suspension of the pans cannot be fixed at exactly equal distances from the centre of suspension of the beam4 ,hence two weights which seem to balance each other will never be quite equal in reality4 ,the difference is detected by reversing the weights1 and it may be estimated by adding small weights to the deficient side to restore equilibrium1 and then taking as the true weight the geometric mean of the two apparent weights of the same object4 ,if the difference is small1 the arithmetic mean1 that is half the sum1 may be substituted for the geometric mean1 from which it will not appreciably differ4 ,this method of reversal is most extensively employed in practical astronomy4 ,the apparent elevation of a heavenly body is observed by a telescope moving upon a divided circle1 upon which the inclination of the telescope is read off4 ,now this reading will be erroneous if the circle and the telescope have not accurately the same centre4 ,but if we read off at the same time both ends of the telescope1 the one reading will be about as much too small as the other is too great1 and the mean will be nearly free from error4 ,in practice the observation is differently conducted1 but the principle is the same2 the telescope is fixed to the circle1 which moves with it1 and the angle through which it moves is read off at three1 six1 or more points1 disposed at equal intervals round the circle4 ,the older astronomers1 down even to the time of ,flamsteed1 were accustomed to use portions only of a divided circle1 generally quadrants1 and ,rmer made a vast improvement when he introduced the complete circle4 ,the transit circle1 employed to determine the meridian passage of heavenly bodies1 is so constructed that the telescope and the axis bearing it1 in fact the whole moving part of the instrument1 can be taken out of the bearing sockets and turned over1 so that what was formerly the western pivot becomes the eastern one1 and 99vice vers994 ,it is impossible that the instrument could have been so perfectly constructed1 mounted1 and adjusted that the telescope should point exactly to the meridian1 but the effect of the reversal is that it will point as much to the west in one position as it does to the east in the other1 and the mean result of observations in the two positions must be free from such cause of error4 ,the accuracy with which the inclination of the compass needle can be determined depends almost entirely on the method of reversal4 ,the dip needle consists of a bar of magnetised steel1 suspended somewhat like the beam of a delicate balance on a slender axis passing through the centre of gravity of the bar1 so that it is at liberty to rest in that exact degree of inclination in the magnetic meridian which the magnetism of the earth induces4 ,the inclination is read off upon a vertical divided circle1 but to avoid error arising from the centring of the needle and circle1 both ends are read1 and the mean of the results is taken4 ,the whole instrument is now turned carefully round through #ahj1 which causes the needle to assume a new position relatively to the circle and gives two new readings1 in which any error due to the wrong position of the zero of the division will be reversed4 ,as the axis of the needle may not be exactly horizontal1 it is now reversed in the same manner as the transit instrument1 the end of the axis which formerly pointed east being made to point west1 and a new set of four readings is taken4 ,finally1 error may arise from the axis not passing accurately through the centre of gravity of the bar1 and this error can only be detected and eliminated on changing the magnetic poles of the bar by the application of a strong magnet4 ,the error is thus made to act in opposite directions4 ,to ensure all possible accuracy each reversal ought to be combined with each other reversal1 so that the needle will be observed in eight different positions by sixteen readings1 the mean of the whole of which will give the required inclination free from all eliminable errors4#bfj #bfj ,quetelet1 99,sur la ,physique du ,globe991 p4 #agd4 ,jamin1 99,cours de ,physique991 vol4 i4 p4 #ejd4 ,there are certain cases in which a disturbing cause can with ease be made to act in opposite directions1 in alternate observations1 so that the mean of the results will be free from disturbance4 ,thus in direct experiments upon the velocity of sound in passing through the air between stations two or three miles apart1 the wind is a cause of error4 ,it will be well1 in the first place1 to choose a time for the experiment when the air is very nearly at rest1 and the disturbance slight1 but if at the same moment signal sounds be made at each station and observed at the other1 two sounds will be passing in opposite directions through the same body of air and the wind will accelerate one sound almost exactly as it retards the other4 ,again1 in trigonometrical surveys the apparent height of a point will be affected by atmospheric refraction and the curvature of the earth4 ,but if in the case of two points the apparent elevation of each as seen from the other be observed1 the corrections will be the same in amount1 but reversed in direction1 and the mean between the two apparent differences of altitude will give the true difference of level4 ,in the next two chapters we really pursue the ,method of ,reversal into more complicated applications4 ,,chapter ,,xvi4 ,,the ,,method ,,of ,,means4 ,all results of the measurement of continuous quantity can be only approximately true4 ,were this assertion doubted1 it could readily be proved by direct experience4 ,if any person1 using an instrument of the greatest precision1 makes and registers successive observations in an unbiassed manner1 it will almost invariably be found that the results differ from each other4 ,when we operate with sufficient care we cannot perform so simple an experiment as weighing an object in a good balance without getting discrepant numbers4 ,only the rough and careless experimenter will think that his observations agree1 but in reality he will be found to overlook the differences4 ,the most elaborate researches1 such as those undertaken in connection with standard weights and measures1 always render it apparent that complete coincidence is out of the question1 and that the more accurate our modes of observation are rendered1 the more numerous are the sources of minute error which become apparent4 ,we may look upon the existence of error in all measurements as the normal state of things4 ,it is absolutely impossible to eliminate separately the multitude of small disturbing influences1 except by balancing them off against each other4 ,even in drawing a mean it is to be expected that we shall come near the truth rather than exactly to it4 ,in the measurement of continuous quantity1 absolute coincidence1 if it seems to occur1 must be only apparent1 and is no indication of precision4 ,it is one of the most embarrassing things we can meet when experimental results agree too closely4 ,such coincidences should raise our suspicion that the apparatus in use is in some way restricted in its operation1 so as not really to give the true result at all1 or that the actual results have not been faithfully recorded by the assistant in charge of the apparatus4 ,if then we cannot get twice over exactly the same result1 the question arises1 ,how can we ever attain the truth or select the result which may be supposed to approach most nearly to it8 ,the quantity of a certain phenomenon is expressed in several numbers which differ from each other2 no more than one of them at the most can be true1 and it is more probable that they are all false4 ,it may be suggested1 perhaps1 that the observer should select the one observation which he judged to be the best made1 and there will often doubtless be a feeling that one or more results were satisfactory1 and the others less trustworthy4 ,this seems to have been the course adopted by the early astronomers4 ,flamsteed1 when he had made several observations of a star1 probably chose in an arbitrary manner that which seemed to him nearest to the truth4#bfa #bfa ,baily's 99,account of ,flamsteed991 p4 #cgf4 ,when ,horrocks selected for his estimate of the sun's semi-diameter a mean between the results of ,kepler and ,tycho1 he professed not to do it from any regard to the idle adage1 8,medio tutissimus ibis10 but because he thought it from his own observations to be correct4#bfb ,but this method will not apply at all when the observer has made a number of measurements which are equally good in his opinion1 and it is quite apparent that in using an instrument or apparatus of considerable complication the observer will not necessarily be able to judge whether slight causes have affected its operation or not4 #bfb 99,the ,transit of ,venus across the ,sun991 by ,horrocks1 ,london1 #ahei1 p4 #adf4 ,in this question1 as indeed throughout inductive logic1 we deal only with probabilities4 ,there is no infallible mode of arriving at the absolute truth1 which lies beyond the reach of human intellect1 and can only be the distant object of our long-continued and painful approximations4 ,nevertheless there is a mode pointed out alike by common sense and the highest mathematical reasoning1 which is more likely than any other1 as a general rule1 to bring us near the truth4 ,the 1 or the 99aurea mediocritas991 was highly esteemed in the ancient philosophy of ,greece and ,rome2 but it is not probable that any of the ancients should have been able clearly to analyse and express the reasons why they advocated the 99mean99 as the safest course4 ,but in the last two centuries this apparently simple question of the mean has been found to afford a field for the exercise of the utmost mathematical skill4 ,roger ,cotes1 the editor of the 99,principia991 appears to have had some insight into the value of the mean2 but profound mathematicians such as ,de ,moivre1 ,daniel ,bernoulli1 ,laplace1 ,lagrange1 ,gauss1 ,quetelet1 ,de ,morgan1 ,airy1 ,leslie ,ellis1 ,boole1 ,glaisher1 and others1 have hardly exhausted the subject4 99,several uses of the ,mean ,result499 ,the elimination of errors of unknown sources1 is almost always accomplished by the simple arithmetical process of taking the 99mean991 or1 as it is often called1 the 99average99 of several discrepant numbers4 ,to take an average is to add the several quantities together1 and divide by the number of quantities thus added1 which gives a quotient lying among1 or in the 99middle99 of1 the several quantities4 ,before however inquiring fully into the grounds of this procedure1 it is essential to observe that this one arithmetical process is really applied in at least three different cases1 for different purposes1 and upon different principles1 and we must take great care not to confuse one application of the process with another4 ,a 99mean result991 then1 may have any one of the following significations4 7#a7 ,it may give a merely representative number1 expressing the general magnitude of a series of quantities1 and serving as a convenient mode of comparing them with other series of quantities4 ,such a number is properly called 99,the fictitious mean99 or 99,the average result994 7#b7 ,it may give a result approximately free from disturbing quantities1 which are known to affect some results in one direction1 and other results equally in the opposite direction4 ,we may say that in this case we get a 99,precise mean result994 7#c7 ,it may give a result more or less free from unknown and uncertain errors2 this we may call the 99,probable mean result994 ,of these three uses of the mean the first is entirely different in nature from the two last1 since it does not yield an approximation to any natural quantity1 but furnishes us with an arithmetic result comparing the aggregate of certain quantities with their number4 ,the third use of the mean rests entirely upon the theory of probability1 and will be more fully considered in a later part of this chapter4 ,the second use is closely connected1 or even identical with1 the ,method of ,reversal already described1 but it will be desirable to enter somewhat fully into all the three employments of the same arithmetical process4 99,the ,mean and the ,average499 ,much confusion exists in the popular1 or even the scientific employment of the terms 99mean99 and 99average991 and they are commonly taken as synonymous4 ,it is necessary to ascertain carefully what significations we ought to attach to them4 ,the ,english word 99mean99 is equivalent to 99medium991 being derived1 perhaps through the ,french 99moyen991 from the ,latin 99medius991 which again is undoubtedly kindred with the ,greek 4 ,etymologists believe1 too1 that this ,greek word is connected with the preposition 1 the ,german 99mitte991 and the true ,english 99mid99 or 99middle992 so that after all the 99mean99 is a technical term identical in its root with the more popular equivalent 99middle994 ,if we inquire what is the mean in a mathematical point of view1 the true answer is that there are several or many kinds of means4 ,the old arithmeticians recognised ten kinds1 which are stated by ,boethius1 and an eleventh was added by ,jordanus4#bfc #bfc ,de ,morgan1 ,supplement to the 99,penny ,cyclopdia991 art4 99,old ,appellations of ,numbers994 ,the 99arithmetic mean99 is the one by far the most commonly denoted by the term1 and that which we may understand it to signify in the absence of any qualification4 ,it is the sum of a series of quantities divided by their number1 and may be represented by the formula #a_/b799;a99 99b9974 ,but there is also the 99geometric mean991 which is the square root of the product1 799a99 99b9971 or that quantity the logarithm of which is the arithmetic mean of the logarithms of the quantities4 ,there is also the 99harmonic mean991 which is the reciprocal of the arithmetic mean of the reciprocals of the quantities4 ,thus if 99a99 and 99b99 be the quantities1 as before1 their reciprocals are #a_/99;a99 and #a_/99;b991 the mean of which is #a_/b 7#a_/99;a99 #a_/99;b9971 and the reciprocal again is 7#b99;ab997_/799a99 99b9971 which is the harmonic mean4 ,other kinds of means might no doubt be invented for particular purposes1 and we might apply the term1 as ,de ,morgan pointed out1#bfd to any quantity a function of which is equal to a function of two or more other quantities1 and is such that the interchange of these latter quantities among themselves will make no alteration in the value of the function4 ,symbolically1 if 799y991 99y991 99y99 44447 799x99#a1 99x99#b1 99x99#c 444471 then 99y99 is a kind of mean of the quantities1 99x99#a1 99x99#b1 c4 #bfd 99,penny ,cyclopdia991 art4 99,mean994 ,the geometric mean is necessarily adopted in certain cases4 ,when we estimate the work done against a force which varies inversely as the square of the distance from a fixed point1 the mean force is the geometric mean between the forces at the beginning and end of the path4 ,when in an imperfect balance1 we reverse the weights to eliminate error1 the true weight will be the geometric mean of the two apparent weights4 ,in almost all the calculations of statistics and commerce the geometric mean ought1 strictly speaking1 to be used4 ,if a commodity rises in price #ajj per cent4 and another remains unaltered1 the mean rise of a price is not #ej per cent4 because the ratio #aej 3 #bjj is not the same as #ajj 3 #aej4 ,the mean ratio is as unity to 7#ajj #bjj7 or #a to #ada4 ,the difference between the three kinds of means in such a case#bfe is very considerable2 while the rise of price estimated by the ,arithmetic mean would be #ej per cent4 it would be only #da and #cc per cent4 respectively according to the ,geometric and ,harmonic means4 #bfe ,jevons1 99,journal of the ,statistical ,society991 ,june #ahfe1 vol4 xxviii1 p4 #bif4 ,in all calculations concerning the average rate of progress of a community1 or any of its operations1 the geometric mean should be employed4 ,for if a quantity increases #ajj per cent4 in #ajj years1 it would not on the average increase #aj per cent4 in each ten years1 as the #aj per cent4 would at the end of each decade be calculated upon larger and larger quantities1 and give at the end of #ajj years much more than #ajj per cent41 in fact as much as #aei per cent4 ,the true mean rate in each decade would be #ajb or about #ajg1 that is1 the increase would be about #g per cent4 in each ten years4 ,but when the quantities differ very little1 the arithmetic and geometric means are approximately the same4 ,thus the arithmetic mean of #ajjj and #ajja is #ajjje1 and the geometric mean is about #ajjjdiih1 the difference being of an order inappreciable in almost all scientific and practical matters4 ,even in the comparison of standard weights by ,gauss' method of reversal1 the arithmetic mean may usually be substituted for the geometric mean which is the true result4 ,regarding the mean in the absence of express qualification to the contrary as the common arithmetic mean1 we must still distinguish between its two uses where it gives with more or less accuracy and probability a really existing quantity1 and where it acts as a mere representative of other quantities4 ,if ,i make many experiments to determine the atomic weight of an element1 there is a certain number which ,i wish to approximate to1 and the mean of my separate results will1 in the absence of any reasons to the contrary1 be the most probable approximate result4 ,when we determine the mean density of the earth1 it is not because any part of the earth is of that exact density2 there may be no part exactly corresponding to the mean density1 and as the crust of the earth has only about half the mean density1 the internal matter of the globe must of course be above the mean4 ,even the density of a homogeneous substance like carbon or gold must be regarded as a mean between the real density of its atoms1 and the zero density of the intervening vacuous space4 ,the very different signification of the word 8mean0 in these two uses was fully explained by ,quetelet1#bff and the importance of the distinction was pointed out by ,sir ,john ,herschel in reviewing his work4#bfg ,it is much to be desired that scientific men would mark the difference by using the word 99mean99 only in the former sense when it denotes approximation to a definite existing quantity2 and 99average991 when the mean is only a fictitious quantity1 used for convenience of thought and expression4 ,the etymology of this word 8average0 is somewhat obscure2 but according to ,de ,morgan#bfh it comes from 99averia991 8havings or possessions10 especially applied to farm stock4 ,by the accidents of language 99averagium99 came to mean the labour of farm horses to which the lord was entitled1 and it probably acquired in this manner the notion of distributing a whole into parts1 a sense in which it was early applied to maritime averages or contributions of the other owners of cargo to those whose goods have been thrown overboard or used for the safety of the vessel4 #bff 99,letters on the ,theory of ,probabilities991 transl4 by ,downes1 ,part ii4 #bfg ,herschel's 99,essays991 c4 pp4 #djd1 #dje4 #bfh 99,on the ,theory of ,errors of ,observations1 ,cambridge ,philosophical ,transactions991 vol4 x4 ,part ii4 #daf4 99,on the ,average or ,fictitious ,mean499 ,although the average when employed in its proper sense of a fictitious mean1 represents no really existing quantity1 it is yet of the highest scientific importance1 as enabling us to conceive in a single result a multitude of details4 ,it enables us to make a hypothetical simplification of a problem1 and avoid complexity without committing error4 ,the weight of a body is the sum of the weights of infinitely small particles1 each acting at a different place1 so that a mechanical problem resolves itself1 strictly speaking1 into an infinite number of distinct problems4 ,we owe to ,archimedes the first introduction of the beautiful idea that one point may be discovered in a gravitating body such that the weight of all the particles may be regarded as concentrated in that point1 and yet the behaviour of the whole body will be exactly represented by the behaviour of this heavy point4 ,this ,centre of ,gravity may be within the body1 as in the case of a sphere1 or it may be in empty space1 as in the case of a ring4 ,any two bodies1 whether connected or separate1 may be conceived as having a centre of gravity1 that of the sun and earth lying within the sun and only #bfg miles from its centre4 ,although we most commonly use the notion of a centre or average point with regard to gravity1 the same notion is applicable to other cases4 ,terrestrial gravity is a case of approximately parallel forces1 and the centre of gravity is but a special case of the more general ,centre of ,parallel ,forces4 ,wherever a number of forces of whatever amount act in parallel lines1 it is possible to discover a point at which the algebraic sum of the forces may be imagined to act with exactly the same effect4 ,water in a cistern presses against the side with a pressure varying according to the depth1 but always in a direction perpendicular to the side4 ,we may then conceive the whole pressure as exerted on one point1 which will be one-third from the bottom of the cistern1 and may be called the ,centre of ,pressure4 ,the ,centre of ,oscillation of a pendulum1 discovered by ,huyghens1 is that point at which the whole weight of the pendulum may be considered as concentrated1 without altering the time of oscillation 7p4 #cae74 ,when one body strikes another the ,centre of ,percussion is that point in the striking body at which all its mass might be concentrated without altering the effect of the stroke4 ,in position the ,centre of ,percussion does not differ from the ,centre of ,oscillation4 ,mathematicians have also described the ,centre of ,gyration1 the ,centre of ,conversion1 the ,centre of ,friction1 c4 ,we ought carefully to distinguish between those cases in which an 99invariable99 centre can be assigned1 and those in which it cannot4 ,in perfect strictness1 there is no such thing as a true invariable centre of gravity4 ,as a general rule a body is capable of possessing an invariable centre only for perfectly parallel forces1 and gravity never does act in absolutely parallel lines4 ,thus1 as usual1 we find that our conceptions are only hypothetically correct1 and only approximately applicable to real circumstances4 ,there are indeed certain geometrical forms called 99,centrobaric991#bfi such that a body of that shape would attract another exactly as if the mass were concentrated at the centre of gravity1 whether the forces act in a parallel manner or not4 ,newton showed that uniform spheres of matter have this property1 and this truth proved of the greatest importance in simplifying his calculations4 ,but it is after all a purely hypothetical truth1 because we can nowhere meet with1 nor can we construct1 a perfectly spherical and homogeneous body4 ,the slightest irregularity or protrusion from the surface will destroy the rigorous correctness of the assumption4 ,the spheroid1 on the other hand1 has no invariable centre at which its mass may always be regarded as concentrated4 ,the point from which its resultant attraction acts will move about according to the distance and position of the other attracting body1 and it will only coincide with the centre as regards an infinitely distant body whose attractive forces may be considered as acting in parallel lines4 #bfi ,thomson and ,tait1 99,treatise on ,natural ,philosophy991 vol4 i4 p4 #cid4 ,physicists speak familiarly of the poles of a magnet1 and the term may be used with convenience4 ,but1 if we attach any definite meaning to the word1 the poles are not the ends of the magnet1 nor any fixed points within1 but the variable points from which the resultants of all the forces exerted by the particles in the bar upon exterior magnetic particles may be considered as acting4 ,the poles are1 in short1 ,centres of ,magnetic ,forces2 but as those forces are never really parallel1 these centres will vary in position according to the relative place of the object attracted4 ,only when we regard the magnet as attracting a very distant1 or1 strictly speaking1 infinitely distant particle1 do its centres become fixed points1 situated in short magnets approximately at one-sixth of the whole length from each end of the bar4 ,we have in the above instances of centres or poles of force sufficient examples of the mode in which the ,fictitious ,mean or ,average is employed in physical science4 99,the ,precise ,mean ,result499 ,we now turn to that mode of employing the mean result which is analogous to the method of reversal1 but which is brought into practice in a most extensive manner throughout many branches of physical science4 ,we find the simplest possible case in the determination of the latitude of a place by observations of the ,pole-star4 ,tycho ,brahe suggested that if the elevation of any circumpolar star were observed at its higher and lower passages across the meridian1 half the sum of the elevations would be the latitude of the place1 which is equal to the height of the pole4 ,such a star is as much above the pole at its highest passage1 as it is below at its lowest1 so that the mean must necessarily give the height of the pole itself free from doubt1 except as regards incidental errors4 ,the ,pole-star is usually selected for the purpose of such observations because it describes the smallest circle1 and is thus on the whole least affected by atmospheric refraction4 ,whenever several causes are in action1 each of which at one time increases and at another time decreases the joint effect by equal quantities1 we may apply this method and disentangle the effects4 ,thus the solar and lunar tides roll on in almost complete independence of each other4 ,when the moon is new or full the solar tide coincides1 or nearly so1 with that caused by the moon1 and the joint effect is the sum of the separate effects4 ,when the moon is in quadrature1 or half full1 the two tides are acting in opposition1 one raising and the other depressing the water1 so that we observe only the difference of the effects4 ,we have in fact-- ,spring tide lunar tide solar tide2 ,neap tide lunar tide - solar tide4 ,we have only then to add together the heights of the maximum spring tide and the minimum neap tide1 and half the sum is the true height of the lunar tide4 ,half the difference of the spring and neap tides on the other hand gives the solar tide4 ,effects of very small amount may be detected with great approach to certainty among much greater fluctuations1 provided that we have a series of observations sufficiently numerous and long continued to enable us to balance all the larger effects against each other4 ,for this purpose the observations should be continued over at least one complete cycle1 in which the effects run through all their variations1 and return exactly to the same relative positions as at the commencement4 ,if casual or irregular disturbing causes exist1 we should probably require many such cycles of results to render their effect inappreciable4 ,we obtain the desired result by taking the mean of all the observations in which a cause acts positively1 and the mean of all in which it acts negatively4 ,half the difference of these means will give the effect of the cause in question1 provided that no other effect happens to vary in the same period or nearly so4 ,since the moon causes a movement of the ocean1 it is evident that its attraction must have some effect upon the atmosphere4 ,the laws of atmospheric tides were investigated by ,laplace1 but as it would be impracticable by theory to calculate their amounts we can only determine them by observation1 as ,laplace predicted that they would one day be determined4#bgj ,but the oscillations of the barometer thus caused are far smaller than the oscillations due to several other causes4 ,storms1 hurricanes1 or changes of weather produce movements of the barometer sometimes as much as a thousand times as great as the tides in question4 ,there are also regular daily1 yearly1 or other fluctuations1 all greater than the desired quantity4 ,to detect and measure the atmospheric tide it was desirable that observations should be made in a place as free as possible from irregular disturbances4 ,on this account several long series of observations were made at ,st4 ,helena1 where the barometer is far more regular in its movements than in a continental climate4 ,the effect of the moon's attraction was then detected by taking the mean of all the readings when the moon was on the meridian and the similar mean when she was on the horizon4 ,the difference of these means was found to be only #jjcfe1 yet it was possible to discover even the variation of this tide according as the moon was nearer to or further from the earth1 though this difference was only #jjjef inch4#bga ,it is quite evident that such minute effects could never be discovered in a purely empirical manner4 ,having no information but the series of observations before us1 we could have no clue as to the mode of grouping them which would give so small a difference4 ,in applying this method of means in an extensive manner we must generally then have 99 priori99 knowledge as to the periods at which a cause will act in one direction or the other4 #bgj 99,essai ,philosophique sur les ,probabilits991 pp4 #di1 #ej4 #bga ,grant1 99,history of ,physical ,astronomy991 p4 #afc4 ,we are sometimes able to eliminate fluctuations and take a mean result by purely mechanical arrangements4 ,the daily variations of temperature1 for instance1 become imperceptible one or two feet below the surface of the earth1 so that a thermometer placed with its bulb at that depth gives very nearly the true daily mean temperature4 ,at a depth of twenty feet even the yearly fluctuations are nearly effaced1 and the thermometer stands a little above the true mean temperature of the locality4 ,in registering the rise and fall of the tide by a tide-gauge1 it is desirable to avoid the oscillations arising from surface waves1 which is very readily accomplished by placing the float in a cistern communicating by a small hole with the sea4 ,only a general rise or fall of the level is then perceptible1 just as in the marine barometer the narrow tube prevents any casual fluctuations and allows only a continued change of pressure to manifest itself4 99,determination of the ,zero point499 ,in many important observations the chief difficulty consists in defining exactly the zero point from which we are to measure4 ,we can point a telescope with great precision to a star and can measure to a second of arc the angle through which the telescope is raised or lowered2 but all this precision will be useless unless we know exactly the centre point of the heavens from which we measure1 or1 what comes to the same thing1 the horizontal line #ij distant from it4 ,since the true horizon has reference to the figure of the earth at the place of observation1 we can only determine it by the direction of gravity1 as marked either by the plumb-line or the surface of a liquid4 ,the question resolves itself then into the most accurate mode of observing the direction of gravity1 and as the plumb-line has long been found hopelessly inaccurate1 astronomers generally employ the surface of mercury in repose as the criterion of horizontality4 ,they ingeniously observe the direction of the surface by making a star the index4 ,from the laws of reflection it follows that the angle between the direct ray from a star and that reflected from a surface of mercury will be exactly double the angle between the surface and the direct ray from the star4 ,hence the horizontal or zero point is the mean between the apparent place of any star or other very distant object and its reflection in mercury4 ,a plumb-line is perpendicular1 or a liquid surface is horizontal only in an approximate sense2 for any irregularity of the surface of the earth1 a mountain1 or even a house must cause some deviation by its attracting power4 ,to detect such deviation might seem very difficult1 because every other plumb-line or liquid surface would be equally affected by gravity4 ,nevertheless it can be detected2 for if we place one plumb-line to the north of a mountain1 and another to the south1 they will be about equally deflected in opposite directions1 and if by observations of the same star we can measure the angle between the plumb-lines1 half the inclination will be the deviation of either1 after allowance has been made for the inclination due to the difference of latitude of the two places of observation4 ,by this mode of observation applied to the mountain ,schiehallion the deviation of the plumb-line was accurately measured by ,maskelyne1 and thus a comparison instituted between the attractive forces of the mountain and the whole globe1 which led to a probable estimate of the earth's density4 ,in some cases it is actually better to determine the zero point by the average of equally diverging quantities than by direct observation4 ,in delicate weighings by a chemical balance it is requisite to ascertain exactly the point at which the beam comes to rest1 and when standard weights are being compared the position of the beam is ascertained by a carefully divided scale viewed through a microscope4 ,but when the beam is just coming to rest1 friction1 small impediments or other accidental causes may readily obstruct it1 because it is near the point at which the force of stability becomes infinitely small4 ,hence it is found better to let the beam vibrate and observe the terminal points of the vibrations4 ,the mean between two extreme points will nearly indicate the position of rest4 ,friction and the resistance of air tend to reduce the vibrations1 so that this mean will be erroneous by half the amount of this effect during a half vibration4 ,but by taking several observations we may determine this retardation and allow for it4 ,thus if 99a991 99b991 99c99 be the readings of the terminal points of three excursions of the beam from the zero of the scale1 then #a_/b799;a99 99b997 will be about as much erroneous in one direction as #a_/b799;b99 99c997 in the other1 so that the mean of these two means1 or #a_/d799;a99 #b99;b99 99c9971 will be exceedingly near to the point of rest4#bgb ,a still closer approximation may be made by taking four readings and reducing them by the formula #a_/f799;a99 #b99;b99 #b99;c99 99d9974 #bgb ,gauss1 ,taylor's 99,scientific ,memoirs991 vol4 ii4 p4 #dc1 c4 ,the accuracy of ,baily's experiments1 directed to determine the density of the earth1 entirely depended upon this mode of observing oscillations4 ,the balls whose gravitation was measured were so delicately suspended by a torsion balance that they never came to rest4 ,the extreme points of the oscillations were observed both when the heavy leaden attracting ball was on one side and on the other4 ,the difference of the mean points when the leaden ball was on the right hand and that when it was on the left hand gave double the amount of the deflection4 ,a beautiful instance of avoiding the use of a zero point is found in ,mr4 ,e4 ,j4 ,stone's observations on the radiant heat of the fixed stars4 ,the difficulty of these observations arose from the comparatively great amounts of heat which were sent into the telescope from the atmosphere1 and which were sufficient to disguise almost entirely the feeble heat rays of a star4 ,but ,mr4 ,stone fixed at the focus of his telescope a double thermo-electric pile of which the two parts were reversed in order4 ,now any disturbance of temperature which acted uniformly upon both piles produced no effect upon the galvanometer needle1 and when the rays of the star were made to fall alternately upon one pile and the other1 the total amount of the deflection represented double the heating power of the star4 ,thus ,mr4 ,stone was able to detect with much certainty a heating effect of the star ,arcturus1 which even when concentrated by the telescope amounted only to #jjb ,fahr41 and which represents a heating effect of the direct ray of only about #jjjjjjacg ,fahr41 equivalent to the heat which would be received from a three-inch cubic vessel full of boiling water at the distance of #djj yards4#bgc ,it is probable that ,mr4 ,stone's arrangement of the pile might be usefully employed in other delicate thermometric experiments subject to considerable disturbing influences4 #bgc 99,proceedings of the ,royal ,society991 vol4 xviii4 p4 #aei 7,jan4 #ac1 #ahgj74 99,philosophical ,magazine99 7#dth ,series71 vol4 xxxix4 p4 #cgf4 99,determination of ,maximum ,points499 ,we employ the method of means in a certain number of observations directed to determine the moment at which a phenomenon reaches its highest point in quantity4 ,in noting the place of a fixed star at a given time there is no difficulty in ascertaining the point to be observed1 for a star in a good telescope presents an exceedingly small disc4 ,in observing a nebulous body which from a bright centre fades gradually away on all sides1 it will not be possible to select with certainty the middle point4 ,in many such cases the best method is not to select arbitrarily the supposed middle point1 but points of equal brightness on either side1 and then take the mean of the observations of these two points for the centre4 ,as a general rule1 a variable quantity in reaching its maximum increases at a less and less rate1 and after passing the highest point begins to decrease by insensible degrees4 ,the maximum may indeed be defined as that point at which the increase or decrease is null4 ,hence it will usually be the most indefinite point1 and if we can accurately measure the phenomenon we shall best determine the place of the maximum by determining points on either side at which the ordinates are equal4 ,there is moreover this advantage in the method that several points may be determined with the corresponding ones on the other side1 and the mean of the whole taken as the true place of the maximum4 ,but this method entirely depends upon the existence of symmetry in the curve1 so that of two equal ordinates one shall be as far on one side of the maximum as the other is on the other side4 ,the method fails when other laws of variation prevail4 ,in tidal observations great difficulty is encountered in fixing the moment of high water1 because the rate at which the water is then rising or falling1 is almost imperceptible4 ,whewell proposed1 therefore1 to note the time at which the water passes a fixed point somewhat below the maximum both in rising and falling1 and take the mean time as that of high water4 ,but this mode of proceeding unfortunately does not give a correct result1 because the tide follows different laws in rising and in falling4 ,there is a difficulty again in selecting the highest spring tide1 another object of much importance in tidology4 ,laplace discovered that the tide of the second day preceding the conjunction of the sun and moon is nearly equal to that of the fifth day following2 and1 believing that the increase and decrease of the tides proceeded in a nearly symmetrical manner1 he decided that the highest tide would occur about thirty-six hours after the conjunction1 that is half-way between the second day before and the fifth day after4#bgd #bgd ,airy 99,on ,tides and ,waves991 ,encycl4 ,metrop4 pp4 #cfd99-cff994 ,this method is also employed in determining the time of passage of the middle or densest point of a stream of meteors4 ,the earth takes two or three days in passing completely through the ,november stream2 but astronomers need for their calculations to have some definite point fixed within a few minutes if possible4 ,when near to the middle they observe the numbers of meteors which come within the sphere of vision in each half hour1 or quarter hour1 and then1 assuming that the law of variation is symmetrical1 they select a moment for the passage of the centre equidistant between times of equal frequency4 ,the eclipses of ,jupiter's satellites are not only of great interest as regards the motions of the satellites themselves1 but were1 and perhaps still are1 of use in determining longitudes1 because they are events occurring at fixed moments of absolute time1 and visible in all parts of the planetary system at the same time1 allowance being made for the interval occupied by the light in travelling4 ,but1 as is explained by ,herschel1#bge the moment of the event is wanting in definiteness1 partly because the long cone of ,jupiter's shadow is surrounded by a penumbra1 and partly because the satellite has itself a sensible disc1 and takes time in entering the shadow4 ,different observers using different telescopes would usually select different moments for that of the eclipse4 ,but the increase of light in the emersion will proceed according to a law the reverse of that observed in the immersion1 so that if an observer notes the time of both events with the same telescope1 he will be as much too soon in one observation as he is too late in the other1 and the mean moment of the two observations will represent with considerable accuracy the time when the satellite is in the middle of the shadow4 ,error of judgment of the observer is thus eliminated1 provided that he takes care to act at the emersion as he did at the immersion4 #bge 99,outlines of ,astronomy991 #dth edition1 #ech4 ,,chapter ,,xvii4 ,,the ,,law ,,of ,,error4 ,to bring error itself under law might seem beyond human power4 ,he who errs surely diverges from law1 and it might be deemed hopeless out of error to draw truth4 ,one of the most remarkable achievements of the human intellect is the establishment of a general theory which not only enables us among discrepant results to approximate to the truth1 but to assign the degree of probability which fairly attaches to this conclusion4 ,it would be a mistake indeed to suppose that this law is necessarily the best guide under all circumstances4 ,every measuring instrument and every form of experiment may have its own special law of error2 there may in one instrument be a tendency in one direction and in another in the opposite direction4 ,every process has its peculiar liabilities to disturbance1 and we are never relieved from the necessity of providing against special difficulties4 ,the general ,law of ,error is the best guide only when we have exhausted all other means of approximation1 and still find discrepancies1 which are due to unknown causes4 ,we must treat such residual differences in some way or other1 since they will occur in all accurate experiments1 and as their origin is assumed to be unknown1 there is no reason why we should treat them differently in different cases4 ,accordingly the ultimate ,law of ,error must be a uniform and general one4 ,it is perfectly recognised by mathematicians that in each case a special ,law of ,error may exist1 and should be discovered if possible4 8,nothing can be more unlikely than that the errors committed in all classes of observations should follow the same law10#bgf and the special ,laws of ,error which will apply to certain instruments1 as for instance the repeating circle1 have been investigated by ,bravais4#bgg ,he concludes that every distinct cause of error gives rise to a curve of possibility of errors1 which may have any form1--a curve which we may either be able or unable to discover1 and which in the first case may be determined by 99 priori99 considerations on the peculiar nature of this cause1 or which may be determined 99 posteriori99 by observation4 ,whenever it is practicable and worth the labour1 we ought to investigate these special conditions of error2 nevertheless1 when there are a great number of different sources of minute error1 the general resultant will always tend to obey that general law which we are about to consider4 #bgf 99,philosophical ,magazine991 #crd ,series1 vol4 xxxvii4 p4 #cbd4 #bgg 99,letters on the ,theory of ,probabilities991 by ,quetelet1 translated by ,o4 ,g4 ,downes1 ,notes to ,letter ,,xxvi4 pp4 #bhf-bie4 99,establishment of the ,law of ,error499 ,mathematicians agree far better as to the form of the ,law of ,error than they do as to the manner in which it can be deduced and proved4 ,they agree that among a number of discrepant results of observation1 that mean quantity is probably the best approximation to the truth which makes the sum of the squares of the errors as small as possible4 ,but there are three principal ways in which this law has been arrived at respectively by ,gauss1 by ,laplace and ,quetelet1 and by ,sir ,john ,herschel4 ,gauss proceeds much upon assumption2 ,herschel rests upon geometrical considerations2 while ,laplace and ,quetelet regard the ,law of ,error as a development of the doctrine of combinations4 ,a number of other mathematicians1 such as ,adrain of ,new ,brunswick1 ,bessel1 ,ivory1 ,donkin1 ,leslie ,ellis1 ,tait1 and ,crofton have either attempted independent proofs or have modified or commented on those here to be described4 ,for full accounts of the literature of the subject the reader should refer either to ,mr4 ,todhunter's 99,history of the ,theory of ,probability99 or to the able memoir of ,mr4 ,j4 ,w4 ,l4 ,glaisher4#bgh #bgh 99,on the ,law of ,facility of ,errors of ,observations1 and on the ,method of ,least ,squares991 ,memoirs of the ,royal ,astronomical ,society1 vol4 xxxix4 p4 #ge4 ,according to ,gauss the ,law of ,error expresses the comparative probability of errors of various magnitude1 and partly from experience1 partly from 99 priori99 considerations1 we may readily lay down certain conditions to which the law will certainly conform4 ,it may fairly be assumed as a first principle to guide us in the selection of the law1 that large errors will be far less frequent and probable than small ones4 ,we know that very large errors are almost impossible1 so that the probability must rapidly decrease as the amount of the error increases4 ,a second principle is that positive and negative errors shall be equally probable1 which may certainly be assumed1 because we are supposed to be devoid of any knowledge as to the causes of the residual errors4 ,it follows that the probability of the error must be a function of an even power of the magnitude1 that is of the square1 or the fourth power1 or the sixth power1 otherwise the probability of the same amount of error would vary according as the error was positive or negative4 ,the even powers 99x99#b1 99x99#d1 99x99#f1 c41 are always intrinsically positive1 whether 99x99 be positive or negative4 ,there is no 99 priori99 reason why one rather than another of these even powers should be selected4 ,gauss himself allows that the fourth or sixth power would fulfil the conditions as well as the second2#bgi but in the absence of any theoretical reasons we should prefer the second power1 because it leads to formul of great comparative simplicity4 ,did the ,law of ,error necessitate the use of the higher powers of the error1 the complexity of the necessary calculations would much reduce the utility of the theory4 #bgi 99,mthode des ,moindres ,carrs4 ,mmoires sur la ,combinaison des ,observations1 par ,ch4 ,fr4 ,gauss4 ,traduit en ,franais par ,j4 ,bertrand991 ,paris1 #ahee1 pp4 #f1 #acc1 c4 ,by mathematical reasoning which it would be undesirable to attempt to follow in this book1 it is shown that under these conditions1 the facility of occurrence1 or in other1 words1 the probability of error is expressed by a function of the general form -99h99#b 99x99#b1 in which 99x99 represents the variable amount of errors4 ,from this law1 to be more fully described in the following sections1 it at once follows that the most probable result of any observations is that which makes the sum of the squares of the consequent errors the least possible4 ,let 99a991 99b991 99c991 c41 be the results of observation1 and 99x99 the quantity selected as the most probable1 that is the most free from unknown errors3 then we must determine 99x99 so that 799a99 - 99x997#b 799b99 - 99x997#b 799c99 - 99x997#b 444 shall be the least possible quantity4 ,thus we arrive at the celebrated 99,method of ,least ,squares991 as it is usually called1 which appears to have been first distinctly put in practice by ,gauss in #agie1 while ,legendre first published in #ahjf an account of the process in his work1 entitled1 99,nouvelles ,mthodes pour la ,dtermination des ,orbites des ,comtes994 ,it is worthy of notice1 however1 that ,roger ,cotes had long previously recommended a method of equivalent nature in his tract1 8,estimatio ,erroris in ,mixta ,mathesi40#bhj #bhj ,de ,morgan1 99,penny ,cyclopdia991 art4 99,least ,squares994 99,herschel's ,geometrical ,proof499 ,a second way of arriving at the ,law of ,error was proposed by ,herschel1 and although only applicable to geometrical cases1 it is remarkable as showing that from whatever point of view we regard the subject1 the same principle will be detected4 ,after assuming that some general law must exist1 and that it is subject to the principles of probability1 he supposes that a ball is dropped from a high point with the intention that it shall strike a given mark on a horizontal plane4 ,in the absence of any known causes of deviation it will either strike that mark1 or1 as is infinitely more probable1 diverge from it by an amount which we must regard as error of unknown origin4 ,now1 to quote the words of ,herschel1#bha 8the probability of that error is the unknown function of its square1 99i4e499 of the sum of the squares of its deviations in any two rectangular directions4 ,now1 the probability of any deviation depending solely on its magnitude1 and not on its direction1 it follows that the probability of each of these rectangular deviations must be the same function of 99its99 square4 ,and since the observed oblique deviation is equivalent to the two rectangular ones1 supposed concurrent1 and which are essentially independent of one another1 and is1 therefore1 a compound event of which they are the simple independent constituents1 therefore its probability will be the product of their separate probabilities4 ,thus the form of our unknown function comes to be determined from this condition1 viz41 that the product of such functions of two independent elements is equal to the same function of their sum4 ,but it is shown in every work on algebra that this property is the peculiar characteristic of1 and belongs only to1 the exponential or antilogarithmic function4 ,this1 then1 is the function of the square of the error1 which expresses the probability of committing that error4 ,that probability decreases1 therefore1 in geometrical progression1 as the square of the error increases in arithmetical40 #bha 99,edinburgh ,review991 ,july #ahej1 vol4 xcii4 p4 #ag4 ,reprinted 99,essays991 p4 #cii4 ,this method of demonstration is discussed by ,boole1 99,transactions of ,royal ,society of ,edinburgh991 vol4 xxi4 pp4 #fbg-fcj4 99,laplace's and ,quetelet's ,proof of the ,law499 ,however much presumption the modes of determining the ,law of ,error1 already described1 may give in favour of the law usually adopted1 it is difficult to feel that the arguments are satisfactory4 ,the law adopted is chosen rather on the grounds of convenience and plausibility1 than because it can be seen to be the necessary law4 ,we can however approach the subject from an entirely different point of view1 and yet get to the same result4 ,let us assume that a particular observation is subject to four chances of error1 each of which will increase the result one inch if it occurs4 ,each of these errors is to be regarded as an event independent of the rest and we can therefore assign1 by the theory of probability1 the comparative probability and frequency of each conjunction of errors4 ,from the ,arithmetical ,triangle 7pp4 #ahb-ahh7 we learn that no error at all can happen only in one way2 an error of one inch can happen in #d ways2 and the ways of happening of errors of #b1 #c and #d inches respectively1 will be #f1 #d and #a in number4 ,we may infer that the error of two inches is the most likely to occur1 and will occur in the long run in six cases out of sixteen4 ,errors of one and three inches will be equally likely1 but will occur less frequently2 while no error at all1 or one of four inches will be a comparatively rare occurrence4 ,if we now suppose the errors to act as often in one direction as the other1 the effect will be to alter the average error by the amount of two inches1 and we shall have the following results3-- ,negative error of #b inches #a way4 ,negative error of #a inch #d ways4 ,no error at all #f ways4 ,positive error of #a inch #d ways4 ,positive error of #b inches #a way4 ,we may now imagine the number of causes of error increased and the amount of each error decreased1 and the arithmetical triangle will give us the frequency of the resulting errors4 ,thus if there be five positive causes of error and five negative causes1 the following table shows the numbers of errors of various amount which will be the result3-- ---------------------------------------- ----------------------- ,direction of ,error4 ,positive ,error4 ,negative ,error4 ---------------------------------------- ----------------------- ,amount of ,error4 #e1 #d1 #c1 #b1 #a #j #a1 #b1 #c1 #d1 #e ---------------------------------------- ----------------------- ,number of such ,errors4#a1 #aj1 #de1 #abj1 #bajbebbaj1 #abj1 #de1 #aj1 #a ---------------------------------------- ----------------------- ,it is plain that from such numbers ,i can ascertain the probability of any particular amount of error under the conditions supposed4 ,the probability of a positive error of exactly one inch is #baj_/ajbd1 in which fraction the numerator is the number of combinations giving one inch positive error1 and the denominator the whole number of possible errors of all magnitudes4 ,i can also1 by adding together the appropriate numbers get the probability of an error not exceeding a certain amount4 ,thus the probability of an error of three inches or less1 positive or negative1 is a fraction whose numerator is the sum of #de #abj #baj #beb #baj #abj #de1 and the denominator1 as before1 giving the result #ajjb_/ajbd4 ,we may see at once that1 according to these principles1 the probability of small errors is far greater than of large ones3 the odds are #ajjb to #bb1 or more than #de to #a1 that the error will not exceed three inches2 and the odds are #ajbb to #b against the occurrence of the greatest possible error of five inches4 ,if any case should arise in which the observer knows the number and magnitude of the chief errors which may occur1 he ought certainly to calculate from the ,arithmetical ,triangle the special ,law of ,error which would apply4 ,but the general law1 of which we are in search1 is to be used in the dark1 when we have no knowledge whatever of the sources of error4 ,to assume any special number of causes of error is then an arbitrary proceeding1 and mathematicians have chosen the least arbitrary course of imagining the existence of an infinite number of infinitely small errors1 just as1 in the inverse method of probabilities1 an infinite number of infinitely improbable hypotheses were submitted to calculation 7p4 #bee74 ,the reasons in favour of this choice are of several different kinds4 #a4 ,it cannot be denied that there may exist infinitely numerous causes of error in any act of observation4 #b4 ,the law resulting from the hypothesis of a moderate number of causes of error1 does not appreciably differ from that given by the hypothesis of an infinite number of causes of error4 #c4 ,we gain by the hypothesis of infinity a general law capable of ready calculation1 and applicable by uniform rules to all problems4 #d4 ,this law1 when tested by comparison with extensive series of observations1 is strikingly verified1 as will be shown in a later section4 ,when we imagine the existence of any large number of causes of error1 for instance one hundred1 the numbers of combinations become impracticably large1 as may be seen to be the case from a glance at the ,arithmetical ,triangle1 which proceeds only up to the seventeenth line4 ,quetelet1 by suitable abbreviating processes1 calculated out a table of probability of errors on the hypothesis of one thousand distinct causes2#bhb but mathematicians have generally proceeded on the hypothesis of infinity1 and then1 by the devices of analysis1 have substituted a general law of easy treatment4 ,in mathematical works upon the subject1 it is shown that the standard ,law of ,error is expressed in the formula 99y99 99,y99-99cx99#b1 #bhb 99,letters on the ,theory of ,probabilities991 ,letter ,,xv4 and ,appendix1 note pp4 #bef-bff4 in which 99x99 is the amount of the error1 99,y99 the maximum ordinate of the curve of error1 and 99c99 a number constant for each series of observations1 and expressing the amount of the tendency to error1 varying between one series of observations and another4 ,the letter is the mathematical constant1 the sum of ratios between the numbers of permutations and combinations1 previously referred to 7p4 #ccj74 ,illustration ,to show the close correspondence of this general law with the special law which might be derived from the supposition of a moderate number of causes of error1 ,i have in the accompanying figure drawn a curved line representing accurately the variation of 99y99 when 99x99 in the above formula is taken equal #j1 #a_/b1 #a1 #c_/b1 #b1 c41 positive or negative1 the arbitrary quantities 99,y99 and 99c99 being each assumed equal to unity1 in order to simplify the calculations4 ,in the same figure are inserted eleven dots1 whose heights above the base line are proportional to the numbers in the eleventh line of the ,arithmetical ,triangle1 thus representing the comparative probabilities of errors of various amounts arising from ten equal causes of error4 ,the correspondence of the general and the special ,law of ,error is almost as close as can be exhibited in the figure1 and the assumption of a greater number of equal causes of error would render the correspondence far more close4 ,it may be explained that the ordinates ,,nm1 99nm991 99nm991 represent values of 99y99 in the equation expressing the ,law of ,error4 ,the occurrence of any one definite amount of error is infinitely improbable1 because an infinite number of such ordinates might be drawn4 ,but the probability of an error occurring between certain limits is finite1 and is represented by a portion of the 99area99 of the curve4 ,thus the probability that an error1 positive or negative1 not exceeding unity will occur1 is represented by the area ,m99mnnm991 in short1 by the area standing upon the line 99nn994 ,since every observation must either have some definite error or none at all1 it follows that the whole area of the curve should be considered as the unit expressing certainty1 and the probability of an error falling between particular limits will then be expressed by the ratio which the area of the curve between those limits bears to the whole area of the curve4 ,the mere fact that the ,law of ,error allows of the possible existence of errors of every assignable amount shows that it is only approximately true4 ,we may fairly say that in measuring a mile it would be impossible to commit an error of a hundred miles1 and the length of life would never allow of our committing an error of one million miles4 ,nevertheless the general ,law of ,error would assign a probability for an error of that amount or more1 but so small a probability as to be utterly inconsiderable and almost inconceivable4 ,all that can1 or in fact need1 be said in defence of the law is1 that it may be made to represent the errors in any special case to a very close approximation1 and that the probability of large and practically impossible errors1 as given by the law1 will be so small as to be entirely inconsiderable4 ,and as we are dealing with error itself1 and our results pretend to nothing more than approximation and probability1 an indefinitely small error in our process of approximation is of no importance whatever4 99,logical ,origin of the ,law of ,error499 ,it is worthy of notice that this ,law of ,error1 abstruse though the subject may seem1 is really founded upon the simplest principles4 ,it arises entirely out of the difference between permutations and combinations1 a subject upon which ,i may seem to have dwelt with unnecessary prolixity in previous pages 7pp4 #agj1 #ahi74 ,the order in which we add quantities together does not affect the amount of the sum1 so that if there be three positive and five negative causes of error in operation1 it does not matter in which order they are considered as acting4 ,they may be intermixed in any arrangement1 and yet the result will be the same4 ,the reader should not fail to notice how laws or principles which appeared to be absurdly simple and evident when first noticed1 reappear in the most complicated and mysterious processes of scientific method4 ,the fundamental ,laws of ,identity and ,difference gave rise to the ,logical ,alphabet which1 after abstracting the character of the differences1 led to the ,arithmetical ,triangle4 ,the ,law of ,error is defined by an infinitely high line of that triangle1 and the law proves that the mean is the most probable result1 and that divergencies from the mean become much less probable as they increase in amount4 ,now the comparative greatness of the numbers towards the middle of each line of the ,arithmetical ,triangle is entirely due to the indifference of order in space or time1 which was first prominently pointed out as a condition of logical relations1 and the symbols indicating them 7pp4 #cb-ce71 and which was afterwards shown to attach equally to numerical symbols1 the derivatives of logical terms 7p4 #afj74 99,verification of the ,law of ,error499 ,the theory of error which we have been considering rests entirely upon an assumption1 namely that when known sources of disturbances are allowed for1 there yet remain an indefinite1 possibly an infinite number of other minute sources of error1 which will as often produce excess as deficiency4 ,granting this assumption1 the ,law of ,error must be as it is usually taken to be1 and there is no more need to verify it empirically than to test the truth of one of ,euclid's propositions mechanically4 ,nevertheless1 it is an interesting occupation to verify even the propositions of geometry1 and it is still more instructive to try whether a large number of observations will justify our assumption of the ,law of ,error4 ,encke has given an excellent instance of the correspondence of theory with experience1 in the case of observations of the differences of ,right ,ascension of the sun and two stars1 namely ,aquil and ,canis minoris4 ,the observations were #dgj in number1 and were made by ,bradley and reduced by ,bessel1 who found the probable error of the final result to be only about one-fourth part of a second 7#jbfcg74 ,he then compared the numbers of errors of each magnitude from #ja second upwards1 as actually given by the observations1 with what should occur according to the ,law of ,error4 ,the results were as follow3--#bhc ---------------------------------------- ----------- ,number of errors of each ,magnitude of the errors magnitude according to in parts of a second4 ------------------------- ,observation4 ,theory4 ---------------------------------------- ---------- #jj to #ja #id #ie #a #b #hh #hi #b #c #gh #gh #c #d #eh #fd #d #e #ea #ej #e #f #cf #cf #f #g #bf #bd #g #h #ad #ae #h #i #aj #i #i #aj #g #e above #aj #h #e ---------------------------------------- ---------- #bhc ,encke1 99,on the ,method of ,least ,squares991 ,taylor's 99,scientific ,memoirs991 vol4 ii4 pp4 #cch1 #cci4 ,the reader will remark that the correspondence is very close1 except as regards larger errors1 which are excessive in practice4 ,it is one objection1 indeed1 to the theory of error1 that1 being expressed in a continuous mathematical function1 it contemplates the existence of errors of every magnitude1 such as could not practically occur2 yet in this case the theory seems to under-estimate the number of large errors4 ,another comparison of the law with observation was made by ,quetelet1 who investigated the errors of #dhg determinations in time of the ,right ,ascension of the ,pole-,star made at ,greenwich during the four years #ahcf-ci4 ,these observations1 although carefully corrected for all known causes of error1 as well as for nutation1 precession1 c41 are yet of course found to differ1 and being classified as regards intervals of one-half second of time1 and then proportionately increased in number1 so that their sum may be one thousand1 give the following results as compared with what ,quetelet's theory would lead us to expect3--#bhd ---------------------------------------- ------------------------ ,magnitude of ,number of ,errors ,magnitude of ,number of ,errors error ------------------- error ------------------- in tenths by by in tenths by by of a second4,observation4,theory4of a second4,observation4,theory4 ---------------------------------------- ---------------------- #jj #afh #afc -- -- -- #je #adh #adg -#je #aej #aeb #aj #abi #aab -#aj #abf #aba #ae #gh #gb -#ae #gd #hb #bj #cc #dj -#bj #dc #df #be #aj #ai -#be #be #bb #cj #b #aj -#cj #ab #aj -- -- -- -#ce #b #d ---------------------------------------- ---------------------- #bhd ,quetelet1 99,letters on the ,theory of ,probabilities991 translated by ,downes1 ,letter ,,xix4 p4 #hh4 ,see also ,galton's 99,hereditary ,genius991 p4 #cgi4 ,in this instance also the correspondence is satisfactory1 but the divergence between theory and fact is in the opposite direction to that discovered in the former comparison1 the larger errors being less frequent than theory would indicate4 ,it will be noticed that ,quetelet's theoretical results are not symmetrical4 99,the ,probable ,mean ,result499 ,one immediate result of the ,law of ,error1 as thus stated1 is that the mean result is the most probable one2 and when there is only a single variable this mean is found by the familiar arithmetical process4 ,an unfortunate error has crept into several works which allude to this subject4 ,mill1 in treating of the 8,elimination of ,chance10 remarks in a note#bhe that 8the mean is spoken of as if it were exactly the same thing as the average4 ,but the mean1 for purposes of inductive inquiry1 is not the average1 or arithmetical mean1 though in a familiar illustration of the theory the difference may be disregarded40 ,he goes on to say that1 according to mathematical principles1 the most probable result is that for which the sums of the squares of the deviations is the least possible4 ,it seems probable that ,mill and other writers were misled by ,whewell1 who says#bhf that 8,the method of least squares is in fact a method of means1 but with some peculiar characters4444 ,the method proceeds upon this supposition3 that all errors are not equally probable1 but that small errors are more probable than large ones40 ,he adds that this method 8removes much that is arbitrary in the method of means40 ,it is strange to find a mathematician like ,whewell making such remarks1 when there is no doubt whatever that the ,method of ,means is only an application of the ,method of ,least ,squares4 ,they are1 in fact1 the same method1 except that the latter method may be applied to cases where two or more quantities have to be determined at the same time4 ,lubbock and ,drinkwater say1#bhg 8,if only one quantity has to be determined1 this method evidently resolves itself into taking the mean of all the values given by observation40 ,encke says1#bhh that the expression for the probability of an error 8not only contains in itself the principle of the arithmetical mean1 but depends so immediately upon it1 that for all those magnitudes for which the arithmetical mean holds good in the simple cases in which it is principally applied1 no other law of probability can be assumed than that which is expressed by this formula40 #bhe 99,system of ,logic991 bk4 iii4 chap4 #ag1 #c4 #eth ed4 vol4 ii4 p4 #ef4 #bhf 99,philosophy of the ,inductive ,sciences991 #bnd ed4 vol4 ii4 pp4 #djh1 #dji4 #bhg 99,essay on ,probability991 ,useful ,knowledge ,society1 #ahcc1 p4 #da4 #bhh ,taylor's 99,scientific ,memoirs991 vol4 ii4 p4 #ccc4 99,the ,probable ,error of ,results499 ,when we draw a conclusion from the numerical results of observations we ought not to consider it sufficient1 in cases of importance1 to content ourselves with finding the simple mean and treating it as true4 ,we ought also to ascertain what is the degree of confidence we may place in this mean1 and our confidence should be measured by the degree of concurrence of the observations from which it is derived4 ,in some cases the mean may be approximately certain and accurate4 ,in other cases it may really be worth little or nothing4 ,the ,law of ,error enables us to give exact expression to the degree of confidence proper in any case2 for it shows how to calculate the probability of a divergence of any amount from the mean1 and we can thence ascertain the probability that the mean in question is within a certain distance from the true number4 ,the 99probable error99 is taken by mathematicians to mean the limits within which it is as likely as not that the truth will fall4 ,thus if #ede be the mean of all the determinations of the density of the earth1 and #bj be approximately the probable error1 the meaning is that the probability of the real density of the earth falling between #ebe and #efe is #a_/b4 ,any other limits might have been selected at will4 ,we might calculate the limits within which it was one hundred or one thousand to one that the truth would fall2 but there is a convention to take the even odds one to one1 as the quantity of probability of which the limits are to be estimated4 ,many books on probability give rules for making the calculations1 but as1 in the progress of science1 persons ought to become more familiar with these processes1 ,i propose to repeat the rules here and illustrate their use4 ,the calculations1 when made in accordance with the directions1 involve none but arithmetic or logarithmic operations4 ,the following are the rules for treating a mean result1 so as thoroughly to ascertain its trustworthiness4 #a4 ,draw the mean of all the observed results4 #b4 ,find the excess or defect1 that is1 the error of each result from the mean4 #c4 ,square each of these reputed errors4 #d4 ,add together all these squares of the errors1 which are of course all positive4 #e4 ,divide by one less than the number of observations4 ,this gives the 99square of the mean error994 #f4 ,take the square root of the last result2 it is the 99mean error of a single observation994 #g4 ,divide now by the square root of the number of observations1 and we get the 99mean error of the mean result994 #h4 ,lastly1 multiply by the natural constant #jfgde 7or approximately by #jfgd1 or even by #b_/c71 and we arrive at the 99probable error of the mean result994 ,suppose1 for instance1 that five measurements of the height of a hill1 by the barometer or otherwise1 have given the numbers of feet as #bic1 #cja1 #cjf1 #cjg1 #cac2 we want to know the probable error of the mean1 namely #cjd4 ,now the differences between this mean and the above numbers1 99paying no regard to direction991 are #aa1 #c1 #b1 #c1 #i2 their squares are #aba1 #i1 #d1 #i1 #ha1 and the sum of the squares of the errors consequently #bbd4 ,the number of observations being #e1 we divide by #a less1 or #d1 getting #ef4 ,this is the square of the mean error1 and taking its square root we have #gdh 7say #g-a_/b71 the mean error of a single observation4 ,dividing by #bbcf1 the square root of #e1 the number of observations1 we find the mean error of the 99mean99 result to be #cce1 or say #c-a_/c1 and lastly1 multiplying by #fgde1 we arrive at the 99probable error of the mean result991 which is found to be #bbei1 or say #b-a_/d4 ,the meaning of this is that the probability is one half1 or the odds are even that the true height of the mountain lies between #cja-c_/d and #cjf-a_/d feet4 ,we have thus an exact measure of the degree of credibility of our mean result1 which mean indicates the most likely point for the truth to fall upon4 ,the reader should observe that as the object in these calculations is only to gain a notion of the degree of confidence with which we view the mean1 there is no real use in carrying the calculations to any great degree of precision2 and whenever the neglect of decimal fractions1 or even the slight alteration of a number1 will much abbreviate the computations1 it may be fearlessly done1 except in cases of high importance and precision4 ,brodie has shown how the law of error may be usefully applied in chemical investigations1 and some illustrations of its employment may be found in his paper4#bhi #bhi 99,philosophical ,transactions991 #ahgc1 p4 #hc4 ,the experiments of ,benzenberg to detect the revolution of the earth1 by the deviation of a ball from the perpendicular line in falling down a deep pit1 have been cited by ,encke#bij as an interesting illustration of the ,law of ,error4 ,the mean deviation was #ejhf lines1 and its probable error was calculated by ,encke to be not more than #iej line1 that is1 the odds were even that the true result lay between #dacf and #fjcf4 ,as the deviation1 according to astronomical theory1 should be #df lines1 which lies well within the limits1 we may consider that the experiments are consistent with the ,copernican system of the universe4 #bij ,taylor's 99,scientific ,memoirs991 vol4 ii4 pp4 #ccj1 #cdg1 c4 ,it will of course be understood that the probable error has regard only to those causes of errors which in the long run act as much in one direction as another2 it takes no account of constant errors4 ,the true result accordingly will often fall far beyond the limits of probable error1 owing to some considerable constant error or errors1 of the existence of which we are unaware4 99,rejection of the ,mean ,result499 ,we ought always to bear in mind that the mean of any series of observations is the best1 that is1 the most probable approximation to the truth1 only in the absence of knowledge to the contrary4 ,the selection of the mean rests entirely upon the probability that unknown causes of error will in the long run fall as often in one direction as the opposite1 so that in drawing the mean they will balance each other4 ,if we have any reason to suppose that there exists a tendency to error in one direction rather than the other1 then to choose the mean would be to ignore that tendency4 ,we may certainly approximate to the length of the circumference of a circle1 by taking the mean of the perimeters of inscribed and circumscribed polygons of an equal and large number of sides4 ,the length of the circular line undoubtedly lies between the lengths of the two perimeters1 but it does not follow that the mean is the best approximation4 ,it may in fact be shown that the circumference of the circle is 99very nearly99 equal to the perimeter of the inscribed polygon1 together with one-third part of the difference between the inscribed and circumscribed polygons of the same number of sides4 ,having this knowledge1 we ought of course to act upon it1 instead of trusting to probability4 ,we may often perceive that a series of measurements tends towards an extreme limit rather than towards a mean4 ,in endeavouring to obtain a correct estimate of the apparent diameter of the brightest fixed stars1 we find a continuous diminution in estimates as the powers of observation increased4 ,kepler assigned to ,sirius an apparent diameter of #bdj seconds2 ,tycho ,brahe made it #abf2 ,gassendi #aj seconds2 ,galileo1 ,hevelius1 and ,j4 ,cassini1 #e or #f seconds4 ,halley1 ,michell1 and subsequently ,sir ,w4 ,herschel came to the conclusion that the brightest stars in the heavens could not have real discs of a second1 and were probably much less in diameter4 ,it would of course be absurd to take the mean of quantities which differ more than #bdj times2 and as the tendency has always been to smaller estimates1 there is a considerable presumption in favour of the smallest4#bia #bia ,quetelet1 99,letters991 c4 p4 #aaf4 ,in many experiments and measurements we know that there is a preponderating tendency to error in one direction4 ,the readings of a thermometer tend to rise as the age of the instrument increases1 and no drawing of means will correct this result4 ,barometers1 on the other hand1 are likely to read too low instead of too high1 owing to the imperfection of the vacuum and the action of capillary attraction4 ,if the mercury be perfectly pure and no appreciable error be due to the measuring apparatus1 the best barometer will be that which gives the highest result4 ,in determining the specific gravity of a solid body the chief danger of error arises from bubbles of air adhering to the body1 which would tend to make the specific gravity too small4 ,much attention must always be given to one-sided errors of this kind1 since the multiplication of experiments does not remove the error4 ,in such cases one very careful experiment is better than any number of careless ones4 ,when we have reasonable grounds for supposing that certain experimental results are liable to grave errors1 we should exclude them in drawing a mean4 ,if we want to find the most probable approximation to the velocity of sound in air1 it would be absurd to go back to the old experiments which made the velocity from #abjj to #adgd feet per second2 for we know that the old observers did not guard against errors arising from wind and other causes4 ,old chemical experiments are valueless as regards quantitative results4 ,the old chemists found the atmosphere in different places to differ in composition nearly ten per cent41 whereas modern accurate experimenters find very slight variations4 ,any method of measurement which we know to avoid a source of error is far to be preferred to others which trust to probabilities for the elimination of the error4 ,as ,flamsteed says1#bib 8,one good instrument is of as much worth as a hundred indifferent ones40 ,but an instrument is good or bad only in a comparative sense1 and no instrument gives invariable and truthful results4 ,hence we must always ultimately fall back upon probabilities for the selection of the final mean1 when other precautions are exhausted4 #bib ,baily1 99,account of ,flamsteed991 p4 #ef4 ,legendre1 the discoverer of the method of ,least ,squares1 recommended that observations differing very much from the results of his method should be rejected4 ,the subject has been carefully investigated by ,professor ,pierce1 who has proposed a criterion for the rejection of doubtful observations based on the following principle3#bic8--observations should be rejected when the probability of the system of errors obtained by retaining them is less than that of the system of errors obtained by their rejection multiplied by the probability of making so many and no more abnormal observations40 ,professor ,pierce's investigation is given nearly in his own words in ,professor ,w4 ,chauvenet's 8,manual of ,spherical and ,practical ,astronomy10 which contains a full and excellent discussion of the methods of treating numerical observations4#bid #bic ,gould's 99,astronomical ,journal991 ,cambridge1 ,mass41 vol4 ii4 p4 #afa4 #bid ,philadelphia 7,london1 ,trbner7 #ahfc4 ,appendix1 vol4 ii4 p4 #eeh4 ,very difficult questions sometimes arise when one or more results of a method of experiment diverge widely from the mean of the rest4 ,are we or are we not to exclude them in adopting the supposed true mean result of the method8 ,the drawing of a mean result rests1 as ,i have frequently explained1 upon the assumption that every error acting in one direction will probably be balanced by other errors acting in an opposite direction4 ,if then we know or can possibly discover any causes of error not agreeing with this assumption1 we shall be justified in excluding results which seem to be affected by this cause4 ,in reducing large series of astronomical observations1 it is not uncommon to meet with numbers differing from others by a whole degree or half a degree1 or some considerable integral quantity4 ,these are errors which could hardly arise in the act of observation or in instrumental irregularity2 but they might readily be accounted for by misreading of figures or mistaking of division marks4 ,it would be absurd to trust to chance that such mistakes would balance each other in the long run1 and it is therefore better to correct arbitrarily the supposed mistake1 or better still1 if new observations can be made1 to strike out the divergent numbers altogether4 ,when results come sometimes too great or too small in a regular manner1 we should suspect that some part of the instrument slips through a definite space1 or that a definite cause of error enters at times1 and not at others4 ,we should then make it a point of prime importance to discover the exact nature and amount of such an error1 and either prevent its occurrence for the future or else introduce a corresponding correction4 ,in many researches the whole difficulty will consist in this detection and avoidance of sources of error4 ,professor ,roscoe found that the presence of phosphorus caused serious and almost unavoidable errors in the determination of the atomic weight of vanadium4#bie ,herschel1 in reducing his observations of double stars at the ,cape of ,good ,hope1 was perplexed by an unaccountable difference of the angles of position as measured by the seven-feet equatorial and the twenty-feet reflector telescopes1 and after a careful investigation was obliged to be contented with introducing a correction experimentally determined4#bif #bie ,bakerian ,lecture1 99,philosophical ,transactions99 7#ahfh71 vol4 clviii4 p4 #f4 #bif 99,results of ,observations at the ,cape of ,good ,hope991 p4 #bhc4 ,when observations are sufficiently numerous it seems desirable to project the apparent errors into a curve1 and then to observe whether this curve exhibits the symmetrical and characteristic form of the curve of error4 ,if so1 it may be inferred that the errors arise from many minute independent sources1 and probably compensate each other in the mean result4 ,any considerable irregularity will indicate the existence of one-sided or large causes of error1 which should be made the subject of investigation4 ,even the most patient and exhaustive investigations will sometimes fail to disclose any reason why some results diverge from others4 ,the question again recurs--,are we arbitrarily to exclude them8 ,the answer should be in the negative as a general rule4 ,the mere fact of divergence ought not to be taken as conclusive against a result1 and the exertion of arbitrary choice would open the way to the fatal influence of bias1 and what is commonly known as the 8cooking0 of figures4 ,it would amount to judging fact by theory instead of theory by fact4 ,the apparently divergent number may prove in time to be the true one4 ,it may be an exception of that valuable kind which upsets our false theories1 a real exception1 exploding apparent coincidences1 and opening a way to a new view of the subject4 ,to establish this position for the divergent fact will require additional research2 but in the meantime we should give it some weight in our mean conclusions1 and should bear in mind the discrepancy as one demanding attention4 ,to neglect a divergent result is to neglect the possible clue to a great discovery4 99,method of ,least ,squares499 ,when two or more unknown quantities are so involved that they cannot be separately determined by the ,simple ,method of ,means1 we can yet obtain their most probable values by the ,method of ,least ,squares1 without more difficulty than arises from the length of the arithmetical computations4 ,if the result of each observation gives an equation between two unknown quantities of the form 99ax99 99by99 99c99 then1 if the observations were free from error1 we should need only two observations giving two equations2 but for the attainment of greater accuracy1 we may take many observations1 and reduce the equations so as to give only a pair with mean coefficients4 ,this reduction is effected by 7#a471 multiplying the coefficients of each equation by the first coefficient1 and adding together all the similar coefficients thus resulting for the coefficients of a new equation2 and 7#b471 by repeating this process1 and multiplying the coefficients of each equation by the coefficient of the second term4 ,meaning by 7sum of 99a99#b7 the sum of all quantities of the same kind1 and having the same place in the equations as 99a99#b1 we may briefly describe the two resulting mean equations as follows3-- 7sum of 99a99#b7 4 99x99 7sum of 99ab997 4 99y99 7sum of 99ac9971 7sum of 99ab997 4 99x99 7sum of 99b99#b7 4 99y99 7sum of 99bc9974 ,when there are three or more unknown quantities the process is exactly the same in nature1 and we get additional mean equations by multiplying by the third1 fourth1 c41 coefficients4 ,as the numbers are in any case approximate1 it is usually unnecessary to make the computations with accuracy1 and places of decimals may be freely cut off to save arithmetical work4 ,the mean equations having been computed1 their solution by the ordinary methods of algebra gives the most probable values of the unknown quantities4 99,works upon the ,theory of ,probability499 ,regarding the ,theory of ,probability and the ,law of ,error as most important subjects of study for any one who desires to obtain a complete comprehension of scientific method as actually applied in physical investigations1 ,i will briefly indicate the works in one or other of which the reader will best pursue the study4 ,the best popular1 and at the same time profound ,english work on the subject is ,de ,morgan's 8,essay on ,probabilities and on their ,application to ,life ,contingencies and ,insurance ,offices10 published in the 99,cabinet ,cyclopdia991 and to be obtained 7in print7 from ,messrs4 ,longman4 ,mr4 ,venn's work on 99,the ,logic of ,chance99 can now be procured in a greatly enlarged second edition2#big it contains a most interesting and able discussion of the metaphysical basis of probability and of related questions concerning causation1 belief1 design1 testimony1 c42 but ,i cannot always agree with ,mr4 ,venn's opinions4 ,no mathematical knowledge beyond that of common arithmetic is required in reading these works4 ,quetelet's 99,letters99 form a good introduction to the subject1 and the mathematical notes are of value4 ,sir ,george ,airy's brief treatise 99,on the ,algebraical and ,numerical ,theory of ,errors of ,observations and the ,combination of ,observations991 contains a complete explanation of the ,law of ,error and its practical applications4 ,de ,morgan's treatise 8,on the ,theory of ,probabilities0 in the 99,encyclopdia ,metropolitana991 presents an abstract of the more abstruse investigations of ,laplace1 together with a multitude of profound and original remarks concerning the theory generally4 ,in ,lubbock and ,drinkwater's work on 99,probability991 in the ,library of ,useful ,knowledge1 we have a concise but good statement of a number of important problems4 ,the ,rev4 ,w4 ,a4 ,whitworth has given1 in a work entitled 99,choice and ,chance991 a number of good illustrations of calculations both in combinations and probabilities4 ,in ,mr4 ,todhunter's admirable ,history we have an exhaustive critical account of almost all writings upon the subject of probability down to the culmination of the theory in ,laplace's works4 ,the ,memoir of ,mr4 ,j4 ,w4 ,l4 ,glaisher has already been mentioned 7p4 #cge74 ,in spite of the existence of these and some other good ,english works1 there seems to be a want of an easy and yet pretty complete mathematical introduction to the study of the theory4 #big 99,the ,logic of ,chance991 an ,essay on the ,foundations and ,province of the ,theory of ,probability1 with especial reference to its ,logical ,bearings and its ,application to ,moral and ,social ,science4 7,macmillan71 #ahgf4 ,among ,french works the ,trait 99lmentaire du ,calcul des ,probabilits991 by ,s4 ,e4 ,lacroix1 of which several editions have been published1 and which is not difficult to obtain1 forms probably the best elementary treatise4 ,poisson's 99,recherches sur la ,probabilit des ,jugements99 7,paris #ahcg71 commence with an admirable investigation of the grounds and methods of the theory4 ,while ,laplace's great 99,thorie ,analytique des ,probabilits99 is of course the 8,principia0 of the subject2 his 99,essai ,philosophique sur les ,probabilits99 is a popular discourse1 and is one of the most profound and interesting essays ever published4 ,it should be familiar to every student of logical method1 and has lost little or none of its importance by lapse of time4 99,detection of ,constant ,errors499 ,the ,method of ,means is absolutely incapable of eliminating any error which is always the same1 or which always lies in one direction4 ,we sometimes require to be roused from a false feeling of security1 and to be urged to take suitable precautions against such occult errors4 8,it is to the observer10 says ,gauss1#bih 8that belongs the task of carefully removing the causes of constant errors10 and this is quite true when the error is absolutely constant4 ,when we have made a number of determinations with a certain apparatus or method of measurement1 there is a great advantage in altering the arrangement1 or even devising some entirely different method of getting estimates of the same quantity4 ,the reason obviously consists in the improbability that the same error will affect two or more different methods of experiment4 ,if a discrepancy is found to exist1 we shall at least be aware of the existence of error1 and can take measures for finding in which way it lies4 ,if we can try a considerable number of methods1 the probability becomes great that errors constant in one method will be balanced or nearly so by errors of an opposite effect in the others4 ,suppose that there be three different methods each affected by an error of equal amount4 ,the probability that this error will in all fall in the same direction is only #a_/d2 and with four methods similarly #a_/h4 ,if each method be affected1 as is always the case1 by several independent sources of error1 the probability becomes much greater that in the mean result of all the methods some of the errors will partially compensate the others4 ,in this case as in all others1 when human vigilance has exhausted itself1 we must trust the theory of probability4 #bih ,gauss1 translated by ,bertrand1 p4 #be4 ,in the determination of a zero point1 of the magnitude of the fundamental standards of time and space1 in the personal equation of an astronomical observer1 we have instances of fixed errors2 but as a general rule a change of procedure is likely to reverse the character of the error1 and many instances may be given of the value of this precaution4 ,if we measure over and over again the same angular magnitude by the same divided circle1 maintained in exactly the same position1 it is evident that the same mark in the circle will be the criterion in each case1 and any error in the position of that mark will equally affect all our results4 ,but if in each measurement we use a different part of the circle1 a new mark will come into use1 and as the error of each mark cannot be in the same direction1 the average result will be nearly free from errors of division4 ,it will be better still to use more than one divided circle4 ,even when we have no perception of the points at which error is likely to enter1 we may with advantage vary the construction of our apparatus in the hope that we shall accidentally detect some latent cause of error4 ,baily's purpose in repeating the experiments of ,michell and ,cavendish on the density of the earth was not merely to follow the same course and verify the previous numbers1 but to try whether variations in the size and substance of the attracting balls1 the mode of suspension1 the temperature of the surrounding air1 c41 would yield different results4 ,he performed no less than #fb distinct series1 comprising #baec experiments1 and he carefully classified and discussed the results so as to disclose the utmost differences4 ,again1 in experimenting upon the resistance of the air to the motion of a pendulum1 ,baily employed no less than #hj pendulums of various forms and materials1 in order to ascertain exactly upon what conditions the resistance depends4 ,regnault1 in his exact researches upon the dilatation of gases1 made arbitrary changes in the magnitude of parts of his apparatus4 ,he thinks that if1 in spite of such modification1 the results are unchanged1 the errors are probably of inconsiderable amount2#bii but in reality it is always possible1 and usually likely1 that we overlook sources of error which a future generation will detect4 ,thus the pendulum experiments of ,baily and ,sabine were directed to ascertain the nature and amount of a correction for air resistance1 which had been entirely misunderstood in the experiments by means of the seconds pendulum1 upon which was founded the definition of the standard yard1 in the ,act of #eth ,george ,,iv4 c4 #gd4 ,it has already been mentioned that a considerable error was discovered in the determination of the standard metre as the ten-millionth part of the distance from the pole to the equator 7p4 #cad74 #bii ,jamin1 99,cours de ,physique991 vol4 ii4 p4 #fj4 ,we shall return in ,chapter ,,xxv4 to the further consideration of the methods by which we may as far as possible secure ourselves against permanent and undetected sources of error4 ,in the meantime1 having completed the consideration of the special methods requisite for treating quantitative phenomena1 we must pursue our principal subject1 and endeavour to trace out the course by which the physicist1 from observation and experiment1 collects the materials of knowledge1 and then proceeds by hypothesis and inverse calculation to induce from them the laws of nature4 ,book ,,iv4 ,,inductive ,,investigation4 ,,chapter ,,xviii4 ,,observation4 ,,all knowledge proceeds originally from experience4 ,using the name in a wide sense1 we may say that experience comprehends all that we 99feel991 externally or internally--the aggregate of the impressions which we receive through the various apertures of perception--the aggregate consequently of what is in the mind1 except so far as some portions of knowledge may be the reasoned equivalents of other portions4 ,as the word experience expresses1 we 99go through99 much in life1 and the impressions gathered intentionally or unintentionally afford the materials from which the active powers of the mind evolve science4 ,no small part of the experience actually employed in science is acquired without any distinct purpose4 ,we cannot use the eyes without gathering some facts which may prove useful4 ,a great science has in many cases risen from an accidental observation4 ,erasmus ,bartholinus thus first discovered double refraction in ,iceland spar2 ,galvani noticed the twitching of a frog's leg2 ,oken was struck by the form of a vertebra2 ,malus accidentally examined light reflected from distant windows with a double refracting substance2 and ,sir ,john ,herschel's attention was drawn to the peculiar appearance of a solution of quinine sulphate4 ,in earlier times there must have been some one who first noticed the strange behaviour of a loadstone1 or the unaccountable motions produced by amber4 ,as a general rule we shall not know in what direction to look for a great body of phenomena widely different from those familiar to us4 ,chance then must give us the starting point2 but one accidental observation well used may lead us to make thousands of observations in an intentional and organised manner1 and thus a science may be gradually worked out from the smallest opening4 99,distinction of ,observation and ,experiment499 ,it is usual to say that the two sources of experience are ,observation and ,experiment4 ,when we merely note and record the phenomena which occur around us in the ordinary course of nature we are said 99to observe994 ,when we change the course of nature by the intervention of our muscular powers1 and thus produce unusual combinations and conditions of phenomena1 we are said 99to experiment994 ,herschel justly remarked#cjj that we might properly call these two modes of experience 99passive and active observation994 ,in both cases we must certainly employ our senses to observe1 and an experiment differs from a mere observation in the fact that we more or less influence the character of the events which we observe4 ,experiment is thus observation 99plus99 alteration of conditions4 #cjj 99,preliminary ,discourse on the ,study of ,natural ,philosophy991 p4 #gg4 ,it may readily be seen that we pass upwards by insensible gradations from pure observation to determinate experiment4 ,when the earliest astronomers simply noticed the ordinary motions of the sun1 moon1 and planets upon the face of the starry heavens1 they were pure observers4 ,but astronomers now select precise times and places for important observations of stellar parallax1 or the transits of planets4 ,they make the earth's orbit the basis of a well arranged 99natural experiment991 as it were1 and take well considered advantage of motions which they cannot control4 ,meteorology might seem to be a science of pure observation1 because we cannot possibly govern the changes of weather which we record4 ,nevertheless we may ascend mountains or rise in balloons1 like ,gay-,lussac and ,glaisher1 and may thus so vary the points of observation as to render our procedure experimental4 ,we are wholly unable either to produce or prevent earth-currents of electricity1 but when we construct long lines of telegraph1 we gather such strong currents during periods of disturbance as to render them capable of easy observation4 ,the best arranged systems of observation1 however1 would fail to give us a large part of the facts which we now possess4 ,many processes continually going on in nature are so slow and gentle as to escape our powers of observation4 ,lavoisier remarked that the decomposition of water must have been constantly proceeding in nature1 although its possibility was unknown till his time4#cja ,no substance is wholly destitute of magnetic or diamagnetic powers2 but it required all the experimental skill of ,faraday to prove that iron and a few other metals had no monopoly of these powers4 ,accidental observation long ago impressed upon men's minds the phenomena of lightning1 and the attractive properties of amber4 ,experiment only could have shown that phenomena so diverse in magnitude and character were manifestations of the same agent4 ,to observe with accuracy and convenience we must have agents under our control1 so as to raise or lower their intensity1 to stop or set them in action at will4 ,just as ,smeaton found it requisite to create an artificial and governable supply of wind for his investigation of windmills1 so we must have governable supplies of light1 heat1 electricity1 muscular force1 or whatever other agents we are examining4 #cja ,lavoisier's 99,elements of ,chemistry991 translated by ,kerr1 #crd ed4 p4 #adh4 ,it is hardly needful to point out too that on the earth's surface we live under nearly constant conditions of gravity1 temperature1 and atmospheric pressure1 so that if we are to extend our inferences to other parts of the universe where conditions are widely different1 we must be prepared to imitate those conditions on a small scale here4 ,we must have intensely high and low temperatures2 we must vary the density of gases from approximate vacuum upwards2 we must subject liquids and solids to pressures or strains of almost unlimited amount4 99,mental ,conditions of ,correct ,observation499 ,every observation must in a certain sense be true1 for the observing and recording of an event is in itself an event4 ,but before we proceed to deal with the supposed meaning of the record1 and draw inferences concerning the course of nature1 we must take care to ascertain that the character and feelings of the observer are not to a great extent the phenomena recorded4 ,the mind of man1 as ,francis ,bacon said1 is like an uneven mirror1 and does not reflect the events of nature without distortion4 ,we need hardly take notice of intentionally false observations1 nor of mistakes arising from defective memory1 deficient light1 and so forth4 ,even where the utmost fidelity and care are used in observing and recording1 tendencies to error exist1 and fallacious opinions arise in consequence4 ,it is difficult to find persons who can with perfect fairness register facts for and against their own peculiar views4 ,among uncultivated observers the tendency to remark favourable and forget unfavourable events is so great1 that no reliance can be placed upon their supposed observations4 ,thus arises the enduring fallacy that the changes of the weather coincide in some way with the changes of the moon1 although exact and impartial registers give no countenance to the fact4 ,the whole race of prophets and quacks live on the overwhelming effect of one success1 compared with hundreds of failures which are unmentioned and forgotten4 ,as ,bacon says1 8,men mark when they hit1 and never mark when they miss40 ,and we should do well to bear in mind the ancient story1 quoted by ,bacon1 of one who in ,pagan times was shown a temple with a picture of all the persons who had been saved from shipwreck1 after paying their vows4 ,when asked whether he did not now acknowledge the power of the gods1 8,ay10 he answered2 8but where are they painted that were drowned after their vows80 ,if indeed we could estimate the amount of 99bias99 existing in any particular observations1 it might be treated like one of the forces of the problem1 and the true course of external nature might still be rendered apparent4 ,but the feelings of an observer are usually too indeterminate1 so that when there is reason to suspect considerable bias1 rejection is the only safe course4 ,as regards facts casually registered in past times1 the capacity and impartiality of the observer are so little known that we should spare no pains to replace these statements by a new appeal to nature4 ,an indiscriminate medley of truth and absurdity1 such as ,francis ,bacon collected in his 99,natural ,history991 is wholly unsuited to the purposes of science4 ,but of course when records relate to past events like eclipses1 conjunctions1 meteoric phenomena1 earthquakes1 volcanic eruptions1 changes of sea margins1 the existence of now extinct animals1 the migrations of tribes1 remarkable customs1 c41 we must make use of statements however unsatisfactory1 and must endeavour to verify them by the comparison of independent records or traditions4 ,when extensive series of observations have to be made1 as in astronomical1 meteorological1 or magnetical observatories1 trigonometrical surveys1 and extensive chemical or physical researches1 it is an advantage that the numerical work should be executed by assistants who are not interested in1 and are perhaps unaware of1 the expected results4 ,the record is thus rendered perfectly impartial4 ,it may even be desirable that those who perform the purely routine work of measurement and computation should be unacquainted with the principles of the subject4 ,the great table of logarithms of the ,french ,revolutionary ,government was worked out by a staff of sixty or eighty computers1 most of whom were acquainted only with the rules of arithmetic1 and worked under the direction of skilled mathematicians2 yet their calculations were usually found more correct than those of persons more deeply versed in mathematics4#cjb ,in the ,indian ,ordnance ,survey the actual measurers were selected so that they should not have sufficient skill to falsify their results without detection4 #cjb ,babbage1 99,economy of ,manufactures991 p4 #aid4 ,both passive observation and experimentation must1 however1 be generally conducted by persons who know for what they are to look4 ,it is only when excited and guided by the hope of verifying a theory that the observer will notice many of the most important points2 and1 where the work is not of a routine character1 no assistant can supersede the mind-directed observations of the philosopher4 ,thus the successful investigator must combine diverse qualities2 he must have clear notions of the result he expects and confidence in the truth of his theories1 and yet he must have that candour and flexibility of mind which enable him to accept unfavourable results and abandon mistaken views4 99,instrumental and ,sensual ,conditions of ,observation499 ,in every observation one or more of the senses must be employed1 and we should ever bear in mind that the extent of our knowledge may be limited by the power of the sense concerned4 ,what we learn of the world only forms the lower limit of what is to be learned1 and1 for all that we can tell1 the processes of nature may infinitely surpass in variety and complexity those which are capable of coming within our means of observation4 ,in some cases inference from observed phenomena may make us indirectly aware of what cannot be directly felt1 but we can never be sure that we thus acquire any appreciable fraction of the knowledge that might be acquired4 ,it is a strange reflection that space may be filled with dark wandering stars1 whose existence could not have yet become in any way known to us4 ,the planets have already cooled so far as to be no longer luminous1 and it may well be that other stellar bodies of various size have fallen into the same condition4 ,from the consideration1 indeed1 of variable and extinguished stars1 ,laplace inferred that there probably exist opaque bodies as great and perhaps as numerous as those we see4#cjc ,some of these dark stars might ultimately become known to us1 either by reflecting light1 or more probably by their gravitating effects upon luminous stars4 ,thus if one member of a double star were dark1 we could readily detect its existence1 and even estimate its size1 position1 and motions1 by observing those of its visible companion4 ,it was a favourite notion of ,huyghens that there may exist stars and vast universes so distant that their light has never yet had time to reach our eyes2 and we must also bear in mind that light may possibly suffer slow extinction in space1 so that there is more than one way in which an absolute limit to the powers of telescopic discovery may exist4 #cjc 99,system of the ,world991 translated by ,harte1 vol4 ii4 p4 #cce4 ,there are natural limits again to the power of our senses in detecting undulations of various kinds4 ,it is commonly said that vibrations of more than #ch1jjj strokes per second are not audible as sound2 and as some ears actually do hear sounds of much higher pitch1 even two octaves higher than what other ears can detect1 it is exceedingly probable that there are incessant vibrations which we cannot call sound because they are never heard4 ,insects may communicate by such acute sounds1 constituting a language inaudible to us2 and the remarkable agreement apparent among bodies of ants or bees might thus perhaps be explained4 ,nay1 as ,fontenelle long ago suggested in his scientific romance1 there may exist unlimited numbers of senses or modes of perception which we can never feel1 though ,darwin's theory would render it probable that any useful means of knowledge in an ancestor would be developed and improved in the descendants4 ,we might doubtless have been endowed with a sense capable of feeling electric phenomena with acuteness1 so that the positive or negative state of charge of a body could be at once estimated4 ,the absence of such a sense is probably due to its comparative uselessness4 ,heat undulations are subject to the same considerations4 ,it is now apparent that what we call light is the affection of the eye by certain vibrations1 the less rapid of which are invisible and constitute the dark rays of radiant heat1 in detecting which we must substitute the thermometer or the thermopile for the eye4 ,at the other end of the spectrum1 again1 the ultra-violet rays are invisible1 and only indirectly brought to our knowledge in the phenomena of fluorescence or photo-chemical action4 ,there is no reason to believe that at either end of the spectrum an absolute limit has yet been reached4 ,just as our knowledge of the stellar universe is limited by the power of the telescope and other conditions1 so our knowledge of the minute world has its limit in the powers and optical conditions of the microscope4 ,there was a time when it would have been a reasonable induction that vegetables are motionless1 and animals alone endowed with power of locomotion4 ,we are astonished to discover by the microscope that minute plants are if anything more active than minute animals4 ,we even find that mineral substances seem to lose their inactive character and dance about with incessant motion when reduced to sufficiently minute particles1 at least when suspended in a non-conducting medium4#cjd ,microscopists will meet a natural limit to observation when the minuteness of the objects examined becomes comparable to the length of light undulations1 and the extreme difficulty already encountered in determining the forms of minute marks on ,diatoms appears to be due to this cause4 ,according to ,helmholtz the smallest distance which can be accurately defined depends upon the interference of light passing through the centres of the bright spaces4 ,with a theoretically perfect microscope and a dry lense the smallest visible object would not be less than one #hj1jjjth part of an inch in red light4 #cjd ,this curious phenomenon1 which ,i propose to call 99pedesis991 or the 99pedetic movement991 from 1 to jump1 is carefully described in my paper published in the 99,quarterly ,journal of ,science99 for ,april1 #ahgh1 vol4 viii4 7,n4,s47 p4 #afg4 ,see also 99,proceedings of the ,literary and ,philosophical ,society of ,manchester991 #beth ,january1 #ahgj1 vol4 ix4 p4 #gh1 99,nature991 #bbnd ,august1 #ahgh1 vol4 xviii4 p4 #ddj1 or the 99,quarterly ,journal of ,science991 vol4 viii4 7,n4,s47 p4 #ead4 ,of the errors likely to arise in estimating quantities by the senses ,i have already spoken1 but there are some cases in which we actually see things differently from what they are4 ,a jet of water appears to be a continuous thread1 when it is really a wonderfully organised succession of small and large drops1 oscillating in form4 ,the drops fall so rapidly that their impressions upon the eye run into each other1 and in order to see the separate drops we require some device for giving an instantaneous view4 ,one insuperable limit to our powers of observation arises from the impossibility of following and identifying the ultimate atoms of matter4 ,one atom of oxygen is probably undistinguishable from another atom2 only by keeping a certain volume of oxygen safely inclosed in a bottle can we assure ourselves of its identity2 allow it to mix with other oxygen1 and we lose all power of identification4 ,accordingly we seem to have no means of directly proving that every gas is in a constant state of diffusion of every part into every part4 ,we can only infer this to be the case from observing the behaviour of distinct gases which we can distinguish in their course1 and by reasoning on the grounds of molecular theory4#cje #cje ,maxwell1 99,theory of ,heat991 p4 #cja4 99,external ,conditions of ,correct ,observation499 ,before we proceed to draw inferences from any series of recorded facts1 we must take care to ascertain perfectly1 if possible1 the external conditions under which the facts are brought to our notice4 ,not only may the observing mind be prejudiced and the senses defective1 but there may be circumstances which cause one kind of event to come more frequently to our notice than another4 ,the comparative numbers of objects of different kinds existing may in any degree differ from the numbers which come to our notice4 ,this difference must if possible be taken into account before we make any inferences4 ,there long appeared to be a strong presumption that all comets moved in elliptic orbits1 because no comet had been proved to move in any other kind of path4 ,the theory of gravitation admitted of the existence of comets moving in hyperbolic orbits1 and the question arose whether they were really non-existent or were only beyond the bounds of easy observation4 ,from reasonable suppositions ,laplace calculated that the probability was at least #fjjj to #a against a comet which comes within the planetary system sufficiently to be visible at the earth's surface1 presenting an orbit which could be discriminated from a very elongated ellipse or parabola in the part of its orbit within the reach of our telescopes4#cjf ,in short1 the chances are very much in favour of our seeing elliptic rather than hyperbolic comets4 ,laplace's views have been confirmed by the discovery of six hyperbolic comets1 which appeared in the years #agbi1 #agga1 #aggd1 #ahah1 #ahdj1 and #ahdc1cjg and as only about #hjj comets altogether have been recorded1 the proportion of hyperbolic ones is quite as large as should be expected4 #cjf ,laplace1 99,essai ,philosophique991 p4 #ei4 ,todhunter's 99,history991 pp4 #dia-did4 #cjg ,chambers' 99,astronomy991 #ast ed4 p4 #bjc4 ,when we attempt to estimate the numbers of objects which may have existed1 we must make large allowances for the limited sphere of our observations4 ,probably not more than #djjj or #ejjj comets have been seen in historical times1 but making allowance for the absence of observers in the southern hemisphere1 and