,,memorabilia ,,mathematica ,,the ,,macmillan ,,company ,,new ,,york4 ,,boston4 ,,chicago4 ,,dallas ,,atlanta4 ,,san ,,francisco ,,macmillan ,,co41 ,,limited ,,london4 ,,bombay4 ,,calcutta ,,melbourne ,,the ,,macmillan ,,co4 ,,of ,,canada1 ,,ltd4 ,,toronto ,,memorabilia ,,mathematica ,,or ,,the ,,philomath',s ,,quotation-,,book ,,by ,,robert ,,edouard ,,moritz1 ,,ph4 ,d41 ,,ph4 ,n4 ,d4 ,,professor ,,of ,,mathematics ,,in ,,the ,,university ,,of ,,washington ,new ,york ,,the ,,macmillan ,,company #aiad ,all rights reserved ,,copyright1 #aiad1 ,,by ,,robert ,,edouard ,,moritz ,,preface ,every one knows that the fine phrase 8,god geometrizes0 is attributed to ,plato1 but few know where this famous passage is found1 or the exact words in which it was first expressed4 ,those who1 like the author1 have spent hours and even days in the search of the exact statements1 or the exact references1 of similar famous passages1 will not question the timeliness and usefulness of a book whose distinct purpose it is to bring together into a single volume exact quotations1 with their exact references1 bearing on one of the most time-honored1 and even today the most active and most fruitful of all the sciences1 the queen-mother of all the sciences1 that is1 mathematics4 ,it is hoped that the present volume will prove indispensable to every teacher of mathematics1 to every writer on mathematics1 and that the student of mathematics and the related sciences will find its perusal not only a source of pleasure but of encouragement and inspiration as well4 ,the layman will find it a repository of useful information covering a field of knowledge which1 owing to the unfamiliar and hence repellant character of the language employed by mathematicians1 is peculiarly inaccessible to the general reader4 ,no technical processes or technical facility is required to understand and appreciate the wealth of ideas here set forth in the words of the world's great thinkers4 ,no labor has been spared to make the present volume worthy of a place among collections of a like kind in other fields4 ,ten years have been devoted to its preparation1 years1 which if they could have been more profitably1 could scarcely have been more pleasurably employed4 ,as a result there have been brought together over one thousand more or less familiar passages pertaining to mathematics1 by poets1 philosophers1 historians1 statesmen1 scientists1 and mathematicians4 ,these have been gathered from over three hundred authors1 and have been grouped under twenty heads1 and cross indexed under nearly seven hundred topics4 ,the author's original plan was to give foreign quotations both in the original and in translation1 but with the growth of material this plan was abandoned as infeasible4 ,it was thought to serve the best interest of the greater number of ,english readers to give translations only1 while preserving the references to the original sources1 so that the student or critical reader may readily consult the original of any given extract4 ,in cases where the translation is borrowed the translator's name is inserted in brackets immediately after the author's name4 ,brackets are also used to indicate inserted words or phrases made necessary to bring out the context4 ,the absence of similar ,english works has made the author's work largely that of the pioneer4 ,rebire's 8,mathmatiques et ,mathmaticiens0 and ,ahrens' 8,scherz und ,ernst in der ,mathematik0 have indeed been frequently consulted but rather with a view to avoid overlapping than to receive aid4 ,thus certain topics as the correspondence of ,german and ,french mathematicians1 so excellently treated by ,ahrens1 have purposely been omitted4 ,the repetitions are limited to a small number of famous utterances whose absence from a work of this kind could scarcely be defended on any grounds4 ,no one can be more keenly aware of the shortcomings of a work than its author1 for none can have so intimate an acquaintance with it4 ,among those of the present work is its incompleteness1 but it should be borne in mind that incompleteness is a necessary concomitant of every collection of whatever kind4 ,much less can completeness be expected in a first collection1 made by a single individual1 in his leisure hours1 and in a field which is already boundless and is yet expanding day by day4 ,a collection of great thoughts1 even if complete today1 would be incomplete tomorrow4 ,again1 if some authors are quoted more