for the small probability that we see any considerable fraction of those which are in the neighbourhood of our system1 we must accept ,kepler's opinion1 that there are more comets in the regions of space than fishes in the depths of the ocean4 ,when like calculations are made concerning the numbers of meteors visible to us1 it is astonishing to find that the number of meteors entering the earth's atmosphere in every twenty-four hours is probably not less than #djj1jjj1jjj1 of which #ac1jjj exist in every portion of space equal to that filled by the earth4 ,serious fallacies may arise from overlooking the inevitable conditions under which the records of past events are brought to our notice4 ,thus it is only the durable objects manufactured by former races of men1 such as flint implements1 which can have come to our notice as a general rule4 ,the comparative abundance of iron and bronze articles used by an ancient nation must not be supposed to be coincident with their comparative abundance in our museums1 because bronze is far the more durable4 ,there is a prevailing fallacy that our ancestors built more strongly than we do1 arising from the fact that the more fragile structures have long since crumbled away4 ,we have few or no relics of the habitations of the poorer classes among the ,greeks or ,romans1 or in fact of any past race2 for the temples1 tombs1 public buildings1 and mansions of the wealthier classes alone endure4 ,there is an immense expanse of past events necessarily lost to us for ever1 and we must generally look upon records or relics as exceptional in their character4 ,the same considerations apply to geological relics4 ,we could not generally expect that animals would be preserved unless as regards the bones1 shells1 strong integuments1 or other hard and durable parts4 ,all the infusoria and animals devoid of mineral framework have probably perished entirely1 distilled perhaps into oils4 ,it has been pointed out that the peculiar character of some extinct floras may be due to the unequal preservation of different families of plants4 ,by various accidents1 however1 we gain glimpses of a world that is usually lost to us--as by insects embedded in amber1 the great mammoth preserved in ice1 mummies1 casts in solid material like that of the ,roman soldier at ,pompeii1 and so forth4 ,we should also remember1 that just as there may be conjunctions of the heavenly bodies that can have happened only once or twice in the period of history1 so remarkable terrestrial conjunctions may take place4 ,great storms1 earthquakes1 volcanic eruptions1 landslips1 floods1 irruptions of the sea1 may1 or rather must1 have occurred1 events of such unusual magnitude and such extreme rarity that we can neither expect to witness them nor readily to comprehend their effects4 ,it is a great advantage of the study of probabilities1 as ,laplace himself remarked1 to make us mistrust the extent of our knowledge1 and pay proper regard to the probability that events would come within the sphere of our observations4 99,apparent ,sequence of ,events499 ,de ,morgan has excellently pointed out#cjh that there are no less than four modes in which one event may seem to follow or be connected with another1 without being really so4 ,these involve mental1 sensual1 and external causes of error1 and ,i will briefly state and illustrate them4 #cjh 99,essay on ,probabilities991 ,cabinet ,cyclopdia1 p4 #aba4 ,instead of ,a causing ,b1 it may be 99our perception of ,a that causes ,b994 ,thus it is that prophecies1 presentiments1 and the devices of sorcery and witchcraft often work their own ends4 ,a man dies on the day which he has always regarded as his last1 from his own fears of the day4 ,an incantation effects its purpose1 because care is taken to frighten the intended victim1 by letting him know his fate4 ,in all such cases the mental condition is the cause of apparent coincidence4 ,in a second class of cases1 99the event ,a may make our perception of ,b follow1 which would otherwise happen without being perceived994 ,thus it was believed to be the result of investigation that more comets appeared in hot than cold summers4 ,no account was taken of the fact that hot summers would be comparatively cloudless1 and afford better opportunities for the discovery of comets4 ,here the disturbing condition is of a purely external character4 ,certain ancient philosophers held that the moon's rays were cold-producing1 mistaking the cold caused by radiation into space for an effect of the moon1 which is more likely to be visible at a time when the absence of clouds permits radiation to proceed4 ,in a third class of cases1 99our perception of ,a may make our perception of ,b follow994 ,the event ,b may be constantly happening1 but our attention may not be drawn to it except by our observing ,a4 ,this case seems to be illustrated by the fallacy of the moon's influence on clouds4 ,the origin of this fallacy is somewhat complicated4 ,in the first place1 when the sky is densely clouded the moon would not be visible at all2 it would be necessary for us to see the full moon in order that our attention should be strongly drawn to the fact1 and this would happen most often on those nights when the sky is cloudless4 ,mr4 ,w4 ,ellis1#cji moreover1 has ingeniously pointed out that there is a general tendency for clouds to disperse at the commencement of night1 which is the time when the full moon rises4 ,thus the change of the sky and the rise of the full moon are likely to attract attention mutually1 and the coincidence in time suggests the relation of cause and effect4 ,mr4 ,ellis proves from the results of observations at the ,greenwich ,observatory that the moon possesses no appreciable power of the kind supposed1 and yet it is remarkable that so sound an observer as ,sir ,john ,herschel was convinced of the connection4 ,in his 8,results of ,observations at the ,cape of ,good ,hope10#caj he mentions many evenings when a full moon occurred with a peculiarly clear sky4 #cji 99,philosophical ,magazine991 #dth ,series 7#ahfg71 vol4 xxxiv4 p4 #fd4 #caj ,see 99,notes to ,measures of ,double ,stars991 #abjd1 #accf1 #adgg1 #afhf1 #aghf1 #ahaf1 #ahce1 #aibi1 #bjha1 #bahf1 pp4 #bfe1 c4 ,see also ,herschel's 99,familiar ,lectures on ,scientific ,subjects991 p4 #adg1 and 99,outlines of ,astronomy991 #gth ed4 p4 #bhe4 ,there is yet a fourth class of cases1 in which 99,b is really the antecedent event1 but our perception of ,a1 which is a consequence of ,b1 may be necessary to bring about our perception of ,b994 ,there can be no doubt1 for instance1 that upward and downward currents are continually circulating in the lowest stratum of the atmosphere during the day-time2 but owing to the transparency of the atmosphere we have no evidence of their existence until we perceive cumulous clouds1 which are the consequence of such currents4 ,in like manner an interfiltration of bodies of air in the higher parts of the atmosphere is probably in nearly constant progress1 but unless threads of cirrous cloud indicate these motions we remain ignorant of their occurrence4#caa ,the highest strata of the atmosphere are wholly imperceptible to us1 except when rendered luminous by auroral currents of electricity1 or by the passage of meteoric stones4 ,most of the visible phenomena of comets probably arise from some substance which1 existing previously invisible1 becomes condensed or electrified suddenly into a visible form4 ,sir ,john ,herschel attempted to explain the production of comet tails in this manner by evaporation and condensation4#cab #caa ,jevons1 99,on the ,cirrous ,form of ,cloud991 ,philosophical ,magazine1 ,july1 #aheg1 #dth ,series1 vol4 xiv4 p4 #bb4 #cab 99,astronomy991 #dth ed4 p4 #ceh4 99,negative ,arguments from ,non-observation499 ,from what has been suggested in preceding sections1 it will plainly appear that the non-observation of a phenomenon is not generally to be taken as proving its non-occurrence4 ,as there are sounds which we cannot hear1 rays of heat which we cannot feel1 multitudes of worlds which we cannot see1 and myriads of minute organisms of which not the most powerful microscope can give us a view1 we must as a general rule interpret our experience in an affirmative sense only4 ,accordingly when inferences have been drawn from the non-occurrence of particular facts or objects1 more extended and careful examination has often proved their falsity4 ,not many years since it was quite a well credited conclusion in geology that no remains of man were found in connection with those of extinct animals1 or in any deposit not actually at present in course of formation4 ,even ,babbage accepted this conclusion as strongly confirmatory of the ,mosaic accounts4#cac ,while the opinion was yet universally held1 flint implements had been found disproving such a conclusion1 and overwhelming evidence of man's long-continued existence has since been forthcoming4 ,at the end of the last century1 when ,herschel had searched the heavens with his powerful telescopes1 there seemed little probability that planets yet remained unseen within the orbit of ,jupiter4 ,but on the first day of this century such an opinion was overturned by the discovery of ,ceres1 and more than a hundred other small planets have since been added to the lists of the planetary system4 #cac ,babbage1 99,ninth ,bridgewater ,treatise991 p4 #fg4 ,the discovery of the ,eozon ,canadense in strata of much greater age than any previously known to contain organic remains1 has given a shock to groundless opinions concerning the origin of organic forms2 and the oceanic dredging expeditions under ,dr4 ,carpenter and ,sir ,wyville ,thomson have modified some opinions of geologists by disclosing the continued existence of forms long supposed to be extinct4 ,these and many other cases which might be quoted show the extremely unsafe character of negative inductions4 ,but it must not be supposed that negative arguments are of no force and value4 ,the earth's surface has been sufficiently searched to render it highly improbable that any terrestrial animals of the size of a camel remain to be discovered4 ,it is believed that no new large animal has been encountered in the last eighteen or twenty centuries1#cad and the probability that if existent they would have been seen1 increases the probability that they do not exist4 ,we may with somewhat less confidence discredit the existence of any large unrecognised fish1 or sea animals1 such as the alleged sea-serpent4 ,but1 as we descend to forms of smaller size negative evidence loses weight from the less probability of our seeing smaller objects4 ,even the strong induction in favour of the four-fold division of the animal kingdom into ,vertebrata1 ,annulosa1 ,mollusca1 and ,clenterata1 may break down by the discovery of intermediate or anomalous forms4 ,as civilisation spreads over the surface of the earth1 and unexplored tracts are gradually diminished1 negative conclusions will increase in force2 but we have much to learn yet concerning the depths of the ocean1 almost wholly unexamined as they are1 and covering three-fourths of the earth's surface4 #cad ,cuvier1 99,essay on the ,theory of the ,earth991 translation1 p4 #fa1 c4 ,in geology there are many statements to which considerable probability attaches on account of the large extent of the investigations already made1 as1 for instance1 that true coal is found only in rocks of a particular geological epoch2 that gold occurs in secondary and tertiary strata only in exceedingly small quantities1#cae probably derived from the disintegration of earlier rocks4 ,in natural history negative conclusions are exceedingly treacherous and unsatisfactory4 ,the utmost patience will not enable a microscopist or the observer of any living thing to watch the behaviour of the organism under all circumstances continuously for a great length of time4 ,there is always a chance therefore that the critical act or change may take place when the observer's eyes are withdrawn4 ,this certainly happens in some cases2 for though the fertilisation of orchids by agency of insects is proved as well as any fact in natural history1 ,mr4 ,darwin has never been able by the closest watching to detect an insect in the performance of the operation4 ,mr4 ,darwin has himself adopted one conclusion on negative evidence1 namely1 that the 99,orchis pyramidalis99 and certain other orchidaceous flowers secrete no nectar4 ,but his caution and unwearying patience in verifying the conclusion give an impressive lesson to the observer4 ,for twenty-three consecutive days1 as he tells us1 he examined flowers in all states of the weather1 at all hours1 in various localities4 ,as the secretion in other flowers sometimes takes place rapidly and might happen at early dawn1 that inconvenient hour of observation was specially adopted4 ,flowers of different ages were subjected to irritating vapours1 to moisture1 and to every condition likely to bring on the secretion2 and only after invariable failure of this exhaustive inquiry was the barrenness of the nectaries assumed to be proved4#caf #cae ,murchison's 99,siluria991 #ast ed4 p4 #dcb4 #caf ,darwin's 99,fertilisation of ,orchids991 p4 #dh4 ,in order that a negative argument founded on the non-observation of an object shall have any considerable force1 it must be shown to be probable that the object if existent would have been observed1 and it is this probability which defines the value of the negative conclusion4 ,the failure of astronomers to see the planet ,vulcan1 supposed by some to exist within ,mercury's orbit1 is no sufficient disproof of its existence4 ,similarly it would be very difficult1 or even impossible1 to disprove the existence of a second satellite of small size revolving round the earth4 ,but if any person make a particular assertion1 assigning place and time1 then observation will either prove or disprove the alleged fact4 ,if it is true that when a ,french observer professed to have seen a planet on the sun's face1 an observer in ,brazil was carefully scrutinising the sun and failed to see it1 we have a negative proof4 ,false facts in science1 it has been well said1 are more mischievous than false theories4 ,a false theory is open to every person's criticism1 and is ever liable to be judged by its accordance with facts4 ,but a false or grossly erroneous assertion of a fact often stands in the way of science for a long time1 because it may be extremely difficult or even impossible to prove the falsity of what has been once recorded4 ,in other sciences the force of a negative argument will often depend upon the number of possible alternatives which may exist4 ,it was long believed that the quality of a musical sound as distinguished from its pitch1 must depend upon the form of the undulation1 because no other cause of it had ever been suggested or was apparently possible4 ,the truth of the conclusion was proved by ,helmholtz1 who applied a microscope to luminous points attached to the strings of various instruments1 and thus actually observed the different modes of undulation4 ,in mathematics negative inductive arguments have seldom much force1 because the possible forms of expression1 or the possible combinations of lines and circles in geometry1 are quite unlimited in number4 ,an enormous number of attempts were made to trisect the angle by the ordinary methods of ,euclid's geometry1 but their invariable failure did not establish the impossibility of the task4 ,this was shown in a totally different manner1 by proving that the problem involves an irreducible cubic equation to which there could be no corresponding plane geometrical solution4#cag ,this is a case of 99reductio ad absurdum991 a form of argument of a totally different character4 ,similarly no number of failures to obtain a general solution of equations of the fifth degree would establish the impossibility of the task1 but in an indirect mode1 equivalent to a 99reductio ad absurdum991 the impossibility is considered to be proved4#cah #cag ,peacock1 99,algebre991 vol4 ii4 p4 #cdd4 #cah ,ibid1 p4 #cei4 ,serret1 99,algbre ,suprieure991 #bnd ed4 p4 #cjd4 ,,chapter ,,xix4 ,,experiment4 ,we may now consider the great advantages which we enjoy in examining the combinations of phenomena when things are within our reach and capable of being experimented on4 ,we are said 99to experiment99 when we bring substances together under various conditions of temperature1 pressure1 electric disturbance1 chemical action1 c41 and then record the changes observed4 ,our object in inductive investigation is to ascertain exactly the group of circumstances or conditions which being present1 a certain other group of phenomena will follow4 ,if we denote by ,a the antecedent group1 and by ,x subsequent phenomena1 our object will usually be to discover a law of the form ,a ,,ax1 the meaning of which is that where ,a is ,x will happen4 ,the circumstances which might be enumerated as present in the simplest experiment are very numerous1 in fact almost infinite4 ,rub two sticks together and consider what would be an exhaustive statement of the conditions4 ,there are the form1 hardness1 organic structure1 and all the chemical qualities of the wood2 the pressure and velocity of the rubbing2 the temperature1 pressure1 and all the chemical qualities of the surrounding air2 the proximity of the earth with its attractive and electric powers2 the temperature and other properties of the persons producing motion2 the radiation from the sun1 and to and from the sky2 the electric excitement possibly existing in any overhanging cloud2 even the positions of the heavenly bodies must be mentioned4 ,on 99 priori99 grounds it is unsafe to assume that any one of these circumstances is without effect1 and it is only by experience that we can single out those precise conditions from which the observed heat of friction proceeds4 ,the great method of experiment consists in removing1 one at a time1 each of those conditions which may be imagined to have an influence on the result4 ,our object in the experiment of rubbing sticks is to discover the exact circumstances under which heat appears4 ,now the presence of air may be requisite2 therefore prepare a vacuum1 and rub the sticks in every respect as before1 except that it is done 99in vacuo994 ,if heat still appears we may say that air is not1 in the presence of the other circumstances1 a requisite condition4 ,the conduction of heat from neighbouring bodies may be a condition4 ,prevent this by making all the surrounding bodies ice cold1 which is what ,davy aimed at in rubbing two pieces of ice together4 ,if heat still appears we have eliminated another condition1 and so we may go on until it becomes apparent that the expenditure of energy in the friction of two bodies is the sole condition of the production of heat4 ,the great difficulty of experiment arises from the fact that we must not assume the conditions to be independent4 ,previous to experiment we have no right to say that the rubbing of two sticks will produce heat in the same way when air is absent as before4 ,we may have heat produced in one way when air is present1 and in another when air is absent4 ,the inquiry branches out into two lines1 and we ought to try in both cases whether cutting off a supply of heat by conduction prevents its evolution in friction4 ,the same branching out of the inquiry occurs with regard to every circumstance which enters into the experiment4 ,regarding only four circumstances1 say ,a1 ,b1 ,c1 ,d1 we ought to test not only the combinations ,,abcd1 ,,abc99d991 ,,ab99c99,d1 ,a99b99,,cd1 99a99,,bcd1 but we ought really to go through the whole of the combinations given in the fifth column of the ,logical ,alphabet4 ,the effect of the absence of each condition should be tried both in the presence and absence of every other condition1 and every selection of those conditions4 ,perfect and exhaustive experimentation would1 in short1 consist in examining natural phenomena in all their possible combinations and registering all relations between conditions and results which are found capable of existence4 ,it would thus resemble the exclusion of contradictory combinations carried out in the ,indirect ,method of ,inference1 except that the exclusion of combinations is grounded not on prior logical premises1 but on 99 posteriori99 results of actual trial4 ,the reader will perceive1 however1 that such exhaustive investigation is practically impossible1 because the number of requisite experiments would be immensely great4 ,four antecedents only would require sixteen experiments2 twelve antecedents would require #djif1 and the number increases as the powers of two4 ,the result is that the experimenter has to fall back upon his own tact and experience in selecting those experiments which are most likely to yield him significant facts4 ,it is at this point that logical rules and forms begin to fail in giving aid4 ,the logical rule is--,try all possible combinations2 but this being impracticable1 the experimentalist necessarily abandons strict logical method1 and trusts to his own insight4 ,analogy1 as we shall see1 gives some assistance1 and attention should be concentrated on those kinds of conditions which have been found important in like cases4 ,but we are now entirely in the region of probability1 and the experimenter1 while he is confidently pursuing what he thinks the right clue1 may be overlooking the one condition of importance4 ,it is an impressive lesson1 for instance1 that ,newton pursued all his exquisite researches on the spectrum unsuspicious of the fact that if he reduced the hole in the shutter to a narrow slit1 all the mysteries of the bright and dark lines were within his grasp1 provided of course that his prisms were sufficiently good to define the rays4 ,in like manner we know not what slight alteration in the most familiar experiments may not open the way to realms of new discovery4 ,practical difficulties1 also1 encumber the progress of the physicist4 ,it is often impossible to alter one condition without altering others at the same time2 and thus we may not get the pure effect of the condition in question4 ,some conditions may be absolutely incapable of alteration2 others may be with great difficulty1 or only in a certain degree1 removable4 ,a very treacherous source of error is the existence of unknown conditions1 which of course we cannot remove except by accident4 ,these difficulties we will shortly consider in succession4 ,it is beautiful to observe how the alteration of a single circumstance sometimes conclusively explains a phenomenon4 ,an instance is found in ,faraday's investigation of the behaviour of ,lycopodium spores scattered on a vibrating plate4 ,it was observed that these minute spores collected together at the points of greatest motion1 whereas sand and all heavy particles collected at the nodes1 where the motion was least4 ,it happily occurred to ,faraday to try the experiment in the exhausted receiver of an air-pump1 and it was then found that the light powder behaved exactly like heavy powder4 ,a conclusive proof was thus obtained that the presence of air was the condition of importance1 doubtless because it was thrown into eddies by the motion of the plate1 and carried the ,lycopodium to the points of greatest agitation4 ,sand was too heavy to be carried by the air4 99,exclusion of ,indifferent ,circumstances499 ,from what has been already said it will be apparent that the detection and exclusion of indifferent circumstances is a work of importance1 because it allows the concentration of attention upon circumstances which contain the principal condition4 ,many beautiful instances may be given where all the most obvious antecedents have been shown to have no part in the production of a phenomenon4 ,a person might suppose that the peculiar colours of mother-of-pearl were due to the chemical qualities of the substance4 ,much trouble might have been spent in following out that notion by comparing the chemical qualities of various iridescent substances4 ,but ,brewster accidentally took an impression from a piece of mother-of-pearl in a cement of resin and bees'-wax1 and finding the colours repeated upon the surface of the wax1 he proceeded to take other impressions in balsam1 fusible metal1 lead1 gum arabic1 isinglass1 c41 and always found the iridescent colours the same4 ,he thus proved that the chemical nature of the substance is a matter of indifference1 and that the form of the surface is the real condition of such colours4#cai ,nearly the same may be said of the colours exhibited by thin plates and films4 ,the rings and lines of colour will be nearly the same in character whatever may be the nature of the substance2 nay1 a void space1 such as a crack in glass1 would produce them even though the air were withdrawn by an air-pump4 ,the conditions are simply the existence of two reflecting surfaces separated by a very small space1 though it should be added that the refractive index of the intervening substance has some influence on the exact nature of the colour produced4 #cai 99,treatise on ,optics991 by ,brewster1 ,cab4 ,cyclo4 p4 #aag4 ,when a ray of light passes close to the edge of an opaque body1 a portion of the light appears to be bent towards it1 and produces coloured fringes within the shadow of the body4 ,newton attributed this inflexion of light to the attraction of the opaque body for the supposed particles of light1 although he was aware that the nature of the surrounding medium1 whether air or other pellucid substance1 exercised no apparent influence on the phenomena4 ,gravesande proved1 however1 that the character of the fringes is exactly the same1 whether the body be dense or rare1 compound or elementary4 ,a wire produces exactly the same fringes as a hair of the same thickness4 ,even the form of the obstructing edge was subsequently shown to be a matter of indifference by ,fresnel1 and the interference spectrum1 or the spectrum seen when light passes through a fine grating1 is absolutely the same whatever be the form or chemical nature of the bars making the grating4 ,thus it appears that the stoppage of a portion of a beam of light is the sole necessary condition for the diffraction or inflexion of light1 and the phenomenon is shown to bear no analogy the refraction of light1 in which the form and nature of the substance are all important4 ,it is interesting to observe how carefully ,newton1 in his researches on the spectrum1 ascertained the indifference of many circumstances by actual trial4 ,he says3#cbj 8,now the different magnitude of the hole in the window-shut1 and different thickness of the prism where the rays passed through it1 and different inclinations of the prism to the horizon1 made no sensible changes in the length of the image4 ,neither did the different matter of the prisms make any3 for in a vessel made of polished plates of glass cemented together in the shape of a prism1 and filled with water1 there is the like success of the experiment according to the quantity of the refraction40 ,but in the latter statement1 as ,i shall afterwards remark 7p4 #dcb71 ,newton assumed an indifference which does not exist1 and fell into an unfortunate mistake4 #cbj 99,opticks991 #crd4 ed4 p4 #be4 ,in the science of sound it is shown that the pitch of a sound depends solely upon the number of impulses in a second1 and the material exciting those impulses is a matter of indifference4 ,whatever fluid1 air or water1 gas or liquid1 be forced into the ,siren1 the sound produced is the same2 and the material of which an organ-pipe is constructed does not at all affect the pitch of its sound4 ,in the science of statical electricity it is an important principle that the nature of the interior of a conducting body is a matter of no importance4 ,the electrical charge is confined to the conducting surface1 and the interior remains in a neutral state4 ,a hollow copper sphere takes exactly the same charge as a solid sphere of the same metal4 ,some of ,faraday's most elegant and successful researches were devoted to the exclusion of conditions which previous experimenters had thought essential for the production of electrical phenomena4 ,davy asserted that no known fluids1 except such as contain water1 could be made the medium of connexion between the poles of a battery2 and some chemists believed that water was an essential agent in electro-chemical decomposition4 ,faraday gave abundant experiments to show that other fluids allowed of electrolysis1 and he attributed the erroneous opinion to the very general use of water as a solvent1 and its presence in most natural bodies4#cba ,it was1 in fact1 upon the weakest kind of negative evidence that the opinion had been founded4 ,many experimenters attributed peculiar powers to the poles of a battery1 likening them to magnets1 which1 by their attractive powers1 tear apart the elements of a substance4 ,by a beautiful series of experiments1#cbb ,faraday proved conclusively that1 on the contrary1 the substance of the poles is of no importance1 being merely the path through which the electric force reaches the liquid acted upon4 ,poles of water1 charcoal1 and many diverse substances1 even air itself1 produced similar results2 if the chemical nature of the pole entered at all into the question1 it was as a disturbing agent4 #cba 99,experimental ,researches in ,electricity991 vol4 i4 pp4 #acc1 #acd4 #cbb ,ibid4 vol i4 pp4 #abg1 #afb1 c4 ,it is an essential part of the theory of gravitation that the proximity of other attracting particles is without effect upon the attraction existing between any two molecules4 ,two pound weights weigh as much together as they do separately4 ,every pair of molecules in the world have1 as it were1 a private communication1 apart from their relations to all other molecules4 ,another undoubted result of experience pointed out by ,newton#cbc is that the weight of a body does not in the least depend upon its form or texture4 ,it may be added that the temperature1 electric condition1 pressure1 state of motion1 chemical qualities1 and all other circumstances concerning matter1 except its mass1 are indifferent as regards its gravitating power4 #cbc 99,principia991 bk4 iii4 ,prop4 vi4 ,corollary i4 ,as natural science progresses1 physicists gain a kind of insight and tact in judging what qualities of a substance are likely to be concerned in any class of phenomena4 ,the physical astronomer treats matter in one point of view1 the chemist in another1 and the students of physical optics1 sound1 mechanics1 electricity1 c41 make a fair division of the qualities among them4 ,but errors will arise if too much confidence be placed in this independence of various kinds of phenomena1 so that it is desirable from time to time1 especially when any unexplained discrepancies come into notice1 to question the indifference which is assumed to exist1 and to test its real existence by appropriate experiments4 99,simplification of ,experiments499 ,one of the most requisite precautions in experimentation is to vary only one circumstance at a time1 and to maintain all other circumstances rigidly unchanged4 ,there are two distinct reasons for this rule1 the first and most obvious being that if we vary two conditions at a time1 and find some effect1 we cannot tell whether the effect is due to one or the other condition1 or to both jointly4 ,a second reason is that if no effect ensues we cannot safely conclude that either of them is indifferent2 for the one may have neutralised the effect of the other4 ,in our symbolic logic ,,ab ,a99b99 was shown to be identical with ,a 7p4 #ig71 so that ,b denotes a circumstance which is indifferently present or absent4 ,but if ,b always goes together with another antecedent ,c1 we cannot show the same independence1 for ,,abc ,a99bc99 is not identical with ,a and none of our logical processes enables us to reduce it to ,a4 ,if we want to prove that oxygen is necessary to life1 we must not put a rabbit into a vessel from which the oxygen has been exhausted by a burning candle4 ,we should then have not only an absence of oxygen1 but an addition of carbonic acid1 which may have been the destructive agent4 ,for a similar reason ,lavoisier avoided the use of atmospheric air in experiments on combustion1 because air was not a simple substance1 and the presence of nitrogen might impede or even alter the effect of oxygen4 ,as ,lavoisier remarks1#cbd 8,in performing experiments1 it is a necessary principle1 which ought never to be deviated from1 that they be simplified as much as possible1 and that every circumstance capable of rendering their results complicated be carefully removed40 ,it has also been well said by ,cuvier#cbe that the method of physical inquiry consists in isolating bodies1 reducing them to their utmost simplicity1 and bringing each of their properties separately into action1 either mentally or by experiment4 #cbd ,lavoisier's 99,chemistry991 translated by ,kerr1 p4 #ajc4 #cbe ,cuvier's 99,animal ,kingdom991 introduction1 pp4 #a1 #b4 ,the electro-magnet has been of the utmost service in the investigation of the magnetic properties of matter1 by allowing of the production or removal of a most powerful magnetic force without disturbing any of the other arrangements of the experiment4 ,many of ,faraday's most valuable experiments would have been impossible had it been necessary to introduce a heavy permanent magnet1 which could not be suddenly moved without shaking the whole apparatus1 disturbing the air1 producing currents by changes of temperature1 c4 ,the electro-magnet is perfectly under control1 and its influence can be brought into action1 reversed1 or stopped by merely touching a button4 ,thus ,faraday was enabled to prove the rotation of the plane of circularly polarised light by the fact that certain light ceased to be visible when the electric current of the magnet was cut off1 and re-appeared when the current was made4 8,these phenomena10 he says1 8could be reversed at pleasure1 and at any instant of time1 and upon any occasion1 showing a perfect dependence of cause and effect40#cbf #cbf 99,experimental ,researches in ,electricity991 vol4 iii4 p4 #d4 ,it was ,newton's omission to obtain the solar spectrum under the simplest conditions which prevented him from discovering the dark lines4 ,using a broad beam of light which had passed through a round hole or a triangular slit1 he obtained a brilliant spectrum1 but one in which many different coloured rays overlapped each other4 ,in the recent history of the science of the spectrum1 one main difficulty has consisted in the mixture of the lines of several different substances1 which are usually to be found in the light of any flame or spark4 ,it is seldom possible to obtain the light of any element in a perfectly simple manner4 ,angstrm greatly advanced this branch of science by examining the light of the electric spark when formed between poles of various metals1 and in the presence of various gases4 ,by varying the pole alone1 or the gaseous medium alone1 he was able to discriminate correctly between the lines due to the metal and those due to the surrounding gas4#cbg #cbg 99,philosophical ,magazine991 #dth ,series1 vol4 ix4 p4 #cbg4 99,failure in the ,simplification of ,experiments499 ,in some cases it seems to be impossible to carry out the rule of varying one circumstance at a time4 ,when we attempt to obtain two instances or two forms of experiment in which a single circumstance shall be present in one case and absent in another1 it may be found that this single circumstance entails others4 ,benjamin ,franklin's experiment concerning the comparative absorbing powers of different colours is well known4 8,i took10 he says1 8a number of little square pieces of broadcloth from a tailor's pattern card1 of various colours4 ,they were black1 deep blue1 lighter blue1 green1 purple1 red1 yellow1 white1 and other colours and shades of colour4 ,i laid them all out upon the snow on a bright sunshiny morning4 ,in a few hours the black1 being most warmed by the sun1 was sunk so low as to be below the stroke of the sun's rays2 the dark blue was almost as low2 the lighter blue not quite so much as the dark2 the other colours less as they were lighter4 ,the white remained on the surface of the snow1 not having entered it at all40 ,this is a very elegant and apparently simple experiment2 but when ,leslie had completed his series of researches upon the nature of heat1 he came to the conclusion that the colour of a surface has very little effect upon the radiating power1 the mechanical nature of the surface appearing to be more influential4 ,he remarks#cbh that 8the question is incapable of being positively resolved1 since no substance can be made to assume different colours without at the same time changing its internal structure40 ,recent investigation has shown that the subject is one of considerable complication1 because the absorptive power of a surface may be different according to the character of the rays which fall upon it2 but there can be no doubt as to the acuteness with which ,leslie points out the difficulty4 ,in ,well's investigations concerning the nature of dew1 we have1 again1 very complicated conditions4 ,if we expose plates of various material1 such as rough iron1 glass1 polished metal1 to the midnight sky1 they will be dewed in various degrees2 but since these plates differ both in the nature of the surface and the conducting power of the material1 it would not be plain whether one or both circumstances were of importance4 ,we avoid this difficulty by exposing the same material polished or varnished1 so as to present different conditions of surface2#cbi and again by exposing different substances with the same kind of surface4 #cbh 99,inquiry into the ,nature of ,heat991 p4 #ie4 #cbi ,herschel1 99,preliminary ,discourse991 p4 #afa4 ,when we are quite unable to isolate circumstances we must resort to the procedure described by ,mill under the name of the ,joint ,method of ,agreement and ,difference4 ,we must collect as many instances as possible in which a given circumstance produces a given result1 and as many as possible in which the absence of the circumstance is followed by the absence of the result4 ,to adduce his example1 we cannot experiment upon the cause of double refraction in ,iceland spar1 because we cannot alter its crystalline condition without altering it altogether1 nor can we find substances exactly like calc spar in every circumstance except one4 ,we resort therefore to the method of comparing together all known substances which have the property of doubly-refracting light1 and we find that they agree in being crystalline4#ccj ,this indeed is nothing but an ordinary process of perfect or probable induction1 already partially described1 and to be further discussed under ,classification4 ,it may be added that the subject does admit of perfect experimental treatment1 since glass1 when compressed in one direction1 becomes capable of doubly-refracting light1 and as there is probably no alteration in the glass but change of elasticity1 we learn that the power of double refraction is probably due to a difference of elasticity in different directions4 #ccj 99,system of ,logic991 bk4 iii4 chap4 viii4 #d1 #eth ed4 vol4 i4 p4 #dcc4 99,removal of ,usual ,conditions499 ,one of the great objects of experiment is to enable us to judge of the behaviour of substances under conditions widely different from those which prevail upon the surface of the earth4 ,we live in an atmosphere which does not vary beyond certain narrow limits in temperature or pressure4 ,many of the powers of nature1 such as gravity1 which constantly act upon us1 are of almost fixed amount4 ,now it will afterwards be shown that we cannot apply a quantitative law to circumstances much differing from those in which it was observed4 ,in the other planets1 the sun1 the stars1 or remote parts of the ,universe1 the conditions of existence must often be widely different from what we commonly experience here4 ,hence our knowledge of nature must remain restricted and hypothetical1 unless we can subject substances to unusual conditions by suitable experiments4 ,the electric arc is an invaluable means of exposing metals or other conducting substances to the highest known temperature4 ,by its aid we learn not only that all the metals can be vaporised1 but that they all give off distinctive rays of light4 ,at the other extremity of the scale1 the intensely powerful freezing mixture devised by ,faraday1 consisting of solid carbonic acid and ether mixed 99in vacuo991 enables us to observe the nature of substances at temperatures immensely below any we meet with naturally on the earth's surface4 ,we can hardly realise now the importance of the invention of the air-pump1 previous to which invention it was exceedingly difficult to experiment except under the ordinary pressure of the atmosphere4 ,the ,torricellian vacuum had been employed by the philosophers of the ,accademia del ,cimento to show the behaviour of water1 smoke1 sound1 magnets1 electric substances1 c41 99in vacuo991 but their experiments were often unsuccessful from the difficulty of excluding air4#cca #cca 99,essayes of ,natural ,experiments made in the ,accademia del ,cimento499 ,englished by ,richard ,waller1 #afhd1 p4 #dj1 c4 ,among the most constant circumstances under which we live is the force of gravity1 which does not vary1 except by a slight fraction of its amount1 in any part of the earth's crust or atmosphere to which we can attain4 ,this force is sufficient to overbear and disguise various actions1 for instance1 the mutual gravitation of small bodies4 ,it was an interesting experiment of ,plateau to neutralise the action of gravity by placing substances in liquids of exactly the same specific gravity4 ,thus a quantity of oil poured into the middle of a suitable mixture of alcohol and water assumes a spherical shape2 on being made to rotate it becomes spheroidal1 and then successively separates into a ring and a group of spherules4 ,thus we have an illustration of the mode in which the planetary system may have been produced1#ccb though the extreme difference of scale prevents our arguing with confidence from the experiment to the conditions of the nebular theory4 #ccb ,plateau1 99,taylor's ,scientific ,memoirs991 vol4 iv4 pp4 #af-dc4 ,it is possible that the so-called elements are elementary only to us1 because we are restricted to temperatures at which they are fixed4 ,lavoisier carefully defined an element as a substance which cannot be decomposed 99by any known means992 but it seems almost certain that some series of elements1 for instance ,iodine1 ,bromine1 and ,chlorine1 are really compounds of a simpler substance4 ,we must look to the production of intensely high temperatures1 yet quite beyond our means1 for the decomposition of these so-called elements4 ,possibly in this age and part of the universe the dissipation of energy has so far proceeded that there are no sources of heat sufficiently intense to effect the decomposition4 99,interference of ,unsuspected ,conditions499 ,it may happen that we are not aware of all the conditions under which our researches are made4 ,some substance may be present or some power may be in action1 which escapes the most vigilant examination4 ,not being aware of its existence1 we are unable to take proper measures to exclude it1 and thus determine the share which it has in the results of our experiments4 ,there can be no doubt that the alchemists were misled and encouraged in their vain attempts by the unsuspected presence of traces of gold and silver in the substances they proposed to transmute4 ,lead1 as drawn from the smelting furnace1 almost always contains some silver1 and gold is associated with many other metals4 ,thus small quantities of noble metal would often appear as the result of experiment and raise delusive hopes4 ,in more than one case the unsuspected presence of common salt in the air has caused great trouble4 ,in the early experiments on electrolysis it was found that when water was decomposed1 an acid and an alkali were produced at the poles1 together with oxygen and hydrogen4 ,in the absence of any other explanation1 some chemists rushed to the conclusion that electricity must have the power of 99generating99 acids and alkalies1 and one chemist thought he had discovered a new substance called 99electric acid994 ,but ,davy proceeded to a systematic investigation of the circumstances1 by varying the conditions4 ,changing the glass vessel for one of agate or gold1 he found that far less alkali was produced2 excluding impurities by the use of carefully distilled water1 he found that the quantities of acid and alkali were still further diminished2 and having thus obtained a clue to the cause1 he completed the exclusion of impurities by avoiding contact with his fingers1 and by placing the apparatus under an exhausted receiver1 no acid or alkali being then detected4 ,it would be difficult to meet with a more elegant case of the detection of a condition previously unsuspected4#ccc #ccc 99,philosophical ,transactions99 #ahbf1 vol4 cxvi4 pp4 #chh1 #chi4 ,works of ,sir ,humphry ,davy1 vol4 v4 pp4 #a-ab4 ,it is remarkable that the presence of common salt in the air1 proved to exist by ,davy1 nevertheless continued a stumbling-block in the science of spectrum analysis1 and probably prevented men1 such as ,brewster1 ,herschel1 and ,talbot1 from anticipating by thirty years the discoveries of ,bunsen and ,kirchhoff4 ,as ,i pointed out1#ccd the utility of the spectrum was known in the middle of the last century to ,thomas ,melvill1 a talented ,scotch physicist1 who died at the early age of #bg years4#cce ,but ,melvill was struck in his examination of coloured flames by the extraordinary predominance of homogeneous yellow light1 which was due to some circumstance escaping his attention4 ,wollaston and ,fraunhofer were equally struck by the prominence of the yellow line in the spectrum of nearly every kind of light4 ,talbot expressly recommended the use of the prism for detecting the presence of substances by what we now call spectrum analysis1 but he found that all substances1 however different the light they yielded in other respects1 were identical as regards the production of yellow light4 ,talbot knew that the salts of soda gave this coloured light1 but in spite of ,davy's previous difficulties with salt in electrolysis1 it did not occur to him to assert that where the light is1 there sodium must be4 ,he suggested water as the most likely source of the yellow light1 because of its frequent presence2 but even substances which were apparently devoid of water gave the same yellow light4#ccf ,brewster and ,herschel both experimented upon flames almost at the same time as ,talbot1 and ,herschel unequivocally enounced the principle of spectrum analysis4#ccg ,nevertheless ,brewster1 after numerous experiments attended with great trouble and disappointment1 found that yellow light might be obtained from the combustion of almost any substance4 ,it was not until #ahef that ,swan discovered that an almost infinitesimal quantity of sodium chloride1 say a millionth part of a grain1 was sufficient to tinge a flame of a bright yellow colour4 ,the universal diffusion of the salts of sodium1 joined to this unique light-producing power1 was thus shown to be the unsuspected condition which had destroyed the confidence of all previous experimenters in the use of the prism4 ,some references concerning the history of this curious point are given below4#cch #ccd 99,national ,review991 ,july1 #ahfa1 p4 #ac4 #cce ,his published works are contained in 99,the ,edinburgh ,physical and ,literary ,essays991 vol4 ii4 p4 #cd2 99,philosophical ,transactions99 #agec1 vol4 xlviii4 p4 #bfa2 see also ,morgan's ,papers in 99,philosophical ,transactions99 #aghe1 vol4 lxxv4 p4 #aij4 #ccf 99,edinburgh ,journal of ,science991 vol4 v4 p4 #gi4 #ccg 99,encyclopdia ,metropolitana991 art4 99,light991 #ebd2 ,herschel's 99,familiar ,lectures991 p4 #bff4 #cch ,talbot1 99,philosophical ,magazine991 #crd ,series1 vol4 ix4 p4 #a 7#ahcf72 ,brewster1 99,transactions of the ,royal ,society of ,edinburgh99 #ahbc1 vol4 ix4 pp4 #dcc1 #dee2 ,swan1 ibid4 #ahef vol4 xxi4 p4 #daa2 99,philosophical ,magazine991 #dth ,series1 vol4 xx4 p4 #agc ,sept4 #ahfj2 ,roscoe1 99,spectrum ,analysis991 ,lecture ,,iii4 ,in the science of radiant heat1 early inquirers were led to the conclusion that radiation proceeded only from the surface of a solid1 or from a very small depth below it4 ,but they happened to experiment upon surfaces covered by coats of varnish1 which is highly athermanous or opaque to heat4 ,had they properly varied the character of the surface1 using a highly diathermanous substance like rock salt1 they would have obtained very different results4#cci ,one of the most extraordinary instances of an erroneous opinion due to overlooking interfering agents is that concerning the increase of rainfall near to the earth's surface4 ,more than a century ago it was observed that rain-gauges placed upon church steeples1 house tops1 and other elevated places1 gave considerably less rain than if they were on the ground1 and it has been recently shown that the variation is most rapid in the close neighbourhood of the ground4#cdj ,all kinds of theories have been started to explain this phenomenon2 but ,i have shown#cda that it is simply due to the interference of wind1 which deflects more or less rain from all the gauges which are exposed to it4 #cci ,balfour ,stewart1 99,elementary ,treatise on ,heat991 p4 #aib4 #cdj ,british ,association1 ,liverpool1 #ahgj4 99,report on ,rainfall991 p4 #agf4 #cda 99,philosophical ,magazine4991 ,dec4 #ahfa4 #dth ,series1 vol4 xxii4 p4 #dba4 ,the great magnetic power of iron renders it a source of disturbance in magnetic experiments4 ,in building a magnetic observatory great care must therefore be taken that no iron is employed in the construction1 and that no masses of iron are near at hand4 ,in some cases magnetic observations have been seriously disturbed by the existence of masses of iron ore in the neighbourhood4 ,in ,faraday's experiments upon feebly magnetic or diamagnetic substances he took the greatest precautions against the presence of disturbing substances in the copper wire1 wax1 paper1 and other articles used in suspending the test objects4 ,it was his custom to try the effect of the magnet upon the apparatus in the absence of the object of experiment1 and without this preliminary trial no confidence could be placed in the results4#cdb ,tyndall has also employed the same mode for testing the freedom of electro-magnetic coils from iron1 and was thus enabled to obtain them devoid of any cause of disturbance4#cdc ,it is worthy of notice that in the very infancy of the science of magnetism1 the acute experimentalist ,gilbert correctly accounted for the opinion existing in his day that magnets would attract silver1 by pointing out that the silver contained iron4 #cdb 99,experimental ,researches in ,electricity991 vol4 iii4 p4 #hd1 c4 #cdc 99,lectures on ,heat991 p4 #ba4 ,even when we are not aware by previous experience of the probable presence of a special disturbing agent1 we ought not to assume the absence of unsuspected interference4 ,if an experiment is of really high importance1 so that any considerable branch of science rests upon it1 we ought to try it again and again1 in as varied conditions as possible4 ,we should intentionally disturb the apparatus in various ways1 so as if possible to hit by accident upon any weak point4 ,especially when our results are more regular than we have fair grounds for anticipating1 ought we to suspect some peculiarity in the apparatus which causes it to measure some other phenomenon than that in question1 just as ,foucault's pendulum almost always indicates the movement of the axes of its own elliptic path instead of the rotation of the globe4 ,it was in this cautious spirit that ,baily acted in his experiments on the density of the earth4 ,the accuracy of his results depended upon the elimination of all disturbing influences1 so that the oscillation of his torsion balance should measure gravity alone4 ,hence he varied the apparatus in many ways1 changing the small balls subject to attraction1 changing the connecting rod1 and the means of suspension4 ,he observed the effect of disturbances1 such as the presence of visitors1 the occurrence of violent storms1 c41 and as no real alteration was produced in the results1 he confidently attributed them to gravity4#cdd #cdd ,baily1 99,memoirs of the ,royal ,astronomical ,society991 vol4 xiv4 pp4 #bi1 #cj4 ,newton would probably have discovered the mode of constructing achromatic lenses1 but for the unsuspected effect of some sugar of lead which he is supposed to have dissolved in the water of a prism4 ,he tried1 by means of a glass prism combined with a water prism1 to produce dispersion of light without refraction1 and if he had succeeded there would have been an obvious mode of producing refraction without dispersion4 ,his failure is attributed to his adding lead acetate to the water for the purpose of increasing its refractive power1 the lead having a high dispersive power which frustrated his purpose4#cde ,judging from ,newton's remarks1 in the 99,philosophical ,transactions991 it would appear as if he had not1 without many unsuccessful trials1 despaired of the construction of achromatic glasses4#cdf #cde ,grant1 99,history of ,physical ,astronomy991 p4 #eca4 #cdf 99,philosophical ,transactions991 abridged by ,lowthorp1 #dth edition1 vol4 i4 p4 #bjb4 ,the ,academicians of ,cimento1 in their early and ingenious experiments upon the vacuum1 were often misled by the mechanical imperfections of their apparatus4 ,they concluded that the air had nothing to do with the production of sounds1 evidently because their vacuum was not sufficiently perfect4 ,otto von ,guericke fell into a like mistake in the use of his newly-constructed air-pump1 doubtless from the unsuspected presence of air sufficiently dense to convey the sound of the bell4 ,it is hardly requisite to point out that the doctrine of spontaneous generation is due to the unsuspected presence of germs1 even after the most careful efforts to exclude them1 and in the case of many diseases1 both of animals and plants1 germs which we have no means as yet of detecting are doubtless the active cause4 ,it has long been a subject of dispute1 again1 whether the plants which spring from newly turned land grow from seeds long buried in that land1 or from seeds brought by the wind4 ,argument is unphilosophical when direct trial can readily be applied2 for by turning up some old ground1 and covering a portion of it with a glass case1 the conveyance of seeds by the wind can be entirely prevented1 and if the same plants appear within and without the case1 it will become clear that the seeds are in the earth4 ,by gross oversight some experimenters have thought before now that crops of rye had sprung up where oats had been sown4 99,blind or ,test ,experiments499 ,every conclusive experiment necessarily consists in the comparison of results between two different combinations of circumstances4 ,to give a fair probability that ,a is the cause of ,x1 we must maintain invariable all surrounding objects and conditions1 and we must then show that where ,a is ,x is1 and where ,a is not ,x is not4 ,this cannot really be accomplished in a single trial4 ,if1 for instance1 a chemist places a certain suspected substance in ,marsh's test apparatus1 and finds that it gives a small deposit of metallic arsenic1 he cannot be sure that the arsenic really proceeds from the suspected substance2 the impurity of the zinc or sulphuric acid may have been the cause of its appearance4 ,it is therefore the practice of chemists to make what they call a 99blind experiment991 that is to try whether arsenic appears in the absence of the suspected substance4 ,the same precaution ought to be taken in all important analytical operations4 ,indeed1 it is not merely a precaution1 it is an essential part of any experiment4 ,if the blind trial be not made1 the chemist merely assumes that he knows what would happen4 ,whenever we assert that because ,a and ,x are found together ,a is the cause of ,x1 we assume that if ,a were absent ,x would be absent4 ,but wherever it is possible1 we ought not to take this as a mere assumption1 or even as a matter of inference4 ,experience is ultimately the basis of all our inferences1 but if we can bring immediate experience to bear upon the point in question we should not trust to anything more remote and liable to error4 ,when ,faraday examined the magnetic properties of the bearing apparatus1 in the absence of the substance to be experimented on1 he really made a blind experiment 7p4 #dca74 ,we ought1 also1 to test the accuracy of a method of experiment whenever we can1 by introducing known amounts of the substance or force to be detected4 ,a new analytical process for the quantitative estimation of an element should be tested by performing it upon a mixture compounded so as to contain a known quantity of that element4 ,the accuracy of the gold assay process greatly depends upon the precaution of assaying alloys of gold of exactly known composition4#cdg ,gabriel ,plattes' works give evidence of much scientific spirit1 and when discussing the supposed merits of the divining rod for the discovery of subterranean treasure1 he sensibly suggests that the rod should be tried in places where veins of metal are known to exist4#cdh #cdg ,jevons in ,watts' 99,dictionary of ,chemistry991 vol4 ii4 pp4 #icf1 #icg4 #cdh 99,discovery of ,subterraneal ,treasure499 ,london1 #afci1 p4 #dh4 99,negative ,results of ,experiment499 ,when we pay proper regard to the imperfection of all measuring instruments and the possible minuteness of effects1 we shall see much reason for interpreting with caution the negative results of experiments4 ,we may fail to discover the existence of an expected effect1 not because that effect is really non-existent1 but because it is of a magnitude inappreciable to our senses1 or confounded with other effects of much greater amount4 ,as there is no limit on 99 priori99 grounds to the smallness of a phenomenon1 we can never1 by a single experiment1 prove the non-existence of a supposed effect4 ,we are always at liberty to assume that a certain amount of effect might have been detected by greater delicacy of measurement4 ,we cannot safely affirm that the moon has no atmosphere at all4 ,we may doubtless show that the atmosphere1 if present1 is less dense than the air in the so-called vacuum of an air-pump1 as did ,du ,sejour4 ,it is equally impossible to prove that gravity occupies 99no time99 in transmission4 ,laplace indeed ascertained that the velocity of propagation of the influence was at least fifty million times greater than that of light2#cdi but it does not really follow that it is instantaneous2 and were there any means of detecting the action of one star upon another exceedingly distant star1 we might possibly find an appreciable interval occupied in the transmission of the gravitating impulse4 ,newton could not demonstrate the absence of all resistance to matter moving through empty space2 but he ascertained by an experiment with the pendulum 7p4 #ddc71 that if such resistance existed1 it was in amount less than one five-thousandth part of the external resistance of the air4#cej #cdi ,laplace1 99,system of the ,world991 translated by ,harte1 vol4 ii4 p4 #cbb4 #cej 99,principia991 bk4 ii4 sect4 #f1 ,prop4 xxxi4 ,motte's translation1 vol4 ii4 p4 #ajh4 ,a curious instance of false negative inference is furnished by experiments on light4 ,euler rejected the corpuscular theory on the ground that particles of matter moving with the immense velocity of light would possess momentum1 of which there was no evidence4 ,bennet had attempted to detect the momentum of light by concentrating the rays of the sun upon a delicately balanced body4 ,observing no result1 it was considered to be proved that light had no momentum4 ,mr4 ,crookes1 however1 having suspended thin vanes1 blacked on one side1 in a nearly vacuous globe1 found that they move under the influence of light4 ,it is now allowed that this effect can be explained in accordance with the undulatory theory of light1 and the molecular theory of gases4 ,it comes to this--that ,bennet failed to detect an effect which he might have detected with a better method of experimenting2 but if he had found it1 the phenomenon would have confirmed1 not the corpuscular theory of light1 as was expected1 but the rival undulatory theory4 ,the conclusion drawn from ,bennet's experiment was falsely drawn1 but it was nevertheless true in matter4 ,many incidents in the history of science tend to show that phenomena1 which one generation has failed to discover1 may become accurately known to a succeeding generation4 ,the compressibility of water which the ,academicians of ,florence could not detect1 because at a low pressure the effect was too small to perceive1 and at a high pressure the water oozed through their silver vessel1#cea has now become the subject of exact measurement and precise calculation4 ,independently of ,newton1 ,hooke entertained very remarkable notions concerning the nature of gravitation4 ,in this and other subjects he showed1 indeed1 a genius for experimental investigation which would have placed him in the first rank in any other age than that of ,newton4 ,he correctly conceived that the force of gravity would decrease as we recede from the centre of the earth1 and he boldly attempted to prove it by experiment4 ,having exactly counterpoised two weights in the scales of a balance1 or rather one weight against another weight and a long piece of fine cord1 he removed his balance to the top of the dome of ,st4 ,paul's1 and tried whether the balance remained in equilibrium after one weight was allowed to hang down to a depth of #bdj feet4 ,no difference could be perceived when the weights were at the same and at different levels1 but ,hooke rightly held that the failure arose from the insufficient elevation4 ,he says1 8,yet ,i am apt to think some difference might be discovered in greater heights40#ceb ,the radius of the earth being about #bj1ibb1jjj feet1 we can now readily calculate from the law of gravity that a height of #bdj would not make a greater difference than one part in #dj1jjj of the weight4 ,such a difference would doubtless be inappreciable in the balances of that day1 though it could readily be detected by balances now frequently constructed4 ,again1 the mutual gravitation of bodies at the earth's surface is so small that ,newton appears to have made no attempt to demonstrate its existence experimentally1 merely remarking that it was too small to fall under the observation of our senses4#cec ,it has since been successfully detected and measured by ,cavendish1 ,baily1 and others4 #cea 99,essayes of ,natural ,experiments991 c4 p4 #aag4 #ceb ,hooke's 99,posthumous ,works991 p4 #ahb4 #cec 99,principia991 bk4 iii4 ,prop4 vii4 ,corollary #a4 ,the smallness of the quantities which we can sometimes observe is astonishing4 ,a balance will weigh to one millionth part of the load4 ,whitworth can measure to the millionth part of an inch4 ,a rise of temperature of the #hhjjth part of a degree centigrade has been detected by ,dr4 ,joule4 ,the spectroscope has revealed the presence of the #aj1jjj1jjjth part of a gram4 ,it is said that the eye can observe the colour produced in a drop of water by the #ej1jjj1jjjth part of a gram of fuschine1 and about the same quantity of cyanine4 ,by the sense of smell we can probably feel still smaller quantities of odorous matter4#ced ,we must nevertheless remember that quantitative effects of far less amount than these must exist1 and we should state our negative results with corresponding caution4 ,we can only disprove the existence of a quantitative phenomenon by showing deductively from the laws of nature1 that if present it would amount to a perceptible quantity4 ,as in the case of other negative arguments 7p4 #dad71 we must demonstrate that the effect would appear1 where it is by experiment found not to appear4 #ced ,keill's 99,introduction to ,natural ,philosophy991 #crd ed41 ,london1 #agcc1 pp4 #dh-ed4 99,limits of ,experiment499 ,it will be obvious that there are many operations of nature which we are quite incapable of imitating in our experiments4 ,our object is to study the conditions under which a certain effect is produced2 but one of those conditions may involve a great length of time4 ,there are instances on record of experiments extending over five or ten years1 and even over a large part of a lifetime2 but such intervals of time are almost nothing to the time during which nature may have been at work4 ,the contents of a mineral vein in ,cornwall may have been undergoing gradual change for a hundred million years4 ,all metamorphic rocks have doubtless endured high temperature and enormous1 pressure for inconceivable periods of time1 so that chemical geology is generally beyond the scope of experiment4 ,arguments have been brought against ,darwin's theory1 founded upon the absence of any clear instance of the production of a new species4 ,during an historical interval of perhaps four thousand years1 no animal1 it is said1 has been so much domesticated as to become different in species4 ,it might as well be argued that no geological changes are taking place1 because no new mountain has risen in ,great ,britain within the memory of man4 ,our actual experience of geological changes is like a point in the infinite progression of time4 ,when we know that rain water falling on limestone will carry away a minute portion of the rock in solution1 we do not hesitate to multiply that quantity by millions1 and infer that in course of time a mountain may be dissolved away4 ,we have actual experience concerning the rise of land in some parts of the globe and its fall in others to the extent of some feet4 ,do we hesitate to infer what may thus be done in course of geological ages8 ,as ,gabriel ,plattes long ago remarked1 8,the sea never resting1 but perpetually winning land in one place and losing in another1 doth show what may be done in length of time by a continual operation1 not subject unto ceasing or intermission40#cee ,the action of physical circumstances upon the forms and characters of animals by natural selection is subject to exactly the same remarks4 ,as regards animals living in a state of nature1 the change of circumstances which can be ascertained to have occurred is so slight1 that we could not expect to observe any change in those animals whatever4 ,nature has made no experiment at all for us within historical times4 ,man1 however1 by taming and domesticating dogs1 horses1 oxen1 pigeons1 c41 has made considerable change in their circumstances1 and we find considerable change also in their forms and characters4 ,supposing the state of domestication to continue unchanged1 these new forms would continue permanent so far as we know1 and in this sense they are permanent4 ,thus the arguments against ,darwin's theory1 founded on the non-observation of natural changes within the historical period1 are of the weakest character1 being purely negative4 #cee 99,discovery of ,subterraneal ,treasure991 #afci1 p4 #eb4 ,,chapter ,,xx4 ,,method ,,of ,,variations4 ,experiments may be of two kinds1 experiments of simple fact1 and experiments of quantity4 ,in the first class of experiments we combine certain conditions1 and wish to ascertain whether or not a certain effect of any quantity exists4 ,hooke wished to ascertain whether or not there was any difference in the force of gravity at the top and bottom of ,st4 ,paul's ,cathedral4 ,the chemist continually performs analyses for the purpose of ascertaining whether or not a given element exists in a particular mineral or mixture2 all such experiments and analyses are qualitative rather than quantitative1 because though the result may be more or less1 the particular amount of the result is not the object of the inquiry4 ,so soon1 however1 as a result is known to be discoverable1 the scientific man ought to proceed to the quantitative inquiry1 how great a result follows from a certain amount of the conditions which are supposed to constitute the cause8 ,the possible numbers of experiments are now infinitely great1 for every variation in a quantitative condition will usually produce a variation in the amount of the effect4 ,the method of variation which thus arises is no narrow or special method1 but it is the general application of experiment to phenomena capable of continuous variation4 ,as ,mr4 ,fowler has well remarked1#cef the observation of variations is really an integration of a supposed infinite number of applications of the so-called method of difference1 that is of experiment in its perfect form4 #cef 99,elements of ,inductive ,logic991 #ast edit4 p4 #age4 ,in induction we aim at establishing a general law1 and if we deal with quantities that law must really be expressed more or less obviously in the form of an equation1 or equations4 ,we treat as before of conditions1 and of what happens under those conditions4 ,but the conditions will now vary1 not in quality1 but quantity1 and the effect will also vary in quantity1 so that the result of quantitative induction is always to arrive at some mathematical expression involving the quantity of each condition1 and expressing the quantity of the result4 ,in other words1 we wish to know what function the effect is of its conditions4 ,we shall find that it is one thing to obtain the numerical results1 and quite another thing to detect the law obeyed by those results1 the latter being an operation of an inverse and tentative character4 99,the ,variable and the ,variant499 ,almost every series of quantitative experiments is directed to obtain the relation between the different values of one quantity which is varied at will1 and another quantity which is caused thereby to vary4 ,we may conveniently distinguish these as respectively the 99variable99 and the 99variant994 ,when we are examining the effect of heat in expanding bodies1 heat1 or one of its dimensions1 temperature1 is the variable1 length the variant4 ,if we compress a body to observe how much it is thereby heated1 pressure1 or it may be the dimensions of the body1 forms the variable1 heat the variant4 ,in the thermo-electric pile we make heat the variable and measure electricity as the variant4 ,that one of the two measured quantities which is an antecedent condition of the other will be the variable4 ,it is always convenient to have the variable entirely under our command4 ,experiments may indeed be made with accuracy1 provided we can exactly measure the variable at the moment when the quantity of the effect is determined4 ,but if we have to trust to the action of some capricious force1 there may be great difficulty in making exact measurements1 and those results may not be disposed over the whole range of quantity in a convenient manner4 ,it is one prime object of the experimenter1 therefore1 to obtain a regular and governable supply of the force which he is investigating4 ,to determine correctly the efficiency of windmills1 when the natural winds were constantly varying in force1 would be exceedingly difficult4 ,smeaton1 therefore1 in his experiments on the subject1 created a uniform wind of the required force by moving his models against the air on the extremity of a revolving arm4#ceg ,the velocity of the wind could thus be rendered greater or less1 it could be maintained uniform for any length of time1 and its amount could be exactly ascertained4 ,in determining the laws of the chemical action of light it would be out of the question to employ the rays of the sun1 which vary in intensity with the clearness of the atmosphere1 and with every passing cloud4 ,one great difficulty in photometry and the investigation of the chemical action of light consists in obtaining a uniform and governable source of light rays4#ceh #ceg 99,philosophical ,transactions991 vol4 li4 p4 #ach2 abridgment1 vol4 xi4 p4 #cee4 #ceh ,see ,bunsen and ,roscoe's researches1 in 99,philosophical ,transactions99 7#ahei71 vol4 cxlix4 p4 #hhj1 c41 where they describe a constant flame of carbon monoxide gas4 ,fizeau's method of measuring the velocity of light enabled him to appreciate the time occupied by light in travelling through a distance of eight or nine thousand metres4 ,but the revolving mirror of ,wheatstone subsequently enabled ,foucault and ,fizeau to measure the velocity in a space of four metres4 ,in this latter method there was the advantage that various media could be substituted for air1 and the temperature1 density1 and other conditions of the experiment could be accurately governed and measured4 99,measurement of the ,variable499 ,there is little use in obtaining exact measurements of an effect unless we can also exactly measure its conditions4 ,it is absurd to measure the electrical resistance of a piece of metal1 its elasticity1 tenacity1 density1 or other physical qualities1 if these vary1 not only with the minute impurities of the metal1 but also with its physical condition4 ,if the same bar changes its properties by being heated and cooled1 and we cannot exactly define the state in which it is at any moment1 our care in measuring will be wasted1 because it can lead to no law4 ,it is of little use to determine very exactly the electric conductibility of carbon1 which as graphite or gas carbon conducts like a metal1 as diamond is almost a non-conductor1 and in several other forms possesses variable and intermediate powers of conduction4 ,it will be of use only for immediate practical applications4 ,before measuring these we ought to have something to measure of which the conditions are capable of exact definition1 and to which at a future time we can recur4 ,similarly the accuracy of our measurement need not much surpass the accuracy with which we can define the conditions of the object treated4 ,the speed of electricity in passing through a conductor mainly depends upon the inductive capacity of the surrounding substances1 and1 except for technical or special purposes1 there is little use in measuring velocities which in some cases are one hundred times as great as in other cases4 ,but the maximum speed of electric conduction is probably a constant quantity of great scientific importance1 and according to ,prof4 ,clerk ,maxwell's determination in #ahfh is #agd1hjj miles per second1 or little less than that of light4 ,the true boiling point of water is a point on which practical thermometry depends1 and it is highly important to determine that point in relation to the absolute thermometric scale4 ,but when water free from air and impurity is heated there seems to be no definite limit to the temperature it may reach1 a temperature of #ahj ,cent4 having been actually observed4 ,such temperatures1 therefore1 do not require accurate measurement4 ,all meteorological measurements depending on the accidental condition of the sky are of far less importance than physical measurements in which such accidental conditions do not intervene4 ,many profound investigations depend upon our knowledge of the radiant energy continually poured upon the earth by the sun2 but this must be measured when the sky is perfectly clear1 and the absorption of the atmosphere at its minimum4 ,the slightest interference of cloud destroys the value of such a measurement1 except for meteorological purposes1 which are of vastly less generality and importance4 ,it is seldom useful1 again1 to measure the height of a snow-covered mountain within a foot1 when the thickness of the snow alone may cause it to vary #be feet or more1 when in short the height itself is indefinite to that extent4#cei #cei ,humboldt's 99,cosmos99 7,bohn71 vol4 i4 p4 #g4 99,maintenance of ,similar ,conditions499 ,our ultimate object in induction must be to obtain the complete relation between the conditions and the effect1 but this relation will generally be so complex that we can only attack it in detail4 ,we must1 as far as possible1 confine the variation to one condition at a time1 and establish a separate relation between each condition and the effect4 ,this is at any rate the first step in approximating to the complete law1 and it will be a subsequent question how far the simultaneous variation of several conditions modifies their separate actions4 ,in many experiments1 indeed1 it is only one condition which we wish to study1 and the others are interfering forces which we would avoid if possible4 ,one of the conditions of the motion of a pendulum is the resistance of the air1 or other medium in which it swings2 but when ,newton was desirous of proving the equal gravitation of all substances1 he had no interest in the air4 ,his object was to observe a single force only1 and so it is in a great many other experiments4 ,accordingly1 one of the most important precautions in investigation consists in maintaining all conditions constant except that which is to be studied4 ,as that admirable experimental philosopher1 ,gilbert1 expressed it1#cfj 8,there is always need of similar preparation1 of similar figure1 and of equal magnitude1 for in dissimilar and unequal circumstances the experiment is doubtful40 #cfj ,gilbert1 99,de ,magnete991 p4 #aji4 ,in ,newton's decisive experiment similar conditions were provided for1 with the simplicity which characterises the highest art4 ,the pendulums of which the oscillations were compared consisted of equal boxes of wood1 hanging by equal threads1 and filled with different substances1 so that the total weights should be equal and the centres of oscillation at the same distance from the points of suspension4 ,hence the resistance of the air became approximately a matter of indifference2 for the outward size and shape of the pendulums being the same1 the absolute force of resistance would be the same1 so long as the pendulums vibrated with equal velocity2 and the weights being equal the resistance would diminish the velocity equally4 ,hence if any inequality were observed in the vibrations of the two pendulums1 it must arise from the only circumstance which was different1 namely the chemical nature of the matter within the boxes4 ,no inequality being observed1 the chemical nature of substances can have no appreciable influence upon the force of gravitation4#cfa #cfa 99,principia991 bk4 iii4 ,prop4 vi4 ,a beautiful experiment was devised by ,dr4 ,joule for the purpose of showing that the gain or loss of heat by a gas is connected1 not with the mere change of its volume and density1 but with the energy received or given out by the gas4 ,two strong vessels1 connected by a tube and stopcock1 were placed in water after the air had been exhausted from one vessel and condensed in the other to the extent of twenty atmospheres4 ,the whole apparatus having been brought to a uniform temperature by agitating the water1 and the temperature having been exactly observed1 the stopcock was opened1 so that the air at once expanded and filled the two vessels uniformly4 ,the temperature of the water being again noted was found to be almost unchanged4 ,the experiment was then repeated in an exactly similar manner1 except that the strong vessels were placed in separate portions of the water4 ,now cold was produced in the vessel from which the air rushed1 and an almost exactly equal quantity of heat appeared in that to which it was conducted4 ,thus ,dr4 ,joule clearly proved that rarefaction produces as much heat as cold1 and that only when there is disappearance of mechanical energy will there be production of heat4#cfb ,what we have to notice1 however1 is not so much the result of the experiment1 as the simple manner in which a single change in the apparatus1 the separation of the portions of water surrounding the air vessels1 is made to give indications of the utmost significance4 #cfb 99,philosophical ,magazine991 #crd ,series1 vol4 xxvi4 p4 #cge4 99,collective ,experiments499 ,there is an interesting class of experiments which enable us to observe a number of quantitative results in one act4 ,generally speaking1 each experiment yields us but one number1 and before we can approach the real processes of reasoning we must laboriously repeat measurement after measurement1 until we can lay out a curve of the variation of one quantity as depending on another4 ,we can sometimes abbreviate this labour1 by making a quantity vary in different parts of the same apparatus through every required amount4 ,in observing the height to which water rises by the capillary attraction of a glass vessel1 we may take a series of glass tubes of different bore1 and measure the height through which it rises in each4 ,but if we take two glass plates1 and place them vertically in water1 so as to be in contact at one vertical side1 and slightly separated at the other side1 the interval between the plates varies through every intermediate width1 and the water rises to a corresponding height1 producing at its upper surface a hyperbolic curve4 ,the absorption of light in passing through a coloured liquid may be beautifully shown by enclosing the liquid in a wedge-shaped glass1 so that we have at a single glance an infinite variety of thicknesses in view4 ,as ,newton himself remarked1 a red liquid viewed in this manner is found to have a pale yellow colour at the thinnest part1 and it passes through orange into red1 which gradually becomes of a deeper and darker tint4#cfc ,the effect may be noticed in a conical wine-glass4 ,the prismatic analysis of light from such a wedge-shaped vessel discloses the reason1 by exhibiting the progressive absorption of different rays of the spectrum as investigated by ,dr4 ,j4 ,h4 ,gladstone4#cfd #cfc 99,opticks991 #crd edit4 p4 #aei4 #cfd ,watts1 99,dictionary of ,chemistry991 vol4 iii4 p4 #fcg4 ,a moving body may sometimes be made to mark out its own course1 like a shooting star which leaves a tail behind it4 ,thus an inclined jet of water exhibits in the clearest manner the parabolic path of a projectile4 ,in ,wheatstone's ,kaleidophone the curves produced by the combination of vibrations of different ratios are shown by placing bright reflective buttons on the tops of wires of various forms4 ,the motions are performed so quickly that the eye receives the impression of the path as a complete whole1 just as a burning stick whirled round produces a continuous circle4 ,the laws of electric induction are beautifully shown when iron filings are brought under the influence of a magnet1 and fall into curves corresponding to what ,faraday called the ,lines of ,magnetic ,force4 ,when ,faraday tried to define what he meant by his lines of force1 he was obliged to refer to the filings4 8,by magnetic curves10 he says1#cfe 8,i mean lines of magnetic forces which would be depicted by iron filings40 ,robison had previously produced similar curves by the action of frictional electricity1 and from a mathematical investigation of the forms of such curves we may infer that magnetic and electric attractions obey the general law of emanation1 that of the inverse square of the distance4 ,in the electric brush we have a similar exhibition of the laws of electric attraction4 #cfe 99,faraday's ,life991 by ,bence ,jones1 vol4 ii4 p4 #e4 ,there are several branches of science in which collective experiments have been used with great advantage4 ,lichtenberg's electric figures1 produced by scattering electrified powder on an electrified resin cake1 so as to show the condition of the latter1 suggested to ,chladni the notion of discovering the state of vibration of plates by strewing sand upon them4 ,the sand collects at the points where the motion is least1 and we gain at a glance a comprehension of the undulations of the plate4 ,to this method of experiment we owe the beautiful observations of ,savart4 ,the exquisite coloured figures exhibited by plates of crystal1 when examined by polarised light1 afford a more complicated example of the same kind of investigation4 ,they led ,brewster and ,fresnel to an explanation of the properties of the optic axes of crystals4 ,the unequal conduction of heat in crystalline substances has also been shown in a similar manner1 by spreading a thin layer of wax over the plate of crystal1 and applying heat to a single point4 ,the wax then melts in a circular or elliptic area according as the rate of conduction is uniform or not4 ,nor should we forget that ,newton's rings were an early and most important instance of investigations of the same kind1 showing the effects of interference of light undulations of all magnitudes at a single view4 ,herschel gave to all such opportunities of observing directly the results of a general law1 the name of 99,collective ,instances991#cff and ,i propose to adopt the name 99,collective ,experiments994 #cff 99,preliminary ,discourse991 c41 p4 #ahe4 ,such experiments will in many subjects only give the first hint of the nature of the law in question1 but will not admit of any exact measurements4 ,the parabolic form of a jet of water may well have suggested to ,galileo his views concerning the path of a projectile2 but it would not serve now for the exact investigation of the laws of gravity4 ,it is unlikely that capillary attraction could be exactly measured by the use of inclined plates of glass1 and tubes would probably be better for precise investigation4 ,as a general rule1 these collective experiments would be most useful for popular illustration4 ,but when the curves are of a precise and permanent character1 as in the coloured figures produced by crystalline plates1 they may admit of exact measurement4 ,newton's rings and diffraction fringes allow of very accurate measurements4 ,under collective experiments we may perhaps place those in which we render visible the motions of gas or liquid by diffusing some opaque substance in it4 ,the behaviour of a body of air may often be studied in a beautiful way by the use of smoke1 as in the production of smoke rings and jets4 ,in the case of liquids lycopodium powder is sometimes employed4 ,to detect the mixture of currents or strata of liquid1 ,i employed very dilute solutions of common salt and silver nitrate1 which produce a visible cloud wherever they come into contact4#cfg ,atmospheric clouds often reveal to us the movements of great volumes of air which would otherwise be quite unapparent4 #cfg 99,philosophical ,magazine991 ,july1 #aheg1 #dth ,series1 vol4 xiv4 p4 #bd4 99,periodic ,variations499 ,a large class of investigations is concerned with ,periodic ,variations4 ,we may define a periodic phenomenon as one which1 with the uniform change of the variable1 returns time after time to the same value4 ,if we strike a pendulum it presently returns to the point from which we disturbed it1 and while time1 the variable1 progresses uniformly1 it goes on making excursions and returning1 until stopped by the dissipation of its energy4 ,if one body in space approaches by gravity towards another1 they will revolve round each other in elliptic orbits1 and return for an indefinite number of times to the same relative positions4 ,on the other hand a single body projected into empty space1 free from the action of any extraneous force1 would go on moving for ever in a straight line1 according to the first law of motion4 ,in the latter case the variation is called 99secular991 because it proceeds during ages in a similar manner1 and suffers no or going round4 ,it may be doubted whether there really is any motion in the universe which is not periodic4 ,mr4 ,herbert ,spencer long since adopted the doctrine that all motion is ultimately rhythmical1#cfh and abundance of evidence may be adduced in favour of his view4 #cfh 99,first ,principles991 #crd edit4 chap4 x4 p4 #bec4 ,the so-called secular acceleration of the moon's motion is certainly periodic1 and as1 so far as we can tell1 no body is beyond the attractive power of other bodies1 rectilinear motion becomes purely hypothetical1 or at least infinitely improbable4 ,all the motions of all the stars must tend to become periodic4 ,though certain disturbances in the planetary system seem to be uniformly progressive1 ,laplace is considered to have proved that they really have their limits1 so that after an immense time1 all the planetary bodies might return to the same places1 and the stability of the system be established4 ,such a theory of periodic stability is really hypothetical1 and does not take into account phenomena resulting in the dissipation of energy1 which may be a really secular process4 ,for our present purposes we need not attempt to form an opinion on such questions4 ,any change which does not present the appearance of a periodic character will be empirically regarded as a secular change1 so that there will be plenty of non-periodic variations4 ,the variations which we produce experimentally will often be non-periodic4 ,when we communicate heat to a gas it increases in bulk or pressure1 and as far as we can go the higher the temperature the higher the pressure4 ,our experiments are of course restricted in temperature both above and below1 but there is every reason to believe that the bulk being the same1 the pressure would never return to the same point at any two different temperatures4 ,we may of course repeatedly raise and lower the temperature at regular or irregular intervals entirely at our will1 and the pressure of the gas will vary in like manner and exactly at the same intervals1 but such an arbitrary series of changes would not constitute ,periodic ,variation4 ,it would constitute a succession of distinct experiments1 which would place beyond reasonable doubt the connexion of cause and effect4 ,whenever a phenomenon recurs at equal or nearly equal intervals1 there is1 according to the theory of probability1 considerable evidence of connexion1 because if the recurrences were entirely casual it is unlikely that they would happen at equal intervals4 ,the fact that a brilliant comet had appeared in the years #acja1 #acgh1 #adef1 #aeca1 #afjg1 and #afhb gave considerable presumption in favour of the identity of the body1 apart from similarity of the orbit4 ,there is nothing which so fascinates the attention of men as the recurrence time after time of some unusual event4 ,things and appearances which remain ever the same1 like mountains and valleys1 fail to excite the curiosity of a primitive people4 ,it has been remarked by ,laplace that even in his day the rising of ,venus in its brightest phase never failed to excite surprise and interest4 ,so there is little doubt that the first germ of science arose in the attention given by ,eastern people to the changes of the moon and the motions of the planets4 ,perhaps the earliest astronomical discovery consisted in proving the identity of the morning and evening stars1 on the grounds of their similarity of aspect and invariable alternation4#cfi ,periodical changes of a somewhat complicated kind must have been understood by the ,chaldeans1 because they were aware of the cycle of #fehe days or #ai years which brings round the new and full moon upon the same days1 hours1 and even minutes of the year4 ,the earliest efforts of scientific prophecy were founded upon this knowledge1 and if at present we cannot help wondering at the precise anticipations of the nautical almanack1 we may imagine the wonder excited by such predictions in early times4 #cfi ,laplace1 99,system of the ,world991 vol4 i4 pp4 #ej1 #ed1 c4 99,combined ,periodic ,changes499 ,we shall seldom find a body subject to a single periodic variation1 and free from other disturbances4 ,we may expect the periodic variation itself to undergo variation1 which may possibly be secular1 but is more likely to prove periodic2 nor is there any limit to the complication of periods beyond periods1 or periods within periods1 which may ultimately be disclosed4 ,in studying a phenomenon of rhythmical character we have a succession of questions to ask4 ,is the periodic variation uniform8 ,if not1 is the change uniform8 ,if not1 is the change itself periodic8 ,is that new period uniform1 or subject to any other change1 or not8 and so on 99ad infinitum994 ,in some cases there may be many distinct causes of periodic variations1 and according to the principle of the superposition of small effects1 to be afterwards considered1 these periodic effects will be simply added together1 or at least approximately so1 and the joint result may present a very complicated subject of investigation4 ,the tides of the ocean consist of a series of superimposed undulations4 ,not only are there the ordinary semi-diurnal tides caused by sun and moon1 but a series of minor tides1 such as the lunar diurnal1 the solar diurnal1 the lunar monthly1 the lunar fortnightly1 the solar annual and solar semi-annual are gradually being disentangled by the labours of ,sir ,w4 ,thomson1 ,professor ,haughton and others4 ,variable stars present interesting periodic phenomena2 while some stars1 ,cephei for instance1 are subject to very regular variations1 others1 like ,mira ,ceti1 are less constant in the degrees of brilliancy which they attain or the rapidity of the changes1 possibly on account of some longer periodic variation4#cgj ,the star ,lyr presents a double maximum and minimum in each of its periods of nearly #ac days1 and since the discovery of this variation the period in a period has probably been on the increase4 8,at first the variability was more rapid1 then it became gradually slower2 and this decrease in the length of time reached its limit between the years #ahdj and #ahdd4 ,during that time its period was nearly invariable2 at present it is again decidedly on the decrease40#cga ,the tracing out of such complicated variations presents an unlimited field for interesting investigation4 ,the number of such variable stars already known is considerable1 and there is no reason to suppose that any appreciable fraction of the whole number has yet been detected4 #cgj ,herschel's 99,outlines of ,astronomy991 #dth edit4 pp4 #eee-eeg4 #cga ,humboldt's 99,cosmos99 7,bohn71 vol4 iii4 p4 #bbi4 99,principle of ,forced ,vibrations499 ,investigations of the connection of periodic causes and effects rest upon a principle1 which has been demonstrated by ,sir ,john ,herschel for some special cases1 and clearly explained by him in several of his works4#cgb ,the principle may be formally stated in the following manner3 8,if one part of any system connected together either by material ties1 or by the mutual attractions of its members1 be continually maintained by any cause1 whether inherent in the constitution of the system or external to it1 in a state of regular periodic motion1 that motion will be propagated throughout the whole system1 and will give rise1 in every member of it1 and in every part of each member1 to periodic movements executed in equal periods1 with that to which they owe their origin1 though not necessarily synchronous with them in their maxima and minima40 ,the meaning of the proposition is that the effect of a periodic cause will be periodic1 and will recur at intervals equal to those of the cause4 ,accordingly when we find two phenomena which do proceed1 time after time1 through changes of the same period1 there is much probability that they are connected4 ,in this manner1 doubtless1 ,pliny correctly inferred that the cause of the tides lies in the sun and the moon1 the intervals between successive high tides being equal to the intervals between the moon's passage across the meridian4 ,kepler and ,descartes too admitted the connection previous to ,newton's demonstration of its precise nature4 ,when ,bradley discovered the apparent motion of the stars arising from the aberration of light1 he was soon able to attribute it to the earth's annual motion1 because it went through its phases in a year4 #cgb 99,encyclopdia ,metropolitana991 art4 99,sound991 #cbc2 99,outlines of ,astronomy991 #dth edit41 #fej4 pp4 #daj1 #dhg-hh2 99,meteorology1 ,encyclopdia ,britannica991 ,reprint1 p4 #aig4 ,the most beautiful instance of induction concerning periodic changes which can be cited1 is the discovery of an eleven-year period in various meteorological phenomena4 ,it would be difficult to mention any two things apparently more disconnected than the spots upon the sun and auroras4 ,as long ago as #ahbf1 ,schwabe commenced a regular series of observations of the spots upon the sun1 which has been continued to the present time1 and he was able to show that at intervals of about eleven years the spots increased much in size and number4 ,hardly was this discovery made known1 when ,lamont pointed out a nearly equal period of variation in the declination of the magnetic needle4 ,magnetic storms or sudden disturbances of the needle were next shown to take place most frequently at the times when sun-spots were prevalent1 and as auroras are generally coincident with magnetic storms1 these phenomena were brought into the cycle4 ,it has since been shown by ,professor ,piazzi ,smyth and ,mr4 ,e4 ,j4 ,stone1 that the temperature of the earth's surface as indicated by sunken thermometers gives some evidence of a like period4 ,the existence of a periodic cause having once been established1 it is quite to be expected1 according to the principle of forced vibrations1 that its influence will be detected in all meteorological phenomena4 99,integrated ,variations499 ,in considering the various modes in which one effect may depend upon another1 we must set in a distinct class those which arise from the accumulated effects of a constantly acting cause4 ,when water runs out of a cistern1 the velocity of motion depends1 according to ,torricelli's theorem1 on the height of the surface of the water above the vent2 but the amount of water which leaves the cistern in a given time depends upon the aggregate result of that velocity1 and is only to be ascertained by the mathematical process of integration4 ,when one gravitating body falls towards another1 the force of gravity varies according to the inverse square of the distance2 to obtain the velocity produced we must integrate or sum the effects of that law2 and to obtain the space passed over by the body in a given time1 we must integrate again4 ,in periodic variations the same distinction must be drawn4 ,the heating power of the sun's rays at any place on the earth varies every day with the height attained1 and is greatest about noon2 but the temperature of the air will not be greatest at the same time4 ,this temperature is an integrated effect of the sun's heating power1 and as long as the sun is able to give more heat to the air than the air loses in other ways1 the temperature continues to rise1 so that the maximum is deferred until about #c ,p4,m4 ,similarly the hottest day of the year falls1 on an average1 about one month later than the summer solstice1 and all the seasons lag about a month behind the motions of the sun4 ,in the case of the tides1 too1 the effect of the moon's attractive power is never greatest when the power is greatest2 the effect always lags more or less behind the cause4 ,yet the intervals between successive tides are equal1 in the absence of disturbance1 to the intervals between the passages of the moon across the meridian4 ,thus the principle of forced vibrations holds true4 ,in periodic phenomena1 however1 curious results sometimes follow from the integration of effects4 ,if we strike a pendulum1 and then repeat the stroke time after time at the same part of the vibration1 all the strokes concur in adding to the momentum1 and we can thus increase the extent and violence of the vibrations to any degree4 ,we can stop the pendulum again by strokes applied when it is moving in the opposite direction1 and the effects being added together will soon bring it to rest4 ,now if we alter the intervals of the strokes so that each two successive strokes act in opposite manners they will neutralise each other1 and the energy expended will be turned into heat or sound at the point of percussion4 ,similar effects occur in all cases of rhythmical motion4 ,if a musical note is sounded in a room containing a piano1 the string corresponding to it will be thrown into vibration1 because every successive stroke of the air-waves upon the string finds it in like position as regards the vibration1 and thus adds to its energy of motion4 ,but the other strings being incapable of vibrating with the same rapidity are struck at various points of their vibrations1 and one stroke will soon be opposed by one contrary in effect4 ,all phenomena of 99resonance99 arise from this coincidence in time of undulation4 ,the air in a pipe closed at one end1 and about #ab inches in length1 is capable of vibrating #eab times in a second4 ,if1 then1 the note ,c is sounded in front of the open end of the pipe1 every successive vibration of the air is treasured up as it were in the motion of the air4 ,in a pipe of different length the pulses of air would strike each other1 and the mechanical energy being transmuted into heat would become no longer perceptible as sound4 ,accumulated vibrations sometimes become so intense as to lead to unexpected results4 ,a glass vessel if touched with a violin bow at a suitable point may be fractured with the violence of vibration4 ,a suspension bridge may be broken down if a company of soldiers walk across it in steps the intervals of which agree with the vibrations of the bridge itself4 ,but if they break the step or march in either quicker or slower pace1 they may have no perceptible effect upon the bridge4 ,in fact if the impulses communicated to any vibrating body are synchronous with its vibrations1 the energy of those vibrations will be unlimited1 and may fracture any body4 ,let us now consider what will happen if the strokes be not exactly at the same intervals as the vibrations of the body1 but1 say1 a little slower4 ,then a succession of strokes will meet the body in nearly but not quite the same position1 and their efforts will be accumulated4 ,afterwards the strokes will begin to fall when the body is in the opposite phase4 ,imagine that one pendulum moving from one extreme point to another in a second1 should be struck by another pendulum which makes #fa beats in a minute2 then1 if the pendulums commence together1 they will at the end of #cj-a_/b beats be moving in opposite directions4 ,hence whatever energy was communicated in the first half minute will be neutralised by the opposite effect of that given in the second half4 ,the effect of the strokes of the second pendulum will therefore be alternately to increase and decrease the vibrations of the first1 so that a new kind of vibration will be produced running through its phases in #fa seconds4 ,an effect of this kind was actually observed by ,ellicott1 a member of the ,royal ,society1 in the case of two clocks4#cgc ,he found that through the wood-work by which the clocks were connected a slight impulse was transmitted1 and each pendulum alternately lost and gained momentum4 ,each clock1 in fact1 tended to stop the other at regular intervals1 and in the intermediate times to be stopped by the other4 #cgc 99,philosophical ,transactions991 7#agci71 vol4 xli4 p4 #abf4 ,many disturbances in the planetary system depend upon the same principle2 for if one planet happens always to pull another in the same direction in similar parts of their orbits1 the effects1 however slight1 will be accumulated1 and a disturbance of large ultimate amount and of long period will be produced4 ,the long inequality in the motions of ,jupiter and ,saturn is thus due to the fact that five times the mean motion of ,saturn is very nearly equal to twice the mean motion of ,jupiter1 causing a coincidence in their relative positions and disturbing powers4 ,the rolling of ships depends mainly upon the question whether the period of vibration of the ship corresponds or not with the intervals at which the waves strike her4 ,much which seems at first sight unaccountable in the behaviour of vessels is thus explained1 and the loss of the 99,captain99 is a sad case in point4 ,,chapter ,,xxi4 ,,theory ,,of ,,approximation4 ,in order that we may gain a true understanding of the kind1 degree1 and value of the knowledge which we acquire by experimental investigation1 it is requisite that we should be fully conscious of its approximate character4 ,we must learn to distinguish between what we can know and cannot know--between the questions which admit of solution1 and those which only seem to be solved4 ,many persons may be misled by the expression 99exact science991 and may think that the knowledge acquired by scientific methods admits of our reaching absolutely true laws1 exact to the last degree4 ,there is even a prevailing impression that when once mathematical formul have been successfully applied to a branch of science1 this portion of knowledge assumes a new nature1 and admits of reasoning of a higher character than those sciences which are still unmathematical4 ,the very satisfactory degree of accuracy attained in the science of astronomy gives a certain plausibility to erroneous notions of this kind4 ,some persons no doubt consider it to be 99proved99 that planets move in ellipses1 in such a manner that all ,kepler's laws hold exactly true2 but there is a double error in any such notions4 ,in the first place1 ,kepler's laws are 99not proved991 if by proof we mean certain demonstration of their exact truth4 ,in the next place1 even assuming ,kepler's laws to be exactly true in a theoretical point of view1 the planets never move according to those laws4 ,even if we could observe the motions of a planet1 of a perfect globular form1 free from all perturbing or retarding forces1 we could never prove that it moved in a perfect ellipse4 ,to prove the elliptical form we should have to measure infinitely small angles1 and infinitely small fractions of a second2 we should have to perform impossibilities4 ,all we can do is to show that the motion of an unperturbed planet approaches 99very nearly99 to the form of an ellipse1 and more nearly the more accurately our observations are made4 ,but if we go on to assert that the path 99is99 an ellipse we pass beyond our data1 and make an assumption which cannot be verified by observation4 ,but1 secondly1 as a matter of fact no planet does move in a perfect ellipse1 or manifest the truth of ,kepler's laws exactly4 ,the law of gravity prevents its own results from being clearly exhibited1 because the mutual perturbations of the planets distort the elliptical paths4 ,those laws1 again1 hold exactly true only of infinitely small bodies1 and when two great globes1 like the sun and ,jupiter1 attract each other1 the law must be modified4 ,the periodic time is then shortened in the ratio of the square root of the number expressing the sun's mass1 to that of the sum of the numbers expressing the masses of the sun and planet1 as was shown by ,newton4#cgd ,even at the present day discrepancies exist between the observed dimensions of the planetary orbits and their theoretical magnitudes1 after making allowance for all disturbing causes4#cge ,nothing is more certain in scientific method than that approximate coincidence alone can be expected4 ,in the measurement of continuous quantity perfect correspondence must be accidental1 and should give rise to suspicion rather than to satisfaction4 #cgd 99,principia991 bk4 iii4 ,prop4 #ae4 #cge ,lockyer's 99,lessons in ,elementary ,astronomy991 p4 #cja4 ,one remarkable result of the approximate character of our observations is that we could never prove the existence of perfectly circular or parabolic movement1 even if it existed4 ,the circle is a singular case of the ellipse1 for which the eccentricity is zero2 it is infinitely improbable that any planet1 even if undisturbed by other bodies1 would have a circle for its orbit2 but if the orbit were a circle we could never prove the entire absence of eccentricity4 ,all that we could do would be to declare the divergence from the circular form to be inappreciable4 ,delambre was unable to detect the slightest ellipticity in the orbit of ,jupiter's first satellite1 but he could only infer that the orbit was 99nearly99 circular4 ,the parabola is the singular limit between the ellipse and the hyperbola4 ,as there are elliptic and hyperbolic comets1 so we might conceive the existence of a parabolic comet4 ,indeed if an undisturbed comet fell towards the sun from an infinite distance it would move in a parabola2 but we could never prove that it so moved4 99,substitution of ,simple ,hypotheses499 ,in truth men never can solve problems fulfilling the complex circumstances of nature4 ,all laws and explanations are in a certain sense hypothetical1 and apply exactly to nothing which we can know to exist4 ,in place of the actual objects which we see and feel1 the mathematician substitutes imaginary objects1 only partially resembling those represented1 but so devised that the discrepancies are not of an amount to alter seriously the character of the solution4 ,when we probe the matter to the bottom physical astronomy is as hypothetical as ,euclid's elements4 ,there may exist in nature perfect straight lines1 triangles1 circles1 and other regular geometrical figures2 to our science it is a matter of indifference whether they do or do not exist1 because in any case they must be beyond our powers of perception4 ,if we submitted a perfect circle to the most rigorous scrutiny1 it is impossible that we should discover whether it were perfect or not4 ,nevertheless in geometry we argue concerning perfect curves1 and rectilinear figures1 and the conclusions apply to existing objects so far as we can assure ourselves that they agree with the hypothetical conditions of our reasoning4 ,this is in reality all that we can do in the most perfect of the sciences4 ,doubtless in astronomy we meet with the nearest approximation to actual conditions4 ,the law of gravity is not a complex one in itself1 and we believe it with much probability to be exactly true2 but we cannot calculate out in any real case its accurate results4 ,the law asserts that every particle of matter in the universe attracts every other particle1 with a force depending on the masses of the particles and their distances4 ,we cannot know the force acting on any particle unless we know the masses and distances and positions of all other particles in the universe4 ,the physical astronomer has made a sweeping assumption1 namely1 that all the millions of existing systems exert no perturbing effects on our planetary system1 that is to say1 no effects in the least appreciable4 ,the problem at once becomes hypothetical1 because there is little doubt that gravitation between our sun and planets and other systems does exist4 ,even when they consider the relations of our planetary bodies 99inter se991 all their processes are only approximate4 ,in the first place they assume that each of the planets is a perfect ellipsoid1 with a smooth surface and a homogeneous interior4 ,that this assumption is untrue every mountain and valley1 every sea1 every mine affords conclusive evidence4 ,if astronomers are to make their calculations perfect1 they must not only take account of the ,himalayas and the ,andes1 but must calculate separately the attraction of every hill1 nay1 of every ant-hill4 ,so far are they from having considered any local inequality of the surface1 that they have not yet decided upon the general form of the earth2 it is still a matter of speculation whether or not the earth is an ellipsoid with three unequal axes4 ,if1 as is probable1 the globe is irregularly compressed in some directions1 the calculations of astronomers will have to be repeated and refined1 in order that they may approximate to the attractive power of such a body4 ,if we cannot accurately learn the form of our own earth1 how can we expect to ascertain that of the moon1 the sun1 and other planets1 in some of which probably are irregularities of greater proportional amount8 ,in a further way the science of physical astronomy is merely approximate and hypothetical4 ,given homogeneous ellipsoids acting upon each other according to the law of gravity1 the best mathematicians have never and perhaps never will determine exactly the resulting movements4 ,even when three bodies simultaneously attract each other the complication of effects is so great that only approximate calculations can be made4 ,astronomers have not even attempted the general problem of the simultaneous attractions of four1 five1 six1 or more bodies2 they resolve the general problem into so many different problems of three bodies4 ,the principle upon which the calculations of physical astronomy proceed1 is to neglect every quantity which does not seem likely to lead to an effect appreciable in observation1 and the quantities rejected are far more numerous and complex than the few larger terms which are retained4 ,all then is merely approximate4 ,concerning other branches of physical science the same statements are even more evidently true4 ,we speak and calculate about inflexible bars1 inextensible lines1 heavy points1 homogeneous substances1 uniform spheres1 perfect fluids and gases1 and we deduce a great number of beautiful theorems2 but all is hypothetical4 ,there is no such thing as an inflexible bar1 an inextensible line1 nor any one of the other perfect objects of mechanical science2 they are to be classed with those mythical existences1 the straight line1 triangle1 circle1 c41 about which ,euclid so freely reasoned4 ,take the simplest operation considered in statics--the use of a crowbar in raising a heavy stone1 and we shall find1 as ,thomson and ,tait have pointed out1 that we neglect far more than we observe4#cgf ,if we suppose the bar to be quite rigid1 the fulcrum and stone perfectly hard1 and the points of contact real points1 we may give the true relation of the forces4 ,but in reality the bar must bend1 and the extension and compression of different parts involve us in difficulties4 ,even if the bar be homogeneous in all its parts1 there is no mathematical theory capable of determining with accuracy all that goes on2 if1 as is infinitely more probable1 the bar is not homogeneous1 the complete solution will be immensely more complicated1 but hardly more hopeless4 ,no sooner had we determined the change of form according to simple mechanical principles1 than we should discover the interference of thermodynamic principles4 ,compression produces heat and extension cold1 and thus the conditions of the problem are modified throughout4 ,in attempting a fourth approximation we should have to allow for the conduction of heat from one part of the bar to another4 ,all these effects are utterly inappreciable in a practical point of view1 if the bar be a good stout one2 but in a theoretical point of view they entirely prevent our saying that we have solved a natural problem4 ,the faculties of the human mind1 even when aided by the wonderful powers of abbreviation conferred by analytical methods1 are utterly unable to cope with the complications of any real problem4 ,and had we exhausted all the known phenomena of a mechanical problem1 how can we tell that hidden phenomena1 as yet undetected1 do not intervene in the commonest actions8 ,it is plain that no phenomenon comes within the sphere of our senses unless it possesses a momentum capable of irritating the appropriate nerves4 ,there may then be worlds of phenomena too slight to rise within the scope of our consciousness4 #cgf 99,treatise on ,natural ,philosophy991 vol4 i4 pp4 #ccg1 c4 ,all the instruments with which we perform our measurements are faulty4 ,we assume that a plumb-line gives a vertical line2 but this is never true in an absolute sense1 owing to the attraction of mountains and other inequalities in the surface of the earth4 ,in an accurate trigonometrical survey1 the divergencies of the plumb-line must be approximately determined and allowed for4 ,we assume a surface of mercury to be a perfect plane1 but even in the breadth of #e inches there is a calculable divergence from a true plane of about one ten-millionth part of an inch2 and this surface further diverges from true horizontality as the plumb-line does from true verticality4 ,that most perfect instrument1 the pendulum1 is not theoretically perfect1 except for infinitely small arcs of vibration1 and the delicate experiments performed with the torsion balance proceed on the assumption that the force of torsion of a wire is proportional to the angle of torsion1 which again is only true for infinitely small angles4 ,such is the purely approximate character of all our operations that it is not uncommon to find the theoretically worse method giving truer results than the theoretically perfect method4 ,the common pendulum which is not isochronous is better for practical purposes than the cycloidal pendulum1 which is isochronous in theory but subject to mechanical difficulties4 ,the spherical form is not the correct form for a speculum or lense1 but it differs so slightly from the true form1 and is so much more easily produced mechanically1 that it is generally best to rest content with the spherical surface4 ,even in a six-feet mirror the difference between the parabola and the sphere is only about one ten-thousandth part of an inch1 a thickness which would be taken off in a few rubs of the polisher4 ,watts' ingenious parallel motion was intended to produce rectilinear movement of the piston-rod4 ,in reality the motion was always curvilinear1 but for his purposes a certain part of the curve approximated sufficiently to a straight line4 99,approximation to ,exact ,laws499 ,though we can not prove numerical laws with perfect accuracy1 it would be a great mistake to suppose that there is any inexactness in the laws of nature4 ,we may even discover a law which we believe to represent the action of forces with perfect exactness4 ,the mind may seem to pass in advance of its data1 and choose out certain numerical results as absolutely true4 ,we can never really pass beyond our data1 and so far as assumption enters in1 so far want of certainty will attach to our conclusions2 nevertheless we may sometimes rightly prefer a probable assumption of a precise law to numerical results1 which are at the best only approximate4 ,we must accordingly draw a strong distinction between the laws of nature which we believe to be accurately stated in our formulas1 and those to which our statements only make an approximation1 so that at a future time the law will be differently stated4 ,the law of gravitation is expressed in the form ,f ,mm_/,d#b1 meaning that gravity is proportional directly to the product of the gravitating masses1 and indirectly to the square of their distance4 ,the latent heat of steam is expressed by the equation log ,f 99a99 99b99t 99c99t1 in which are five quantities 99a991 99b991 99c991 1 1 to be determined by experiment4 ,there is every reason to believe that in the progress of science the law of gravity will remain entirely unaltered1 and the only effect of further inquiry will be to render it a more and more probable expression of the absolute truth4 ,the law of the latent heat of steam on the other hand1 will be modified by every new series of experiments1 and it may not improbably be shown that the assumed law can never be made to agree exactly with the results of experiment4 ,philosophers have not always supposed that the law of gravity was exactly true4 ,newton1 though he had the highest confidence in its truth1 admitted that there were motions in the planetary system which he could not reconcile with the law4 ,euler and ,clairaut who were1 with ,d',alembert1 the first to apply the full powers of mathematical analysis to the theory of gravitation as explaining the perturbations of the planets1 did not think the law sufficiently established to attribute all discrepancies to the errors of calculation and observation4 ,they did not feel certain that the force of gravity exactly obeyed the well-known rule4 ,the law might involve other powers of the distance4 ,it might be expressed in the form ,f 444 99a99_/,d 99b99_/,d#b 99c99_/,d#c 444 and the coefficients 99a99 and 99c99 might be so small that those terms would become apparent only in very accurate comparisons with fact4 ,attempts have been made to account for difficulties1 by attributing value to such neglected terms4 ,gauss at one time thought the even more fundamental principle of gravity1 that the force is dependent only on mass and distance1 might not be exactly true1 and he undertook accurate pendulum experiments to test this opinion4 ,only as repeated doubts have time after time been resolved in favour of the law of ,newton1 has it been assumed as precisely correct4 ,but this belief does not rest on experiment or observation only4 ,the calculations of physical astronomy1 however accurate1 could never show that the other terms of the above expression were absolutely devoid of value4 ,it could only be shown that they had such slight value as never to become apparent4 ,there are1 however1 other reasons why the law is probably complete and true as commonly stated4 ,whatever influence spreads from a point1 and expands uniformly through space1 will doubtless vary inversely in intensity as the square of the distance1 because the area over which it is spread increases as the square of the radius4 ,this part of the law of gravity may be considered as due to the properties of space1 and there is a perfect analogy in this respect between gravity and all other 99emanating99 forces1 as was pointed out by ,keill4#cgg ,thus the undulations of light1 heat1 and sound1 and the attractions of electricity and magnetism obey the very same law so far as we can ascertain4 ,if the molecules of a gas or the particles of matter constituting odour were to start from a point and spread uniformly1 their distances would increase and their density decrease according to the same principle4 #cgg 99,an ,introduction to ,natural ,philosophy991 #crd edit4 #agcc1 p4 #e4 ,other laws of nature stand in a similar position4 ,dalton's laws of definite combining proportions never have been1 and never can be1 exactly proved2 but chemists having shown1 to a considerable degree of approximation1 that the elements combine together as if each element had atoms of an invariable mass1 assume that this is exactly true4 ,they go even further4 ,prout pointed out in #ahae that the equivalent weights of the elements appeared to be simple numbers2 and the researches of ,dumas1 ,pelouze1 ,marignac1 ,erdmann1 ,stas1 and others have gradually rendered it likely that the atomic weights of hydrogen1 carbon1 oxygen1 nitrogen1 chlorine1 and silver1 are in the ratios of the numbers #a1 #ab1 #af1 #ad1 #cee1 and #ajh4 ,chemists then step beyond their data2 they throw aside their actual experimental numbers1 and assume that the true ratios are not those exactly indicated by any weighings1 but the simple ratios of these numbers4 ,they boldly assume that the discrepancies are due to experimental errors1 and they are justified by the fact that the more elaborate and skilful the researches on the subject1 the more nearly their assumption is verified4 ,potassium is the only element whose atomic weight has been determined with great care1 but which has not shown an approach to a simple ratio with the other elements4 ,this exception may be due to some unsuspected cause of error4#cgh ,a similar assumption is made in the law of definite combining volumes of gases1 and ,brodie has clearly pointed out the line of argument by which the chemist1 observing that the discrepancies between the law and fact are within the limits of experimental error1 assumes that they are due to error4#cgi #cgh ,watts1 99,dictionary of ,chemistry991 vol4 i4 p4 #dee4 #cgi 99,philosophical ,transactions991 7#ahff7 vol4 clvi4 p4 #hji4 ,faraday1 in one of his researches1 expressly makes an assumption of the same kind4 ,having shown1 with some degree of experimental precision1 that there exists a simple proportion between quantities of electrical energy and the quantities of chemical substances which it can decompose1 so that for every atom dissolved in the battery cell an atom ought theoretically1 that is without regard to dissipation of some of the energy1 to be decomposed in the electrolytic cell1 he does not stop at his numerical results4 8,i have not hesitated10 he says1#chj 8to apply the more strict results of chemical analysis to correct the numbers obtained as electrolytic results4 ,this1 it is evident1 may be done in a great number of cases1 without using too much liberty towards the due severity of scientific research40 #chj 99,experimental ,researches in ,electricity991 vol4 i4 p4 #bdf4 ,the law of the conservation of energy1 one of the widest of all physical generalisations1 rests upon the same footing4 ,the most that we can do by experiment is to show that the energy entering into any experimental combination is almost equal to what comes out of it1 and more nearly so the more accurately we perform the measurements4 ,absolute equality is always a matter of assumption4 ,we cannot even prove the indestructibility of matter2 for were an exceedingly minute fraction of existing matter to vanish in any experiment1 say one part in ten millions1 we could never detect the loss4 99,successive ,approximations to ,natural ,conditions499 ,when we examine the history of scientific problems1 we find that one man or one generation is usually able to make but a single step at a time4 ,a problem is solved for the first time by making some bold hypothetical simplification1 upon which the next investigator makes hypothetical modifications approaching more nearly to the truth4 ,errors are successively pointed out in previous solutions1 until at last there might seem little more to be desired4 ,careful examination1 however1 will show that a series of minor inaccuracies remain to be corrected and explained1 were our powers of reasoning sufficiently great1 and the purpose adequate in importance4 ,newton's successful solution of the problem of the planetary movements entirely depended at first upon a great simplification4 ,the law of gravity only applies directly to two infinitely small particles1 so that when we deal with vast globes like the earth1 ,jupiter1 and the sun1 we have an immense aggregate of separate attractions to deal with1 and the law of the aggregate need not coincide with the law of the elementary particles4 ,but ,newton1 by a great effort of mathematical reasoning1 was able to show that two homogeneous spheres of matter act as if the whole of their masses were concentrated at the centres2 in short1 that such spheres are centrobaric bodies 7p4 #cfd74 ,he was then able with comparative ease to calculate the motions of the planets on the hypothesis of their being spheres1 and to show that the results roughly agreed with observation4 ,newton1 indeed1 was one of the few men who could make two great steps at once4 ,he did not rest contented with the spherical hypothesis2 having reason to believe that the earth was really a spheroid with a protuberance around the equator1 he proceeded to a second approximation1 and proved that the attraction of the protuberant matter upon the moon accounted for the precession of the equinoxes1 and led to various complicated effects4 ,but1 7p4 #dei71 even the spheroidal hypothesis is far from the truth4 ,it takes no account of the irregularities of surface1 the great protuberance of land in ,central ,asia and ,south ,america1 and the deficiency in the bed of the ,atlantic4 ,to determine the law according to which a projectile1 such as a cannon ball1 moves through the atmosphere is a problem very imperfectly solved at the present day1 but in which many successive advances have been made4 ,so little was known concerning the subject three or four centuries ago that a cannon ball was supposed to move at first in a straight line1 and after a time to be deflected into a curve4 ,tartaglia ventured to maintain that the path was curved throughout1 as by the principle of continuity it should be2 but the ingenuity of ,galileo was required to prove this opinion1 and to show that the curve was approximately a parabola4 ,it is only1 however1 under forced hypotheses that we can assert the path of a projectile to be truly a parabola3 the path must be through a perfect vacuum1 where there is no resisting medium of any kind2 the force of gravity must be uniform and act in parallel lines2 or else the moving body must be either a mere point1 or a perfect centrobaric body1 that is a body possessing a definite centre of gravity4 ,these conditions cannot be really fulfilled in practice4 ,the next great step in the problem was made by ,newton and ,huyghens1 the latter of whom asserted that the atmosphere would offer a resistance proportional to the velocity of the moving body1 and concluded that the path would have in consequence a logarithmic character4 ,newton investigated in a general manner the subject of resisting media1 and came to the conclusion that the resistance is more nearly proportional to the square of the velocity4 ,the subject then fell into the hands of ,daniel ,bernoulli1 who pointed out the enormous resistance of the air in cases of rapid movement1 and calculated that a cannon ball1 if fired vertically in a vacuum1 would rise eight times as high as in the atmosphere4 ,in recent times an immense amount both of theoretical and experimental investigation has been spent upon the subject1 since it is one of importance in the art of war4 ,successive approximations to the true law have been made1 but nothing like a complete and final solution has been achieved or even hoped for4#cha #cha ,hutton's 99,mathematical ,dictionary991 vol4 ii4 pp4 #bhg-bib4 ,it is quite to be expected that the earliest experimenters in any branch of science will overlook errors which afterwards become most apparent4 ,the ,arabian astronomers determined the meridian by taking the middle point between the places of the sun when at equal altitudes on the same day4 ,they overlooked the fact that the sun has its own motion in the time between the observations4 ,newton thought that the mutual disturbances of the planets might be disregarded1 excepting perhaps the effect of the mutual attraction of the greater planets1 ,jupiter and ,saturn1 near their conjunction4#chb ,the expansion of quicksilver was long used as the measure of temperature1 no clear idea being possessed of temperature apart from some of its more obvious effects4 ,rumford1 in the first experiment leading to a determination of the mechanical equivalent of heat1 disregarded the heat absorbed by the apparatus1 otherwise he would1 in ,dr4 ,joule's opinion1 have come nearly to the correct result4 #chb 99,principia991 bk4 iii4 ,prop4 #ac4 ,it is surprising to learn the number of causes of error which enter into the simplest experiment1 when we strive to attain rigid accuracy4 ,we cannot accurately perform the simple experiment of compressing gas in a bent tube by a column of mercury1 in order to test the truth of ,boyle's ,law1 without paying regard to--7#a7 the variations of atmospheric pressure1 which are communicated to the gas through the mercury2 7#b7 the compressibility of mercury1 which causes the column of mercury to vary in density2 7#c7 the temperature of the mercury throughout the column2 7#d7 the temperature of the gas1 which is with difficulty maintained invariable2 7#e7 the expansion of the glass tube containing the gas4 ,although ,regnault took all these circumstances into account in his examination of the law1#chc there is no reason to suppose that he exhausted the sources of inaccuracy4 #chc ,jamin1 99,cours de ,physique991 vol4 i4 pp4 #bhb1 #bhc4 ,the early investigations concerning the nature of waves in elastic media proceeded upon the assumption that waves of different lengths would travel with equal speed4 ,newton's theory of sound led him to this conclusion1 and observation 7p4 #bie7 had verified the inference4 ,when the undulatory theory came to be applied at the commencement of this century to explain the phenomena of light1 a great difficulty was encountered4 ,the angle at which a ray of light is refracted in entering a denser medium depends1 according to that theory1 on the velocity with which the wave travels1 so that if all waves of light were to travel with equal velocity in the same medium1 the dispersion of mixed light by the prism and the production of the spectrum could not take place4 ,some most striking phenomena were thus in direct conflict with the theory4 ,cauchy first pointed out the explanation1 namely1 that all previous investigators had made an arbitrary assumption for the sake of simplifying the calculations4 ,they had assumed that the particles of the vibrating medium are so close together that the intervals are inconsiderable compared with the length of the wave4 ,this hypothesis happened to be approximately true in the case of air1 so that no error was discovered in experiments on sound4 ,had it not been so1 the earlier analysts would probably have failed to give any solution1 and the progress of the subject might have been retarded4 ,cauchy was able to make a new approximation under the more difficult supposition1 that the particles of the vibrating medium are situated at considerable distances1 and act and react upon the neighbouring particles by attractive and repulsive forces4 ,to calculate the rate of propagation of disturbance in such a medium is a work of excessive difficulty4 ,the complete solution of the problem appears indeed to be beyond human power1 so that we must be content1 as in the case of the planetary motions1 to look forward to successive approximations4 ,all that ,cauchy could do was to show that certain quantities1 neglected in previous theories1 became of considerable amount under the new conditions of the problem1 so that there will exist a relation between the length of the wave1 and the velocity at which it travels4 ,to remove1 then1 the difficulties in the way of the undulatory theory of light1 a new approach to probable conditions was needed4#chd #chd ,lloyd's 99,lectures on the ,wave ,theory991 pp4 #bb1 #bc4 ,in a similar manner ,fourier's theory of the conduction and radiation of heat was based upon the hypothesis that the quantity of heat passing along any line is simply proportional to the rate of change of temperature4 ,but it has since been shown by ,forbes that the conductivity of a body diminishes as its temperature increases4 ,all the details of ,fourier's solution therefore require modification1 and the results are in the meantime to be regarded as only approximately true4#che #che ,tait's 99,thermodynamics991 p4 #aj4 ,we ought to distinguish between those problems which are physically and those which are merely mathematically incomplete4 ,in the latter case the physical law is correctly seized1 but the mathematician neglects1 or is more often unable to follow out the law in all its results4 ,the law of gravitation and the principles of harmonic or undulatory movement1 even supposing the data to be correct1 can never be followed into all their ultimate results4 ,young explained the production of ,newton's rings by supposing that the rays reflected from the upper and lower surfaces of a thin film of a certain thickness were in opposite phases1 and thus neutralised each other4 ,it was pointed out1 however1 that as the light reflected from the nearer surface must be undoubtedly a little brighter than that from the further surface1 the two rays ought not to neutralise each other so completely as they are observed to do4 ,it was finally shown by ,poisson that the discrepancy arose only from incomplete solution of the problem2 for the light which has once got into the film must be to a certain extent reflected backwards and forwards 99ad infinitum992 and if we follow out this course of the light by perfect mathematical analysis1 absolute darkness may be shown to result from the interference of the rays4#chf ,in this case the natural laws concerned1 those of reflection and refraction1 are accurately known1 and the only difficulty consists in developing their full consequences4 #chf ,lloyd's 99,lectures on the ,wave ,theory991 pp4 #hb1 #hc4 99,discovery of ,hypothetically ,simple ,laws499 ,in some branches of science we meet with natural laws of a simple character which are in a certain point of view exactly true and yet can never be manifested as exactly true in natural phenomena4 ,such1 for instance1 are the laws concerning what is called a 99perfect gas994 ,the gaseous state of matter is that in which the properties of matter are exhibited in the simplest manner4 ,there is much advantage accordingly in approaching the question of molecular mechanics from this side4 ,but when we ask the question--,what is a gas8 the answer must be a hypothetical one4 ,finding that gases 99nearly99 obey the law of ,boyle and ,mariotte2 that they 99nearly99 expand by heat at the uniform rate of one part in #bgbi of their volume at #j for each degree centigrade2 and that they 99more nearly99 fulfil these conditions the more distant the point of temperature at which we examine them from the liquefying point1 we pass by the principle of continuity to the conception of a perfect gas4 ,such a gas would probably consist of atoms of matter at so great a distance from each other as to exert no attractive forces upon each other2 but for this condition to be fulfilled the distances must be infinite1 so that an absolutely perfect gas cannot exist4 ,but the perfect gas is not merely a limit to which we may approach1 it is a limit passed by at least one real gas4 ,it has been shown by ,despretz1 ,pouillet1 ,dulong1 ,arago1 and finally ,regnault1 that all gases diverge from the ,boylean law1 and in nearly all cases the density of the gas increases in a somewhat greater ratio than the pressure1 indicating a tendency on the part of the molecules to approximate of their own accord4 ,in the more condensable gases such as sulphurous acid1 ammonia1 and cyanogen1 this tendency is strongly apparent near the liquefying point4 ,hydrogen1 on the contrary1 diverges from the law of a perfect gas in the opposite direction1 that is1 the density increases less than in the ratio of the pressure4#chg ,this is a singular exception1 the bearing of which ,i am unable to comprehend4 #chg ,jamin1 99,cours de ,physique991 vol4 i4 pp4 #bhc-bhh4 ,all gases diverge again from the law of uniform expansion by heat1 but the divergence is less as the gas in question is less condensable1 or examined at a temperature more removed from its liquefying point4 ,thus the perfect gas must have an infinitely high temperature4 ,according to ,dalton's law each gas in a mixture retains its own properties unaffected by the presence of any other gas4#chh ,this law is probably true only by approximation1 but it is obvious that it would be true of the perfect gas with infinitely distant particles4#chi #chh ,joule and ,thomson1 99,philosophical ,transactions991 #ahed1 vol4 cxliv4 p4 #ccg4 #chi ,the properties of a perfect gas have been described by ,rankine1 99,transactions of the ,royal ,society of ,edinburgh991 vol4 xxv4 p4 #efa4 99,mathematical ,principles of ,approximation499 ,the approximate character of physical science will be rendered more plain if we consider it from a mathematical point of view4 ,throughout quantitative investigations we deal with the relation of one quantity to other quantities1 of which it is a function2 but the subject is sufficiently complicated if we view one quantity as a function of one other4 ,now1 as a general rule1 a function can be developed or expressed as the sum of quantities1 the values of which depend upon the successive powers of the variable quantity4 ,if 99y99 be a function of 99x99 then we may say that 99y99 ,a ,b99x99 ,c99x99#b ,d99x99#c ,e99x99#d 4444 ,in this equation1 ,a1 ,b1 ,c1 ,d1 c41 are fixed quantities1 of different values in different cases4 ,the terms may be infinite in number or after a time may cease to have any value4 ,any of the coefficients ,a1 ,b1 ,c1 c41 may be zero or negative2 but whatever they be they are fixed4 ,the quantity 99x99 on the other hand may be made what we like1 being variable4 ,suppose1 in the first place1 that 99x99 and 99y99 are both lengths4 ,let us assume that #a_/aj1jjj part of an inch is the least that we can take note of4 ,then when 99x99 is one hundredth of an inch1 we have 99x99#b #a_/aj1jjj1 and if ,c be less than unity1 the term ,c99x99#b will be inappreciable1 being less than we can measure4 ,unless any of the quantities ,d1 ,e1 c41 should happen to be very great1 it is evident that all the succeeding terms will also be inappreciable1 because the powers of 99x99 become rapidly smaller in geometrical ratio4 ,thus when 99x99 is made small enough the quantity 99y99 seems to obey the equation 99y99 ,a ,b99x994 ,if 99x99 should be still less1 if it should become as small1 for instance1 as #a_/a1jjj1jjj of an inch1 and ,b should not be very great1 then 99y99 would appear to be the fixed quantity ,a1 and would not seem to vary with 99x99 at all4 ,on the other hand1 were x to grow greater1 say equal to #a_/aj inch1 and ,c not be very small1 the term ,c99x99#b would become appreciable1 and the law would now be more complicated4 ,we can invert the mode of viewing this question1 and suppose that while the quantity 99y99 undergoes variations depending on many powers of 99x991 our power of detecting the changes of value is more or less acute4 ,while our powers of observation remain very rude we may be unable to detect any change in the quantity at all1 that is to say1 ,b99x99 may always be too small to come within our notice1 just as in former days the fixed stars were so called because they remained at apparently fixed distances from each other4 ,with the use of telescopes and micrometers we become able to detect the existence of some motion1 so that the distance of one star from another may be expressed by ,a ,b99x991 the term including 99x99#b being still inappreciable4 ,under these circumstances the star will seem to move uniformly1 or in simple proportion to the time 99x994 ,with much improved means of measurement it will probably be found that this uniformity of motion is only apparent1 and that there exists some acceleration or retardation4 ,more careful investigation will show the law to be more and more complicated than was previously supposed4 ,there is yet another way of explaining the apparent results of a complicated law4 ,if we take any curve and regard a portion of it free from any kind of discontinuity1 we may represent the character of such portion by an equation of the form 99y99 ,a ,b99x99 ,c99x99#b ,d99x99#c 4444 ,restrict the attention to a very small portion of the curve1 and the eye will be unable to distinguish its difference from a straight line1 which amounts to saying that in the portion examined the term ,c99x99#b has no value appreciable by the eye4 ,take a larger portion of the curve and it will be apparent that it possesses curvature1 but it will be possible to draw a parabola or ellipse so that the curve shall apparently coincide with a portion of that parabola or ellipse4 ,in the same way if we take larger and larger arcs of the curve it will assume the character successively of a curve of the third1 fourth1 and perhaps higher degrees2 that is to say1 it corresponds to equations involving the third1 fourth1 and higher powers of the variable quantity4 ,we have arrived then at the conclusion that every phenomenon1 when its amount can only be rudely measured1 will either be of fixed amount1 or will seem to vary uniformly like the distance between two inclined straight lines4 ,more exact measurement may show the error of this first assumption1 and the variation will then appear to be like that of the distance between a straight line and a parabola or ellipse4 ,we may afterwards find that a curve of the third or higher degrees is really required to represent the variation4 ,i propose to call the variation of a quantity 99linear991 99elliptic991 99cubic991 99quartic991 99quintic991 c41 according as it is discovered to involve the first1 second1 third1 fourth1 fifth1 or higher powers of the variable4 ,it is a general rule in quantitative investigation that we commence by discovering linear1 and afterwards proceed to elliptic or more complicated laws of variation4 ,the approximate curves which we employ are all1 according to ,de ,morgan's use of the name1 parabolas of some order or other2 and since the common parabola of the second order is approximately the same as a very elongated ellipse1 and is in fact an infinitely elongated ellipse1 it is convenient and proper to call variation of the second order 99elliptic994 ,it might also be called 99quadric99 variation4 ,as regards many important phenomena we are yet only in the first stage of approximation4 ,we know that the sun and many so-called fixed stars1 especially #fa ,cygni1 have a proper motion through space1 and the direction of this motion at the present time is known with some degree of accuracy4 ,but it is hardly consistent with the theory of gravity that the path of any body should really be a straight line4 ,hence1 we must regard a rectilinear path as only a provisional description of the motion1 and look forward to the time when its curvature will be detected1 though centuries perhaps must first elapse4 ,we are accustomed to assume that on the surface of the earth the force of gravity is uniform1 because the variation is of so slight an amount that we are scarcely able to detect it4 ,but supposing we could measure the variation1 we should find it simply proportional to the height4 ,taking the earth's radius to be unity1 let 99h99 be the height at which we measure the force of gravity4 ,then by the well-known law of the inverse square1 that force will be proportional to 99g99_/7#a 99h997#b1 or to 99g997#a - #b99;h99 #c99;h99#b - #d99;h99#c 44474 ,but at all heights to which we can attain 99h99 will be so small a fraction of the earth's radius that #c99;h99#b will be inappreciable1 and the force of gravity will seem to follow the law of linear variation1 being proportional to #a - #b99;h994 ,when the circumstances of an experiment are much altered1 different powers of the variable may become prominent4 ,the resistance of a liquid to a body moving through it may be approximately expressed as the sum of two terms respectively involving the first and second powers of the velocity4 ,at very low velocities the first power is of most importance1 and the resistance1 as ,professor ,stokes has shown1 is nearly in simple proportion to the velocity4 ,when the motion is rapid the resistance increases in a still greater degree1 and is more nearly proportional to the square of the velocity4 99,approximate ,independence of ,small ,effects499 ,one result of the theory of approximation possesses such importance in physical science1 and is so often applied1 that we may consider it separately4 ,the investigation of causes and effects is immensely simplified when we may consider each cause as producing its own effect invariably1 whether other causes are acting or not4 ,thus1 if the body ,p produces 99x991 and ,q produces 99y991 the question is whether ,p and ,q acting together will produce the sum of the separate effects1 99x99 99y994 ,it is under this supposition that we treated the methods of eliminating error 7,chap4 ,,xv471 and errors of a less amount would still remain if the supposition was a forced one4 ,there are probably some parts of science in which the supposition of independence of effects holds rigidly true4 ,the mutual gravity of two bodies is entirely unaffected by the presence of other gravitating bodies4 ,people do not usually consider that this important principle is involved in such a simple thing as putting two pound weights in the scale of a balance4 ,how do we know that two pounds together will weigh twice as much as one8 ,do we know it to be exactly so8 ,like other results founded on induction we cannot prove it absolutely1 but all the calculations of physical astronomy proceed upon the assumption1 so that we may consider it proved to a very high degree of approximation4 ,had not this been true1 the calculations of physical astronomy would have been infinitely more complex than they actually are1 and the progress of knowledge would have been much slower4 ,it is a general principle of scientific method that if effects be of small amount1 comparatively to our means of observation1 all joint effects will be of a higher order of smallness1 and may therefore be rejected in a first approximation4 ,this principle was employed by ,daniel ,bernoulli in the theory of sound1 under the title of 99,the ,principle of the ,coexistence of ,small ,vibrations994 ,he showed that if a string is affected by two kinds of vibrations1 we may consider each to be going on as if the other did not exist4 ,we cannot perceive that the sounding of one musical instrument prevents or even modifies the sound of another1 so that all sounds would seem to travel through the air1 and act upon the ear in independence of each other4 ,a similar assumption is made in the theory of tides1 which are great waves4 ,one wave is produced by the attraction of the moon1 and another by the attraction of the sun1 and the question arises1 whether when these waves coincide1 as at the time of spring tides1 the joint wave will be simply the sum of the separate waves4 ,on the principle of ,bernoulli this will be so1 because the tides on the ocean are very small compared with the depth of the ocean4 ,the principle of ,bernoulli1 however1 is only approximately true4 ,a wave never is exactly the same when another wave is interfering with it1 but the less the displacement of particles due to each wave1 the less in a still higher degree is the effect of one wave upon the other4 ,in recent years ,helmholtz was led to suspect that some of the phenomena of sound might after all be due to resultant effects overlooked by the assumption of previous physicists4 ,he investigated the secondary waves which would arise from the interference of considerable disturbances1 and was able to show that certain summation of resultant tones ought to be heard1 and experiments subsequently devised for the purpose showed that they might be heard4 ,illustration ,throughout the mechanical sciences the 99,principle of the ,superposition of ,small ,motions99 is of fundamental importance1#cij and it may be thus explained4 ,suppose that two forces1 acting from the points ,b and ,c1 are simultaneously moving a body ,a4 ,let the force acting from ,b be such that in one second it would move ,a to 99p991 and similarly let the second force1 acting alone1 move ,a to 99r994 ,the question arises1 then1 whether their joint action will urge ,a to 99q99 along the diagonal of the parallelogram4 ,may we say that ,a will move the distance ,a99p99 in the direction ,,ab1 and ,a99r99 in the direction ,,ac1 or1 what is the same thing1 along the parallel line 99pq998 ,in strictness we cannot say so2 for when ,a has moved towards 99p991 the force from ,c will no longer act along the line ,,ac1 and similarly the motion of ,a towards 99r99 will modify the action of the force from ,b4 ,this interference of one force with the line of action of the other will evidently be greater the larger is the extent of motion considered2 on the other hand1 as we reduce the parallelogram ,a99pqr991 compared with the distances ,,ab and ,,ac1 the less will be the interference of the forces4 ,accordingly mathematicians avoid all error by considering the motions as infinitely small1 so that the interference becomes of a still higher order of infinite smallness1 and may be entirely neglected4 ,by the resources of the differential calculus it is possible to calculate the motion of the particle ,a1 as if it went through an infinite number of infinitely small diagonals of parallelograms4 ,the great discoveries of ,newton really arose from applying this method of calculation to the movements of the moon round the earth1 which1 while constantly tending to move onward in a straight line1 is also deflected towards the earth by gravity1 and moves through an elliptic curve1 composed as it were of the infinitely small diagonals of infinitely numerous parallelograms4 ,the mathematician1 in his investigation of a curve1 always treats it as made up of a great number of straight lines1 and it may be doubted whether he could treat it in any other manner4 ,there is no error in the final results1 because having obtained the formul flowing from this supposition1 each straight line is then regarded as becoming infinitely small1 and the polygonal line becomes undistinguishable from a perfect curve4#cia #cij ,thomson and ,tait's 99,natural ,philosophy991 vol4 i4 p4 #fj4 #cia ,challis1 99,notes on the ,principles of ,pure and ,applied ,calculation991 #ahfi1 p4 #hc4 ,in abstract mathematical theorems the approximation to absolute truth is perfect1 because we can treat of infinitesimals4 ,in physical science1 on the contrary1 we treat of the least quantities which are perceptible4 ,nevertheless1 while carefully distinguishing between these two different cases1 we may fearlessly apply to both the principle of the superposition of small effects4 ,in physical science we have only to take care that the effects really are so small that any joint effect will be unquestionably imperceptible4 ,suppose1 for instance1 that there is some cause which alters the dimensions of a body in the ratio of #a to #a 1 and another cause which produces an alteration in the ratio of #a to #a 4 ,if they both act at once the change will be in the ratio of #a to 7#a 77#a 71 or as #a to #a 4 ,but if and be both very small fractions of the total dimensions1 will be yet far smaller and may be disregarded2 the ratio of change is then approximately that of #a to #a 1 or the joint effect is the sum of the separate effects4 ,thus if a body were subjected to three strains1 at right angles to each other1 the total change in the volume of the body would be approximately equal to the sum of the changes produced by the separate strains1 provided that these are very small4 ,in like manner not only is the expansion of every solid and liquid substance by heat approximately proportional to the change of temperature1 when this change is very small in amount1 but the cubic expansion may also be considered as being three times as great as the linear expansion4 ,for if the increase of temperature expands a bar of metal in the ratio of #a to #a 1 and the expansion be equal in all directions1 then a cube of the same metal would expand as #a to 7#a 7#c1 or as #a to #a #c #cb #c4 ,when is a very small quantity the third term #cb will be imperceptible1 and still more so the fourth term #c4 ,the coefficients of expansion of solids are in fact so small1 and so imperfectly determined1 that physicists seldom take into account their second and higher powers4 ,it is a result of these principles that all small errors may be assumed to vary in simple proportion to their causes--a new reason why1 in eliminating errors1 we should first of all make them as small as possible4 ,let us suppose that there is a right-angled triangle of which the two sides containing the right angle are really of the lengths #c and #d1 so that the hypothenuse is 7#cb #db7 or #e4 ,now1 if in two measurements of the first side we commit slight errors1 making it successively #djja and #djjb1 then calculation will give the lengths of the hypothenuse as almost exactly #ejjjh and #ejjaf1 so that the error in the hypothenuse will seem to vary in simple proportion to that of the side1 although it does not really do so with perfect exactness4 ,the logarithm of a number does not vary in proportion to that number--nevertheless we find the difference between the logarithms of the numbers #ajjjjj and #ajjjja to be almost exactly equal to that between the numbers #ajjjja and #ajjjjb4 ,it is thus a general rule that very small differences between successive values of a function are approximately proportional to the small differences of the variable quantity4 ,on these principles it is easy to draw up a series of rules such as those given by ,kohlrausch#cib for performing calculations in an abbreviated form when the variable quantity is very small compared with unity4 ,thus for #a 7#a 7 we may substitute #a - 2 for #a 7#a - 7 we may put #a 2 #a 7#a 7 becomes #a - 7#a_/b71 and so forth4 #cib 99,an ,introduction to ,physical ,measurements991 translated by ,waller and ,procter1 #ahgc1 p4 #aj4 99,four ,meanings of ,equality499 ,although it might seem that there are few terms more free from ambiguity than the term 99equal991 yet scientific men do employ it with at least four meanings1 which it is desirable to distinguish4 ,these meanings ,i may describe as 7#a7 ,absolute ,equality4 7#b7 ,sub-equality4 7#c7 ,apparent ,equality4 7#d7 ,probable ,equality4 ,by 99absolute equality99 we signify that which is complete and perfect to the last degree2 but it is obvious that we can only know such equality in a theoretical or hypothetical manner4 ,the areas of two triangles standing upon the same base and between the same parallels are absolutely equal4 ,hippocrates beautifully proved that the area of a lunula or figure contained between two segments of circles was absolutely equal to that of a certain right-angled triangle4 ,as a general rule all geometrical and other elementary mathematical theorems involve absolute equality4 ,de ,morgan proposed to describe as 99sub-equal99 those quantities which are equal within an infinitely small quantity1 so that 99x99 is sub-equal to 99x99 99dx994 ,the differential calculus may be said to arise out of the neglect of infinitely small quantities1 and in mathematical science other subtle distinctions may have to be drawn between kinds of equality1 as ,de ,morgan has shown in a remarkable memoir 8,on ,infinity2 and on the sign of ,equality40#cic #cic 99,cambridge ,philosophical ,transactions99 7#ahfe71 vol4 xi4 ,part ,i4 99,apparent equality99 is that with which physical science deals4 ,those magnitudes are apparently equal which differ only by an imperceptible quantity4 ,to the carpenter anything less than the hundredth part of an inch is non-existent2 there are few arts or artists to which the hundred-thousandth of an inch is of any account4 ,since all coincidence between physical magnitudes is judged by one or other sense1 we must be restricted to a knowledge of apparent equality4 ,in reality even apparent equality is rarely to be expected4 ,more commonly experiments will give only 99probable equality991 that is results will come so near to each other that the difference may be ascribed to unimportant disturbing causes4 ,physicists often assume quantities to be equal provided that they fall within the limits of probable error of the processes employed4 ,we cannot expect observations to agree with theory more closely than they agree with each other1 as ,newton remarked of his investigations concerning ,halley's ,comet4 99,arithmetic of ,approximate ,quantities499 ,considering that almost all the quantities which we treat in physical and social science are approximate only1 it seems desirable that attention should be paid in the teaching of arithmetic to the correct interpretation and treatment of approximate numerical statements4 ,we seem to need notation for expressing the approximateness or exactness of decimal numbers4 ,the fraction #jbe may mean either precisely one #djth part1 or it may mean anything between #jbde and #jbee4 ,i propose that when a decimal fraction is completely and exactly given1 a 99small cipher99 or circle should be added to indicate that there is nothing more to come1 as in #jbe4 ,when the first figure of the decimals rejected is #e or more1 the first figure retained should be raised by a unit1 according to a rule approved by ,de ,morgan1 and now generally recognised4 ,to indicate that the fraction thus retained is more than the truth1 a point has been placed over the last figure in some tables of logarithms2 but a similar point is used to denote the period of a repeating decimal1 and ,i should therefore propose to employ a colon 99after99 the figure2 thus #jbe3 would mean that the true quantity lies between #jbde and #jbe inclusive of the lower but not the higher limit4 ,when the fraction is less than the truth1 two dots might be placed horizontally as in #jbe44 which would mean anything between #jbe and #jbee not inclusive4 ,when approximate numbers are added1 subtracted1 multiplied1 or divided1 it becomes a matter of some complexity to determine the degree of accuracy of the result4 ,there are few persons who could assert off-hand that the sum of the approximate numbers #cdgj1 #ebfic1 #hja1 is #afge 99within less than99 #jg4 ,mr4 ,sandeman has traced out the rules of approximate arithmetic in a very thorough manner1 and his directions are worthy of careful attention4#cid ,the third part of ,sonnenschein and ,nesbitt's excellent book on arithmetic#cie describes fully all kinds of approximate calculations1 and shows both how to avoid needless labour and how to take proper account of inaccuracy in operating with approximate decimal fractions4 ,a simple investigation of the subject is to be found in ,sonnet's 99,algbre ,elmentaire99 7,paris1 #ahdh7 chap4 xiv41 8,des ,approximations ,absolues et ,relatives40 ,there is also an ,american work on the subject4#cif #cid ,sandeman1 99,pelicotetics991 p4 #bad4 #cie 99,the ,science and ,art of ,arithmetic for the ,use of ,schools499 7,whitaker and ,co47 #cif 99,principles of ,approximate ,calculations991 by ,j4 ,j4 ,skinner1 ,c4,e4 7,new ,york1 ,henry ,holt71 #ahgf4 ,although the accuracy of measurement has so much advanced since the time of ,leslie1 it is not superfluous to repeat his protest against the unfairness of affecting by a display of decimal fractions a greater degree of accuracy than the nature of the case requires and admits4#cig ,i have known a scientific man to register the barometer to a second of time when the nearest quarter of an hour would have been amply sufficient4 ,chemists often publish results of analysis to the ten-thousandth or even the millionth part of the whole1 when in all probability the processes employed cannot be depended on beyond the hundredth part4 ,it is seldom desirable to give more than one place of figures of uncertain amount2 but it must be allowed that a nice perception of the degree of accuracy possible and desirable is requisite to save misapprehension and needless computation on the one hand1 and to secure all attainable exactness on the other hand4 #cig ,leslie1 99,inquiry into the ,nature of ,heat991 p4 #eje4 ,,chapter ,,xxii4 ,,quantitative ,,induction4 ,we have not yet formally considered any processes of reasoning which have for their object to disclose laws of nature expressed in quantitative equations4 ,we have been inquiring into the modes by which a phenomenon may be measured1 and1 if it be a composite phenomenon1 may be resolved1 by the aid of several measurements1 into its component parts4 ,we have also considered the precautions to be taken in the performance of observations and experiments in order that we may know what phenomena we really do measure1 but we must remember that1 no number of facts and observations can by themselves constitute science4 ,numerical facts1 like other facts1 are but the raw materials of knowledge1 upon which our reasoning faculties must be exerted in order to draw forth the principles of nature4 ,it is by an inverse process of reasoning that we can alone discover the mathematical laws to which varying quantities conform4 ,by well-conducted experiments we gain a series of values of a variable1 and a corresponding series of values of a variant1 and we now want to know what mathematical function the variant is as regards the variable4 ,in the usual progress of a science three questions will have to be answered as regards every important quantitative phenomenon3-- 7#a7 ,is there any constant relation between a variable and a variant8 7#b7 ,what is the empirical formula expressing this relation8 7#c7 ,what is the rational formula expressing the law of nature involved8 99,probable ,connection of ,varying ,quantities499 ,we find it stated by ,mill1#cih that 8,whatever phenomenon varies in any manner whenever another phenomenon varies in some particular manner1 is either a cause or an effect of that phenomenon1 or is connected with it through some fact of causation40 ,this assertion may be considered true when it is interpreted with sufficient caution2 but it might otherwise lead us into error4 ,there is nothing whatever in the nature of things to prevent the existence of two variations which should apparently follow the same law1 and yet have no connection with each other4 ,one binary star might be going through a revolution which1 so far as we could tell1 was of equal period with that of another binary star1 and according to the above rule the motion of one would be the cause of the motion of the other1 which would not be really the case4 ,two astronomical clocks might conceivably be made so nearly perfect that1 for several years1 no difference could be detected1 and we might then infer that the motion of one clock was the cause or effect of the motion of the other4 ,this matter requires careful discrimination4 ,we must bear in mind that the continuous quantities of space1 time1 force1 c41 which we measure1 are made up of an infinite number of infinitely small units4 ,we may then meet with two variable phenomena which follow laws so nearly the same1 that in no part of the variations open to our observation can any discrepancy be discovered4 ,i grant that if two clocks could be shown to have kept 99exactly99 the same time during any finite interval1 the probability would become infinitely high that there was a connection between their motions4 ,but we can never absolutely prove such coincidences to exist4 ,allow that we may observe a difference of one-tenth of a second in their time1 yet it is possible that they were independently regulated so as to go together within less than that quantity of time4 ,in short1 it would require either an infinitely long time of observation1 or infinitely acute powers of measuring discrepancy1 to decide positively whether two clocks were or were not in relation with each other4 #cih 99,system of ,logic991 bk4 iii4 chap4 viii #f4 ,a similar question actually occurs in the case of the moon's motion4 ,we have no record that any other portion of the moon was ever visible to men than such as we now see4 ,this fact sufficiently proves that within the historical period the rotation of the moon on its own axis has coincided with its revolutions round the earth4 ,does this coincidence prove a relation of cause and effect to exist8 ,the answer must be in the negative1 because there might have been so slight a discrepancy between the motions that there has not yet been time to produce any appreciable effect4 ,there may nevertheless be a high probability of connection4 ,the whole question of the relation of quantities thus resolves itself into one of probability4 ,when we can only rudely measure a quantitative result1 we can assign but slight importance to any correspondence4 ,because the brightness of two stars seems to vary in the same manner1 there is no considerable probability that they have any relation with each other4 ,could it be shown that their periods of variation were the same to infinitely small quantities it would be certain1 that is infinitely probable1 that they were connected1 however unlikely this might be on other grounds4 ,the general mode of estimating such probabilities is identical with that applied to other inductive problems4 ,that any two periods of variation should by chance become 99absolutely equal99 is infinitely improbable2 hence if1 in the case of the moon or other moving bodies1 we could prove absolute coincidence we should have certainty of connection4#cii ,with approximate measurements1 which alone are within our power1 we must hope for approximate certainty at the most4 #cii ,laplace1 99,system of the ,world991 translated by ,harte1 vol4 ii4 p4 #cff4 ,the principles of inference and probability1 according to which we treat causes and effects varying in amount1 are exactly the same as those by which we treated simple experiments4 ,continuous quantity1 however1 affords us an infinitely more extensive sphere of observation1 because every different amount of cause1 however little different1 ought to be followed by a different amount of effect4 ,if we can measure temperature to the one-hundredth part of a degree centigrade1 then between #j and #ajj we have #aj1jjj possible trials4 ,if the precision of our measurements is increased1 so that the one-thousandth part of a degree can be appreciated1 our trials may be increased tenfold4 ,the probability of connection will be proportional to the accuracy of our measurements4 ,when we can vary the quantity of a cause at will it is easy to discover whether a certain effect is due to that cause or not4 ,we can then make as many irregular changes as we like1 and it is quite incredible that the supposed effect should by chance go through exactly the corresponding series of changes except by dependence4 ,if we have a bell ringing 99in vacuo991 the sound increases as we let in the air1 and it decreases again as we exhaust the air4 ,tyndall's singing flames evidently obeyed the directions of his own voice2 and ,faraday when he discovered the relation of magnetism and light found that1 by making or breaking or reversing the current of the electro-magnet1 he had complete command over a ray of light1 proving beyond all reasonable doubt the dependence of cause and effect4 ,in such cases it is the perfect coincidence in time between the change in the effect and that in the cause which raises a high improbability of casual coincidence4 ,it is by a simple case of variation that we infer the existence of a material connection between two bodies moving with exactly equal velocity1 such as the locomotive engine and the train which follows it4 ,elaborate observations were requisite before astronomers could all be convinced that the red hydrogen flames seen during solar eclipses belonged to the sun1 and not to the moon's atmosphere as ,flamsteed assumed4 ,as early as #agjf1 ,stannyan noticed a blood-red streak in an eclipse which he witnessed at ,berne1 and he asserted that it belonged to the sun2 but his opinion was not finally established until photographs of the eclipse in #ahfj1 taken by ,mr4 ,de la ,rue1 showed that the moon's dark body gradually covered the red prominences on one side1 and uncovered those on the other2 in short1 that these prominences moved precisely as the sun moved1 and not as the moon moved4 ,even when we have no means of accurately measuring the variable quantities we may yet be convinced of their connection1 if one always varies perceptibly at the same time as the other4 ,fatigue increases with exertion2 hunger with abstinence from food2 desire and degree of utility decrease with the quantity of commodity consumed4 ,we know that the sun's heating power depends upon his height of the sky2 that the temperature of the air falls in ascending a mountain2 that the earth's crust is found to be perceptibly warmer as we sink mines into it2 we infer the direction in which a sound comes from the change of loudness as we approach or recede4 ,the facility with which we can time after time observe the increase or decrease of one quantity with another sufficiently shows the connection1 although we may be unable to assign any precise law of relation4 ,the probability in such cases depends upon frequent coincidence in time4 99,empirical ,mathematical ,laws499 ,it is important to acquire a clear comprehension of the part which is played in scientific investigation by empirical formul and laws4 ,if we have a table containing certain values of a variable and the corresponding values of the variant1 there are mathematical processes by which we can infallibly discover a mathematical formula yielding numbers in more or less exact agreement with the table4 ,we may generally assume that the quantities will approximately conform to a law of the form 99y99 ,a ,b99x99 ,c99x99#b1 in which 99x99 is the variable and 99y99 the variant4 ,we can then select from the table three values of 99y991 and the corresponding values of 99x992 inserting them in the equation1 we obtain three equations by the solution of which we gain the values of ,a1 ,b1 and ,c4 ,it will be found as a general rule that the formula thus obtained yields the other numbers of the table to a considerable degree of approximation4 ,in many cases even the second power of the variable will be unnecessary2 ,regnault found that the results of his elaborate inquiry into the latent heat of steam at different pressures were represented with sufficient accuracy by the empirical formula #fjfe #jcje 99t991 in which is the total heat of the steam1 and 99t99 the temperature4#djj ,in other cases it may be requisite to include the third power of the variable4 ,thus physicists assume the law of the dilatation of liquids to be of the form t 99at99 99bt99#b 99ct99#c1 #djj 99,chemical ,reports and ,memoirs991 ,cavendish ,society1 p4 #bid4 and they calculate from results of observation the values of the three constants 99a991 99b991 99c991 which are usually small quantities not exceeding one-hundredth part of a unit1 but requiring to be determined with great accuracy4#dja ,theoretically speaking1 this process of empirical representation might be applied with any degree of accuracy2 we might include still higher powers in the formula1 and with sufficient labour obtain the values of the constants1 by using an equal number of experimental results4 ,the method of least squares may also be employed to obtain the most probable values of the constants4 #dja ,jamin1 99,cours de ,physique991 vol4 ii4 p4 #ch4 ,in a similar manner all periodic variations may be represented with any required degree of accuracy by formul involving the sines and cosines of angles and their multiples4 ,the form of any tidal or other wave may thus be expressed1 as ,sir ,g4 ,b4 ,airy has explained4#djb ,almost all the phenomena registered by meteorologists are periodic in character1 and when freed from disturbing causes may be embodied in empirical formul4 ,bessel has given a rule by which from any regular series of observations we may1 on the principle of the method of least squares1 calculate out with a moderate amount of labour a formula expressing the variation of the quantity observed1 in the most probable manner4 ,in meteorology three or four terms are usually sufficient for representing any periodic phenomenon1 but the calculation might be carried to any higher degree of accuracy4 ,as the details of the process have been described by ,herschel in his treatise on ,meteorology1#djc ,i need not further enter into them4 #djb 99,on ,tides and ,waves991 ,encyclopdia ,metropolitana1 p4 #cff994 #djc 99,encyclopdia ,britannica991 art4 99,meteorology994 ,reprint1 #aeb-aef4 ,the reader might be tempted to think that in these processes of calculation we have an infallible method of discovering inductive laws1 and that my previous statements 7,chap4 ,,vii47 as to the purely tentative and inverse character of the inductive process are negatived4 ,were there indeed any general method of inferring laws from facts it would overturn my statement1 but it must be carefully observed that these empirical formul do not coincide with natural laws4 ,they are only approximations to the results of natural laws founded upon the general principles of approximation4 ,it has already been pointed out that however complicated be the nature of a curve1 we may examine so small a portion of it1 or we may examine it with such rude means of measurement1 that its divergence from an elliptic curve will not be apparent4 ,as a still ruder approximation a portion of a straight line will always serve our purpose2 but if we need higher precision a curve of the third or fourth degree will almost certainly be sufficient4 ,now empirical formul really represent these approximate curves1 but they give us no information as to the precise nature of the curve itself to which we are approximating4 ,we do not learn what function the variant is of the variable1 but we obtain another function which1 within the bounds of observation1 gives nearly the same values4 99,discovery of ,rational ,formul499 ,let us now proceed to consider the modes in which from numerical results we can establish the actual relation between the quantity of the cause and that of the effect4 ,what we want is a 99rational99 formula or function1 which will exhibit the 99reason99 or exact nature and origin of the law in question4 ,there is no word more frequently used by mathematicians than the word 99function991 and yet it is difficult to define its meaning with perfect accuracy4 ,originally it meant performance or execution1 being equivalent to the ,greek or 4 ,mathematicians at first used it to mean 99any power of a quantity991 but afterwards generalised it so as to include 8any quantity formed in any manner whatsoever from another quantity40#djd ,any quantity1 then1 which depends upon and varies with another quantity may be called a function of it1 and either may be considered a function of the other4 #djd ,lagrange1 99,leons sur le ,calcul des ,fonctions991 #ahjf1 p4 #d4 ,given the quantities1 we want the function of which they are the values4 ,simple inspection of the numbers cannot as a general rule disclose the function4 ,in an earlier chapter 7p4 #abd7 ,i put before the reader certain numbers1 and requested him to point out the law which they obey1 and the same question will have to be asked in every case of quantitative induction4 ,there are perhaps three methods1 more or less distinct1 by which we may hope to obtain an answer3 7#a7 ,by purely haphazard trial4 7#b7 ,by noting the general character of the variation of the quantities1 and trying by preference functions which give a similar form of variation4 7#c7 ,by deducing from previous knowledge the form of the function which is most likely to suit4 ,having numerical results we are always at liberty to invent any kind of mathematical formula we like1 and then try whether1 by the suitable selection of values for the unknown constant quantities1 we can make it give the required results4 ,if ever we fall upon a formula which does so1 to a fair degree of approximation1 there is a presumption in favour of its being the true function1 although there is no certainty whatever in the matter4 ,in this way ,i discovered a simple mathematical law which closely agreed with the results of my experiments on muscular exertion4 ,this law was afterwards shown by ,professor ,haughton to be the true rational law according to his theory of muscular action4#dje #dje ,haughton1 99,principles of ,animal ,mechanics991 #ahgc1 pp4 #ddd-dej4 ,jevons1 99,nature991 #cjth of ,june1 #ahgj1 vol4 ii4 p4 #aeh4 ,see also the experiments of ,professor ,nipher1 of ,washington ,university1 ,st4 ,louis1 in 99,american ,journal of ,science991 vol4 ix4 p4 #acj1 vol4 x4 p4 #a2 99,nature991 vol4 xi4 pp4 #bef1 #bgf4 ,but the chance of succeeding in this manner is small4 ,the number of possible functions is infinite1 and even the number of comparatively simple functions is so large that the probability of falling upon the correct one by mere chance is very slight4 ,even when we obtain the law it is by a deductive process1 not by showing that the numbers give the law1 but that the law gives the numbers4 ,in the second way1 we may1 by a survey of the numbers1 gain a general notion of the kind of law they are likely to obey1 and we may be much assisted in this process by drawing them out in the form of a curve4 ,we can in this way ascertain with some probability whether the curve is likely to return into itself1 or whether it has infinite branches2 whether such branches are asymptotic1 that is1 approach infinitely towards straight lines2 whether it is logarithmic in character1 or trigonometric4 ,this indeed we can only do if we remember the results of previous investigations4 ,the process is still inversely deductive1 and consists in noting what laws give particular curves1 and then inferring inversely that such curves belong to such laws4 ,if we can in this way discover the class of functions to which the required law belongs1 our chances of success are much increased1 because our haphazard trials are now reduced within a narrower sphere4 ,but1 unless we have almost the whole curve before us1 the identification of its character must be a matter of great uncertainty2 and if1 as in most physical investigations1 we have a mere fragment of the curve1 the assistance given would be quite illusory4 ,curves of almost any character can be made to approximate to each other for a limited extent1 so that it is only by a kind of 99divination99 that we fall upon the actual function1 unless we have theoretical knowledge of the kind of function applicable to the case4 ,when we have once obtained what we believe to be the correct form of function1 the remainder of the work is mere mathematical computation to be performed infallibly according to fixed rules1#djf which include those employed in the determination of empirical formul 7p4 #dhg74 ,the function will involve two or three or more unknown constants1 the values of which we need to determine by our experimental results4 ,selecting some of our results widely apart and nearly equidistant1 we form by means of them as many equations as there are constant quantities to be determined4 ,the solution of these equations will then give us the constants required1 and having now the actual function we can try whether it gives with sufficient accuracy the remainder of our experimental results4 ,if not1 we must either make a new selection of results to give a new set of equations1 and thus obtain a new set of values for the constants1 or we must acknowledge that our form of function has been wrongly chosen4 ,if it appears that the form of function has been correctly ascertained1 we may regard the constants as only approximately accurate and may proceed by the ,method of ,least ,squares 7p4 #cic7 to determine the most probable values as given by the whole of the experimental results4 #djf ,jamin1 99,cours de ,physique991 vol4 ii4 p4 #ej4 ,in most cases we shall find ourselves obliged to fall back upon the third mode1 that is1 anticipation of the form of the law to be expected on the ground of previous knowledge4 ,theory and analogical reasoning must be our guides4 ,the general nature of the phenomenon will often indicate the kind of law to be looked for4 ,if one form of energy or one kind of substance is being converted into another1 we may expect the law of direct simple proportion4 ,in one distinct class of cases the effect already produced influences the amount of the ensuing effect1 as for instance in the cooling of a heated body1 when the law will be of an exponential form4 ,when the direction of a force influences its action1 trigonometrical functions enter4 ,any influence which spreads freely through tridimensional space will be subject to the law of the inverse square of the distance4 ,from such considerations we may sometimes arrive deductively and analogically at the general nature of the mathematical law required4 99,the ,graphical ,method499 ,in endeavouring to discover the mathematical law obeyed by experimental results it is often desirable to call in the aid of space-representations4 ,every equation involving two variable quantities corresponds to some kind of plane curve1 and every plane curve may be represented symbolically in an equation containing two unknown quantities4 ,now in an experimental research we obtain a number of values of the variant corresponding to an equal number of values of the variable2 but all the numbers are affected by more or less error1 and the values of the variable will often be irregularly disposed4 ,even if the numbers were absolutely correct and disposed at regular intervals1 there is1 as we have seen1 no direct mode of discovering the law1 but the difficulty of discovery is much increased by the uncertainty and irregularity of the results4 ,under such circumstances1 the best mode of proceeding is to prepare a paper divided into equal rectangular spaces1 a convenient size for the spaces being one-tenth of an inch square4 ,the values of the variable being marked off on the lowest horizontal line1 a point is marked for each corresponding value of the variant perpendicularly above that of the variable1 and at such a height as corresponds to the value of the variant4 ,the exact scale of the drawing is not of much importance1 but it may require to be adjusted according to circumstances1 and different values must often be attributed to the upright and horizontal divisions1 so as to make the variations conspicuous but not excessive4 ,if a curved line be drawn through all the points or ends of the ordinates1 it will probably exhibit irregular inflections1 owing to the errors which affect the numbers4 ,but1 when the results are numerous1 it becomes apparent which results are more divergent than others1 and guided by a so-called 99sense of continuity991 it is possible to trace a line among the points which will approximate to the true law more nearly than the points themselves4 ,the accompanying figure sufficiently explains itself4 ,illustration ,perkins employed this graphical method with much care in exhibiting the results of his experiments on the compression of water4#djg ,the numerical results were marked upon a sheet of paper very exactly ruled at intervals of one-tenth of an inch1 and the original marks were left in order that the reader might judge of the correctness of the curve drawn1 or choose another for himself4 ,regnault carried the method to perfection by laying off the points with a screw dividing engine2#djh and he then formed a table of results by drawing a continuous curve1 and measuring its height for equidistant values of the variable4 ,not only does a curve drawn in this manner enable us to infer numerical results more free from accidental errors than any of the numbers obtained directly from experiment1 but the form of the curve sometimes indicates the class of functions to which our results belong4 #djg 99,philosophical ,transactions991 #ahbf1 p4 #edd4 #djh ,jamin1 99,cours de ,physique991 vol4 ii4 p4 #bd1 c4 ,engraved sheets of paper prepared for the drawing of curves may be obtained from ,mr4 ,stanford at ,charing ,cross1 ,messrs4 ,w4 and ,a4 ,k4 ,johnston1 of ,london and ,edinburgh1 ,waterlow and ,sons1 ,letts and ,co41 and probably other publishers4 ,when we do not require great accuracy1 paper ruled by the common machine-ruler into equal squares of about one-fifth or one-sixth of an inch square will serve well enough4 ,i have met with engineers' and surveyors' memorandum books ruled with one-twelfth inch squares4 ,when a number of curves have to be drawn1 ,i have found it best to rule a good sheet of drawing paper with lines carefully adjusted at the most convenient distances1 and then to prick the points of the curve through it upon another sheet fixed underneath4 ,in this way we obtain an accurate curve upon a blank sheet1 and need only introduce such division lines as are requisite to the understanding of the curve4 ,in some cases our numerical results will correspond1 not to the height of single ordinates1 but to the area of the curve between two ordinates1 or the average height of ordinates between certain limits4 ,if we measure1 for instance1 the quantities of heat absorbed by water when raised in temperature from #j to #e1 from #e to #aj1 and so on1 these quantities will really be represented by 99areas99 of the curve denoting the specific heat of water2 and since the specific heat varies continuously between every two points of temperature1 we shall not get the correct curve by simply laying off the quantities of heat at the mean temperatures1 namely #b-a_/b1 and #g-a_/b1 and so on4 ,lord ,rayleigh has shown that if we have drawn such an incorrect curve1 we can with little trouble correct it by a simple geometrical process1 and obtain to a close approximation the true ordinates instead of those denoting areas4#dji #dji ,j4 ,w4 ,strutt1 99,on a correction sometimes required in curves professing to represent the connexion between two physical magnitudes994 ,philosophical ,magazine1 #dth ,series1 vol4 xlii4 p4 #dda4 99,interpolation and ,extrapolation499 ,when we have by experiment obtained two or more numerical results1 and endeavour1 without further experiment1 to calculate intermediate results1 we are said to 99interpolate994 ,if we wish to assign by reasoning results lying beyond the limits of experiment1 we may be said1 using an expression of ,sir ,george ,airy1 to 99extrapolate994 ,these two operations are the same in principle1 but differ in practicability4 ,it is a matter of great scientific importance to apprehend precisely how far we can practise interpolation or extrapolation1 and on what grounds we proceed4 ,in the first place1 if the interpolation is to be more than empirical1 we must have not only the experimental results1 but the laws which they obey--we must in fact go through the complete process of scientific investigation4 ,having discovered the laws of nature applying to the case1 and verified them by showing that they agree with the experiments in question1 we are then in a position to anticipate the results of similar experiments4 ,our knowledge even now is not certain1 because we cannot completely prove the truth of any assumed law1 and we cannot possibly exhaust all the circumstances which may affect the result4 ,at the best then our interpolations will partake of the want of certainty and precision attaching to all our knowledge of nature4 ,yet1 having the supposed laws1 our results will be as sure and accurate as any we can attain to4 ,but such a complete procedure is more than we commonly mean by interpolation1 which usually denotes some method of estimating in a merely approximate manner the results which might have been expected independently of a theoretical investigation4 ,regarded in this light1 interpolation is in reality an indeterminate problem4 ,from given values of a function it is impossible to determine that function2 for we can invent an infinite number of functions which will give those values if we are not restricted by any conditions1 just as through a given series of points we can draw an infinite number of curves1 if we may diverge between or beyond the points into bends and cusps as we think fit4#daj ,in interpolation we must in fact be guided more or less by 99 priori99 considerations2 we must know1 for instance1 whether or not periodical fluctuations are to be expected4 ,supposing that the phenomenon is non-periodic1 we proceed to assume that the function can be expressed in a limited series of the powers of the variable4 ,the number of powers which can be included depends upon the number of experimental results available1 and must be at least one less than this number4 ,by processes of calculation1 which have been already alluded to in the section on empirical formul1 we then calculate the coefficients of the powers1 and obtain an empirical formula which will give the required intermediate results4 ,in reality1 then1 we return to the methods treated under the head of approximation and empirical formul2 and interpolation1 as commonly understood1 consists in assuming that a curve of simple character is to pass through certain determined points4 ,if we have1 for instance1 two experimental results1 and only two1 we assume that the curve is a straight line2 for the parabolas which can be passed through two points are infinitely various in magnitude1 and quite indeterminate4 ,one straight line alone can pass through two points1 and it will have an equation of the form1 99y99 99mx99 99n991 the constant quantities of which can be determined from two results4 ,thus1 if the two values for 99x991 #g and #aa1 give the values for 99y991 #ce and #ec1 the solution of two equations gives 99y99 #de 99x99 #ce as the equation1 and for any other value of 99x991 for instance #aj1 we get a value of 99y991 that is #dhe4 ,when we take a mean value of 99x991 namely #i1 this process yields a simple mean result1 namely #dd4 ,three experimental results being given1 we assume that they fall upon a portion of a parabola and algebraic calculation gives the position of any intermediate point upon the parabola4 ,concerning the process of interpolation as practised in the science of meteorology the reader will find some directions in the ,french edition of ,kamtz's ,meteorology4#daa #daj ,herschel3 ,lacroix' 99,differential ,calculus991 p4 #eea4 #daa 99,cours complet de ,mtorologie991 ,note ,a1 p4 #ddi4 ,when we have1 either by direct experiment or by the use of a curve1 a series of values of the variant for equidistant values of the variable1 it is instructive to take the differences between each value of the variant and the next1 and then the differences between those differences1 and so on4 ,if any series of differences approaches closely to zero it is an indication that the numbers may be correctly represented by a finite empirical formula2 if the 99n99th differences are zero1 then the formula will contain only the first 99n99 - #a powers of the variable4 ,indeed we may sometimes obtain by the calculus of differences a correct empirical formula2 for if 99p99 be the first term of the series of values1 and 99p991 #b99p991 #c99p991 be the first number in each column of differences1 then the 99m99th term of the series of values will be 99p99 99m9999p99 99m99799m99 - #a7_/bb99p99 99m99799m99 - #a7_/b799m99 - #b7_/cc99p99 c4 ,a closely equivalent but more practicable formula for interpolation by differences1 as devised by ,lagrange1 will be found in ,thomson and ,tait's 99,elements of ,natural ,philosophy991 p4 #aae4 ,if no column of differences shows any tendency to become zero throughout1 it is an indication that the law is of a more complicated1 for instance of an exponential character1 so that it requires different treatment4 ,dr4 ,j4 ,hopkinson has suggested a method of arithmetical interpolation1#dab which is intended to avoid much that is arbitrary in the graphical method4 ,his process will yield the same results in all hands4 #dab 99,on the ,calculation of ,empirical ,formul4 ,the ,messenger of ,mathematics991 ,new ,series1 ,no4 #ag1 #ahgb4 ,so far as we can infer the results likely to be obtained by variations beyond the limits of experiment1 we must proceed upon the same principles4 ,if possible we must detect the exact laws in action1 and then trust to them as a guide when we have no experience4 ,if not1 an empirical formula of the same character as those employed in interpolation is our only resource4 ,but to extend our inference far beyond the limits of experience is exceedingly unsafe4 ,our knowledge is at the best only approximate1 and takes no account of small tendencies4 ,now it usually happens that tendencies small within our limits of observation become perceptible or great under extreme circumstances4 ,when the variable in our empirical formula is small1 we are justified in overlooking the higher powers1 and taking only two or three lower powers4 ,but as the variable increases1 the higher powers gain in importance1 and in time yield the principal part of the value of the function4 ,this is no mere theoretical inference4 ,excepting the few primary laws of nature1 such as the law of gravity1 of the conservation of energy1 c41 there is hardly any natural law which we can trust in circumstances widely different from those with which we are practically acquainted4 ,from the expansion or contraction1 fusion or vaporisation of substances by heat at the surface of the earth1 we can form a most imperfect notion of what would happen near the centre of the earth1 where the pressure almost infinitely exceeds anything possible in our experiments4 ,the physics of the earth give us a feeble1 and probably a misleading1 notion of a body like the sun1 in which an inconceivably high temperature is united with an inconceivably high pressure4 ,if there are in the realms of space nebul consisting of incandescent and unoxidised vapours of metals and other elements1 so highly heated perhaps that chemical composition is out of the question1 we are hardly able to treat them as subjects of scientific inference4 ,hence arises the great importance of experiments in which we investigate the properties of substances under extreme circumstances of cold or heat1 density or rarity1 intense electric excitation1 c4 ,this insecurity in extending our inferences arises from the approximate character of our measurements4 ,had we the power of appreciating infinitely small quantities1 we should by the principle of continuity discover some trace of every change which a substance could undergo under unattainable circumstances4 ,by observing1 for instance1 the tension of aqueous vapour between #j and #ajj ,c41 we ought theoretically to be able to infer its tension at every other temperature2 but this is out of the question practically because we cannot really ascertain the law precisely between those temperatures4 ,many instances might be given to show that laws which appear to represent correctly the results of experiments within certain limits altogether fail beyond those limits4 ,the experiments of ,roscoe and ,dittmar1 on the absorption of gases in water#dac afford interesting illustrations1 especially in the case of hydrochloric acid1 the quantity of which dissolved in water under different pressures follows very closely a linear law of variation1 from which however it diverges widely at low pressures4#dad ,herschel1 having deduced from observations of the double star ,virginis an elliptic orbit for the motion of one component round the centre of gravity of both1 found that for a time the motion of the star agreed very well with this orbit4 ,nevertheless divergence began to appear and after a time became so great that an entirely new orbit1 of more than double the dimensions of the old one1 had ultimately to be adopted4#dae #dac ,watts' 99,dictionary of ,chemistry991 vol4 ii4 p4 #gij4 #dad 99,quarterly ,journal of the ,chemical ,society991 vol4 viii4 p4 #ae4 #dae 99,results of ,observations at the ,cape of ,good ,hope991 p4 #bic4 99,illustrations of ,empirical ,quantitative ,laws499 ,although our object in quantitative inquiry is to discover the exact or rational formul1 expressing the laws which apply to the subject1 it is instructive to observe in how many important branches of science1 no precise laws have yet been detected4 ,the tension of aqueous vapour at different temperatures has been determined by a succession of eminent experimentalists--,dalton1 ,kamtz1 ,dulong1 ,arago1 ,magnus1 and ,regnault--and by the last mentioned the measurements were conducted with extraordinary care4 ,yet no incontestable general law has been established4 ,several functions have been proposed to express the elastic force of the vapour as depending on the temperature4 ,the first form is that of ,young1 namely ,f 799a99 99b t997m1 in which 99a991 99b991 and 99m99 are unknown quantities to be determined by observation4 ,roche proposed1 on theoretical grounds1 a complicated formula of an exponential form1 and a third form of function is that of ,biot1#daf as follows--log ,f 99a99 99b99t 99c99t4 ,i mention these formul1 because they well illustrate the feeble powers of empirical inquiry4 ,none of the formul can be made to correspond closely with experimental results1 and the two last forms correspond almost equally well4 ,there is very little probability that the real law has been reached1 and it is unlikely that it will be discovered except by deduction from mechanical theory4 #daf ,jamin1 99,cours de ,physique991 vol4 ii4 p4 #ach4 ,much ingenious labour has been spent upon the discovery of some general law of atmospheric refraction4 ,tycho ,brahe and ,kepler commenced the inquiry3 ,cassini first formed a table of refractions1 calculated on theoretical grounds3 ,newton entered into some profound investigations upon the subject3 ,brooke ,taylor1 ,bouguer1 ,simpson1 ,bradley1 ,mayer1 and ,kramp successively attacked the question1 which is of the highest practical importance as regards the correction of astronomical observations4 ,laplace next laboured on the subject without exhausting it1 and ,brinkley and ,ivory have also treated it4 ,the true law is yet undiscovered4 ,a closely connected problem1 that regarding the relation between the pressure and elevation in different strata of the atmosphere1 has received the attention of a long succession of physicists and was most carefully investigated by ,laplace4 ,yet no invariable and general law has been detected4 ,the same may be said concerning the law of human mortality2 abundant statistics on this subject are available1 and many hypotheses more or less satisfactory have been put forward as to the form of the curve of mortality1 but it seems to be impossible to discover more than an approximate law4 ,it may perhaps be urged that in such subjects no single invariable law can be expected4 ,the atmosphere may be divided into several variable strata which by their unconnected changes frustrate the exact calculations of astronomers4 ,human life may be subject at different ages to a succession of different influences incapable of reduction under any one law4 ,the results observed may in fact be aggregates of an immense number of separate results each governed by its own separate laws1 so that the subjects may be complicated beyond the possibility of complete resolution by empirical methods4 ,this is certainly true of the mathematical functions which must some time or other be introduced into the science of political economy4 99,simple ,proportional ,variation499 ,when we first treat numerical results in any novel kind of investigation1 our impression will probably be that one quantity varies in 99simple proportion99 to another1 so as to obey the law 99y99 99mx99 99n994 ,we must learn to distinguish carefully between the cases where this proportionality is really1 and where it is only apparently true4 ,in considering the principles of approximation we found that a small portion of any curve will appear to be a straight line4 ,when our modes of measurement are comparatively rude1 we must expect to be unable to detect the curvature4 ,kepler made meritorious attempts to discover the law of refraction1 and he approximated to it when he observed that the angles of incidence and refraction 99if small99 bear a constant ratio to each other4 ,angles when small are nearly as their sines1 so that he reached an approximate result of the true law4 ,cardan assumed1 probably as a mere guess1 that the force required to sustain a body on an inclined plane was simply proportional to the angle of elevation of the plane4 ,this is approximately the case when the angle is small1 but in reality the law is much more complicated1 the power required being proportional to the sine of the angle4 ,the early thermometer-makers were unaware whether the expansion of mercury was proportional or not to the heat communicated to it1 and it is only in the present century that we have learnt it to be not so4 ,we now know that even gases obey the law of uniform expansion by heat only in an approximate manner4 ,until reason to the contrary is shown1 we should do well to look upon every law of simple proportion as only provisionally true4 ,nevertheless many important laws of nature are in the form of simple proportions4 ,wherever a cause acts in independence of its previous effects1 we may expect this relation4 ,an accelerating force acts equally upon a moving and a motionless body4 ,hence the velocity produced is in simple proportion to the force1 and to the duration of its uniform action4 ,as gravitating bodies never interfere with each other's gravity1 this force is in direct simple proportion to the mass of each of the attracting bodies1 the mass being measured by1 or proportional to inertia4 ,similarly1 in all cases of 8direct unimpeded action10 as ,herschel has remarked1#dag we may expect simple proportion to manifest itself4 ,in such cases the equation expressing the relation may have the simple form 99y99 99mx994 #dag 99,preliminary ,discourse991 c41 p4 #aeb4 ,a similar relation holds true when there is conversion of one substance or form of energy into another4 ,the quantity of a compound is equal to the quantity of the elements which combine4 ,the heat produced in friction is exactly proportional to the mechanical energy absorbed4 ,it was experimentally proved by ,faraday that 8the chemical power of the current of electricity is in direct proportion to the quantity of electricity which passes40 ,when an electric current is produced1 the quantity of electric energy is simply proportional to the weight of metal dissolved4 ,if electricity is turned into heat1 there is again simple proportion4 ,wherever1 in fact1 one thing is but another thing with a new aspect1 we may expect to find the law of simple proportion4 ,but it is only in the most elementary cases that this simple relation will hold true4 ,simple conditions do not1 generally speaking1 produce simple results4 ,the planets move in approximate circles round the sun1 but the apparent motions1 as seen from the earth1 are very various4 ,all those motions1 again1 are summed up in the law of gravity1 of no great complexity2 yet men never have been1 and never will be1 able to exhaust the complications of action and reaction arising from that law1 even among a small number of planets4 ,we should be on our guard against a tendency to assume that the connection of cause and effect is one of direct proportion4 ,bacon reminds us of the woman in sop's fable1 who expected that her hen1 with a double measure of barley1 would lay two eggs a day instead of one1 whereas it grew fat1 and ceased to lay any eggs at all4 ,it is a wise maxim that the half is often better than the whole4 ,,chapter ,,xxiii4 ,,the ,,use ,,of ,,hypothesis4 ,if the views upheld in this work be correct1 all inductive investigation consists in the marriage of hypothesis and experiment4 ,when facts are in our possession1 we frame an hypothesis to explain their relations1 and by the success of this explanation is the value of the hypothesis to be judged4 ,in the invention and treatment of such hypotheses1 we must avail ourselves of the whole body of science already accumulated1 and when once we have obtained a probable hypothesis1 we must not rest until we have verified it by comparison with new facts4 ,we must endeavour by deductive reasoning to anticipate such phenomena1 especially those of a singular and exceptional nature1 as would happen if the hypothesis be true4 ,out of the infinite number of experiments which are possible1 theory must lead us to select those critical ones which are suitable for confirming or negativing our anticipations4 ,this work of inductive investigation cannot be guided by any system of precise and infallible rules1 like those of deductive reasoning4 ,there is1 in fact1 nothing to which we can apply rules of method1 because the laws of nature must be in our possession before we can treat them4 ,if there were any rule of inductive method1 it would direct us to make an exhaustive arrangement of facts in all possible orders4 ,given the specimens in a museum1 we might arrive at the best classification by going systematically through all possible classifications1 and1 were we endowed with infinite time and patience1 this would be an effective method4 ,it is the method by which the first simple steps are taken in an incipient branch of science4 ,before the dignified name of science is applicable1 some coincidences will force themselves upon the attention4 ,before there was a science of meteorology observant persons learned to associate clearness of the atmosphere with coming rain1 and a colourless sunset with fine weather4 ,knowledge of this kind is called 99empirical991 as seeming to come directly from experience2 and there is a considerable portion of knowledge which bears this character4 ,we may be obliged to trust to the casual detection of coincidences in those branches of knowledge where we are deprived of the aid of any guiding notions2 but a little reflection will show the utter insufficiency of haphazard experiment1 when applied to investigations of a complicated nature4 ,at the best1 it will be the simple identity1 or partial identity1 of classes1 as illustrated in pages #abg or #acd1 which can be thus detected4 ,it was pointed out that1 even when a law of nature involves only two circumstances1 and there are one hundred distinct circumstances which may possibly be connected1 there will be no less than #d1iej pairs of circumstances between which coincidence may exist4 ,when a law involves three or more circumstances1 the possible number of relations becomes vastly greater4 ,when considering the subject of combinations and permutations1 it became apparent that we could never cope with the possible variety of nature4 ,an exhaustive examination of the possible metallic alloys1 or chemical compounds1 was found to be out of the question 7p4 #aia74 ,it is on such considerations that we can explain the very small additions made to our knowledge by the alchemists4 ,many of them were men of the greatest acuteness1 and their indefatigable labours were pursued through many centuries4 ,a few things were discovered by them1 but a true insight into nature1 now enables chemists to discover more useful facts in a year than were yielded by the alchemists during many centuries4 ,there can be no doubt that ,newton was an alchemist1 and that he often laboured night and day at alchemical experiments4 ,but in trying to discover the secret by which gross metals might be rendered noble1 his lofty powers of deductive investigation were wholly useless4 ,deprived of all guiding clues1 his experiments were like those of all the alchemists1 purely tentative and haphazard4 ,while his hypothetical and deductive investigations have given us the true system of the ,universe1 and opened the way in almost all the great branches of natural philosophy1 the whole results of his tentative experiments are comprehended in a few happy guesses1 given in his celebrated 8,queries40 ,even when we are engaged in apparently passive observation of a phenomenon1 which we cannot modify experimentally1 it is advantageous that our attention should be guided by theoretical anticipations4 ,a phenomenon which seems simple is1 in all probability1 really complex1 and unless the mind is actively engaged in looking for particular details1 it is likely that the critical circumstances will be passed over4 ,bessel regretted that no distinct theory of the constitution of comets had guided his observations of ,halley's comet2#dah in attempting to verify or refute a hypothesis1 not only would there be a chance of establishing a true theory1 but if confuted1 the confutation would involve a store of useful observations4 #dah ,tyndall1 99,on ,cometary ,theory991 ,philosophical ,magazine1 ,april #ahfi4 #dth ,series1 vol4 xxxvii4 p4 #bdc4 ,it would be an interesting work1 but one which ,i cannot undertake1 to trace out the gradual reaction which has taken place in recent times against the purely empirical or ,baconian theory of induction4 ,francis ,bacon1 seeing the futility of the scholastic logic1 which had long been predominant1 asserted that the accumulation of facts and the orderly abstraction of axioms1 or general laws from them1 constituted the true method of induction4 ,even ,bacon was not wholly unaware of the value of hypothetical anticipation4 ,in one or two places he incidentally acknowledges it1 as when he remarks that the subtlety of nature surpasses that of reason1 adding that 8axioms abstracted from particular facts in a careful and orderly manner1 readily suggest and mark out new particulars40 ,nevertheless ,bacon's method1 as far as we can gather the meaning of the main portions of his writings1 would correspond to the process of empirically collecting facts and exhaustively classifying them1 to which ,i alluded4 ,the value of this method may be estimated historically by the fact that it has not been followed by any of the great masters of science4 ,whether we look to ,galileo1 who preceded ,bacon1 to ,gilbert1 his contemporary1 or to ,newton and ,descartes1 ,leibnitz and ,huyghens1 his successors1 we find that discovery was achieved by the opposite method to that advocated by ,bacon4 ,throughout ,newton's works1 as ,i shall show1 we find deductive reasoning wholly predominant1 and experiments are employed1 as they should be1 to confirm or refute hypothetical anticipations of nature4 ,in my 8,elementary ,lessons in ,logic0 7p4 #beh71 ,i stated my belief that there was no kind of reference to ,bacon in ,newton's works4 ,i have since found that ,newton does once or twice employ the expression 99experimentum crucis99 in his 8,opticks10 but this is the only expression1 so far as ,i am aware1 which could indicate on the part of ,newton direct or indirect acquaintance with ,bacon's writings4#dai #dai ,see 99,philosophical ,transactions991 abridged by ,lowthorp4 #dth edit4 vol4 i4 p4 #acj4 ,i find that opinions similar to those in the text have been briefly expressed by ,de ,morgan in his remarkable preface to 99,from ,matter to ,spirit991 by ,c4,d41 pp4 xxi4 xxii4 ,other great physicists of the same age were equally prone to the use of hypotheses rather than the blind accumulation of facts in the ,baconian manner4 ,hooke emphatically asserts in his posthumous work on ,philosophical ,method1 that the first requisite of the ,natural ,philosopher is readiness at guessing the solution of phenomena and making queries4 8,he ought to be very well skilled in those several kinds of philosophy already known1 to understand their several hypotheses1 suppositions1 collections1 observations1 c41 their various ways of ratiocinations and proceedings1 the several failings and defects1 both in their way of raising and in their way of managing their several theories3 for by this means the mind will be somewhat more ready at guessing at the solution of many phenomena almost at first sight1 and thereby be much more prompt at making queries1 and at tracing the subtlety of ,nature1 and in discovering and searching into the true reason of things40 ,we find ,horrocks1 again1 than whom no one was more filled with the scientific spirit1 telling us how he tried theory after theory in order to discover one which was in accordance with the motions of ,mars4#dbj ,huyghens1 who possessed one of the most perfect philosophical intellects1 followed the deductive process combined with continual appeal to experiment1 with a skill closely analogous to that of ,newton4 ,as to ,descartes and ,leibnitz1 they fell into excess in the use of hypothesis1 since they sometimes adopted hypothetical reasoning to the exclusion of experimental verification4 ,throughout the eighteenth century science was supposed to be advancing by the pursuance of the ,baconian method1 but in reality hypothetical investigation was the main instrument of progress4 ,it is only in the present century that physicists began to recognise this truth4 ,so much opprobrium had been attached by ,bacon to the use of hypotheses1 that we find ,young speaking of them in an apologetic tone4 8,the practice of advancing general principles and applying them to particular instances is so far from being fatal to truth in all sciences1 that when those principles are advanced on sufficient grounds1 it constitutes the essence of true philosophy20#dba and he quotes cases in which ,davy trusted to his theories rather than his experiments4 #dbj ,horrocks1 99,opera ,posthuma99 7#afgc71 p4 #bgf4 #dba ,young's 99,works991 vol4 i4 p4 #eic4 ,herschel1 who was both a practical physicist and an abstract logician1 entertained the deepest respect for ,bacon1 and made the 8,novum ,organum0 as far as possible the basis of his own admirable 99,discourse on the ,study of ,natural ,philosophy994 ,yet we find him in ,chapter ,,vii4 recognising the part which the formation and verification of theories takes in the higher and more general investigations of physical science4 ,j4 ,s4 ,mill carried on the reaction by describing the ,deductive ,method in which ratiocination1 that is deductive reasoning1 is employed for the discovery of new opportunities of testing and verifying an hypothesis4 ,nevertheless throughout the other parts of his system he inveighed against the value of the deductive process1 and even asserted that empirical inference from particulars to particulars is the true type of reasoning4 ,the irony of fate will probably decide that the most original and valuable part of ,mill's ,system of ,logic is irreconcilable with those views of the syllogism and of the nature of inference which occupy the main part of the treatise1 and are said to have effected a revolution in logical science4 ,mill would have been saved from much confusion of thought had he not failed to observe that the inverse use of deduction constitutes induction4 ,in later years ,professor ,huxley has strongly insisted upon the value of hypothesis4 ,when he advocates the use of 8working hypotheses0 he means no doubt that any hypothesis is better that none1 and that we cannot avoid being guided in our observations by some hypothesis or other4 ,professor ,tyndall's views as to the use of the ,imagination in the pursuit of ,science put the same truth in another light4 ,it ought to be pointed out that ,neil in his 99,art of ,reasoning991 a popular but able exposition of the principles of ,logic1 published in #ahec1 fully recognises in ,chapter ,,xi4 the value and position of hypothesis in the discovery of truth4 ,he endeavours to show1 too 7p4 #aji71 that ,francis ,bacon did not object to the use of hypothesis4 ,the true course of inductive procedure is that which has yielded all the more lofty results of science4 ,it consists in 99,anticipating ,nature991 in the sense of forming hypotheses as to the laws which are probably in operation2 and then observing whether the combinations of phenomena are such as would follow from the laws supposed4 ,the investigator begins with facts and ends with them4 ,he uses facts to suggest probable hypotheses2 deducing other facts which would happen if a particular hypothesis is true1 he proceeds to test the truth of his notion by fresh observations4 ,if any result prove different from what he expects1 it leads him to modify or to abandon his hypothesis2 but every new fact may give some new suggestion as to the laws in action4 ,even if the result in any case agrees with his anticipations1 he does not regard it as finally confirmatory of his theory1 but proceeds to test the truth of the theory by new deductions and new trials4 ,in such a process the investigator is assisted by the whole body of science previously accumulated4 ,he may employ analogy1 as ,i shall point out1 to guide him in the choice of hypotheses4 ,the manifold connections between one science and another give him clues to the kind of laws to be expected1 and out of the infinite number of possible hypotheses he selects those which are1 as far as can be foreseen at the moment1 most probable4 ,each experiment1 therefore1 which he performs is that most likely to throw light upon his subject1 and even if it frustrate his first views1 it tends to put him in possession of the correct clue4 99,requisites of a good ,hypothesis499 ,there is little difficulty in pointing out to what condition an hypothesis must conform in order to be accepted as probable and valid4 ,that condition1 as ,i conceive1 is the single one of enabling us to infer the existence of phenomena which occur in our experience4 99,agreement with fact is the sole and sufficient test of a true hypothesis499 ,hobbes has named two conditions which he considers requisite in an hypothesis1 namely 7#a7 ,that it should be conceivable and not absurd2 7#b7 ,that it should allow of phenomena being necessarily inferred4 ,boyle1 in noticing ,hobbes' views1 proposed to add a third condition1 to the effect that the hypothesis should not be inconsistent with any other truth on phenomenon of nature4#dbb ,i think that of these three conditions1 the first cannot be accepted1 unless by 99inconceivable99 and 99absurd99 we mean self-contradictory or inconsistent with the laws of thought and nature4 ,i shall have to point out that some satisfactory theories involve suppositions which are wholly 99inconceivable99 in a certain sense of the word1 because the mind cannot sufficiently extend its ideas to frame a notion of the actions supposed to take place4 ,that the force of gravity should act instantaneously between the most distant parts of the planetary system1 or that a ray of violet light should consist of about #gjj billions of vibrations in a second1 are statements of an inconceivable and absurd character in one sense2 but they are so far from being opposed to fact that we cannot on any other suppositions account for phenomena observed4 ,but if an hypothesis involve self-contradiction1 or is inconsistent with known laws of nature1 it is self-condemned4 ,we cannot even apply deductive reasoning to a self-contradictory notion2 and being opposed to the most general and certain laws known to us1 the primary laws of thought1 it thereby conspicuously fails to agree with facts4 ,since nature1 again1 is never self-contradictory1 we cannot at the same time accept two theories which lead to contradictory results4 ,if the one agrees with nature1 the other cannot4 ,hence if there be a law which we believe with high probability to be verified by observation1 we must not frame an hypothesis in conflict with it1 otherwise the hypothesis will necessarily be in disagreement with observation4 ,since no law or hypothesis is proved1 indeed1 with absolute certainty1 there is always a chance1 however slight1 that the new hypothesis may displace the old one2 but the greater the probability which we assign to that old hypothesis1 the greater must be the evidence required in favour of the new and conflicting one4 #dbb ,boyle's 99,physical ,examen991 p4 #hd4 ,i assert1 then1 that there is but one test of a good hypothesis1 namely1 99its conformity with observed facts992 but this condition may be said to involve three constituent conditions1 nearly equivalent to those suggested by ,hobbes and ,boyle1 namely3-- 7#a7 ,that it allow of the application of deductive reasoning and the inference of consequences capable of comparison with the results of observation4 7#b7 ,that it do not conflict with any laws of nature1 or of mind1 which we hold to be true4 7#c7 ,that the consequences inferred do agree with facts of observation4 99,possibility of ,deductive ,reasoning499 ,as the truth of an hypothesis is to be proved by its conformity with fact1 the first condition is that we be able to apply methods of deductive reasoning1 and learn what should happen according to such an hypothesis4 ,even if we could imagine an object acting according to laws hitherto wholly unknown it would be useless to do so1 because we could never decide whether it existed or not4 ,we can only infer what would happen under supposed conditions by applying the knowledge of nature we possess to those conditions4 ,hence1 as ,boscovich truly said1 we are to understand by hypotheses 8not fictions altogether arbitrary1 but suppositions conformable to experience or analogy40 ,it follows that every hypothesis worthy of consideration must suggest some likeness1 analogy1 or common law1 acting in two or more things4 ,if1 in order to explain certain facts1 99a991 99a991 99a991 c41 we invent a cause ,a1 then we must in some degree appeal to experience as to the mode in which ,a will act4 ,as the laws of nature are not known to the mind intuitively1 we must point out some other cause1 ,b1 which supplies the requisite notions1 and all we do is to invent a fourth term to an analogy4 ,as ,b is to its effects 99b991 99b991 99b991 c41 so is ,a to its effects 99a991 99a991 99a991 c4 ,when we attempt to explain the passage of light and heat radiations through space unoccupied by matter1 we imagine the existence of the so-called 99ether994 ,but if this ether were wholly different from anything else known to us1 we should in vain try to reason about it4 ,we must apply to it at least the laws of motion1 that is we must so far liken it to matter4 ,and as1 when applying those laws to the elastic medium air1 we are able to infer the phenomena of sound1 so by arguing in a similar manner concerning ether we are able to infer the existence of light phenomena corresponding to what do occur4 ,all that we do is to take an elastic substance1 increase its elasticity immensely1 and denude it of gravity and some other properties of matter1 but we must retain sufficient likeness to matter to allow of deductive calculations4 ,the force of gravity is in some respects an incomprehensible existence1 but in other respects entirely conformable to experience4 ,we observe that the force is proportional to mass1 and that it acts in entire independence of other matter which may be present or intervening4 ,the law of the decrease of intensity1 as the square of the distance increases1 is observed to hold true of light1 sound1 and other influences emanating from a point1 and spreading uniformly through space4 ,the law is doubtless connected with the properties of space1 and is so far in agreement with our necessary ideas4 ,it may be said1 however1 that no hypothesis can be so much as framed in the mind unless it be more or less conformable to experience4 ,as the material of our ideas is derived from sensation we cannot figure to ourselves any agent1 but as endowed with some of the properties of matter4 ,all that the mind can do in the creation of new existences is to alter combinations1 or the intensity of sensuous properties4 ,the phenomenon of motion is familiar to sight and touch1 and different degrees of rapidity are also familiar2 we can pass beyond the limits of sense1 and imagine the existence of rapid motion1 such as our senses could not observe4 ,we know what is elasticity1 and we can therefore in a way figure to ourselves elasticity a thousand or a million times greater than any which is sensuously known to us4 ,the waves of the ocean are many times higher than our own bodies2 other waves1 are many times less2 continue the proportion1 and we ultimately arrive at waves as small as those of light4 ,thus it is that the powers of mind enable us from a sensuous basis to reason concerning agents and phenomena different in an unlimited degree4 ,if no hypothesis then can be absolutely opposed to sense1 accordance with experience must always be a question of degree4 ,in order that an hypothesis may allow of satisfactory comparison with experience1 it must possess definiteness and in many cases mathematical exactness allowing of the precise calculation of results4 ,we must be able to ascertain whether it does or does not agree with facts4 ,the theory of vortices is an instance to the contrary1 for it did not present any mode of calculating the exact relations between the distances and periods of the planets and satellites2 it could not1 therefore1 undergo that rigorous testing to which ,newton scrupulously submitted his theory of gravity before its promulgation4 ,vagueness and incapability of precise proof or disproof often enable a false theory to live2 but with those who love truth1 vagueness should excite suspicion4 ,the upholders of the ancient doctrine of ,nature's abhorrence of a vacuum1 had