frequently than others of greater fame and authority1 the reason may be sought not only in the fact that the writings of some authors peculiarly lent themselves to quotation1 a quality singularly absent in other writers of the greatest merit and authority1 but also in this1 that the greatest freedom has been exercised in the choice of selections4 ,the author has followed the bent of his own fancy in collecting whatever seemed to him sufficiently valuable because of its content1 its beauty1 its originality1 or its terseness1 to deserve a place in a 8,memorabilia40 ,great pains has been taken to furnish exact readings and references4 ,in some cases where a passage could not be traced to its first source1 the secondary source has been given rather than the reputed source4 ,for the same reason many references are to later editions rather than to inaccessible first editions4 ,the author feels confident that this work will be of assistance to his co-workers in the field of mathematics and allied fields4 ,if in addition it should aid in a better appreciation of mathematicians and their work on the part of laymen and students in other fields1 the author's foremost aim in the preparation of this work will have been achieved4 ,,robert ,,edouard ,,moritz1 ,september1 #aiac4 ,,contents ,,chapter ,,page ,i4 ,,definitions ,,and ,,object ,,of ,,mathematics #a ,,ii4 ,,the ,,nature ,,of ,,mathematics #aj ,,iii4 ,,estimates ,,of ,,mathematics #ci ,,iv4 ,,the ,,value ,,of ,,mathematics #di ,v4 ,,the ,,teaching ,,of ,,mathematics #gb ,,vi4 ,,study ,,and ,,research ,,in ,,mathematics #hf ,,vii4 ,,modern ,,mathematics #ajh ,,viii4 ,,the ,,mathematician #aba ,,ix4 ,,persons ,,and ,,anecdotes 7,a-,m7 #ace ,x4 ,,persons ,,and ,,anecdotes 7,n-,z7 #aff ,,xi4 ,,mathematics ,,as ,a ,,fine ,,art #aha ,,xii4 ,,mathematics ,,as ,a ,,language #aid ,,xiii4 ,,mathematics ,,and ,,logic #bja ,,xiv4 ,,mathematics ,,and ,,philosophy #bji ,,xv4 ,,mathematics ,,and ,,science #bbd ,,xvi4 ,,arithmetic #bfa ,,xvii4 ,,algebra #bge ,,xviii4 ,,geometry #bib ,,xix4 ,,the ,,calculus ,,and ,,allied ,,topics #cbc ,,xx4 ,,the ,,fundamental ,,concepts ,,of ,,time ,,and ,,space #cde ,,xxi4 ,,paradoxes ,,and ,,curiosities #cfd ,,index #che ,alles ,gescheite ist schon gedacht worden2 man muss nur versuchen1 es noch einmal zu denken4--,,goethe4 ,sprche in ,prosa1 ,ethisches1 ,i4 #a4 ,a great man quotes bravely1 and will not draw on his invention when his memory serves him with a word as good4--,,emerson4 ,letters and ,social ,aims1 ,quotation and ,originality4 ,,memorabilia ,,mathematica ,,memorabilia ,,mathematica ,,chapter ,i ,,definitions ,,and ,,object ,,of ,,mathematics #aja4 ,i think it would be desirable that this form of word mathematics should be reserved for the applications of the science1 and that we should use mathematic in the singular to denote the science itself1 in the same way as we speak of logic1 rhetoric1 or 7own sister to algebra7 music4--,,sylvester1 ,j4 ,j4 ,presidential ,address to the ,british ,association1 ,exeter ,british ,association ,report 7#ahfi72 ,collected ,mathematical ,papers1 ,vol4 #b1 p4 #fei4 #ajb4 444 all the sciences which have for their end investigations concerning order and measure1 are related to mathematics1 it being of small importance whether this measure be sought in numbers1 forms1 stars1 sounds1 or any other object2 that1 accordingly1 there ought to exist a general science which should explain all that can be known about order and measure1 considered independently of any application to a particular subject1 and that1 indeed1 this science has its own proper name1 consecrated by long usage1 to wit1 mathematics4 ,and a proof that it far surpasses in facility and importance the sciences which depend upon it is that it embraces at once all the objects to which these are devoted and a great many others besides2 4444 --,,descartes4 ,rules for the ,direction of the ,mind1 ,philosophy of ,d4 ,torrey 7,new ,york1 #ahib71 p4 #gb4 #ajc4 ,mathematics has for its object the indirect measurement of magnitudes1 and it purposes to determine magnitudes by each other1 according to the precise relations which exist between them4--,,comte4 ,positive ,philosophy ,martineau1 ,bk4 #a1 chap4 #a4 #ajd4 ,the business of concrete mathematics is to discover the equations which express the mathematical laws of the phenomenon under consideration2 and these equations are the starting-point of the calculus1 which must obtain from