been unable to anticipate the important fact that water would not rise more than #cc feet in a common suction pump4 ,nor when the fact was pointed out could they explain it1 except by introducing a special alteration of the theory to the effect that ,nature's abhorrence of a vacuum was limited to #cc feet4 99,consistency with the ,laws of ,nature499 ,in the second place an hypothesis must not be contradictory to what we believe to be true concerning ,nature4 ,it must not involve self-inconsistency which is opposed to the highest and simplest laws1 namely1 those of ,logic4 ,neither ought it to be irreconcilable with the simple laws of motion1 of gravity1 of the conservation of energy1 nor any parts of physical science which we consider to be established beyond reasonable doubt4 ,not that we are absolutely forbidden to entertain such an hypothesis1 but if we do so we must be prepared to disprove some of the best demonstrated truths in the possession of mankind4 ,the fact that conflict exists means that the consequences of the theory are not verified if previous discoveries are correct1 and we must therefore show that previous discoveries are incorrect before we can verify our theory4 ,an hypothesis will be exceedingly improbable1 not to say absurd1 if it supposes a substance to act in a manner unknown in other cases2 for it then fails to be verified in our knowledge of that substance4 ,several physicists1 especially ,euler and ,grove1 have supposed that we might dispense with an ethereal basis of light1 and infer from the interstellar passage of rays that there was a kind of rare gas occupying space4 ,but if so1 that gas must be excessively rare1 as we may infer from the apparent absence of an atmosphere around the moon1 and from other facts known to us concerning gases and the atmosphere2 yet it must possess an elastic force at least a billion times as great as atmospheric air at the earth's surface1 in order to account for the extreme rapidity of light rays4 ,such an hypothesis then is inconsistent with our knowledge concerning gases4 ,provided that there be no clear and absolute conflict with known laws of nature1 there is no hypothesis so improbable or apparently inconceivable that it may not be rendered probable1 or even approximately certain1 by a sufficient number of concordances4 ,in fact the two best founded and most successful theories in physical science involve the most absurd suppositions4 ,gravity is a force which appears to act between bodies through vacuous space2 it is in positive contradiction to the old dictum that nothing can act but through some medium4 ,it is even more puzzling that the force acts in perfect indifference to intervening obstacles4 ,light in spite of its extreme velocity shows much respect to matter1 for it is almost instantaneously stopped by opaque substances1 and to a considerable extent absorbed and deflected by transparent ones4 ,but to gravity all media are1 as it were1 absolutely transparent1 nay non-existent2 and two particles at opposite points of the earth affect each other exactly as if the globe were not between4 ,the action is1 so far as we can observe1 instantaneous1 so that every particle of the universe is at every moment in separate cognisance1 as it were1 of the relative position of every other particle throughout the universe at that same moment of time4 ,compared with such incomprehensible conditions1 the theory of vortices deals with commonplace realities4 ,newton's celebrated saying 99hypotheses non fingo991 bears the appearance of irony2 and it was not without apparent grounds that ,leibnitz and the continental philosophers charged ,newton with re-introducing occult powers and qualities4 ,the undulatory theory of light presents almost equal difficulties of conception4 ,we are asked by physical philosophers to give up our prepossessions1 and to believe that interstellar space which seems empty is not empty at all1 but filled with 99something99 immensely more solid and elastic than steel4 ,as ,young himself remarked1#dbc 8the luminiferous ether1 pervading all space1 and penetrating almost all substances1 is not only highly elastic1 but absolutely solid6660 ,herschel calculated the force which may be supposed1 according to the undulatory theory of light1 to be constantly exerted at each point in space1 and finds it to be #a1adh1jjj1jjj1jjj times the elastic force of ordinary air at the earth's surface1 so that the pressure of ether per square inch must be about seventeen billions of pounds4#dbd ,yet we live and move without appreciable resistance through this medium1 immensely harder and more elastic than adamant4 ,all our ordinary notions must be laid aside in contemplating such an hypothesis2 yet it is no more than the observed phenomena of light and heat force us to accept4 ,we cannot deny even the strange suggestion of ,young1 that there may be independent worlds1 some possibly existing in different parts of space1 but others perhaps pervading each other unseen and unknown in the same space4#dbe ,for if we are bound to admit the conception of this adamantine firmament1 it is equally easy to admit a plurality of such4 ,we see1 then1 that mere difficulties of conception must not discredit a theory which otherwise agrees with facts1 and we must only reject hypotheses which are inconceivable in the sense of breaking distinctly the primary laws of thought and nature4 #dbc ,young's 99,works991 vol4 i4 p4 #dae4 #dbd 99,familiar ,lectures on ,scientific ,subjects991 p4 #bhb4 #dbe ,young's 99,works991 vol4 i4 p4 #dag4 99,conformity with ,facts499 ,before we accept a new hypothesis it must be shown to agree not only with the previously known laws of nature1 but also with the particular facts which it is framed to explain4 ,assuming that these facts are properly established1 it must agree with all of them4 ,a single absolute conflict between fact and hypothesis1 is fatal to the hypothesis2 99falsa in uno1 falsa in omnibus994 ,seldom1 indeed1 shall we have a theory free from difficulties and apparent inconsistency with facts4 ,though one real inconsistency would overturn the most plausible theory1 yet there is usually some probability that the fact may be misinterpreted1 or that some supposed law of nature1 on which we are relying1 may not be true4 ,it may be expected1 moreover1 that a good hypothesis1 besides agreeing with facts already noticed1 will furnish us with distinct credentials by enabling us to anticipate deductively series of facts which are not already connected and accounted for by any equally probable hypothesis4 ,we cannot lay down any precise rule as to the number of accordances which can establish the truth of an hypothesis1 because the accordances will vary much in value4 ,while1 on the one hand1 no finite number of accordances will give entire certainty1 the probability of the hypothesis will increase very rapidly with the number of accordances4 ,almost every problem in science thus takes the form of a balance of probabilities4 ,it is only when difficulty after difficulty has been successfully explained away1 and decisive 99experimenta crucis99 have1 time after time1 resulted in favour of our theory1 that we can venture to assert the falsity of all objections4 ,the sole real test of an hypothesis is its accordance with fact4 ,descartes' celebrated system of vortices is exploded1 not because it was intrinsically absurd and inconceivable1 but because it could not give results in accordance with the actual motions of the heavenly bodies4 ,the difficulties of conception involved in the apparatus of vortices1 are child's play compared with those of gravitation and the undulatory theory already described4 ,vortices are on the whole plausible suppositions2 for planets and satellites bear at first sight much resemblance to objects carried round in whirlpools1 an analogy which doubtless suggested the theory4 ,the failure was in the first and third requisites2 for1 as already remarked1 the theory did not allow of precise calculation of planetary motions1 and was thus incapable of rigorous verification4 ,but so far as we can institute a comparison1 facts are entirely against the vortices4 ,newton did not ridicule the theory as absurd1 but showed#dbf that it was 8pressed with many difficulties40 ,he carefully pointed out that the ,cartesian theory was inconsistent with the laws of ,kepler1 and would represent the planets as moving more rapidly at their aphelia than at their perihelia4#dbg ,the rotatory motion of the sun and planets on their own axes is in striking conflict with the revolutions of the satellites carried round them2 and comets1 the most flimsy of bodies1 calmly pursue their courses in elliptic paths1 irrespective of the vortices which they pass through4 ,we may now also point to the interlacing orbits of the minor planets as a new and insuperable difficulty in the way of the ,cartesian ideas4 #dbf 99,principia991 bk4 iii4 ,prop4 #dc4 ,general ,scholium4 #dbg ,ibid4 bk4 ii4 ,sect4 ix4 ,prop4 #ec4 ,newton1 though he established the best of theories1 was also capable of proposing one of the worst2 and if we want an instance of a theory decisively contradicted by facts1 we have only to turn to his views concerning the origin of natural colours4 ,having analysed1 with incomparable skill1 the origin of the colours of thin plates1 he suggests that the colours of all bodies are determined in like manner by the size of their ultimate particles4 ,a thin plate of a definite thickness will reflect a definite colour2 hence1 if broken up into fragments it will form a powder of the same colour4 ,but1 if this be a sufficient explanation of coloured substances1 then every coloured fluid ought to reflect the complementary colour of that which it transmits4 ,colourless transparency arises1 according to ,newton1 from particles being too minute to reflect light2 but if so1 every black substance should be transparent4 ,newton himself so acutely felt this last difficulty as to suggest that true blackness is due to some internal refraction of the rays to and fro1 and an ultimate stifling of them1 which he did not attempt to explain further4 ,unless some other process comes into operation1 neither refraction nor reflection1 however often repeated1 will destroy the energy of light4 ,the theory therefore gives no account1 as ,brewster shows1 of #bd parts out of #be of the light which falls upon a black coal1 and the remaining part which is reflected from the lustrous surface is equally inconsistent with the theory1 because fine coal-dust is almost entirely devoid of reflective power4#dbh ,it is now generally believed that the colours of natural bodies are due to the unequal absorption of rays of light of different refrangibility4 #dbh ,brewster's 99,life of ,newton991 #ast edit4 chap4 vii4 99,experimentum ,crucis499 ,as we deduce more and more conclusions from a theory1 and find them verified by trial1 the probability of the theory increases in a rapid manner2 but we never escape the risk of error altogether4 ,absolute certainty is beyond the powers of inductive investigation1 and the most plausible supposition may ultimately be proved false4 ,such is the groundwork of similarity in nature1 that two very different conditions may often give closely similar results4 ,we sometimes find ourselves therefore in possession of two or more hypotheses which both agree with so many experimental facts as to have great appearance of truth4 ,under such circumstances we have need of some new experiment1 which shall give results agreeing with one hypothesis but not with the other4 ,any such experiment which decides between two rival theories may be called an 99,experimentum ,crucis991 an ,experiment of the ,finger ,post4 ,whenever the mind stands1 as it were1 at cross-roads and knows not which way to select1 it needs some decisive guide1 and ,bacon therefore assigned great importance and authority to instances which serve in this capacity4 ,the name given by ,bacon has become familiar2 it is almost the only one of ,bacon's figurative expressions which has passed into common use4 ,even ,newton1 as ,i have mentioned 7p4 #ejg71 used the name4 ,i do not think1 indeed1 that the common use of the word at all agrees with that intended by ,bacon4 ,herschel says that 8we make an experiment of the crucial kind when we form combinations1 and put in action causes from which some particular one shall be deliberately excluded1 and some other purposely admitted40#dbi ,this1 however1 seems to be the description of any special experiment not made at haphazard4 ,pascal's experiment of causing a barometer to be carried to the top of the ,puy-de-,dme has often been considered as a perfect 99experimentum crucis991 if not the first distinct one on record2#dcj but if so1 we must dignify the doctrine of ,nature's abhorrence of a vacuum with the position of a rival theory4 ,a crucial experiment must not simply confirm one theory1 but must negative another2 it must decide a mind which is in equilibrium1 as ,bacon says1#dca between two equally plausible views4 8,when in search of any nature1 the understanding comes to an equilibrium1 as it were1 or stands suspended as to which of two or more natures the cause of nature inquired after should be attributed or assigned1 by reason of the frequent and common occurrence of several natures1 then these ,crucial ,instances show the true and inviolable association of one of these natures to the nature sought1 and the uncertain and separable alliance of the other1 whereby the question is decided1 the former nature admitted for the cause1 and the other rejected4 ,these instances1 therefore1 afford great light1 and have a kind of overruling authority1 so that the course of interpretation will sometimes terminate in them1 or be finished by them40 #dbi 99,discourse on the ,study of ,natural ,philosophy991 p4 #aea4 #dcj ,ibid4 p4 #bbi4 #dca 99,novum ,organum991 bk4 ii4 ,aphorism #cf4 ,the long-continued strife between the ,corpuscular and ,undulatory theories of light forms the best possible illustration of an ,experimentum ,crucis4 ,it is remarkable in how plausible a manner both these theories agreed with the ordinary laws of geometrical optics1 relating to reflection and refraction4 ,according to the first law of motion a moving particle proceeds in a perfectly straight line1 when undisturbed by extraneous forces4 ,if the particle being perfectly elastic1 strike a perfectly elastic plane1 it will bound off in such a path that the angles of incidence and reflection will be equal4 ,now a ray of light proceeds in a straight line1 or appears to do so1 until it meets a reflecting body1 when its path is altered in a manner exactly similar to that of the elastic particle4 ,here is a remarkable correspondence which probably suggested to ,newton's mind the hypothesis that light consists of minute elastic particles moving with excessive rapidity in straight lines4 ,the correspondence was found to extend also to the law of simple refraction2 for if particles of light be supposed capable of attracting matter1 and being attracted by it at insensibly small distances1 then a ray of light1 falling on the surface of a transparent medium1 will suffer an increase in its velocity perpendicular to the surface1 and the law of sines is the consequence4 ,this remarkable explanation of the law of refraction had doubtless a very strong effect in leading ,newton to entertain the corpuscular theory1 and he appears to have thought that the analogy between the propagation of rays of light and the motion of bodies was perfectly exact1 whatever might be the actual nature of light4#dcb ,it is highly remarkable1 again1 that ,newton was able to give by his corpuscular theory1 a plausible explanation of the inflection of light as discovered by ,grimaldi4 ,the theory would indeed have been a very probable one could ,newton's own law of gravity have applied2 but this was out of the question1 because the particles of light1 in order that they may move in straight lines1 must be devoid of any influence upon each other4 #dcb 99,principia991 bk4 i4 ,sect4 xiv4 ,prop4 #if4 ,scholium4 99,opticks991 ,prop4 vi4 #crd edit4 p4 #gj4 ,the ,huyghenian or ,undulatory theory of light was also able to explain the same phenomena1 but with one remarkable difference4 ,if the undulatory theory be true1 light must move more slowly in a dense refracting medium than in a rarer one2 but the ,newtonian theory assumed that the attraction of the dense medium caused the particles of light to move more rapidly than in the rare medium4 ,on this point1 then1 there was complete discrepancy between the theories1 and observation was required to show which theory was to be preferred4 ,now by simply cutting a uniform plate of glass into two pieces1 and slightly inclining one piece so as to increase the length of the path of a ray passing through it1 experimenters were able to show that light does move more slowly in glass than in air4#dcc ,more recently ,fizeau and ,foucault independently measured the velocity of light in air and in water1 and found that the velocity is greater in air4#dcd #dcc ,airy's 99,mathematical ,tracts991 #crd edit4 pp4 #bhf-bhh4 #dcd ,jamin1 99,cours de ,physique991 vol4 iii4 p4 #cgb4 ,there are a number of other points at which experience decides against ,newton1 and in favour of ,huyghens and ,young4 ,laplace pointed out that the attraction supposed to exist between matter and the corpuscular particles of light would cause the velocity of light to vary with the size of the emitting body1 so that if a star were #bej times as great in diameter as our sun1 its attraction would prevent the emanation of light altogether4#dce ,but experience shows that the velocity of light is uniform1 and independent of the magnitude of the emitting body1 as it should be according to the undulatory theory4 ,lastly1 ,newton's explanation of diffraction or inflection fringes of colours was only 99plausible991 and not true2 for ,fresnel ascertained that the dimensions of the fringes are not what they would be according to ,newton's theory4 #dce ,young's 99,lectures on ,natural ,philosophy99 7#ahde71 vol4 i4 p4 #cfa4 ,although the ,science of ,light presents us with the most beautiful examples of crucial experiments and observations1 instances are not wanting in other branches of science4 ,copernicus asserted1 in opposition to the ancient ,ptolemaic theory1 that the earth moved round the sun1 and he predicted that if ever the sense of sight could be rendered sufficiently acute and powerful1 we should see phases in ,mercury and ,venus4 ,galileo with his telescope was able1 in #afaj to verify the prediction as regards ,venus1 and subsequent observations of ,mercury led to a like conclusion4 ,the discovery of the aberration of light added a new proof1 still further strengthened by the more recent determination of the parallax of fixed stars4 ,hooke proposed to prove the existence of the earth's diurnal motion by observing the deviation of a falling body1 an experiment successfully accomplished by ,benzenberg2 and ,foucault's pendulum has since furnished an additional indication of the same motion1 which is indeed also apparent in the trade winds4 ,all these are crucial facts in favour of the ,copernican theory4 99,descriptive ,hypotheses499 ,there are hypotheses which we may call 99descriptive hypotheses991 and which serve for little else than to furnish convenient names4 ,when a phenomenon is of an unusual kind1 we cannot even speak of it without using some analogy4 ,every word implies some resemblance between the thing to which it is applied1 and some other thing1 which fixes the meaning of the word4 ,if we are to speak of what constitutes electricity1 we must search for the nearest analogy1 and as electricity is characterised by the rapidity and facility of its movements1 the notion of a fluid of a very subtle character presents itself as appropriate4 ,there is the single-fluid and the double-fluid theory of electricity1 and a great deal of discussion has been uselessly spent upon them4 ,the fact is1 that if these theories be understood as more than convenient modes of describing the phenomena1 they are altogether invalid4 ,the analogy extends only to the rapidity of motion1 or rather the fact that a phenomenon occurs successively at different points of the body4 ,the so-called electric fluid adds nothing to the weight of the conductor1 and to suppose that it really consists of particles of matter is even more absurd than to reinstate the corpuscular theory of light4 ,a far closer analogy exists between electricity and light undulations1 which are about equally rapid in propagation4 ,we shall probably continue for a long time to talk of the 99electric fluid991 but there can be no doubt that this expression represents merely a phase of molecular motion1 a wave of disturbance4 ,the invalidity of these fluid theories is shown moreover in the fact that they have not led to the invention of a single new experiment4 ,among these merely descriptive hypotheses ,i should place ,newton's theory of ,fits of ,easy ,reflection and ,refraction4 ,that theory did not do more than describe what took place4 ,it involved no analogy to other phenomena of nature1 for ,newton could not point to any other substance which went through these extraordinary fits4 ,we now know that the true analogy would have been waves of sound1 of which ,newton had acquired in other respects so complete a comprehension4 ,but though the notion of interference of waves had distinctly occurred to ,hooke1 ,newton failed to see how the periodic phenomena of light could be connected with the periodic character of waves4 ,his hypothesis fell because it was out of analogy with everything else in nature1 and it therefore did not allow him1 as in other cases1 to descend by mathematical deduction to consequences which could be verified or refuted4 ,we are at freedom to imagine the existence of a new agent1 and to give it an appropriate name1 provided there are phenomena incapable of explanation from known causes4 ,we may speak of 99vital force99 as occasioning life1 provided that we do not take it to be more than a name for an undefined something giving rise to inexplicable facts1 just as the ,french chemists called ,iodine the ,substance ,x1 so long as they were unaware of its real character and place in chemistry4#dcf ,encke was quite justified in speaking of the 99resisting medium99 in space so long as the retardation of his comet could not be otherwise accounted for4 ,but such hypotheses will do much harm whenever they divert us from attempts to reconcile the facts with known laws1 or when they lead us to mix up discrete things4 ,because we speak of vital force we must not assume that it is a really existing physical force like electricity2 we do not know what it is4 ,we have no right to confuse ,encke's supposed resisting medium with the basis of light without distinct evidence of identity4 ,the name protoplasm1 now so familiarly used by physiologists1 is doubtless legitimate so long as we do not mix up different substances under it1 or imagine that the name gives us any knowledge of the obscure origin of life4 ,to name a substance protoplasm no more explains the infinite variety of forms of life which spring out of the substance1 than does the 99vital force99 which may be supposed to reside in the protoplasm4 ,both expressions are mere names for an inexplicable series of causes which out of apparently similar conditions produce the most diverse results4 #dcf ,paris1 99,life of ,davy991 p4 #bgd4 ,hardly to be distinguished from descriptive hypotheses are certain imaginary objects which we frame for the ready comprehension of a subject4 ,the mathematician1 in treating abstract questions of probability1 finds it convenient to represent the conditions by a concrete hypothesis in the shape of a ballot-box4 ,poisson proved the principle of the inverse method of probabilities by imagining a number of ballot-boxes to have their contents mixed in one great ballot-box 7p4 #bdd74 ,many such devices are used by mathematicians4 ,the ,ptolemaic theory of 99cycles99 and 99epi-cycles99 was no grotesque and useless work of the imagination1 but a perfectly valid mode of analysing the motions of the heavenly bodies2 in reality it is used by mathematicians at the present day4 ,newton employed the pendulum as a means of representing the nature of an undulation4 ,centres of gravity1 oscillation1 c41 poles of the magnet1 lines of force1 are other imaginary existences employed to assist our thoughts 7p4 #cfd74 ,such devices may be called 99,representative ,hypotheses991 and they are only permissible so far as they embody analogies4 ,their further consideration belongs either to the subject of ,analogy1 or to that of language and representation1 founded upon analogy4 ,,chapter ,,xxiv4 ,,empirical ,,knowledge1 ,,explanation1 ,,and ,,prediction4 ,inductive investigation1 as we have seen1 consists in the union of hypothesis and experiment1 deductive reasoning being the link by which experimental results are made to confirm or confute the hypothesis4 ,now when we consider this relation between hypothesis and experiment it is obvious that we may classify our knowledge under four heads4 7#a7 ,we may be acquainted with facts which have not yet been brought into accordance with any hypothesis4 ,such facts constitute what is called 99,empirical ,knowledge994 7#b7 ,another extensive portion of our knowledge consists of facts which having been first observed empirically1 have afterwards been brought into accordance with other facts by an hypothesis concerning the general laws applying to them4 ,this portion of our knowledge may be said to be 99explained991 99reasoned991 or 99generalised994 7#c7 ,in the third place comes the collection of facts1 minor in number1 but most important as regards their scientific interest1 which have been anticipated by theory and afterwards verified by experiment4 7#d7 ,lastly1 there exists knowledge which is accepted solely on the ground of theory1 and is incapable of experimental confirmation1 at least with the instrumental means in our possession4 ,it is a work of much interest to compare and illustrate the relative extent and value of these four groups of knowledge4 ,we shall observe that as a general rule a great branch of science originates in facts observed accidentally1 or without distinct consciousness of what is to be expected4 ,as a science progresses1 its power of foresight rapidly increases1 until the mathematician in his library acquires the power of anticipating nature1 and predicting what will happen in circumstances which the eye of man has never examined4 99,empirical ,knowledge499 ,by empirical knowledge we mean such as is derived directly from the examination of detached facts1 and rests entirely on those facts1 without corroboration from other branches of knowledge4 ,it is contrasted with generalised and theoretical knowledge1 which embraces many series of facts under a few comprehensive principles1 so that each series serves to throw light upon each other series of facts4 ,just as1 in the map of a half-explored country1 we see detached bits of rivers1 isolated mountains1 and undefined plains1 not connected into any complete plan1 so a new branch of knowledge consists of groups of facts1 each group standing apart1 so as not to allow us to reason from one to another4 ,before the time of ,descartes1 and ,newton1 and ,huyghens1 there was much empirical knowledge of the phenomena of light4 ,the rainbow had always struck the attention of the most careless observers1 and there was no difficulty in perceiving that its conditions of occurrence consisted in rays of the sun shining upon falling drops of rain4 ,it was impossible to overlook the resemblance of the ordinary rainbow to the comparatively rare lunar rainbow1 to the bow which appears upon the spray of a waterfall1 or even upon beads of dew suspended on grass and spiders' webs4 ,in all these cases the uniform conditions are rays of light and round drops of water4 ,roger ,bacon had noticed these conditions1 as well as the analogy of the rainbow colours to those produced by crystals4#dcg ,but the knowledge was empirical until ,descartes and ,newton showed how the phenomena were connected with facts concerning the refraction of light4 #dcg 99,opus ,majus499 ,edit4 #agcc4 ,cap4 x4 p4 #dfj4 ,there can be no better instance of an empirical truth than that detected by ,newton concerning the high refractive powers of combustible substances4 ,newton's chemical notions were almost as vague as those prevalent in his day1 but he observed that certain 8fat1 sulphureous1 unctuous bodies10 as he calls them1 such as camphor1 oils spirit of turpentine1 amber1 c41 have refractive powers two or three times greater than might be anticipated from their densities4#dch ,the enormous refractive index of diamond1 led him with great sagacity to regard this substance as of the same unctuous or inflammable nature1 so that he may be regarded as predicting the combustibility of the diamond1 afterwards demonstrated by the ,florentine ,academicians in #afid4 ,brewster having entered into a long investigation of the refractive powers of different substances1 confirmed ,newton's assertions1 and found that the three elementary combustible substances1 diamond1 phosphorus1 and sulphur1 have1 in comparison with their densities1 by far the highest known refractive indices1#dci and there are only a few substances1 such as chromate of lead or glass of antimony1 which exceed them in absolute power of refraction4 ,the oils and hydrocarbons generally possess excessive indices4 ,but all this knowledge remains to the present day purely empirical1 no connection having been pointed out between this coincidence of inflammability and high refractive power1 with other laws of chemistry or optics4 ,it is worth notice1 as pointed out by ,brewster1 that if ,newton had argued concerning two minerals1 ,greenockite and ,octahedrite1 as he did concerning diamond1 his predictions would have proved false1 showing sufficiently that he did not make any sure induction on the subject4 ,in the present day1 the relation of the refractive index to the density and atomic weight of a substance is becoming a matter of theory2 yet there remain specific differences of refracting power known only on empirical grounds1 and it is curious that in hydrogen an abnormally high refractive power has been found to be joined to inflammability4 #dch ,newton's 99,opticks994 ,third edit4 p4 #bdi4 #dci ,brewster4 99,treatise on ,new ,philosophical ,instruments991 p4 #bff1 c4 ,the science of chemistry1 however much its theory may have progressed1 still presents us with a vast body of empirical knowledge4 ,not only is it as yet hopeless to attempt to account for the particular group of qualities belonging to each element1 but there are multitudes of particular facts of which no further account can be given4 ,why should the sulphides of many metals be intensely black8 ,why should a slight amount of phosphoric acid have so great a power of interference with the crystallisation of vanadic acid8#ddj ,why should the compound silicates of alkalies and alkaline metals be transparent8 ,why should gold be so highly ductile1 and gold and silver the only two sensibly translucent metals8 ,why should sulphur be capable of so many peculiar changes into allotropic modifications8 #ddj ,roscoe1 ,bakerian ,lecture1 99,philosophical ,transactions99 7#ahfh71 vol4 clviii4 p4 #f4 ,there are whole branches of chemical knowledge which are mere collections of disconnected facts4 ,the properties of alloys are often remarkable2 but no laws have yet been detected1 and the laws of combining proportions seem to have no clear application4#dda ,not the slightest explanation can be given of the wonderful variations of the qualities of iron1 according as it contains more or less carbon and silicon1 nay1 even the facts of the case are often involved in uncertainty4 ,why1 again1 should the properties of steel be remarkably affected by the presence of a little tungsten or manganese8 ,all that was determined by ,matthiessen concerning the conducting powers of copper1 was of a purely empirical character4#ddb ,many animal substances cannot be shown to obey the laws of combining proportions4 ,thus for the most part chemistry is yet an empirical science occupied with the registration of immense numbers of disconnected facts1 which may at some future time become the basis of a greatly extended theory4 #dda 99,life of ,faraday991 vol4 ii4 p4 #ajd4 #ddb ,watts1 99,dictionary of ,chemistry991 vol4 ii1 p4 #ci1 c4 ,we must not indeed suppose that any science will ever entirely cease to be empirical4 ,multitudes of phenomena have been explained by the undulatory theory of light2 but there yet remain many facts to be treated4 ,the natural colours of bodies and the rays given off by them when heated1 are unexplained1 and yield few empirical coincidences4 ,the theory of electricity is partially understood1 but the conditions of the production of frictional electricity defy explanation1 although they have been studied for two centuries4 ,i shall subsequently point out that even the establishment of a wide and true law of nature is but the starting-point for the discovery of exceptions and divergences giving a new scope to empirical discovery4 ,there is probably no science1 ,i have said1 which is entirely free from empirical and unexplained facts4 ,logic approaches most nearly to this position1 as it is merely a deductive development of the laws of thought and the principle of substitution4 ,yet some of the facts established in the investigation of the inverse logical problem may be considered empirical4 ,that a proposition of the form ,a ,,bc 99b c99 possesses the least number of distinct logical variations1 and the greatest number of logical equivalents of the same form among propositions involving three classes 7p4 #ada71 is a case in point4 ,so also is the fact discovered by ,professor ,clifford that in regard to statements involving four classes1 there is only one example of two dissimilar statements having the same distances 7p4 #add74 ,mathematical science often yields empirical truths4 ,why1 for instance1 should the value of 1 when expressed to a great number of figures1 contain the digit #g much less frequently than any other digit8#ddc ,even geometry may allow of empirical truths1 when the matter does not involve quantities of space1 but numerical results and the positive or negative character of quantities1 as in ,de ,morgan's theorem concerning negative areas4 #ddc ,de ,morgan's 99,budget of ,paradoxes991 p4 #bia4 99,accidental ,discovery499 ,there are not a few cases where almost pure accident has determined the moment when a new branch of knowledge was to be created4 ,the laws of the structure of crystals were not discovered until ,hay happened to drop a beautiful crystal of calc-spar upon a stone pavement4 ,his momentary regret at destroying a choice specimen was quickly removed when1 in attempting to join the fragments together1 he observed regular geometrical faces1 which did not correspond with the external facets of the crystals4 ,a great many more crystals were soon broken intentionally1 to observe the planes of cleavage1 and the discovery of the internal structure of crystalline substances was the result4 ,here we see how much more was due to the reasoning power of the philosopher1 than to an accident which must often have happened to other persons4 ,in a similar manner1 a fortuitous occurrence led ,malus to discover the polarisation of light by reflection4 ,the phenomena of double refraction had been long known1 and when engaged in ,paris in #ahjh1 in investigating the character of light thus polarised1 ,malus chanced to look through a double refracting prism at the light of the setting sun1 reflected from the windows of the ,luxembourg ,palace4 ,in turning the prism round1 he was surprised to find that the ordinary image disappeared at two opposite positions of the prism4 ,he remarked that the reflected light behaved like light which had been polarised by passing through another prism4 ,he was induced to test the character of light reflected under other circumstances1 and it was eventually proved that polarisation is invariably connected with reflection4 ,some of the general laws of optics1 previously unsuspected1 were thus discovered by pure accident4 ,in the history of electricity1 accident has had a large part4 ,for centuries some of the more common effects of magnetism and of frictional electricity had presented themselves as unaccountable deviations from the ordinary course of ,nature4 ,accident must have first directed attention to such phenomena1 but how few of those who witnessed them had any conception of the all-pervading character of the power manifested4 ,the very existence of galvanism1 or electricity of low tension1 was unsuspected until ,galvani accidentally touched the leg of a frog with pieces of metal4 ,the decomposition of water by voltaic electricity also was accidentally discovered by ,nicholson in #ahja1 and ,davy speaks of this discovery as the foundation of all that had since been done in electro-chemical science4 ,it is otherwise with the discovery of electro-magnetism4 ,oersted1 in common with many others1 had suspected the existence of some relation between the magnet and electricity1 and he appears to have tried to detect its exact nature4 ,once1 as we are told by ,hansteen1 he had employed a strong galvanic battery during a lecture1 and at the close it occurred to him to try the effect of placing the conducting wire parallel to a magnetic needle1 instead of at right angles1 as he had previously done4 ,the needle immediately moved and took up a position nearly at right angles to the wire2 he inverted the direction of the current1 and the needle deviated in a contrary direction4 ,the great discovery was made1 and if by accident1 it was such an accident as happens1 as ,lagrange remarked of ,newton1 only to those who deserve it4#ddd ,there was1 in fact1 nothing accidental1 except that1 as in all totally new discoveries1 ,oersted did not know what to look for4 ,he could not infer from previous knowledge the nature of the relation1 and it was only repeated trial in different modes which could lead him to the right combination4 ,high and happy powers of inference1 and not accident1 subsequently led ,faraday to reverse the process1 and to show that the motion of the magnet would occasion an electric current in the wire4 #ddd 99,life of ,faraday991 vol4 ii p4 #cif4 ,sufficient investigation would probably show that almost every branch of art and science had an accidental beginning4 ,in historical times almost every important new instrument as the telescope1 the microscope1 or the compass1 was probably suggested by some accidental occurrence4 ,in pre-historic times the germs of the arts must have arisen still more exclusively in the same way4 ,cultivation of plants probably arose1 in ,mr4 ,darwin's opinion1 from some such accident as the seeds of a fruit falling upon a heap of refuse1 and producing an unusually fine variety4 ,even the use of fire must1 some time or other1 have been discovered in an accidental manner4 ,with the progress of a branch of science1 the element of chance becomes much reduced4 ,not only are laws discovered which enable results to be predicted1 as we shall see1 but the systematic examination of phenomena and substances often leads to discoveries which can in no sense be said to be accidental4 ,it has been asserted that the ansthetic properties of chloroform were disclosed by a little dog smelling at a saucerful of the liquid in a chemist's shop in ,linlithgow1 the singular effects upon the dog being reported to ,simpson1 who turned the incident to good account4 ,this story1 however1 has been shown to be a fabrication1 the fact being that ,simpson had for many years been endeavouring to discover a better ansthetic than those previously employed1 and that he tested the properties of chloroform1 among other substances1 at the suggestion of ,waldie1 a ,liverpool chemist4 ,the valuable powers of chloral hydrate have since been discovered in a like manner1 and systematic inquiries are continually being made into the therapeutic or economic values of new chemical compounds4 ,if we must attempt to draw a conclusion concerning the part which chance plays in scientific discovery1 it must be allowed that it more or less affects the success of all inductive investigation1 but becomes less important with the progress of science4 ,accident may bring a new and valuable combination to the notice of some person who had never expressly searched for a discovery of the kind1 and the probabilities are certainly in favour of a discovery being occasionally made in this manner4 ,but the greater the tact and industry with which a physicist applies himself to the study of nature1 the greater is the probability that he will meet with fortunate accidents1 and will turn them to good account4 ,thus it comes to pass that1 in the refined investigations of the present day1 genius united to extensive knowledge1 cultivated powers1 and indomitable industry1 constitute the characteristics of the successful discoverer4 99,empirical ,observations subsequently ,explained499 ,the second great portion of scientific knowledge consists of facts which have been first learnt in a purely empirical manner1 but have afterwards been shown to follow from some law of nature1 that is1 from some highly probable hypothesis4 ,facts are said to be explained when they are thus brought into harmony with other facts1 or bodies of general knowledge4 ,there are few words more familiarly used in scientific phraseology than this word 99explanation991 and it is necessary to decide exactly what we mean by it1 since the question touches the deepest points concerning the nature of science4 ,like most terms referring to mental actions1 the verbs 99to explain991 or 99to explicate991 involve material similes4 ,the action is 99ex plicis plana reddere991 to take out the folds1 and render a thing plain or even4 ,explanation thus renders a thing clearly comprehensible in all its points1 so that there is nothing left outstanding or obscure4 ,every act of explanation consists in pointing out a resemblance between facts1 or in showing that similarity exists between apparently diverse phenomena4 ,this similarity may be of any extent and depth2 it may be a general law of nature1 which harmonises the motions of all the heavenly bodies by showing that there is a similar force which governs all those motions1 or the explanation may involve nothing more than a single identity1 as when we explain the appearance of shooting stars by showing that they are identical with portions of a comet4 ,wherever we detect resemblance1 there is a more or less explanation4 ,the mind is disquieted when it meets a novel phenomenon1 one which is 99sui generis992 it seeks at once for parallels which may be found in the memory of past sensations4 ,the so-called sulphurous smell which attends a stroke of lightning often excited attention1 and it was not explained until the exact similarity of the smell to that of ozone was pointed out4 ,the marks upon a flagstone are explained when they are shown to correspond with the feet of an extinct animal1 whose bones are elsewhere found4 ,explanation1 in fact1 generally commences by the discovery of some simple resemblance2 the theory of the rainbow began as soon as ,antonio de ,dominis pointed out the resemblance between its colours and those presented by a ray of sunlight passing through a glass globe full of water4 ,the nature and limits of explanation can only be fully considered1 after we have entered upon the subjects of generalisation and analogy4 ,it must suffice to remark1 in this place1 that the most important process of explanation consists in showing that an observed fact is one case of a general law or tendency4 ,iron is always found combined with sulphur1 when it is in contact with coal1 whereas in other parts of the carboniferous strata it always occurs as a carbonate4 ,we explain this empirical fact as being due to the reducing power of carbon and hydrogen1 which prevents the iron from combining with oxygen1 and leaves it open to the affinity of sulphur4 ,the uniform strength and direction of the trade-winds were long familiar to mariners1 before they were explained by ,halley on hydrostatical principles4 ,the winds were found to arise from the action of gravity1 which causes a heavier body to displace a lighter one1 while the direction from east to west was explained as a result of the earth's rotation4 ,whatever body in the northern hemisphere changes its latitude1 whether it be a bird1 or a railway train1 or a body of air1 must tend towards the right hand4 ,dove's law of the winds is that the winds tend to veer in the northern hemisphere in the direction ,n4,e4,s4,w41 and in the southern hemisphere in the direction ,n4,w4,s4,e4 ,this tendency was shown by him to be the necessary effect of the same conditions which apply to the trade winds4 ,whenever1 then1 any fact is connected by resemblance1 law1 theory1 or hypothesis1 with other facts1 it is explained4 ,although the great mass of recorded facts must be empirical1 and awaiting explanation1 such knowledge is of minor value1 because it does not admit of safe and extensive inference4 ,each recorded result informs us exactly what will be experienced again in the same circumstances1 but has no bearing upon what will happen in other circumstances4 99,overlooked ,results of ,theory499 ,we must by no means suppose that1 when a scientific truth is in our possession1 all its consequences will be foreseen4 ,deduction is certain and infallible1 in the sense that each step in deductive reasoning will lead us to some result1 as certain as the law itself4 ,but it does not follow that deduction will lead the reasoner to every result of a law or combination of laws4 ,whatever road a traveller takes1 he is sure to arrive somewhere1 but unless he proceeds in a systematic manner1 it is unlikely that he will reach every place to which a network of roads will conduct him4 ,in like manner there are many phenomena which were virtually within the reach of philosophers by inference from their previous knowledge1 but were never discovered until accident or systematic empirical observation disclosed their existence4 ,that light travels with a uniform high velocity was proved by ,roemer from observations of the eclipses of ,jupiter's satellites4 ,corrections were thenceforward made in all astronomical observations requiring it1 for the difference of absolute time at which an event happened1 and that at which it would be seen on the earth4 ,but no person happened to remark that the motion of light compounded with that of the earth in its orbit would occasion a small apparent displacement of the greater part of the heavenly bodies4 ,fifty years elapsed before ,bradley empirically discovered this effect1 called by him aberration1 when reducing his observations of the fixed stars4 ,when once the relation between an electric current and a magnet had been detected by ,oersted and ,faraday1 it ought to have been possible for them to foresee the diverse results which must ensue in different circumstances4 ,if1 for instance1 a plate of copper were placed beneath an oscillating magnetic needle1 it should have been seen that the needle would induce currents in the copper1 but as this could not take place without a certain reaction against the needle1 it ought to have been seen that the needle would come to rest more rapidly than in the absence of the copper4 ,this peculiar effect was accidentally discovered by ,gambey in #ahbd4 ,arago acutely inferred from ,gambey's experiment that if the copper were set in rotation while the needle was stationary the motion would gradually be communicated to the needle4 ,the phenomenon nevertheless puzzled the whole scientific world1 and it required the deductive genius of ,faraday to show that it was a result of the principles of electro-magnetism4#dde #dde 99,experimental ,researches in ,electricity991 #ast ,series1 pp4 #bd-dd4 ,many other curious facts might be mentioned which when once noticed were explained as the effects of well-known laws4 ,it was accidentally discovered that the navigation of canals of small depth could be facilitated by increasing the speed of the boats1 the resistance being actually reduced by this increase of speed1 which enables the boat to ride as it were upon its own forced wave4 ,now mathematical theory might have predicted this result had the right application of the formul occurred to any one4#ddf ,giffard's injector for supplying steam boilers with water by the force of their own steam1 was1 ,i believe1 accidentally discovered1 but no new principles of mechanics are involved in it1 so that it might have been theoretically invented4 ,the same may be said of the curious experiment in which a stream of air or steam issuing from a pipe is made to hold a free disc upon the end of the pipe and thus obstruct its own outlet4 ,the possession then of a true theory does not by any means imply the foreseeing of all the results4 ,the effects of even a few simple laws may be manifold1 and some of the most curious and useful effects may remain undetected until accidental observation brings them to our notice4 #ddf ,airy1 99,on ,tides and ,waves991 ,encyclopdia ,metropolitana1 p4 #cdh994 99,predicted ,discoveries499 ,the most interesting of the four classes of facts specified in p4 #ebe1 is probably the third1 containing those the occurrence of which has been first predicted by theory and then verified by observation4 ,there is no more convincing proof of the soundness of knowledge than that it confers the gift of foresight4 ,auguste ,comte said that 8,prevision is the test of true theory20 ,i should say that it is 99one test99 of true theory1 and that which is most likely to strike the public attention4 ,coincidence with fact is the test of true theory1 but when the result of theory is announced before-hand1 there can be no doubt as to the unprejudiced spirit in which the theorist interprets the results of his own theory4 ,the earliest instance of scientific prophecy is naturally furnished by the science of ,astronomy1 which was the earliest in development4 ,herodotus#ddg narrates that1 in the midst of a battle between the ,medes and ,lydians1 the day was suddenly turned into night1 and the event had been foretold by ,thales1 the ,father of ,philosophy4 ,a cessation of the combat and peace confirmed by marriages were the consequences of this happy scientific effort4 ,much controversy has taken place concerning the date of this occurrence1 ,baily assigning the year #faj ,b4,c41 but ,airy has calculated that the exact day was the #bhth of ,may1 #ehd ,b4,c4 ,there can be no doubt that this and other predictions of eclipses attributed to ancient philosophers were due to a knowledge of the ,metonic ,cycle1 a period of #f1ehe days1 or #bbc lunar months1 or about #ai years1 after which a nearly perfect recurrence of the phases and eclipses of the moon takes place2 but if so1 ,thales must have had access to long series of astronomical records of the ,egyptians or the ,chaldeans4 ,there is a well-known story as to the happy use which ,columbus made of the power of predicting eclipses in overawing the islanders of ,jamaica who refused him necessary supplies of food for his fleet4 ,he threatened to deprive them of the moon's light4 8,his threat was treated at first with indifference1 but when the eclipse actually commenced1 the barbarians vied with each other in the production of the necessary supplies for the ,spanish fleet40 #ddg ,lib4 i4 cap4 #gd4 ,exactly the same kind of awe which the ancients experienced at the prediction of eclipses1 has been felt in modern times concerning the return of comets4 ,seneca asserted in distinct terms that comets would be found to revolve in periodic orbits and return to sight4 ,the ancient ,chaldeans and the ,pythagoreans are also said to have entertained a like opinion4 ,but it was not until the age of ,newton and ,halley that it became possible to calculate the path of a comet in future years4 ,a great comet appeared in #afhb1 a few years before the first publication of the 99,principia991 and ,halley showed that its orbit corresponded with that of remarkable comets recorded to have appeared in the years #aeca and #afjg4 ,the intervals of time were not quite equal1 but ,halley conceived the bold idea that this difference might be due to the disturbing power of ,jupiter1 near which the comet had passed in the interval #afjg-afhb4 ,he predicted that the comet would return about the end of #ageh or the beginning of #agei1 and though ,halley did not live to enjoy the sight1 it was actually detected on the night of ,christmas-day1 #ageh4 ,a second return of the comet was witnessed in #ahce nearly at the anticipated time4 ,in recent times the discovery of ,neptune has been the most remarkable instance of prevision in astronomical science4 ,a full account of this discovery may be found in several works1 as for instance ,herschel's 99,outlines of ,astronomy991 and 99,grant's ,history of ,physical ,astronomy991 ,chapters ,,xii and ,,xiii4 99,predictions in the ,science of ,light499 ,next after astronomy the science of physical optics has furnished the most beautiful instances of the prophetic power of correct theory4 ,these cases are the more striking because they proceed from the profound application of mathematical analysis and show an insight into the mysterious workings of matter which is surprising to all1 but especially to those who are unable to comprehend the methods of research employed4 ,by its power of prevision the truth of the undulatory theory of light has been conspicuously proved1 and the contrast in this respect between the undulatory and ,corpuscular theories is remarkable4 ,even ,newton could get no aid from his corpuscular theory in the invention of new experiments1 and to his followers who embraced that theory we owe little or nothing in the science of light4 ,laplace did not derive from the theory a single discovery4 ,as ,fresnel remarks3#ddh #ddh ,taylor's 99,scientific ,memoirs991 vol4 v4 p4 #bda4 8,the assistance to be derived from a good theory is not to be confined to the calculation of the forces when the laws of the phenomena are known4 ,there are certain laws so complicated and so singular1 that observation alone1 aided by analogy1 could never lead to their discovery4 ,to divine these enigmas we must be guided by theoretical ideas founded on a 99true99 hypothesis4 ,the theory of luminous vibrations presents this character1 and these precious advantages2 for to it we owe the discovery of optical laws the most complicated and most difficult to divine40 ,physicists who embraced the corpuscular theory had nothing but their own quickness of observation to rely upon4 ,fresnel having once seized the conditions of the true undulatory theory1 as previously stated by ,young1 was enabled by the mere manipulation of his mathematical symbols to foresee many of the complicated phenomena of light4 ,who could possibly suppose1 that by stopping a portion of the rays passing through a circular aperture1 the illumination of a point upon a screen behind the aperture might be many times multiplied4 ,yet this paradoxical effect was predicted by ,fresnel1 and verified both by himself1 and in a careful repetition of the experiment1 by ,billet4 ,few persons are aware that in the middle of the shadow of an opaque circular disc is a point of light sensibly as bright as if no disc had been interposed4 ,this startling fact was deduced from ,fresnel's theory by ,poisson1 and was then verified experimentally by ,arago4 ,airy1 again1 was led by pure theory to predict that ,newton's rings would present a modified appearance if produced between a lens of glass and a plate of metal4 ,this effect happened to have been observed fifteen years before by ,arago1 unknown to ,airy4 ,another prediction of ,airy1 that there would be a further modification of the rings when made between two substances of very different refractive indices1 was verified by subsequent trial with a diamond4 ,a reversal of the rings takes place when the space intervening between the plates is filled with a substance of intermediate refractive power1 another phenomenon predicted by theory and verified by experiment4 ,there is hardly a limit to the number of other complicated effects of the interference of rays of light under different circumstances which might be deduced from the mathematical expressions1 if it were worth while1 or which1 being previously observed1 can be explained4 ,an interesting case was observed by ,herschel and explained by ,airy4#ddi #ddi ,airy's 99,mathematical ,tracts991 #crd edit4 p4 #cab4 ,by a somewhat different effort of scientific foresight1 ,fresnel discovered that any solid transparent medium might be endowed with the power of double refraction by mere compression4 ,as he attributed the double refracting power of crystals to unequal elasticity in different directions1 he inferred that unequal elasticity1 if artificially produced1 would give similar phenomena4 ,with a powerful screw and a piece of glass1 he then produced not only the colours due to double refraction1 but the actual duplication of images4 ,thus1 by a great scientific generalisation1 are the remarkable properties of ,iceland spar shown to belong to all transparent substances under certain conditions4#dej #dej ,young's 99,works991 vol4 i4 p4 #dab4 ,all other predictions in optical science are1 however1 thrown into the shade by the theoretical discovery of conical refraction by the late ,sir ,w4 ,r4 ,hamilton1 of ,dublin4 ,in investigating the passage of light through certain crystals1 ,hamilton found that ,fresnel had slightly misinterpreted his own formul1 and that1 when rightly understood1 they indicated a phenomenon of a kind never witnessed4 ,a small ray of light sent into a crystal of arragonite in a particular direction1 becomes spread out into an infinite number of rays1 which form a hollow cone within the crystal1 and a hollow cylinder when emerging from the opposite side4 ,in another case1 a different1 but equally strange1 effect is produced1 a ray of light being spread out into a hollow cone at the point where it quits the crystal4 ,these phenomena are peculiarly interesting1 because cones and cylinders of light are not produced in any other cases4 ,they are opposed to all analogy1 and constitute singular exceptions1 of a kind which we shall afterwards consider more fully4 ,their strangeness rendered them peculiarly fitted to test the truth of the theory by which they were discovered2 and when ,professor ,lloyd1 at ,hamilton's request1 succeeded1 after considerable difficulty1 in witnessing the new appearances1 no further doubt could remain of the validity of the wave theory which we owe to ,huyghens1 ,young1 and ,fresnel4#dea #dea ,lloyd's 99,wave ,theory991 ,part ii4 pp4 #eb-eh4 ,babbage1 99,ninth ,bridgewater ,treatise991 p4 #ajd1 quoting ,lloyd1 99,transactions of the ,royal ,irish ,academy991 vol4 xvii4 ,clifton1 99,quarterly ,journal of ,pure and ,applied ,mathematics991 ,january #ahfj4 99,predictions from the ,theory of ,undulations499 ,it is curious that the undulations of light1 although inconceivably rapid and small1 admit of more accurate measurement than waves of any other kind4 ,but so far as we can carry out exact experiments on other kinds of waves1 we find the phenomena of interference repeated1 and analogy gives considerable power of prediction4 ,herschel was perhaps the first to suggest that two sounds might be made to destroy each other by interference4#deb ,for if one-half of a wave travelling through a tube could be separated1 and conducted by a longer passage1 so as1 on rejoining the other half1 to be one-quarter of a vibration behind-hand1 the two portions would exactly neutralise each other4 ,this experiment has been performed with success4 ,the interference arising between the waves from the two prongs of a tuning-fork was also predicted by theory1 and proved to exist by ,weber2 indeed it may be observed by merely holding a vibrating fork close to the ear and turning it round4#dec #deb 99,encyclopdia ,metropolitana991 art4 99,sound991 p4 #gec4 #dec ,tyndall's 99,sound991 pp4 #bfa1 #bgc4 ,it is a result of the theory of sound that1 if we move rapidly towards a sounding body1 or if it move rapidly towards us1 the pitch of the sound will be a little more acute2 and1 99vice vers991 when the relative motion is in the opposite direction1 the pitch will be more grave4 ,this arises from the less or greater intervals of time elapsing between the successive strokes of waves upon the auditory nerve1 according as the ear moves towards or from the source of sound relatively speaking4 ,this effect was predicted by theory1 and afterwards verified by the experiments of ,buys ,ballot1 on ,dutch railways1 and of ,scott ,russell1 in ,england4 ,whenever one railway train passes another1 on the locomotive of which the whistle is being sounded1 the drop in the acuteness of the sound may be noticed at the moment of passing4 ,this change gives the sound a peculiar howling character1 which many persons must have noticed4 ,i have calculated that with two trains travelling thirty miles an hour1 the effect would amount to rather more than half a tone1 and with some express trains it would amount to a tone4 ,a corresponding effect is produced in the case of light undulations1 when the eye and the luminous body approach or recede from each other4 ,it is shown by a slight change in the refrangibility of the rays of light1 and a consequent change in the place of the lines of the spectrum1 which has been made to give important and unexpected information concerning the relative approach or recession of stars4 ,tides are vast waves1 and were the earth's surface entirely covered by an ocean of uniform depth1 they would admit of exact theoretical investigation4 ,the irregular form of the seas introduces unknown quantities and complexities with which theory cannot cope4 ,nevertheless1 ,whewell1 observing that the tides of the ,german ,ocean consist of interfering waves1 which arrive partly round the ,north of ,scotland and partly through the ,british ,channel1 was enabled to predict that at a point about midway between ,brill on the coast of ,holland1 and ,lowestoft no tides would be found to exist4 ,at that point the two waves would be of the same amount1 but in opposite phases1 so as to neutralise each other4 ,this prediction was verified by a surveying vessel of the ,british navy4#ded #ded ,whewell's 99,history of the ,inductive ,sciences991 vol4 ii4 p4 #dga4 ,herschel's 99,physical ,geography991 #gg4 99,prediction in other ,sciences499 ,generations1 or even centuries1 may elapse before mankind are in possession of a mathematical theory of the constitution of matter as complete as the theory of gravitation4 ,nevertheless1 mathematical physicists have in recent years acquired a hold of some of the relations of the physical forces1 and the proof is found in anticipations of curious phenomena which had never been observed4 ,professor ,james ,thomson deduced from ,carnot's theory of heat that the application of pressure would lower the melting-point of ice4 ,he even ventured to assign the amount of this effect1 and his statement was afterwards verified by ,sir ,w4 ,thomson4#dee 8,in this very remarkable speculation1 an entirely novel physical phenomenon was 99predicted991 in anticipation of any direct experiments on the subject2 and the actual observation of the phenomenon was pointed out as a highly interesting object for experimental research40 ,just as liquids which expand in solidifying will have the temperature of solidification lowered by pressure1 so liquids which contract in solidifying will exhibit the reverse effect4 ,they will be assisted in solidifying1 as it were1 by pressure1 so as to become solid at a higher temperature1 as the pressure is greater4 ,this latter result was verified by ,bunsen and ,hopkins1 in the case of paraffin1 spermaceti1 wax1 and stearin4 ,the effect upon water has more recently been carried to such an extent by ,mousson1 that under the vast pressure of #acjj atmospheres1 water did not freeze until cooled down to -#ah;,c4 ,another remarkable prediction of ,professor ,thomson was to the effect that1 if a metallic spring be weakened by a rise of temperature1 work done against the spring in bending it will cause a cooling effect4 ,although the effect to be expected in a certain apparatus was only about four-thousandths of a degree ,centigrade1 ,dr4 ,joule#def succeeded in measuring it to the extent of three-thousandths of a degree1 such is the delicacy of modern heat measurements4 ,i cannot refrain from quoting ,dr4 ,joule's reflections upon this fact4 8,thus even in the above delicate case10 he says1 8is the formula of ,professor ,thomson completely verified4 ,the mathematical investigation of the thermo-elastic qualities of metals has enabled my illustrious friend to predict with certainty a whole class of highly interesting phenomena4 ,to him especially do we owe the important advance which has been recently made to a new era in the history of science1 when the famous philosophical system of ,bacon will be to a great extent superseded1 and when1 instead of arriving at discovery by induction from experiment1 we shall obtain our largest accessions of new facts by reasoning deductively from fundamental principles40 #dee ,maxwell's 99,theory of ,heat991 p4 #agd4 99,philosophical ,magazine991 ,august #ahej4 ,third ,series1 vol4 xxxvii4 p4 #abc4 #def 99,philosophical ,transactions991 #aheh1 vol4 cxlviii4 p4 #abg4 ,the theory of electricity is a necessary part of the general theory of matter1 and is rapidly acquiring the power of prevision4 ,as soon as ,wheatstone had proved experimentally that the conduction of electricity occupies time1 ,faraday remarked in #ahch1 with wonderful sagacity1 that if the conducting wires were connected with the coatings of a large ,leyden jar1 the rapidity of conduction would be lessened4 ,this prediction remained unverified for sixteen years1 until the submarine cable was laid beneath the ,channel4 ,a considerable retardation of the electric spark was then detected1 and ,faraday at once pointed out that the wire surrounded by water resembles a ,leyden jar on a large scale1 so that each message sent through the cable verified his remark of #ahch4deg #deg ,tyndall's 99,faraday991 pp4 #gc1 #gd2 99,life of ,faraday991 vol4 ii4 pp4 #hb1 #hc4 ,the joint relations of heat and electricity to the metals constitute a new science of thermo-electricity by which ,sir ,w4 ,thomson was enabled to anticipate the following curious effect1 namely1 that an electric current passing in an iron bar from a hot to a cold part produces a cooling effect1 but in a copper bar the effect is exactly opposite in character1 that is1 the bar becomes heated4#deh ,the action of crystals with regard to heat and electricity was partly foreseen on the grounds of theory by ,poisson4 #deh ,tait's 99,thermodynamics991 p4 #gg4 ,chemistry1 although to a great extent an empirical science1 has not been without prophetic triumphs4 ,the existence of the metals potassium and sodium was foreseen by ,lavoisier1 and their elimination by ,davy was one of the chief 99experimenta crucis99 which established ,lavoisier's system4 ,the existence of many other metals which eye had never seen was a natural inference1 and theory has not been at fault4 ,in the above cases the compounds of the metal were well known1 and it was the result of decomposition that was foretold4 ,the discovery in #ahgf of the metal gallium is peculiarly interesting because the existence of this metal1 previously wholly unknown1 had been inferred from theoretical considerations by ,m4 ,mendelief1 and some of its properties had been correctly predicted4 ,no sooner1 too1 had a theory of organic compounds been conceived by ,professor ,a4 ,w4 ,williamson than he foretold the formation of a complex substance consisting of water in which both atoms of hydrogen are replaced by atoms of acetyle4 ,this substance1 known as the acetic anhydride1 was afterwards produced by ,gerhardt4 ,in the subsequent progress of organic chemistry occurrences of this kind have become common4 ,the theoretical chemist by the classification of his specimens and the manipulation of his formul can plan out whole series of unknown oils1 acids1 and alcohols1 just as a designer might draw out a multitude of patterns4 ,professor ,cayley has even calculated for certain cases the possible numbers of chemical compounds4#dei ,the formation of many such substances is a matter of course2 but there is an interesting prediction given by ,hofmann1 concerning the possible existence of new compounds of sulphur and selenium1 and even oxides of ammonium1 which it remains for chemists to verify4#dfj #dei 99,on the ,analytical ,forms called ,trees1 with ,application to the ,theory of ,chemical ,combinations499 ,report of the ,british ,association1 #ahge1 p4 #beg4 #dfj ,hofmann's 99,introduction to ,chemistry991 pp4 #bbd1 #bbe4 99,prediction by ,inversion of ,cause and ,effect499 ,there is one process of experiment which has so often led to important discoveries as to deserve separate illustration--,i mean the inversion of ,cause and ,effect4 ,thus if ,a and ,b in one experiment produce ,c as a consequent1 then antecedents of the nature of ,b and ,c may usually be made to produce a consequent of the nature of ,a inverted in direction4 ,when we apply heat to a gas it tends to expand2 hence if we allow the gas to expand by its own elastic force1 cold is the result2 that is1 ,b 7air7 and ,c 7expansion7 produce the negative of ,a 7heat74 ,again1 ,b 7air7 and compression1 the negative of ,c1 produce ,a 7heat74 ,similar results may be expected in a multitude of cases4 ,it is a familiar law that heat expands iron4 ,what may be expected1 then1 if instead of increasing the length of an iron bar by heat we use mechanical force and stretch the bar8 ,having the bar and the former consequent1 expansion1 we should expect the negative of the former antecedent1 namely cold4 ,the truth of this inference was proved by ,dr4 ,joule1 who investigated the amount of the effect with his usual skill4#dfa #dfa 99,philosophical ,transactions99 7#ahee71 vol4 cxlv4 pp4 #ajj1 c4 ,this inversion of cause and effect in the case of heat may be itself inverted in a highly curious manner4 ,it happens that there are a few substances which are unexplained exceptions to the general law of expansion by heat4 ,india-rubber especially is remarkable for 99contracting99 when heated4 ,since1 then1 iron and india-rubber are oppositely related to heat1 we may expect that as distension of the iron produced cold1 distension of the india-rubber will produce heat4 ,this is actually found to be the case1 and anyone may detect the effect by suddenly stretching an india-rubber band while the middle part is in the mouth4 ,when being stretched it grows slightly warm1 and when relaxed cold4 ,the reader will see that some of the scientific predictions mentioned in preceding sections were due to the principle of inversion2 for instance1 ,thomson's speculations on the relation between pressure and the melting-point4 ,but many other illustrations could be adduced4 ,the usual agent by which we melt a substance is heat2 but if we can melt a substance without heat1 then we may expect the negative of heat as an effect4 ,this is the foundation of all freezing mixtures4 ,the affinity of salt for water causes it to melt ice1 and we may thus reduce the temperature to ,fahrenheit's zero4 ,calcium chloride has so much higher an attraction for water that a temperature of -#de ,c4 may be attained by its use4 ,even the solution of a certain alloy of lead1 tin1 and bismuth in mercury1 may be made to reduce the temperature through #bg ,c4 ,all the other modes of producing cold are inversions of more familiar uses of heat4 ,carr's freezing machine is an inverted distilling apparatus1 the distillation being occasioned by chemical affinity instead of heat4 ,another kind of freezing machine is the exact inverse of the steam-engine4 ,a very paradoxical effect is due to another inversion4 ,it is hard to believe that a current of steam at #ajj ,c4 can raise a body of liquid to a higher temperature than the steam itself possesses4 ,but ,mr4 ,spence has pointed out that if the boiling-point of a saline solution be above #ajj1 it will continue1 on account of its affinity for water1 to condense steam when above #ajj in temperature4 ,it will condense the steam until heated to the point at which the tension of its vapour is equal to that of the atmosphere1 that is1 its own boiling-point4#dfb ,again1 since heat melts ice1 we might expect to produce heat by the inverse change from water into ice4 ,this is accomplished in the phenomenon of suspended freezing4 ,water may be cooled in a clean glass vessel many degrees below the freezing-point1 and yet retained in the liquid condition4 ,but if disturbed1 and especially if brought into contact with a small particle of ice1 it instantly solidifies and rises in temperature to #j ,c4 ,the effect is still better displayed in the lecture-room experiment of the suspended crystallisation of a solution of sodium sulphate1 in which a sudden rise of temperature of #ae or #bj ,c4 is often manifested4 #dfb 99,proceedings of the ,manchester ,philosophical ,society991 ,feb4 #ahgj4 ,the science of electricity is full of most interesting cases of inversion4 ,as ,professor ,tyndall has remarked1 ,faraday had a profound belief in the reciprocal relations of the physical forces4 ,the great starting-point of his researches1 the discovery of electro-magnetism1 was clearly an inversion4 ,oersted and ,ampre had proved that with an electric current and a magnet in a particular position as antecedents1 motion is the consequent4 ,if then a magnet1 a wire and motion be the antecedents1 an 99opposite99 electric current will be the consequent4 ,it would be an endless task to trace out the results of this fertile relationship4 ,another part of ,faraday's researches was occupied in ascertaining the direct and inverse relations of magnetic and diamagnetic1 amorphous and crystalline substances in various circumstances4 ,in all other relations of electricity the principle of inversion holds4 ,the voltameter or the electro-plating cell is the inverse of the galvanic battery4 ,as heat applied to a junction of antimony and bismuth bars produces electricity1 it follows that an electric current passed through such a junction will produce cold4 ,but it is now sufficiently apparent that inversion of cause and effect is a most fertile means of discovery and prediction4 99,facts known only by ,theory499 ,of the four classes of facts enumerated in p4 #ebe the last remains unconsidered4 ,it includes the unverified predictions of science4 ,scientific prophecy arrests the attention of the world when it refers to such striking events as an eclipse1 the appearance of a great comet1 or any phenomenon which people can verify with their own eyes4 ,but it is surely a matter for greater wonder that a physicist describes and measures phenomena which eye cannot see1 nor sense of any kind detect4 ,in most cases this arises from the effect being too small in amount to affect our organs of sense1 or come within the powers of our instruments as at present constructed4 ,but there is a class of yet more remarkable cases1 in which a phenomenon cannot possibly be observed1 and yet we can say what it would be if it were observed4 ,in astronomy1 systematic aberration is an effect of the sun's proper motion almost certainly known to exist1 but which we have no hope of detecting by observation in the present age of the world4 ,as the earth's motion round the sun combined with the motion of light causes the stars to deviate apparently from their true positions to the extent of about #ah at the most1 so the motion of the whole planetary system through space must occasion a similar displacement of at most #e4 ,the ordinary aberration can be readily detected with modern astronomical instruments1 because it goes through a yearly change in direction or amount2 but systematic aberration is constant so long as the planetary system moves uniformly in a sensibly straight line4 ,only then in the course of ages1 when the curvature of the sun's path becomes apparent1 can we hope to verify the existence of this kind of aberration4 ,a curious effect must also be produced by the sun's proper motion upon the apparent periods of revolution of the binary stars4 ,to my mind1 some of the most interesting truths in the whole range of science are those which have not been1 and in many cases probably never can be1 verified by trial4 ,thus the chemist assigns1 with a very high degree of probability1 the vapour densities of such elements as carbon and silicon1 which have never been observed separately in a state of vapour4 ,the chemist is also familiar with the vapour densities of elements at temperatures at which the elements in question never have been1 and probably never can be1 submitted to experiment in the form of vapour4 ,joule and others have calculated the actual velocity of the molecules of a gas1 and even the number of collisions which must take place per second during their constant circulation4 ,physicists have not yet given us the exact magnitudes of the particles of matter1 but they have ascertained by several methods the limits within which their magnitudes must lie4 ,such scientific results must be for ever beyond the power of verification by the senses4 ,i have elsewhere had occasion to remark that waves of light1 the intimate processes of electrical changes1 the properties of the ether which is the base of all phenomena1 are necessarily determined in a hypothetical1 but not therefore a less certain manner4 ,though only two of the metals1 gold and silver1 have ever been observed to be transparent1 we know on the grounds of theory that they are all more or less so2 we can even estimate by theory their refractive indices1 and prove that they are exceedingly high4 ,the phenomena of elliptic polarisation1 and perhaps also those of internal radiation1#dfc depend upon the refractive index1 and thus1 even when we cannot observe any refracted rays1 we can indirectly learn how they would be refracted4 #dfc ,balfour ,stewart1 99,elementary ,treatise on ,heat991 #ast edit4 p4 #aih4 ,in many cases large quantities of electricity must be produced1 which we cannot observe because it is instantly discharged4 ,in the common electric machine the cylinder and rubber are made of non-conductors1 so that we can separate and accumulate the electricity4 ,but a little damp1 by serving as a conductor1 prevents this separation from enduring any sensible time4 ,hence there is no doubt that when we rub two good conductors against each other1 for instance two pieces of metals1 much electricity is produced1 but instantaneously converted into some other form of energy4 ,joule believes that all the heat of friction is transmuted electricity4 ,as regards phenomena of insensible amount1 nature is absolutely full of them4 ,we must regard those changes which we can observe as the comparatively rare aggregates of minuter changes4 ,on a little reflection we must allow that no object known to us remains for two instants of exactly the same temperature4 ,if so1 the dimensions of objects must be in a perpetual state of variation4 ,the minor planetary and lunar perturbations are infinitely numerous1 but usually too small to be detected by observation1 although their amounts may be assigned by theory4 ,there is every reason to believe that chemical and electric actions of small amount are constantly in progress4 ,the hardest substances1 if reduced to extremely small particles1 and diffused in pure water1 manifest oscillatory movements which must be due to chemical and electric changes1 so slight that they go on for years without affecting appreciably the weight of the particles4#dfd ,the earth's magnetism must more or less affect every object which we handle4 ,as ,tyndall remarks1 8,an upright iron stone influenced by the earth's magnetism becomes a magnet1 with its bottom a north and its top a south pole4 ,doubtless1 though in an immensely feebler degree1 every erect marble statue is a true diamagnet1 with its head a north pole and its feet a south pole4 ,the same is certainly true of man as he stands upon the earth's surface1 for all the tissues of the human body are diamagnetic40#dfe ,the sun's light produces a very quick and perceptible effect upon the photographic plate2 in all probability it has a less effect upon a great variety of substances4 ,we may regard every phenomenon as an exaggerated and conspicuous case of a process which is1 in infinitely numerous cases1 beyond the means of observation4 #dfd ,jevons1 99,proceedings of the ,manchester ,literary and ,philosophical ,society991 #beth ,january1 #ahgj1 vol4 ix4 p4 #gh4 #dfe 99,philosophical ,transactions991 vol4 cxlvi4 p4 #bdi4 ,,chapter ,,xxv4 ,,accordance ,,of ,,quantitative ,,theories4 ,in the preceding chapter we found that facts may be classed under four heads as regards their connection with theory1 and our powers of explanation or prediction4 ,the facts hitherto considered were generally of a qualitative rather than a quantitative nature2 but when we look exclusively to the quantity of a phenomenon1 and the various modes in which we may determine its amount1 nearly the same system of classification will hold good4 ,there will1 however1 be five possible cases3-- 7#a7 ,we may directly and empirically measure a phenomenon1 without being able to explain why it should have any particular quantity1 or to connect it by theory with other quantities4 7#b7 ,in a considerable number of cases we can theoretically predict the existence of a phenomenon1 but are unable to assign its amount1 except by direct measurement1 or to explain the amount theoretically when thus ascertained4 7#c7 ,we may measure a quantity1 and afterwards explain it as related to other quantities1 or as governed by known quantitative laws4 7#d7 ,we may predict the quantity of an effect on theoretical grounds1 and afterwards confirm the prediction by direct measurement4 7#e7 ,we may indirectly determine the quantity of an effect without being able to verity it by experiment4 ,these classes of quantitative facts might be illustrated by an immense number of interesting points in the history of physical science4 ,only a few instances of each class can be given here4 99,empirical ,measurements499 ,under the first head of purely empirical measurements1 which have not been brought under any theoretical system1 may be placed the great bulk of quantitative facts recorded by scientific observers4 ,the tables of numerical results which abound in books on chemistry and physics1 the huge quartos containing the observations of public observatories1 the multitudinous tables of meteorological observations1 which are continually being published1 the more abstruse results concerning terrestrial magnetism--such results of measurement1 for the most part1 remain empirical1 either because theory is defective1 or the labour of calculation and comparison is too formidable4 ,in the ,greenwich ,observatory1 indeed1 the salutary practice has been maintained by the present ,astronomer ,royal1 of always reducing the observations1 and comparing them with the theories of the several bodies4 ,the divergences from theory thus afford material for the discovery of errors or of new phenomena2 in short1 the observations have been turned to the use for which they were intended4 ,but it is to be feared that other establishments are too often engaged in merely recording numbers of which no real use is made1 because the labour of reduction and comparison with theory is too great for private inquirers to undertake4 ,in meteorology1 especially1 great waste of labour and money is taking place1 only a small fraction of the results recorded being ever used for the advancement of the science4 ,for one meteorologist like ,quetelet1 ,dove1 or ,baxendell1 who devotes himself to the truly useful labour of reducing other people's observations1 there are hundreds who labour under the delusion that they are advancing science by loading our book-shelves with numerical tables4 ,it is to be feared1 in like manner1 that almost the whole bulk of statistical numbers1 whether commercial1 vital1 or moral1 is of little scientific value4 ,purely empirical measurements may have a direct practical value1 as when tables of the specific gravity1 or strength of materials1 assist the engineer2 the specific gravities of mixtures of water with acids1 alcohols1 salts1 c41 are useful in chemical manufactories1 custom-house gauging1 c42 observations of rainfall are requisite for questions of water supply2 the refractive index of various kinds of glass must be known in making achromatic lenses2 but in all such cases the use made of the measurements is not scientific but practical4 ,it may be asserted1 that no number which remains isolated1 and uncompared by theory with other numbers1 is of scientific value4 ,having tried the tensile strength of a piece of iron in a particular condition1 we know what will be the strength of the same kind of iron in a similar condition1 provided we can ever meet with that exact kind of iron again2 but we cannot argue from piece to piece1 nor lay down any laws exactly connecting the strength of iron with the quantity of its impurities4 99,quantities indicated by ,theory1 but ,empirically ,measured499 ,in many cases we are able to foresee the existence of a quantitative effect1 on the ground of general principles1 but are unable1 either from the want of numerical data1 or from the entire absence of any mathematical theory1 to assign the amount of such effect4 ,we then have recourse to direct experiment to determine its amount4 ,whether we argued from the oceanic tides by analogy1 or deductively from the theory of gravitation1 there could be no doubt that atmospheric tides of some amount must occur in the atmosphere4 ,theory1 however1 even in the hands of ,laplace1 was not able to overcome the complicated mechanical conditions of the atmosphere1 and predict the amounts of such tides2 and1 on the other hand1 these amounts were so small1 and were so masked by far larger undulations arising from the heating power of the sun1 and from other meteorological disturbances1 that they would probably have never been discovered by purely empirical observations4 ,theory having1 however1 indicated their existence and their periods1 it was easy to make series of barometrical observations in places selected so as to be as free as possible from casual fluctuations1 and then1 by the suitable application of the method of means1 to detect the small effects in question4 ,the principal lunar atmospheric tide was thus proved to amount to between #jjc and #jjd inch4#dff #dff ,grant's 99,history of ,physical ,astronomy991 p4 #afb4 ,theory yields the greatest possible assistance in applying the method of means4 ,for if we have a great number of empirical measurements1 each representing the joint effect of a number of causes1 our object will be to take the mean of all those in which the effect to be measured is present1 and compare it with the mean of the remainder in which the effect is absent1 or acts in the opposite direction4 ,the difference will then represent the amount of the effect1 or double the amount respectively4 ,thus1 in the case of the atmospheric tides1 we take the mean of all the observations when the moon was on the meridian1 and compare it with the mean of all observations when she was on the horizon4 ,in this case we trust to chance that all other effects will lie about as often in one direction as the other1 and will neutralise themselves in the drawing of each mean4 ,it is a great advantage1 however1 to be able to decide by theory when each principal disturbing effect is present or absent2 for the means may then be drawn so as to separate each such effect1 leaving only minor and casual divergences to the law of error4 ,thus1 if there be three principal effects1 and we draw means giving respectively the sum of all three1 the sum of the first two1 and the sum of the last two1 then we gain three simple equations1 by the solution of which each quantity is determined4 99,explained ,results of ,measurement499 ,the second class of measured phenomena contains those which1 after being determined in a direct and purely empirical application of measuring instruments1 are afterwards shown to agree with some hypothetical explanation4 ,such results are turned to their proper use1 and several advantages may arise from the comparison4 ,the correspondence with theory will seldom or never be precise2 and1 even if it be so1 the coincidence must be regarded as accidental4 ,if the divergences between theory and experiment be comparatively small1 and variable in amount and direction1 they may often be safely attributed to inconsiderable sources of error in the experimental processes4 ,the strict method of procedure is to calculate the probable error of the mean of the observed results 7p4 #chg71 and then observe whether the theoretical result falls within the limits of probable error4 ,if it does1 and if the experimental results agree as well with theory as they agree with each other1 then the probability of the theory is much increased1 and we may employ the theory with more confidence in the anticipation of further results4 ,the probable error1 it should be remembered1 gives a measure only of the effects of incidental and variable sources of error1 but in no degree indicates the amount of fixed causes of error4 ,thus1 if the mean results of two modes of determining a quantity are so far apart that the limits of probable error do not overlap1 we may infer the existence of some overlooked source of fixed error in one or both modes4 ,we will further consider in a subsequent section the discordance of measurements4 99,quantities determined by ,theory and verified by ,measurement499 ,one of the most satisfactory tests of a theory consists in its application not only to predict the nature of a phenomenon1 and the circumstances in which it may be observed1 but also to assign the precise quantity of the phenomenon4 ,if we can subsequently apply accurate instruments and measure the amount of the phenomenon witnessed1 we have an excellent opportunity of verifying or negativing the theory4 ,it was in this manner that ,newton first attempted to verify his theory of gravitation4 ,he knew approximately the velocity produced in falling bodies at the earth's surface1 and if the law of the inverse square of the distance held true1 and the reputed distance of the moon was correct1 he could infer that the moon ought to fall towards the earth at the rate of fifteen feet in one minute4 ,now1 the actual divergence of the moon from the tangent of its orbit appeared to amount only to thirteen feet in one minute1 and there was a discrepancy of two feet in fifteen1 which caused ,newton to lay 8aside at that time any further thoughts of this matter40 ,many years afterwards1 probably fifteen or sixteen years1 ,newton obtained more precise data from which he could calculate the size of the moon's orbit1 and he then found the discrepancy to be inconsiderable4 ,his theory of gravitation was thus verified as far as the moon was concerned2 but this was to him only the beginning of a long course of deductive calculations1 each ending in a verification4 ,if the earth and moon attract each other1 and also the sun and the earth1 there is reason to expect that the sun and moon should attract each other4 ,newton followed out the consequences of this inference1 and showed that the moon would not move as if attracted by the earth only1 but sometimes faster and sometimes slower4 ,comparison with ,flamsteed's observations of the moon showed that such was the case4 ,newton argued again1 that as the waters of the ocean are not rigidly attached to the earth1 they might attract the moon1 and be attracted in return1 independently of the rest of the earth4 ,certain daily motions resembling the tides would then be caused1 and there were the tides to verify the reasoning4 ,it was the extraordinary power with which ,newton traced out geometrically the consequences of his theory1 and submitted them to repeated comparison with experience1 which constitutes his pre-eminence over all physicists4 99,quantities determined by ,theory and not verified499 ,it will continually happen that we are able1 from certain measured phenomena and a correct theory1 to determine the amount of some other phenomenon which we may either be unable to measure at all1 or to measure with an accuracy corresponding to that required to verify the prediction4 ,thus ,laplace having worked out a theory of the motions of ,jupiter's satellites on the hypothesis of gravitation1 found that these motions were greatly affected by the spheroidal form of ,jupiter4 ,the motions of the satellites can be observed with great accuracy owing to their frequent eclipses and transits1 and from these motions he was able to argue inversely1 and assign the ellipticity of the planet4 ,the ratio of the polar and equatorial axes thus determined was very nearly that of #ac to #ad2 and it agrees well with such direct micrometrical measurements of the planet as have been made2 but ,laplace believed that the theory gave a more accurate result than direct observation could yield1 so that the theory could hardly be said to admit of direct verification4 ,the specific heat of air was believed on the grounds of direct experiment to amount to #jbffi1 the specific heat of water being taken as unity2 but the methods of experiment were open to considerable causes of error4 ,rankine showed in #ahej that it was possible to calculate from the mechanical equivalent of heat and other thermodynamic data1 what this number should be1 and he found it to be #jbcgh4 ,this determination was at the time accepted as the most satisfactory result1 although not verified2 subsequently in #ahec ,regnault obtained by direct experiment the number #jbcgg1 proving that the prediction had been well grounded4 ,it is readily seen that in quantitative questions verification is a matter of degree and probability4 ,a less accurate method of measurement cannot verify the results of a more accurate method1 so that if we arrive at a determination of the same physical quantity in several distinct modes it is often a delicate matter to decide which result is most reliable1 and should be used for the indirect determination of other quantities4 ,for instance1 ,joule's and ,thomson's ingenious experiments upon the thermal phenomena of fluids in motion#dfg involved1 as one physical constant1 the mechanical equivalent of heat2 if requisite1 then1 they might have been used to determine that important constant4 ,but if more direct methods of experiment give the mechanical equivalent of heat with superior accuracy1 then the experiments on fluids will be turned to a better use in determining various quantities relating to the theory of fluids4 ,we will further consider questions of this kind in succeeding sections4 #dfg 99,philosophical ,transactions99 7#ahed71 vol4 cxliv4 p4 #cfd4 ,there are of course many quantities assigned on theoretical grounds which we are quite unable to verify with corresponding accuracy4 ,the thickness of a film of gold leaf1 the average depths of the oceans1 the velocity of a star's approach to or regression from the earth as inferred from spectroscopic data 7pp4 #bif-ii71 are cases in point2 but many others might be quoted where direct verification seems impossible4 ,newton and subsequent physicists have measured light undulations1 and by several methods we learn the velocity with which light travels4 ,since an undulation of the middle green is about five ten-millionths of a metre in length1 and travels at the rate of nearly #cjj1jjj1jjj of metres per second1 it follows that about #fjj1jjj1jjj1jjj1jjj undulations must strike in one second the retina of an eye which perceives such light4 ,but how are we to verify such an astounding calculation by directly counting pulses which recur six hundred billions of times in a second8 99,discordance of ,theory and ,experiment499 ,when a distinct want of accordance is found to exist between the results of theory and direct measurement1 interesting questions arise as to the mode in which we can account for this discordance4 ,the ultimate explanation of the discrepancy may be accomplished in at least four ways as follows3-- 7#a7 ,the direct measurement may be erroneous owing to various sources of casual error4 7#b7 ,the theory may be correct as far as regards the general form of the supposed laws1 but some of the constant numbers or other quantitative data employed in the theoretical calculations may be inaccurate4 7#c7 ,the theory may be false1 in the sense that the forms of the mathematical equations assumed to express the laws of nature are incorrect4 7#d7 ,the theory and the involved quantities may be approximately accurate1 but some regular unknown cause may have interfered1 so that the divergence may be regarded as a 99residual effect99 representing possibly a new and interesting phenomenon4 ,no precise rules can be laid down as to the best mode of proceeding to explain the divergence1 and the experimentalist will have to depend upon his own insight and knowledge2 but the following recommendations may be made4 ,if the experimental measurements are not numerous1 repeat them and take a more extensive mean result1 the probable accuracy of which1 as regards casual errors1 will increase as the square root of the number of experiments4 ,supposing that no considerable modification of the result is thus effected1 we may suspect the existence of more deep-seated sources of error in our method of measurement4 ,the next resource will be to change the size and form of the apparatus employed1 and to introduce various modifications in the materials employed or the course of procedure1 in the hope 7p4 #cif7 that some cause of constant error may thus be removed4 ,if the inconsistency with theory still remains unreduced we may attempt to invent some widely different mode of arriving at the same physical quantity1 so that we may be almost sure that the same cause of error will not affect both the new and old results4 ,in some cases it is possible to find five or six essentially different modes of arriving at the same determination4 ,supposing that the discrepancy still exists we may begin to suspect that our direct measurements are correct1 and that the data employed in the theoretical calculations are inaccurate4 ,we must now review the grounds on which these data depend1 consisting as they must ultimately do of direct measurements4 ,a comparison of the recorded data will show the degree of probability attaching to the mean result employed2 and if there is any ground for imagining the existence of error1 we should repeat the observations1 and vary the forms of experiment just as in the case of the previous direct measurements4 ,the continued existence of the discrepancy must show that we have not attained to a complete acquaintance with the theory of the causes in action1 but two different cases still remain4 ,we may have misunderstood the action of those causes which we know to exist1 or we may have overlooked the existence of one or more other causes4 ,in the first case our hypothesis appears to be wrongly chosen and inapplicable2 but whether we are to reject it will depend upon whether we can form another hypothesis which yields a more accurate accordance4 ,the probability of an hypothesis1 it will be remembered 7p4 #bdc71 is to be judged1 in the absence of 99 priori99 grounds of judgment1 by the probability that if the supposed causes exist the observed result follows2 but as there is now little probability of reconciling the original hypothesis with our direct measurements the field is open for new hypotheses1 and any one which gives a closer accordance with measurement will so far have better claims to attention4 ,of course we must never estimate the probability of an hypothesis merely by its accordance with a few results only4 ,its general analogy and accordance with other known laws of nature1 and the fact that it does not conflict with other probable theories1 must be taken into account1 as we shall see in the next book4 ,the requisite condition of a good hypothesis1 that it must admit of the deduction of facts verified in observation1 must be interpreted in the widest manner1 as including all ways in which there may be accordance or discordance4 ,all our attempts at reconciliation having failed1 the only conclusion we can come to is that some unknown cause of a new character exists4 ,if the measurements be accurate and the theory probable1 then there remains a 99residual phenomenon991 which1 being devoid of theoretical explanation1 must be set down as a new empirical fact worthy of further investigation4 ,outstanding residual discrepancies have often been found to involve new discoveries of the greatest importance4 99,accordance of ,measurements of ,astronomical ,distances499 ,one of the most instructive instances which we can meet1 of the manner in which different measurements confirm or check each other1 is furnished by the determination of the velocity of light1 and the dimensions of the planetary system4 ,roemer first discovered that light requires time to travel1 by observing that the eclipses of ,jupiter's satellites1 although they occur at fixed moments of absolute time1 are visible at different moments in different parts of the earth's orbit1 according to the distance between the earth and ,jupiter4 ,the time occupied by light in traversing the mean semi-diameter of the earth's orbit is found to be about eight minutes4 ,the mean distance of the sun and earth was long assumed by astronomers as being about #ie1bgd1jjj miles1 this result being deduced by ,bessel from the observations of the transit of ,venus1 which occurred in #agfi1 and which were found to give the solar parallax1 or which is the same thing1 the apparent angular magnitude of the earth seen from the sun1 as equal to #hegh4 ,dividing the mean distance of the sun and earth by the number of seconds in #hm4 #acs4#c we find the velocity of light to be about #aib1jjj miles per second4 ,nearly the same result was obtained in what seems a different manner4 ,the aberration of light is the apparent change in the direction of a ray of light owing to the composition of its motion with that of the earth's motion round the sun4 ,if we know the amount of aberration and the mean velocity of the earth1 we can estimate that of light1 which is thus found to be #aia1ajj miles per second4 ,now this determination depends upon a new physical quantity1 that of aberration1 which is ascertained by direct observation of the stars1 so that the close accordance of the estimates of the velocity of light as thus arrived at by different methods might seem to leave little room for doubt1 the difference being less than one per cent4 ,nevertheless1 experimentalists were not satisfied until they had succeeded in measuring the velocity of light by direct experiments performed upon the earth's surface4 ,fizeau1 by a rapidly revolving toothed wheel1 estimated the velocity at #aie1ibj miles per second4 ,as this result differed by about one part in sixty from estimates previously accepted1 there was thought to be room for further investigation4 ,the revolving mirror1 used by ,wheatstone in measuring the velocity of electricity1 was now applied in a more refined manner by ,fizeau and by ,foucault to determine the velocity of light4 ,the latter physicist came to the startling conclusion that the velocity was not really more than #ahe1agb miles per second4 ,no repetition of the experiment would shake this result1 and there was accordingly a discrepancy between the astronomical and the experimental results of about #g1jjj miles per second4 ,the latest experiments1 those of ,m4 ,cornu1 only slightly raise the estimate1 giving #ahf1ffj miles per second4 ,a little consideration shows that both the astronomical determinations involve the magnitude of the earth's orbit as one datum1 because our estimate of the earth's velocity in its orbit depends upon our estimate of the sun's mean distance4 ,accordingly as regards this quantity the two astronomical results count only for one4 ,though the transit of ,venus had been considered to give the best data for the calculation of the sun's parallax1 yet astronomers had not neglected less favourable opportunities4 ,hansen1 calculating from certain inequalities in the moon's motion1 had estimated it at #hiaf2 ,winneke1 from observations of ,mars1 at #hifd2 ,leverrier1 from the motions of ,mars1 ,venus1 and the moon1 at #hiej4 ,these independent results agree much better with each other than with that of ,bessel 7#hegh7 previously received1 or that of ,encke 7#heh7 deduced from the transits of ,venus in #agfa and #agfi1 and though each separately might be worthy of less credit1 yet their close accordance renders their mean result 7#hidc7 comparable in probability with that of ,bessel4 ,it was further found that if ,foucault's value for the velocity of light were assumed to be correct1 and the sun's distance were inversely calculated from that1 the sun's parallax would be #hifj1 which closely agreed with the above mean result4 ,this further correspondence of independent results threw the balance of probability strongly against the results of the transit of ,venus1 and rendered it desirable to reconsider the observations made on that occasion4 ,mr4 ,e4 ,j4 ,stone1 having re-discussed those observations1#dfh found that grave oversights had been made in the calculations1 which being corrected would alter the estimate of parallax to #hia1 a quantity in such comparatively close accordance with the other results that astronomers did not hesitate at once to reduce their estimate of the sun's mean distance from #ie1bgd1jjj to #ia1gga1jjj1 miles1 although this alteration involved a corresponding correction in the assumed magnitudes and distances of most of the heavenly bodies4 ,the solar parallax is now 7#ahge7 believed to be about #hhgh1 the number deduced from ,cornu's experiments on the velocity of light4 ,this result agrees very closely with #hhgi1 the estimate obtained from new observations on the transit of ,venus1 by the ,french observers1 and with #hhgc1 the result of ,galle's observations of the planet ,flora4 ,when all the observations of the late transit of ,venus are fully discussed the sun's distance will probably be known to less than one part in a thousand1 if not one part in ten thousand4#dfi #dfh 99,monthly ,notices of the ,royal ,astronomical ,society991 vol4 xxviii4 p4 #bfd4 #dfi ,it would seem to be absurd to repeat the profuse expenditure of #ahgd at the approaching transit in #ahhb4 ,the aggregate sum spent in #ahgd by various governments and individuals can hardly be less than #bjj1jjj1 a sum which1 wisely expended on scientific investigations1 would give a hundred important results4 ,in this question the theoretical relations between the velocity of light1 the constant of aberration1 the sun's parallax1 and the sun's mean distance1 are of the simplest character1 and can hardly be open to any doubt1 so that the only doubt was as to which result of observation was the most reliable4 ,eventually the chief discrepancy was found to arise from misapprehension in the reduction of observations1 but we have a satisfactory example of the value of different methods of estimation in leading to the detection of a serious error4 ,is it not surprising that ,foucault by measuring the velocity of light when passing through the space of a few yards1 should lead the way to a change in our estimates of the magnitudes of the whole universe8 99,selection of the best ,mode of ,measurement499 ,when we once obtain command over a question of physical science by comprehending the theory of the subject1 we often have a wide choice opened to us as regards the methods of measurement1 which may thenceforth be made to give the most accurate results4 ,if we can measure one fundamental quantity very precisely we may be able by theory to determine accurately many other quantitative results4 ,thus1 if we determine satisfactorily the atomic weights of certain elements1 we do not need to determine with equal accuracy the composition and atomic weights of their several compounds4 ,having learnt the relative atomic weights of oxygen and sulphur1 we can calculate the composition by weight of the several oxides of sulphur4 ,chemists accordingly select with the greatest care that compound of two elements which seems to allow of the most accurate analysis1 so as to give the ratio of their atomic weights4 ,it is obvious that we only need the ratio of the atomic weight of each element to that of some common element1 in order to calculate1 that of each to each4 ,moreover the atomic weight stands in simple relation to other quantitative facts4 ,the weights of equal volumes of elementary gases at equal temperature and pressure have the same ratios as the atomic weights2 now1 as nitrogen under such circumstances weighs #adjf times as much as hydrogen1 we may infer that the atomic weight of nitrogen is about #adjf1 or more probably #adjj1 that of hydrogen being unity4 ,there is much evidence1 again1 that the specific heats of elements are inversely as their atomic weights1 so that these two classes of quantitative data throw light mutually upon each other4 ,in fact the atomic weight1 the atomic volume1 and the atomic heat of an element1 are quantities so closely connected that the determination of one will lead to that of the others4 ,the chemist has to solve a complicated problem in deciding in the case of each of #fj or #gj elements which mode of determination is most accurate4 ,modern chemistry presents us with an almost infinitely extensive web of numerical ratios developed out of a few fundamental ratios4 ,in hygrometry we have a choice among at least four modes of measuring the quantity of aqueous vapour contained in a given bulk of air4 ,we can extract the vapour by absorption in sulphuric acid1 and directly weigh its amount2 we can place the air in a barometer tube and observe how much the absorption of the vapour alters the elastic force of the air2 we can observe the dew-point of the air1 that is the temperature at which the vapour becomes saturated2 or1 lastly1 we can insert a dry and wet bulb thermometer and observe the temperature of an evaporating surface4 ,the results of each mode can be connected by theory with those of the other modes1 and we can select for each experiment that mode which is most accurate or most convenient4 ,the chemical method of direct measurement is capable of the greatest accuracy1 but is troublesome2 the dry and wet bulb thermometer is sufficiently exact for meteorological purposes and is most easy to use4 99,agreement of ,distinct ,modes of ,measurement499 ,many illustrations might be given of the accordance which has been found to exist in some cases between the results of entirely different methods of arriving at the measurement of a physical quantity4 ,while such accordance must1 in the absence of information to the contrary1 be regarded as the best possible proof of the approximate correctness of the mean result1 yet instances have occurred to show that we can never take too much trouble in confirming results of great importance4 ,when three or even more distinct methods have given nearly coincident numbers1 a new method has sometimes disclosed a discrepancy which it is yet impossible to explain4 ,the ellipticity of the earth is known with considerable approach to certainty and accuracy1 for it has been estimated in three independent ways4 ,the most direct mode is to measure long arcs extending north and south upon the earth's surface1 by means of trigonometrical surveys1 and then to compare the lengths of these arcs with their curvature as determined by observations of the altitude of certain stars at the terminal points4 ,the most probable ellipticity of the earth deduced from all measurements of this kind was estimated by ,bessel at #a_/cjj1 though subsequent measurements might lead to a slightly different estimate4 ,the divergence from a globular form causes a small variation in the force of gravity at different parts of the earth's surface1 so that exact pendulum observations give the data for an independent estimate of the ellipticity1 which is thus found to be #a_/cbj4 ,in the third place the spheroidal protuberance about the earth's equator leads to a certain inequality in the moon's motion1 as shown by ,laplace2 and from the amount of that inequality1 as given by observations1 ,laplace was enabled to calculate back to the amount of its cause4 ,he thus inferred that the ellipticity is #a_/cje1 which lies between the two numbers previously given1 and was considered by him the most satisfactory determination4 ,in this case the accordance is undisturbed by subsequent results1 so that we are obliged to accept ,laplace's result as a highly probable one4 ,the mean density of the earth is a constant of high importance1 because it is necessary for the determination of the masses of all the other heavenly bodies4 ,astronomers and physicists accordingly have bestowed a great deal of labour upon the exact estimation of this constant4 ,the method of procedure consists in comparing the gravitation of the globe with that of some body of matter of which the mass is known in terms of the assumed unit of mass4 ,this body of matter1 serving as an intermediate term of comparison1 may be variously chosen2 it may consist of a mountain1 or a portion of the earth's crust1 or a heavy ball of metal4 ,the method of experiment varies so much according as we select one body or the other1 that we may be said to have three independent modes of arriving at the desired result4 ,the mutual gravitation of two balls is so exceedingly small compared with their gravitation towards the immense mass of the earth1 that it is usually quite imperceptible1 and although asserted by ,newton to exist1 on the ground of theory1 was never observed until the end of the #ahth century4 ,michell attached two small balls to the extremities of a delicately suspended torsion balance1 and then bringing heavy balls of lead alternately to either side of these small balls was able to detect a slight deflection of the torsion balance4 ,he thus furnished a new verification of the theory of gravitation4 ,cavendish carried out the experiment with more care1 and estimated the gravitation of the balls by treating the torsion balance as a pendulum2 then taking into account the respective distances of the balls from each other and from the centre of the earth1 he was able to assign #edh 7or as re-computed by ,baily1 #eddh7 as the probable mean density of the earth4 ,newton's sagacious guess to the effect that the density of the earth was between five and six times that of water1 was thus remarkably confirmed4 ,the same kind of experiment repeated by ,reich gave #edch4 ,baily having again performed the experiment with every possible refinement obtained a slightly higher number1 #effj4 ,a different method of procedure consisted in ascertaining the effect of a mountain mass in deflecting the plumb-line2 for1 assuming that we can determine the dimensions and mean density of the mountain1 the plumb-line enables us to compare its mass with that of the whole earth4 ,the mountain ,schehallien was selected for the experiment1 and observations and calculations performed by ,maskelyne1 ,hutton1 and ,playfair1 gave as the most probable result #dgac4 ,the difference from the experimental results already mentioned is considerable and is important1 because the instrumental operations are of an entirely different character from those of ,cavendish and ,baily's experiments4 ,sir ,henry ,james' similar determination from the attraction of ,arthur's ,seat gave #ead4 ,a third distinct method consists in determining the force of gravity at points elevated above the surface of the earth on mountain ranges1 or sunk below it in mines4 ,carlini experimented with a pendulum at the hospice of ,mont ,cenis1 #f1cge feet above the sea1 and by comparing the attractive forces of the earth and the ,alps1 found the density to be still smaller1 namely1 #dci1 or as corrected by ,giulio1 #diej4 ,lastly1 the ,astronomer ,royal has on two occasions adopted the opposite method of observing a pendulum at the bottom of a deep mine1 so as to compare the density of the strata penetrated with the density of the whole earth4 ,on the second occasion he carried his method into effect at the ,harton ,colliery1 #a1bfj feet deep2 all that could be done by skill in measurement and careful consideration of all the causes of error1 was accomplished in this elaborate series of observations#dgj 7p4 #bia74 ,no doubt ,sir ,george ,airy was much perplexed when he found that his new result considerably exceeded that obtained by any other method1 being no less than #feff1 or #ffbc as finally corrected4 ,in this case we learn an impressive lesson concerning the value of repeated determinations by distinct methods in disabusing our minds of the reliance which we are only too apt to place in results which show a certain degree of coincidence4 #dgj 99,philosophical ,transactions99 7#ahef71 vol4 cxlvi4 p4 #cdb4 ,in #ahdd ,herschel remarked in his memoir of ,francis ,baily1#dga 8that the mean specific gravity of this our planet is1 in all human probability1 quite as well determined as that of an ordinary hand-specimen in a mineralogical cabinet1--a marvellous result1 which should teach us to despair of nothing which lies within the compass of number1 weight and measure40 ,but at the same time he pointed out that ,baily's final result1 of which the probable error was only #jjjcb1 was the highest of all determinations then known1 and ,airy's investigation has since given a much higher result1 quite beyond the limits of probable error of any of the previous experiments4 ,if we treat all determinations yet made as of equal weight1 the simple mean is about #ede1 the mean error nearly #je1 and the probable error almost #jb1 so that it is as likely as not that the truth lies between #efe and #ebe on this view of the matter4 ,but it is remarkable that the two most recent and careful series of observations by ,baily and ,airy1#dgb lie beyond these limits1 and as with the increase of care the estimate rises1 it seems requisite to reject the earlier results1 and look upon the question as still requiring further investigation4 ,physicists often take #e-b_/c or #efg as the best guess at the truth1 but it is evident that new experiments are much required4 ,i cannot help thinking that a portion of the great sums of money which many governments and private individuals spent upon the transit of ,venus expeditions in #ahgd1 and which they will probably spend again in #ahhb 7p4 #efb71 would be better appropriated to new determinations of the earth's density4 ,it seems desirable to repeat ,baily's experiment in a vacuous case1 and with the greater mechanical refinements which the progress of the last forty years places at the disposal of the experimentalist4 ,it would be desirable1 also1 to renew the pendulum experiments of ,airy in some other deep mine4 ,it might even be well to repeat upon some suitable mountain the observations performed at ,schehallien4 ,all these operations might be carried out for the cost of one of the superfluous transit expeditions4 #dga 99,monthly ,notices of the ,royal ,astronomical ,society991 for #hth ,nov4 #ahdd1 ,no4 ,x4 vol4 vi4 p4 #hi4 #dgb 99,philosophical ,magazine991 #bnd ,series1 vol4 xxvi4 p4 #fa4 ,since the establishment of the dynamical theory of heat it has become a matter of the greatest importance to determine with accuracy the mechanical equivalent of heat1 or the quantity of energy which must be given1 or received1 in a definite change of temperature effected in a definite quantity of a standard substance1 such as water4 ,no less than seven almost entirely distinct modes of determining this constant have been tried4 ,dr4 ,joule first ascertained by the friction of water that to raise the temperature of one kilogram of water through one degree centigrade1 we must employ energy sufficient to raise #dbd kilograms through the height of one metre against the force of gravity at the earth's surface4 ,joule1 ,mayer1 ,clausius1#dgc ,favre and other experimentalists have made determinations by less direct methods4 ,experiments on the mechanical properties of gases give #dbf kilogrammetres as the constant2 the work done by a steam-engine gives #dac2 from the heat evolved in electrical experiments several determinations have been obtained2 thus from induced electric currents we get #deb2 from the electro-magnetic engine #ddc2 from the circuit of a battery #dbj2 and1 from an electric current1 the lowest result of all1 namely1 #djj4dgd #dgc ,clausius in 99,philosophical ,magazine991 #dth ,series1 vol4 ii4 p4 #aai4 #dgd ,watts' 99,dictionary of ,chemistry991 vol4 iii4 p4 #abi4 ,considering the diverse and in many cases difficult methods of observation1 these results exhibit satisfactory accordance1 and their mean 7#dbci7 comes very close to the number derived by ,dr4 ,joule from the apparently most accurate method4 ,the constant generally assumed as the most probable result is #dbcee kilogrammetres4 99,residual ,phenomena499 ,even when the experimental data employed in the verification of a theory are sufficiently accurate1 and the theory itself is sound1 there may exist discrepancies demanding further investigation4 ,herschel pointed out the importance of such outstanding quantities1 and called them 99residual phenomena994#dge ,now if the observations and the theory be really correct1 such discrepancies must be due to the incompleteness of our knowledge of the causes in action1 and the ultimate explanation must consist in showing that there is in action1 either #dge 99,preliminary ,discourse991 #aeh1 #agd4 99,outlines of ,astronomy991 #dth edit4 #hef4 7#a7 ,some agent of known nature whose presence was not suspected2 ,or 7#b7 ,some new agent of unknown nature4 ,in the first case we can hardly be said to make a new discovery1 for our ultimate success consists merely in reconciling the theory with known facts when our investigation is more comprehensive4 ,but in the second case we meet with a totally new fact1 which may lead us to realms of new discovery4 ,take the instance adduced by ,herschel4 ,the theory of ,newton and ,halley concerning comets was that they were gravitating bodies revolving round the sun in elliptic orbits1 and the return of ,halley's ,comet1 in #ageh1 verified this theory4 ,but1 when accurate observations of ,encke's ,comet came to be made1 the verification was not found to be exact4 ,encke's ,comet returned each time a little sooner than it ought to do1 the period regularly decreasing from #ababgi days1 between #aghf and #aghi1 to #abajdd between #ahee and #aheh2 and the hypothesis has been started that there is a resisting medium filling the space through which the comet passes4 ,this hypothesis is a 99deus ex machin99 for explaining this solitary phenomenon1 and cannot possess much probability unless it can be shown that other phenomena are deducible from it4 ,many persons have identified this medium with that through which light undulations pass1 but ,i am not aware that there is anything in the undulatory theory of light to show that the medium would offer resistance to a moving body4 ,if ,professor ,balfour ,stewart can prove that a rotating disc would experience resistance in a vacuous receiver1 here is an experimental fact which distinctly supports the hypothesis4 ,but in the mean time it is open to question whether other known agents1 for instance electricity1 may not be brought in1 and ,i have tried to show that if1 as is believed1 the tail of a comet is an electrical phenomenon1 it is a necessary result of the conservation of energy that the comet shall exhibit a loss of energy manifested in a diminution of its mean distance from the sun and its period of revolution4#dgf ,it should be added that if ,professor ,tait's theory be correct1 as seems very probable1 and comets consist of swarms of small meteors1 there is no difficulty in accounting for the retardation4 ,it has long been known that a collection of small bodies travelling together in an orbit round a central body will tend to fall towards it4 ,in either case1 then1 this residual phenomenon seems likely to be reconciled with known laws of nature4 #dgf 99,proceedings of the ,manchester ,literary and ,philosophical ,society991 #bhth ,november1 #ahga1 vol4 xi4 p4 #cc4 ,since the above remarks were written1 ,professor ,balfour ,stewart has pointed out to me his paper in the 99,proceedings of the ,manchester ,literary and ,philosophical ,society99 for #aeth ,november1 #ahgj 7vol4 x4 p4 #cb71 in which he shows that a body moving in an enclosure of uniform temperature would probably experience resistance independently of the presence of a ponderable medium1 such as gas1 between the moving body and the enclosure4 ,the proof is founded on the theory of the dissipation of energy1 and this view is said to be accepted by ,professors ,thomson and ,tait4 ,the enclosure is used in this case by ,professor ,stewart simply as a means of obtaining a proof1 just as it was used by him on a previous occasion to obtain a proof of certain consequences of the ,theory of ,exchanges4 ,he is of opinion that in both of these cases when once the proof has been obtained1 the enclosure may be dispensed with4 ,we know1 for instance1 that the relation between the inductive and absorptive powers of bodies--although this relation may have been proved by means of an enclosure1 does not depend upon its presence1 and ,professor ,stewart thinks that in like manner two bodies1 or at least two bodies possessing heat such as the sun and the earth in motion relative to each other1 will have the differential motion retarded until perhaps it is ultimately destroyed4 ,in other cases residual phenomena have involved important inferences not recognised at the time4 ,newton showed how the velocity of sound in the atmosphere could be calculated by a theory of pulses or undulations from the observed tension and density of the air4 ,he inferred that the velocity in the ordinary state of the atmosphere at the earth's surface would be #ifh feet per second1 and rude experiments made by him in the cloisters of ,trinity ,college seemed to show that this was not far from the truth4 ,subsequently it was ascertained by other experimentalists that the velocity of sound was more nearly #a1adb feet1 and the discrepancy being one-sixth part of the whole was far too much to attribute to casual errors in the numerical data4 ,newton attempted to explain away this discrepancy by hypotheses as to the reactions of the molecules of air1 but without success4 ,new investigations having been made from time to time concerning the velocity of sound1 both as observed experimentally and as calculated from theory1 it was found that each of ,newton's results was inaccurate1 the theoretical velocity being #iaf feet per second1 and the real velocity about #a1jij feet4 ,the discrepancy1 nevertheless1 remained as serious as ever1 and it was not until the year #ahaf that ,laplace showed it to be due to the heat developed by the sudden compression of the air in the passage of the wave1 this heat having the effect of increasing the elasticity of the air and accelerating the impulse4 ,it is now perceived that this discrepancy really involves the doctrine of the equivalence of heat and energy1 and it was applied by ,mayer1 at least by implication1 to give an estimate of the mechanical equivalent of heat4 ,the estimate thus derived agrees satisfactorily with direct determinations by ,dr4 ,joule and other physicists1 so that the explanation of the residual phenomenon which exercised ,newton's ingenuity is now complete1 and forms an important part of the new science of thermodynamics4 ,as ,herschel observed1 almost all great astronomical discoveries have been disclosed in the form of residual differences4 ,it is the practice at well-conducted observatories to compare the positions of the heavenly bodies as actually observed with what might have been expected theoretically4 ,this practice was introduced by ,halley when ,astronomer ,royal1 and his reduction of the lunar observations gave a series of residual errors from #agbb to #agci1 by the examination of which the lunar theory was improved4 ,most of the greater astronomical variations arising from nutation1 aberration1 planetary perturbation were discovered in the same manner4 ,the precession of the equinox was perhaps the earliest residual difference observed2 the systematic divergence of ,uranus from its calculated places was one of the latest1 and was the clue to the remarkable discovery of ,neptune4 ,we may also class under residual phenomena all the so-called 99proper motions99 of the stars4 ,a complete star catalogue1 such as that of the ,british ,association1 gives a greater or less amount of proper motion for almost every star1 consisting in the apparent difference of position of the star as derived from the earliest and latest good observations4 ,but these apparent motions are often due1 as explained by ,baily1#dgg the author of the catalogue1 to errors of observation and reduction4 ,in many cases the best astronomical authorities have differed as to the very direction of the supposed proper motion of stars1 and as regards the amount of the motion1 for instance of ,polaris1 the most different estimates have been formed4 ,residual quantities will often be so small that their very existence is doubtful4 ,only the gradual progress of theory and of measurement will show clearly whether a discrepancy is to be referred to casual errors of observation or to some new phenomenon4 ,but nothing is more requisite for the progress of science than the careful recording and investigation of such discrepancies4 ,in no part of physical science can we be free from exceptions and outstanding facts1 of which our present knowledge can give no account4 ,it is among such anomalies that we must look for the clues to new realms of facts worthy of discovery4 ,they are like the floating waifs which led ,columbus to suspect the existence of the new world4 #dgg 99,british ,association ,catalogue of ,stars991 p4 #di4 ,,chapter ,,xxvi4 ,,character ,,of ,,the ,,experimentalist4 ,in the present age there seems to be a tendency to believe that the importance of individual genius is less than it was-- 8,the individual withers1 and the world is more and more40 ,society1 it is supposed1 has now assumed so highly developed a form1 that what was accomplished in past times by the solitary exertions of a great intellect1 may now be worked out by the united labours of an army of investigators4 ,just as the well-organised power of a modern army supersedes the single-handed bravery of the medival knights1 so we are to believe that the combination of intellectual labour has superseded the genius of an ,archimedes1 a ,newton1 or a ,laplace4 ,so-called original research is now regarded as a profession1 adopted by hundreds of men1 and communicated by a system of training4 ,all that we need to secure additions to our knowledge of nature is the erection of great laboratories1 museums1 and observatories1 and the offering of pecuniary rewards to those who can invent new chemical compounds1 detect new species1 or discover new comets4 ,doubtless this is not the real meaning of the eminent men who are now urging upon ,government the endowment of physical research4 ,they can only mean that the greater the pecuniary and material assistance given to men of science1 the greater the result which the available genius of the country may be expected to produce4 ,money and opportunities of study can no more produce genius than sunshine and moisture can generate living beings2 the inexplicable germ is wanting in both cases4 ,but as1 when the germ is present1 the plant will grow more or less vigorously according to the circumstances in which it is placed1 so it may be allowed that pecuniary assistance may favour development of intellect4 ,public opinion however is not discriminating1 and is likely to interpret the agitation for the endowment of science as meaning that science can be had for money4 ,all such notions are erroneous4 ,in no branch of human affairs1 neither in politics1 war1 literature1 industry1 nor science1 is the influence of genius less considerable than it was4 ,it is possible that the extension and organisation of scientific study1 assisted by the printing-press and the accelerated means of communication1 has increased the rapidity with which new discoveries are made known1 and their details worked out by many heads and hands4 ,a ,darwin now no sooner propounds original ideas concerning the evolution of living creatures1 than those ideas are discussed and illustrated1 and applied by naturalists in every part of the world4 ,in former days his discoveries would have been hidden for decades of years in scarce manuscripts1 and generations would have passed away before his theory had enjoyed the same amount of criticism and corroboration as it has already received4 ,the result is that the genius of ,darwin is more valuable1 not less valuable1 than it would formerly have been4 ,the advance of military science and the organisation of enormous armies has not decreased the value of a skilful general2 on the contrary1 the rank and file are still more in need than they used to be of the guiding power of a far-seeing intellect4 ,the swift destruction of the ,french military power was not due alone to the perfection of the ,german army1 nor to the genius of ,moltke2 it was due to the combination of a well-disciplined multitude with a leader of the highest powers4 ,so in every branch of human affairs the influence of the individual is not withering1 but is growing with the extent of the material resources which are at his command4 ,turning to our own subject1 it is a work of undiminished interest to reflect upon those qualities of mind which lead to great advances in natural knowledge4 ,nothing1 indeed1 is less amenable than genius to scientific analysis and explanation4 ,even definition is out of the question4 ,buffon said that 8genius is patience10 and certainly patience is one of its most requisite components4 ,but no one can suppose that patient labour alone will invariably lead to those conspicuous results which we attribute to genius4 ,in every branch of science1 literature1 art1 or industry1 there are thousands of men and women who work with unceasing patience1 and thereby ensure moderate success2 but it would be absurd to suppose that equal amounts of intellectual labour yield equal results4 ,a ,newton may modestly attribute his discoveries to industry and patient thought1 and there is reason to believe that genius is unconscious and unable to account for its own peculiar powers4 ,as genius is essentially creative1 and consists in divergence from the ordinary grooves of thought and action1 it must necessarily be a phenomenon beyond the domain of the laws of nature4 ,nevertheless1 it is always an interesting and instructive work to trace out1 as far as possible1 the characteristics of mind by which great discoveries have been achieved1 and we shall find in the analysis much to illustrate the principles of scientific method4 99,error of the ,baconian ,method499 ,hundreds of investigators may be constantly engaged in experimental inquiry2 they may compile numberless note-books full of scientific facts1 and endless tables of numerical results2 but1 if the views of induction here maintained be true1 they can never by such work alone rise to new and great discoveries4 ,by a system of research they may work out deductively the details of a previous discovery1 but to arrive at a new principle of nature is another matter4 ,francis ,bacon spread abroad the notion that to advance science we must begin by accumulating facts1 and then draw from them1 by a process of digestion1 successive laws of higher and higher generality4 ,in protesting against the false method of the scholastic logicians1 he exaggerated a partially true philosophy1 until it became as false as that which preceded it4 ,his notion of scientific method was a kind of scientific bookkeeping4 ,facts were to be indiscriminately gathered from every source1 and posted in a ledger1 from which would emerge in time a balance of truth4 ,it is difficult to imagine a less likely way of arriving at great discoveries4 ,the greater the array of facts1 the less is the probability that they will by any routine system of classification disclose the laws of nature they embody4 ,exhaustive classification in all possible orders is out of the question1 because the possible orders are practically infinite in number4 ,it is before the glance of the philosophic mind that facts must display their meaning1 and fall into logical order4 ,the natural philosopher must therefore have1 in the first place1 a mind of impressionable character1 which is affected by the slightest exceptional phenomenon4 ,his associating and identifying powers must be great1 that is1 a strange fact must suggest to his mind whatever of like nature has previously come within his experience4 ,his imagination must be active1 and bring before his mind multitudes of relations in which the unexplained facts may possibly stand with regard to each other1 or to more common facts4 ,sure and vigorous powers of deductive reasoning must then come into play1 and enable him to infer what will happen under each supposed condition4 ,lastly1 and above all1 there must be the love of certainty leading him diligently and with perfect candour1 to compare his speculations with the test of fact and experiment4 99,freedom of ,theorising499 ,it would be an error to suppose that the great discoverer seizes at once upon the truth1 or has any unerring method of divining it4 ,in all probability the errors of the great mind exceed in number those of the less vigorous one4 ,fertility of imagination and abundance of guesses at truth are among the first requisites of discovery2 but the erroneous guesses must be many times as numerous as those which prove well founded4 ,the weakest analogies1 the most whimsical notions1 the most apparently absurd theories1 may pass through the teeming brain1 and no record remain of more than the hundredth part4 ,there is nothing really absurd except that which proves contrary to logic and experience4 ,the truest theories involve suppositions which are inconceivable1 and no limit can really be placed to the freedom of hypothesis4 ,kepler is an extraordinary instance to this effect4 ,no minor laws of nature are more firmly established than those which he detected concerning the orbits and motions of planetary masses1 and on these empirical laws the theory of gravitation was founded4 ,did we not learn from his own writings the multitude of errors into which he fell1 we might have imagined that he had some special faculty of seizing on the truth4 ,but1 as is well known1 he was full of chimerical notions2 his favourite and long-studied theory was founded on a fanciful analogy between the planetary orbits and the regular solids4 ,his celebrated laws were the outcome of a lifetime of speculation1 for the most part vain and groundless4 ,we know this because he had a curious pleasure in dwelling upon erroneous and futile trains of reasoning1 which most persons consign to oblivion4 ,but ,kepler's name was destined to be immortal1 on account of the patience with which he submitted his hypotheses to comparison with observation1 the candour with which he acknowledged failure after failure1 and the perseverance and ingenuity with which he renewed his attack upon the riddles of nature4 ,next after ,kepler perhaps ,faraday is the physical philosopher who has given us the best insight into the progress of discovery1 by recording erroneous as well as successful speculations4 ,the recorded notions1 indeed1 are probably but a tithe of the fancies which arose in his active brain4 ,as ,faraday himself said--8,the world little knows how many of the thoughts and theories which have passed through the mind of a scientific investigator1 have been crushed in silence and secrecy by his own severe criticism and adverse examination2 that in the most successful instances not a tenth of the suggestions1 the hopes1 the wishes1 the preliminary conclusions have been realised40 ,nevertheless1 in ,faraday's researches1 published in the 99,philosophical ,transactions991 in minor papers1 in manuscript note-books1 or in other materials1 made known in his interesting life by ,dr4 ,bence ,jones1 we find invaluable lessons for the experimentalist4 ,these writings are full of speculations which we must not judge by the light of subsequent discovery4 ,it may perhaps be said that ,faraday committed to the printing press crude ideas which a friend would have counselled him to keep back4 ,there was occasionally even a wildness and vagueness in his notions1 which in a less careful experimentalist would have been fatal to the attainment of truth4 ,this is especially apparent in a curious paper concerning ,ray-vibrations2 but fortunately ,faraday was aware of the shadowy character of his speculations1 and expressed the feeling in words which must be quoted4 8,i think it likely10 he says1#dgh 8that ,i have made many mistakes in the preceding pages1 for even to myself my ideas on this point appear only as the shadow of a speculation1 or as one of those impressions upon the mind1 which are allowable for a time as guides to thought and research4 ,he who labours in experimental inquiries knows how numerous these are1 and how often their apparent fitness and beauty vanish before the progress and development of real natural truth40 ,if1 then1 the experimentalist has no royal road to the discovery of the truth1 it is an interesting matter to consider by what logical procedure he attains the truth4 #dgh 99,experimental ,researches in ,chemistry and ,physics991 p4 #cgb4 99,philosophical ,magazine991 #crd ,series1 ,may #ahdf1 vol4 xxviii4 p4 #cej4 ,if ,i have taken a correct view of logical method1 there is really no such thing as a distinct process of induction4 ,the probability is infinitely small that a collection of complicated facts will fall into an arrangement capable of exhibiting directly the laws obeyed by them4 ,the mathematician might as well expect to integrate his functions by a ballot-box1 as the experimentalist to draw deep truths from haphazard trials4 ,all induction is but the inverse application of deduction1 and it is by the inexplicable action of a gifted mind that a multitude of heterogeneous facts are ranged in luminous order as the results of some uniformly acting law4 ,so different1 indeed1 are the qualities of mind required in different branches of science1 that it would be absurd to attempt to give an exhaustive description of the character of mind which leads to discovery4 ,the labours of ,newton could not have been accomplished except by a mind of the utmost mathematical genius2 ,faraday1 on the other hand1 has made the most extensive additions to human knowledge without passing beyond common arithmetic4 ,i do not remember meeting in ,faraday's writings with a single algebraic formula or mathematical problem of any complexity4 ,professor ,clerk ,maxwell1 indeed1 in the preface to his new 99,treatise on ,electricity991 has strongly recommended the reading of ,faraday's researches by all students of science1 and has given his opinion that though ,faraday seldom or never employed mathematical formul1 his methods and conceptions were not the less mathematical in their nature4#dgi ,i have myself protested against the prevailing confusion between a mathematical and an exact science1#dhj yet ,i certainly think that ,faraday's experiments were for the most part qualitative1 and that his mathematical ideas were of a rudimentary character4 ,it is true that he could not possibly investigate such a subject as magne-crystallic action without involving himself in geometrical relations of some complexity4 ,nevertheless ,i think that he was deficient in mathematical deductive power1 that power which is so highly developed by the modern system of mathematical training at ,cambridge4 #dgi ,see also 99,nature991 ,september #ah1 #ahgc2 vol4 viii4 p4 #cih4 #dhj 99,theory of ,political ,economy991 pp4 #c-ad4 ,faraday was acquainted with the forms of his celebrated lines of force1 but ,i am not aware that he ever entered into the algebraic nature of those curves1 and ,i feel sure that he could not have explained their forms as depending on the resultant attractions of all the magnetic particles4 ,there are even occasional indications that he did not understand some of the simpler mathematical doctrines of modern physical science4 ,although he so clearly foresaw the correlation of the physical forces1 and laboured so hard with his own hands to connect gravity with other forces1 it is doubtful whether he understood the doctrine of the conservation of energy as applied to gravitation4 ,faraday was probably equal to ,newton in experimental skill1 and in that peculiar kind of deductive power which leads to the invention of simple qualitative experiments2 but it must be allowed that he exhibited little of that mathematical power which enabled ,newton to follow out intuitively the quantitative results of a complicated problem with such wonderful facility4 ,two instances1 ,newton and ,faraday1 are sufficient to show that minds of widely different conformation will meet with suitable regions of research4 ,nevertheless1 there are certain traits which we may discover in all the highest scientific minds4 99,the ,newtonian ,method1 the ,true ,organum499 ,laplace was of opinion that the 99,principia99 and the 99,opticks99 of ,newton furnished the best models then available of the delicate art of experimental and theoretical investigation4 ,in these1 as he says1 we meet with the most happy illustrations of the way in which1 from a series of inductions1 we may rise to the causes of phenomena1 and thence descend again to all the resulting details4 ,the popular notion concerning ,newton's discoveries is that in early life1 when driven into the country by the ,great ,plague1 a falling apple accidentally suggested to him the existence of gravitation1 and that1 availing himself of this hint1 he was led to the discovery of the law of gravitation1 the explanation of which constitutes the 99,principia994 ,it is difficult to imagine a more ludicrous and inadequate picture of ,newton's labours4 ,no originality1 or at least priority1 was claimed by ,newton as regards the discovery of the law of the inverse square1 so closely associated with his name4 ,in a well-known ,scholium#dha he acknowledges that ,sir ,christopher ,wren1 ,hooke1 and ,halley1 had severally observed the accordance of ,kepler's third law of motion with the principle of the inverse square4 #dha 99,principia991 bk4 i4 ,prop4 iv4 ,newton's work was really that of developing the methods of deductive reasoning and experimental verification1 by which alone great hypotheses can be brought to the touchstone of fact4 ,archimedes was the greatest of ancient philosophers1 for he showed how mathematical theory could be wedded to physical experiments2 and his works are the first true ,organum4 ,newton is the modern ,archimedes1 and the 99,principia99 forms the true ,novum ,organum of scientific method4 ,the laws which he established are great1 but his example of the manner of establishing them is greater still4 ,excepting perhaps chemistry and electricity1 there is hardly a progressive branch of physical and mathematical science1 which has not been developed from the germs of true scientific procedure which he disclosed in the 99,principia99 or the 99,opticks994 ,overcome by the success of his theory of universal gravitation1 we are apt to forget that in his theory of sound he originated the mathematical investigation of waves and the mutual action of particles2 that in his corpuscular theory of light1 however mistaken1 he first ventured to apply mathematical calculation to molecular attractions and repulsions2 that in his prismatic experiments he showed how far experimental verification could be pushed2 that in his examination of the coloured rings named after him1 he accomplished the most remarkable instance of minute measurement yet known1 a mere practical application of which by ,fizeau was recently deemed worthy of a medal by the ,royal ,society4 ,we only learn by degrees how complete was his scientific insight2 a few words in his third law of motion display his acquaintance with the fundamental principles of modern thermodynamics and the conservation of energy1 while manuscripts long overlooked prove that in his inquiries concerning atmospheric refraction he had overcome the main difficulties of applying theory to one of the most complex of physical problems4 ,after all1 it is only by examining the way in which he effected discoveries1 that we can rightly appreciate his greatness4 ,the 99,principia99 treats not of gravity so much as of forces in general1 and the methods of reasoning about them4 ,he investigates not one hypothesis only1 but mechanical hypotheses in general4 ,nothing so much strikes the reader of the work as the exhaustiveness of his treatment1 and the unbounded power of his insight4 ,if he treats of central forces1 it is not one law of force which he discusses1 but many1 or almost all imaginable laws1 the results of each of which he sketches out in a few pregnant words4 ,if his subject is a resisting medium1 it is not air or water alone1 but resisting media in general4 ,we have a good example of his method in the scholium to the twenty-second proposition of the second book1 in which he runs rapidly over many suppositions as to the laws of the compressing forces which might conceivably act in an atmosphere of gas1 a consequence being drawn from each case1 and that one hypothesis ultimately selected which yields results agreeing with experiments upon the pressure and density of the terrestrial atmosphere4 ,newton said that he did not frame hypotheses1 but1 in reality1 the greater part of the 99,principia99 is purely hypothetical1 endless varieties of causes and laws being imagined which have no counterpart in nature4 ,the most grotesque hypotheses of ,kepler or ,descartes were not more imaginary4 ,but ,newton's comprehension of logical method was perfect2 no hypothesis was entertained unless it was definite in conditions1 and admitted of unquestionable deductive reasoning2 and the value of each hypothesis was entirely decided by the comparison of its consequences with facts4 ,i do not entertain a doubt that the general course of his procedure is identical with that view of the nature of induction1 as the inverse application of deduction1 which ,i advocate throughout this book4 ,francis ,bacon held that science should be founded on experience1 but he mistook the true mode of using experience1 and1 in attempting to apply his method1 ludicrously failed4 ,newton did not less found his method on experience1 but he seized the true method of treating it1 and applied it with a power and success never since equalled4 ,it is a great mistake to say that modern science is the result of the ,baconian philosophy2 it is the ,newtonian philosophy and the ,newtonian method which have led to all the great triumphs of physical science1 and ,i repeat that the 99,principia99 forms the true 8,novum ,organum40 ,in bringing his theories to a decisive experimental verification1 ,newton showed1 as a general rule1 exquisite skill and ingenuity4 ,in his hands a few simple pieces of apparatus were made to give results involving an unsuspected depth of meaning4 ,his most beautiful experimental inquiry was that by which he proved the differing refrangibility of rays of light4 ,to suppose that he originally discovered the power of a prism to break up a beam of white light would be a mistake1 for he speaks of procuring a glass prism to try the 8celebrated phenomena of colours40 ,but we certainly owe to him the theory that white light is a mixture of rays differing in refrangibility1 and that lights which differ in colour1 differ also in refrangibility4 ,other persons might have conceived this theory2 in fact1 any person regarding refraction as a quantitative effect must see that different parts of the spectrum have suffered different amounts of refraction4 ,but the power of ,newton is shown in the tenacity with which he followed his theory into every consequence1 and tested each result by a simple but conclusive experiment4 ,he first shows that different coloured spots are displaced by different amounts when viewed through a prism1 and that their images come to a focus at different distances from the lense1 as they should do1 if the refrangibility differed4 ,after excluding by many experiments a variety of indifferent circumstances1 he fixes his attention upon the question whether the rays are merely shattered1 disturbed1 and spread out in a chance manner1 as ,grimaldi supposed1 or whether there is a constant relation between the colour and the refrangibility4 ,if ,grimaldi was right1 it might be expected that a part of the spectrum taken separately1 and subjected to a second refraction1 would suffer a new breaking up1 and produce some new spectrum4 ,newton inferred from his own theory that a particular ray of the spectrum would have a constant refrangibility1 so that a second prism would merely bend it more or less1 but not further disperse it in any considerable degree4 ,by simply cutting off most of the rays of the spectrum by a screen1 and allowing the remaining narrow ray to fall on a second prism1 he proved the truth of this conclusion2 and then slowly turning the first prism1 so as to vary the colour of the ray falling on the second one1 he found that the spot of light formed by the twice-refracted ray travelled up and down1 a palpable proof that the amount of refrangibility varies with the colour4 ,for his further satisfaction1 he sometimes refracted the light a third or fourth time1 and he found that it might be refracted upwards or downwards or sideways1 and yet for each colour there was a definite amount of refraction through each prism4 ,he completed the proof by showing that the separated rays may again be gathered together into white light by an inverted prism1 so that no number of refractions alters the character of the light4 ,the conclusion thus obtained serves to explain the confusion arising in the use of a common lense2 he shows that with homogeneous light there is one distinct focus1 with mixed light an infinite number of foci1 which prevent a clear view from being obtained at any point4 ,what astonishes the reader of the 99,opticks99 is the persistence with which ,newton follows out the consequences of a preconceived theory1 and tests the one notion by a wonderful variety of simple comparisons with fact4 ,the ease with which he invents new combinations1 and foresees the results1 subsequently verified1 produces an insuperable conviction in the reader that he has possession of the truth4 ,and it is certainly the theory which leads him to the experiments1 most of which could hardly be devised by accident4 ,newton actually remarks that it was by mathematically determining all kinds of phenomena of colours which could be produced by refraction that he had 8invented0 almost all the experiments in the book1 and he promises that others who shall 8argue truly10 and try the experiments with care1 will not be disappointed in the results4#dhb #dhb 99,opticks991 bk4 i4 part ii4 ,prop4 #c4 #crd ed4 p4 #aae4 ,the philosophic method of ,huyghens was the same as that of ,newton1 and ,huyghens' investigation of double refraction furnishes almost equally beautiful instances of theory guiding experiment4 ,so far as we know double refraction was first discovered by accident1 and was described by ,erasmus ,bartholinus in #affi4 ,the phenomenon then appeared to be entirely exceptional1 and the laws governing the two paths of the refracted rays were so unapparent and complicated1 that ,newton altogether misunderstood the phenomenon1 and it was only at the latter end of the last century that scientific men began to comprehend its laws4 ,nevertheless1 ,huyghens had1 with rare genius1 arrived at the true theory as early as #afgh4 ,he regarded light as an undulatory motion of some medium1 and in his 99,trait de la ,lumire99 he pointed out that1 in ordinary refraction1 the velocity of propagation of the wave is equal in all directions1 so that the front of an advancing wave is spherical1 and reaches equal distances in equal times4 ,but in crystals1 as he supposed1 the medium would be of unequal elasticity in different directions1 so that a disturbance would reach unequal distances in equal times1 and the wave produced would have a spheroidal form4 ,huyghens was not satisfied with an unverified theory4 ,he calculated what might be expected to happen when a crystal of calc-spar was cut in various directions1 and he says3 8,i have examined in detail the properties of the extraordinary refraction of this crystal1 to see if each phenomenon which is deduced from theory would agree with what is really observed4 ,and this being so1 it is no slight proof of the truth of our suppositions and principles2 but what ,i am going to add here confirms them still more wonderfully2 that is1 the different modes of cutting this crystal1 in which the surfaces produced give rise to refraction exactly such as they ought to be1 and as ,i had foreseen them1 according to the preceding theory40 ,newton's mistaken corpuscular theory of light caused the theories and experiments of ,huyghens to be disregarded for more than a century2 but it is not easy to imagine a more beautiful or successful application of the true method of inductive investigation1 theory guiding experiment1 and yet wholly relying on experiment for confirmation4 99,candour and ,courage of the ,philosophic ,mind499 ,perfect readiness to reject a theory inconsistent with fact is a primary requisite of the philosophic mind4 ,but it would be a mistake to suppose that this candour has anything akin to fickleness2 on the contrary1 readiness to reject a false theory may be combined with a peculiar pertinacity and courage in maintaining an hypothesis as long as its falsity is not actually apparent4 ,there must1 indeed1 be no prejudice or bias distorting the mind1 and causing it to pass over the unwelcome results of experiment4 ,there must be that scrupulous honesty and flexibility of mind1 which assigns adequate value to all evidence2 indeed1 the more a man loves his theory1 the more scrupulous should be his attention to its faults4 ,it is common in life to meet with some theorist1 who1 by long cogitation over a single theory1 has allowed it to mould his mind1 and render him incapable of receiving anything but as a contribution to the truth of his one theory4 ,a narrow and intense course of thought may sometimes lead to great results1 but the adoption of a wrong theory at the outset is in such a mind irretrievable4 ,the man of one idea has but a single chance of truth4 ,the fertile discoverer1 on the contrary1 chooses between many theories1 and is never wedded to any one1 unless impartial and repeated comparison has convinced him of its validity4 ,he does not choose and then compare2 but he compares time after time1 and then chooses4 ,having once deliberately chosen1 the philosopher may rightly entertain his theory with the strongest fidelity4 ,he will neglect no objection2 for he may chance at any time to meet a fatal one2 but he will bear in mind the inconsiderable powers of the human mind compared with the tasks it has to undertake4 ,he will see that no theory can at first be reconciled with all objections1 because there may be many interfering causes1 and the very consequences of the theory may have a complexity which prolonged investigation by successive generations of men may not exhaust4 ,if1 then1 a theory exhibit a number of striking coincidences with fact1 it must not be thrown aside until at least one 99conclusive discordance99 is proved1 regard being had to possible error in establishing that discordance4 ,in science and philosophy something must be risked4 ,he who quails at the least difficulty will never establish a new truth1 and it was not unphilosophic in ,leslie to remark concerning his own inquiries into the nature of heat-- 8,in the course of investigation1 ,i have found myself compelled to relinquish some preconceived notions2 but ,i have not abandoned them hastily1 nor1 till after a warm and obstinate defence1 ,i was driven from every post40#dhc #dhc 99,experimental ,inquiry into the ,nature of ,heat499 ,preface1 p4 xv4 ,faraday's life1 again1 furnishes most interesting illustrations of this tenacity of the philosophic mind4 ,though so candid in rejecting some theories1 there were others to which he clung through everything4 ,one of his favourite notions resulted in a brilliant discovery2 another remains in doubt to the present day4 99,the ,philosophic ,character of ,faraday499 ,in ,faraday's researches concerning the connection of magnetism and light1 we find an excellent instance of the pertinacity with which a favourite theory may be pursued1 so long as the results of experiment do not clearly negative the notions entertained4 ,in purely quantitative questions1 as we have seen1 the absence of apparent effect can seldom be regarded as proving the absence of all effect4 ,now ,faraday was convinced that some mutual relation must exist between magnetism and light4 ,as early as #ahbb1 he attempted to produce an effect upon a ray of polarised light1 by passing it through water placed between the poles of a voltaic battery2 but he was obliged to record that not the slightest effect was observable4 ,during many years the subject1 we are told1#dhd rose again and again to his mind1 and no failure could make him relinquish his search after this unknown relation4 ,it was in the year #ahde that he gained the first success2 on ,august #cjth he began to work with common electricity1 vainly trying glass1 quartz1 ,iceland spar1 c4 ,several days of labour gave no result2 yet he did not desist4 ,heavy glass1 a transparent medium of great refractive powers1 composed of borate of lead1 was now tried1 being placed between the poles of a powerful electro-magnet while a ray of polarised light was transmitted through it4 ,when the poles of the electro-magnet were arranged in certain positions with regard to the substance under trial1 no effects were apparent2 but at last ,faraday happened fortunately to place a piece of heavy glass so that contrary magnetic poles were on the same side1 and now an effect was witnessed4 ,the glass was found to have the power of twisting the plane of polarisation of the ray of light4 #dhd ,bence ,jones1 99,life of ,faraday991 vol4 i4 p4 #cfb4 ,all ,faraday's recorded thoughts upon this great experiment are replete with curious interest4 ,he attributes his success to the opinion1 almost amounting to a conviction1 that the various forms1 under which the forces of matter are made manifest1 have one common origin1 and are so directly related and mutually dependent that they are convertible4 8,this strong persuasion10 he says1#dhe 8extended to the powers of light1 and led to many exertions having for their object the discovery of the direct relation of light and electricity4 ,these ineffectual exertions could not remove my strong persuasion1 and ,i have at last succeeded40 ,he describes the phenomenon in somewhat figurative language as 99the magnetisation of a ray of light991 and also as 99the illumination of a magnetic curve or line of force994 ,he has no sooner got the effect in one case1 than he proceeds1 with his characteristic comprehensiveness of research1 to test the existence of a like phenomenon in all the substances available4 ,he finds that not only heavy glass1 but solids and liquids1 acids and alkalis1 oils1 water1 alcohol1 ether1 all possess this power2 but he was not able to detect its existence in any gaseous substance4 ,his thoughts cannot be restrained from running into curious speculations as to the possible results of the power in certain cases4 8,what effect10 he says1 8does this force have in the earth where the magnetic curves of the earth traverse its substance8 ,also what effect in a magnet80 ,and then he falls upon the strange notion that perhaps this force tends to make iron and oxide of iron transparent1 a phenomenon never observed4 ,we can meet with nothing more instructive as to the course of mind by which great discoveries are made1 than these records of ,faraday's patient labours1 and his varied success and failure4 ,nor are his unsuccessful experiments upon the relation of gravity and electricity less interesting1 or less worthy of study4 #dhe ,ibid4 vol4 ii4 p4 #aii4 ,throughout a large part of his life1 ,faraday was possessed by the idea that gravity cannot be unconnected with the other forces of nature4 ,on ,march #aith1 #ahdi1 he wrote in his laboratory book1--8,gravity4 ,surely this force must be capable of an experimental relation to electricity1 magnetism1 and the other forces1 so as to bind it up with them in reciprocal action and equivalent effect80#dhf ,he filled twenty paragraphs or more with reflections and suggestions1 as to the mode of treating the subject by experiment4 ,he anticipated that the mutual approach of two bodies would develop electricity in them1 or that a body falling through a conducting helix would excite a current changing in direction as the motion was reversed4 899,all this is a dream9910 he remarks2 8still examine it by a few experiments4 ,nothing is too wonderful to be true1 if it be consistent with the laws of nature2 and in such things as these1 experiment is the best test of such consistency40 #dhf ,see also his more formal statement in the 99,experimental ,researches in ,electricity991 #bdth ,series1 #bgjb1 vol4 iii4 p4 #afa4 ,he executed many difficult and tedious experiments1 which are described in the #bdth ,series of ,experimental ,researches4 ,the result was 99nil991 and yet he concludes3 8,here end my trials for the present4 ,the results are negative2 they do not shake my strong feeling of the existence of a relation between gravity and electricity1 though they give no proof that such a relation exists40 ,he returned to the work when he was ten years older1 and in #aheh-i recorded many remarkable reflections and experiments4 ,he was much struck by the fact that electricity is essentially a 99dual force991 and it had always been a conviction of ,faraday that no body could be electrified positively without some other body becoming electrified negatively2 some of his researches had been simple developments of this relation4 ,but observing that between two mutually gravitating bodies there was no apparent circumstance to determine which should be positive and which negative1 he does not hesitate to call in question an old opinion4 8,the evolution of 99one99 electricity would be a new and very remarkable thing4 ,the idea throws a doubt on the whole2 but still try1 for who knows what is possible in dealing with gravity80 ,we cannot but notice the candour with which he thus acknowledges in his laboratory book the doubtfulness of the whole thing1 and is yet prepared as a forlorn hope to frame experiments in opposition to all his previous experience of the course of nature4 ,for a time his thoughts flow on as if the strange detection were already made1 and he had only to trace out its consequences throughout the universe4 8,let us encourage ourselves by a little more imagination prior to experiment10 he says2 and then he reflects upon the infinity of actions in nature1 in which the mutual relations of electricity and gravity would come into play2 he pictures to himself the planets and the comets charging themselves as they approach the sun2 cascades1 rain1 rising vapour1 circulating currents of the atmosphere1 the fumes of a volcano1 the smoke in a chimney become so many electrical machines4 ,a multitude of events and changes in the atmosphere seem to be at once elucidated by such actions2 for a moment his reveries have the vividness of fact4 8,i think we have been dull and blind not to have suspected some such results10 and he sums up rapidly the consequences of his great but imaginary theory2 an entirely new mode of exciting heat or electricity1 an entirely new relation of the natural forces1 an analysis of gravitation1 and a justification of the conservation of force4 ,such were ,faraday's fondest dreams of what might be1 and to many a philosopher they would have been sufficient basis for the writing of a great book4 ,but ,faraday's imagination was within his full control2 as he himself says1 8,let the imagination go1 guarding it by judgment and principle1 and holding it in and directing it by experiment40 ,his dreams soon took a very practical form1 and for many days he laboured with ceaseless energy1 on the staircase of the ,royal ,institution1 in the clock tower of the ,houses of ,parliament1 or at the top of the ,shot ,tower in ,southwark1 raising and lowering heavy weights1 and combining electrical helices and wires in every conceivable way4 ,his skill and long experience in experiment were severely taxed to eliminate the effects of the earth's magnetism1 and time after time he saved himself from accepting mistaken indications1 which to another man might have seemed conclusive verifications of his theory4 ,when all was done there remained absolutely no results4 8,the experiments10 he says1 8were well made1 but the results are negative20 and yet1 he adds1 8,i cannot accept them as conclusive40 ,in this position the question remains to the present day2 it may be that the effect was too slight to be detected1 or it may be that the arrangements adopted were not suited to develop the particular relation which exists1 just as ,oersted could not detect electro-magnetism1 so long as his wire was perpendicular to the plane of motion of his needle4 ,but these are not matters which concern us further here4 ,we have only to notice the profound conviction in the unity of natural laws1 the active powers of inference and imagination1 the unbounded licence of theorising1 combined above all with the utmost diligence in experimental verification which this remarkable research exhibits4 99,reservation of ,judgment499 ,there is yet another characteristic needed in the philosophic mind2 it is that of suspending judgment when the data are insufficient4 ,many people will express a confident opinion on almost any question which is put before them1 but they thereby manifest not strength1 but narrowness of mind4 ,to see all sides of a complicated subject1 and to weigh all the different facts and probabilities correctly1 require no ordinary powers of comprehension4 ,hence it is most frequently the philosophic mind which is in doubt1 and the ignorant mind which is ready with a positive decision4 ,faraday has himself said1 in a very interesting lecture3#dhg 8,occasionally and frequently the exercise of the judgment ought to end in 99absolute reservation994 ,it may be very distasteful1 and great fatigue1 to suspend a conclusion2 but as we are not infallible1 so we ought to be cautious2 we shall eventually find our advantage1 for the man who rests in his position is not so far from right as he who1 proceeding in a wrong direction1 is ever increasing his distance40 #dhg ,printed in 99,modern ,culture991 edited by ,youmans1 p4 #bai4 ,arago presented a conspicuous example of this high quality of mind1 as ,faraday remarks2 for when he made known his curious discovery of the relation of a magnetic needle to a revolving copper plate1 a number of supposed men of science in different countries gave immediate and confident explanations of it1 which were all wrong4 ,but ,arago1 who had both discovered the phenomenon and personally investigated its conditions1 declined to put forward publicly any theory at all4 ,at the same time we must not suppose that the truly philosophic mind can tolerate a state of doubt1 while a chance of decision remains open4 ,in science nothing like compromise is possible1 and truth must be one4 ,hence1 doubt is the confession of ignorance1 and involves a painful feeling of incapacity4 ,but doubt lies between error and truth1 so that if we choose wrongly we are further away than ever from our goal4 ,summing up1 then1 it would seem as if the mind of the great discoverer must combine contradictory attributes4 ,he must be fertile in theories and hypotheses1 and yet full of facts and precise results of experience4 ,he must entertain the feeblest analogies1 and the merest guesses at truth1 and yet he must hold them as worthless till they are verified in experiment4 ,when there are any grounds of probability he must hold tenaciously to an old opinion1 and yet he must be prepared at any moment to relinquish it when a clearly contradictory fact is encountered4 8,the philosopher10 says ,faraday1#dhh 8should be a man willing to listen to every suggestion1 but determined to judge for himself4 ,he should not be biased by appearances2 have no favourite hypothesis2 be of no school2 and in doctrine have no master4 ,he should not be a respecter of persons1 but of things4 ,truth should be his primary object4 ,if to these qualities be added industry1 he may indeed hope to walk within the veil of the temple of nature40 #dhh 99,life of ,faraday991 vol4 i4 p4 #bbe4 ,,book ,v4 ,,generalisation1 ,,analogy1 ,,and ,,classification4 ,,chapter ,,xxvii4 ,,generalisation4 ,i have endeavoured to show in preceding chapters that all inductive reasoning is an inverse application of deductive reasoning1 and consists in demonstrating that the consequences of certain assumed laws agree with facts of nature gathered by active or passive observation4 ,the fundamental process of reasoning1 as stated in the outset1 consists in inferring of a thing what we know of similar objects1 and it is on this principle that the whole of deductive reasoning1 whether simply logical or mathematico-logical1 is founded4 ,all inductive reasoning must be founded on the same principle4 ,it might seem that by a plain use of this principle we could avoid the complicated processes of induction and deduction1 and argue directly from one particular case to another1 as ,mill proposed4 ,if the ,earth1 ,venus1 ,mars1 ,jupiter1 and other planets move in elliptic orbits1 cannot we dispense with elaborate precautions1 and assert that ,neptune1 ,ceres1 and the last discovered planet must do so likewise8 ,do we not know that ,mr4 ,gladstone must die1 because he is like other men8 ,may we not argue that because some men die therefore he must8 ,is it requisite to ascend by induction to the general proposition 8all men must die10 and then descend by deduction from that general proposition to the case of ,mr4 ,gladstone8 ,my answer undoubtedly is that we must ascend to general propositions4 ,the fundamental principle of the substitution of similars gives us no warrant in affirming of ,mr4 ,gladstone what we know of other men1 because we cannot be sure that ,mr4 ,gladstone is exactly similar to other men4 ,until his death we cannot be perfectly sure that he possesses all the attributes of other men2 it is a question of probability1 and ,i have endeavoured to explain the mode in which the theory of probability is applied to calculate the probability that from a series of similar events we may infer the recurrence of like events under identical circumstances4 ,there is then no such process as that of inferring from particulars to particulars4 ,a careful analysis of the conditions under which such an inference appears to be made1 shows that the process is really a general one1 and that what is inferred of a particular case might be inferred of all similar cases4 ,all reasoning is essentially general1 and all science implies generalisation4 ,in the very birth-time of philosophy this was held to be so3 8,nulla scientia est de individuis1 sed de solis universalibus10 was the doctrine of ,plato1 delivered by ,porphyry4 ,and ,aristotle#dhi held a like opinion-- ' 444 ' 4 8,no art treats of particular cases2 for particulars are infinite and cannot be known40 ,no one who holds the doctrine that reasoning may be from particulars to particulars1 can be supposed to have the most rudimentary notion of what constitutes reasoning and scence4 #dhi ,aristotle's 99,rhetoric991 ,liber ,i4 #b4 #aa4 ,at the same time there can be no doubt that practically what we find to be true of many similar objects will probably be true of the next similar object4 ,this is the result to which an analysis of the ,inverse ,method of ,probabilities leads us1 and1 in the absence of precise data from which we may calculate probabilities1 we are usually obliged to make a rough assumption that similars in some respects are similars in other respects4 ,thus it comes to pass that a large part of the reasoning processes in which scientific men are engaged1 consists in detecting similarities between objects1 and then rudely assuming that the like similarities will be detected in other cases4 99,distinction of ,generalisation and ,analogy499 ,there is no distinction but that of degree between what is known as reasoning by 99generalisation99 and reasoning by 99analogy994 ,in both cases from certain observed resemblances we infer1 with more or less probability1 the existence of other resemblances4 ,in generalisation the resemblances have great extension and usually little intension1 whereas in analogy we rely upon the great intension1 the extension being of small amount 7p4 #bf74 ,if we find that the qualities ,a and ,b are associated together in a great many instances1 and have never been found separate1 it is highly probable that on the next occasion when we meet with ,a1 ,b will also be present1 and 99vice vers994 ,thus wherever we meet with an object possessing gravity1 it is found to possess inertia also1 nor have we met with any material objects possessing inertia without discovering that they also possess gravity4 ,the probability has therefore become very great1 as indicated by the rules founded on the ,inverse ,method of ,probabilities 7p4 #beg71 that whenever in the future we meet an object possessing either of the properties of gravity and inertia1 it will be found on examination to possess the other of these properties4 ,this is a clear instance of the employment of generalisation4 ,in analogy1 on the other hand1 we reason from likeness in many points to likeness in other points4 ,the qualities or points of resemblance are now numerous1 not the objects4 ,at the poles of ,mars are two white spots which resemble in many respects the white regions of ice and snow at the poles of the earth4 ,there probably exist no other similar objects with which to compare these1 yet the exactness of the resemblance enables us to infer1 with high probability1 that the spots on ,mars consist of ice and snow4 ,in short1 many points of resemblance imply many more4 ,from the appearance and behaviour of those white spots we infer that they have all the chemical and physical properties of frozen water4 ,the inference is of course only probable1 and based upon the improbability that aggregates of many qualities should be formed in a like manner in two or more cases1 without being due to some uniform condition or cause4 ,in reasoning by analogy1 then1 we observe that two objects ,,abcde4444 and ,a,b,c,d,e4444 have many like qualities1 as indicated by the identity of the letters1 and we infer that1 since the first has another quality1 ,x1 we shall discover this quality in the second case by sufficiently close examination4 ,as ,laplace says1--8,analogy is founded on the probability that similar things have causes of the same kind1 and produce the same effects4 ,the more perfect this similarity1 the greater is this probability40#dij ,the nature of analogical inference is aptly described in the work on ,logic attributed to ,kant1 where the rule of ordinary induction is stated in the words1 899,eines in vielen1 also in allen9910 one quality in many things1 therefore in all2 and the rule of analogy is 899,vieles in einem1 also auch das brige in demselben9910#dia many 7qualities7 in one1 therefore also the remainder in the same4 ,it is evident that there may be intermediate cases in which1 from the identity of a moderate number of objects in several properties1 we may infer to other objects4 ,probability must rest either upon the number of instances or the depth of resemblance1 or upon the occurrence of both in sufficient degrees4 ,what there is wanting in extension must be made up by intension1 and 99vice vers994 #dij 99,essai ,philosophique sur les ,probabilits991 p4 #hf4 #dia ,kant's 99,logik991 #hd1 ,knigsberg1 #ahjj1 p4 #bjg4 99,two ,meanings of ,generalisation499 ,the term generalisation1 as commonly used1 includes two processes which are of different character1 but are often closely associated together4 ,in the first place1 we generalise when we recognise even in two objects a common nature4 ,we cannot detect the slightest similarity without opening the way to inference from one case to the other4 ,if we compare a cubical crystal with a regular octahedron1 there is little apparent similarity2 but1 as soon as we perceive that either can be produced by the symmetrical modification of the other1 we discover a groundwork of similarity in the crystals1 which enables us to infer many things of one1 because they are true of the other4 ,our knowledge of ozone took its rise from the time when the similarity of smell1 attending electric sparks1 strokes of lightning1 and the slow combustion of phosphorus1 was noticed by ,schnbein4 ,there was a time when the rainbow was an inexplicable phenomenon--a portent1 like a comet1 and a cause of superstitious hopes and fears4 ,but we find the true spirit of science in ,roger ,bacon1 who desires us to consider the objects which present the same colours as the rainbow2 he mentions hexagonal crystals from ,ireland and ,india1 but he bids us not suppose that the hexagonal form is essential1 for similar colours may be detected in many transparent stones4 ,drops of water scattered by the oar in the sun1 the spray from a water-wheel1 the dewdrops lying on the grass in the summer morning1 all display a similar phenomenon4 ,no sooner have we grouped together these apparently diverse instances1 than we have begun to generalise1 and have acquired a power of applying to one instance what we can detect of others4 ,even when we do not apply the knowledge gained to new objects1 our comprehension of those already observed is greatly strengthened and deepened by learning to view them as particular cases of a more general property4 ,a second process1 to which the name of generalisation is often given1 consists in passing from a fact or partial law to a multitude of unexamined cases1 which we believe to be subject to the same conditions4 ,instead of merely recognising similarity as it is brought before us1 we predict its existence before our senses can detect it1 so that generalisation of this kind endows us with a prophetic power of more or less probability4 ,having observed that many substances assume1 like water and mercury1 the three states of solid1 liquid1 and gas1 and having assured ourselves by frequent trial that the greater the means we possess of heating and cooling1 the more substances we can vaporise and freeze1 we pass confidently in advance of fact1 and assume that all substances are capable of these three forms4 ,such a generalisation was accepted by ,lavoisier and ,laplace before many of the corroborative facts now in our possession were known4 ,the reduction of a single comet beneath the sway of gravity was considered sufficient indication that all comets obey the same power4 ,few persons doubted that the law of gravity extended over the whole heavens2 certainly the fact that a few stars out of many millions manifest the action of gravity1 is now held to be sufficient evidence of its general extension over the visible universe4 99,value of ,generalisation499 ,it might seem that if we know particular facts1 there can be little use in connecting them together by a general law4 ,the particulars must be more full of useful information than an abstract general statement4 ,if we know1 for instance1 the properties of an ellipse1 a circle1 a parabola1 and hyperbola1 what is the use of learning all these properties over again in the general theory of curves of the second degree8 ,if we understand the phenomena of sound and light and water-waves separately1 what is the need of erecting a general theory of waves1 which1 after all1 is inapplicable to practice until resolved again into particular cases8 ,but1 in reality1 we never do obtain an adequate knowledge of particulars until we regard them as cases of the general4 ,not only is there a singular delight in discovering the many in the one1 and the one in the many1 but there is a constant interchange of light and knowledge4 ,properties which are unapparent in the hyperbola may be readily observed in the ellipse4 ,most of the complex relations which old geometers discovered in the circle will be reproduced 99mutatis mutandis99 in the other conic sections4 ,the undulatory theory of light might have been unknown at the present day1 had not the theory of sound supplied hints by analogy4 ,the study of light has made known many phenomena of interference and polarisation1 the existence of which had hardly been suspected in the case of sound1 but which may now be sought out1 and perhaps found to possess unexpected interest4 ,the careful study of water-waves shows how waves alter in form and velocity with varying depth of water4 ,analogous changes may some time be detected in sound waves4 ,thus there is mutual interchange of aid4 8,every study of a generalisation or extension10 ,de ,morgan has well said1#dib 8gives additional power over the particular form by which the generalisation is suggested4 ,nobody who has ever returned to quadratic equations after the study of equations of all degrees1 or who has done the like1 will deny my assertion that may be predicated of any one who studies a branch or a case1 without afterwards making it part of a larger whole4 ,accordingly it is always worth while to generalise1 were it only to give power over the 99particular994 ,this principle1 of daily familiarity to the mathematician1 is almost unknown to the logician40 #dib 99,syllabus of a ,proposed ,system of ,logic991 p4 #cd4 99,comparative ,generality of ,properties499 ,much of the value of science depends upon the knowledge which we gradually acquire of the different degrees of generality of properties and phenomena of various kinds4 ,the use of science consists in enabling us to act with confidence1 because we can foresee the result4 ,now this foresight must rest upon the knowledge of the powers which will come into play4 ,that knowledge1 indeed1 can never be certain1 because it rests upon imperfect induction1 and the most confident beliefs and predictions of the physicist may be falsified4 ,nevertheless1 if we always estimate the probability of each belief according to the due teaching of the data1 and bear in mind that probability when forming our anticipations1 we shall ensure the minimum of disappointment4 ,even when he cannot exactly apply the theory of probabilities1 the physicist may acquire the habit of making judgments in general agreement with its principles and results4 ,such is the constitution of nature1 that the physicist learns to distinguish those properties which have wide and uniform extension1 from those which vary between case and case4 ,not only are certain laws distinctly laid down1 with their extension carefully defined1 but a scientific training gives a kind of tact in judging how far other laws are likely to apply under any particular circumstances4 ,we learn by degrees that crystals exhibit phenomena depending upon the directions of the axes of elasticity1 which we must not expect in uniform solids4 ,liquids1 compared even with non-crystalline solids1 exhibit laws of far less complexity and variety2 and gases assume1 in many respects1 an aspect of nearly complete uniformity4 ,to trace out the branches of science in which varying degrees of generality prevail1 would be an inquiry of great interest and importance2 but want of space1 if there were no other reason1 would forbid me to attempt it1 except in a very slight manner4 ,gases1 so far as they are really gaseous1 not only have exactly the same properties in all directions of space1 but one gas exactly resembles other gases in many qualities4 ,all gases expand by heat1 according to the same law1 and by nearly the same amount2 the specific heats of equivalent weights are equal1 and the densities are exactly proportional to the atomic weights4 ,all such gases obey the general law1 that the volume multiplied by the pressure1 and divided by the absolute temperature1 is constant or nearly so4 ,the laws of diffusion and transpiration are the same in all cases1 and1 generally speaking1 all physical laws1 as distinguished from chemical laws1 apply equally to all gases4 ,even when gases differ in chemical or physical properties1 the differences are minor in degree4 ,thus the differences of viscosity are far less marked than in the liquid and solid states4 ,nearly all gases1 again1 are colourless1 the exceptions being chlorine1 the vapours of iodine1 bromine1 and a few other substances4 ,only in one single point1 so far as ,i am aware1 do gases present distinguishing marks unknown or nearly so1 in the solid and liquid states4 ,i mean as regards the light given off when incandescent4 ,each gas when sufficiently heated1 yields its own peculiar series of rays1 arising from the free vibrations of the constituent parts of the molecules4 ,hence the possibility of distinguishing gases by the spectroscope4 ,but the molecules of solids and liquids appear to be continually in conflict with each other1 so that only a confused 99noise99 of atoms is produced1 instead of a definite series of luminous chords4 ,at the same temperature1 accordingly1 all solids and liquids give off nearly the same rays when strongly heated1 and we have in this case an exception to the greater generality of properties in gases4 ,liquids are in many ways intermediate in character between gases and solids4 ,while incapable of possessing different elasticity in different directions1 and thus denuded of the rich geometrical complexity of solids1 they retain the variety of density1 colour degrees of transparency1 great diversity in surface tension1 viscosity1 coefficients of expansion1 compressibility1 and many other properties which we observe in solids1 but not for the most part in gases4 ,though our knowledge of the physical properties of liquids is much wanting in generality at present1 there is ground to hope that by degrees laws connecting and explaining the variations may be traced out4 ,solids are in every way contrasted to gases4 ,each solid substance has its own peculiar degree of density1 hardness1 compressibility1 transparency1 tenacity1 elasticity1 power of conducting heat and electricity1 magnetic properties1 capability of producing frictional electricity1 and so forth4 ,even different specimens of the same kind of substance will differ widely1 according to the accidental treatment received4 ,and not only has each substance its own specific properties1 but1 when crystallised1 its properties vary in each direction with regard to the axes of crystallisation4 ,the velocity of radiation1 the rate of conduction of heat1 the coefficients of expansibility and compressibility1 the thermo-electric properties1 all vary in different crystallographic directions4 ,it is probable that many apparent differences between liquids1 and even between solids1 will be explained when we learn to regard them under exactly corresponding circumstances4 ,the extreme generality of the properties of gases is in reality only true at an infinitely high temperature1 when they are all equally remote from their condensing points4 ,now1 it is found that if we compare liquids--for instance1 different kinds of alcohols--not at equal temperatures1 but at points equally distant from their respective boiling points1 the laws and coefficients of expansion are nearly equal4 ,the vapour-tensions of liquids also are more nearly equal1 when compared at corresponding points1 and the boiling-points appear in many cases to be simply related to the chemical composition4 ,no doubt the progress of investigation will enable us to discover generality1 where at present we only see variety and puzzling complexity4 ,in some cases substances exhibit the same physical properties in the liquid as in the solid state4 ,lead has a high refractive power1 whether in solution1 or in solid salts1 crystallised or vitreous4 ,the magnetic power of iron is conspicuous1 whatever be its chemical condition2 indeed1 the magnetic properties of substances1 though varying with temperature1 seem not to be greatly affected by other physical changes4 ,colour1 absorptive power for heat or light rays1 and a few other properties are also often the same in liquids and gases4 ,iodine and bromine possess a deep colour whenever they are chemically uncombined4 ,nevertheless1 we can seldom argue safely from the properties of a substance in one condition to those in another condition4 ,ice is an insulator1 water a conductor of electricity1 and the same contrast exists in most other substances4 ,the conducting power of a liquid for electricity increases with the temperature1 while that of a solid decreases4 ,by degrees we may learn to distinguish between those properties of matter which depend upon the intimate construction of the chemical molecule1 and those which depend upon the contact1 conflict1 mutual attraction1 or other relations of distinct molecules4 ,the properties of a substance with respect to light seem generally to depend upon the molecule2 thus1 the power of certain substances to cause the plane of polarisation of a ray of light to rotate1 is exactly the same whatever be its degree of density1 or the diluteness of the solution in which it is contained4 ,taken as a whole1 the physical properties of substances and their quantitative laws1 present a problem of infinite complexity1 and centuries must elapse before any moderately complete generalisations on the subject become possible4 99,uniform ,properties of all ,matter499 ,some laws are held to be true of all matter in the universe absolutely1 without exception1 no instance to the contrary having ever been noticed4 ,this is the case with the laws of motion1 as laid down by ,galileo and ,newton4 ,it is also conspicuously true of the law of universal gravitation4 ,the rise of modern physical science may perhaps be considered as beginning at the time when ,galileo showed1 in opposition to the ,aristotelians1 that matter is equally affected by gravity1 irrespective of its form1 magnitude1 or texture4 ,all objects fall with equal rapidity1 when disturbing causes1 such as the resistance of the air1 are removed or allowed for4 ,that which was rudely demonstrated by ,galileo from the leaning tower of ,pisa1 was proved by ,newton to a high degree of approximation1 in an experiment which has been mentioned 7p4 #ddc74 ,newton formed two pendulums1 as nearly as possible the same in outward shape and size by taking two equal round wooden boxes1 and suspending them by equal threads1 eleven feet long4 ,the pendulums were therefore equally subject to the resistance of the air4 ,he filled one box with wood1 and in the centre of oscillation of the other he placed an equal weight of gold4 ,the pendulums were then equal in weight as well as in size2 and1 on setting them simultaneously in motion1 ,newton found that they vibrated for a length of time with equal vibrations4 ,he tried the same experiment with silver1 lead1 glass1 sand1 common salt1 water1 and wheat1 in place of the gold1 and ascertained that the motion of his pendulum was exactly the same whatever was the kind of matter inside4#dic ,he considered that a difference of a thousandth part would have been apparent4 ,the reader must observe that the pendulums were made of equal weight only in order that they might suffer equal retardation from the air4 ,the meaning of the experiment is that all substances manifest exactly equal acceleration from the force of gravity1 and that therefore the inertia or resistance of matter to force1 which is the only independent measure of mass known to us1 is always proportional to gravity4 #dic 99,principia991 bk4 iii4 ,prop4 ,,vi4 ,motte's translation1 vol4 ii4 p4 #bbj4 ,these experiments of ,newton were considered conclusive up to very recent times1 when certain discordances between the theory and observations of the movements of planets led ,nicolai1 in #ahbf1 to suggest that the equal gravitation of different kinds of matter might not be absolutely exact4 ,it is perfectly philosophical thus to call in question1 from time to time1 some of the best accepted laws4 ,on this occasion ,bessel carefully repeated the experiments of ,newton with pendulums composed of ivory1 glass1 marble1 quartz1 meteoric stones1 c41 but was unable to detect the least difference4 ,this conclusion is also confirmed by the ultimate agreement of all the calculations of physical astronomy based upon it4 ,whether the mass of ,jupiter be calculated from the motion of its own satellites1 from the effect upon the small planets1 ,vesta1 ,juno1 c41 or from the perturbation of ,encke's ,comet1 the results are closely accordant1 showing that precisely the same law of gravity applies to the most different bodies which we can observe4 ,the gravity of a body1 again1 appears to be entirely independent of its other physical conditions1 being totally unaffected by any alteration in the temperature1 density1 electric or magnetic condition1 or other physical properties of the substance4 ,one paradoxical result of the law of equal gravitation is the theorem of ,torricelli1 to the effect that all liquids of whatever density fall or flow with equal rapidity4 ,if there be two equal cisterns respectively filled with mercury and water1 the mercury1 though thirteen times as heavy1 would flow from an aperture neither more rapidly nor more slowly than the water1 and the same would be true of ether1 alcohol1 and other liquids1 allowance being made1 however1 for the resistance of the air1 and the differing viscosities of the liquids4 ,in its exact equality and its perfect independence of all circumstances1 except mass and distance1 the force of gravity stands apart from all the other forces and phenomena of nature1 and has not yet been brought into any relation with them except through the general principle of the conservation of energy4 ,magnetic attraction1 as remarked by ,newton1 follows very different laws1 depending upon the chemical quality and molecular structure of each particular substance4 ,we must remember that in saying 8all matter gravitates10 we exclude from the term matter the basis of light-undulations1 which is immensely more extensive in amount1 and obeys in many respects the laws of mechanics4 ,this adamantine substance appears1 so far as can be ascertained1 to be perfectly uniform in its properties when existing in space unoccupied by matter4 ,light and heat are conveyed by it with equal velocity in all directions1 and in all parts of space so far as observation informs us4 ,but the presence of gravitating matter modifies the density and mechanical properties of the so-called ether in a way which is yet quite unexplained4#did ,leaving gravity1 it is somewhat difficult to discover other laws which are equally true of all matter4 ,boerhaave was considered to have established that all bodies expand by heat2 but not only is the expansion very different in different substances1 but we now know positive exceptions4 ,many liquids and a few solids contract by heat at certain temperatures4 ,there are indeed other relations of heat to matter which seem to be universal and uniform2 all substances begin to give off rays of light at the same temperature1 according to the law of ,draper2 and gases will not be an exception if sufficiently condensed1 as in the experiments of ,frankland4 ,grove considers it to be universally true that all bodies in combining produce heat2 with the doubtful exception of sulphur and selenium1 all solids in becoming liquids1 and all liquids in becoming gases1 absorb heat2 but the quantities of heat absorbed vary with the chemical qualities of the matter4 ,carnot's ,thermodynamic ,law is held to be exactly true of all matter without distinction2 it expresses the fact that the amount of mechanical energy which might be theoretically obtained from a certain amount of heat energy depends only upon the change of the temperatures1 so that whether an engine be worked by water1 air1 alcohol1 ammonia1 or any other substance1 the result would theoretically be the same1 if the boiler and condenser were maintained at similar temperatures4 #did ,professor ,lovering has pointed out how obscure and uncertain the ideas of scientific men about this ether are1 in his interesting ,presidential ,address before the ,american ,association at ,hartford1 #ahgd4 99,silliman's ,journal991 ,october #ahgd1 p4 #big4 99,philosophical ,magazine991 vol4 xlviii4 p4 #dic4 99,variable ,properties of ,matter499 ,i have enumerated some of the few properties of matter1 which are manifested in exactly the same manner by all substances1 whatever be their differences of chemical or physical constitution4 ,but by far the greater number of qualities vary in degree2 substances are more or less dense1 more or less transparent1 more or less compressible1 more or less magnetic1 and so on4 ,one common result of the progress of science is to show that qualities supposed to be entirely absent from many substances are present only in so low a degree of intensity that the means of detection were insufficient4 ,newton believed that most bodies were quite unaffected by the magnet2 ,faraday and ,tyndall have rendered it very doubtful whether any substance whatever is wholly devoid of magnetism1 including under that term diamagnetism4 ,we are rapidly learning to believe that there are no substances absolutely opaque1 or non-conducting1 non-electric1 non-elastic1 non-viscous1 non-compressible1 insoluble1 infusible1 or non-volatile4 ,all tends to become a matter of degree1 or sometimes of direction4 ,there may be some substances oppositely affected to others1 as ferro-magnetic substances are oppositely affected to diamagnetics1 or as substances which contract by heat are opposed to those which expand2 but the tendency is certainly for every affection of one kind of matter to be represented by something similar in other kinds4 ,on this account one of ,newton's rules of philosophising seems to lose all validity2 he said1 8,those qualities of bodies which are not capable of being heightened1 and remitted1 and which are found in all bodies on which experiment can be made1 must be considered as universal qualities of all bodies40 ,as far as ,i can see1 the contrary is more probable1 namely1 that qualities variable in degree will be found in every substance in a greater or less degree4 ,it is remarkable that ,newton whose method of investigation was logically perfect1 seemed incapable of generalising and describing his own procedure4 ,his celebrated 8,rules of ,reasoning in ,philosophy10 described at the commencement of the third book of the 99,principia991 are of questionable truth1 and still more questionable value4 99,extreme ,instances of ,properties499 ,although substances usually differ only in degree1 great interest may attach to particular substances which manifest a property in a conspicuous and intense manner4 ,every branch of physical science has usually been developed from the attention forcibly drawn to some singular substance4 ,just as the loadstone disclosed magnetism and amber frictional electricity1 so did ,iceland spar show the existence of double refraction1 and sulphate of quinine the phenomenon of fluorescence4 ,when one such startling instance has drawn the attention of the scientific world1 numerous less remarkable cases of the phenomenon will be detected1 and it will probably prove that the property in question is actually universal to all matter4 ,nevertheless1 the extreme instances retain their interest1 partly in a historical point of view1 partly because they furnish the most convenient substances for experiment4 ,francis ,bacon was fully aware of the value of such examples1 which he called 99,ostensive ,instances99 or ,light-giving1 ,free and ,predominant ,instances4 8,they are those10 he says1#die 8which show the nature under investigation naked1 in an exalted condition1 or in the highest degree of power2 freed from impediments1 or at least by its strength predominating over and suppressing them40 ,he mentions quicksilver as an ostensive instance of weight or density1 thinking it not much less dense than gold1 and more remarkable than gold as joining density to liquidity4 ,the magnet is mentioned as an ostensive instance of attraction4 ,it would not be easy to distinguish clearly between these ostensive instances and those which he calls 99,instantiae ,monodicae991 or 99,irregulares991 or 99,heteroclitae991 under which he places whatever is extravagant in its properties or magnitude1 or exhibits least similarity to other things1 such as the sun and moon among the heavenly bodies1 the elephant among animals1 the letter 99s99 among letters1 or the magnet among stones4#dif #die 99,novum ,organum991 bk4 ii4 ,aphorisms1 #bd1 #be4 #dif ,ibid4 ,aph4 #bh4 ,in optical science great use has been made of the high dispersive power of the transparent compounds of lead1 that is1 the power of giving a long spectrum 7p4 #dcb74 ,dollond1 having noticed this peculiar dispersive power in lenses made of flint glass1 employed them to produce an achromatic arrangement4 ,the element strontium presents a contrast to lead in this respect1 being characterised by a remarkably low dispersive power2 but ,i am not aware that this property has yet been turned to account4 ,compounds of lead have both a high dispersive and a high refractive index1 and in the latter respect they proved very useful to ,faraday4 ,having spent much labour in preparing various kinds of optical glass1 ,faraday happened to form a compound of lead1 silica1 and boracic acid1 now known as 99heavy glass991 which possessed an intensely high refracting power4 ,many years afterwards in attempting to discover the action of magnetism upon light he failed to detect any effect1 as has been already mentioned1 7p4 #ehh71 until he happened to test a piece of the heavy glass4 ,the peculiar refractive power of this medium caused the magnetic strain to be apparent1 and the rotation of the plane of polarisation was discovered4 ,in almost every part of physical science there is some substance of powers pre-eminent for the special purpose to which it is put4 ,rock-salt is invaluable for its extreme diathermancy or transparency to the least refrangible rays of the spectrum4 ,quartz is equally valuable for its transparency1 as regards the ultra-violet or most refrangible rays4 ,diamond is the most highly refracting substance which is at the same time transparent2 were it more abundant and easily worked it would be of great optical importance4 ,cinnabar is distinguished by possessing a power of rotating the plane of polarisation of light1 from #ae to #ag times as much as quartz4 ,in electric experiments copper is employed for its high conducting powers and exceedingly low magnetic properties2 iron is of course indispensable for its enormous magnetic powers2 while bismuth holds a like place as regards its diamagnetic powers1 and was of much importance in ,tyndall's decisive researches upon the polar character of the diamagnetic force4#dig ,in regard to magne-crystallic action the mineral cyanite is highly remarkable1 being so powerfully affected by the earth's magnetism1 that1 when delicately suspended1 it assumes a constant position with regard to the magnetic meridian1 and may almost be used like the compass needle4 ,sodium is distinguished by its unique light-giving powers1 which are so extraordinary that probably one half of the whole number of stars in the heavens have a yellow tinge in consequence4 #dig 99,philosophical ,transactions99 7#ahef7 vol4 cxlvi4 p4 #bdf4 ,it is remarkable that water1 though the most common of all fluids1 is distinguished in almost every respect by extreme qualities4 ,of all known substances water has the highest specific heat1 being thus peculiarly fitted for the purpose of warming and cooling1 to which it is often put4 ,it rises by capillary attraction to a height more than twice that of any other liquid4 ,in the state of ice it is nearly twice as dilatable by heat as any other known solid substance4#dih ,in proportion to its density it has a far higher surface tension than any other substance1 being surpassed in absolute tension only by mercury2 and it would not be difficult to extend considerably the list of its remarkable and useful properties4 #dih 99,philosophical ,magazine991 #dth ,series1 ,january #ahgj1 vol4 xxxix4 p4 #b4 ,under extreme instances we may include cases of remarkably low powers or qualities4 ,such cases seem to correspond to what ,bacon calls 99,clandestine ,instances991 which exhibit a given nature in the least intensity1 and as it were in a rudimentary state4#dii ,they may often be important1 he thinks1 as allowing the detection of the cause of the property by difference4 ,i may add that in some cases they may be of use in experiments4 ,thus hydrogen is the least dense of all known substances1 and has the least atomic weight4 ,liquefied nitrous oxide has the lowest refractive index of all known fluids4#ejj ,the compounds of strontium have the lowest dispersive power4 ,it is obvious that a property of very low degree may prove as curious and valuable a phenomenon as a property of very high degree4 #dii 99,novum ,organum991 bk4 ii4 ,aphorism #be4 #ejj ,faraday's 99,experimental ,researches in ,chemistry and ,physics991 p4 #ic4 99,the ,detection of ,continuity499 ,we should bear in mind that phenomena which are in reality of a closely similar or even identical nature1 may present to the senses very different appearances4 ,without a careful analysis of the changes which take place1 we may often be in danger of widely separating facts and processes1 which are actually instances of the same law4 ,extreme difference of degree or magnitude is a frequent cause of error4 ,it is truly difficult at the first moment to recognise any similarity between the gradual rusting of a piece of iron1 and the rapid combustion of a heap of straw4 ,yet ,lavoisier's chemical theory was founded upon the similarity of the oxydising process in one case and the other4 ,we have only to divide the iron into excessively small particles to discover that it is really the more combustible of the two1 and that it actually takes fire spontaneously and burns like tinder4 ,it is the excessive slowness of the process in the case of a massive piece of iron which disguises its real character4 ,if ,xenophon reports truly1 ,socrates was misled by not making sufficient allowance for extreme differences of degree and quantity4 ,anaxagoras held that the sun is a fire1 but ,socrates rejected this opinion1 on the ground that we can look at a fire1 but not at the sun1 and that plants grow by sunshine while they are killed by fire4 ,he also pointed out that a stone heated in a fire is not luminous1 and soon cools1 whereas the sun ever remains equally luminous and hot4#eja ,all such mistakes evidently arise from not perceiving that difference of quantity may be so extreme as to assume the appearance of difference of quality4 ,it is the least creditable thing we know of ,socrates1 that after pointing out these supposed mistakes of earlier philosophers1 he advised his followers not to study astronomy4 #eja 99,memorabilia991 iv4 #g4 ,masses of matter of very different size may be expected to exhibit apparent differences of conduct1 arising from the various intensity of the forces brought into play4 ,many persons have thought it requisite to imagine occult forces producing the suspension of the clouds1 and there have even been absurd theories representing cloud particles as minute water-balloons buoyed up by the warm air within them4 ,but we have only to take proper account of the enormous comparative resistance which the air opposes to the fall of minute particles1 to see that all cloud particles are probably constantly falling through the air1 but so slowly that there is no apparent effect4 ,mineral matter again is always regarded as inert and incapable of spontaneous movement4 ,we are struck by astonishment on observing in a powerful microscope1 that every kind of solid matter suspended in extremely minute particles in pure water1 acquires an oscillatory movement1 often so marked as to resemble dancing or skipping4 ,i conceive that this movement is due to the comparatively vast intensity of chemical action when exerted upon minute particles1 the effect being #e1jjj or #aj1jjj greater in proportion to the mass than in fragments of an inch diameter 7p4 #djf74 ,much that was formerly obscure in the science of electricity arose from the extreme differences of intensity and quantity in which this form of energy manifests itself4 ,between the brilliant explosive discharge of a thunder-cloud and the gentle continuous current produced by two pieces of metal and some dilute acid1 there is no apparent analogy whatever4 ,it was therefore a work of great importance when ,faraday demonstrated the identity of the forces in action1 showing that common frictional electricity would decompose water like that from the voltaic battery4 ,the relation of the phenomena became plain when he succeeded in showing that it would require #hjj1jjj discharges of his large ,leyden battery to decompose one single grain of water4 ,lightning was now seen to be electricity of excessively high tension1 but extremely small quantity1 the difference being somewhat analogous to that between the force of one million gallons of water falling through one foot1 and one gallon of water falling through one million feet4 ,faraday estimated that one grain of water acting on four grains of zinc1 would yield electricity enough for a great thunderstorm4 ,it was long believed that electrical conductors and insulators belonged to two opposed classes of substances4 ,between the inconceivable rapidity with which the current passes through pure copper wire1 and the apparently complete manner in which it is stopped by a thin partition of gutta-percha or gum-lac1 there seemed to be no resemblance4 ,faraday again laboured successfully to show that these were but the extreme cases of a chain of substances varying in all degrees in their powers of conduction4 ,even the best conductors1 such as pure copper or silver1 offer resistance to the electric current4 ,the other metals have considerably higher powers of resistance1 and we pass gradually down through oxides and sulphides4 ,the best insulators1 on the other hand1 allow of an atomic induction which is the necessary antecedent of conduction4 ,hence ,faraday inferred that whether we can measure the effect or not1 all substances discharge electricity more or less4#ejb ,one consequence of this doctrine must be1 that every discharge of electricity produces an induced current4 ,in the case of the common galvanic current we can readily detect the induced current in any parallel wire or other neighbouring conductor1 and can separate the opposite currents which arise at the moments when the original current begins and ends4 ,but a discharge of high tension electricity like lightning1 though it certainly occupies time and has a beginning and an end1 yet lasts so minute a fraction of a second1 that it would be hopeless to attempt to detect and separate the two opposite induced currents1 which are nearly simultaneous and exactly neutralise each other4 ,thus an apparent failure of analogy is explained away1 and we are furnished with another instance of a phenomenon incapable of observation and yet theoretically known to exist4#ejc #ejb 99,experimental ,researches in ,electricity991 ,series xii4 vol4 i4 p4 #dbj4 #ejc 99,life of ,faraday991 vol4 ii4 p4 #g4 ,perhaps the most extraordinary case of the detection of unsuspected continuity is found in the discovery of ,cagniard de la ,tour and ,professor ,andrews1 that the liquid and gaseous conditions of matter are only remote points in a continuous course of change4 ,nothing is at first sight more apparently distinct than the physical condition of water and aqueous vapour4 ,at the boiling-point there is an entire breach of continuity1 and the gas produced is subject to laws incomparably more simple than the liquid from which it arose4 ,but ,cagniard de la ,tour showed that if we maintain a liquid under sufficient pressure its boiling point may be indefinitely raised1 and yet the liquid will ultimately assume the gaseous condition with but a small increase of volume4 ,professor ,andrews1 recently following out this course of inquiry1 has shown that liquid carbonic acid may1 at a particular temperature 7#cjib ,c471 and under the pressure of #gd atmospheres1 be at the same time in a state indistinguishable from that of liquid and gas4 ,at higher pressures carbonic acid may be made to pass from a palpably liquid state to a truly gaseous state without any abrupt change whatever4 ,as the pressure is greater the abruptness of the change from liquid to gas gradually decreases1 and finally vanishes4 ,similar phenomena or an approximation to them have been observed in other liquids1 and there is little doubt that we may make a wide generalisation1 and assert that1 under adequate pressure1 every liquid might be made to pass into a gas without breach of continuity4#ejd ,the liquid state1 moreover1 is considered by ,professor ,andrews to be but an intermediate step between the solid and gaseous conditions4 ,there are various indications that the process of melting is not perfectly abrupt2 and could experiments be made under adequate pressures1 it is believed that every solid could be made to pass by insensible degrees into the state of liquid1 and subsequently into that of gas4 #ejd 99,nature991 vol4 ii4 p4 #bgh4 ,these discoveries appear to open the way to most important and fundamental generalisations1 but it is probable that in many other cases phenomena now regarded as discrete may be shown to be different degrees of the same process4 ,graham was of opinion that chemical affinity differs but in degree from the ordinary attraction which holds different particles of a body together4 ,he found that sulphuric acid continued to evolve heat when mixed even with the fiftieth equivalent of water1 so that there seemed to be no distinct limit to chemical affinity4 ,he concludes1 8,there is reason to believe that chemical affinity passes in its lowest degree into the attraction of aggregation40#eje #eje 99,journal of the ,chemical ,society991 vol4 viii4 p4 #ea4 ,the atomic theory is well established1 but its limits are not marked out4 ,as ,grove points out1 we may by selecting sufficiently high multipliers express any combination or mixture of elements in terms of their equivalent weights4#ejf ,sir ,w4 ,thomson has suggested that the power which vegetable fibre1 oatmeal1 and other substances possess of attracting and condensing aqueous vapour is probably continuous1 or1 in fact1 identical with capillary attraction1 which is capable of interfering with the pressure of aqueous vapour and aiding its condensation4#ejg ,there are many cases of so-called catalytic or surface action1 such as the extraordinary power of animal charcoal for attracting organic matter1 or of spongy platinum for condensing hydrogen1 which can only be considered as exalted cases of a more general power of attraction4 ,the number of substances which are decomposed by light in a striking manner is very limited2 but many other substances1 such as vegetable colours1 are affected by long exposure2 on the principle of continuity we might expect to find that all kinds of matter are more or less susceptible of change by the incidence of light rays4#ejh ,it is the opinion of ,grove that wherever an electric current passes there is a tendency to decomposition1 a strain on the molecules1 which when sufficiently intense leads to disruption4 ,even a metallic conducting wire may be regarded as tending to decomposition4 ,davy was probably correct in describing electricity as chemical affinity acting on masses1 or rather1 as ,grove suggests1 creating a disturbance through a chain of particles4#eji ,laplace went so far as to suggest that all chemical phenomena may be results of the ,newtonian law of attraction1 applied to atoms of various mass and position2 but the time is probably far distant when the progress of molecular philosophy and of mathematical methods will enable such a generalisation to be verified or refuted4 #ejf 99,correlation of ,physical ,forces991 #crd edit4 p4 #ahd4 #ejg 99,philosophical ,magazine991 #dth ,series1 vol4 xlii4 p4 #dea4 #ejh ,grove1 99,correlation of ,physical ,forces991 #crd edit4 p4 #aah4 #eji ,ibid4 pp4 #aff1 #aii1 c4 99,the ,law of ,continuity499 ,under the title of the ,law of ,continuity we may place many applications of the general principle of reasoning1 that what is true of one case will be true of similar cases1 and probably true of what are probably similar4 ,whenever we find that a law or similarity is rigorously fulfilled up to a certain point in time or space1 we expect with a high degree of probability that it will continue to be fulfilled at least a little further4 ,if we see part only of a circle1 we naturally expect that the circular form will be continued in the part hidden from us4 ,if a body has moved uniformly over a certain space1 we expect that it will continue to move uniformly4 ,the ground of such inferences is doubtless identical with that of other inductive inferences4 ,in continuous motion every infinitely small space passed over constitutes a separate constituent fact1 and had we perfect powers of observation the smallest finite motion would include an infinity of information1 which1 by the principles of the inverse method of probabilities1 would enable us to infer with certainty to the next infinitely small portion of the path4 ,but when we attempt to infer from one finite portion of a path to another finite portion1 inference will be only more or less probable1 according to the comparative lengths of the portions and the accuracy of observation2 the longer our experience is1 the more probable our inference will be2 the greater the length of time or space over which the inference extends1 the less probable4 ,this principle of continuity presents itself in nature in a great variety of forms and cases4 ,it is familiarly expressed in the dictum 99,natura non agit per saltum994 ,as ,graham expressed the maxim1 there are in nature no abrupt transitions1 and the distinctions of class are never absolute4#eaj ,there is always some notice--some forewarning of every phenomenon1 and every change begins by insensible degrees1 could we observe it with perfect accuracy4 ,the cannon ball1 indeed1 is forced from the cannon in an inappreciable portion of time2 the trigger is pulled1 the fuze fired1 the powder inflamed1 the ball expelled1 all simultaneously to our senses4 ,but there is no doubt that time is occupied by every part of the process1 and that the ball begins to move at first with infinite slowness4 ,captain ,noble is able to measure by his chronoscope the progress of the shot in a #cjj-pounder gun1 and finds that the whole motion within the barrel takes place in something less than one #bjjth part of a second4 ,it is certain that no finite force can produce motion1 except in a finite space of time4 ,the amount of momentum communicated to a body is proportional to the accelerating force multiplied by the time during which it acts uniformly4 ,thus a slight force produces a great velocity only by long-continued action4 ,in a powerful shock1 like that of a railway collision1 the stroke of a hammer on an anvil1 or the discharge of a gun1 the time is very short1 and therefore the accelerating forces brought into play are exceedingly great1 but never infinite4 ,in the case of a large gun the powder in exploding is said to exert for a moment a force equivalent to at least #b1hjj1jjj horses4 #eaj 99,philosophical ,transactions991 #ahfa4 99,chemical and ,physical ,researches991 p4 #eih4 ,our belief in some of the fundamental laws of nature rests upon the principle of continuity4 ,galileo is held to be the first philosopher who consciously employed this principle in his arguments concerning the nature of motion1 and it is certain that we can never by mere experience assure ourselves of the truth even of the first law of motion4 99,a material particle991 we are told1 99when not acted on by extraneous forces will continue in the same state of rest or motion499 ,this may be true1 but as we can find no body which is free from the action of extraneous causes1 how are we to prove it8 ,only by observing that the less the amount of those forces the more nearly is the law found to be true4 ,a ball rolled along rough ground is soon stopped2 along a smooth pavement it continues longer in movement4 ,a delicately suspended pendulum is almost free from friction against its supports1 but it is gradually stopped by the resistance of the air2 place it in the vacuous receiver of an air-pump and we find the motion much prolonged4 ,a large planet like ,jupiter experiences almost infinitely less friction1 in comparison to its vast momentum1 than we can produce experimentally1 and we find in such a case that there is not the least evidence of the falsity of the law4 ,experience1 then1 informs us that we may approximate indefinitely to a uniform motion by sufficiently decreasing the disturbing forces4 ,it is an act of inference which enables us to travel on beyond experience1 and assert that1 in the total absence of any extraneous force1 motion would be absolutely uniform4 ,the state of rest1 again1 is a limiting case in which motion is infinitely small or zero1 to which we may attain1 on the principle of continuity1 by successively considering cases of slower and slower motion4 ,there are many classes of phenomena1 in which1 by gradually passing from the apparent to the obscure1 we can assure ourselves of the nature of phenomena which would otherwise be a matter of great doubt4 ,thus we can sufficiently prove in the manner of ,galileo1 that a musical sound consists of rapid uniform pulses1 by causing strokes to be made at intervals which we gradually diminish until the separate strokes coalesce into a uniform hum or note4 ,with great advantage we approach1 as ,tyndall says1 the sonorous through the grossly mechanical4 ,in listening to a great organ we cannot fail to perceive that the longest pipes1 or their partial tones1 produce a tremor and fluttering of the building4 ,at the other extremity of the scale1 there is no fixed limit to the acuteness of sounds which we can hear2 some individuals can hear sounds too shrill for other ears1 and as there is nothing in the nature of the atmosphere to prevent the existence of undulations far more rapid than any of which we are conscious1 we may infer1 by the principle of continuity1 that such undulations probably exist4 ,there are many habitual actions which we perform we know not how4 ,so rapidly are acts of minds accomplished that analysis seems impossible4 ,we can only investigate them when in process of formation1 observing that the best formed habit is slowly and continuously acquired1 and it is in the early stages that we can perceive the rationale of the process4 ,let it be observed that this principle of continuity must be held of much weight only in exact physical laws1 those which doubtless repose ultimately upon the simple laws of motion4 ,if we fearlessly apply the principle to all kinds of phenomena1 we may often be right in our inferences1 but also often wrong4 ,thus1 before the development of spectrum analysis1 astronomers had observed that the more they increased the powers of their telescopes the more nebul they could resolve into distinct stars4 ,this result had been so often found true that they almost irresistibly assumed that all nebul would be ultimately resolved by telescopes of sufficient power2 yet ,huggins has in recent years proved by the spectroscope1 that certain nebul are actually gaseous1 and in a truly nebulous state4 ,the principle of continuity must have been continually employed in the inquiries of ,galileo1 ,newton1 and other experimental philosophers1 but it appears to have been distinctly formulated for the first time by ,leibnitz4 ,he at least claims to have first spoken of 8the law of continuity0 in a letter to ,bayle1 printed in the 99,nouvelles de la ,rpublique des ,lettres991 an extract from which is given in ,erdmann's edition of ,leibnitz's works1 p4 #ajd1 under the title 8,sur un ,principe ,gnral utile l'explication des ,lois de la ,nature40#eaa ,it has indeed been asserted that the doctrine of the 99latens processus99 of ,francis ,bacon involves the principle of continuity1#eab but ,i think that this doctrine1 like that of the 99natures99 of substances1 is merely a vague statement of the principle of causation4 #eaa 99,life of ,sir ,w4 ,hamilton991 p4 #dci4 #eab ,powell's 99,history of ,natural ,philosophy991 p4 #bja4 99,novum ,organum991 bk4 ii4 ,aphorisms #e-g4 99,failure of the ,law of ,continuity499 ,there are certain cautions which must be given as to the application of the principle of continuity4 ,in the first place1 where this principle really holds true1 it may seem to fail owing to our imperfect means of observation4 ,though a physical law may not admit of perfectly abrupt change1 there is no limit to the approach which it may make to abruptness4 ,when we warm a piece of very cold ice1 the absorption of heat1 the temperature1 and the dilatation of the ice vary according to apparently simple laws until we come to the zero of the ,centigrade scale4 ,everything is then changed2 an enormous absorption of heat takes place without any rise of temperature1 and the volume of the ice decreases as it changes into water4 ,unless carefully investigated1 this change appears to be perfectly abrupt2 but accurate observation seems to show that there is a certain forewarning2 the ice does not turn into water all at once1 but through a small fraction of a degree the change is gradual4 ,all the phenomena concerned1 if measured very exactly1 would be represented not by angular lines1 but continuous curves1 undergoing rapid flexures2 and we may probably assert with safety that between whatever points of temperature we examine ice1 there would be found some indication1 though almost infinitesimally small1 of the apparently abrupt change which was to occur at a higher temperature4 ,it might also be pointed out that the important and apparently simple physical laws1 such as those of ,boyle and ,mariotte1 ,dalton and ,gay-,lussac1 c41 are only approximately true1 and the divergences from the simple laws are forewarnings of abrupt changes1 which would otherwise break the law of continuity4 ,secondly1 it must be remembered that mathematical laws of some complexity will probably present singular cases or negative results1 which may bear the appearance of discontinuity1 as when the law of retraction suddenly yields us with perfect abruptness the phenomenon of total internal reflection4 ,in the undulatory theory1 however1 there is no real change of law between refraction and reflection4 ,faraday in the earlier part of his career found so many substances possessing magnetic power1 that he ventured on a great generalisation1 and asserted that all bodies shared in the magnetic property of iron4 ,his mistake1 as he afterwards discovered1 consisted in overlooking the fact that though magnetic in a certain sense1 some substances have negative magnetism1 and are repelled instead of being attracted by the magnet4 ,thirdly1 where we might expect to find a uniform mathematical law prevailing1 the law may undergo abrupt change at singular points1 and actual discontinuity may arise4 ,we may sometimes be in danger of treating under one law phenomena which really belong to different laws4 ,for instance1 a spherical shell of uniform matter attracts an external particle of matter with a force varying inversely as the square of the distance from the centre of the sphere4 ,but this law only holds true so long as the particle is external to the shell4 ,within the shell the law is wholly different1 and the aggregate gravity of the sphere becomes zero1 the force in every direction being neutralised by an exactly equal opposite force4 ,if an infinitely small particle be in the superficies of a sphere1 the law is again different1 and the attractive power of the shell is half what it would be with regard to particles infinitely close to the surface of the shell4 ,thus in approaching the centre of a shell from a distance1 the force of gravity shows double discontinuity in passing through the shell4#eac #eac ,thomson and ,tait1 99,treatise on ,natural ,philosophy991 vol4 i4 pp4 #cdf-cea4 ,it may admit of question1 too1 whether discontinuity is really unknown in nature4 ,we perpetually do meet with events which are real breaks upon the previous law1 though the discontinuity may be a sign that some independent cause has come into operation4 ,if the ordinary course of the tides is interrupted by an enormous irregular wave1 we attribute it to an earthquake1 or some gigantic natural disturbance4 ,if a meteoric stone falls upon a person and kills him1 it is clearly a discontinuity in his life1 of which he could have had no anticipation4 ,a sudden sound may pass through the air neither preceded nor followed by any continuous effect4 ,although1 then1 we may regard the ,law of ,continuity as a principle of nature holding rigorously true in many of the relations of natural forces1 it seems to be a matter of difficulty to assign the limits within which the law is verified4 ,much caution is required in its application4 99,negative ,arguments on the ,principle of ,continuity499 ,upon the principle of continuity we may sometimes found arguments of great force which prove an hypothesis to be impossible1 because it would involve a continual repetition of a process 99ad infinitum991 or else a purely arbitrary breach at some point4 ,bonnet's famous theory of reproduction represented every living creature as containing germs which were perfect representatives of the next generation1 so that on the same principle they necessarily included germs of the next generation1 and so on indefinitely4 ,the theory was sufficiently refuted when once clearly stated1 as in the following poem called the ,universe1#ead by ,henry ,baker3-- 8,each seed includes a plant3 that plant1 again1 ,has other seeds1 which other plants contain3 ,those other plants have all their seeds1 and those ,more plants again1 successively inclose4 8,thus1 ev'ry single berry that we find1 ,has1 really1 in itself whole forests of its kind1 ,empire and wealth one acorn may dispense1 ,by fleets to sail a thousand ages hence40 #ead 99,philosophical ,transactions99 7#agdj71 vol4 xli4 p4 #ded4 ,the general principle of inference1 that what we know of one case must be true of similar cases1 so far as they are similar1 prevents our asserting anything which we cannot apply time after time under the same circumstances4 ,on this principle ,stevinus beautifully demonstrated that weights resting on two inclined planes and balancing each other must be proportional to the lengths of the planes between their apex and a horizontal plane4 ,he imagined a uniform endless chain to be hung over the planes1 and to hang below in a symmetrical festoon4 ,if the chain were ever to move by gravity1 there would be the same reason for its moving on for ever1 and thus producing a perpetual motion4 ,as this is absurd1 the portions of the chain lying on the planes1 and equal in length to the planes1 must balance each other4 ,on similar grounds we may disprove the existence of any 99self-moving machine992 for if it could once alter its own state of motion or rest1 in however small a degree1 there is no reason why it should not do the like time after time 99ad infinitum994 ,newton's proof of his third law of motion1 in the case of gravity1 is of this character4 ,for he remarks that if two gravitating bodies do not exert exactly equal forces in opposite directions1 the one exerting the strongest pull will carry both away1 and the two bodies will move off into space together with velocity increasing 99ad infinitum994 ,but though the argument might seem sufficiently convincing1 ,newton in his characteristic way made an experiment with a loadstone and iron floated upon the surface of water4#eae ,in recent years the very foundation of the principle of conservation of energy has been placed on the assumption that it is impossible by any combination of natural bodies to produce force continually from nothing4#eaf ,the principle admits of application in various subtle forms4 #eae 99,principia991 bk4 i4 ,law iii4 ,corollary #f4 #eaf ,helmholtz1 ,taylor's 99,scientific ,memoirs99 7#ahec71 vol4 vi4 p4 #aah4 ,lucretius attempted to prove1 by a most ingenious argument of this kind1 that matter must be indestructible4 ,for if a finite quantity1 however small1 were to fall out of existence in any finite time1 an equal quantity might be supposed to lapse in every equal interval of time1 so that in the infinity of past time the universe must have ceased to exist4#eag ,but the argument1 however ingenious1 seems to fail at several points4 ,if past time be infinite1 why may not matter have been created infinite also8 ,it would be most reasonable1 again1 to suppose the matter destroyed in any time to be proportional to the matter then remaining1 and not to the original quantity2 under this hypothesis even a finite quantity of original matter could never wholly disappear from the universe4 ,for like reasons we cannot hold that the doctrine of the conservation of energy is really proved1 or can ever be proved to be absolutely true1 however probable it may be regarded4 #eag 99,lucretius991 bk4 i4 lines #bcb-bfd4 99,tendency to ,hasty ,generalisation499 ,in spite of all the powers and advantages of generalisation1 men require no incitement to generalise2 they are too apt to draw hasty and ill-considered inferences4 ,as ,francis ,bacon said1 our intellects want not wings1 but rather weights of lead to moderate their course4#eah ,the process is inevitable to the human mind2 it begins with childhood and lasts through the second childhood4 ,the child that has once been hurt fears the like result on all similar occasions1 and can with difficulty be made to distinguish between case and case4 ,it is caution and discrimination in the adoption of conclusions that we have chiefly to learn1 and the whole experience of life is one continued lesson to this effect4 ,baden ,powell has excellently described this strong natural propensity to hasty inference1 and the fondness of the human mind for tracing resemblances real or fanciful4 8,our first inductions10 he says1#eai 8are always imperfect and inconclusive2 we advance towards real evidence by successive approximations2 and accordingly we find false generalisation the besetting error of most first attempts at scientific research4 ,the faculty to generalise accurately and philosophically requires large caution and long training1 and is not fully attained1 especially in reference to more general views1 even by some who may properly claim the title of very accurate scientific observers in a more limited field4 ,it is an intellectual habit which acquires immense and accumulating force from the contemplation of wider analogies40 #eah 99,novum ,organum991 bk4 #a ,aphorism #ajd4 #eai 99,the ,unity of ,worlds and of ,nature991 #bnd edit4 p4 #aaf4 ,hasty and superficial generalisations have always been the bane of science1 and there would be no difficulty in finding endless illustrations4 ,between things which are the same in number there is a certain resemblance1 namely in number2 but in the infancy of science men could not be persuaded that there was not a deeper resemblance implied in that of number4 ,pythagoras was not the inventor of a mystical science of number4 ,in the ancient ,oriental religions the seven metals were connected with the seven planets1 and in the seven days of the week we still have1 and probably always shall have1 a relic of the septiform system ascribed by ,dio ,cassius to the ancient ,egyptians4 ,the disciples of ,pythagoras carried the doctrine of the number seven into great detail4 ,seven days are mentioned in ,genesis2 infants acquire their teeth at the end of seven months2 they change them at the end of seven years2 seven feet was the limit of man's height2 every seventh year was a climacteric or critical year1 at which a change of disposition took place4 ,then again there were the seven sages of ,greece1 the seven wonders of the world1 the seven rites of the ,grecian games1 the seven gates of ,thebes1 and the seven generals destined to conquer that city4 ,in natural science there were not only the seven planets1 and the seven metals1 but also the seven primitive colours1 and the seven tones of music4 ,so deep a hold did this doctrine take that we still have its results in many customs1 not only in the seven days of the week1 but the seven years' apprenticeship1 puberty at fourteen years1 the second climacteric1 and legal majority at twenty-one years1 the third climacteric4 ,the idea was reproduced in the seven sacraments of the ,roman ,catholic ,church1 and the seven year periods of ,comte's grotesque system of domestic worship4 ,even in scientific matters the loftiest intellects have occasionally yielded1 as when ,newton was misled by the analogy between the seven tones of music and the seven colours of his spectrum4 ,other numerical analogies1 though rejected by ,galileo1 held ,kepler in thraldom2 no small part of ,kepler's labours during seventeen years was spent upon numerical and geometrical analogies of the most baseless character2 and he gravely held that there could not be more than six planets1 because there were not more than five regular solids4 ,even the genius of ,huyghens did not prevent him from inferring that but one satellite could belong to ,saturn1 because1 with those of ,jupiter and the ,earth1 it completed the perfect number of six4 ,a whole series of other superstitions and fallacies attach to the numbers six and nine4 ,it is by false generalisation1 again1 that the laws of nature have been supposed to possess that perfection which we attribute to simple forms and relations4 ,the heavenly bodies1 it was held1 must move in circles1 for the circle was the perfect figure4 ,newton seemed to adopt the questionable axiom that nature always proceeds in the simplest way2 in stating his first rule of philosophising1 he adds3#ebj 8,to this purpose the philosophers say1 that nature does nothing in vain1 when less will serve2 for nature is pleased with simplicity1 and affects not the pomp of superfluous causes40 ,keill lays down#eba as an axiom that 8,the causes of natural things are such1 as are the most simple1 and are sufficient to explain the phenomena3 for nature always proceeds in the simplest and most expeditious method2 because by this manner of operating the ,divine ,wisdom displays itself the more40 ,if this axiom had any clear grounds of truth1 it would not apply to proximate laws2 for even when the ultimate law is simple the results may be infinitely diverse1 as in the various elliptic1 hyperbolic1 parabolic1 or circular orbits of the heavenly bodies4 ,simplicity is naturally agreeable to a mind of limited powers1 but to an infinite mind all things are simple4 #ebj 99,principia991 bk4 iii1 99ad initium994 #eba ,keill1 99,introduction to ,natural ,philosophy991 p4 #hi4 ,every great advance in science consists in a great generalisation1 pointing out deep and subtle resemblances4 ,the ,copernican system was a generalisation1 in that it classed the earth among the planets2 it was1 as ,bishop ,wilkins expressed it1 8the discovery of a new planet10 but it was opposed by a more shallow generalisation4 ,those who argued from the condition of things upon the earth's surface1 thought that every object must be attached to and rest upon something else4 ,shall the earth1 they said1 alone be free8 ,accustomed to certain special results of gravity they could not conceive its action under widely different circumstances4#ebb ,no hasty thinker could seize the deep analogy pointed out by ,horrocks between a pendulum and a planet1 true in substance though mistaken in some details4 ,all the advances of modern science rise from the conception of ,galileo1 that in the heavenly bodies1 however apparently different their condition1 we shall ultimately recognise the same fundamental principles of mechanical science which are true on earth4 #ebb ,jeremi ,horroccii 99,opera ,posthuma99 7#afgc71 pp4 #bf1 #bg4 ,generalisation is the great prerogative of the intellect1 but it is a power only to be exercised safely with much caution and after long training4 ,every mind must generalise1 but there are the widest differences in the depth of the resemblances discovered and the care with which the discovery is verified4 ,there seems to be an innate power of insight which a few men have possessed pre-eminently1 and which enabled them1 with no exemption indeed from labour or temporary error1 to discover the one in the many4 ,minds of excessive acuteness may exist1 which have yet only the powers of minute discrimination1 and of storing up1 in the treasure-house of memory1 vast accumulations of words and incidents4 ,but the power of discovery belongs to a more restricted class of minds4 ,laplace said that1 of all inventors who had contributed the most to the advancement of human knowledge1 ,newton and ,lagrange appeared to possess in the highest degree the happy tact of distinguishing general principles among a multitude of objects enveloping them1 and this tact he conceived to be the true characteristic of scientific genius4#ebc #ebc ,young's 99,works991 vol4 ii4 p4 #efd4 ,,chapter ,,xxviii4 ,,analogy4 ,as we have seen in the previous chapter1 generalisation passes insensibly into reasoning by analogy1 and the difference is one of degree4 ,we are said to generalise when we view many objects as agreeing in a few properties1 so that the resemblance is extensive rather than deep4 ,when we have only a few objects of thought1 but are able to discover many points of resemblance1 we argue by analogy that the correspondence will be even deeper than appears4 ,it may not be true that the words are always used in such distinct senses1 and there is great vagueness in the employment of these and many logical terms2 but if any clear discrimination can be drawn between generalisation and analogy1 it is as indicated above4 ,it has been said1 indeed1 that analogy denotes not a resemblance between things1 but between the relations of things4 ,a pilot is a very different man from a prime minister1 but he bears the same relation to a ship that the minister does to the state1 so that we may analogically describe the prime minister as the pilot of the state4 ,a man differs still more from a horse1 nevertheless four men bear to three men the same relation as four horses bear to three horses4 ,there is a real analogy between the tones of the ,monochord1 the ,sages of ,greece1 and the ,gates of ,thebes1 but it does not extend beyond the fact that they were all seven in number4 ,between the most discrete notions1 as1 for instance1 those of time and space1 analogy may exist1 arising from the fact that the mathematical conditions of the lapse of time and of motion along a line are similar4 ,there is no identity of nature between a word and the thing it signifies2 the substance 99iron99 is a heavy solid1 the word 99iron99 is either a momentary disturbance of the air1 or a film of black pigment on white paper2 but there is analogy between words and their significates4 ,the substance iron is to the substance iron-carbonate1 as the name iron is to the name iron-carbonate1 when these names are used according to their scientific definitions4 ,the whole structure of language and the whole utility of signs1 marks1 symbols1 pictures1 and representations of various kinds1 rest upon analogy4 ,i may hope perhaps to enter more fully upon this important subject at some future time1 and to attempt to show how the invention of signs enables us to express1 guide1 and register our thoughts4 ,it will be sufficient to observe here that the use of words constantly involves analogies of a subtle kind2 we should often be at a loss how to describe a notion1 were we not at liberty to employ in a metaphorical sense the name of anything sufficiently resembling it4 ,there would be no expression for the sweetness of a melody1 or the brilliancy of an harangue1 unless it were furnished by the taste of honey and the brightness of a torch4 ,a cursory examination of the way in which we popularly use the word analogy1 shows that it includes all degrees of resemblance or similarity4 ,the analogy may consist only in similarity of number or ratio1 or in like relations of time and space4 ,it may also consist in simple resemblance between physical properties4 ,we should not be using the word inconsistently with custom1 if we said that there was an analogy between iron1 nickel1 and cobalt1 manifested in the strength of their magnetic powers4 ,there is a still more perfect analogy between iodine and chlorine2 not that every property of iodine is identical with the corresponding property of chlorine2 for then they would be one and the same kind of substance1 and not two substances2 but every property of iodine resembles in all but degree some property of chlorine4 ,for almost every substance in which iodine forms a component1 a corresponding substance may be discovered containing chlorine1 so that we may confidently infer from the compounds of the one to the compounds of the other substance4 ,potassium iodide crystallises in cubes2 therefore it is to be expected that potassium chloride will also crystallise in cubes4 ,the science of chemistry as now developed rests almost entirely upon a careful and extensive comparison of the properties of substances1 bringing deep-lying analogies to light4 ,when any new substance is encountered1 the chemist is guided in his treatment of it by the analogies which it seems to present with previously known substances4 ,in this chapter ,i cannot hope to illustrate the all-pervading influence of analogy in human thought and science4 ,all science1 it has been said1 at the outset1 arises from the discovery of identity1 and analogy is but one name by which we denote the deeper-lying cases of resemblance4 ,i shall only try to point out at present how analogy between apparently diverse classes of phenomena often serves as a guide in discovery4 ,we thus commonly gain the first insight into the nature of an apparently unique object1 and thus1 in the progress of a science1 we often discover that we are treating over again1 in a new form1 phenomena which were well known to us in another form4 99,analogy as a ,guide in ,discovery499 ,there can be no doubt that discovery is most frequently accomplished by following up hints received from analogy1 as ,jeremy ,bentham remarked4#ebd ,whenever a phenomenon is perceived1 the first impulse of the mind is to connect it with the most nearly similar phenomenon4 ,if we could ever meet a thing wholly 99sui generis991 presenting no analogy to anything else1 we should be incapable of investigating its nature1 except by purely haphazard trial4 ,the probability of success by such a process is so slight1 that it is preferable to follow up the faintest clue4 ,as ,i have pointed out already 7p4 #dah71 the possible experiments are almost infinite in number1 and very numerous also are the hypotheses upon which we may proceed4 ,now it is self-evident that1 however slightly superior the probability of success by one course of procedure may be over another1 the most probable one should always be adopted first4 #ebd 99,essay on ,logic991 99,works991 vol4 viii4 p4 #bgf4 ,the chemist having discovered what he believes to be a new element1 will have before him an infinite variety of modes of treating and investigating it4 ,if in any of its qualities the substance displays a resemblance to an alkaline metal1 for instance1 he will naturally proceed to try whether it possesses other properties of the alkaline metals4 ,even the simplest phenomenon presents so many points for notice that we have a choice from among many hypotheses4 ,it would be difficult to find a more instructive instance of the way in which the mind is guided by analogy than in the description by ,sir ,john ,herschel of the course of thought by which he was led to anticipate in theory one of ,faraday's greatest discoveries4 ,herschel noticed that a screw-like form1 technically called helicoidal dissymmetry1 was observed in three cases1 namely1 in electrical helices1 plagihedral quartz crystals1 and the rotation of the plane of polarisation of light4 ,as he said1#ebe 8,i reasoned thus3 ,here are three phenomena agreeing in a 99very strange peculiarity994 ,probably1 this peculiarity is a connecting link1 physically speaking1 among them4 ,now1 in the case of the crystals and the light1 this probability has been turned into certainty by my own experiments4 ,therefore1 induction led me to conclude that a similar connection exists1 and must turn up1 somehow or other1 between the electric current and polarised light1 and that the plane of polarisation would be deflected by magneto-electricity40 ,by this course of analogical thought ,herschel had actually been led to anticipate ,faraday's great discovery of the influence of magnetic strain upon polarised light4 ,he had tried in #ahbb-be to discover the influence of electricity on light1 by sending a ray of polarised light through a helix1 or near a long wire conveying an electric current4 ,such a course of inquiry1 followed up with the persistency of ,faraday1 and with his experimental resources1 would doubtless have effected the discovery4 ,herschel also suggests that the plagihedral form of quartz crystals must be due to a screw-like strain during crystallisation2 but the notion remains unverified by experiment4 #ebe 99,life of ,faraday991 by ,bence ,jones1 vol4 ii4 p4 #bjf4 99,analogy in the ,mathematical ,sciences499 ,whoever wishes to acquire a deep acquaintance with ,nature must observe that there are analogies which connect whole branches of science in a parallel manner1 and enable us to infer of one class of phenomena what we know of another4 ,it has thus happened on several occasions that the discovery of an unsuspected analogy between two branches of knowledge has been the starting-point for a rapid course of discovery4 ,the truths readily observed in the one may be of a different character from those which present themselves in the other4 ,the analogy1 once pointed out1 leads us to discover regions of one science yet undeveloped1 to which the key is furnished by the corresponding truths in the other science4 ,an interchange of aid most wonderful in its results may thus take place1 and at the same time the mind rises to a higher generalisation1 and a more comprehensive view of nature4 ,no two sciences might seem at first sight more different in their subject matter than geometry and algebra4 ,the first deals with circles1 squares1 parallelograms1 and other forms in space2 the latter with mere symbols of number4 ,prior to the time of ,descartes1 the sciences were developed slowly and painfully in almost entire independence of each other4 ,the ,greek philosophers indeed could not avoid noticing occasional analogies1 as when ,plato in the ,thetetus describes a square number as 99equally equal991 and a number produced by multiplying two unequal factors as 99oblong994 ,euclid1 in the #gth and #hth books of his ,elements1 continually uses expressions displaying a consciousness of the same analogies1 as when he calls a number of two factors a 99plane number991 1 and distinguishes a square number of which the two factors are equal as an equal-sided and plane number1 4 ,he also calls the root of a cubic number its side1 4 ,in the ,diophantine algebra many problems of a geometrical character were solved by algebraic or numerical processes2 but there was no general system1 so that the solutions were of an isolated character4 ,in general the ancients were far more advanced in geometric than symbolic methods2 thus ,euclid in his #dth book gives the means of dividing a circle by purely geometric means into #b1 #c1 #d1 #e1 #f1 #h1 #aj1 #ab1 #ae1 #af1 #bj1 #bd1 #cj parts1 but he was totally unacquainted with the theory of the roots of unity exactly corresponding to this division of the circle4 ,during the middle ages1 on the contrary1 algebra advanced beyond geometry1 and modes of solving equations were gradually discovered by those who had no notion that at every step they were implicitly solving geometric problems4 ,it is true that ,regiomontanus1 ,tartaglia1 ,bombelli1 and possibly other early algebraists1 solved isolated geometrical problems by the aid of algebra1 but particular numbers were always used1 and no consciousness of a general method was displayed4 ,vieta in some degree anticipated the final discovery1 and occasionally represented the roots of an equation geometrically1 but it was reserved for ,descartes to show1 in the most general manner1 that every equation may be represented by a curve or figure in space1 and that every bend1 point1 cusp1 or other peculiarity in the curve indicates some peculiarity in the equation4 ,it is impossible to describe in any adequate manner the importance of this discovery4 ,the advantage was two-fold3 algebra aided geometry1 and geometry gave reciprocal aid to algebra4 ,curves such as the well-known sections of the cone were found to correspond to quadratic equations2 and it was impossible to manipulate the equations without discovering properties of those all-important curves4 ,the way was thus opened for the algebraic treatment of motions and forces1 without which ,newton's 99,principia99 could never have been worked out4 ,newton indeed was possessed by a strong infatuation in favour of the ancient geometrical methods2 but it is well known that he employed symbolic methods to discover his theorems1 and he now and then1 by some accidental use of algebraic expression1 confessed its greater power and generality4 ,geometry1 on the other hand1 gave great assistance to algebra1 by affording concrete representations of relations which would otherwise be too abstract for easy comprehension4 ,a curve of no great complexity may give the whole history of the variations of value of a troublesome mathematical expression4 ,as soon as we know1 too1 that every regular geometrical curve represents some algebraic equation1 we are presented by observation of mechanical movements with abundant suggestions towards the discovery of mathematical problems4 ,every particle of a carriage-wheel when moving on a level road is constantly describing a cycloidal curve1 the curious properties of which exercised the ingenuity of all the most skilful mathematicians of the seventeenth century1 and led to important advancements in algebraic power4 ,it may be held that the discovery of the ,differential ,calculus was mainly due to geometrical analogy1 because mathematicians1 in attempting to treat algebraically the tangent of a curve1 were obliged to entertain the notion of infinitely small quantities4#ebf ,there can be no doubt that ,newton's fluxional1 that is1 geometrical mode of stating the differential calculus1 however much it subsequently retarded its progress in ,england1 facilitated its apprehension at first1 and ,i should think it almost certain that ,newton discovered the principles of the calculus geometrically4 #ebf ,lacroix1 99,trait lmentaire de ,calcul ,diffrentiel et de ,calcul ,intgral991 #eme dit4 p4 #fii4 ,we may accordingly look upon this discovery of analogy1 this happy alliance1 as ,bossut calls it1#ebg between geometry and algebra1 as the chief source of discoveries which have been made for three centuries past in mathematical methods4 ,this is certainly the opinion of ,lagrange1 who says1 8,so long as algebra and geometry have been separate1 their progress was slow1 and their employment limited2 but since these two sciences have been united1 they have lent each other mutual strength1 and have marched together with a rapid step towards perfection40 #ebg 99,histoire des ,mathmatiques991 vol4 i4 p4 #bih4 ,the advancement of mechanical science has also been greatly aided by analogy4 ,an abstract and intangible existence like force demands much power of conception1 but it has a perfect concrete representative in a line1 the end of which may denote the point of application1 and the direction the line of action of the force1 while the length can be made arbitrarily to denote the amount of the force4 ,nor does the analogy end here2 for the moment of the force about any point1 or its product into the perpendicular distance of its line of action from the point1 is found to be represented by an area1 namely twice the area of the triangle contained between the point and the ends of the line representing the force4 ,of late years a great generalisation has been effected2 the ,double ,algebra of ,de ,morgan is true not only of space relations1 but of forces1 so that the triangle of forces is reduced to a case of pure geometrical addition4 ,nay1 the triangle of lines1 the triangle of velocities1 the triangle of forces1 the triangle of couples1 and perhaps other cognate theorems1 are reduced by analogy to one simple theorem1 which amounts to this1 that there are two ways of getting from one angular point of a triangle to another1 which ways1 though different in length1 are identical in their final results4#ebh ,in the system of quaternions of the late ,sir ,w4 ,r4 ,hamilton1 these analogies are embodied and carried out in the most general manner1 so that whatever problem involves the threefold dimensions of space1 or relations analogous to those of space1 is treated by a symbolic method of the most comprehensive simplicity4 #ebh ,see ,goodwin1 99,cambridge ,philosophical ,transactions99 7#ahde71 vol4 viii4 p4 #bfi4 ,o',brien1 8,on ,symbolical ,statics10 99,philosophical ,magazine991 #dth ,series1 vol4 i4 pp4 #dia1 c4 ,see also ,professor ,clerk ,maxwell's delightful 99,manual of ,elementary ,science991 called 99,matter and ,motion991 published by the ,society for ,promoting ,christian ,knowledge4 ,in this admirable little work some of the most advanced results of mechanical and physical science are explained according to the method of quaternions1 but with hardly any use of algebraic symbols4 ,it ought to be added that to the discovery of analogy between the forms of mathematical and logical expressions1 we owe the greatest advance in logical science4 ,boole based his extension of logical processes upon the notion that logic is an algebra of two quantities #j and #a4 ,his profound genius for symbolic investigation led him to perceive by analogy that there must exist a general system of logical deduction1 of which the old logicians had seized only a few fragments4 ,mistaken as he was in placing algebra as a higher science than logic1 no one can deny that the development of the more complex and dependent science had advanced far beyond that of the simpler science1 and that ,boole1 in drawing attention to the connection1 made one of the most important discoveries in the history of science4 ,as ,descartes had wedded algebra and geometry1 so did ,boole accomplish the marriage of logic and algebra4 99,analogy in the ,theory of ,undulations499 ,there is no class of phenomena which more thoroughly illustrates alike the power and weakness of analogy than the waves which agitate every kind of medium4 ,all waves1 whatsoever be the matter through which they pass1 obey the principles of rhythmical or harmonic motion1 and the subject therefore presents a fine field for mathematical generalisation4 ,each kind of medium may allow of waves peculiar in their conditions1 so that it is a beautiful exercise in analogical reasoning to decide how1 in making inferences from one kind of medium to another1 we must make allowance for difference of circumstances4 ,the waves of the ocean are large and visible1 and there are the yet greater tidal waves which extend around the globe4 ,from such palpable cases of rhythmical movement we pass to waves of sound1 varying in length from about #cb feet to a small fraction of an inch4 ,we have but to imagine1 if we can1 the fortieth octave of the middle ,c of a piano1 and we reach the undulations of yellow light1 the ultra-violet being about the forty-first octave4 ,thus we pass from the palpable and evident to that which is obscure1 if not incomprehensible4 ,yet the same phenomena of reflection1 interference1 and refraction1 which we find in some kinds of waves1 may be expected to occur1 99mutatis mutandis991 in other kinds4 ,from the great to the small1 from the evident to the obscure1 is not only the natural order of inference1 but it is the historical order of discovery4 ,the physical science of the ,greek philosophers must have remained incomplete1 and their theories groundless1 because they did not understand the nature of undulations4 ,their systems were based upon the notion of movement of translation from place to place4 ,modern science tends to the opposite notion that all motion is alternating or rhythmical1 energy flowing onwards but matter remaining comparatively fixed in position4 ,diogenes ,laertius indeed correctly compared the propagation of sound with the spreading of waves on the surface of water when disturbed by a stone1 and ,vitruvius displayed a more complete comprehension of the same analogy4 ,it remained for ,newton to create the theory of undulatory motion in showing by mathematical deductive reasoning that the particles of an elastic fluid by vibrating backwards and forwards1 might carry a pulse or wave moving from the source of disturbance1 while the disturbed particles return to their place of rest4 ,he was even able to make a first approximation by theoretical calculation to the velocity of sound-waves in the atmosphere4 ,his theory of sound formed a hardly less important epoch in science than his far more celebrated theory of gravitation4 ,it opened the way to all the subsequent applications of mechanical principles to the insensible motion of molecules4 ,he seems to have been1 too1 upon the brink of another application of the same principles which would have advanced science by a century of progress1 and made him the undisputed founder of all the theories of matter4 ,he expressed opinions at various times that light might be due to undulatory movements of a medium occupying space1 and in one intensely interesting sentence remarks#ebi that colours are probably vibrations of different lengths1 8much after the manner that1 in the sense of hearing1 nature makes use of arial vibrations of several bignesses to generate sounds of divers tones1 for the analogy of nature is to be observed40 ,he correctly foresaw that red and yellow light would consist of the longer undulations1 and blue and violet of the shorter1 while white light would be composed of an indiscriminate mixture of waves of various lengths4 ,newton almost overcame the strongest apparent difficulty of the undulatory theory of light1 namely1 the propagation of light in straight lines4 ,for he observed that though waves of sound bend round an obstacle to some extent1 they do not do so in the same degree as water-waves4#ecj ,he had but to extend the analogy proportionally to light-waves1 and not only would the difficulty have vanished1 but the true theory of diffraction would have been open to him4 ,unfortunately he had a preconceived theory that rays of light are bent from and not towards the shadow of a body1 a theory which for once he did not sufficiently compare with observation to detect its falsity4 ,i am not aware1 too1 that ,newton has1 in any of his works1 displayed an understanding of the phenomena of interference without which his notion of waves must have been imperfect4 #ebi ,birch1 99,history of the ,royal ,society991 vol4 iii4 p4 #bfb1 quoted by ,young1 99,works991 vol4 i4 p4 #bdf4 #ecj 99,opticks991 ,query #bh1 #crd edit4 p4 #ccg4 ,while the general principles of undulatory motion will be the same in whatever medium the motion takes place1 the circumstances may be excessively different4 ,between light travelling #ahf1jjj miles per second and sound travelling in air only about #a1ajj feet in the same time1 or almost #ijj1jjj times as slowly1 we cannot expect a close outward resemblance4 ,there are great differences1 too1 in the character of the vibrations4 ,gases scarcely admit of transverse vibration1 so that sound travelling in air is a longitudinal wave1 the particles of air moving backwards and forwards in the same line in which the wave moves onwards4 ,light1 on the other hand1 appears to consist entirely in the movement of points of force transversely to the direction of propagation of the ray4 ,the light-wave is partially analogous to the bending of a rod or of a stretched cord agitated at one end4 ,now this bending motion may take place in any one of an infinite number of planes1 and waves of which the planes are perpendicular to each other cannot interfere any more than two perpendicular forces can interfere4 ,the complicated phenomena of polarised light arise out of this transverse character of the luminous wave1 and we must not expect to meet analogous phenomena in atmospheric sound-waves4 ,it is conceivable that in solids we might produce transverse sound undulations1 in which phenomena of polarisation might be reproduced4 ,but it would appear that even between transverse sound and light-waves the analogy holds true rather of the principles of harmonic motion than the circumstances of the vibrating medium2 from experiment and theory it is inferred that the plane of polarisation in plane polarised light is perpendicular to instead of being coincident with the direction of vibration1 as it would be in the case of transverse sound undulations4 ,if so the laws of elastic forces are essentially different in application to the luminiferous ether and to ordinary solid bodies4#eca #eca ,rankine1 99,philosophical ,transactions99 7#ahef71 vol4 cxlvi4 p4 #bhb4 99,analogy in ,astronomy499 ,we shall be much assisted in gaining a true appreciation of the value of analogy in its feebler degrees1 by considering how much it has contributed to the progress of astronomical science4 ,our point of observation is so fixed with regard to the universe1 and our means of examining distant bodies are so restricted1 that we are necessarily guided by limited and apparently feeble resemblances4 ,in many cases the result has been confirmed by subsequent direct evidence of the most forcible character4 ,while the scientific world was divided in opinion between the ,copernican and ,ptolemaic systems1 it was analogy which furnished the most satisfactory argument4 ,galileo discovered1 by the use of his new telescope1 the four small satellites which circulate round ,jupiter1 and make a miniature planetary world4 ,these four ,medicean ,stars1 as they were called1 were plainly seen to revolve round ,jupiter in various periods1 but approximately in one plane1 and astronomers irresistibly inferred that what might happen on the smaller scale might also be found true of the greater planetary system4 ,this discovery gave 8the holding turn10 as ,herschel expressed it1 to the opinions of mankind4 ,even ,francis ,bacon1 who1 little to the credit of his scientific sagacity1 had previously opposed the ,copernican views1 now became convinced1 saying 8,we affirm the solisequium of ,venus and ,mercury2 since it has been found by ,galileo that ,jupiter also has attendants40 ,nor did ,huyghens think it superfluous to adopt the analogy as a valid argument4#ecb ,even in an advanced stage of physical astronomy1 the ,jovian system has not lost its analogical interest2 for the mutual perturbations of the four satellites pass through all their phases within a few centuries1 and thus enable us to verify in a miniature case the principles of stability1 which ,laplace established for the great planetary system4 ,oscillations or disturbances which in the motions of the planets appear to be secular1 because their periods extend over millions of years1 can be watched1 in the case of ,jupiter's satellites1 through complete revolutions within the historical period of astronomy4#ecc #ecb 99,cosmotheoros99 7#afii71 p4 #af4 #ecc ,laplace1 99,system of the ,world991 vol4 ii4 p4 #caf4 ,in obtaining a knowledge of the stellar universe we must sometimes depend upon precarious analogies4 ,we still hold upon this ground the opinion1 entertained by ,bruno as long ago as #aeia1 that the stars may be suns attended by planets like our earth4 ,this is the most probable first assumption1 and it is supported by spectrum observations1 which show the similarity of light derived from many stars with that of the sun4 ,but at the same time we learn by the prism that there are nebul and stars in conditions widely different from anything known in our system4 ,in the course of time the analogy may perhaps be restored to comparative completeness by the discovery of suns in various stages of nebulous condensation4 ,the history of the evolution of our own world may be traced back in bodies less developed1 or traced forwards in systems more advanced towards the dissipation of energy1 and the extinction of life4 ,as in a great workshop1 we may perhaps see the material work of ,creation as it has progressed through thousands of millions of years4 ,in speculations concerning the physical condition of the planets and their satellites1 we depend upon analogies of a weak character4 ,we may be said to know that the moon has mountains and valleys1 plains and ridges1 volcanoes and streams of lava1 and1 in spite of the absence of air and water1 the rocky surface of the moon presents so many familiar appearances that we do not hesitate to compare them with the features of our globe4 ,we infer with high probability that ,mars has polar snow and an atmosphere absorbing blue rays like our own2 ,jupiter undoubtedly possesses a cloudy atmosphere1 possibly not unlike a magnified copy of that surrounding the earth1 but our tendency to adopt analogies receives a salutary correction in the recently discovered fact that the atmosphere of ,uranus contains hydrogen4 ,philosophers have not stopped at these comparatively safe inferences1 but have speculated on the existence of living creatures in other planets4 ,huyghens remarked that as we infer by analogy from the dissected body of a dog to that of a pig and ox or other animal of the same general form1 and as we expect to find the same viscera1 the heart1 stomach1 lungs1 intestines1 c41 in corresponding positions1 so when we notice the similarity of the planets in many respects1 we must expect to find them alike in other respects4#ecd ,he even enters into an inquiry whether the inhabitants of other planets would possess reason and knowledge of the same sort as ours1 concluding in the affirmative4 ,although the power of intellect might be different1 he considers that they would have the same geometry if they had any at all1 and that what is true with us would be true with them4#ece ,as regards the sun1 he wisely observes that every conjecture fails4 ,laplace entertained a strong belief in the existence of inhabitants on other planets4 ,the benign influence of the sun gives birth to animals and plants upon the surface of the earth1 and analogy induces us to believe that his rays would tend to have a similar effect elsewhere4 ,it is not probable that matter which is here so fruitful of life would be sterile upon so great a globe as ,jupiter1 which1 like the earth1 has its days and nights and years1 and changes which indicate active forces4 ,man indeed is formed for the temperature and atmosphere in which he lives1 and1 so far as appears1 could not live upon the other planets4 ,but there might be an infinity of organisations relative to the diverse constitutions of the bodies of the universe4 ,the most active imagination cannot form any idea of such various creatures1 but their existence is not unlikely4#ecf #ecd 99,cosmotheoros99 7#afii71 p4 #ag4 #ece ,ibid4 p4 #cf4 #ecf 99,system of the ,world991 vol4 ii4 p4 #cbf4 99,essai ,philosophique991 p4 #hg4 ,we now know that many metals and other elements never found in organic structures are yet capable of forming compounds with substances of vegetable or animal origin4 ,it is therefore just possible that at different temperatures creatures formed of different yet analogous compounds might exist1 but it would seem indispensable that carbon should form the basis of organic structures4 ,we have no analogies to lead us to suppose that in the absence of that complex element life can exist4 ,could we find globes surrounded by atmospheres resembling our own in temperature and composition1 we should be almost forced to believe them inhabited1 but the probability of any analogical argument decreases rapidly as the condition of a globe diverges from that of our own4 ,the ,cardinal ,nicholas de ,cusa held long ago that the moon was inhabited1 but the absence of any appreciable atmosphere renders the existence of inhabitants highly improbable4 ,speculations resting upon weak analogies hardly belong to the scope of true science1 and can only be tolerated as an antidote to the far worse dogmas which assert that the thousand million of persons on earth1 or rather a small fraction of them1 are the sole objects of care of the ,power which designed this limitless ,universe4 99,failures of ,analogy499 ,so constant is the aid which we derive from the use of analogy in all attempts at discovery or explanation1 that it is most important to observe in what cases it may lead us into difficulties4 ,that which we expect by analogy to exist 7#a7 ,may be found to exist2 7#b7 ,may seem not to exist1 but nevertheless may really exist2 7#c7 ,may actually be non-existent4 ,in the second case the failure is only apparent1 and arises from our obtuseness of perception1 the smallness of the phenomenon to be noticed1 or the disguised character in which it appears4 ,i have already pointed out that the analogy of sound and light seems to fail because light does not apparently bend round a corner1 the fact being that it does so bend in the phenomena of diffraction1 which present the effect1 however1 in such an unexpected and minute form1 that even ,newton was misled1 and turned from the correct hypothesis of undulations which he had partially entertained4 ,in the third class of cases analogy fails us altogether1 and we expect that to exist which really does not exist4 ,thus we fail to discover the phenomena of polarisation in sound travelling through the atmosphere1 since air is not capable of any appreciable transverse undulations4 ,these failures of analogy are of peculiar interest1 because they make the mind aware of its superior powers4 ,there have been many philosophers who said that we can conceive nothing in the intellect which we have not previously received through the senses4 ,this is true in the sense that we cannot 99image99 them to the mind in the concrete form of a shape or a colour2 but we can speak of them and reason concerning them2 in short1 we often know them in everything but a sensuous manner4 ,accurate investigation shows that all material substances retard the motion of bodies through them by subtracting energy by impact4 ,by the law of continuity we can frame the notion of a vacuous space in which there is no resistance whatever1 nor need we stop there2 for we have only to proceed by analogy to the case where a medium should accelerate the motion of bodies passing through it1 somewhat in the mode which ,aristotelians attributed falsely to the air4 ,thus we can frame the notion of 99negative density991 and ,newton could reason exactly concerning it1 although no such thing exists4#ecg #ecg 99,principia991 bk4 ii4 ,section ii4 ,prop4 x4 ,in every direction of thought we may meet ultimately with similar failures of analogy4 ,a moving point generates a line1 a moving line generates a surface1 a moving surface generates a solid1 but what does a moving solid generate8 ,when we compare a polyhedron1 or many-sided solid1 with a polygon1 or plane figure of many sides1 the volume of the first is analogous to the area of the second2 the face of the solid answers to the side of the polygon2 the edge of the solid to the point of the figure2 but the corner1 or junction of edges in the polyhedron1 is left wholly unrepresented in the plane of the polygon4 ,even if we attempted to draw the analogies in some other manner1 we should still find a geometrical notion embodied in the solid which has no representative in the figure of two dimensions4#ech #ech ,de ,morgan1 99,cambridge ,philosophical ,transactions991 vol4 xi4 ,part ii4 p4 #bdf4 ,faraday was able to frame some notion of matter in a fourth condition1 which should be to gas what gas is to liquid4#eci ,such substance1 he thought1 would not fall far short of 99radiant matter991 by which apparently he meant the supposed caloric or matter assumed to constitute heat1 according to the corpuscular theory4 ,even if we could frame the notion1 matter in such a state cannot be known to exist1 and recent discoveries concerning the continuity of the solid1 liquid1 and gaseous states remove the basis of the speculation4 #eci 99,life of ,faraday991 vol4 i4 p4 #baf4 ,from these and many other instances which might be adduced1 we learn that analogical reasoning leads us to the conception of many things which1 so far as we can ascertain1 do not exist4 ,in this way great perplexities have arisen in the use of language and mathematical symbols4 ,all language depends upon analogy2 for we join and arrange words so that they may represent the corresponding junctions or arrangements of things and their equalities4 ,but in the use of language we are obviously capable of forming many combinations of words to which no corresponding meaning apparently exists4 ,the same difficulty arises in the use of mathematical signs1 and mathematicians have needlessly puzzled themselves about the square root of a negative quantity1 which is1 in many applications of algebraic calculation1 simply a sign without any analogous meaning1 there being a failure of analogy4 ,,chapter ,,xxix4 ,,exceptional ,,phenomena4 ,if science consists in the detection of identity and the recognition of uniformity existing in many objects1 it follows that the progress of science depends upon the study of exceptional phenomena4 ,such new phenomena are the raw material upon which we exert our faculties of observation and reasoning1 in order to reduce the new facts beneath the sway of the laws of nature1 either those laws already well known1 or those to be discovered4 ,not only are strange and inexplicable facts those which are on the whole most likely to lead us to some novel and important discovery1 but they are also best fitted to arouse our attention4 ,so long as events happen in accordance with our anticipations1 and the routine of every-day observation is unvaried1 there is nothing to impress upon the mind the smallness of its knowledge1 and the depth of mystery1 which may be hidden in the commonest sights and objects4 ,in early times the myriads of stars which remained in apparently fixed relative positions upon the heavenly sphere1 received less notice from astronomers than those few planets whose wandering and inexplicable motions formed a riddle4 ,hipparchus was induced to prepare the first catalogue of stars1 because a single new star had been added to those nightly visible2 and in the middle ages two brilliant but temporary stars caused more popular interest in astronomy than any other events1 and to one of them we owe all the observations of ,tycho ,brahe1 the medival ,hipparchus4 ,in other sciences1 as well as in that of the heavens1 exceptional events are commonly the points from which we start to explore new regions of knowledge4 ,it has been beautifully said that ,wonder is the daughter of ,ignorance1 but the mother of ,invention2 and though the most familiar and slight events1 if fully examined1 will afford endless food for wonder and for wisdom1 yet it is the few peculiar and unlooked-for events which most often lead to a course of discovery4 ,it is true1 indeed1 that it requires much philosophy to observe things which are too near to us4 ,the high scientific importance attaching1 then1 to exceptions1 renders it desirable that we should carefully consider the various modes in which an exception may be disposed of2 while some new facts will be found to confirm the very laws to which they seem at first sight clearly opposed1 others will cause us to limit the generality of our previous statements4 ,in some cases the exception may be proved to be no exception2 occasionally it will prove fatal to our previous most confident speculations2 and there are some new phenomena which1 without really destroying any of our former theories1 open to us wholly new fields of scientific investigation4 ,the study of this subject is especially interesting and important1 because1 as ,i have before said 7p4 #ehg71 no important theory can be built up complete and perfect all at once4 ,when unexplained phenomena present themselves as objections to the theory1 it will often demand the utmost judgment and sagacity to assign to them their proper place and force4 ,the acceptance or rejection of a theory will depend upon discriminating the one insuperable contradictory fact from many1 which1 however singular and inexplicable at first sight1 may afterwards be shown to be results of different causes1 or possibly the most striking results of the very law with which they stand in apparent conflict4 ,i can enumerate at least eight classes or kinds of exceptional phenomena1 to one or other of which any supposed exception to the known laws of nature can usually be referred2 they may be briefly described as below1 and will be sufficiently illustrated in the succeeding sections4 7#a7 ,imaginary1 or false exceptions1 that is1 facts1 objects1 or events which are not really what they are supposed to be4 7#b7 ,apparent1 but congruent exceptions1 which1 though apparently in conflict with a law of nature1 are really in agreement with it4 7#c7 ,singular exceptions1 which really agree with a law of nature1 but exhibit remarkable and unique results of it4 7#d7 ,divergent exceptions1 which really proceed from the ordinary action of known processes of nature1 but which are excessive in amount or monstrous in character4 7#e7 ,accidental exceptions1 arising from the interference of some entirely distinct but known law of nature4 7#f7 ,novel and unexplained exceptions1 which lead to the discovery of a new series of laws and phenomena1 modifying or disguising the effects of previously known laws1 without being inconsistent with them4 7#g7 ,limiting exceptions showing the falsity of a supposed law in some cases to which it had been extended1 but not affecting its truth in other cases4 7#h7 ,contradictory or real exceptions which lead us to the conclusion that a supposed hypothesis or theory is in opposition to the phenomena of nature1 and must therefore be abandoned4 ,it ought to be clearly understood that in no case is a law of nature really thwarted or prevented from being fulfilled4 ,the effects of a law may be disguised and hidden from our view in some instances3 in others the law itself may be rendered inapplicable altogether2 but if a law is applicable it must be carried out4 ,every law of nature must therefore be stated with the utmost generality of all the instances really coming under it4 ,babbage proposed to distinguish between 99universal principles991 which do not admit of a single exception1 such as that every number ending in #e is divisible by five1 and 99general principles99 which are more frequently obeyed than violated1 as that 8men will be governed by what they believe to be their interest40#edj ,but in a scientific point of view general principles must be universal as regards some distinct class of objects1 or they are not principles at all4 ,if a law to which exceptions exist is stated without allusion to those exceptions1 the statement is erroneous4 ,i have no right to say that 8,all liquids expand by heat10 if ,i know that water below #d ,c4 does not2 ,i ought to say1 8,all liquids1 except water below #d ,c41 expand by heat20 and every new exception discovered will falsify the statement until inserted in it4 ,to speak of some laws as being 99generally99 true1 meaning not universally but in the majority of cases1 is a hurtful abuse of the word1 but is quite usual4 99,general99 should mean that which is true of a whole 99genus99 or class1 and every true statement must be true of some assigned or assignable class4 #edj ,babbage1 99,the ,exposition of #ahea991 p4 #a4 99,imaginary or ,false ,exceptions499 ,when a supposed exception to a law of nature is brought to our notice1 the first inquiry ought properly to be--,is there any breach of the law at all8 ,it may be that the supposed exceptional fact is not a fact at all1 but a mere figment of the imagination4 ,when ,king ,charles requested the ,royal ,society to investigate the curious fact that a live fish put into a bucket of water does not increase the weight of the bucket and its contents1 the ,royal ,society wisely commenced their deliberations by inquiring whether the fact was so or not4 ,every statement1 however false1 must have some cause or prior condition1 and the real question for the ,royal ,society to investigate was1 how the ,king came to think that the fact was so4 ,mental conditions1 as we have seen1 enter into all acts of observation1 and are often a worthy subject of inquiry4 ,but there are many instances in the history of science1 in which trouble and error have been caused by false assertions carelessly made1 and carelessly accepted without verification4 ,the reception of the ,copernican theory was much impeded by the objection1 that if the earth were moving1 a stone dropped from the top of a high tower should be left behind1 and should appear to move towards the west1 just as a stone dropped from the mast-head of a moving ship would fall behind1 owing to the motion of the ship4 ,the ,copernicans attempted to meet this grave objection in every way but the true one1 namely1 showing by trial that the asserted facts are not correct4 ,in the first place1 if a stone had been dropped with suitable precautions from the mast-head of a moving ship1 it would have fallen close to the foot of the mast1 because1 by the first law of motion1 it would remain in the same state of horizontal motion communicated to it by the mast4 ,as the anti-,copernicans had assumed the contrary result as certain to ensue1 their argument would of course have fallen through4 ,had the ,copernicans next proceeded to test with great care the other assertion involved1 they would have become still better convinced of the truth of their own theory4 ,a stone dropped from the top of a high tower1 or into a deep well1 would certainly not have been deflected from the vertical direction in the considerable degree required to agree with the supposed consequences of the ,copernican views2 but1 with very accurate observation1 they might have discovered1 as ,benzenberg subsequently did1 a very small deflection towards the east1 showing that the eastward velocity is greater at the top than the bottom4 ,had the ,copernicans then been able to detect and interpret the meaning of the small divergence thus arising1 they would have found in it corroboration of their own views4 ,multitudes of cases might be cited in which laws of nature seem to be evidently broken1 but in which the apparent breach arises from a misapprehension of the case4 ,it is a general law1 absolutely true of all crystals yet submitted to examination1 that no crystal has a re-entrant angle1 that is an angle which towards the axis of the crystal is greater than two right angles4 ,wherever the faces of a crystal meet they produce a projecting edge1 and wherever edges meet they produce a corner4 ,many crystals1 however1 when carelessly examined1 present exceptions to this law1 but closer observation always shows that the apparently re-entrant angle really arises from the oblique union of two distinct crystals4 ,other crystals seem to possess faces contradicting all the principles of crystallography2 but careful examination shows that the supposed faces are not true faces1 but surfaces produced by the orderly junction of an immense number of distinct thin crystalline plates1 each plate being in fact a separate crystal1 in which the laws of crystallography are strictly observed4 ,the roughness of the supposed face1 the stri detected by the microscope1 or inference by continuity from other specimens where the true faces of the plates are clearly seen1 prove the mistaken character of the supposed exceptions4 ,again1 four of the faces of a regular octahedron may become so enlarged in the crystallisation of iron pyrites and some other substances1 that the other four faces become imperceptible and a regular tetrahedron appears to be produced1 contrary to the laws of crystallographic symmetry4 ,many other crystalline forms are similarly modified1 so as to produce a series of what are called 99hemihedral99 forms4 ,in tracing out the isomorphic relations of the elements1 great perplexity has often been caused by mistaking one substance for another4 ,it was pointed out that though arsenic was supposed to be isomorphous with phosphorus1 the arseniate of soda crystallised in a form distinct from that of the corresponding phosphate4 ,some chemists held this to be a fatal objection to the doctrine of isomorphism2 but it was afterwards pointed out by ,clarke1 that the arseniate and phosphate in question were not corresponding compounds1 as they differed in regard to the water of crystallisation4#eda ,vanadium again appeared to be an exception to the laws of isomorphism1 until it was proved by ,professor ,roscoe1 that what ,berzelius supposed to be metallic vanadium was really an oxide of vanadium4#edb #eda ,daubeny's 99,atomic ,theory991 p4 #gf4 #edb 99,bakerian ,lecture1 ,philosophical ,transactions99 7#ahfh71 vol4 clviii4 p4 #b4 99,apparent but ,congruent ,exceptions499 ,not unfrequently a law of nature will present results in certain circumstances which appear to be entirely in conflict with the law itself4 ,not only may the action of the law be much complicated and disguised1 but it may in various ways be reversed or inverted1 so that careless observers are misled4 ,ancient philosophers generally believed that while some bodies were heavy by nature1 others1 such as flame1 smoke1 bubbles1 clouds1 c41 were essentially light1 or possessed a tendency to move upwards4 ,so acute an inquirer as ,aristotle failed to perceive the true nature of buoyancy1 and the doctrine of intrinsic lightness1 expounded in his works1 became the accepted view for many centuries4 ,it is true that ,lucretius was aware why flame tends to rise1 holding that-- 8,the flame has weight1 though highly rare1 ,nor mounts but when compelled by heavier air40 ,archimedes also was so perfectly acquainted with the buoyancy of bodies immersed in water1 that he could not fail to perceive the existence of a parallel effect in air4 ,yet throughout the early middle ages the light of true science could not contend with the glare of the ,peripatetic doctrine4 ,the genius of ,galileo and ,newton was required to convince people of the simple truth that all matter is heavy1 but that the gravity of one substance may be overborne by that of another1 as one scale of a balance is carried up by the preponderating weight in the opposite scale4 ,it is curious to find ,newton gravely explaining the difference of absolute and relative gravity1 as if it were a new discovery proceeding from his theory4#edc ,more than a century elapsed before other apparent exceptions to the ,newtonian philosophy were explained away4 #edc 99,principia991 bk4 ii4 ,prop4 #bj4 ,corollaries1 #e and #f4 ,newton himself allowed that the motion of the apsides of the moon's orbit appeared to be irreconcilable with the law of gravity1 and it remained for ,clairaut to remove the difficulty by more complete mathematical analysis4 ,there must always remain1 in the motions of the heavenly bodies1 discrepancies of some amount between theory and observation2 but such discrepancies have so often yielded in past times to prolonged investigation that physicists now regard them as merely apparent exceptions1 which will afterwards be found to agree with the law of gravity4 ,the most beautiful instance of an apparent exception1 is found in the total reflection of light1 which occurs when a beam of light within a medium falls very obliquely upon the boundary separating it from a rarer medium4 ,the general law is that when a ray strikes the limit between two media of different refractive indices1 part of the light is reflected and part is refracted2 but when the obliquity of the ray within the denser medium passes beyond a certain point1 there is a sudden apparent breach of continuity1 and the whole of the light is reflected4 ,a clear reason can be given for this exceptional conduct of the light4 ,according to the law of refraction1 the sine of the angle of incidence bears a fixed ratio to the sine of the angle of refraction1 so that the greater of the two angles1 which is always that in the less dense medium1 may increase up to a right angle2 but when the media differ in refractive power1 the less angle cannot become a right angle1 as this would require the sine of an angle to be greater than the radius4 ,it might seem that this is an exception of the kind described below as a limiting exception1 by which a law is shown to be inapplicable beyond certain limits2 but in the explanation of the exception according to the undulatory theory1 we find that there is really no breach of the general law4 ,when an undulation strikes a point in a bounding surface1 spherical waves are produced and spread from the point4 ,the refracted ray is the resultant of an infinite number of such spherical waves1 and the bending of the ray at the common surface of two media depends upon the comparative velocities of propagation of the undulations in those media4 ,but if a ray falls very obliquely upon the surface of a rarer medium1 the waves proceeding from successive points of the surface spread so rapidly as never to intersect1 and no resultant wave will then be produced4 ,we thus perceive that from similar mathematical conditions arise distinct apparent effects4 ,there occur from time to time failures in our best grounded predictions4 ,a comet1 of which the orbit has been well determined1 may fail1 like ,lexell's ,comet1 to appear at the appointed time and place in the heavens4 ,in the present day we should not allow such an exception to our successful predictions to weigh against our belief in the theory of gravitation1 but should assume that some unknown body had through the action of gravitation deflected the comet4 ,as ,clairaut remarked1 in publishing his calculations concerning the expected reappearance of ,halley's ,comet1 a body which passes into regions so remote1 and which is hidden from our view during such long periods1 might be exposed to the influence of forces totally unknown to us1 such as the attraction of other comets1 or of planets too far removed from the sun to be ever perceived by us4 ,in the case of ,lexell's ,comet it was afterwards shown1 curiously enough1 that its appearance was not one of a regular series of periodical returns within the sphere of our vision1 but a single exceptional visit never to be repeated1 and probably due to the perturbing powers of ,jupiter4 ,this solitary visit became a strong confirmation of the law of gravity with which it seemed to be in conflict4 99,singular ,exceptions499 ,among the most interesting of apparent exceptions are those which ,i call 99singular exceptions991 because they are more or less analogous to the singular cases or solutions which occur in mathematical science4 ,a general mathematical law embraces an infinite multitude of cases which perfectly agree with each other in a certain respect4 ,it may nevertheless happen that a single case1 while really obeying the general law1 stands out as apparently different from all the rest4 ,the rotation of the earth upon its axis gives to all the stars an apparent motion of rotation from east to west2 but while countless thousands obey the rule1 the ,pole ,star alone seems to break it4 ,exact observations indeed show that it also revolves in a small circle1 but a star might happen for a short time to exist so close to the pole that no appreciable change of place would be caused by the earth's rotation4 ,it would then constitute a perfect singular exception2 while really obeying the law1 it would break the terms in which it is usually stated4 ,in the same way the poles of every revolving body are singular points4 ,whenever the laws of nature are reduced to a mathematical form we may expect to meet with singular cases1 and1 as all the physical sciences will meet in the mathematical principles of mechanics1 there is no part of nature where we may not encounter them4 ,in mechanical science the motion of rotation may be considered an exception to the motion of translation4 ,it is a general law that any number of parallel forces1 whether acting in the same or opposite directions1 will have a resultant which may be substituted for them with like effect4 ,this resultant will be equal to the algebraic sum of the forces1 or the difference of those acting in one direction and the other2 it will pass through a point which is determined by a simple formula1 and which may be described as the mean point of all the points of application of the parallel forces 7p4 #cfd74 ,thus we readily determine the resultant of parallel forces except in one peculiar case1 namely1 when two forces are equal and opposite but not in the same straight line4 ,being equal and opposite the amount of the resultant is nothing1 yet1 as the forces are not in the same straight line1 they do not balance each other4 ,examining the formula for the point of application of the resultant1 we find that it gives an infinitely great magnitude1 so that the resultant is nothing at all1 and acts at an infinite distance1 which is practically the same as to say that there is no resultant4 ,two such forces constitute what is known in mechanical science as a 99couple991 which occasions rotatory instead of rectilinear motion1 and can only be neutralised by an equal and opposite couple of forces4 ,the best instances of singular exceptions are furnished by the science of optics4 ,it is a general law that in passing through transparent media the plane of vibration of polarised light remains unchanged4 ,but in certain liquids1 some peculiar crystals of quartz1 and transparent solid media subjected to a magnetic strain1 as in ,faraday's experiment 7pp4 #ehh1 #fcj71 the plane of polarisation is rotated in a screw-like manner4 ,this effect is so entirely 99sui generis991 so unlike any other phenomena in nature1 as to appear truly exceptional2 yet mathematical analysis shows it to be only a single case of much more general laws4 ,as stated by ,thomson and ,tait1#edd it arises from the composition of two uniform circular motions4 ,if while a point is moving round a circle1 the centre of that circle move upon another circle1 a great variety of curious curves will be produced according as we vary the dimensions of the circles1 the rapidity or the direction of the motions4 ,when the two circles are exactly equal1 the rapidities nearly so1 and the directions opposite1 the point will be found to move gradually round the centre of the stationary circle1 and describe a curious star-like figure connected with the molecular motions out of which the rotational power of the media rises4 ,among other singular exceptions in optics may be placed the conical refraction of light1 already noticed 7p4 #edj71 arising from the peculiar form assumed by a wave of light when passing through certain double-refracting crystals4 ,the laws obeyed by the wave are exactly the same as in other cases1 yet the results are entirely 99sui generis994 ,so far are such cases from contradicting the law of ordinary cases1 that they afford the best opportunities for verification4 #edd 99,treatise on ,natural ,philosophy991 vol4 i4 p4 #ej4 ,in astronomy singular exceptions might occur1 and in an approximate manner they do occur4 ,we may point to the rings of ,saturn as objects which1 though undoubtedly obeying the law of gravity1 are yet unique1 as far as our observation of the universe has gone4 ,they agree1 indeed1 with the other bodies of the planetary system in the stability of their movements1 which never diverge far from the mean position4 ,there seems to be little doubt that these rings are composed of swarms of small meteoric stones2 formerly they were thought to be solid continuous rings1 and mathematicians proved that if so constituted an entirely exceptional event might have happened under certain circumstances4 ,had the rings been exactly uniform all round1 and with a centre of gravity coinciding for a moment with that of ,saturn1 a singular case of unstable equilibrium would have arisen1 necessarily resulting in the sudden collapse of the rings1 and the fall of their debris upon the surface of the planet4 ,thus in one single case the theory of gravity would give a result wholly unlike anything else known in the mechanism of the heavens4 ,it is possible that we might meet with singular exceptions in crystallography4 ,if a crystal of the second or dimetric system1 in which the third axis is usually unequal to either of the other two1 happened to have the three axes equal1 it might be mistaken for a crystal of the cubic system1 but would exhibit different faces and dissimilar properties4 ,there is1 again1 a possible class of diclinic crystals in which two axes are at right angles and the third axis inclined to the other two4 ,this class is chiefly remarkable for its non-existence1 since no crystals have yet been proved to have such axes4 ,it seems likely that the class would constitute only a singular case of the more general triclinic system1 in which all three axes are inclined to each other at various angles4 ,now if the diclinic form were merely accidental1 and not produced by any general law of molecular constitution1 its actual occurrence would be infinitely improbable1 just as it is infinitely improbable that any star should indicate the ,north ,pole with perfect exactness4 ,in the curves denoting the relation between the temperature and pressure of water there is1 as shown by ,professor ,j4 ,thomson1 one very remarkable point entirely unique1 at which alone water can remain in the three conditions of gas1 liquid1 and solid in the same vessel4 ,it is the triple point at which three lines meet1 namely 7#a7 the steam line1 which shows at what temperatures and pressures water is just upon the point of becoming gaseous2 7#b7 the ice line1 showing when ice is just about to melt2 and 7#c7 the hoar-frost line1 which similarly indicates the pressures and temperatures at which ice is capable of passing directly into the state of gaseous vapour4#ede #ede ,maxwell's 99,theory of ,heat991 7#ahga71 p4 #age4 99,divergent ,exceptions499 ,closely analogous to singular exceptions are those divergent exceptions1 in which a phenomenon manifests itself in unusual magnitude or character1 without becoming subject to peculiar laws4 ,thus in throwing ten coins1 it happened in four cases out of #b1jdh throws1 that all the coins fell with heads uppermost 7p4 #bjh72 these would usually be regarded as very singular events1 and1 according to the theory of probabilities1 they would be rare2 yet they proceed only from an unusual conjunction of accidental events1 and from no really exceptional causes4 ,in all classes of natural phenomena we may expect to meet with similar divergencies from the average1 sometimes due merely to the principles of probability1 sometimes to deeper reasons4 ,among every large collection of persons1 we shall probably find some persons who are remarkably large or remarkably small1 giants or dwarfs1 whether in bodily or mental conformation4 ,such cases appear to be not mere 99lusus natur991 since they occur with a frequency closely accordant with the law of error or divergence from an average1 as shown by ,quetelet and ,mr4 ,galton4#edf ,the rise of genius1 and the occurrence of extraordinary musical or mathematical faculties1 are attributed by ,mr4 ,galton to the same principle of divergence4 #edf ,galton1 on the ,height and ,weight of ,boys4 99,journal of the ,anthropological ,institute991 #ahge1 p4 #agd4 ,when several distinct forces happen to concur together1 we may have surprising or alarming results4 ,great storms1 floods1 droughts1 and other extreme deviations from the average condition of the atmosphere thus arise4 ,they must be expected to happen from time to time1 and will yet be very infrequent compared with minor disturbances4 ,they are not anomalous but only extreme events1 analogous to extreme runs of luck4 ,there seems1 indeed1 to be a fallacious impression in the minds of many persons1 that the theory of probabilities necessitates uniformity in the happening of events1 so that in the same space of time there will always be nearly the same number of railway accidents and murders4 ,buckle has superficially remarked upon the constancy of such events as ascertained by ,quetelet1 and some of his readers acquire the false notion that there is a mysterious inexorable law producing uniformity in human affairs4 ,but nothing can be more opposed to the teachings of the theory of probability1 which always contemplates the occurrence of unusual runs of luck4 ,that theory shows the great improbability that the number of railway accidents per month should be always equal1 or nearly so4 ,the public attention is strongly attracted to any unusual conjunction of events1 and there is a fallacious tendency to suppose that such conjunction must be due to a peculiar new cause coming into operation4 ,unless it can be clearly shown that such unusual conjunctions occur more frequently than they should do according to the theory of probabilities1 we should regard them as merely divergent exceptions4 ,eclipses and remarkable conjunctions of the heavenly bodies may also be regarded as results of ordinary laws which nevertheless appear to break the regular course of nature1 and never fail to excite surprise4 ,such events vary greatly in frequency4 ,one or other of the satellites of ,jupiter is eclipsed almost every day1 but the simultaneous eclipse of three satellites can only take place1 according to the calculations of ,wargentin1 after the lapse of #a1cag1ijj years4 ,the relations of the four satellites are so remarkable1 that it is actually impossible1 according to the theory of gravity1 that they should all suffer eclipse simultaneously4 ,but it may happen that while some of the satellites are really eclipsed by entering ,jupiter's shadow1 the others are either occulted or rendered invisible by passing over his disk4 ,thus on four occasions1 in #afha1 #ahjb1 #ahbf1 and #ahdc1 ,jupiter has been witnessed in the singular condition of being apparently deprived of satellites4 ,a close conjunction of two planets always excites admiration1 though such conjunctions must occur at intervals in the ordinary course of their motions4 ,we cannot wonder that when three or four planets approach each other closely1 the event is long remembered4 ,a most remarkable conjunction of ,mars1 ,jupiter1 ,saturn1 and ,mercury1 which took place in the year #bddf ,b4,c41 was adopted by the ,chinese ,emperor1 ,chuen ,hio1 as a new epoch for the chronology of his ,empire1 though there is some doubt whether the conjunction was really observed1 or was calculated from the supposed laws of motion of the planets4 ,it is certain that on the #aath ,november1 #aebd1 the planets ,venus1 ,jupiter1 ,mars1 and ,saturn were seen very close together1 while ,mercury was only distant by about #af or thirty apparent diameters of the sun1 this conjunction being probably the most remarkable which has occurred in historical times4 ,among the perturbations of the planets we find divergent exceptions arising from the peculiar accumulation of effects1 as in the case of the long inequality of ,jupiter and ,saturn 7p4 #dee74 ,leverrier has shown that there is one place between the orbits of ,mercury and ,venus1 and another between those of ,mars and ,jupiter1 in either of which1 if a small planet happened to exist1 it would suffer comparatively immense disturbance in the elements of its orbit4 ,now between ,mars and ,jupiter there do occur the minor planets1 the orbits of which are in many cases exceptionally divergent4#edg #edg ,grant's 99,history of ,physical ,astronomy991 p4 #aaf4 ,under divergent exceptions we might place all or nearly all the instances of substances possessing physical properties in a very high or low degree1 which were described in the chapter on ,generalisation 7p4 #fjg74 ,quicksilver is divergent among metals as regards its melting point1 and potassium and sodium as regards their specific gravities4 ,monstrous productions and variations1 whether in the animal or vegetable kingdoms1 should probably be assigned to this class of exceptions4 ,it is worthy of notice that even in such a subject as formal logic1 divergent exceptions seem to occur1 not of course due to chance1 but exhibiting in an unusual degree a phenomenon which is more or less manifested in all other cases4 ,i pointed out in p4 #ada that propositions of the general type ,a ,,bc 99bc99 are capable of expression in six equivalent logical forms1 so that they manifest in a higher degree than any other proposition yet discovered the phenomenon of logical equivalence4 99,accidental ,exceptions499 ,the third and largest class of exceptions contains those which arise from the casual interference of extraneous causes4 ,a law may be in operation1 and1 if so1 must be perfectly fulfilled2 but1 while we conceive that we are examining its results1 we may have before us the effects of a different cause1 possessing no connexion with the subject of our inquiry4 ,the law is not really broken1 but at the same time the supposed exception is not illusory4 ,it may be a phenomenon which cannot occur but under the condition of the law in question1 yet there has been such interference that there is an apparent failure of science4 ,there is1 for instance1 no subject in which more rigorous and invariable laws have been established than in crystallography4 ,as a general rule1 each chemical substance possesses its own definite form1 by which it can be infallibly recognised2 but the mineralogist has to be on his guard against what are called 99pseudomorphic99 crystals4 ,in some circumstances a substance1 having assumed its proper crystalline form1 may afterwards undergo chemical change2 a new ingredient may be added1 a former one removed1 or one element may be substituted for another4 ,in calcium carbonate the carbonic acid is sometimes replaced by sulphuric acid1 so that we find gypsum in the form of calcite2 other cases are known where the change is inverted and calcite is found in the form of gypsum4 ,mica1 talc1 steatite1 hematite1 are other minerals subject to these curious transmutations4 ,sometimes a crystal embedded in a matrix is entirely dissolved away1 and a new mineral is subsequently deposited in the cavity as in a mould4 ,quartz is thus found cast in many forms wholly unnatural to it4 ,a still more perplexing case sometimes occurs4 ,calcium carbonate is capable of assuming two distinct forms of crystallisation1 in which it bears respectively the names of calcite and arragonite4 ,now arragonite1 while retaining its outward form unchanged1 may undergo an internal molecular change into calcite1 as indicated by the altered cleavage4 ,thus we may come across crystals apparently of arragonite1 which seem to break all the laws of crystallography1 by possessing the cleavage of a different system of crystallisation4 ,some of the most invariable laws of nature are disguised by interference of unlooked-for causes4 ,while the barometer was yet a new and curious subject of investigation1 its theory1 as stated by ,torricelli and ,pascal1 seemed to be contradicted by the fact that in a well-constructed instrument the mercury would often stand far above #ca inches in height4 ,boyle showed#edh that mercury could be made to stand as high as #ge inches in a perfectly cleansed tube1 or about two and a half times as high as could be due to the pressure of the atmosphere4 ,many theories about the pressure of imaginary fluids were in consequence put forth1#edi and the subject was involved in much confusion until the adhesive or cohesive force between glass and mercury1 when brought into perfect contact1 was pointed out as the real interfering cause4 ,it seems to me1 however1 that the phenomenon is not thoroughly understood as yet4 #edh 99,discourse to the ,royal ,society991 #bhth ,may1 #afhd4 #edi ,robert ,hooke's 99,posthumous ,works991 p4 #cfe4 ,gay-,lussac observed that the temperature of boiling water was very different in some kinds of vessels from what it was in others4 ,it is only when in contact with metallic surfaces or sharply broken edges that the temperature is fixed at #ajj ,c4 ,the suspended freezing of liquids is another case where the action of a law of nature appears to be interrupted4 ,spheroidal ebullition was at first sight a most anomalous phenomenon2 it was almost incredible that water should not boil in a red-hot vessel1 or that ice could actually be produced in a red-hot crucible4 ,these paradoxical results are now fully explained as due to the interposition of a non-conducting film of vapour between the globule of liquid and the sides of the vessel4 ,the feats of conjurors who handle liquid metals are accounted for in the same manner4 ,at one time the 99passive state99 of steel was regarded as entirely anomalous4 ,it may be assumed as a general law that when pieces of electro-negative and electro-positive metal are placed in nitric acid1 and made to touch each other1 the electro-negative metal will undergo rapid solution4 ,but when iron is the electro-negative and platinum the electro-positive1 the solution of the iron entirely and abruptly ceases4 ,faraday ingeniously proved that this effect is due to a thin film of oxide of iron1 which forms upon the surface of the iron and protects it4#eej #eej 99,experimental ,researches in ,electricity991 vol4 ii4 pp4 #bdj-bde4 ,the law of gravity is so simple1 and disconnected from the other laws of nature1 that it never suffers any disturbance1 and is in no way disguised1 but by the complication of its own effects4 ,it is otherwise with those secondary laws of the planetary system which have only an empirical basis4 ,the fact that all the long known planets and satellites have a similar motion from west to east is not necessitated by any principles of mechanics1 but points to some common condition existing in the nebulous mass from which our system has been evolved4 ,the retrograde motions of the satellites of ,uranus constituted a distinct breach in this law of uniform direction1 which became all the more interesting when the single satellite of ,neptune was also found to be retrograde4 ,it now became probable1 as ,baden ,powell well observed1 that the anomaly would cease to be singular1 and become a case of another law1 pointing to some general interference which has taken place on the bounds of the planetary system4 ,not only have the satellites suffered from this perturbance1 but ,uranus is also anomalous in having an axis of rotation lying nearly in the ecliptic2 and ,neptune constitutes a partial exception to the empirical law of ,bode concerning the distances of the planets1 which circumstance may possibly be due to the same disturbance4 ,geology is a science in which accidental exceptions are likely to occur4 ,only when we find strata in their original relative positions can we surely infer that the order of succession is the order of time4 ,but it not uncommonly happens that strata are inverted by the bending and doubling action of extreme pressure4 ,landslips may carry one body of rock into proximity with an unrelated series1 and produce results apparently inexplicable4#eea ,floods1 streams1 icebergs1 and other casual agents1 may lodge remains in places where they would be wholly unexpected4 ,though such interfering causes have been sometimes wrongly supposed to explain important discoveries1 the geologist must bear the possibility of interference in mind4 ,scarcely more than a century ago it was held that fossils were accidental productions of nature1 mere forms into which minerals had been shaped by no peculiar cause4 ,voltaire appears not to have accepted such an explanation2 but fearing that the occurrence of fossil fishes on the ,alps would support the ,mosaic account of the deluge1 he did not hesitate to attribute them to the remains of fishes accidentally brought there by pilgrims4 ,in archological investigations the greatest caution is requisite in allowing for secondary burials in ancient tombs and tumuli1 for imitations1 forgeries1 casual coincidences1 disturbance by subsequent races or by other archologists4 ,in common life extraordinary events will happen from time to time1 as when a shepherdess in ,france was astonished at an iron chain falling out of the sky close to her1 the fact being that ,gay-,lussac had thrown it out of his balloon1 which was passing over her head at the time4 #eea ,murchison's 99,silurian ,system991 vol4 ii4 p4 #gcc1 c4 99,novel and ,unexplained ,exceptions499 ,when a law of nature appears to fail because some other law has interfered with its action1 two cases may present themselves2--the interfering law may be a known one1 or it may have been previously undetected4 ,in the first case1 which we have sufficiently considered in the preceding section1 we have nothing to do but calculate as exactly as possible the amount of interference1 and make allowance for it2 the apparent failure of the law under examination should then disappear4 ,but in the second case the results may be much more important4 ,a phenomenon which cannot be explained by any known laws may indicate the interference of undiscovered natural forces4 ,the ancients could not help perceiving that the general tendency of bodies downwards failed in the case of the loadstone1 nor would the doctrine of essential lightness explain the exception1 since the substance drawn upwards by the loadstone is a heavy metal4 ,we now see that there was no breach in the perfect generality of the law of gravity1 but that a new form of energy manifested itself in the loadstone for the first time4 ,other sciences show us that laws of nature1 rigorously true and exact1 may be developed by those who are ignorant of more complex phenomena involved in their application4 ,newton's comprehension of geometrical optics was sufficient to explain all the ordinary refractions and reflections of light4 ,the simple laws of the bending of rays apply to all rays1 whatever the character of the undulations composing them4 ,newton suspected the existence of other classes of phenomena when he spoke of rays as 99having sides992 but it remained for later experimentalists to show that light is a transverse undulation1 like the bending of a rod or cord4 ,dalton's atomic theory is doubtless true of all chemical compounds1 and the essence of it is that the same compound will always be found to contain the same elements in the same definite proportions4 ,pure calcium carbonate contains #dh parts by weight of oxygen to #dj of calcium and #ab of carbon4 ,but when careful analyses were made of a great many minerals1 this law appeared to fail4 ,what was unquestionably the same mineral1 judging by its crystalline form and physical properties1 would give varying proportions of its components1 and would sometimes contain unusual elements which yet could not be set down as mere impurities4 ,dolomite1 for instance1 is a compound of the carbonates of magnesia and lime1 but specimens from different places do not exhibit any fixed ratio between the lime and magnesia4 ,such facts could be reconciled with the laws of ,dalton only by supposing the interference of a new law1 that of ,isomorphism4 ,it is now established that certain elements are related to each other1 so that they can1 as it were1 step into each other's places without apparently altering the shapes of the crystals which they constitute4 ,the carbonates of iron1 calcium1 and magnesium1 are nearly identical in their crystalline forms1 hence they may crystallise together in harmony1 producing mixed minerals of considerable complexity1 which nevertheless perfectly verify the laws of equivalent proportions4 ,this principle of isomorphism once established1 not only explains what was formerly a stumbling-block1 but gives valuable aid to chemists in deciding upon the constitution of new salts1 since compounds of isomorphous elements which have identical crystalline forms must possess corresponding chemical formul4 ,we may expect that from time to time extraordinary phenomena will be discovered1 and will lead to new views of nature4 ,the recent observation1 for instance1 that the resistance of a bar of selenium to a current of electricity is affected in an extraordinary degree by rays of light falling upon the selenium1 points to a new relation between light and electricity4 ,the allotropic changes which sulphur1 selenium1 and phosphorus undergo by an alteration in the amount of latent heat which they contain1 will probably lead at some future time to important inferences concerning the molecular constitution of solids and liquids4 ,the curious substance ozone has perplexed many chemists1 and ,andrews and ,tait thought that it afforded evidence of the decomposition of oxygen by the electric discharge4 ,the researches of ,sir ,b4 ,c4 ,brodie negative this notion1 and afford evidence of the real constitution of the substance1#eeb which still1 however1 remains exceptional in its properties and relations1 and affords a hope of important discoveries in chemical theory4 #eeb 99,philosophical ,transactions99 7#ahgb71 vol4 clxii4 ,no4 #bc4 99,limiting ,exceptions499 ,we pass to cases where exceptional phenomena are actually irreconcilable with a law of nature previously regarded as true4 ,error must now be allowed to have been committed1 but the error may be more or less extensive4 ,it may happen that a law holding rigorously true of the facts actually under notice had been extended by generalisation to other series of facts then unexamined4 ,subsequent investigation may show the falsity of this generalisation1 and the result must be to limit the law for the future to those objects of which it is really true4 ,the contradiction to our previous opinions is partial and not total4 ,newton laid down as a result of experiment that every ray of homogeneous light has a definite refrangibility1 which it preserves throughout its course until extinguished4 ,this is one case of the general principle of undulatory movement1 which ,herschel stated under the title 8,principle of ,forced ,vibrations0 7p4 #dea71 and asserted to be absolutely without exception4 ,but ,herschel himself described in the 99,philosophical ,transactions99 for #ahde a curious appearance in a solution of quinine2 as viewed by transmitted light the solution appeared colourless1 but in certain aspects it exhibited a beautiful celestial blue tint4 ,curiously enough the colour is seen only in the first portion of liquid which the light enters4 ,similar phenomena in fluor-spar had been described by ,brewster in #ahch4 ,professor ,stokes1 having minutely investigated the phenomena1 discovered that they were more or less present in almost all vegetable infusions1 and in a number of mineral substances4 ,he came to the conclusion that this phenomenon1 called by him ,fluorescence1 could only be explained by an alteration in the refrangibility of the rays of light2 he asserts that light-rays of very short length of vibration in falling upon certain atoms excite undulations of greater length1 in opposition to the principle of forced vibrations4 ,no complete explanation of the mode of change is yet possible1 because it depends upon the intimate constitution of the atoms of the substances concerned2 but ,professor ,stokes believes that the principle of forced vibrations is true only so long as the excursions of an atom are very small compared with the magnitude of the complex molecules4#eec #eec 99,philosophical ,transactions99 7#aheb71 vol4 cxlii4 pp4 #dfe1 #edh1 c4 ,it is well known that in ,calorescence the refrangibility of rays is increased and the wave-length diminished4 ,rays of obscure heat and low refrangibility may be concentrated so as to heat a solid substance1 and make it give out rays belonging to any part of the spectrum1 and it seems probable that this effect arises from the impact of distinct but conflicting atoms4 ,nor is it in light only that we discover limiting exceptions to the law of forced vibrations2 for if we notice gentle waves lapping upon the stones at the edge of a lake we shall see that each larger wave in breaking upon a stone gives rise to a series of smaller waves4 ,thus there is constantly in progress a degradation in the magnitude of water-waves4 ,the principle of forced vibrations seems then to be too generally stated by ,herschel1 but it must be a difficult question of mechanical theory to discriminate the circumstances in which it does and does not hold true4 ,we sometimes foresee the possible existence of exceptions yet unknown by experience1 and limit the statement of our discoveries accordingly4 ,extensive inquiries have shown that all substances yet examined fall into one of two classes2 they are all either ferro-magnetic1 that is1 magnetic in the same way as iron1 or they are diamagnetic like bismuth4 ,but it does not follow that every substance must be ferro-magnetic or diamagnetic4 ,the magnetic properties are shown by ,sir ,w4 ,thomson#eed to depend upon the specific inductive capacities of the substance in three rectangular directions4 ,if these inductive capacities are all positive1 we have a ferro-magnetic substance2 if negative1 a diamagnetic substance2 but if the specific inductive capacity were positive in one direction and negative in the others1 we should have an exception to previous experience1 and could not place the substance under either of the present recognised classes4 #eed 99,philosophical ,magazine991 #dth ,series1 vol4 i4 p4 #ahb4 ,so many gases have been reduced to the liquid state1 and so many solids fused1 that scientific men rather hastily adopted the generalisation that all substances could exist in all three states4 ,a certain number of gases1 such as oxygen1 hydrogen1 and nitrogen1 have resisted all efforts to liquefy them1 and it now seems probable from the experiments of ,dr4 ,andrews that they are limiting exceptions4 ,he finds that above #ca ,c4 carbonic acid cannot be liquefied by any pressure he could apply1 whereas below this temperature liquefaction is always possible4 ,by analogy it becomes probable that even hydrogen might be liquefied if cooled to a very low temperature4 ,we must modify our previous views1 and either assert that 99below a certain critical temperature99 every gas may be liquefied1 or else we must assume that a highly condensed gas is1 when above the critical temperature1 undistinguishable from a liquid4 ,at the same time we have an explanation of a remarkable exception presented by liquid carbonic acid to the general rule that gases expand more by heat than liquids4 ,liquid carbonic acid was found by ,thilorier in #ahce to expand more than four times as much as air2 but by the light of ,andrews' experiments we learn to regard the liquid as rather a highly condensed gas than an ordinary liquid1 and it is actually possible to reduce the gas to the apparently liquid condition without any abrupt condensation4#eee #eee ,maxwell1 99,theory of ,heat991 p4 #abc4 ,limiting exceptions occur most frequently in the natural sciences of ,botany1 ,zoology1 ,geology1 c41 the laws of which are empirical4 ,in innumerable instances the confident belief of one generation has been falsified by the wider observation of a succeeding one4 ,aristotle confidently held that all swans are white1#eef and the proposition seemed true until not a hundred years ago black swans were discovered in ,western ,australia4 ,in zoology and physiology we may expect a fundamental identity to exist in the vital processes1 but continual discoveries show that there is no limit to the apparently anomalous expedients by which life is reproduced4 ,alternate generation1 fertilisation for several successive generations1 hermaphroditism1 are opposed to all we should expect from induction founded upon the higher animals4 ,but such phenomena are only limiting exceptions showing that what is true of one class is not true of another4 ,in certain of the cephalopoda we meet the extraordinary fact that an arm of the male is cast off and lives independently until it encounters the female4 #eef 99,prior ,analytics991 ii4 #b1 #h1 and elsewhere4 99,real ,exceptions to ,supposed ,laws499 ,the exceptions which we have lastly to consider are the most important of all1 since they lead to the entire rejection of a law or theory before accepted4 ,no law of nature can fail2 there are no such things as real exceptions to real laws4 ,where contradiction exists it must be in the mind of the experimentalist4 ,either the law is imaginary or the phenomena which conflict with it2 if1 then1 by our senses we satisfy ourselves of the actual occurrence of the phenomena1 the law must be rejected as illusory4 ,the followers of ,aristotle held that nature abhors a vacuum1 and thus accounted for the rise of water in a pump4 ,when ,torricelli pointed out the visible fact that water would not rise more than #cc feet in a pump1 nor mercury more than about #cj inches in a glass tube1 they attempted to represent these facts as limiting exceptions1 saying that nature abhorred a vacuum to a certain extent and no further4 ,but the ,academicians del ,cimento completed their discomfiture by showing that if we remove the pressure of the surrounding air1 and in proportion as we remove it1 nature's feelings of abhorrence decrease and finally disappear altogether4 ,even ,aristotelian doctrines could not stand such direct contradiction4 ,lavoisier's ideas concerning the constitution of acids received complete refutation4 ,he named oxygen the 99acid generator991 because he believed that all acids were compounds of oxygen1 a generalisation based on insufficient data4 ,berthollet1 as early as #aghi1 proved by analysis that hydrogen sulphide and prussic acid1 both clearly acting the part of acids1 were devoid of oxygen2 the former might perhaps have been interpreted as a limiting exception1 but when so powerful an acid as hydrogen chloride 7muriatic acid7 was found to contain no oxygen the theory had to be relinquished4 ,berzelius' theory of the dual formation of chemical compounds met a similar fate4 ,it is obvious that all conclusive 99experimenta crucis99 constitute real exceptions to the supposed laws of the theory which is overthrown4 ,newton's corpuscular theory of light was not rejected on account of its absurdity or inconceivability1 for in these respects it is1 as we have seen1 far superior to the undulatory theory4 ,it was rejected because certain small fringes of colour did not appear in the exact place and of the exact size in which calculation showed that they ought to appear according to the theory 7pp4 #eaf-eba74 ,one single fact clearly irreconcilable with a theory involves its rejection4 ,in the greater number of cases1 what appears to be a fatal exception may be afterwards explained away as a singular or disguised result of the laws with which it seems to conflict1 or as due to the interference of extraneous causes2 but if we fail thus to reduce the fact to congruity1 it remains more powerful than any theories or any dogmas4 ,of late years not a few of the favourite doctrines of geologists have been rudely destroyed4 ,it was the general belief that human remains were to be found only in those deposits which are actually in progress at the present day1 so that the creation of man appeared to have taken place in this geological age4 ,the discovery of a single worked flint in older strata and in connexion with the remains of extinct mammals was sufficient to explode such a doctrine4 ,similarly1 the opinions of geologists have been altered by the discovery of the ,eozon in the ,laurentian rocks of ,canada2 it was previously held that no remains of life occurred in any older strata than those of the ,cambrian system4 ,as the examination of the strata of the globe becomes more complete1 our views of the origin and succession of life upon the globe must undergo many changes4 99,unclassed ,exceptions499 ,at every period of scientific progress there will exist a multitude of unexplained phenomena which we know not how to regard4 ,they are the outstanding facts upon which the labours of investigators must be exerted1--the ore from which the gold of future discovery is to be extracted4 ,it might be thought that1 as our knowledge of the laws of nature increases1 the number of such exceptions should decrease2 but1 on the contrary1 the more we know the more there is yet to explain4 ,this arises from several reasons2 in the first place1 the principal laws and forces in nature are numerous1 so that he who bears in mind the wonderfully large numbers developed in the doctrine of combinations1 will anticipate the existence of immensely numerous relations of one law to another4 ,when we are once in possession of a law1 we are potentially in possession of all its consequences2 but it does not follow that the mind of man1 so limited in its powers and capacities1 can actually work them all out in detail4 ,just as the aberration of light was discovered empirically1 though it should have been foreseen1 so there are multitudes of unexplained facts1 the connexion of which with laws of nature already known to us1 we should perceive1 were we not hindered by the imperfection of our deductive powers4 ,but1 in the second place1 as will be more fully pointed out1 it is not to be supposed that we have approximated to an exhaustive knowledge of nature's powers4 ,the most familiar facts may teem with indications of forces1 now secrets hidden from us1 because we have not mind-directed eyes to discriminate them4 ,the progress of science will consist in the discovery from time to time of new exceptional phenomena1 and their assignment by degrees to one or other of the heads already described4 ,when a new fact proves to be merely a false1 apparent1 singular1 divergent1 or accidental exception1 we gain a more minute and accurate acquaintance with the effects of laws already known to exist4 ,we have indeed no addition to what was implicitly in our possession1 but there is much difference between knowing the laws of nature and perceiving all their complicated effects4 ,should a new fact prove to be a limiting or real exception1 we have to alter1 in part or in whole1 our views of nature1 and are saved from errors into which we had fallen4 ,lastly1 the new fact may come under the sixth class1 and may eventually prove to be a novel phenomenon1 indicating the existence of new laws and forces1 complicating but not otherwise interfering with the effects of laws and forces previously known4 ,the best instance which ,i can find of an unresolved exceptional phenomenon1 consists in the anomalous vapour-densities of phosphorus1 arsenic1 mercury1 and cadmium4 ,it is one of the most important laws of chemistry1 discovered by ,gay-,lussac1 that equal volumes of gases exactly correspond to equivalent weights of the substances4 ,nevertheless phosphorus and arsenic give vapours exactly twice as dense as they should do by analogy1 and mercury and cadmium diverge in the other direction1 giving vapours half as dense as we should expect4 ,we cannot treat these anomalies as limiting exceptions1 and say that the law holds true of substances generally but not of these2 for the properties of gases 7p4 #fja71 usually admit of the widest generalisations4 ,besides1 the preciseness of the ratio of divergence points to the real observance of the law in a modified manner4 ,we might endeavour to reduce the exceptions by doubling the atomic weights of phosphorus and arsenic1 and halving those of mercury and cadmium4 ,but this step has been maturely considered by chemists1 and is found to conflict with all the other analogies of the substances and with the principle of isomorphism4 ,one of the most probable explanations is1 that phosphorus and arsenic produce vapour in an allotropic condition1 which might perhaps by intense heat be resolved into a simpler gas of half the density2 but facts are wanting to support this hypothesis1 and it cannot be applied to the other two exceptions without supposing that gases and vapours generally are capable of resolution into something simpler4 ,in short1 chemists can at present make nothing of these anomalies4 ,as ,hofmann says1 8,their philosophical interpretation belongs to the future4444 ,they may turn out to be typical facts1 round which many others of the like kind may come hereafter to be grouped2 and they may prove to be allied with special properties1 or dependent on particular conditions as yet unsuspected40#eeg #eeg ,hofmann's 99,introduction to ,chemistry991 p4 #aih4 ,it would be easy to point out a great number of other unexplained anomalies4 ,physicists assert1 as an absolutely universal law1 that in liquefaction heat is absorbed2#eeh yet sulphur is at least an apparent exception4 ,the two substances1 sulphur and selenium1 are1 in fact1 very anomalous in their relations to heat4 ,sulphur may be said to have two melting points1 for1 though liquid like water at #abj ,c41 it becomes quite thick and tenacious between #bba and #bdi1 and melts again at a higher temperature4 ,both sulphur and selenium may be thrown into several curious states1 which chemists conveniently dispose of by calling them 99allotropic991 a term freely used when they are puzzled to know what has happened4 ,the chemical and physical history of iron1 again1 is full of anomalies2 not only does it undergo inexplicable changes of hardness and texture in its alloys with carbon and other elements1 but it is almost the only substance which conveys sound with greater velocity at a higher than at a lower temperature1 the velocity increasing from #bj to #ajj ,c41 and then decreasing4 ,silver also is anomalous in regard to sound4 ,these are instances of inexplicable exceptions1 the bearing of which must be ascertained in the future progress of science4 #eeh ,stewart's 99,elementary ,treatise on ,heat991 p4 #hj4 ,when the discovery of new and peculiar phenomena conflicting with our theories of the constitution of nature is reported to us1 it becomes no easy task to steer a philosophically correct course between credulity and scepticism4 ,we are not to assume1 on the one hand1 that there is any limit to the wonders which nature can present to us4 ,nothing except the contradictory is really impossible1 and many things which we now regard as common-place were considered as little short of the miraculous when first perceived4 ,the electric telegraph was a visionary dream among medival physicists2#eei it has hardly yet ceased to excite our wonder2 to our descendants centuries hence it will probably appear inferior in ingenuity to some inventions which they will possess4 ,now every strange phenomenon may be a secret spring which1 if rightly touched1 will open the door to new chambers in the palace of nature4 ,to refuse to believe in the occurrence of anything strange would be to neglect the most precious chances of discovery4 ,we may say with ,hooke1 that 8the believing strange things possible may perhaps be an occasion of taking notice of such things as another would pass by without regard as useless40 ,we are not1 therefore1 to shut our ears even to such apparently absurd stories as those concerning second-sight1 clairvoyance1 animal magnetism1 ode force1 table-turning1 or any of the popular delusions which from time to time are current4 ,the facts recorded concerning these matters are facts in some sense or other1 and they demand explanation1 either as new natural phenomena1 or as the results of credulity and imposture4 ,most of the supposed phenomena referred to have been1 or by careful investigation would doubtless be1 referred to the latter head1 and the absence of scientific ability in many of those who describe them is sufficient to cast a doubt upon their value4 #eei ,jevons1 99,proceedings of the ,manchester ,literary and ,philosophical ,society991 #fth ,march1 #ahgg1 vol4 xvi4 p4 #afd4 ,see also ,mr4 ,w4 ,e4 ,a4 ,axon's note on the same subject1 ibid4 p4 #aff4 ,it is to be remembered that according to the principle of the inverse method of probability1 the probability of any hypothetical explanation is affected by the probability of each other possible explanation4 ,if no other reasonable explanation could be suggested1 we should be forced to look upon spiritualist manifestations as indicating mysterious causes4 ,but as soon as it is shown that fraud has been committed in several important cases1 and that in other cases persons in a credulous and excited state of mind have deceived themselves1 the probability becomes very considerable that similar explanations may apply to most like manifestations4 ,the performances of conjurors sufficiently prove that it requires no very great skill to perform tricks the 99modus operandi99 of which shall entirely escape the notice of spectators4 ,it is on these grounds of probability that we should reject the so-called spiritualist stories1 and not simply because they are strange4 ,certainly in the obscure phenomena of mind1 those relating to memory1 dreams1 somnambulism1 and other peculiar states of the nervous system1 there are many inexplicable and almost incredible facts1 and it is equally unphilosophical to believe or to disbelieve without clear evidence4 ,there are many facts1 too1 concerning the instincts of animals1 and the mode in which they find their way from place to place1 which are at present quite inexplicable4 ,no doubt there are many strange things not dreamt of in our philosophy1 but this is no reason why we should believe in every strange thing which is reported to have happened4 ,,chapter ,,xxx4 ,,classification4 ,the extensive subject of ,classification has been deferred to a late part of this treatise1 because it involves questions of difficulty1 and did not seem naturally to fall into an earlier place4 ,but it must not be supposed that1 in now formally taking up the subject1 we are for the first time entertaining the notion of classification4 ,all logical inference involves classification1 which is indeed the necessary accompaniment of the action of judgment4 ,it is impossible to detect similarity between objects without thereby joining them together in thought1 and forming an incipient class4 ,nor can we bestow a common name upon objects without implying the existence of a class4 ,every common name is the name of a class1 and every name of a class is a common name4 ,it is evident also that to speak of a general notion or concept is but another way of speaking of a class4 ,usage leads us to employ the word classification in some cases and not in others4 ,we are said to form the 99general notion99 parallelogram when we regard an infinite number of possible four-sided rectilinear figures as resembling each other in the common property of possessing parallel sides4 ,we should be said to form a 99class991 ,trilobite1 when we place together in a museum a number of specimens resembling each other in certain defined characters4 ,but the logical nature of the operation is the same in both cases4 ,we form a 99class99 of figures called parallelograms and we form a 99general notion99 of trilobites4 ,science1 it was said at the outset1 is the detection of identify1 and classification is the placing together1 either in thought or in actual proximity of space1 those objects between which identity has been detected4 ,accordingly1 the value of classification is co-extensive with the value of science and general reasoning4 ,whenever we form a class we reduce multiplicity to unity1 and detect1 as ,plato said1 the one in the many4 ,the result of such classification is to yield generalised knowledge1 as distinguished from the direct and sensuous knowledge of particular facts4 ,of every class1 so far as it is correctly formed1 the principle of substitution is true1 and whatever we know of one object in a class we know of the other objects1 so far as identity has been detected between them4 ,the facilitation and abbreviation of mental labour is at the bottom of all mental progress4 ,the reasoning faculties of ,newton were not different in nature from those of a ploughman2 the difference lay in the extent to which they were exerted1 and the number of facts which could be treated4 ,every thinking being generalises more or less1 but it is the depth and extent of his generalisations which distinguish the philosopher4 ,now it is the exertion of the classifying and generalising powers which enables the intellect of man to cope in some degree with the infinite number of natural phenomena4 ,in the chapters upon combinations and permutations it was made evident1 that from a few elementary differences immense numbers of combinations can be produced4 ,the process of classification enables us to resolve these combinations1 and refer each one to its place according to one or other of the elementary circumstances out of which it was produced4 ,we restore nature to the simple conditions out of which its endless variety was developed4 ,as ,professor ,bowen has said1#efj 8,the first necessity which is imposed upon us by the constitution of the mind itself1 is to break up the infinite wealth of ,nature into groups and classes of things1 with reference to their resemblances and affinities1 and thus to enlarge the grasp of our mental faculties1 even at the expense of sacrificing the minuteness of information which can be acquired only by studying objects in detail4 ,the first efforts in the pursuit of knowledge1 then1 must be directed to the business of classification4 ,perhaps it will be found in the sequel1 that classification is not only the beginning1 but the culmination and the end1 of human knowledge40 #efj 99,a ,treatise on ,logic1 or1 the ,laws of ,pure ,thought991 by ,francis ,bowen1 ,professor of ,moral ,philosophy in ,harvard ,college1 ,cambridge1 ,united ,states1 #ahff1 p4 #cae4 99,classification ,involving ,induction499 ,the purpose of classification is the detection of the laws of nature4 ,however much the process may in some cases be disguised1 classification is not really distinct from the process of perfect induction1 whereby we endeavour to ascertain the connexions existing between properties of the objects under treatment4 ,there can be no use in placing an object in a class unless something more than the fact of being in the class is implied4 ,if we arbitrarily formed a class of metals and placed therein a selection from the list of known metals made by ballot1 we should have no reason to expect that the metals in question would resemble each other in any points except that they are metals1 and have been selected by the ballot4 ,but when chemists select from the list the five metals1 potassium1 sodium1 csium1 rubidium1 and lithium and call them the ,alkaline metals1 a great deal is implied in this classification4 ,on comparing the qualities of these substances they are all found to combine very energetically with oxygen1 to decompose water at all temperatures1 and to form strongly basic oxides1 which are highly soluble in water1 yielding powerfully caustic and alkaline hydrates from which water cannot be expelled by heat4 ,their carbonates are also soluble in water1 and each metal forms only one chloride4 ,it may also be expected that each salt of one of the metals will correspond to a salt of each other metal1 there being a general analogy between the compounds of these metals and their properties4 ,now in forming this class of alkaline metals1 we have done more than merely select a convenient order of statement4 ,we have arrived at a discovery of certain empirical laws of nature1 the probability being very considerable that a metal which exhibits some of the properties of alkaline metals will also possess the others4 ,if we discovered another metal whose carbonate was soluble in water1 and which energetically combined with water at all temperatures1 producing a strongly basic oxide1 we should infer that it would form only a single chloride1 and that generally speaking1 it would enter into a series of compounds corresponding to the salts of the other alkaline metals4 ,the formation of this class of alkaline metals then1 is no mere matter of convenience2 it is an important and successful act of inductive discovery1 enabling us to register many undoubted propositions as results of perfect induction1 and to make a great number of inferences depending upon the principles of imperfect induction4 ,an excellent instance as to what classification can do1 is found in ,mr4 ,lockyer's researches on the sun4#efa ,wanting some guide as to what more elements to look for in the sun's photosphere1 he prepared a classification of the elements according as they had or had not been traced in the sun1 together with a detailed statement of the chief chemical characters of each element4 ,he was then able to observe that the elements found in the sun were for the most part those forming stable compounds with oxygen4 ,he then inferred that other elements forming stable oxides would probably exist in the sun1 and he was rewarded by the discovery of five such metals4 ,here we have empirical and tentative classification leading to the detection of the correlation between existence in the sun1 and the power of forming stable oxides and then leading by imperfect induction to the discovery of more coincidences between these properties4 #efa 99,proceedings of the ,royal ,society991 ,november1 #ahgc1 vol4 xxi4 p4 #eab4 ,professor ,huxley has defined the process of classification in the following terms4#efb 8,by the classification of any series of objects1 is meant the actual or ideal arrangement together of those which are like and the separation of those which are unlike2 the purpose of this arrangement being to facilitate the operations of the mind in clearly conceiving and retaining in the memory the characters of the objects in question40 #efb 99,lectures on the ,elements of ,comparative ,anatomy991 #ahfd1 p4 #a4 ,this statement is doubtless correct1 so far as it goes1 but it does not include all that ,professor ,huxley himself implicitly treats under classification4 ,he is fully aware that deep correlations1 or in other terms deep uniformities or laws of nature1 will be disclosed by any well chosen and profound system of classification4 ,i should therefore propose to modify the above statement1 as follows3--8,by the classification of any series of objects1 is meant the actual or ideal arrangement together of those which are like and the separation of those which are unlike1 the purpose of this arrangement being1 primarily1 to disclose the correlations or laws of union of properties and circumstances1 and1 secondarily1 to facilitate the operations of the mind in clearly conceiving and retaining in the memory the characters of the objects in question40 99,multiplicity of ,modes of ,classification499 ,in approaching the question how any given group of objects may be best classified1 let it be remarked that there must generally be an unlimited number of modes of classifying a group of objects4 ,misled1 as we shall see1 by the problem of classification in the natural sciences1 philosophers seem to think that in each subject there must be one essentially natural system of classification which is to be selected1 to the exclusion of all others4 ,this erroneous notion probably arises also in part from the limited powers of thought and the inconvenient mechanical conditions under which we labour4 ,if we arrange the books in a library catalogue1 we must arrange them in some one order2 if we compose a treatise on mineralogy1 the minerals must be successively described in some one arrangement2 if we treat such simple things as geometrical figures1 they must be taken in some fixed order4 ,we shall naturally select that arrangement which appears to be most convenient and instructive for our principal purpose4 ,but it does not follow that this method of arrangement possesses any exclusive excellence1 and there will be usually many other possible arrangements1 each valuable in its own way4 ,a perfect intellect would not confine itself to one order of thought1 but would simultaneously regard a group of objects as classified in all the ways of which they are capable4 ,thus the elements may be classified according to their atomicity into the groups of monads1 dyads1 triads1 tetrads1 pentads1 and hexads1 and this is probably the most instructive classification2 but it does not prevent us from also classifying them according as they are metallic or non-metallic1 solid1 liquid or gaseous at ordinary temperatures1 useful or useless1 abundant or scarce1 ferro-magnetic or diamagnetic1 and so on4 ,mineralogists have spent a great deal of labour in trying to discover the supposed natural system of classification for minerals4 ,they have constantly encountered the difficulty that the chemical composition does not run together with the crystallographic form1 and the various physical properties of the mineral4 ,substances identical in the forms of their crystals1 especially those belonging to the first or cubical system of crystals1 are often found to have no resemblance in chemical composition4 ,the same substance1 again1 is occasionally found crystallised in two essentially different crystallographic forms2 calcium carbonate1 for instance1 appearing as calc-spar and arragonite4 ,the simple truth is that if we are unable to discover any correspondence1 or1 as we may call it1 any 99correlation99 between the properties of minerals1 we cannot make any one arrangement which will enable us to treat all these properties in a single system of classification4 ,we must classify minerals in as many different ways as there are different groups of unrelated properties of sufficient importance4 ,even if1 for the purpose of describing minerals successively in a treatise1 we select one chief system1 that1 for instance1 having regard to chemical composition1 we ought mentally to regard the minerals as classified in all other useful modes4 ,exactly the same may be said of the classification of plants4 ,an immense number of different modes of classifying plants have been proposed at one time or other1 an exhaustive account of which will be found in the article on classification in ,rees's 8,cyclopdia10 or in the introduction to ,lindley's 8,vegetable ,kingdom40 ,there have been the ,fructists1 such as ,csalpinus1 ,morison1 ,hermann1 ,boerhaave or ,gaertner1 who arranged plants according to the form of the fruit4 ,the ,corollists1 ,rivinus1 ,ludwig1 and ,tournefort1 paid attention chiefly to the number and arrangement of the parts of the corolla4 ,magnol selected the calyx as the critical part1 while ,sauvage arranged plants according to their leaves2 nor are these instances more than a small selection from the actual variety of modes of classification which have been tried4 ,of such attempts it may be said that every system will probably yield some information concerning the relations of plants1 and it is only after trying many modes that it is possible to approximate to the best4 99,natural and ,artificial ,systems of ,classification499 ,it has been usual to distinguish systems of classification as natural and artificial1 those being called natural which seemed to express the order of existing things as determined by nature4 ,artificial methods of classification1 on the other hand1 included those formed for the mere convenience of men in remembering or treating natural objects4 ,the difference1 as it is commonly regarded1 has been well described by ,ampre1#efc as follows3 8,we can distinguish two kinds of classifications1 the natural and the artificial4 ,in the latter kind1 some characters1 arbitrarily chosen1 serve to determine the place of each object2 we abstract all other characters1 and the objects are thus found to be brought near to or to be separated from each other1 often in the most bizarre manner4 ,in natural systems of classification1 on the contrary1 we employ concurrently all the characters essential to the objects with which we are occupied1 discussing the importance of each of them2 and the results of this labour are not adopted unless the objects which present the closest analogy are brought most near together1 and the groups of the several orders which are formed from them are also approximated in proportion as they offer more similar characters4 ,in this way it arises that there is always a kind of connexion1 more or less marked1 between each group and the group which follows it40 #efc 99,essai sur la ,philosophie des ,sciences991 p4 #i4 ,there is much1 however1 that is vague and logically false in this and other definitions which have been proposed by naturalists to express their notion of a natural system4 ,we are not informed how the 99importance99 of a resemblance is to be determined1 nor what is the measure of the 99closeness99 of analogy4 ,until all the words employed in a definition are made clear in meaning1 the definition itself is worse than useless4 ,now if the views concerning classification here upheld are true1 there can be no sharp and precise distinction between natural and artificial systems4 ,all arrangements which serve any purpose at all must be more or less natural1 because1 if closely enough scrutinised1 they will involve more resemblances than those whereby the class was defined4 ,it is true that in the biological sciences there would be one arrangement of plants or animals which would be conspicuously instructive1 and in a certain sense natural1 if it could be attained1 and it is that after which naturalists have been in reality striving for nearly two centuries1 namely1 that 99arrangement which would display the genealogical descent of every form from the original life germ994 ,those morphological resemblances upon which the classification of living beings is almost always based are inherited resemblances1 and it is evident that descendants will usually resemble their parents and each other in a great many points4 ,i have said that a natural is distinguished from an arbitrary or artificial system only in degree4 ,it will be found almost impossible to arrange objects according to any circumstance without finding that some correlation of other circumstances is thus made apparent4 ,no arrangement could seem more arbitrary than the common alphabetical arrangement according to the initial letter of the name4 ,but we cannot scrutinise a list of names of persons without noticing a predominance of ,evans's and ,jones's1 under the letters ,e and ,j1 and of names beginning with ,mac under the letter ,m4 ,the predominance is so great that we could not attribute it to chance1 and inquiry would of course show that it arose from important facts concerning the nationality of the persons4 ,it would appear that the ,evans's and ,jones's were of ,welsh descent1 and those whose names bear the prefix ,mac of ,keltic descent4 ,with the nationality would be more or less strictly correlated many peculiarities of physical constitution1 language1 habits1 or mental character4 ,in other cases ,i have been interested in noticing the empirical inferences which are displayed in the most arbitrary arrangements4 ,if a large register of the names of ships be examined it will often be found that a number of ships bearing the same name were built about the same time1 a correlation due to the occurrence of some striking incident shortly previous to the building of the ships4 ,the age of ships or other structures is usually correlated with their general form1 nature of materials1 c41 so that ships of the same name will often resemble each other in many points4 ,it is impossible to examine the details of some of the so-called artificial systems of classification of plants1 without finding that many of the classes are natural in character4 ,thus in ,tournefort's arrangement1 depending almost entirely on the formation of the corolla1 we find the natural orders of the ,labiat1 ,crucifer1 ,rosace1 ,umbellifer1 ,liliace1 and ,papilionace1 recognised in his #dth1 #eth1 #fth1 #gth1 #ith1 and #ajth classes4 ,many of the classes in ,linnus' celebrated sexual system also approximate to natural classes4 99,correlation of ,properties499 ,habits and usages of language are apt to lead us into the error of imagining that when we employ different words we always mean different things4 ,in introducing the subject of classification nominally ,i was careful to draw the reader's attention to the fact that all reasoning and all operations of scientific method really involve classification1 though we are accustomed to use the name in some cases and not in others4 ,the name 99correlation99 requires to be used with the same qualification4 ,things are correlated 799con991 99relata997 when they are so related or bound to each other that 99where one is the other is1 and where one is not the other is not994 ,throughout this work we have then been dealing with correlations4 ,in geometry the occurrence of three equal angles in a triangle is correlated with the existence of three equal sides2 in physics gravity is correlated with inertia2 in botany exogenous growth is correlated with the possession of two cotyledons1 or the production of flowers with that of spiral vessels4 ,wherever a proposition of the form ,a ,b is true there correlation exists4 ,but it is in the classificatory sciences especially that the word correlation has been employed4 ,we find it stated that in the class ,mammalia the possession of two occipital condyles1 with a well-ossified basi-occipital1 is correlated with the possession of mandibles1 each ramus of which is composed of a single piece of bone1 articulated with the squamosal element of the skull1 and also with the possession of mamm and non-nucleated red blood-corpuscles4 ,professor ,huxley remarks#efd that this statement of the character of the class mammalia is something more than an arbitrary definition2 it is a statement of a law of correlation or co-existence of animal structures1 from which most important conclusions are deducible4 ,it involves a generalisation to the effect that in nature the structures mentioned are always found associated together4 ,this amounts to saying that the formation of the class mammalia involves an act of inductive discovery1 and results in the establishment of certain empirical laws of nature4 ,professor ,huxley has excellently expressed the mode in which discoveries of this kind enable naturalists to make deductions or predictions with considerable confidence1 but he has also pointed out that such inferences are likely from time to time to prove mistaken4 ,i will quote his own words3 #efd 99,lectures on the ,elements of ,comparative ,anatomy1 and on the ,classification of ,animals991 #ahfd1 p4 #c4 8,if a fragmentary fossil be discovered1 consisting of no more than a ramus of a mandible1 and that part of the skull with which it articulated1 a knowledge of this law may enable the palontologist to affirm1 with great confidence1 that the animal of which it formed a part suckled its young1 and had non-nucleated red blood-corpuscles2 and to predict that should the back part of that skull be discovered1 it will exhibit two occipital condyles and a well-ossified basi-occipital bone4 8,deductions of this kind1 such as that made by ,cuvier in the famous case of the fossil opossum of ,montmartre1 have often been verified1 and are well calculated to impress the vulgar imagination2 so that they have taken rank as the triumphs of the anatomist4 ,but it should carefully be borne in mind1 that1 like all merely empirical laws1 which rest upon a comparatively narrow observational basis1 the reasoning from them may at any time break down4 ,if ,cuvier1 for example1 had had to do with a fossil ,thylacinus instead of a fossil ,opossum1 he would not have found the marsupial bones1 though the inflected angle of the jaw would have been obvious enough4 ,and so1 though1 practically1 any one who met with a characteristically mammalian jaw would be justified in expecting to find the characteristically mammalian occiput associated with it2 yet1 he would be a bold man indeed1 who should strictly assert the belief which is implied in this expectation1 viz41 that at no period of the world's history did animals exist which combined a mammalian occiput with a reptilian jaw1 or 99vice vers9940 ,one of the most distinct and remarkable instances of correlation in the animal world is that which occurs in ruminating animals1 and which could not be better stated than in the following extract from the classical work of ,cuvier3#efe #efe 99,ossemens ,fossiles991 #dth edit4 vol4 i4 p4 #afd4 ,quoted by ,huxley1 99,lectures991 c41 p4 #e4 8,i doubt if any one would have divined1 if untaught by observation1 that all ruminants have the foot cleft1 and that they alone have it4 ,i doubt if any one would have divined that there are frontal horns only in this class3 that those among them which have sharp canines for the most part lack horns4 8,however1 since these relations are constant1 they must have some sufficient cause2 but since we are ignorant of it1 we must make good the defect of the theory by means of observation3 it enables us to establish empirical laws which become almost as certain as rational laws when they rest on sufficiently repeated observations2 so that now whoso sees merely the print of a cleft foot may conclude that the animal which left this impression ruminated1 and this conclusion is as certain as any other in physics or morals4 ,this footprint alone then1 yields1 to him who observes it1 the form of the teeth1 the form of the jaws1 the form of the vertebr1 the form of all the bones of the legs1 of the thighs1 of the shoulders1 and of the pelvis of the animal which has passed by3 it is a surer mark than all those of ,zadig40 ,we meet with a good instance of the purely empirical correlation of circumstances when we classify the planets according to their densities and periods of axial rotation4#eff ,if we examine a table specifying the usual astronomical elements of the solar system1 we find that four planets resemble each other very closely in the period of axial rotation1 and the same four planets are all found to have high densities1 thus3-- #eff ,chambers1 99,descriptive ,astronomy991 #ast edit4 p4 #bc4 ,name of ,period of ,axial ,planet4 ,rotation4 ,density4 ,mercury #bd hours #e minutes #gid ,venus #bc #ba #ecc ,earth #bc #ef #efg ,mars #bd #cg #ehd ,a similar table for the other larger planets1 is as follows3-- ,jupiter #i hours #ee minutes #acf ,saturn #aj #bi #gd ,uranus #i #cj #ig ,neptune -- -- #ajb ,it will be observed that in neither group is the equality of the rotational period or the density more than rudely approximate2 nevertheless the difference of the numbers in the first and second group is so very well marked1 the periods of the first being at least double and the densities four or five times those of the second1 that the coincidence cannot be attributed to accident4 ,the reader will also notice that the first group consists of the planets nearest to the sun2 that with the exception of the earth none of them possess satellites2 and that they are all comparatively small4 ,the second group are furthest from the sun1 and all of them possess several satellites1 and are comparatively great4 ,therefore1 with but slight exceptions1 the following correlations hold true3-- ,interior planets4 ,long period4 ,small size4 ,high ,density4 ,no satellites4 ,exterior ,short ,great ,low ,many ,these coincidences point with much probability to a difference in the origin of the two groups1 but no further explanation of the matter is yet possible4 ,the classification of comets according to their periods by ,mr4 ,hind and ,mr4 ,a4 ,s4 ,davies1 tends to establish the conclusion that distinct groups of comets have been brought into the solar system by the attractive powers of ,jupiter1 ,uranus1 or other planets4#efg ,the classification of nebul as commenced by the two ,herschels1 and continued by ,lord ,rosse1 ,mr4 ,huggins1 and others1 will probably lead at some future time to the discovery of important empirical laws concerning the constitution of the universe4 ,the minute examination and classification of meteorites1 as carried on by ,mr4 ,sorby and others1 seems likely to afford us an insight into the formation of the heavenly bodies4 #efg 99,philosophical ,magazine991 #dth ,series1 vol4 xxxix4 p4 #cif2 vol4 xl4 p4 #ahc2 vol4 xli4 p4 #dd4 ,see also ,proctor1 99,popular ,science ,review991 ,october #ahgd1 p4 #cej4 ,we should never fail to remember the slightest and most inexplicable correlations1 for they may prove of importance in the future4 ,discoveries begin when we are least expecting them4 ,it is a significant fact1 for instance1 that the greater number of variable stars are of a reddish colour4 ,not all variable stars are red1 nor all red stars variable2 but considering that only a small fraction of the observed stars are known to be variable1 and only a small fraction are red1 the number which fall into both classes is too great to be accidental4#efh ,it is also remarkable that the greater number of stars possessing great proper motion are double stars1 the star #fa ,cygni being especially noticeable in this respect4#efi ,the correlation in these cases is not without exception1 but the preponderance is so great as to point to some natural connexion1 the exact nature of which must be a matter for future investigation4 ,herschel remarked that the two double stars #fa ,cygni and ,centauri of which the orbits were well ascertained1 evidently belonged to the same family or genus4#egj #efh ,humboldt1 99,cosmos99 7,bohn71 vol4 iii4 p4 #bbd4 #efi ,baily1 ,british 99,association ,catalogue991 p4 #dh4 #egj 99,outlines of ,astronomy991 #hej1 #dth edit4 p4 #egh4 99,classification in ,crystallography499 ,perhaps the most perfect and instructive instance of classification which we can find is furnished by the science of crystallography 7p4 #acc74 ,the system of arrangement now generally adopted is conspicuously natural1 and is even mathematically perfect4 ,a crystal consists in every part of similar molecules similarly related to the adjoining molecules1 and connected with them by forces the nature of which we can only learn by their apparent effects4 ,but these forces are exerted in space of three dimensions1 so that there is a limited number of suppositions which can be entertained as to the relations of these forces4 ,in one case each molecule will be similarly related to all those which are next to it2 in a second case1 it will be similarly related to those in a certain plane1 but differently related to those not in that plane4 ,in the simpler cases the arrangement of molecules is rectangular2 in the remaining cases oblique either in one or two planes4 ,in order to simplify the explanation and conception of the complicated phenomena which crystals exhibit1 an hypothesis has been invented which is an excellent instance of the ,descriptive ,hypotheses before mentioned 7p4 #ebb74 ,crystallographers imagine that there are within each crystal certain axes1 or lines of direction1 by the comparative length and the mutual inclination of which the nature of the crystal is determined4 ,in one class of crystals there are three such axes lying in one plane1 and a fourth perpendicular to that plane2 but in all the other classes there are imagined to be only three axes4 ,now these axes can be varied in three ways as regards length3 they may be 7#a7 all equal1 or 7#b7 two equal and one unequal1 or 7#c7 all unequal4 ,they may also be varied in four ways as regards direction3 7#a7 they may be all at right angles to each other2 7#b7 two axes may be oblique to each other and at right angles to the third2 7#c7 two axes may be at right angles to each other and the third oblique to both2 7#d7 the three axes may be all oblique4 ,now1 if all the variations as regards length were combined with those regarding direction1 it would seem to be possible to have twelve classes of crystals in all1 the enumeration being then logically and geometrically complete4 ,but as a matter of empirical observation1 many of these classes are not found to occur1 oblique axes being seldom or never equal4 ,there remain seven recognised classes of crystals1 but even of these one class is not positively known to be represented in nature4 ,the first class of crystals is defined by possessing three equal rectangular axes1 and equal elasticity in all directions4 ,the primary or simple form of the crystals is the cube1 but by the removal of the corners of the cube by planes variously inclined to the axes1 we have the regular octohedron1 the dodecahedron1 and various combinations of these forms4 ,now it is a law of this class of crystals that as each axis is exactly like each other axis1 every modification of any corner of a crystal must be repeated symmetrically with regard to the other axes2 thus the forms produced are symmetrical or regular1 and the class is called the 99,regular ,system99 of crystals4 ,it includes a great variety of substances1 some of them being elements1 such as carbon in the form of diamond1 others more or less complex compounds1 such as rock-salt1 potassium iodide and bromide1 the several kinds of alum1 fluor-spar1 iron bisulphide1 garnet1 spinelle1 c4 ,no correlation then is apparent between the form of crystallisation and the chemical composition4 ,but what we have to notice is that the physical properties of the crystallised substances with regard to light1 heat1 electricity1 c41 are closely similar4 ,light and heat undulations1 wherever they enter a crystal of the regular system1 spread with equal rapidity in all directions1 just as they would in a uniform fluid4 ,crystals of the regular system accordingly do not in any case exhibit the phenomena of double refraction1 unless by mechanical compression we alter the conditions of elasticity4 ,these crystals1 again1 expand equally in all directions when heated1 and if we could cut a sufficiently large plate from a cubical crystal1 and examine the sound vibrations of which it is capable1 we should find that they indicated an equal elasticity in every direction4 ,thus we see that a great number of important properties are correlated with that of crystallisation in the regular system1 and as soon as we know that the primary form of a substance is the cube1 we are able to infer with approximate certainty that it possesses all these properties4 ,the class of regular crystals is then an eminently natural class1 one disclosing many general laws connecting together the physical and mechanical properties of the substances classified4 ,in the second class of crystals1 called the dimetric1 square prismatic1 or pyramidal system1 there are also three axes at right angles to each other2 two of the axes are equal1 but the third or principal axis is unequal1 being either greater or less than either of the other two4 ,in such crystals accordingly the elasticity and other properties are alike in all directions perpendicular to the principal axis1 but vary in all other directions4 ,if a point within a crystal of this system be heated1 the heat spreads with equal rapidity in planes perpendicular to the principal axis1 but more or less rapidly in the direction of this axis1 so that the isothermal surface is an ellipsoid of revolution round that axis4 ,nearly the same statement may be made concerning the third or hexagonal or rhombohedral system of crystals1 in which there are three axes lying in one plane and meeting at angles of #fj1 while the fourth axis is perpendicular to the other three4 ,the hexagonal prism and rhombohedron are the commonest forms assumed by crystals of this system1 and in ice1 quartz1 and calc-spar1 we have abundance of beautiful specimens of the various shapes produced by the modification of the primitive form4 ,calc-spar alone is said to crystallise in at least #gjj varieties of form4 ,now of all the crystals belonging both to this and the dimetric class1 we know that a ray of light passing in the direction of the principal axis will be refracted singly as in a crystal of the regular system2 but in every other direction the light will suffer double refraction being separated into two rays1 one of which obeys the ordinary law of refraction1 but the other a much more complicated law4 ,the other physical properties vary in an analogous manner4 ,thus calc-spar expands by heat in the direction of the principal axis1 but contracts a little in directions perpendicular to it4 ,so closely are the physical properties correlated that ,mitscherlich1 having observed the law of expansion in calc-spar1 was enabled to predict that the double refracting power of the substance would be decreased by a rise of temperature1 as was proved by experiment to be the case4 ,in the fourth system1 called the trimetric1 rhombic1 or right prismatic system1 there are three axes1 at right angles1 but all unequal in length4 ,it may be asserted in general terms that the mechanical properties vary in such crystals in every direction1 and heat spreads so that the isothermal surface is an ellipsoid with three unequal axes4 ,in the remaining three classes1 called the monoclinic1 diclinic1 and triclinic1 the axes are more or less oblique1 and at the same time unequal4 ,the complication of phenomena is therefore greatly increased1 and it need only be stated that there are always two directions in which a ray is singly refracted1 but that in all other directions double refraction takes place4 ,the conduction of heat is unequal in all directions1 the isothermal surface being an ellipsoid of three unequal axes4 ,the relations of such crystals to other phenomena are often very complicated1 and hardly yet reduced to law4 ,some crystals1 called pyro-electric1 manifest vitreous electricity at some points of their surface1 and resinous electricity at other points when rising in temperature1 the character of the electricity being changed when the temperature sinks again4 ,this production of electricity is believed to be connected with the hemihedral character of the crystals exhibiting it4 ,the crystalline structure of a substance again influences its magnetic behaviour1 the general law being that the direction in which the molecules of a crystal are most approximated tends to place itself axially or equatorially between the poles of a magnet1 respectively as the body is magnetic or diamagnetic4 ,further questions arise if we apply pressure to crystals4 ,thus doubly refracting crystals with one principal axis acquire two axes when the pressure is perpendicular in direction to the principal axis4 ,all the phenomena peculiar to crystalline bodies are thus closely correlated with the formation of the crystal1 or will almost certainly be found to be so as investigation proceeds4 ,it is upon empirical observation indeed that the laws of connexion are in the first place founded1 but the simple hypothesis that the elasticity and approximation of the particles vary in the directions of the crystalline axes allows of the application of deductive reasoning4 ,the whole of the phenomena are gradually being proved to be consistent with this hypothesis1 so that we have in this subject of crystallography a beautiful instance of successful classification1 connected with a nearly perfect physical hypothesis4 ,moreover this hypothesis was verified experimentally as regards the mechanical vibrations of sound by ,savart1 who found that the vibrations in a plate of biaxial crystal indicated the existence of varying elasticity in varying directions4 99,classification an ,inverse and ,tentative ,operation499 ,if attempts at so-called natural classification are really attempts at perfect induction1 it follows that they are subject to the remarks which were made upon the inverse character of the inductive process1 and upon the difficulty of every inverse operation 7pp4 #aa1 #ab1 #abb1 c474 ,there will be no royal road to the discovery of the best system1 and it will even be impossible to lay down rules of procedure to assist those who are in search of a good arrangement4 ,the only logical rule would be as follows3--,having given certain objects1 group them in every way in which they can be grouped1 and then observe in which method of grouping the correlation of properties is most conspicuously manifested4 ,but this method of exhaustive classification will in almost every case be impracticable1 owing to the immensely great number of modes in which a comparatively small number of objects may be grouped together4 ,about sixty-three elements have been classified by chemists in six principal groups as monad1 dyad1 triad1 c41 elements1 the numbers in the classes varying from three to twenty elements4 ,now if we were to calculate the whole number of ways in which sixty-three objects can be arranged in six groups1 we should find the number to be so great that the life of the longest lived man would be wholly inadequate to enable him to go through these possible groupings4 ,the rule of exhaustive arrangement1 then1 is absolutely impracticable4 ,it follows that mere haphazard trial cannot as a general rule give any useful result4 ,if we were to write the names of the elements in succession upon sixty-three cards1 throw them into a ballot-box1 and draw them out haphazard in six handfuls time after time1 the probability is excessively small that we should take them out in a specified order1 that for instance at present adopted by chemists4 ,the usual mode in which an investigator proceeds to form a classification of a new group of objects seems to consist in tentatively arranging them according to their most obvious similarities4 ,any two objects which present a close resemblance to each other will be joined and formed into the rudiment of a class1 the definition of which will at first include all the apparent points of resemblance4 ,other objects as they come to our notice will be gradually assigned to those groups with which they present the greatest number of points of resemblance1 and the definition of a class will often have to be altered in order to admit them4 ,the early chemists could hardly avoid classing together the common metals1 gold1 silver1 copper1 lead1 and iron1 which present such conspicuous points of similarity as regards density1 metallic lustre1 malleability1 c4 ,with the progress of discovery1 however1 difficulties began to present themselves in such a grouping4 ,antimony1 bismuth1 and arsenic are distinctly metallic as regards lustre1 density1 and some chemical properties1 but are wanting in malleability4 ,the recently discovered tellurium presents greater difficulties1 for it has many of the physical properties of metal1 and yet all its chemical properties are analogous to those of sulphur and selenium1 which have never been regarded as metals4 ,great chemical differences again are discovered by degrees between the five metals mentioned2 and the class1 if it is to have any chemical validity1 must be made to include other elements1 having none of the original properties on which the class was founded4 ,hydrogen is a transparent colourless gas1 and the least dense of all substances2 yet in its chemical analogies it is a metal1 as suggested by ,faraday#ega in #ahch1 and almost proved by ,graham2#egb it must be placed in the same class as silver4 ,in this way it comes to pass that almost every classification which is proposed in the early stages of a science will be found to break down as the deeper similarities of the objects come to be detected4 ,the most obvious points of difference will have to be neglected4 ,chlorine is a gas1 bromine a liquid1 and iodine a solid1 and at first sight these might have seemed formidable circumstances to overlook2 but in chemical analogy the substances are closely united4 ,the progress of organic chemistry1 again1 has yielded wholly new ideas of the similarities of compounds4 ,who1 for instance1 would recognise without extensive research a close similarity between glycerine and alcohol1 or between fatty substances and ether8 ,the class of paraffins contains three substances gaseous at ordinary temperatures1 several liquids1 and some crystalline solids4 ,it required much insight to detect the analogy which exists between such apparently different substances4 #ega 99,life of ,faraday991 vol4 ii4 p4 #hg4 #egb 99,proceedings of the ,royal ,society991 vol4 xvii4 p4 #bab4 99,chemical and ,physical ,researches991 reprint1 by ,young and ,angus ,smith1 p4 #bij4 ,the science of chemistry now depends to a great extent on a correct classification of the elements1 as will be learnt by consulting the able article on ,classification by ,professor ,g4 ,c4 ,foster in ,watts' 99,dictionary of ,chemistry994 ,but the present system of chemical classification was not reached until at least three previous false systems had been long entertained4 ,and though there is much reason to believe that the present mode of classification according to atomicity is substantially correct1 errors may yet be discovered in the details of the grouping4 99,symbolic ,statement of the ,theory of ,classification499 ,the theory of classification can be explained in the most complete and general manner1 by reverting for a time to the use of the ,logical ,alphabet1 which was found to be of supreme importance in ,formal ,logic4 ,that form expresses the necessary classification of all objects and ideas as depending on the laws of thought1 and there is no point concerning the purpose and methods of classification which may not be stated precisely by the use of letter combinations1 the only inconvenience being the abstract form in which the subject is thus represented4 ,if we pay regard only to three qualities in which things may resemble each other1 namely1 the qualities ,a1 ,b1 ,c1 there are according to the laws of thought eight possible classes of objects1 shown in the fourth column of the ,logical ,alphabet 7p4 #id74 ,if there exist objects belonging to all these eight classes1 it follows that the qualities ,a1 ,b1 ,c1 are subject to no conditions except the primary laws of thought and things 7p4 #e74 ,there is then no special law of nature to discover1 and1 if we arrange the objects in any one order rather than another1 it must be for the purpose of showing that the combinations are logically complete4 ,suppose1 however1 that there are but four kinds of objects possessing the qualities ,a1 ,b1 ,c1 and that these kinds are represented by the combinations ,,abc1 ,a99b99,c1 99a99,b99c991 99abc994 ,the order of arrangement will now be of importance2 for if we place them in the order ,,abc ,a99b99,c 99a99,b99c99 99abc99 placing the ,b's first and those which are 99b99's last1 we shall perhaps overlook the law of correlation of properties involved4 ,but if we arrange the combinations as follows ,,abc 99a99,b99c99 ,a99b99,c 99abc99 it becomes apparent at once that where ,a is1 and only where ,a is1 the property ,c is to be found1 ,b being indifferently present and absent4 ,the second arrangement then would be called a natural one1 as rendering manifest the conditions under which the combinations exist4 ,as a further instance1 let us suppose that eight objects are presented to us for classification1 which exhibit combinations of the five properties1 ,a1 ,b1 ,c1 ,d1 ,e1 in the following manner3-- ,,abc99d99,e 99a99,,bc99d99,e ,,ab99cde99 99a99,b99cde99 ,a99b99,,cde 99ab99,,cde ,a99bc99,d99e99 99abc99,d99e99 ,they are now classified1 so that those containing ,a stand first1 and those devoid of ,a second1 but no other property seems to be correlated with ,a4 ,let us alter this arrangement and group the combinations thus3-- ,,abc99d99,e ,a99b99,,cde ,,ab99cde99 ,a99bc99,d99e99 99a99,,bc99d99,e 99ab99,,cde 99a99,b99cde99 99abc99,d99e99 ,it requires little examination to discover that in the first group ,b is always present and ,d absent1 whereas in the second group1 ,b is always absent and ,d present4 ,this is the result which follows from a law of the form ,b d 7p4 #acf71 so that in this mode of arrangement we readily discover correlation between two letters4 ,altering the groups again as follows3-- ,,abc99d99,e ,,ab99cde99 99a99,,bc99d99,e 99a99,b99cde99 ,a99b99,,cde ,a99bc99,d99e99 99ab99,,cde 99abc99,d99e991 we discover another evident correlation between ,c and ,e4 ,between ,a and the other letters1 or between the two pairs of letters ,b1 ,d and ,c1 ,e1 there is no logical connexion4 ,this example may seem tedious1 but it will be found instructive in this way4 ,we are classifying only eight objects or combinations1 in each of which only five qualities are considered4 ,there are only two laws of correlation between four of those five qualities1 and those laws are of the simplest logical character4 ,yet the reader would hardly discover what those laws are1 and confidently assign them by rapid contemplation of the combinations1 as given in the first group4 ,several tentative classifications must probably be made before we can resolve the question4 ,let us now suppose that instead of eight objects and five qualities1 we have1 say1 five hundred objects and fifty qualities4 ,if we were to attempt the same method of exhaustive grouping which we before employed1 we should have to arrange the five hundred objects in fifty different ways1 before we could be sure that we had discovered even the simpler laws of correlation4 ,but even the successive grouping of all those possessing each of the fifty properties would not necessarily give us all the laws4 ,there might exist complicated relations between several properties simultaneously1 for the detection of which no rule of procedure whatever can be given4 99,bifurcate ,classification499 ,every system of classification ought to be formed on the principles of the ,logical ,alphabet4 ,each superior class should be divided into two inferior classes1 distinguished by the possession and non-possession of a single specified difference4 ,each of these minor classes1 again1 is divisible by any other quality whatever which can be suggested1 and thus every classification logically consists of an infinitely extended series of subaltern genera and species4 ,the classifications which we form are in reality very small fragments of those which would correctly and fully represent the relations of existing things4 ,but if we take more than four or five qualities into account1 the number of subdivisions grows impracticably large4 ,our finite minds are unable to treat any complex group exhaustively1 and we are obliged to simplify and generalise scientific problems1 often at the risk of overlooking particular conditions and exceptions4 ,every system of classes displayed in the manner of the ,logical ,alphabet may be called 99bifurcate991 because every class branches out at each step into two minor classes1 existent or imaginary4 ,it would be a great mistake to regard this arrangement as in any way a peculiar or special method2 it is not only a natural and important one1 but it is the inevitable and only system which is logically perfect1 according to the fundamental laws of thought4 ,all other arrangements of classes correspond to the bifurcate arrangement1 with the implication that some of the minor classes are not represented among existing things4 ,if we take the genus ,a and divide it into the species ,,ab and ,,ac1 we imply two propositions1 namely that in the class ,a1 the properties of ,b and ,c never occur together1 and that they are never both absent2 these propositions are logically equivalent to one1 namely ,,ab ,a99c994 ,our classification is then identical with the following bifurcate one3-- ,a -------------------- ,,ab ,a99b99 ------------ ------------ ,,abc #j ,,ab99c99 ,a99b99,c ,a99bc99 #j ,if1 again1 we divide the genus ,a into three species1 ,,ab1 ,,ac1 ,,ad1 we are either logically in error1 or else we must be understood to imply that1 as regards the other letters1 there exist only three combinations containing ,a1 namely ,,ab99cd991 ,a99b99,c99d991 and ,a99bc99,d4 ,the logical necessity of bifurcate classification has been clearly and correctly stated in the 99,outline of a ,new ,system of ,logic99 by ,george ,bentham1 the eminent botanist1 a work of which the logical value has been quite overlooked until lately4 ,mr4 ,bentham points out1 in p4 #aac1 that every classification must be essentially bifurcate1 and takes1 as an example1 the division of vertebrate animals into four sub-classes1 as follows3-- ,mammifera--endowed with mamm and lungs4 ,birds without mamm but with lungs and wings4 ,fish deprived of lungs4 ,reptiles deprived of mamm and wings but with lungs4 ,we have1 then1 as ,mr4 ,bentham says1 three bifid divisions1 thus represented3-- ,vertebrata ---------------------- ,endowed with lungs deprived of lungs ,fish4 ------------------------ ,endowed with deprived of mamm mamm ,mammifera4 ------------ with wings without wings ,birds4 ,reptiles4 ,it is quite evident that according to the laws of thought even this arrangement is incomplete4 ,the sub-class mammifera must either have wings or be deprived of them2 we must either subdivide this class1 or assume that none of the mammifera have wings1 which is1 as a matter of fact1 the case1 the wings of bats not being true wings in the meaning of the term as applied to birds4 ,fish1 again1 ought to be considered with regard to the possession of mamm and wings2 and in leaving them undivided we really imply that they never have mamm nor wings1 the wings of the flying-fish1 again1 being no exception4 ,if we resort to the use of our letters and define them as follows-- ,a vertebrata1 ,b having lungs1 ,c having mamm1 ,d having wings1 then there are four existent classes of vertebrata which appear to be thus described-- ,,abc ,,ab99c99,d ,,ab99cd99 ,a99b994 ,but in reality the combinations are implied to be ,,abc99d99 ,mammifera1 ,,ab99c99,d ,birds1 ,,ab99cd99 ,reptiles1 ,a99bcd99 ,fish1 and we imply at the same time that the other four conceivable combinations containing ,b1 ,c1 or ,d1 namely ,,abcd1 ,a99b99,,cd1 ,a99b99,c99d991 and ,a99bc99,d1 do not exist in nature4 ,mr4 ,bentham points out#egc that it is really this method of classification which was employed by ,lamarck and ,de ,candolle in their so-called analytical arrangement of the ,french ,flora4 ,he gives as an example a table of the principal classes of ,de ,candolle's system1 as also a bifurcate arrangement of animals after the method proposed by ,dumril in his 99,zoologie ,analytique991 this naturalist being distinguished by his clear perception of the logical importance of the method4 ,a bifurcate classification of the animal kingdom may also be found in ,professor ,reay ,greene's 99,manual of the ,clenterata991 p4 #ah4 #egc 99,essai sur la ,nomenclature et la ,classification991 ,paris1 #ahbc1 pp4 #ajg1 #ajh4 ,the bifurcate form of classification seems to be needless when the quality according to which we classify any group of things admits of numerical discrimination4 ,it would seem absurd to arrange things according as they have one degree of the quality or not one degree1 two degrees or not two degrees1 and so on4 ,the elements are classified according as the atom of each saturates one1 two1 three1 or more atoms of a monad element1 such as chlorine1 and they are called accordingly monad1 dyad1 triad1 tetrad elements1 and so on4 ,it would be useless to apply the bifid arrangement1 thus3-- ,element ------------ ,monad not-,monad ------------------ ,dyad not-,dyad ------------------ ,triad not-,triad ----------------- ,tetrad not-,tetrad4 ,the reason of this is that1 by the nature of number 7p4 #aeg7 every number is logically discriminated from every other number4 ,there can thus be no logical confusion in a numerical arrangement1 and the series of numbers indefinitely extended is also exhaustive4 ,every thing admitting of a quality expressible in numbers must find its place somewhere in the series of numbers4 ,the chords in music correspond to the simpler numerical ratios and must admit of complete exhaustive classification in respect to the complexity of the ratios forming them4 ,plane rectilinear figures may be classified according to the numbers of their sides1 as triangles1 quadrilateral figures1 pentagons1 hexagons1 heptagons1 c4 ,the bifurcate arrangement is not false when applied to such series of objects2 it is even necessarily involved in the arrangement which we do apply1 so that its formal statement is needless and tedious4 ,the same may be said of the division of portions of space4 ,reid and ,kames endeavoured to cast ridicule on the bifurcate arrangement#egd by proposing to classify the parts of ,england into ,middlesex and what is not ,middlesex1 dividing the latter again into ,kent and what is not ,kent1 ,sussex and what is not ,sussex2 and so on4 ,this is so far1 however1 from being an absurd proceeding that it is requisite to assure us that we have made an exhaustive enumeration of the parts of ,england4 #egd ,george ,bentham1 99,outline of a ,new ,system of ,logic991 p4 #aae4 99,the ,five ,predicables499 ,as a rule it is highly desirable to consign to oblivion the ancient logical names and expressions1 which have infested the science for many centuries past4 ,if logic is ever to be a useful and progressive science1 logicians must distinguish between logic and the history of logic4 ,as in the case of any other science it may be desirable to examine the course of thought by which logic has1 before or since the time of ,aristotle1 been brought to its present state2 the history of a science is always instructive as giving instances of the mode in which discoveries take place4 ,but at the same time we ought carefully to disencumber the statement of the science itself of all names and other vestiges of antiquity which are not actually useful at the present day4 ,among the ancient expressions which may well be excepted from such considerations and retained in use1 are the 8,five ,words0 or 8,five ,predicables0 which were described by ,porphyry in his introduction to ,aristotle's ,organum4 ,two of them1 99,genus99 and 99,species991 are the most venerable names in philosophy1 having probably been first employed in their present logical meanings by ,socrates4 ,in the present day it requires some mental effort1 as remarked by ,grote1 to see anything important in the invention of notions now so familiar as those of ,genus and ,species4 ,but in reality the introduction of such terms showed the rise of the first germs of logic and scientific method2 it showed that men were beginning to analyse their processes of thought4 ,the ,five ,predicables are ,genus1 ,species1 ,difference1 ,property1 and ,accident1 or in the original ,greek1 1 1 1 1 4 ,of these1 ,genus may be taken to mean any class of objects which is regarded as broken up into two minor classes1 which form ,species of it4 ,the genus is defined by a certain number of qualities or circumstances which belong to all objects included in the class1 and which are sufficient to mark out these objects from all others which we do not intend to include4 ,interpreted as regards intension1 then1 the genus is a group of qualities2 interpreted as regards extension1 it is a group of objects possessing those qualities4 ,if another quality be taken into account which is possessed by some of the objects and not by the others1 this quality becomes a difference which divides the genus into two species4 ,we may interpret the species either in intension or extension2 in the former respect it is more than the genus as containing one more quality1 the difference3 in the latter respect it is less than the genus as containing only a portion of the group constituting the genus4 ,we may say1 then1 with ,aristotle1 that in one sense the genus is in the species1 namely in intension1 and in another sense the species is in the genus1 namely in extension4 ,the difference1 it is evident1 can be interpreted in intension only4 ,a ,property is a quality which belongs to the whole of a class1 but does not enter into the definition of that class4 ,a generic property belongs to every individual object contained in the genus4 ,it is a property of the genus parallelogram that the opposite angles are equal4 ,if we regard a rectangle as a species of parallelogram1 the difference being that 99one99 angle is a right angle1 it follows as a specific property that all the angles are right angles4 ,though a property in the strict logical sense must belong to each of the objects included in the class of which it is a property1 it may or may not belong to other objects4 ,the property of having the opposite angles equal may belong to many figures besides parallelograms1 for instance1 regular hexagons4 ,it is a property of the circle that all triangles constructed upon the diameter with the apex upon the circumference are right-angled triangles1 and 99vice vers991 all curves of which this is true must be circles4 ,a property which thus belongs to the whole of a class and only to that class1 corresponds to the of ,aristotle and ,porphyry2 we might conveniently call it 99a peculiar property994 ,every such property enables us to make a statement in the form of a simple identity 7p4 #cg74 ,thus we know it to be a peculiar property of the circle that for a given length of perimeter it encloses a greater area than any other possible curve2 hence we may say-- ,curve of equal curvature curve of greatest area4 ,it is a peculiar property of equilateral triangles that they are equiangular1 and 99vice vers991 it is a peculiar property of equiangular triangles that they are equilateral4 ,it is a property of crystals of the regular system that they are devoid of the power of double refraction1 but this is not a property peculiar to them1 because liquids and gases are devoid of the same property4 ,an ,accident1 the fifth and last of the ,predicables1 is any quality which may or may not belong to certain objects1 and which has no connexion with the classification adopted4 ,the particular size of a crystal does not in the slightest degree affect the form of the crystal1 nor does the manner in which it is grouped with other crystals2 these1 then1 are accidents as regards a crystallographic classification4 ,with respect to the chemical composition of a substance1 again1 it is an accident whether the substance be crystallised or not1 or whether it be organised or not4 ,as regards botanical classification the absolute size of a plant is an accident4 ,thus we see that a logical accident is any quality or circumstance which is not known to be correlated with those qualities or circumstances forming the definition of the species4 ,the meanings of the ,predicables can be clearly explained by our symbols4 ,let ,a be any definite group of qualities and ,b another quality or group of qualities2 then ,a will constitute a genus1 and ,,ab1 ,a99b99 will be species of it1 ,b being the difference4 ,let ,c1 ,d and ,e be other qualities or groups of qualities1 and on examining the combinations in which ,a1 ,b1 ,c1 ,d1 ,e occur let them be as follows3-- ,,abcde ,a99b99,c99d99,e ,,abcd99e99 ,a99b99,c99de994 ,here we see that wherever ,a is we also find ,c1 so that ,c is a generic property2 ,d occurs always with ,b1 so that it constitutes a specific property1 while ,e is indifferently present and absent1 so as not to be related to any other letter2 it represents1 therefore1 an accident4 ,it will now be seen that the ,logical ,alphabet represents an interminable series of subordinate genera and species2 it is but a concise symbolic statement of what was involved in the ancient doctrine of the ,predicables4 99,summum ,genus and ,infima ,species499 ,as a genus means any class whatever which is regarded as composed of minor classes or species1 it follows that the same class will be a genus in one point of view and a species in another4 ,metal is a genus as regards alkaline metal1 a species as regards element1 and any extensive system of classes consists of a series of subordinate1 or as they are technically called1 99subaltern99 genera and species4 ,the question1 however1 arises1 whether such a chain of classes has a definite termination at either end4 ,the doctrine of the old logicians was to the effect that it terminated upwards in a 99genus generalissimum99 or 99summum genus991 which was not a species of any wider class4 ,some very general notion1 such as substance1 object1 or thing1 was supposed to be so comprehensive as to include all thinkable objects1 and for all practical purposes this might be so4 ,but as ,i have already explained 7p4 #gd71 we cannot really think of any object or class without thereby separating it from what is not that object or class4 ,all thinking is relative1 and implies discrimination1 so that every class and every logical notion must have its negative4 ,if so1 there is no such thing as a 99summum genus992 for we cannot frame the requisite notion of a class forming it without implying the existence of another class discriminated from it2 add this new negative class to the supposed 99summum genus991 and we form a still higher genus1 which is absurd4 ,although there is no absolute summum genus1 nevertheless relatively to any branch of knowledge or any particular argument1 there is always some class or notion which bounds our horizon as it were4 ,the chemist restricts his view to material substances and the forces manifested in them2 the mathematician extends his view so as to comprehend all notions capable of numerical discrimination4 ,the biologist1 on the other hand1 has a narrower sphere containing only organised bodies1 and of these the botanist and the zoologist take parts4 ,in other subjects there may be a still narrower summum genus1 as when the lawyer regards only reasoning beings of his own country together with their property4 ,in the description of the ,logical ,alphabet it was pointed out 7p4 #ic7 that every series of combinations is really the development of a single class1 denoted by ,x1 which letter was accordingly placed in the first column of the table on p4 #id4 ,this is the formal acknowledgment of the principle clearly stated by ,de ,morgan1 that all reasoning proceeds within an assumed summum genus4 ,but at the same time the fact that ,x as a logical term must have its negative 99x991 shows that it cannot be an absolute summum genus4 ,there arises1 again1 the question whether there be any such thing as an 99infima species991 which cannot be divided into minor species4 ,the ancient logicians were of opinion that there always was some assignable class which could only be divided into individuals1 but this doctrine appears to be theoretically incorrect1 as ,mr4 ,george ,bentham long ago stated4#ege ,we may put an arbitrary limit to the subdivision of our classes at any point convenient to our purpose4 ,the crystallographer would not generally treat as different species crystalline forms which differ only in the degree of development of the faces4 ,the naturalist overlooks innumerable slight differences between animals which he refers to the same species4 ,but in a strictly logical point of view classification might be carried on as long as there is a difference1 however minute1 between two objects1 and we might thus go on until we arrive at individual objects which are numerically distinct in the logical sense attributed to that expression in the chapter upon ,number4 ,either1 then1 we must call the individual the 99infima species99 or allow that there is no such thing at all4 #ege 99,outline of a ,new ,system of ,logic991 #ahbg1 p4 #aag4 99,the ,tree of ,porphyry499 ,both ,aristotle and ,plato were acquainted with the value of bifurcate classification1 which they occasionally employed in an explicit manner4 ,it is impossible too that ,aristotle should state the laws of thought1 and employ the predicables without implicitly recognising the logical necessity of that method4 ,it is1 however1 in ,porphyry's remarkable and in many respects excellent 99,introduction to the ,categories of ,aristotle99 that we find the most distinct account of it4 ,porphyry not only fully and accurately describes the ,predicables1 but incidentally introduces an example for illustrating those predicables1 which constitutes a good specimen of bifurcate classification4 ,translating his words#egf freely we may say that he takes ,substance as the genus to be divided1 under which are successively placed as ,species--,body1 ,animated ,body1 ,animal1 ,rational ,animal1 and ,man4 ,under ,man1 again1 come ,socrates1 ,plato1 and other particular men4 ,now of these notions ,substance is the genus generalissimum1 and is a genus only1 not a species4 ,man1 on the other hand1 is the species specialissima 7infima species71 and is a species only1 not a genus4 ,body is a species of substance1 but a genus of animated body1 which1 again1 is a species of body but a genus of animal4 ,animal is a species of animated body1 but a genus of rational animal1 which1 again1 is a species of animal1 but a genus of man4 ,finally1 man is a species of rational animal1 but is a species merely and not a genus1 being divisible only into particular men4 #egf 99,porphyrii ,isagoge991 ,caput ii4 #bd4 ,porphyry proceeds at some length to employ his example in further illustration of the predicables4 ,we do not find in ,porphyry's own work any scheme or diagram exhibiting this curious specimen of classification1 but some of the earlier commentators and epitome writers drew what has long been called the ,tree of ,porphyry4 ,this diagram1 which may be found in most elementary works on ,logic1#egg is also called the ,ramean ,tree1 because ,ramus insisted much upon the value of ,dichotomy4 ,with the exception of ,jeremy ,bentham#egh and ,george ,bentham1 hardly any modern logicians have shown an appreciation of the value of bifurcate classification4 ,the latter author has treated the subject1 both in his 99,outline of a ,new ,system of ,logic99 7pp4 #aje-aah71 and in his earlier work entitled 99,essai sur la ,nomenclature et la ,classification des ,principales ,branches d',art-et-,science99 7,paris1 #ahbc71 which consists of a free translation or improved version of his uncle's ,essay on ,classification in the 99,chrestomathia994 ,some interest attaches to the history of the ,tree of ,porphyry and ,ramus1 because it is the prototype of the ,logical ,alphabet which lies at the basis of logical method4 ,jeremy ,bentham speaks truly of 8the matchless beauty of the ,ramean ,tree40 ,after fully showing its logical value as an exhaustive method of classification1 and refuting the objections of ,reid and ,kames1 on a wrong ground1 as ,i think1 he proceeds to inquire to what length it may be carried4 ,he correctly points out two objections to the extensive use of bifid arrangements1 7#a7 that they soon become impracticably extensive and unwieldy1 and 7#b7 that they are uneconomical4 ,in his day the recorded number of different species of plants was #dj1jjj1 and he leaves the reader to estimate the immense number of branches and the enormous area of a bifurcate table which should exhibit all these species in one scheme4 ,he also points out the apparent loss of labour in making any large bifurcate classification2 but this he considers to be fully recompensed by the logical value of the result1 and the logical training acquired in its execution4 ,jeremy ,bentham1 then1 fully recognises the value of the ,logical ,alphabet under another name1 though he apprehends also the limit to its use placed by the finiteness of our mental and manual powers4 #egg ,jevons1 99,elementary ,lessons in ,logic991 p4 #ajd4 #egh 99,chrestomathia2 being a ,collection of ,papers1 c499 ,london1 #ahaf1 ,appendix ,v4 99,does ,abstraction imply ,generalisation899 ,before we can acquire a sound comprehension of the subject of classification we must answer the very difficult question whether logical abstraction does or does not imply generalisation4 ,it comes to exactly the same thing if we ask whether a species may be coextensive with its genus1 or whether1 on the other hand1 the genus must contain more than the species4 ,to abstract logically is 7p4 #bg71 to overlook or withdraw our notice from some point of difference4 ,whenever we form a class we abstract1 for the time being1 the differences of the objects so united in respect of some common quality4 ,if we class together a great number of objects as dwelling-houses1 we overlook the fact that some dwelling-houses are constructed of stone1 others of brick1 wood1 iron1 c4 ,often at least the abstraction of a circumstance increases the number of objects included under a class according to the law of the inverse relation of the quantities of extension and intension 7p4 #bf74 ,dwelling-house is a wider term than brick-dwelling-house4 ,house is more general than dwelling-house4 ,but the question before us is1 whether abstraction 99always99 increases the number of objects included in a class1 which amounts to asking whether the law of the inverse relation of logical quantities is 99always99 true4 ,the interest of the question partly arises from the fact1 that so high a philosophical authority as ,mr4 ,herbert ,spencer has denied that generalisation is implied in abstraction1#egi making this doctrine the ground for rejecting previous methods of classifying the sciences1 and for forming an ingenious but peculiar method of his own4 ,the question is also a fundamental one of the highest logical importance1 and involves subtle difficulties which have made me long hesitate in forming a decisive opinion4 #egi 99,the ,classification of the ,sciences991 c41 #crd edit4 p4 #g4 99,essays3 ,scientific1 ,political1 and ,speculative991 vol4 iii4 p4 #ac4 ,let us attempt to answer the question by examination of a few examples4 ,compare the two classes 99gun99 and 99iron gun994 ,it is certain that there are many guns which are not made of iron1 so that abstraction of the circumstance 8made of iron0 increases the extent of the notion4 ,next compare 99gun99 and 99metallic gun994 ,all guns made at the present day consist of metal1 so that the two notions seem to be coextensive2 but guns were at first made of pieces of wood bound together like a tub1 and as the logical term gun takes no account of time1 it must include all guns that have ever existed4 ,here again extension increases as intension decreases4 ,compare once more 8steam-locomotive engine0 and 8locomotive engine40 ,in the present day1 as far as ,i am aware1 all locomotives are worked by steam1 so that the omission of that qualification might seem not to widen the term2 but it is quite possible that in some future age a different motive power may be used in locomotives2 and as there is no limitation of time in the use of logical terms1 we must certainly assume that there is a class of locomotives not worked by steam1 as well as a class that is worked by steam4 ,when the natural class of ,euphorbiace was originally formed1 all the plants known to belong to it were devoid of corollas2 it would have seemed therefore that the two classes 8,euphorbiace10 and 8,euphorbiace devoid of ,corollas10 were of equal extent4 ,subsequently a number of plants plainly belonging to the same class were found in tropical countries1 and they possessed bright coloured corollas4 ,naturalists believe with the utmost confidence that 8,ruminants0 and 8,ruminants with cleft feet0 are identical terms1 because no ruminant has yet been discovered without cleft feet4 ,but we can see no impossibility in the conjunction of rumination with uncleft feet1 and it would be too great an assumption to say that we are certain that an example of it will never be met with4 ,instances can be quoted1 without end1 of objects being ultimately discovered combining properties which had never before been seen together4 ,in the animal kingdom the ,black ,swan1 the ,ornithorhynchus ,paradoxus1 and more recently the singular fish called ,ceratodus ,forsteri1 all discovered in ,australia1 have united characters never previously known to coexist4 ,at the present time deep-sea dredging is bringing to light many animals of an unprecedented nature4 ,singular exceptional discoveries may certainly occur in other branches of science4 ,when ,davy first discovered metallic potassium1 it was a well established empirical law that all metallic substances possessed a high specific gravity1 the least dense of the metals then known being zinc1 of which the specific gravity is #ga4 ,yet to the surprise of chemists1 potassium was found to be an undoubted metal of less density than water1 its specific gravity being #jhfe4 ,it is hardly requisite to prove by further examples that our knowledge of nature is incomplete1 so that we cannot safely assume the non-existence of new combinations4 ,logically speaking1 we ought to leave a place open for animals which ruminate but are without cleft feet1 and for every possible intermediate form of animal1 plant1 or mineral4 ,a purely logical classification must take account not only of what certainly does exist1 but of what may in after ages be found to exist4 ,i will go a step further1 and say that we must have places in our scientific classifications for purely imaginary existences4 ,a large proportion of the mathematical functions which are conceivable have no application to the circumstances of this world4 ,physicists certainly do investigate the nature and consequences of forces which nowhere exist4 ,newton's 99,principia99 is full of such investigations4 ,in one chapter of his 99,mcanique ,cleste99 ,laplace indulges in a remarkable speculation as to what the laws of motion would have been if momentum1 instead of varying simply as the velocity1 had been a more complicated function of it4 ,i have already mentioned 7p4 #bbc7 that ,airy contemplated the existence of a world in which the laws of force should be such that a perpetual motion would be possible1 and the ,law of ,conservation of ,energy would not hold true4 ,thought is not bound down to the limits of what is materially existent1 but is circumscribed only by those ,fundamental ,laws of ,identity1 ,contradiction and ,duality1 which were laid down at the outset4 ,this is the point at which ,i should differ from ,mr4 ,spencer4 ,he appears to suppose that a classification is complete if it has a place for every existing object1 and this may perhaps seem to be practically sufficient2 but it is subject to two profound objections4 ,firstly1 we do not know all that exists1 and therefore in limiting our classes we are erroneously omitting multitudes of objects of unknown form and nature which may exist either on this earth or in other parts of space4 ,secondly1 as ,i have explained1 the powers of thought are not limited by material existences1 and we may1 or1 for some purposes1 must imagine objects which probably do not exist1 and if we imagine them we ought to find places for them in the classifications of science4 ,the chief difficulty of this subject1 however1 consists in the fact that mathematical or other certain laws may entirely forbid the existence of some combinations4 ,the circle may be defined as a plane curve of equal curvature1 and it is a property of the circle that it contains the greatest area within the least possible perimeter4 ,may we then contemplate mentally a circle not a figure of greatest possible area8 ,or1 to take a still simpler example1 a parallelogram possesses the property of having the opposite angles equal4 ,may we then mentally divide parallelograms into two classes according as they do or do not have their opposite angles equal8 ,it might seem absurd to do so1 because we know that one of the two species of parallelogram would be non-existent4 ,but1 then1 unless the student had previously contemplated the existence of both species as possible1 what is the meaning of the thirty-fourth proposition of ,euclid's first book8 ,we cannot deny or disprove the existence of a certain combination without thereby in a certain way recognising that combination as an object of thought4 ,the conclusion at which ,i arrive is in opposition to that of ,mr4 ,spencer4 ,i think that whenever we abstract a quality or circumstance we do generalise or widen the notion from which we abstract4 ,whatever the terms ,a1 ,b1 and ,c may be1 ,i hold that in strict logic ,,ab is mentally a wider term than ,,abc1 because ,,ab includes the two species ,,abc and ,,ab99c994 ,the term ,a is wider still1 for it includes the four species ,,abc1 ,,ab99c991 ,a99b99,c1 ,a99bc994 ,the ,logical ,alphabet1 in short1 is the only limit of the classes of objects which we must contemplate in a purely logical point of view4 ,whatever notions be brought before us1 we must mentally combine them in all the ways sanctioned by the laws of thought and exhibited in the ,logical ,alphabet1 and it is a matter for after consideration to determine how many of these combinations exist in outward nature1 or how many are actually forbidden by the conditions of space4 ,a classification is essentially a mental1 not a material thing4 99,discovery of ,marks or ,characteristics499 ,although the chief purpose of classification is to disclose the deepest and most general resemblances of the objects classified1 yet the practical value of a system will depend partly upon the ease with which we can refer an object to its proper class1 and thus infer concerning it all that is known generally of that class4 ,this operation of discovering to which class of a system a certain specimen or case belongs1 is generally called 99,diagnosis991 a technical term familiarly used by physicians1 who constantly require to diagnose or determine the nature of the disease from which a patient is suffering4 ,now every class is defined by certain specified qualities or circumstances1 the whole of which are present in every object contained in the class1 and 99not all present99 in any object excluded from it4 ,these defining circumstances ought to consist of the deepest and most important circumstances1 by which we vaguely mean those probably forming the conditions with which the minor circumstances are correlated4 ,but it will often happen that the so-called important points of an object are not those which can most readily be observed4 ,thus the two great classes of phanerogamous plants are defined respectively by the possession of two cotyledons or seed-leaves1 and one cotyledon4 ,but when a plant comes to our notice and we want to refer it to the right class1 it will often happen that we have no seed at all to examine1 in order to discover whether there be one seed-leaf or two in the germ4 ,even if we have a seed it will often be small1 and a careful dissection under the microscope will be requisite to ascertain the number of cotyledons4 ,occasionally the examination of the germ would mislead us1 for the cotyledons may be obsolete1 as in ,cuscuta1 or united together1 as in ,clintonia4 ,botanists therefore seldom actually refer to the seed for such information4 ,certain other characters of a plant are correlated with the number of seed-leaves2 thus monocotyledonous plants almost always possess leaves with parallel veins like those of grass1 while dicotyledonous plants have leaves with reticulated veins like those of an oak leaf4 ,in monocotyledonous plants1 too1 the parts of the flower are most often three or some multiple of three in number1 while in dicotyledonous plants the numbers four and five and their multiples prevail4 ,botanists1 therefore1 by a glance at the leaves and flowers can almost certainly refer a plant to its right class1 and can infer not only the number of cotyledons which would be found in the seed or young plant1 but also the structure of the stem and other general characters4 ,any conspicuous and easily discriminated property which we thus select for the purpose of deciding to which class an object belongs1 may be called a 99characteristic994 ,the logical conditions of a good characteristic mark are very simple1 namely1 that it should be possessed by all objects entering into a certain class1 and by none others4 ,every characteristic should enable us to assert a simple identity2 if ,a is a characteristic1 and ,b1 viewed intensively1 the class of objects of which it is the mark1 then ,a ,b ought to be true4 ,the characteristic may consist either of a single quality or circumstance1 or of a group of such1 provided that they all be constant and easily detected4 ,thus in the classification of mammals the teeth are of the greatest assistance1 not because a slight variation in the number and form of the teeth is of importance in the general economy of the animal1 but because such variations are proved by empirical observation to coincide with most important differences in the general affinities4 ,it is found that the minor classes and genera of mammals can be discriminated accurately by their teeth1 especially by the foremost molars and the hindmost pre-molars4 ,some teeth1 indeed1 are occasionally missing1 so that zoologists prefer to trust to those characteristic teeth which are most constant1#ehj and to infer from them not only the arrangement of the other teeth1 but the whole conformation of the animal4 #ehj ,owen1 99,essay on the ,classification and ,geographical ,distribution of the ,mammalia991 p4 #bj4 ,it is a very difficult matter to mark out a boundary-line between the animal and vegetable kingdoms1 and it may even be doubted whether a rigorous boundary can be established4 ,the most fundamental and important difference of a vegetable as compared with an animal substance probably consists in the absence of nitrogen from the constituent membranes4 ,supposing this to be the case1 the difficulty arises that in examining minute organisms we cannot ascertain directly whether they contain nitrogen or not4 ,some minor but easily detected circumstance is therefore needed to discriminate between animals and vegetables1 and this is furnished to some extent by the fact that the production of starch granules is restricted to the vegetable kingdom4 ,thus the ,desmidiace may be safely assigned to the vegetable kingdom1 because they contain starch4 ,but we must not employ this characteristic negatively2 the ,diatomace are probably vegetables1 though they do not produce starch4 99,diagnostic ,systems of ,classification499 ,we have seen that diagnosis is the process of discovering the place in any system of classes1 to which an object has been referred by some previous investigation1 the object being to avail ourselves of the information relating to such an object which has been accumulated and recorded4 ,it is obvious that this is a matter of great importance1 for1 unless we can recognise1 from time to time1 objects or substances which have been investigated1 recorded discoveries would lose their value4 ,even a single investigator must have means of recording and systematising his observations of any large groups of objects like the vegetable and animal kingdoms4 ,now whenever a class has been properly formed1 a definition must have been laid down1 stating the qualities and circumstances possessed by all the objects which are intended to be included in the class1 and not possessed 99completely99 by any other objects4 ,diagnosis1 therefore1 consists in comparing the qualities of a certain object with the definitions of a series of classes2 the absence in the object of any one quality stated in the definition excludes it from the class thus defined2 whereas1 if we find every point of a definition exactly fulfilled in the specimen1 we may at once assign it to the class in question4 ,it is of course by no means certain that everything which has been affirmed of a class is true of all objects afterwards referred to the class2 for this would be a case of imperfect inference1 which is never more than matter of probability4 ,a definition can only make known a finite number of the qualities of an object1 and it always remains possible that objects agreeing in those assigned qualities will differ in others4 99,an individual cannot be defined991 and can only be made known by the exhibition of the individual itself1 or by a material specimen exactly representing it4 ,but this and other questions relating to definition must be treated when ,i am able to take up the subject of language in another work4 ,diagnostic systems of classification should1 as a general rule1 be arranged on the bifurcate method explicitly4 ,any quality may be chosen which divides the whole group of objects into two distinct parts1 and each part may be sub-divided successively by any prominent and well-marked circumstance which is present in a large part of the genus and not in the other4 ,to refer an object to its proper place in such an arrangement we have only to note whether it does or does not possess the successive critical differenti4 ,dana devised a classification of this kind#eha by which to refer a crystal to its place in the series of six or seven classes already described4 ,if a crystal has all its edges modified alike or the angles replaced by three or six similar planes1 it belongs to the monometric system2 if not1 we observe whether the number of similar planes at the extremity of the crystal is three or some multiple of three1 in which case it is a crystal of the hexagonal system2 and so we proceed with further successive discriminations4 ,to ascertain the name of a mineral by examination with the blow-pipe1 an arrangement more or less evidently on the bifurcate plan1 has been laid down by ,von ,kobell4#ehb ,minerals are divided according as they possess or do not possess metallic lustre2 as they are fusible or not fusible1 according as they do or do not on charcoal give a metallic bead1 and so on4 #eha ,dana's 99,mineralogy991 vol4 i4 p4 #abc2 quoted in ,watts' 99,dictionary of ,chemistry991 vol4 ii4 p4 #aff4 #ehb 99,instructions for the ,discrimination of ,minerals by ,simple ,chemical ,experiments991 by ,franz von ,kobell1 translated from the ,german by ,r4 ,c4 ,campbell4 ,glasgow1 #ahda4 ,perhaps the best example to be found of an arrangement devised simply for the purpose of diagnosis1 is ,mr4 ,george ,bentham's 99,analytical ,key to the ,natural ,orders and ,anomalous ,genera of the ,british ,flora991 given in his 99,handbook of the ,british ,flora994#ehc ,in this scheme1 the great composite family of plants1 together with the closely approximate genus ,jasione1 are first separated from all other flowering plants by the compound character of their flowers4 ,the remaining plants are sub-divided according as the perianth is double or single4 ,since no plants are yet known in which the perianth can be said to have three or more distinct rings1 this division becomes practically the same as one into double and not-double4 ,flowers with a double perianth are next discriminated according as the corolla does or does not consist of one piece2 according as the ovary is free or not free2 as it is simple or not simple2 as the corolla is regular or irregular2 and so on4 ,on looking over this arrangement1 it will be found that numerical discriminations often occur1 the numbers of petals1 stamens1 capsules1 or other parts being the criteria1 in which cases1 as already explained 7p4 #fig71 the actual exhibition of the bifid division would be tedious4 #ehc ,edition of #ahff1 p4 lxiii4 ,linnus appears to have been perfectly acquainted with the nature and uses of diagnostic classification1 which he describes under the name of ,synopsis1 saying3#ehd--8,synopsis tradit ,divisiones arbitrarias1 longiores aut breviores1 plures aut pauciores3 a ,botanicis in genere non agnoscenda4 ,synopsis est dichotomia arbitraria1 qu instar vi ad ,botanicem ducit4 ,limites autem non determinat40 #ehd 99,philosophia ,botanica99 7#aggj71 #aed1 p4 #ih4 ,the rules and tables drawn out by chemists to facilitate the discovery of the nature of a substance in qualitative analysis are usually arranged on the bifurcate method1 and form excellent examples of diagnostic classification1 the qualities of the substances produced in testing being in most cases merely characteristic properties of little importance in other respects4 ,the chemist does not detect potassium by reducing it to the state of metallic potassium1 and then observing whether it has all the principal qualities belonging to potassium4 ,he selects from among the whole number of compounds of potassium that salt1 namely the compound of platinum tetra-chloride1 and potassium chloride1 which has the most distinctive appearance1 as it is comparatively insoluble and produces a peculiar yellow and highly crystalline precipitate4 ,accordingly1 potassium is present whenever this precipitate can be produced by adding platinum chloride to a solution4 ,the fine purple or violet colour which potassium salts communicate to the blowpipe flame1 had long been used as a characteristic mark4 ,some other elements were readily detected by the colouring of the blowpipe flame1 barium giving a pale yellowish green1 and salts of strontium a bright red4 ,by the use of the spectroscope the coloured light given off by an incandescent vapour is made to give perfectly characteristic marks of the elements contained in the vapour4 ,diagnosis seems to be identical with the process termed by the ancient logicians 99abscissio infiniti991 the cutting off of the infinite or negative part of a genus when we discover by observation that an object possesses a particular difference4 ,at every step in a bifurcate division1 some objects possessing the difference will fall into the affirmative part or species2 all the remaining objects in the world fall into the negative part1 which will be infinite in extent4 ,diagnosis consists in the successive rejection from further notice of those infinite classes with which the specimen in question does not agree4 99,index ,classifications499 ,under classification we may include all arrangements of objects or names1 which we make for saving labour in the discovery of an object4 ,even alphabetical indices are real classifications4 ,no such arrangement can be of use unless it involves some correlation of circumstances1 so that knowing one thing we learn another4 ,if we merely arrange letters in the pigeon-holes of a secretaire we establish a correlation1 for all letters in the first hole will be written by persons1 for instance1 whose names begin with ,a1 and so on4 ,knowing then the initial letter of the writer's name1 we know also the place of the letter1 and the labour of search is thus reduced to one twenty-sixth part of what it would be without arrangement4 ,now the purpose of a catalogue is to discover the place in which an object is to be found2 but the art of cataloguing involves logical considerations of some importance4 ,we want to establish a correlation between the place of an object and some circumstance about the object which shall enable us readily to refer to it2 this circumstance therefore should be that which will most readily dwell in the memory of the searcher4 ,a piece of poetry will be best remembered by the first line of the piece1 and the name of the author will be the next most definite circumstance2 a catalogue of poetry should therefore be arranged alphabetically according to the first word of the piece1 or the name of the author1 or1 still better1 in both ways4 ,it would be impossible to arrange poems according to their subjects1 so vague and mixed are these found to be when the attempt is made4 ,it is a matter of considerable literary importance to decide upon the best mode of cataloguing books1 so that any required book in a library shall be most readily found4 ,books may be classified in a great number of ways1 according to subject1 language1 date1 or place of publication1 size1 the initial words of the text or title-page1 or colophon1 the author's name1 the publisher's name1 the printer's name1 the character of the type1 and so on4 ,every one of these modes of arrangement may be useful1 for we may happen to remember one circumstance about a book when we have forgotten all others2 but as we cannot usually go to the expense of forming more than two or three indices1 we must select those circumstances which will lead to the discovery of a book most frequently4 ,many of the criteria mentioned are evidently inapplicable4 ,the language in which a book is written is definite enough1 provided that the whole book is written in the same language2 but it is obvious that language gives no means for the subdivision and arrangement of the literature of any one people4 ,classification by subjects would be an exceedingly useful method if it were practicable1 but experience shows it to be a logical absurdity4 ,it is a very difficult matter to classify the sciences1 so complicated are the relations between them4 ,but with books the complication is vastly greater1 since the same book may treat of different sciences1 or it may discuss a problem involving many branches of knowledge4 ,a good account of the steam-engine will be antiquarian1 so far as it traces out the earliest efforts at discovery2 purely scientific1 as regards the principles of thermodynamics involved2 technical1 as regards the mechanical means of applying those principles2 economical1 as regards the industrial results of the invention2 biographical1 as regards the lives of the inventors4 ,a history of ,westminster ,abbey might belong either to the history of architecture1 the history of the ,church1 or the history of ,england4 ,if we abandon the attempt to carry out an arrangement according to the natural classification of the sciences1 and form comprehensive practical groups1 we shall be continually perplexed by the occurrence of intermediate cases1 and opinions will differ 99ad infinitum99 as to the details4 ,if1 to avoid the difficulty about ,westminster ,abbey1 we form a class of books devoted to the ,history of ,buildings1 the question will then arise whether ,stonehenge is a building1 and if so1 whether cromlechs1 mounds1 and monoliths are so4 ,we shall be uncertain whether to include lighthouses1 monuments1 bridges1 c4 ,in regard to literary works1 rigorous classification is still less possible4 ,the same work may partake of the nature of poetry1 biography1 history1 philosophy1 or if we form a comprehensive class of ,belles-lettres1 nobody can say exactly what does or does not come under the term4 ,my own experience entirely bears out the opinion of ,de ,morgan1 that classification according to the name of the author is the only one practicable in a large library1 and this method has been admirably carried out in the great catalogue of the ,british ,museum4 ,the name of the author is the most precise circumstance concerning a book1 which usually dwells in the memory4 ,it is a better characteristic of the book than anything else4 ,in an alphabetical arrangement we have an exhaustive classification1 including a place for every name4 ,the following remarks#ehe of ,de ,morgan seem therefore to be entirely correct4 8,from much1 almost daily use1 of catalogues for many years1 ,i am perfectly satisfied that a classed catalogue is more difficult to use than to make4 ,it is one man's theory of the subdivision of knowledge1 and the chances are against its suiting any other man4 ,even if all doubtful works were entered under several different heads1 the frontier of the dubious region would itself be a mere matter of doubt4 ,i never turn from a classed catalogue to an alphabetical one without a feeling of relief and security4 ,with the latter ,i can always1 by taking proper pains1 make a library yield its utmost2 with the former ,i can never be satisfied that ,i have taken proper pains1 until ,i have made it1 in fact1 as many different catalogues as there are different headings1 with separate trouble for each4 ,those to whom bibliographical research is familiar1 know that they have much more frequently to hunt an author than a subject3 they know also that in searching for a subject1 it is never safe to take another person's view1 however good1 of the limits of that subject with reference to their own particular purposes40 #ehe 99,philosophical ,magazine991 #crd ,series 7#ahde71 vol4 xxvi4 p4 #ebb4 ,see also ,de ,morgan's evidence before the ,royal ,commission on the ,british ,museum in #ahdi1 ,report 7#ahej71 ,questions1 #egjd99-ehae991 #fdha-feac4 ,this evidence should be studied by every person who wishes to understand the elements of ,bibliography4 ,it is often desirable1 however1 that a name catalogue should be accompanied by a subordinate subject catalogue1 but in this case no attempt should be made to devise a theoretically complete classification4 ,every principal subject treated in a book should be entered separately in an alphabetical list1 under the name most likely to occur to the searcher1 or under several names4 ,this method was partially carried out in ,watts' 99,bibliotheca ,britannica991 but it was excellently applied in the admirable subject index to the 99,british ,catalogue of ,books991 and equally well in the 99,catalogue of the ,manchester ,free ,library99 at ,campfield1 drawn up under the direction of ,mr4 ,crestadoro1 this latter being the most perfect model of a printed catalogue with which ,i am acquainted4 ,the ,catalogue of the ,london ,library is also in the right form1 and has a useful index of subjects1 though it is too much condensed and abbreviated4 ,the public catalogue of the ,british ,museum is arranged as far as possible according to the alphabetical order of the authors' names1 but in writing the titles for this catalogue several copies are simultaneously produced by a manifold writer1 so that a catalogue according to the order of the books on the shelves1 and another according to the first words of the title-page1 are created by a mere rearrangement of the spare copies4 ,in the 99,english ,cyclopdia99 it is suggested that twenty copies of the book titles might readily have been utilised in forming additional catalogues1 arranged according to the place of publication1 the language of the book1 the general nature of the subject1 and so forth4#ehf ,an excellent suggestion has also been made to the effect that each book when published should have a fly-leaf containing half a dozen printed copies of the title1 drawn up in a form suitable for insertion in catalogues4 ,every owner of a library could then easily make accurate printed catalogues to suit his own purposes1 by merely cutting out these titles and pasting them in books in any desirable order4 #ehf 99,english ,cyclopdia1 ,arts and ,sciences991 vol4 v4 p4 #bcc4 ,it will hardly be a digression to point out the enormous saving of labour1 or1 what comes to the same thing1 the enormous increase in our available knowledge1 both literary and scientific1 which arises from the formation of extensive indices4 ,the 8,state ,papers10 containing the whole history of the nation1 were practically sealed to literary inquirers until the ,government undertook the task of calendaring and indexing them4 ,the ,british ,museum ,catalogue is another national work1 of which the importance in advancing knowledge cannot be overrated4 ,the ,royal ,society is doing great service in publishing a complete catalogue of memoirs upon physical science4 ,the time will perhaps come when our views upon this subject will be extended1 and either ,government or some public society will undertake the systematic cataloguing and indexing of masses of historical and scientific information which are now almost closed against inquiry4 99,classification in the ,biological ,sciences499 ,the great generalisations established in the works of ,herbert ,spencer and ,charles ,darwin have thrown much light upon other sciences1 and have removed several difficulties out of the way of the logician4 ,the subject of classification has long been studied in almost exclusive reference to the arrangement of animals and plants4 ,systematic botany and zoology have been commonly known as the ,classificatory ,sciences1 and scientific men seemed to suppose that the methods of arrangement1 which were suitable for living creatures1 must be the best for all other classes of objects4 ,several mineralogists1 especially ,mohs1 have attempted to arrange minerals in genera and species1 just as if they had been animals capable of reproducing their kind with variations4 ,this confusion of ideas between the relationship of living forms and the logical relationship of things in general prevailed from the earliest times1 as manifested in the etymology of words4 ,we familiarly speak of a 99kind99 of things meaning a class of things1 and the kind consists of those things which are 99akin991 or come of the same race4 ,when ,socrates and his followers wanted a name for a class regarded in a philosophical light1 they adopted the analogy in question1 and called it a 1 or race1 the root - being connected with the notion of generation4 ,so long as species of plants and animals were believed to proceed from distinct acts of ,creation1 there was no apparent reason why methods of classification suitable to them should not be treated as a guide to the classification of other objects generally4 ,but when once we regard these resemblances as hereditary in their origin1 we see that the sciences of systematic botany and zoology have a special character of their own4 ,there is no reason to suppose that the same kind of natural classification which is best in biology will apply also in mineralogy1 in chemistry1 or in astronomy4 ,the logical principles which underlie all classification are of course the same in natural history as in the sciences of lifeless matter1 but the special resemblances which arise from the relation of parent and offspring will not be found to prevail between different kinds of crystals or mineral bodies4 ,the genealogical view of the relations of animals and plants leads us to discard all notions of a regular progression of living forms1 or any theory as to their symmetrical relations4 ,it was at one time a question whether the ultimate scheme of natural classification would lead to arrangement in a simple line1 or a circle1 or a combination of circles4 ,macleay's once celebrated system was a circular one1 and each class-circle was composed of five order-circles1 each of which was composed again of five tribe-circles1 and so on1 the subdivision being at each step into five minor circles4 ,macleay held that in the animal kingdom there are five sub-kingdoms--the ,vertebrata1 ,annulosa1 ,radiata1 ,acrita1 and ,mollusca4 ,each of these was again divided into five--the ,vertebrata1 consisting of ,mammalia1 ,reptilia1 ,pisces1 ,amphibia1 and ,aves4#ehg ,it is evident that in such a symmetrical system the animals were made to suit themselves to the classes instead of the classes being suited to the animals4 #ehg ,swainson1 8,treatise on the ,geography and ,classification of ,animals10 99,cabinet ,cyclopdia991 p4 #bja4 ,we now perceive that the ultimate system will have the form of an immensely extended genealogical tree1 which will be capable of representation by lines on a plane surface of sufficient extent4 ,strictly speaking1 this genealogical tree ought to represent the descent of each individual living form now existing or which has existed4 ,it should be as personal and minute in its detail of relations1 as the ,stemma of the ,kings of ,england4 ,we must not assume that any two forms are exactly alike1 and in any case they are numerically distinct4 ,every parent then must be represented at the apex of a series of divergent lines1 representing the generation of so many children4 ,any complete system of classification must regard individuals as the infim species4 ,but as in the lower races of animals and plants the differences between individuals are slight and apparently unimportant1 while the numbers of such individuals are immensely great1 beyond all possibility of separate treatment1 scientific men have always stopped at some convenient but arbitrary point1 and have assumed that forms so closely resembling each other as to present no constant difference were all of one kind4 ,they have1 in short1 fixed their attention entirely upon the main features of family difference4 ,in the genealogical tree which they have been unconsciously aiming to construct1 diverging lines meant races diverging in character1 and the purpose of all efforts at so-called natural classification was to trace out the descents between existing groups of plants or animals4 ,now it is evident that hereditary descent may have in different cases produced very different results as regards the problem of classification4 ,in some cases the differentiation of characters may have been very frequent1 and specimens of all the characters produced may have been transmitted to the present time4 ,a living form will then have1 as it were1 an almost infinite number of cousins of various degrees1 and there will be an immense number of forms finely graduated in their resemblances4 ,exact and distinct classification will then be almost impossible1 and the wisest course will be not to attempt arbitrarily to distinguish forms closely related in nature1 but to allow that there exist transitional forms of every degree1 to mark out if possible the extreme limits of the family relationship1 and perhaps to select the most generalised form1 or that which presents the greatest number of close resemblances to others of the family1 as the 99type99 of the whole4 ,mr4 ,darwin1 in his most interesting work upon ,orchids1 points out that the tribe of ,malaxe are distinguished from ,epidendre by the absence of a caudicle to the pollinia2 but as some of the ,malaxe have a minute caudicle1 the division really breaks down in the most essential point4 8,this is a misfortune10 he remarks1#ehh 8which every naturalist encounters in attempting to classify a largely developed or so-called natural group1 in which1 relatively to other groups1 there has been little extinction4 ,in order that the naturalist may be enabled to give precise and clear definitions of his divisions1 whole ranks of intermediate or gradational forms must have been utterly swept away3 if here and there a member of the intermediate ranks has escaped annihilation1 it puts an effectual bar to any absolutely distinct definition40 #ehh ,darwin1 99,fertilisation of ,orchids991 p4 #aei4 ,in other cases a particular plant or animal may perhaps have transmitted its form from generation to generation almost unchanged1 or1 what comes to the same result1 those forms which diverged in character from the parent stock may have proved unsuitable to their circumstances1 and perished4 ,we shall then find a particular form standing apart from all others1 and marked by many distinct characters4 ,occasionally we may meet with specimens of a race which was formerly far more common but is now undergoing extinction1 and is nearly the last of its kind4 ,thus we explain the occurrence of exceptional forms such as are found in the ,amphioxus4 ,the ,equisetace perplex botanists by their want of affinity to other orders of ,acrogenous plants4 ,this doubtless indicates that their genealogical connection with other plants must be sought for in the most distant ages of geological development4 ,constancy of character1 as ,mr4 ,darwin has said1#ehi is what is chiefly valued and sought after by naturalists2 that is to say1 naturalists wish to find some distinct family mark1 or group of characters1 by which they may clearly recognise the relationship of descent between a large group of living forms4 ,it is accordingly a great relief to the mind of the naturalist when he comes upon a definitely marked group1 such as the ,diatomace1 which are clearly separated from their nearest neighbours the ,desmidiace by their siliceous framework and the absence of chlorophyll4 ,but we must no longer think that because we fail in detecting constancy of character the fault is in our classificatory sciences4 ,where gradation of character really exists1 we must devote ourselves to defining and registering the degrees and limits of that gradation4 ,the ultimate natural arrangement will often be devoid of strong lines of demarcation4 #ehi 99,descent of ,man991 vol4 i4 p4 #bad4 ,let naturalists1 too1 form their systems of natural classification with all care they can1 yet it will certainly happen from time to time that new and exceptional forms of animals or vegetables will be discovered and will require the modification of the system4 ,a natural system is directed1 as we have seen1 to the discovery of empirical laws of correlation1 but these laws being purely empirical will frequently be falsified by more extensive investigation4 ,from time to time the notions of naturalists have been greatly widened1 especially in the case of ,australian animals and plants1 by the discovery of unexpected combinations of organs1 and such events must often happen in the future4 ,if indeed the time shall come when all the forms of plants are discovered and accurately described1 the science of ,systematic ,botany will then be placed in a new and more favourable position1 as remarked by ,alphonse ,decandolle4#eij #eij 99,laws of ,botanical ,nomenclature991 p4 #af4 ,it ought to be remembered that though the genealogical classification of plants or animals is doubtless the most instructive of all1 it is not necessarily the best for all purposes4 ,there may be correlations of properties important for medicinal1 or other practical purposes1 which do not correspond to the correlations of descent4 ,we must regard the bamboo as a tree rather than a grass1 although it is botanically a grass4 ,for legal purposes we may continue with advantage to treat the whale1 seal1 and other cetace1 as fish4 ,we must also class plants according as they belong to arctic1 alpine1 temperate1 sub-tropical or tropical regions4 ,there are causes of likeness apart from hereditary relationship1 and 99we must not attribute exclusive excellence to any one method of classification994 99,classification by ,types499 ,perplexed by the difficulties arising in natural history from the discovery of intermediate forms1 naturalists have resorted to what they call classification by types4 ,instead of forming one distinct class defined by the invariable possession of certain assigned properties1 and rigidly including or excluding objects according as they do or do not possess all these properties1 naturalists select a typical specimen1 and they group around it all other specimens which resemble this type more than any other selected type4 8,the type of each genus10 we are told1#eia 8should be that species in which the characters of its group are best exhibited and most evenly balanced40 ,it would usually consist of those descendants of a form which had undergone little alteration1 while other descendants had suffered slight differentiation in various directions4 #eia ,waterhouse1 quoted by ,woodward in his 99,rudimentary ,treatise of ,recent and ,fossil ,shells991 p4 #fa4 ,it would be a great mistake to suppose that this classification by types is a logically distinct method4 ,it is either not a real method of classification at all1 or it is merely an abbreviated mode of representing a complicated system of arrangement4 ,a class must be defined by the invariable presence of certain common properties4 ,if1 then1 we include an individual in which one of these properties does not appear1 we either fall into logical contradiction1 or else we form a new class with a new definition4 ,even a single exception constitutes a new class by itself1 and by calling it an exception we merely imply that this new class closely resembles that from which it diverges in one or two points only4 ,thus in the definition of the natural order of ,rosace1 we find that the seeds are one or two in each carpel1 but that in the genus ,spira there are three or four2 this must mean either that the number of seeds is not a part of the fixed definition of the class1 or else that ,spira does not belong to that class1 though it may closely approximate to it4 ,naturalists continually find themselves between two horns of a dilemma2 if they restrict the number of marks specified in a definition so that every form intended to come within the class shall possess all those marks1 it will then be usually found to include too many forms2 if the definition be made more particular1 the result is to produce so-called anomalous genera1 which1 while they are held to belong to the class1 do not in all respects conform to its definition4 ,the practice has hence arisen of allowing considerable latitude in the definition of natural orders4 ,the family of ,crucifer1 for instance1 forms an exceedingly well-marked natural order1 and among its characters we find it specified that the fruit is a pod1 divided into two cells by a thin partition1 from which the valves generally separate at maturity2 but we are also informed that1 in a few genera1 the pod is one-celled1 or indehiscent1 or separates transversely into several joints4#eib ,now this must either mean that the formation of the pod is not an essential point in the definition of the family1 or that there are several closely associated families4 #eib ,bentham's 99,handbook of the ,british ,flora99 7#ahff71 p4 #be4 ,the same holds true of typical classification4 ,the type itself is an individual1 not a class1 and no other object can be exactly like the type4 ,but as soon as we abstract the individual peculiarities of the type and thus specify a finite number of qualities in which other objects may resemble the type1 we immediately constitute a class4 ,if some objects resemble the type in some points1 and others in other points1 then each definite collection of points of resemblance constitutes intensively a separate class4 ,the very notion of classification by types is in fact erroneous in a logical point of view4 ,the naturalist is constantly occupied in endeavouring to mark out definite groups of living forms1 where the forms themselves do not in many cases admit of such rigorous lines of demarcation4 ,a certain laxity of logical method is thus apt to creep in1 the only remedy for which will be the frank recognition of the fact1 that1 according to the theory of hereditary descent1 gradation of characters is probably the rule1 and precise demarcation between groups the exception4 99,natural ,genera and ,species499 ,one important result of the establishment of the theory of evolution is to explode all notions about natural groups constituting separate creations4 ,naturalists long held that every plant belongs to some species1 marked out by invariable characters1 which do not change by difference of soil1 climate1 cross-breeding1 or other circumstances4 ,they were unable to deny the existence of such things as sub-species1 varieties1 and hybrids1 so that a species of plants was often subdivided and classified within itself4 ,but then the differences upon which this sub-classification depended were supposed to be variable1 and thus distinguished from the invariable characters imposed upon the whole species at its creation4 ,similarly a natural genus was a group of species1 and was marked out from other genera by eternal differences of still greater importance4 ,we now1 however1 perceive that the existence of any such groups as genera and species is an arbitrary creation of the naturalist's mind4 ,all resemblances of plants are natural so far as they express hereditary affinities2 but this applies as well to the variations within the species as to the species itself1 or to the larger groups4 ,all is a matter of degree4 ,the deeper differences between plants have been produced by the differentiating action of circumstances during millions of years1 so that it would naturally require millions of years to undo this result1 and prove experimentally that the forms can be approximated again4 ,sub-species may sometimes have arisen within historical times1 and varieties approaching to sub-species may often be produced by the horticulturist in a few years4 ,such varieties can easily be brought back to their original forms1 or1 if placed in the original circumstances1 will themselves revert to those forms2 but according to ,darwin's views all forms are capable of unlimited change1 and it might possibly be1 unlimited reversion if suitable circumstances and sufficient time be granted4 ,many fruitless attempts have been made to establish a rigorous criterion of specific and generic difference1 so that these classes might have a definite value and rank in all branches of biology4 ,linnus adopted the view that the species was to be defined as a distinct creation1 saying1#eic 8,species tot numeramus1 quot divers form in principio sunt creat20 or again1 8,species tot sunt1 quot diversas formas ab initio produxit ,infinitum ,ens2 qu form1 secundum generationis inditas leges1 produxere plures1 at sibi semper similes40 ,of genera he also says1#eid 8,genus omne est naturale1 in primordio tale creatum40 ,it was a common doctrine added to and essential to that of distinct creation that these species could not produce intermediate and variable forms1 so that we find ,linnus obliged by the ascertained existence of hybrids to take a different view in another work2 he says1#eie 8,novas species immo et genera ex copula diversarum specierum in regno vegetabilium oriri primo intuitu paradoxum videtur2 interim observationes sic fieri non ita dissuadent40 ,even supposing in the present day that we could assent to the notion of a certain number of distinct creational acts1 this notion would not help us in the theory of classification4 ,naturalists have never pointed out any method of deciding what are the results of distinct creations1 and what are not4 ,as ,darwin says1#eif 8the definition must not include an element which cannot possibly be ascertained1 such as an act of creation40 ,it is1 in fact1 by investigation of forms and classification that we should ascertain what were distinct creations and what were not2 this information would be a result and not a means of classification4 #eic 99,philosophia ,botanica99 7#aggj71 #aeg1 p4 #ii4 #eid 99,ibid499 #aei1 p4 #ajj4 #eie 99,amnitates ,academic99 7#agdd71 vol4 i4 p4 #gj4 ,quoted in 99,edinburgh ,review991 ,october #ahfh1 vol4 cxxviii4 pp4 #daf1 #dag4 #eif 99,descent of ,man991 vol4 i4 p4 #bbh4 ,agassiz seemed to consider that he had discovered an important principle1 to the effect that general plan or structure is the true ground for the discrimination of the great classes of animals1 which may be called branches of the animal kingdom4#eig ,he also thought that genera are definite and natural groups4 8,genera10 he says1#eih 8are most closely allied groups of animals1 differing neither in form1 nor in complication of structure1 but simply in the ultimate structural peculiarities of some of their parts2 and this is1 ,i believe1 the best definition which can be given of genera40 ,but it is surely apparent that there are endless degrees both of structural peculiarity and of complication of structure4 ,it is impossible to define the amount of structural peculiarity which constitutes the genus as distinguished from the species4 #eig ,agassiz1 99,essay on ,classification991 p4 #bai4 #eih 99,ibid499 p4 #bdi4 ,the form which any classification of plants or animals tends to take is that of an unlimited series of subaltern classes4 ,originally botanists confined themselves for the most part to a small number of such classes4 ,linnus adopted ,class1 ,order1 ,genus1 ,species1 and ,variety1 and even seemed to think that there was something essentially natural in a five-fold arrangement of groups4#eii #eii 99,philosophia ,botanica991 #aee1 p4 #ih4 ,with the progress of botany intermediate and additional groups have gradually been introduced4 ,according to the ,laws of ,botanical ,nomenclature adopted by the ,international ,botanical ,congress1 held at ,paris#fjj in ,august #ahfg1 no less than twenty-one names of classes are recognised--namely1 ,kingdom1 ,division1 ,sub-division1 ,class1 ,sub-class1 ,cohort1 ,sub-cohort1 ,order1 ,sub-order1 ,tribe1 ,sub-tribe1 ,genus1 ,sub-genus1 ,section1 ,sub-section1 ,species1 ,sub-species1 ,variety1 ,sub-variety1 ,variation1 ,sub-variation4 ,it is allowed by the authors of this scheme1 that the rank or degree of importance to be attributed to any of these divisions may vary in a certain degree according to individual opinion4 ,the only point on which botanists are not allowed discretion is as to the order of the successive sub-divisions2 any inversion of the arrangement1 such as division of a genus into tribes1 or of a tribe into orders1 is quite inadmissible4 ,there is no reason to suppose that even the above list is complete and inextensible4 ,the ,botanical ,congress itself recognised the distinction between variations according as they are ,seedlings1 ,half-breeds1 or 99,lusus ,natur994 ,the complication of the inferior classes is increased again by the existence of 99hybrids991 arising from the fertilisation of one species by another deemed a distinct species1 nor can we place any limit to the minuteness of discrimination of degrees of breeding short of an actual pedigree of individuals4 #fjj 99,laws of ,botanical ,nomenclature991 by ,alphonse ,decandolle1 translated from the ,french1 #ahfh1 p4 #ai4 ,it will be evident to the reader that in the remarks upon classification as applied to the ,natural ,sciences1 given in this and the preceding sections1 ,i have not in the least attempted to treat the subject in a manner adequate to its extent and importance4 ,a volume would be insufficient for tracing out the principles of scientific method specially applicable to these branches of science4 ,what more ,i may be able to say upon the subject will be better said1 if ever1 when ,i am able to take up the closely-connected subjects of ,scientific ,nomenclature1 ,terminology1 and ,descriptive ,representation4 ,in the meantime1 ,i have wished to show1 in a negative point of view1 that natural classification in the animal and vegetable kingdoms is a special problem1 and that the particular methods and difficulties to which it gives rise are not those common to all cases of classification1 as so many physicists have supposed4 ,genealogical resemblances are only a special case of resemblances in general4 99,unique or ,exceptional ,objects499 ,in framing a system of classification in almost any branch of science1 we must expect to meet with unique or peculiar objects1 which stand alone1 having comparatively few analogies with other objects4 ,they may also be said to be 99sui generis991 each unique object forming1 as it were1 a genus by itself2 or they are called 99nondescript991 because from thus standing apart it is difficult to find terms in which to describe their properties4 ,the rings of ,saturn1 for instance1 form a unique object among the celestial bodies4 ,we have indeed considered this and many other instances of unique objects in the preceding chapter on ,exceptional ,phenomena4 ,apparent1 ,singular1 and ,divergent ,exceptions especially1 are analogous to unique objects4 ,in the classification of the elements1 ,carbon stands apart as a substance entirely unique in its powers of producing compounds4 ,it is considered to be a quadrivalent element1 and it obeys all the ordinary laws of chemical combination4 ,yet it manifests powers of affinity in such an exalted degree that the substances in which it appears are more numerous than all the other compounds known to chemists4 ,almost the whole of the substances which have been called organic contain carbon1 and are probably held together by the carbon atoms1 so that many chemists are now inclined to abandon the name ,organic ,chemistry1 and substitute the name ,chemistry of the ,carbon ,compounds4 ,it used to be believed that the production of organic compounds could be effected only by the action of vital force1 or of some inexplicable cause involved in the phenomena of life2 but it is now found that chemists are able to commence with the elementary materials1 pure carbon1 hydrogen1 and oxygen1 and by strictly chemical operations to combine these so as to form complicated organic compounds4 ,so many substances have already been formed that we might be inclined to generalise and infer that all organic compounds might ultimately be produced without the agency of living beings4 ,thus the distinction between the organic and the inorganic kingdoms seems to be breaking down1 but our wonder at the peculiar powers of carbon must increase at the same time4 ,in considering generalisation1 the law of continuity was applied chiefly to physical properties capable of mathematical treatment4 ,but in the classificatory sciences1 also1 the same important principle is often beautifully exemplified4 ,many objects or events seem to be entirely exceptional and abnormal1 and in regard to degree or magnitude they may be so termed2 but it is often easy to show that they are connected by intermediate links with ordinary cases4 ,in the organic kingdoms there is a common groundwork of similarity running through all classes1 but particular actions and processes present themselves conspicuously in particular families and classes4 ,tenacity of life is most marked in the ,rotifera1 and some other kinds of microscopic organisms1 which can be dried and boiled without loss of life4 ,reptiles are distinguished by torpidity1 and the length of time they can live without food4 ,birds1 on the contrary1 exhibit ceaseless activity and high muscular power4 ,the ant is as conspicuous for intelligence and size of brain among insects as the quadrumana and man among vertebrata4 ,among plants the ,leguminos are distinguished by a tendency to sleep1 folding their leaves at the approach of night4 ,in the genus ,mimosa1 especially the ,mimosa pudica1 commonly called the sensitive plant1 the same tendency is magnified into an extreme irritability1 almost resembling voluntary motion4 ,more or less of the same irritability probably belongs to vegetable forms of every kind1 but it is of course to be investigated with special ease in such an extreme case4 ,in the ,gymnotus and ,torpedo1 we find that organic structures can act like galvanic batteries4 ,are we to suppose that such animals are entirely anomalous exceptions2 or may we not justly expect to find less intense manifestations of electric action in all animals8 ,some extraordinary differences between the modes of reproduction of animals have been shown to be far less than was at first sight apparent4 ,the lower animals seem to differ entirely from the higher ones in the power of reproducing lost limbs4 ,a kind of crab has the habit of casting portions of its claws when much frightened1 but they soon grow again4 ,there are multitudes of smaller animals which1 like the ,hydra1 may be cut in two and yet live and develop into new complete individuals4 ,no mammalian animal can reproduce a limb1 and in appearance there is no analogy4 ,but it was suggested by ,blumenbach that the healing of a wound in the higher animals really represents in a lower degree the power of reproducing a limb4 ,that this is true may be shown by adducing a multitude of intermediate cases1 each adjoining pair of which are clearly analogous1 so that we pass gradually from one extreme to the other4 ,darwin holds1 moreover1 that any such restoration of parts is closely connected with that perpetual replacement of the particles which causes every organised body to be after a time entirely new as regards its constituent substance4 ,in short1 we approach to a great generalisation under which all the phenomena of growth1 restoration1 and maintenance of organs are effects of one and the same power4#fja ,it is perhaps still more surprising to find that the complicated process of reproduction in the higher animals may be gradually traced down to a simpler and simpler form1 which at last becomes undistinguishable from the budding out of one plant from the stem of another4 ,by a great generalisation we may regard all the modes of reproduction of organic life as alike in their nature1 and varying only in complexity of development4#fjb #fja ,darwin1 99,the ,variation of ,animals and ,plants991 vol4 ii4 pp4 #bic1 #cei1 c42 quoting ,paget1 99,lectures on ,pathology991 #ahec1 pp4 #aeb1 #afd4 #fjb 99,ibid499 vol4 ii4 p4 #cgb4 99,limits of ,classification499 ,science can extend only so far as the power of accurate classification extends4 ,if we cannot detect resemblances1 and assign their exact character and amount1 we cannot have that generalised knowledge which constitutes science2 we cannot infer from case to case4 ,classification is the opposite process to discrimination4 ,if we feel that two tastes differ1 the tastes of two kinds of wine for instance1 the mere fact of difference existing prevents inference4 ,the detection of the difference saves us1 indeed1 from false inference1 because so far as difference exists1 inference is impossible4 ,but classification consists in detecting resemblances of all degrees of generality1 and ascertaining exactly how far such resemblances extend1 while assigning precisely the points at which difference begins4 ,it enables us1 then1 to generalise1 and make inferences where it is possible1 and it saves us at the same time from going too far4 ,a full classification constitutes a complete record of all our knowledge of the objects or events classified1 and the limits of exact knowledge are identical with the limits of classification4 ,it must by no means be supposed that every group of natural objects will be found capable of rigorous classification4 ,there may be substances which vary by insensible degrees1 consisting1 for instance1 in varying mixtures of simpler substances4 ,granite is a mixture of quartz1 felspar1 and mica1 but there are hardly two specimens in which the proportions of these three constituents are alike1 and it would be impossible to lay down definitions of distinct species of granite without finding an infinite variety of intermediate species4 ,the only true classification of granites1 then1 would be founded on the proportions of the constituents present1 and a chemical or microscopic analysis would be requisite1 in order that we might assign a specimen to its true position in the series4 ,granites vary1 again1 by insensible degrees1 as regards the magnitude of the crystals of felspar and mica4 ,precisely similar remarks might be made concerning the classification of other plutonic rocks1 such as syenite1 basalt1 pumice-stone1 lava4 ,the nature of a ray of homogeneous light is strictly defined1 either by its place in the spectrum or by the corresponding wave-length1 but a ray of mixed light admits of no simple classification2 any of the infinitely numerous rays of the continuous spectrum may be present or absent1 or present in various intensities1 so that we can only class and define a mixed colour by defining the intensity and wave-length of each ray of homogeneous light which is present in it4 ,complete spectroscopic analysis and the determination of the intensity of every part of the spectrum yielded by a mixed ray is requisite for its accurate classification4 ,nearly the same may be said of complex sounds4 ,a simple sound undulation1 if we could meet with such a sound1 would admit of precise and exhaustive classification as regards pitch1 the length of wave1 or the number of waves reaching the ear per second being a sufficient criterion4 ,but almost all ordinary sounds1 even those of musical instruments1 consist of complex aggregates of undulations of different pitches1 and in order to classify the sound we should have to measure the intensities of each of the constituent sounds1 a work which has been partially accomplished by ,helmholtz1 as regards the vowel sounds4 ,the different tones of voice distinctive of different individuals must also be due to the intermixture of minute waves of various pitch1 which are yet quite beyond the range of experimental investigation4 ,we cannot1 then1 at present attempt to classify the different kinds or 99timbres99 of sound4 ,the difficulties of classification are still greater when a varying phenomenon cannot be shown to be a mixture of simpler phenomena4 ,if we attempt to classify tastes1 we may rudely group them according as they are sweet1 bitter1 saline1 alkaline1 acid1 astringent or fiery2 but it is evident that these groups are bounded by no sharp lines of definition4 ,tastes of mixed or intermediate character may exist almost 99ad infinitum991 and what is still more troublesome1 the tastes clearly united within one class may differ more or less from each other1 without our being able to arrange them in subordinate genera and species4 ,the same remarks may be made concerning the classification of odours1 which may be roughly grouped according to the arrangement of ,linnus as1 aromatic1 fragrant1 ambrosiac1 alliaceous1 fetid1 virulent1 nauseous4 ,within each of these vague classes1 however1 there would be infinite shades of variety1 and each class would graduate into other classes4 ,the odours which can be discriminated by an acute nose are infinite2 every rock1 stone1 plant1 or animal has some slight smell1 and it is well known that dogs1 or even blind men1 can discriminate persons by a slight distinctive odour which usually passes unnoticed4 ,similar remarks may be made concerning the feelings of the human mind1 called emotions4 ,we know what is anger1 grief1 fear1 hatred1 love2 and many systems for classifying these feelings have been proposed4 ,they may be roughly distinguished according as they are pleasurable or painful1 prospective or retrospective1 selfish or sympathetic1 active or passive1 and possibly in many other ways2 but each mode of arrangement will be indefinite and unsatisfactory when followed into details4 ,as a general rule1 the emotional state of the mind at any moment will be neither pure anger nor pure fear1 nor any one pure feeling1 but an indefinite and complex aggregate of feelings4 ,it may be that the state of mind is really a sum of several distinct modes of agitation1 just as a mixed colour is the sum of the several rays of the spectrum4 ,in this case there may be more hope of some method of analysis being successfully applied at a future time4 ,but it may be found that states of mind really graduate into each other so that rigorous classification would be hopeless4 ,a little reflection will show that there are whole worlds of existences which in like manner are incapable of logical analysis and classification4 ,one friend may be able to single out and identify another friend by his countenance among a million other countenances4 ,faces are capable of infinite discrimination1 but who shall classify and define them1 or say by what particular shades of feature he does judge8 ,there are of course certain distinct types of face1 but each type is connected with each other type by infinite intermediate specimens4 ,we may classify melodies according to the major or minor key1 the character of the time1 and some other distinct points2 but every melody has1 independently of such circumstances1 its own distinctive character and effect upon the mind4 ,we can detect differences between the styles of literary1 musical1 or artistic compositions4 ,we can even in some cases assign a picture to its painter1 or a symphony to its composer1 by a subtle feeling of resemblances or differences which may be felt1 but cannot be described4 ,finally1 it is apparent that in human character there is unfathomable and inexhaustible diversity4 ,every mind is more or less like every other mind2 there is always a basis of similarity1 but there is a superstructure of feelings1 impulses1 and motives which is distinctive for each person4 ,we can sometimes predict the general character of the feelings and actions which will be produced by a given external event in an individual well known to us2 but we also know that we are often inexplicably at fault in our inferences4 ,no one can safely generalise upon the subtle variations of temper and emotion which may arise even in a person of ordinary character4 ,as human knowledge and civilisation progress1 these characteristic differences tend to develop and multiply themselves1 rather than decrease4 ,character grows more many-sided4 ,two well educated ,englishmen are far better distinguished from each other than two common labourers1 and these are better distinguished than two ,australian aborigines4 ,the complexities of existing phenomena probably develop themselves more rapidly than scientific method can overtake them4 ,in spite of all the boasted powers of science1 we cannot really apply scientific method to our own minds and characters1 which are more important to us than all the stars and nebul4 ,,book ,,vi4 ,,chapter ,,xxxi4 ,,reflections ,,on ,,the ,,results ,,and ,,limits ,,of ,,scientific ,,method4 ,before concluding a work on the ,principles of ,science1 it will not be inappropriate to add some remarks upon the limits and ultimate bearings of the knowledge which we may acquire by the employment of scientific method4 ,all science consists1 it has several times been stated1 in the detection of identities in the action of natural agents4 ,the purpose of inductive inquiry is to ascertain the apparent existence of necessary connection between causes and effects1 expressed in the form of natural laws4 ,now so far as we thus learn the invariable course of nature1 the future becomes the necessary sequel of the present1 and we are brought beneath the sway of powers with which nothing can interfere4 ,by degrees it is found1 too1 that the chemistry of organised substances is not entirely separated from1 but is continuous with1 that of earth and stones4 ,life seems to be nothing but a special form of energy which is manifested in heat and electricity and mechanical force4 ,the time may come1 it almost seems1 when the tender mechanism of the brain will be traced out1 and every thought reduced to the expenditure of a determinate weight of nitrogen and phosphorus4 ,no apparent limit exists to the success of scientific method in weighing and measuring1 and reducing beneath the sway of law1 the phenomena both of matter and of mind4 ,and if mental phenomena be thus capable of treatment by the balance and the micrometer1 can we any longer hold that mind is distinct from matter8 ,must not the same inexorable reign of law which is apparent in the motions of brute matter be extended to the subtle feelings of the human heart8 ,are not plants and animals1 and ultimately man himself1 merely crystals1 as it were1 of a complicated form8 ,if so1 our boasted free will becomes a delusion1 moral responsibility a fiction1 spirit a mere name for the more curious manifestations of material energy4 ,all that happens1 whether right or wrong1 pleasurable or painful1 is but the outcome of the necessary relations of time and space and force4 ,materialism seems1 then1 to be the coming religion1 and resignation to the nonentity of human will the only duty4 ,such may not generally be the reflections of men of science1 but ,i believe that we may thus describe the secret feelings of fear which the constant advance of scientific investigation excites in the minds of many4 ,is science1 then1 essentially atheistic and materialistic in its tendency8 ,does the uniform action of material causes1 which we learn with an ever-increasing approximation to certainty1 preclude the hypothesis of a benevolent ,creator1 who has not only designed the existing universe1 but who still retains the power to alter its course from time to time8 ,to enter upon actual theological discussions would be evidently beyond the scope of this work4 ,it is with the scientific method common to all the sciences1 and not with any of the separate sciences1 that we are concerned4 ,theology therefore would be at least as much beyond my scope as chemistry or geology4 ,but ,i believe that grave misapprehensions exist as regards the very nature of scientific method4 ,there are scientific men who assert that the interposition of ,providence is impossible1 and prayer an absurdity1 because the laws of nature are inductively proved to be invariable4 ,inferences are drawn not so much from particular sciences as from the logical nature of science itself1 to negative the impulses and hopes of men4 ,now ,i may state that my own studies in logic lead me to call in question such negative inferences4 ,laws of nature are uniformities observed to exist in the action of certain material agents1 but it is logically impossible to show that all other agents must behave as these do4 ,the too exclusive study of particular branches of physical science seems to generate an over-confident and dogmatic spirit4 ,rejoicing in the success with which a few groups of facts are brought beneath the apparent sway of laws1 the investigator hastily assumes that he is close upon the ultimate springs of being4 ,a particle of gelatinous matter is found to obey the ordinary laws of chemistry2 yet it moves and lives4 ,the world is therefore asked to believe that chemistry can resolve the mysteries of existence4 99,the ,meaning of ,natural ,law499 ,pindar speaks of ,law as the ,ruler of the ,mortals and the ,immortals1 and it seems to be commonly supposed that the so-called ,laws of ,nature1 in like manner1 rule man and his ,creator4 ,the course of nature is regarded as being determined by invariable principles of mechanics which have acted since the world began1 and will act for evermore4 ,even if the origin of all things is attributed to an intelligent creative mind1 that ,being is regarded as having yielded up arbitrary power1 and as being subject like a human legislator to the laws which he has himself enacted4 ,such notions ,i should describe as superficial and erroneous1 being derived1 as ,i think1 from false views of the nature of scientific inference1 and the degree of certainty of the knowledge which we acquire by inductive investigation4 ,a law of nature1 as ,i regard the meaning of the expression1 is not a uniformity which must be obeyed by all objects1 but merely a uniformity which is as a matter of fact obeyed by those objects which have come beneath our observation4 ,there is nothing whatever incompatible with logic in the discovery of objects which should prove exceptions to any law of nature4 ,perhaps the best established law is that which asserts an invariable correlation to exist between gravity and inertia1 so that all gravitating bodies are found to possess inertia1 and all bodies possessing inertia are found to gravitate4 ,but it would be no reproach to our scientific method1 if something were ultimately discovered to possess gravity without inertia4 ,strictly defined and correctly interpreted1 the law itself would acknowledge the possibility2 for with the statement of every law we ought properly to join an estimate of the number of instances in which it has been observed to hold true1 and the probability thence calculated1 that it will hold true in the next case4 ,now1 as we found 7p4 #bei71 no finite number of instances can warrant us in expecting with certainty that the next instance will be of like nature2 in the formulas yielded by the inverse method of probabilities a unit always appears to represent the probability that our inference will be mistaken4 ,i demur to the assumption that there is any necessary truth even in such fundamental laws of nature as the ,indestructibility of ,matter1 the ,conservation of ,energy1 or the ,laws of ,motion4 ,certain it is that men of science have recognised the conceivability of other laws1 and even investigated their mathematical consequences4 ,airy investigated the mathematical conditions of a perpetual motion 7p4 #bbc71 and ,laplace and ,newton discussed imaginary laws of forces inconsistent with those observed to operate in the universe 7pp4 #fdb1 #gjf74 ,the laws of nature1 as ,i venture to regard them1 are simply general propositions concerning the correlation of properties which have been observed to hold true of bodies hitherto observed4 ,on the assumption that our experience is of adequate extent1 and that no arbitrary interference takes place1 we are then able to assign the probability1 always less than certainty1 that the next object of the same apparent nature will conform to the same laws4 99,infiniteness of the ,universe499 ,we may safely accept as a satisfactory scientific hypothesis the doctrine so grandly put forth by ,laplace1 who asserted that a perfect knowledge of the universe1 as it existed at any given moment1 would give a perfect knowledge of what was to happen thenceforth and for ever after4 ,scientific inference is impossible1 unless we may regard the present as the outcome of what is past1 and the cause of what is to come4 ,to the view of perfect intelligence nothing is uncertain4 ,the astronomer can calculate the positions of the heavenly bodies when thousands of generations of men shall have passed away1 and in this fact we have some illustration1 as ,laplace remarks1 of the power which scientific prescience may attain4 ,doubtless1 too1 all efforts in the investigation of nature tend to bring us nearer to the possession of that ideally perfect power of intelligence4 ,nevertheless1 as ,laplace with profound wisdom adds1#fjc we must ever remain at an infinite distance from the goal of our aspirations4 #fjc 99,thorie ,analytique des ,probabilits991 quoted by ,babbage1 99,ninth ,bridgewater ,treatise991 p4 #agc4 ,let us assume1 for a time at least1 as a highly probable hypothesis1 that whatever is to happen must be the outcome of what is2 there then arises the question1 ,what is8 ,now our knowledge of what exists must ever remain imperfect and fallible in two respects4 ,firstly1 we do not know all the matter that has been created1 nor the exact manner in which it has been distributed through space4 ,secondly1 assuming that we had that knowledge1 we should still be wanting in a perfect knowledge of the way in which the particles of matter will act upon each other4 ,the power of scientific prediction extends at the most to the limits of the data employed4 ,every conclusion is purely hypothetical and conditional upon the non-interference of agencies previously undetected4 ,the law of gravity asserts that every body tends to approach towards every other body1 with a certain determinate force2 but1 even supposing the law to hold true1 it does not assert that the body 99will99 approach4 ,no single law of nature can warrant us in making an absolute prediction4 ,we must know all the laws of nature and all the existing agents acting according to those laws before we can say what will happen4 ,to assume1 then1 that scientific method can take everything within its cold embrace of uniformity1 is to imply that the ,creator cannot outstrip the intelligence of his creatures1 and that the existing ,universe is not infinite in extent and complexity1 an assumption for which ,i see no logical basis whatever4 99,the ,indeterminate ,problem of ,creation499 ,a second and very serious misapprehension concerning the import of a law of nature may now be pointed out4 ,it is not uncommonly supposed that a law determines the character of the results which shall take place1 as1 for instance1 that the law of gravity determines what force of gravity shall act upon a given particle4 ,surely a little reflection must render it plain that a law by itself determines nothing4 ,it is 99law plus agents obeying law which has results991 and it is no function of law to govern or define the number and place of its own agents4 ,whether a particle of matter shall gravitate1 depends not only upon the law of ,newton1 but also upon the distribution of surrounding particles4 ,the theory of gravitation may perhaps be true throughout all time and in all parts of space1 and the ,creator may never find occasion to create those possible exceptions to it which ,i have asserted to be conceivable4 ,let this be as it may2 our science cannot certainly determine the question4 ,certain it is1 that the law of gravity does not alone determine the forces which may be brought to bear at any point of space4 ,the force of gravitation acting upon any particle depends upon the mass1 distance1 and relative position of all the other particles of matter within the bounds of space at the instant in question4 ,even assuming that all matter when once distributed through space at the ,creation was thenceforth to act in an invariable manner without subsequent interference1 yet the actual configuration of matter at any moment1 and the consequent results of the law of gravitation1 must have been entirely a matter of free choice4 ,chalmers has most distinctly pointed out that the existing 99collocations99 of the material world are as important as the laws which the objects obey4 ,he remarks that a certain class of writers entirely overlook the distinction1 and forget that mere laws without collocations would have afforded no security against a turbid and disorderly chaos4#fjd ,mill has recognised#fje the truth of ,chalmers' statement1 without drawing the proper inferences from it4 ,he says#fjf of the distribution of matter through space1 8,we can discover nothing regular in the distribution itself2 we can reduce it to no uniformity1 to no law40 ,more lately the ,duke of ,argyll in his well-known work on the 99,reign of ,law99 has drawn attention to the profound distinction between laws and collocations of causes4 #fjd 99,first ,bridgewater ,treatise99 7#ahcd71 pp4 #af-bd4 #fje 99,system of ,logic991 #eth edit4 bk4 ,,iii4 chap4 ,v4 #g2 chap4 ,,xvi4 #c4 #fjf 99,system of ,logic991 vol4 i4 p4 #chd4 ,the original conformation of the material universe1 as far as we can tell1 was free from all restriction4 ,there was unlimited space in which to frame it1 and an unlimited number of material particles1 each of which could be placed in any one of an infinite number of different positions4 ,it should be added1 that each particle might be endowed with any one of an infinite number of quantities of 99vis viva99 acting in any one of an infinite number of different directions4 ,the problem of ,creation was1 then1 what a mathematician would call 99an indeterminate problem991 and it was indeterminate in a great number of ways4 ,infinitely numerous and various universes might then have been fashioned by the various distribution of the original nebulous matter1 although all the particles of matter should obey the law of gravity4 ,lucretius tells us how in the original rain of atoms some of these little bodies diverged from the rectilinear direction1 and coming into contact with other atoms gave rise to the various combinations of substances which exist4 ,he omitted to tell us whence the atoms came1 or by what force some of them were caused to diverge2 but surely these omissions involve the whole question4 ,i accept the ,lucretian conception of creation when properly supplemented4 ,every atom which existed in any point of space must have existed there previously1 or must have been created there by a previously existing ,power4 ,when placed there it must have had a definite mass and a definite energy4 ,now1 as before remarked1 an unlimited number of atoms can be placed in unlimited space in an unlimited number of modes of distribution4 ,out of infinitely infinite choices which were open to the ,creator1 that one choice must have been made which has yielded the ,universe as it now exists4 ,it would be a mistake1 indeed1 to suppose that the law of gravity1 when it holds true1 is no restriction on the distribution of force4 ,that law is a geometrical law1 and it would in many cases be mathematically impossible1 as far as we can see1 that the force of gravity acting on one particle should be small while that on a neighbouring particle is great4 ,we cannot conceive that even ,omnipotent ,power should make the angles of a triangle greater than two right angles4 ,the primary laws of thought and the fundamental notions of the mathematical sciences do not seem to admit of error or alteration4 ,into the metaphysical origin and meaning of the apparent necessity attaching to such laws ,i have not attempted to inquire in this work1 and it is not requisite for my present purpose4 ,if the law of gravity were the only law of nature and the ,creator had chosen to render all matter obedient to that law1 there would doubtless be restrictions upon the effects derivable from any one distribution of matter4 99,hierarchy of ,natural ,laws499 ,a further consideration presents itself4 ,a natural law like that of gravity expresses a certain uniformity in the action of agents submitted to it1 and this produces1 as we have seen1 certain geometrical restrictions upon the effects which those agents may produce4 ,but there are other forces and laws besides gravity4 ,one force may override another1 and two laws may each be obeyed and may each disguise the action of the other4 ,in the intimate constitution of matter there may be hidden springs which1 while acting in accordance with their own fixed laws1 may lead to sudden and unexpected changes4 ,so at least it has been found from time to time in the past1 and so there is every reason to believe it will be found in the future4 ,to the ancients it seemed incredible that one lifeless stone could make another leap towards it4 ,a piece of iron while it obeys the magnetic force of the loadstone does not the less obey the law of gravity4 ,a plant gravitates downwards as regards every constituent cell or fibre1 and yet it persists in growing upwards4 ,life is altogether an exception to the simpler phenomena of mineral substances1 not in the sense of disproving those laws1 but in superadding forces of new and inexplicable character4 ,doubtless no law of chemistry is broken by the action of the nervous cells1 and no law of physics by the pulses of the nervous fibres1 but something requires to be added to our sciences in order that we may explain these subtle phenomena4 ,now there is absolutely nothing in science or in scientific method to warrant us in assigning a limit to this hierarchy of laws4 ,when in many undoubted cases we find law overriding law1 and at certain points in our experience producing unexpected results1 we cannot venture to affirm that we have exhausted the strange phenomena which may have been provided for in the original constitution of matter4 ,the ,universe might have been so designed that it should go for long intervals through the same round of unvaried existence1 and yet that events of exceptional character should be produced from time to time4 ,babbage showed in that most profound and eloquent work1 99,the ,ninth ,bridgewater ,treatise991 that it was theoretically possible for human artists to design a machine1 consisting of metallic wheels and levers1 which should work invariably according to a simple law of action during any finite number of steps1 and yet at a fixed moment1 however distant1 should manifest a single breach of law4 ,such an engine might go on counting1 for instance1 the natural numbers until they would reach a number requiring for its expression a hundred million digits4 8,if every letter in the volume now before the reader's eyes10 says ,babbage1#fjg 8were changed into a figure1 and if all the figures contained in a thousand such volumes were arranged in order1 the whole together would yet fall far short of the vast induction the observer would have had in favour of the truth of the law of natural numbers4444 ,yet shall the engine1 true to the prediction of its director1 after the lapse of myriads of ages1 fulfil its task1 and give that one1 the first and only exception to that time-sanctioned law4 ,what would have been the chances against the appearance of the excepted case1 immediately prior to its occurrence80 #fjg 99,ninth ,bridgewater ,treatise991 p4 #adj4 ,as ,babbage further showed1#fjh a calculating engine1 after proceeding through any required number of motions according to a first law1 may be made suddenly to suffer a change1 so that it shall then commence to calculate according to a wholly new law4 ,after giving the natural numbers for a finite time1 it might suddenly begin to give triangular1 or square1 or cube numbers1 and these changes might be conceived theoretically as occurring time after time4 ,now if such occurrences can be designed and foreseen by a human artist1 it is surely within the capacity of the ,divine ,artist to provide for analogous changes of law in the mechanism of the atom1 or the construction of the heavens4 #fjh 99,ibid499 pp4 #cd-dc4 ,physical science1 so far as its highest speculations can be trusted1 gives some indication of a change of law in the past history of the ,universe4 ,according to ,sir ,w4 ,thomson's deductions from ,fourier's 99,theory of ,heat991 we can trace down the dissipation of heat by conduction and radiation to an infinitely distant time when all things will be uniformly cold4 ,but we cannot similarly trace the heat-history of the ,universe to an infinite distance in the past4 ,for a certain negative value of the time the formul give impossible values1 indicating that there was some initial distribution of heat which could not have resulted1 according to known laws of nature1#fji from any previous distribution4#faj ,there are other cases in which a consideration of the dissipation of energy leads to the conception of a limit to the antiquity of the present order of things4#faa ,human science1 of course1 is fallible1 and some oversight or erroneous simplification in these theoretical calculations may afterwards be discovered2 but as the present state of scientific knowledge is the only ground on which erroneous inferences from the uniformity of nature and the supposed reign of law are founded1 ,i am right in appealing to the present state of science in opposition to these inferences4 ,now the theory of heat places us in the dilemma either of believing in ,creation at an assignable date in the past1 or else of supposing that some inexplicable change in the working of natural laws then took place4 ,physical science gives no countenance to the notion of infinite duration of matter in one continuous course of existence4 ,and if in time past there has been a discontinuity of law1 why may there not be a similar event awaiting the world in the future8 ,infinite ingenuity could have implanted some agency in matter so that it might never yet have made its tremendous powers manifest4 ,we have a very good theory of the conservation of energy1 but the foremost physicists do not deny that there may possibly be forms of energy1 neither kinetic nor potential1 and therefore of unknown nature4#fab #fji ,professor ,clifford1 in his most interesting lecture on 8,the ,first and ,last ,catastrophe0 799,fortnightly ,review991 ,april #ahge1 p4 #dhj1 reprint by the ,sunday ,lecture ,society1 p4 #bd71 objects that ,i have erroneously substituted 8known laws of nature0 for 8known laws of conduction of heat40 ,i quite admit the error1 without admitting all the conclusions which ,professor ,clifford proceeds to draw2 but ,i maintain the paragraph unchanged1 in order that it may be discussed in the ,preface4 #faj ,tait's 99,thermodynamics991 p4 #ch4 99,cambridge ,mathematical ,journal991 vol4 iii4 p4 #agd4 #faa ,clerk ,maxwell's 99,theory of ,heat991 p4 #bde4 #fab ,maxwell's 99,theory of ,heat991 p4 #ib4 ,we can imagine reasoning creatures dwelling in a world where the atmosphere was a mixture of oxygen and inflammable gas like the fire-damp of coal-mines4 ,if devoid of fire1 they might have lived through long ages unconscious of the tremendous forces which a single spark would call into play4 ,in the twinkling of an eye new laws might come into action1 and the poor reasoning creatures1 so confident about their knowledge of the reign of law in their world1 would have no time to speculate upon the overthrow of all their theories4 ,can we with our finite knowledge be sure that such an overthrow of our theories is impossible8 99,the ,ambiguous ,expression1 8,uniformity of ,nature4099 ,i have asserted that serious misconception arises from an erroneous interpretation of the expression ,uniformity of ,nature4 ,every law of nature is the statement of a certain uniformity observed to exist among phenomena1 and since the laws of nature are invariably obeyed1 it seems to follow that the course of nature itself is uniform1 so that we can safely judge of the future by the present4 ,this inference is supported by some of the results of physical astronomy4 ,laplace proved that the planetary system is stable1 so that no perturbation which planet produces upon planet can become so great as to cause disruption and permanent alteration of the planetary orbits4 ,a full comprehension of the law of gravity shows that all such disturbances are essentially periodic1 so that after the lapse of millions of years the planets will return to the same relative positions1 and a new cycle of disturbances will then commence4 ,as other branches of science progress1 we seem to gain assurance that no great alteration of the world's condition is to be expected4 ,conflict with a comet has long been the cause of fear1 but now it is credibly asserted that we have passed through a comet's tail without the fact being known at the time1 or manifested by any more serious a phenomenon than a slight luminosity of the sky4 ,more recently still the earth is said to have touched the comet ,biela1 and the only result was a beautiful and perfectly harmless display of meteors4 ,a decrease in the heating power of the sun seems to be the next most probable circumstance from which we might fear the extinction of life on the earth4 ,but calculations founded on reasonable physical data show that no appreciable change can be going on1 and experimental data to indicate a change are wholly wanting4 ,geological investigations show indeed that there have been extensive variations of climate in past times2 vast glaciers and icebergs have swept over the temperate regions at one time1 and tropical vegetation has flourished near the poles at another time4 ,but here again the vicissitudes of climate assume a periodic character1 so that the stability of the earth's condition does not seem to be threatened4 ,all these statements may be reasonable1 but they do not establish the ,uniformity of ,nature in the sense that extensive alterations or sudden catastrophes are impossible4 ,in the first place1 ,laplace's theory of the stability of the planetary system is of an abstract character1 as paying regard to nothing but the mutual gravitation of the planetary bodies and the sun4 ,it overlooks several physical causes of change and decay in the system which were not so well known in his day as at present1 and it also presupposes the absence of any interruption of the course of things by conflict with foreign astronomical bodies4 ,it is now acknowledged by astronomers that there are at least two ways in which the 99vis viva99 of the planets and satellites may suffer loss4 ,the friction of the tides upon the earth produces a small quantity of heat which is radiated into space1 and this loss of energy must result in a decrease of the rotational velocity1 so that ultimately the terrestrial day will become identical with the year1 just as the periods of revolution of the moon upon its axis and around the earth have already become equal4 ,secondly1 there can be little doubt that certain manifestations of electricity upon the earth's surface depend upon the relative motions of the planets and the sun1 which give rise to periods of increased intensity4 ,such electrical phenomena must result in the production and dissipation of heat1 the energy of which must be drawn1 partially at least1 from the moving bodies4 ,this effect is probably identical 7p4 #egj7 with the loss of energy of comets attributed to the so-called resisting medium4 ,but whatever be the theoretical explanation of these phenomena1 it is almost certain that there exists a tendency to the dissipation of the energy of the planetary system1 which will1 in the indefinite course of time1 result in the fall of the planets into the sun4 ,it is hardly probable1 however1 that the planetary system will be left undisturbed throughout the enormous interval of time required for the dissipation of its energy in this way4 ,conflict with other bodies is so far from being improbable1 that it becomes approximately certain when we take very long intervals of time into account4 ,as regards cometary conflicts1 ,i am by no means satisfied with the negative conclusions drawn from the remarkable display on the evening of the #bgth of ,november1 #ahgb4 ,we may often have passed through the tail of a comet1 the light of which is probably an electrical manifestation no more substantial than the aurora borealis4 ,every remarkable shower of shooting stars may also be considered as proceeding from a cometary body1 so that we may be said to have passed through the thinner parts of innumerable comets4 ,but the earth has probably never passed1 in times of which we have any record1 through the nucleus of a comet1 which consists perhaps of a dense swarm of small meteorites4 ,we can only speculate upon the effects which might be produced by such a conflict1 but it would probably be a much more serious event than any yet registered in history4 ,the probability of its occurrence1 too1 cannot be assigned2 for though the probability of conflict with any one cometary nucleus is almost infinitesimal1 yet the number of comets is immensely great 7p4 #djh74 ,it is far from impossible1 again1 that the planetary system may be invaded by bodies of greater mass than comets4 ,the sun seems to be placed in so extensive a portion of empty space that its own proper motion would not bring it to the nearest known star 7 ,centauri7 in less than #aci1bjj years4 ,but in order to be sure that this interval of undisturbed life is granted to our globe1 we must prove that there are no stars moving so as to meet us1 and no dark bodies of considerable size flying through intervening space unknown to us4 ,the intrusion of comets into our system1 and the fact that many of them have hyperbolic paths1 is sufficient to show that the surrounding parts of space are occupied by multitudes of dark bodies of some size4 ,it is quite probable that small suns may have cooled sufficiently to become non-luminous2 for even if we discredit the theory that the variation of brightness of periodic stars is due to the revolution of dark companion stars1 yet there is in our own globe an unquestionable example of a smaller body which has cooled below the luminous point4 ,altogether1 then1 it is a mere assumption that the uniformity of nature involves the unaltered existence of our own globe4 ,there is no kind of catastrophe which is too great or too sudden to be theoretically consistent with the reign of law4 ,for all that our science can tell1 human history may be closed in the next instant of time4 ,the world may be dashed to pieces against a wandering star2 it may be involved in a nebulous atmosphere of hydrogen to be exploded a second afterwards2 it may be scorched up or dissipated into vapour by some great explosion in the sun2 there might even be within the globe itself some secret cause of disruption1 which only needs time for its manifestation4 ,there are some indications1 as already noticed 7p4 #ffj71 that violent disturbances have actually occurred in the history of the solar system4 ,olbers sought for the minor planets on the supposition that they were fragments of an exploded planet1 and he was rewarded with the discovery of some of them4 ,the retrograde motion of the satellites of the more distant planets1 the abnormal position of the poles of ,uranus and the excessive distance of ,neptune1 are other indications of some violent event1 of which we have no other evidence4 ,i adduce all these facts and arguments1 not to show that there is any considerable probability1 as far as we can judge1 of interruption within the scope of human history1 but to prove that the ,uniformity of ,nature is theoretically consistent with the most unexpected events of which we can form a conception4 99,possible ,states of the ,universe499 ,when we give the rein to scientific imagination1 it becomes apparent that conflict of body with body must not be regarded as the rare exception1 but as the general rule and the inevitable fate of each star system4 ,so far as we can trace out the results of the law of gravitation1 and of the dissipation of energy1 the universe must be regarded as undergoing gradual condensation into a single cold solid body of gigantic dimensions4 ,those who so frequently use the expression ,uniformity of ,nature seem to forget that the ,universe might exist consistently with the laws of nature in the most diverse conditions4 ,it might consist1 on the one hand1 of a glowing nebulous mass of gaseous substances4 ,the heat might be so intense that all elements1 even carbon and silicon1 would be in the state of gas1 and all atoms1 of whatever nature1 would be flying about in chemical independence1 diffusing themselves almost uniformly in the neighbouring parts of space4 ,there would then be no life1 unless we can apply that name to the passage through each part of space of similar average trains of atoms1 the particular succession of atoms being governed only by the theory of probability1 and the law of divergence from a mean exhibited in the ,arithmetical ,triangle4 ,such a universe would correspond partially to the ,lucretian rain of atoms1 and to that nebular hypothesis out of which ,laplace proposed philosophically to explain the evolution of the planetary system4 ,according to another extreme supposition1 the intense heat-energy of this nebulous mass might be radiated away into the unknown regions of outer space4 ,the attraction of gravity would exert itself between each two particles1 and the energy of motion thence arising would1 by incessant conflicts1 be resolved into heat and dissipated4 ,inconceivable ages might be required for the completion of this process1 but the dissipation of energy thus proceeding could end only in the production of a cold and motionless universe4 ,the relation of cause and effect1 as we see it manifested in life and growth1 would degenerate into the constant existence of every particle in a fixed position relative to every other particle4 ,logical and geometrical resemblances would still exist between atoms1 and between groups of atoms crystallised in their appropriate forms for evermore4 ,but time1 the great variable1 would bring no variation1 and as to human hopes and troubles1 they would have gone to eternal rest4 ,science is not really adequate to proving that such is the inevitable fate of the universe1 for we can seldom trust our best-established theories far from their data4 ,nevertheless1 the most probable speculations which we can form as to the history1 especially of our own planetary system1 is that it originated in a heated revolving nebulous mass of gas1 and is in a state of excessively slow progress towards the cold and stony condition4 ,other speculative hypotheses might doubtless be entertained4 ,every hypothesis is pressed by difficulties4 ,if the whole universe be cooling1 whither does the heat go8 ,if we are to get rid of it entirely1 outer space must be infinite in extent1 so that it shall never be stopped and reflected back4 ,but not to speak of metaphysical difficulties1 if the medium of heat undulations be infinite in extent1 why should not the material bodies placed in it be infinite also in number and aggregate mass8 ,it is apparent that we are venturing into speculations which surpass our powers of scientific inference4 ,but then ,i am arguing negatively2 ,i wish to show that those who speak of the uniformity of nature1 and the reign of law1 misinterpret the meaning involved in those expressions4 ,law is not inconsistent with extreme diversity1 and1 so far as we can read the history of this planetary system1 it did probably originate in heated nebulous matter1 and man's history forms but a brief span in its progress towards the cold and stony condition4 ,it is by doubtful and speculative hypotheses alone that we can avoid such a conclusion1 and ,i depart least from undoubted facts and well-established laws when ,i assert that1 whatever uniformities may underlie the phenomena of nature1 constant variety and ever-progressing change is the real outcome4 99,speculations on the ,reconcentration of ,energy499 ,there are unequivocal indications1 as ,i have said1 that the material universe1 as we at present see it1 is progressing from some act of creation1 or some discontinuity of existence of which the date may be approximately fixed by scientific inference4 ,it is progressing towards a state in which the available energy of matter will be dissipated through infinite surrounding space1 and all matter will become cold and lifeless4 ,this constitutes1 as it were1 the historical period of physical science1 that over which our scientific foresight may more or less extend4 ,but in this1 as in other cases1 we have no right to interpret our experience negatively1 so as to infer that because the present state of things began at a particular time1 there was no previous existence4 ,it may be that the present period of material existence is but one of an indefinite series of like periods4 ,all that we can see1 and feel1 and infer1 and reason about may be1 as it were1 but a part of one single pulsation in the existence of the universe4 ,after ,sir ,w4 ,thomson had pointed out the preponderating tendency which now seems to exist towards the conversion of all energy into heat-energy1 and its equal diffusion by radiation throughout space1 the late ,professor ,rankine put forth a remarkable speculation4#fac ,he suggested that the ethereal1 or1 as ,i have called it1 the 99adamantine99 medium in which all the stars exist1 and all radiation takes place1 may have bounds1 beyond which only empty space exists4 ,all heat undulations reaching this boundary will be totally reflected1 according to the theory of undulations1 and will be reconcentrated into foci situated in various parts of the medium4 ,whenever a cold and extinct star happens to pass through one of these foci1 it will be instantly ignited and resolved by intense heat into its constituent elements4 ,discontinuity will occur in the history of that portion of matter1 and the star will begin its history afresh with a renewed store of energy4 #fac 99,report of the ,british ,association99 7#aheb71 ,report of ,sections1 p4 #ab4 ,this is doubtless a mere speculation1 practically incapable of verification by observation1 and almost free from restrictions afforded by present knowledge4 ,we might attribute various shapes to the adamantine medium1 and the consequences would be various4 ,but there is this value in such speculations1 that they draw attention to the finiteness of our knowledge4 ,we cannot deny the possible truth of such an hypothesis1 nor can we place a limit to the scientific imagination in the framing of other like hypotheses4 ,it is impossible1 indeed1 to follow out our scientific inferences without falling into speculation4 ,if heat be radiated into outward space1 it must either proceed 99ad infinitum991 or it must be stopped somewhere4 ,in the latter case we fall upon ,rankine's hypothesis4 ,but if the material universe consist of a finite collection of heated matter situated in a finite portion of an infinite adamantine medium1 then either this universe must have existed for a finite time1 or else it must have cooled down during the infinity of past time indefinitely near to the absolute zero of temperature4 ,i objected to ,lucretius' argument against the destructibility of matter1 that we have no knowledge whatever of the laws according to which it would undergo destruction4 ,but we do know the laws according to which the dissipation of heat appears to proceed1 and the conclusion inevitably is that a finite heated material body placed in a perfectly cold infinitely extended medium would in an infinite time sink to zero of temperature4 ,now our own world is not yet cooled down near to zero1 so that physical science seems to place us in the dilemma of admitting either the finiteness of past duration of the world1 or else the finiteness of the portion of medium in which we exist4 ,in either case we become involved in metaphysical and mechanical difficulties surpassing our mental powers4 99,the ,divergent ,scope for ,new ,discovery499 ,in the writings of some recent philosophers1 especially of ,auguste ,comte1 and in some degree ,john ,stuart ,mill1 there is an erroneous and hurtful tendency to represent our knowledge as assuming an approximately complete character4 ,at least these and many other writers fail to impress upon their readers a truth which cannot be too constantly borne in mind1 namely1 that the utmost successes which our scientific method can accomplish will not enable us to comprehend more than an infinitesimal fraction of what there doubtless is to comprehend4#fad ,professor ,tyndall seems to me open to the same charge in a less degree4 ,he remarks#fae that we can probably never bring natural phenomena completely under mathematical laws1 because the approach of our sciences towards completeness may be asymptotic1 so that however far we may go1 there may still remain some facts not subject to scientific explanation4 ,he thus likens the supply of novel phenomena to a convergent series1 the earlier and larger terms of which have been successfully disposed of1 so that comparatively minor groups of phenomena alone remain for future investigators to occupy themselves upon4 #fad ,mr4 ,c4 ,j4 ,monroe objects that in this statement ,i do injustice to ,comte1 who1 he thinks1 did impress upon his readers the inadequacy of our mental powers compared with the vastness of the subject matter of science4 ,the error of ,comte1 he holds1 was in maintaining that science had been carried about as far as it is worth while to carry it1 which is a different matter4 ,in either case1 ,comte's position is so untenable that ,i am content to leave the question undecided4 #fae 99,fragments of ,science991 p4 #cfb4 ,on the contrary1 as it appears to me1 the supply of new and unexplained facts is divergent in extent1 so that the more we have explained1 the more there is to explain4 ,the further we advance in any generalisation1 the more numerous and intricate are the exceptional cases still demanding further treatment4 ,the experiments of ,boyle1 ,mariotte1 ,dalton1 ,gay-,lussac1 and others1 upon the physical properties of gases1 might seem to have exhausted that subject by showing that all gases obey the same laws as regards temperature1 pressure1 and volume4 ,but in reality these laws are only approximately true1 and the divergences afford a wide and quite unexhausted field for further generalisation4 ,the recent discoveries of ,professor ,andrews have summed up some of these exceptional facts under a wider generalisation1 but in reality they have opened to us vast new regions of interesting inquiry1 and they leave wholly untouched the question why one gas behaves differently from another4 ,the science of crystallography is that perhaps in which the most precise and general laws have been detected1 but it would be untrue to assert that it has lessened the area of future discovery4 ,we can show that each one of the seven or eight hundred forms of calcite is derivable by geometrical modifications from an hexagonal prism2 but who has attempted to explain the molecular forces producing these modifications1 or the chemical conditions in which they arise8 ,the law of isomorphism is an important generalisation1 for it establishes a general resemblance between the forms of crystallisation of natural classes of elements4 ,but if we examine a little more closely we find that these forms are only approximately alike1 and the divergence peculiar to each substance is an unexplained exception4 ,by many similar illustrations it might readily be shown that in whatever direction we extend our investigations and successfully harmonise a few facts1 the result is only to raise up a host of other unexplained facts4 ,can any scientific man venture to state that there is less opening now for new discoveries than there was three centuries ago8 ,is it not rather true that we have but to open a scientific book and read a page or two1 and we shall come to some recorded phenomenon of which no explanation can yet be given8 ,in every such fact there is a possible opening for new discoveries1 and it can only be the fault of the investigator's mind if he can look around him and find no scope for the exercise of his faculties4 99,infinite ,incompleteness of the ,mathematical ,sciences499 ,there is one privilege which a certain amount of knowledge should confer2 it is that of becoming aware of the weakness of our powers compared with the tasks which they might undertake if stronger4 ,to the poor savage who cannot count twenty the arithmetical accomplishments of the schoolboy are miraculously great4 ,the schoolboy cannot comprehend the vastly greater powers of the student1 who has acquired facility in algebraic processes4 ,the student can but look with feelings of surprise and reverence at the powers of a ,newton or a ,laplace4 ,but the question at once suggests itself1 ,do the powers of the highest human intellect bear a finite ratio to the things which are to be understood and calculated8 ,how many further steps must we take in the rise of mental ability and the extension of mathematical methods before we begin to exhaust the knowable8 ,i am inclined to find fault with mathematical writers because they often exult in what they can accomplish1 and omit to point out that what they do is but an infinitely small part of what might be done4 ,they exhibit a general inclination1 with few exceptions1 not to do so much as mention the existence of problems of an impracticable character4 ,this may be excusable as far as the immediate practical result of their researches is in question1 but the custom has the effect of misleading the general public into the fallacious notion that mathematics is a 99perfect99 science1 which accomplishes what it undertakes in a complete manner4 ,on the contrary1 it may be said that if a mathematical problem were selected by chance out of the whole number which might be proposed1 the probability is infinitely slight that a human mathematician could solve it4 ,just as the numbers we can count are nothing compared with the numbers which might exist1 so the accomplishments of a ,laplace or a ,lagrange are1 as it were1 the little corner of the multiplication-table1 which has really an infinite extent4 ,i have pointed out that the rude character of our observations prevents us from being aware of the greater number of effects and actions in nature4 ,it must be added that1 if we perceive them1 we should usually be incapable of including them in our theories from want of mathematical power4 ,some persons may be surprised that though nearly two centuries have elapsed since the time of ,newton's discoveries1 we have yet no general theory of molecular action4 ,some approximations have been made towards such a theory4 ,joule and ,clausius have measured the velocity of gaseous atoms1 or even determined the average distance between the collisions of atom and atom4 ,thomson has approximated to the number of atoms in a given bulk of substance4 ,rankine has formed some reasonable hypotheses as to the actual constitution of atoms4 ,it would be a mistake to suppose that these ingenious results of theory and experiment form any appreciable approach to a complete solution of molecular motions4 ,there is every reason to believe1 judging from the spectra of the elements1 their atomic weights and other data1 that chemical atoms are very complicated structures4 ,an atom of pure iron is probably a far more complicated system than that of the planets and their satellites4 ,a compound atom may perhaps be compared with a stellar system1 each star a minor system in itself4 ,the smallest particle of solid substance will consist of a great number of such stellar systems united in regular order1 each bounded by the other1 communicating with it in some manner yet wholly incomprehensible4 ,what are our mathematical powers in comparison with this problem8 ,after two centuries of continuous labour1 the most gifted men have succeeded in calculating the mutual effects of three bodies each upon the other1 under the simple hypothesis of the law of gravity4 ,concerning these calculations we must further remember that they are purely approximate1 and that the methods would not apply where four or more bodies are acting1 and all produce considerable effects upon each other4 ,there is reason to believe that each constituent of a chemical atom goes through an orbit in the millionth part of the twinkling of an eye4 ,in each revolution it is successively or simultaneously under the influence of many other constituents1 or possibly comes into collision with them4 ,it is no exaggeration to say that mathematicians have the least notion of the way in which they could successfully attack so difficult a problem of forces and motions4 ,as ,herschel has remarked1#faf each of these particles is for ever solving differential equations1 which1 if written out in full1 might belt the earth4 #faf 99,familiar ,lectures on ,scientific ,subjects991 p4 #deh4 ,some of the most extensive calculations ever made were those required for the reduction of the measurements executed in the course of the ,trigonometrical ,survey of ,great ,britain4 ,the calculations arising out of the principal triangulation occupied twenty calculators during three or four years1 in the course of which the computers had to solve simultaneous equations involving seventy-seven unknown quantities4 ,the reduction of the levellings required the solution of a system of ninety-one equations4 ,but these vast calculations present no approach whatever to what would be requisite for the complete treatment of any one physical problem4 ,the motion of glaciers is supposed to be moderately well understood in the present day4 ,a glacier is a viscid1 slowly yielding mass1 neither absolutely solid nor absolutely rigid1 but it is expressly remarked by ,forbes1#fag that not even an approximate solution of the mathematical conditions of such a moving mass can yet be possible4 8,every one knows10 he says1 8that such problems are beyond the compass of exact mathematics20 but though mathematicians may know this1 they do not often enough impress that knowledge on other people4 #fag 99,philosophical ,magazine991 #crd ,series1 vol4 xxvi4 p4 #djf4 ,the problems which are solved in our mathematical books consist of a small selection of those which happen from peculiar conditions to be solvable4 ,but the very simplest problem in appearance will often give rise to impracticable calculations4 ,mr4 ,todhunter#fah seems to blame ,condorcet1 because in one of his memoirs he mentions a problem to solve which would require a great and impracticable number of successive integrations4 ,now1 if our mathematical sciences are to cope with the problems which await solution1 we must be prepared to effect an unlimited number of successive integrations2 yet at present1 and almost beyond doubt for ever1 the probability that an integration taken haphazard will come within our powers is exceedingly small4 #fah 99,history of the ,theory of ,probability991 p4 #cih4 ,in some passages of that remarkable work1 the 99,ninth ,bridgewater ,treatise99 7pp4 #aac-aae71 ,babbage has pointed out that if we had power to follow and detect the minutest effects of any disturbance1 each particle of existing matter would furnish a register of all that has happened4 8,the track of every canoe--of every vessel that has yet disturbed the surface of the ocean1 whether impelled by manual force or elemental power1 remains for ever registered in the future movement of all succeeding particles which may occupy its place4 ,the furrow which it left is1 indeed1 instantly filled up by the closing waters2 but they draw after them other and larger portions of the surrounding element1 and these again1 once moved1 communicate motion to others in endless succession40 ,we may even say that 8,the air itself is one vast library1 on whose pages are for ever written all that man has ever said or even whispered4 ,there1 in their mutable but unerring characters1 mixed with the earliest as well as the latest sighs of mortality1 stand for ever recorded1 vows unredeemed1 promises unfulfilled1 perpetuating in the united movements of each particle the testimony of man's changeful will40 ,when we read reflections such as these1 we may congratulate ourselves that we have been endowed with minds which1 rightly employed1 can form some estimate of their incapacity to trace out and account for all that proceeds in the simpler actions of material nature4 ,it ought to be added that1 wonderful as is the extent of physical phenomena open to our investigation1 intellectual phenomena are yet vastly more extensive4 ,of this ,i might present one satisfactory proof were space available by pointing out that the mathematical functions employed in the calculations of physical science form an infinitely small fraction of the functions which might be invented4 ,common trigonometry consists of a great series of useful formul1 all of which arise out of the relation of the sine and cosine expressed in one equation1 sin #b99x99 cos #b99x99 #a4 ,but this is not the only trigonometry which may exist2 mathematicians also recognise hyperbolic trigonometry1 of which the fundamental equation is cos #b99x99 - sin #b99x99 #a4 ,de ,morgan has pointed out that the symbols of ordinary algebra form but three of an interminable series of conceivable systems4#fai ,as the logarithmic operation is to addition or addition to multiplication1 so is the latter to a higher operation1 and so on without limit4 #fai 99,trigonometry and ,double ,algebra991 chap4 ix4 ,we may rely upon it that immense1 and to us inconceivable1 advances will be made by the human intellect1 in the absence of any catastrophe to the species or the globe4 ,within historical periods we can trace the rise of mathematical science from its simplest germs4 ,we can prove our descent from ancestors who counted only on their fingers4 ,how infinitely is a ,newton or a ,laplace above those simple savages4 ,pythagoras is said to have sacrificed a hecatomb when he discovered the forty-seventh proposition of ,euclid1 and the occasion was worthy of the sacrifice4 ,archimedes was beside himself when he first perceived his beautiful mode of determining specific gravities4 ,yet these great discoveries are the commonplaces of our school books4 ,step by step we can trace upwards the acquirement of new mental powers4 ,what could be more wonderful than ,napier's discovery of logarithms1 a new mode of calculation which has multiplied perhaps a hundredfold the working powers of every computer1 and has rendered easy calculations which were before impracticable8 ,since the time of ,newton and ,leibnitz worlds of problems have been solved which before were hardly conceived as matters of inquiry4 ,in our own day extended methods of mathematical reasoning1 such as the system of quaternions1 have been brought into existence4 ,what intelligent man will doubt that the recondite speculations of a ,cayley1 a ,sylvester1 or a ,clifford may lead to some new development of new mathematical power1 at the simplicity of which a future age will wonder1 and yet wonder more that to us they were so dark and difficult4 ,may we not repeat the words of ,seneca3 8,veniet tempus1 quo ista qu nunc latent1 in lucem dies extrahat1 et longioris vi diligentia3 ad inquisitionem tantorum tas una non sufficit4 ,veniet tempus1 quo posteri nostri tam aperta nos nescisse mirentur40 99,the ,reign of ,law in ,mental and ,social ,phenomena499 ,after we pass from the so-called physical sciences to those which attempt to investigate mental and social phenomena1 the same general conclusions will hold true4 ,no one will be found to deny that there are certain uniformities of thinking and acting which can be detected in reasoning beings1 and so far as we detect such laws we successfully apply scientific method4 ,but those who attempt to establish social or moral sciences soon become aware that they are dealing with subjects of enormous perplexity4 ,take as an instance the science of political economy4 ,if a science at all1 it must be a mathematical science1 because it deals with quantities of commodities4 ,but as soon as we attempt to draw out the equations expressing the laws of demand and supply1 we discover that they have a complexity entirely surpassing our powers of mathematical treatment4 ,we may lay down the general form of the equations1 expressing the demand and supply for two or three commodities among two or three trading bodies1 but all the functions involved are so complicated in character that there is not much fear of scientific method making rapid progress in this direction4 ,if such be the prospects of a comparatively formal science1 like political economy1 what shall we say of moral science8 ,any complete theory of morals must deal with quantities of pleasure and pain1 as ,bentham pointed out1 and must sum up the general tendency of each kind of action upon the good of the community4 ,if we are to apply scientific method to morals1 we must have a calculus of moral effects1 a kind of physical astronomy investigating the mutual perturbations of individuals4 ,but as astronomers have not yet fully solved the problem of three gravitating bodies1 when shall we have a solution of the problem of three moral bodies8 ,the sciences of political economy and morality are comparatively abstract and general1 treating mankind from simple points of view1 and attempting to detect general principles of action4 ,they are to social phenomena what the abstract sciences of chemistry1 heat1 and electricity are to the concrete science of meteorology4 ,before we can investigate the actions of any aggregate of men1 we must have fairly mastered all the more abstract sciences applying to them1 somewhat in the way that we have acquired a fair comprehension of the simpler truths of chemistry and physics4 ,but all our physical sciences do not enable us to predict the weather two days hence with any great probability1 and the general problem of meteorology is almost unattempted as yet4 ,what shall we say then of the general problem of social science1 which shall enable us to predict the course of events in a nation8 ,several writers have proposed to lay the foundations of the science of history4 ,buckle undertook to write the 99,history of ,civilisation in ,england991 and to show how the character of a nation could be explained by the nature of the climate and the fertility of the soil4 ,he omitted to explain the contrast between the ancient ,greek nation and the present one2 there must have been an extraordinary revolution in the climate or the soil4 ,auguste ,comte detected the simple laws of the course of development through which nations pass4 ,there are always three phases of intellectual condition1--the theological1 the metaphysical1 and the positive2 applying this general law of progress to concrete cases1 ,comte was enabled to predict that in the hierarchy of ,european nations1 ,spain would necessarily hold the highest place4 ,such are the parodies of science offered to us by the 99positive99 philosophers4 ,a science of history in the true sense of the term is an absurd notion4 ,a nation is not a mere sum of individuals whom we can treat by the method of averages2 it is an organic whole1 held together by ties of infinite complexity4 ,each individual acts and re-acts upon his smaller or greater circle of friends1 and those who acquire a public position exert an influence on much larger sections of the nation4 ,there will always be a few great leaders of exceptional genius or opportunities1 the unaccountable phases of whose opinions and inclinations sway the whole body4 ,from time to time arise critical situations1 battles1 delicate negotiations1 internal disturbances1 in which the slightest incidents may change the course of history4 ,a rainy day may hinder a forced march1 and change the course of a campaign2 a few injudicious words in a despatch may irritate the national pride2 the accidental discharge of a gun may precipitate a collision the effects of which will last for centuries4 ,it is said that the history of ,europe depended at one moment upon the question whether the look-out man upon ,nelson's vessel would or would not descry a ship of ,napoleon's expedition to ,egypt which was passing not far off4 ,in human affairs1 then1 the smallest causes may produce the greatest effects1 and the real application of scientific method is out of the question4 99,the ,theory of ,evolution499 ,profound philosophers have lately generalised concerning the production of living forms and the mental and moral phenomena regarded as their highest development4 ,herbert ,spencer's theory of evolution purports to explain the origin of all specific differences1 so that not even the rise of a ,homer or a ,beethoven would escape from his broad theories4 ,the homogeneous is unstable and must differentiate itself1 says ,spencer1 and hence comes the variety of human institutions and characters4 ,in order that a living form shall continue to exist and propagate its kind1 says ,darwin1 it must be suitable to its circumstances1 and the most suitable forms will prevail over and extirpate those which are less suitable4 ,from these fruitful ideas are developed theories of evolution and natural selection which go far towards accounting for the existence of immense numbers of living creatures--plants1 and animals4 ,apparent adaptations of organs to useful purposes1 which ,paley regarded as distinct products of creative intelligence1 are now seen to follow as natural effects of a constantly acting tendency4 ,even man1 according to these theories1 is no distinct creation1 but rather an extreme case of brain development4 ,his nearest cousins are the apes1 and his pedigree extends backwards until it joins that of the lowliest zoophytes4 ,the theories of ,darwin and ,spencer are doubtless not demonstrated2 they are to some extent hypothetical1 just as all the theories of physical science are to some extent hypothetical1 and open to doubt4 ,judging from the immense numbers of diverse facts which they harmonise and explain1 ,i venture to look upon the theories of evolution and natural selection in their main features as two of the most probable hypotheses ever proposed4 ,i question whether any scientific works which have appeared since the 99,principia99 of ,newton are comparable in importance with those of ,darwin and ,spencer1 revolutionising as they do all our views of the origin of bodily1 mental1 moral1 and social phenomena4 ,granting all this1 ,i cannot for a moment admit that the theory of evolution will destroy theology4 ,that theory embraces several laws or uniformities which are observed to be true in the production of living forms2 but these laws do not determine the size and figure of living creatures1 any more than the law of gravitation determines the magnitudes and distances of the planets4 ,suppose that ,darwin is correct in saying that man is descended from the ,ascidians3 yet the precise form of the human body must have been influenced by an infinite train of circumstances affecting the reproduction1 growth1 and health of the whole chain of intermediate beings4 ,no doubt1 the circumstances being what they were1 man could not be otherwise than he is1 and if in any other part of the universe an exactly similar earth1 furnished with exactly similar germs of life1 existed1 a race must have grown up there exactly similar to the human race4 ,by a different distribution of atoms in the primeval world a different series of living forms on this earth would have been produced4 ,from the same causes acting according to the same laws1 the same results will follow2 but from different causes acting according to the same laws1 different results will follow4 ,so far as we can see1 then1 infinitely diverse living creatures might have been created consistently with the theory of evolution1 and the precise reason why we have a backbone1 two hands with opposable thumbs1 an erect stature1 a complex brain1 about #bbc bones1 and many other peculiarities1 is only to be found in the original act of creation4 ,i do not1 any less than ,paley1 believe that the eye of man manifests design4 ,i believe that the eye was gradually developed1 and we can in fact trace its gradual development from the first germ of a nerve affected by light-rays in some simple zoophyte4 ,in proportion as the eye became a more accurate instrument of vision1 it enabled its possessor the better to escape destruction1 but the ultimate result must have been contained in the aggregate of the causes1 and these causes1 as far as we can see1 were subject to the arbitrary choice of the ,creator4 ,although ,agassiz was clearly wrong in holding that every species of living creature appeared on earth by the immediate intervention of the ,creator1 which would amount to saying that no laws of connection between forms are discoverable1 yet he seems to be right in asserting that living forms are distinct from those produced by purely physical causes4 8,the products of what are commonly called physical agents10 he says1#fbj 8are everywhere the same 799i4e499 upon the whole surface of the earth71 and have always been the same 799i4e499 during all geological periods72 while organised beings are everywhere different and have differed in all ages4 ,between two such series of phenomena there can be no causal or genetic connection40 ,living forms as we now regard them are essentially variable1 but from constant mechanical causes constant effects would ensue4 ,if vegetable cells are formed on geometrical principles being first spherical1 and then by mutual compression dodecahedral1 then all cells should have similar forms4 ,in the ,foraminifera and some other lowly organisms1 we seem to observe the production of complex forms on geometrical principles4 ,but from similar causes acting according to similar laws only similar results could be produced4 ,if the original life germ of each creature is a simple particle of protoplasm1 unendowed with any distinctive forces1 then the whole of the complex phenomena of animal and vegetable life are effects without causes4 ,protoplasm may be chemically the same substance1 and the germ-cell of a man and of a fish may be apparently the same1 so far as the microscope can decide2 but if certain cells produce men1 and others as uniformly produce a species of fish1 there must be a hidden constitution determining the extremely different results4 ,if this were not so1 the generation of every living creature from the uniform germ would have to be regarded as a distinct act of creation4 #fbj ,agassiz1 99,essay on ,classification991 p4 #ge4 ,theologians have dreaded the establishment of the theories of ,darwin and ,huxley and ,spencer1 as if they thought that those theories could explain everything upon the purest mechanical and material principles1 and exclude all notions of design4 ,they do not see that those theories have opened up more questions than they have closed4 ,the doctrine of evolution gives a complete explanation of no single living form4 ,while showing the general principles which prevail in the variation of living creatures1 it only points out the infinite complexity of the causes and circumstances which have led to the present state of things4 ,any one of ,mr4 ,darwin's books1 admirable though they all are1 consists but in the setting forth of a multitude of indeterminate problems4 ,he proves in the most beautiful manner that each flower of an orchid is adapted to some insect which frequents and fertilises it1 and these adaptations are but a few cases of those immensely numerous ones which have occurred in the lives of plants and animals4 ,but why orchids should have been formed so differently from other plants1 why anything1 indeed1 should be as it is1 rather than in some of the other infinitely numerous possible modes of existence1 he can never show4 ,the origin of everything that exists is wrapped up in the past history of the universe4 ,at some one or more points in past time there must have been arbitrary determinations which led to the production of things as they are4 99,possibility of ,divine ,interference499 ,i will now draw the reader's attention to pages #adi to #aeb4 ,i there pointed out that all inductive inference involves the assumption that our knowledge of what exists is complete1 and that the conditions of things remain unaltered between the time of our experience and the time to which our inferences refer4 ,recurring to the illustration of a ballot-box1 employed in the chapter on the inverse method of probabilities1 we assume when predicting the probable nature of the next drawing1 firstly1 that our previous drawings have been sufficiently numerous to give us knowledge of the contents of the box2 and1 secondly1 that no interference with the ballot-box takes place between the previous and the next drawings4 ,the results yielded by the theory of probability are quite plain4 ,no finite number of casual drawings can give us sure knowledge of the contents of the box1 so that1 even in the absence of all disturbance1 our inferences are merely the best which can be made1 and do not approach to infallibility4 ,if1 however1 interference be possible1 even the theory of probability ceases to be applicable1 for1 the amount and nature of that interference being arbitrary and unknown1 there ceases to be any connection between premises and conclusion4 ,many years of reflection have not enabled me to see the way of avoiding this hiatus in scientific certainty4 ,the conclusions of scientific inference appear to be always of a hypothetical and provisional nature4 ,given certain experience1 the theory of probability yields us the true interpretation of that experience and is the surest guide open to us4 ,but the best calculated results which it can give are never absolute probabilities2 they are purely relative to the extent of our information4 ,it seems to be impossible for us to judge how far our experience gives us adequate information of the universe as a whole1 and of all the forces and phenomena which can have place therein4 ,i feel that ,i cannot in the space remaining at my command in the present volume1 sufficiently follow out the lines of thought suggested1 or define with precision my own conclusions4 ,this chapter contains merely 99,reflections99 upon subjects of so weighty a character that ,i should myself wish for many years--nay for more than a lifetime of further reflection4 ,my purpose1 as ,i have repeatedly said1 is the purely negative one of showing that atheism and materialism are no necessary results of scientific method4 ,from the preceding reviews of the value of our scientific knowledge1 ,i draw one distinct conclusion1 that we cannot disprove the possibility of ,divine interference in the course of nature4 ,such interference might arise1 so far as our knowledge extends1 in two ways4 ,it might consist in the disclosure of the existence of some agent or spring of energy previously unknown1 but which effects a given purpose at a given moment4 ,like the pre-arranged change of law in ,babbage's imaginary calculating machine1 there may exist pre-arranged surprises in the order of nature1 as it presents itself to us4 ,secondly1 the same ,power1 which created material nature1 might1 so far as ,i can see1 create additions to it1 or annihilate portions which do exist4 ,such events are in a certain sense inconceivable to us2 yet they are no more inconceivable than the existence of the world as it is4 ,the indestructibility of matter1 and the conservation of energy1 are very probable scientific hypotheses1 which accord satisfactorily with experiments of scientific men during a few years past1 but it would be gross misconception of scientific inference to suppose that they are certain in the sense that a proposition in geometry is certain4 ,philosophers no doubt hold that 99de nihilo nihil fit991 that is to say1 their senses give them no means of imagining to the mind how creation can take place4 ,but we are on the horns of a trilemma2 we must either deny that anything exists1 or we must allow that it was created out of nothing at some moment of past time1 or that it existed from eternity4 ,the first alternative is absurd2 the other two seem to me equally conceivable4 99,conclusion499 ,it may seem that there is one point where our speculations must end1 namely where contradiction begins4 ,the laws of ,identity and ,difference and ,duality were the foundations from which we started1 and they are1 so far as ,i can see1 the foundations which we can never quit without tottering4 ,scientific ,method must begin and end with the laws of thought1 but it does not follow that it will save us from encountering inexplicable1 and at least apparently contradictory results4 ,the nature of continuous quantity leads us into extreme difficulties4 ,any finite space is composed of an infinite number of infinitely small spaces1 each of which1 again1 is composed of an infinite number of spaces of a second order of smallness2 these spaces of the second order are composed1 again1 of infinitely small spaces of the third order4 ,even these spaces of the third order are not absolute geometrical points answering to ,euclid's definition of a point1 as position without magnitude4 ,go on as far as we will1 in the subdivision of continuous quantity1 yet we never get down to the absolute point4 ,thus scientific method leads us to the inevitable conception of an infinite series of successive orders of infinitely small quantities4 ,if so1 there is nothing impossible in the existence of a myriad universes within the compass of a needle's point1 each with its stellar systems1 and its suns and planets1 in number and variety unlimited4 ,science does nothing to reduce the number of strange things that we may believe4 ,when fairly pursued it makes absurd drafts upon our powers of comprehension and belief4 ,some of the most precise and beautiful theorems in mathematical science seem to me to involve apparent contradiction4 ,can we imagine that a point moving along a perfectly straight line towards the west would ever get round to the east and come back again1 having performed1 as it were1 a circuit through infinite space1 yet without ever diverging from a perfectly straight direction8 ,yet this is what happens to the intersecting point of two straight lines in the same plane1 when one line revolves4 ,the same paradox is exhibited in the hyperbola regarded as an infinite ellipse1 one extremity of which has passed to an infinite distance and come back in the opposite direction4 ,a varying quantity may change its sign by passing either through zero or through infinity4 ,in the latter case there must be one intermediate value of the variable for which the variant is indifferently negative infinity and positive infinity4 ,professor ,clifford tells me that he has found a mathematical function which approaches infinity as the variable approaches a certain limit2 yet at the limit the function is finite6 ,mathematicians may shirk difficulties1 but they cannot make such results of mathematical principles appear otherwise than contradictory to our common notions of space4 ,the hypothesis that there is a ,creator at once all-powerful and all-benevolent is pressed1 as it must seem to every candid investigator1 with difficulties verging closely upon logical contradiction4 ,the existence of the smallest amount of pain and evil would seem to show that ,he is either not perfectly benevolent1 or not all-powerful4 ,no one can have lived long without experiencing sorrowful events of which the significance is inexplicable4 ,but if we cannot succeed in avoiding contradiction in our notions of elementary geometry1 can we expect that the ultimate purposes of existence shall present themselves to us with perfect clearness8 ,i can see nothing to forbid the notion that in a higher state of intelligence much that is now obscure may become clear4 ,we perpetually find ourselves in the position of finite minds attempting infinite problems1 and can we be sure that where we see contradiction1 an infinite intelligence might not discover perfect logical harmony8 ,from science1 modestly pursued1 with a due consciousness of the extreme finitude of our intellectual powers1 there can arise only nobler and wider notions of the purpose of ,creation4 ,our philosophy will be an affirmative one1 not the false and negative dogmas of ,auguste ,comte1 which have usurped the name1 and misrepresented the tendencies of a true 99positive philosophy994 ,true science will not deny the existence of things because they cannot be weighed and measured4 ,it will rather lead us to believe that the wonders and subtleties of possible existence surpass all that our mental powers allow us clearly to perceive4 ,the study of logical and mathematical forms has convinced me that even space itself is no requisite condition of conceivable existence4 ,everything1 we are told by materialists1 must be here or there1 nearer or further1 before or after4 ,i deny this1 and point to logical relations as my proof4 ,there formerly seemed to me to be something mysterious in the denominators of the binomial expansion 7p4 #aij71 which are reproduced in the natural constant 1 or #a #a_/a #a_/7a 4 #b7 #a_/7a 4 #b 4 #c7 444 and in many results of mathematical analysis4 ,i now perceive1 as already explained 7pp4 #cc1 #afj1 #chc71 that they arise out of the fact that the relations of space do not apply to the logical conditions governing the numbers of combinations as contrasted to those of permutations4 ,so far am ,i from accepting ,kant's doctrine that space is a necessary form of thought1 that ,i regard it as an accident1 and an impediment to pure logical reasoning4 ,material existences must exist in space1 no doubt1 but intellectual existences may be neither in space nor out of space2 they may have no relation to space at all1 just as space itself has no relation to time4 ,for all that ,i can see1 then1 there may be intellectual existences to which both time and space are nullities4 ,now among the most unquestionable rules of scientific method is that first law that 99whatever phenomenon is1 is994 ,we must ignore no existence whatever2 we may variously interpret or explain its meaning and origin1 but1 if a phenomenon does exist1 it demands some kind of explanation4 ,if then there is to be competition for scientific recognition1 the world without us must yield to the undoubted existence of the spirit within4 ,our own hopes and wishes and determinations are the most undoubted phenomena within the sphere of consciousness4 ,if men do act1 feel1 and live as if they were not merely the brief products of a casual conjunction of atoms1 but the instruments of a far-reaching purpose1 are we to record all other phenomena and pass over these8 ,we investigate the instincts of the ant and the bee and the beaver1 and discover that they are led by an inscrutable agency to work towards a distant purpose4 ,let us be faithful to our scientific method1 and investigate also those instincts of the human mind by which man is led to work as if the approval of a ,higher ,being were the aim of life4 ,,index4 ,abacus1 logical1 #ajd2 arithmetical1 #ajg2 ,panchrestus1 #ahb4 ,aberration of light1 #efa2 systematic1 #edg4 ,abscissio infiniti1 #gi1 #gac4 ,abstract terms1 #bg2 number1 #aei4 ,abstraction1 #gjd2 logical1 #be2 numerical1 #aeh2 of indifferent circumstances1 #ig4 ,accademia del ,cimento1 #dbg1 #dcb1 #dcf1 #ebg4 ,accident1 logical1 #gjj4 ,accidental discovery1 #ebi4 ,achromatic lenses1 #dcb4 ,actinometer1 #ccg4 ,adamantine medium1 #fje1 #gea4 ,adjectives1 #ad1 #cj1 #ca1 #ce2 indeterminate1 #da4 ,adrain1 of ,new ,brunswick1 #cge4 ,affirmation1 #dd4 ,agassiz1 on genera1 #gbf2 on creation of species1 #gfc4 ,agreement1 #dd4 ,airy1 ,sir ,george ,biddell1 on perpetual motion1 #bbc2 new property of sphere1 #bcb2 pendulum experiments1 #bia1 #cjd1 #cdh1 #efg2 standard clock1 #cec2 book on 99,errors of ,observation991 #cie2 tides1 #dhh2 extra-polation1 #die2 ,thales' eclipse1 #ecg2 interference of light1 #eci2 density of earth1 #bia4 ,alchemists1 #eje2 how misled1 #dbh4 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,approximation1 theory of1 #def2 to exact laws1 #dfb2 mathematical principles of1 #dga2 arithmetic of1 #dha4 ,aqueous vapour1 #ejj4 ,aquinas1 on disjunctive propositions1 #fi4 ,arago1 photometer1 #bhh2 rotating disc1 #ece2 his philosophic character1 #eib4 ,archimedes1 99,de ,aren ,numero991 #aie2 centre of gravity1 #cfc4 ,arcual unit1 #cjf1 #ccj4 ,argyll1 ,duke of1 #gda4 ,aristarchus on sun's and moon's distances1 #bid4 ,aristotelian doctrines1 #fff4 ,aristotle1 dictum1 #ba2 singular terms1 #ci2 overlooked simple identities1 #dj2 order of premises1 #aad2 logical error1 #aag2 definition of time1 #cjg2 on science1 #eie2 on white swans1 #fff4 ,arithmetic1 reasoning in1 #afg2 of approximate quantities1 #dha4 ,arithmetical triangle1 #ic1 #adc1 #ahb1 #bjb1 #cgh1 #chc2 diagram of1 #ahd2 connection with ,logical ,alphabet1 #ahi2 in probability1 #bjh4 ,asteroids1 discovery of1 #dab1 #gdh4 ,astronomy1 physical1 #dei4 ,atmospheric tides1 #eec4 ,atomic theory1 #ffb4 ,atomic weights1 #efc4 ,atoms1 size of1 #aie2 impossibility of observing1 #djf4 ,augustin on time1 #cjg4 ,average1 #cei1 #cfj2 divergence from1 #ahh2 etymology of1 #cfc4 ,axes of crystals1 #fhf4 ,axioms of algebra1 #afd4 ,babbage1 ,charles1 calculating machine1 #ajg1 #bca1 #gdc2 lighthouse signals1 #aid2 natural constants1 #cbi2 ,mosaic history1 #dab2 universal and general truths1 #fdf2 change of law1 #bcj2 persistence of effects1 #geg4 ,bacon1 ,francis ,lord1 99,novum ,organum991 #ajg2 on induction1 #aba2 biliteral cipher1 #aic2 ,first ,aphorism1 #bai2 on causes1 #bba2 ,copernican system1 #bdi1 #fch2 deficient powers of senses1 #bgh2 observation1 #djb2 ,natural ,history1 #djc2 use of hypothesis1 #ejf2 his method1 #ejg2 99experimentum crucis991 #eai2 error of his method1 #egf2 ostensive1 clandestine instances1 c41 #fjh1 #faj2 99latens precessus991 #fai4 ,bacon1 ,roger1 on the rainbow1 #ebf1 #eih4 ,baily1 ,francis1 #bgb2 density of earth1 #cdb1 #eff2 experiments with torsion balance1 #cgj1 #cig1 #dcb1 #efg-h2 motions of stars1 #egb4 ,bain1 ,alexander1 on powers of mind1 #d2 ,mill's reform of logic1 #bbg4 ,baker's poem1 99,the ,universe991 #fba4 ,balance1 use of the chemical1 #bib1 #cea1 #ced1 #cfi2 delicacy of1 #cjd2 vibrations of1 #cfi4 ,ballot1 ,buys1 experiment on sound1 #eda4 ,ballot-box1 simile of1 #aej1 #bea-f1 #gfe4 ,barbara1 #ee1 #eg1 #hh1 #aje1 #ada4 ,baroko1 #he4 ,barometer1 #fei2 ,gay ,lussac's standard1 #cdf2 variations1 #ccg1 #cdf1 #cdi4 ,bartholinus on double refraction1 #ehe4 ,base-line1 measurement of1 #cjd4 ,bauhusius1 verses of1 #age4 ,baxendell1 ,joseph1 #eeb4 ,beneke1 on substitution1 #ba4 ,bennet1 momentum of light1 #dce4 ,bentham1 ,george1 #ae2 bifurcate classification1 #fie2 infima species1 #gjb2 works on classification1 #gjc2 analytical key to flora1 #gab4 ,bentham1 ,jeremy1 on analogy1 #fbi2 bifurcate classification1 #gjc4 ,benzenberg's experiment1 #chh4 ,bernoulli1 ,daniel1 planetary orbits1 #bej2 resisting media and projectiles1 #dfg2 vibrations1 #dgf4 ,bernoulli1 ,james1 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