them certain quantities by means of others4--,,comte4 ,positive ,philosophy ,martineau1 ,bk4 #a1 chap4 #b4 #aje4 ,mathematics is the science of the connection of magnitudes4 ,magnitude is anything that can be put equal or unequal to another thing4 ,two things are equal when in every assertion each may be replaced by the other4--,,grassmann1 ,,hermann4 ,stcke aus dem ,lehrbuche der ,arithmetik1 ,werke 7,leipzig1 #aijd71 ,bd4 #b1 p4 #bih4 #ajf4 ,mathematic is either ,pure or ,mixed3 ,to ,pure ,mathematic belong those sciences which handle ,quantity entirely severed from matter and from axioms of natural philosophy4 ,these are two1 ,geometry and ,arithmetic2 the one handling quantity continued1 the other dissevered4444 ,mixed ,mathematic has for its subject some axioms and parts of natural philosophy1 and considers quantity in so far as it assists to explain1 demonstrate and actuate these4 --,,bacon1 ,,francis4 ,de ,augmentis1 ,bk4 #c2 ,advancement of ,learning1 ,bk4 #b4 #ajg4 ,the ideas which these sciences1 ,geometry1 ,theoretical ,arithmetic and ,algebra involve extend to all objects and changes which we observe in the external world2 and hence the consideration of mathematical relations forms a large portion of many of the sciences which treat of the phenomena and laws of external nature1 as ,astronomy1 ,optics1 and ,mechanics4 ,such sciences are hence often termed ,mixed ,mathematics1 the relations of space and number being1 in these branches of knowledge1 combined with principles collected from special observation2 while ,geometry1 ,algebra1 and the like subjects1 which involve no result of experience1 are called ,pure ,mathematics4--,,whewell1 ,,william4 ,the ,philosophy of the ,inductive ,sciences1 ,part #a1 ,bk4 #b1 chap4 ,i1 sect4 #d4 7,london1 #aheh74 #ajh4 ,higher ,mathematics is the art of reasoning about numerical relations between natural phenomena2 and the several sections of ,higher ,mathematics are different modes of viewing these relations4--,,mellor1 ,j4 ,w4 ,higher ,mathematics for ,students of ,chemistry and ,physics 7,new ,york1 #aijb71 ,prologue4 #aji4 ,number1 place1 and combination 444 the three intersecting but distinct spheres of thought to which all mathematical ideas admit of being referred4--,,sylvester1 ,j4 ,j4 ,philosophical ,magazine1 ,vol4 #bd 7#ahdd71 p4 #bhe2 ,collected ,mathematical ,papers1 ,vol4 #a1 p4 #ia4 #aaj4 ,there are three ruling ideas1 three so to say1 spheres of thought1 which pervade the whole body of mathematical science1 to some one or other of which1 or to two or all three of them combined1 every mathematical truth admits of being referred2 these are the three cardinal notions1 of ,number1 ,space and ,order4 ,arithmetic has for its object the properties of number in the abstract4 ,in algebra1 viewed as a science of operations1 order is the predominating idea4 ,the business of geometry is with the evolution of the properties of space1 or of bodies viewed as existing in space4--,,sylvester1 ,j4 ,j4 ,a ,probationary ,lecture on ,geometry1 ,york ,british ,association ,report 7#ahdd71 ,part #b2 ,collected ,mathematical ,papers1 ,vol4 #b1 p4 #e4 #aaa4 ,the object of pure mathematics is those relations which may be conceptually established among any conceived elements whatsoever by assuming them contained in some ordered manifold2 the law of order of this manifold must be subject to our choice2 the latter is the case in both of the only conceivable kinds of manifolds1 in the discrete as well as in the continuous4 --,,papperitz1 ,e4 ber das ,system der rein mathematischen ,wissenschaften1 ,jahresbericht der ,deutschen ,mathematiker-,vereinigung1 ,bd4 #a1 p4 #cf4 #aab4 ,pure mathematics is not concerned with magnitude4 ,it is merely the doctrine of notation of relatively ordered thought operations which have become mechanical4--,,novalis4 ,schriften 7,berlin1 #aija71 ,zweiter ,teil1 p4 #bhb4 #aac4 ,any conception which is definitely and completely determined by means of a finite number of specifications1 say by assigning a finite number of elements1 is a mathematical conception4 ,mathematics has for its function to develop the consequences involved in the definition of a group of mathematical conceptions4 ,interdependence and mutual logical consistency among the members of the group are postulated1 otherwise the group would either have to be treated as several distinct groups1 or would lie beyond the sphere of mathematics4--,,chrystal1 ,,george4 ,encyclopedia ,britannica 7#ith edition71 ,article 8,mathematics40 #aad4 ,the purely formal sciences1 logic and mathematics1 deal with those relations which are1 or can be1 independent of the particular content or the substance of objects4 ,to mathematics in particular fall those relations between objects which involve the concepts of magnitude1 of measure and of number4--,,hankel1 ,,hermann4 ,theorie der ,complexen ,zahlensysteme1 7,leipzig1 #ahfg71 p4 #a4 #aae4 ,quantity is that which is operated with according to fixed mutually consistent laws4 ,both operator and operand must derive their meaning from the laws of operation4 ,in the case of ordinary algebra these are the three laws already indicated the commutative1 associative1 and distributive laws1 in the algebra of quaternions the same save the law of commutation for multiplication and division1 and so on4 ,it may be questioned whether this definition is sufficient1 and it may be objected that it is vague2 but the reader will do well to reflect that any definition must include the linear algebras of ,peirce1 the algebra of logic1 and others that may be easily imagined1 although they have not yet been developed4 ,this general definition of quantity enables us to see how operators may be treated as quantities1 and thus to understand the rationale of the so called symbolical methods4--,,chrystal1 ,,george4 ,encyclopedia ,britannica 7#ith edition71 ,article 8,mathematics40 #aaf4 ,mathematics--in a strict sense--is the abstract science which investigates deductively the conclusions implicit in the elementary conceptions of spatial and numerical relations4 --,,murray1 ,j4 ,a4 ,h4 ,a ,new ,english ,dictionary4 #aag4 ,everything that the greatest minds of all times have accomplished toward the comprehension of forms by means of concepts is gathered into one great science1 mathematics4 --,,herbart1 ,j4 ,f4 ,pestalozzi's ,idee eines ,a ,b ,c der ,anschauung1 ,werke ,kehrbach1 7,langensalza1 #ahij71 ,bd4 #a1 p4 #afc4 #aah4 ,perhaps the least inadequate description of the general scope of modern ,pure ,mathematics--,i will not call it a definition--would be to say that it deals with form1 in a very general sense of the term2 this would include algebraic form1 functional relationship1 the relations of order in any ordered set of entities such as numbers1 and the analysis of the peculiarities of form of groups of operations4--,,hobson1 ,e4 ,w4 ,presidential ,address ,british ,association for the ,advancement of ,science 7#aiaj72 ,nature1 ,vol4 #hd1 p4 #bhg4 #aai4 ,the ideal of mathematics should be to erect a calculus to facilitate reasoning in connection with every province of thought1 or of external experience1 in which the succession of thoughts1 or of events can be definitely ascertained and precisely stated4 ,so that all serious thought which is not philosophy1 or inductive reasoning1 or imaginative literature1 shall be mathematics developed by means of a calculus4 --,,whitehead1 ,a4 ,n4 ,universal ,algebra 7,cambridge1 #ahih71 ,preface4 #abj4 ,mathematics is the science which draws necessary conclusions4--,,peirce1 ,,benjamin4 ,linear ,associative ,algebra1 ,american ,journal of ,mathematics1 ,vol4 #d 7#ahha71 p4 #ig4 #aba4 ,mathematics is the universal art apodictic4--,,smith1 ,w4 ,b4 ,quoted by ,keyser1 ,c4 ,j4 in ,lectures on ,science1 ,philosophy and ,art 7,new ,york1 #aijh71 p4 #ac4 #abb4 ,mathematics in its widest signification is the development of all types of formal1 necessary1 deductive reasoning4 --,,whitehead1 ,a4 ,n4 ,universal ,algebra 7,cambridge1 #ahih71 ,preface1 p4 vi4 #abc4 ,mathematics in general is fundamentally the science of self-evident things4--,,klein1 ,,felix4 ,anwendung der ,differential- und ,integralrechnung auf ,geometrie 7,leipzig1 #aijb71 p4 #bf4 #abd4 ,a mathematical science is any body of propositions which is capable of an abstract formulation and arrangement in such a way that every proposition of the set after a certain one is a formal logical consequence of some or all the preceding propositions4 ,mathematics consists of all such mathematical sciences4--,,young1 ,,charles ,,wesley4 ,fundamental ,concepts of ,algebra and ,geometry 7,new ,york1 #aiaa71 p4 #bbb4 #abe4 ,pure mathematics is a collection of hypothetical1 deductive theories1 each consisting of a definite system of primitive1 undefined1 concepts or symbols and primitive1 unproved1 but self-consistent assumptions 7commonly called axioms7 together with their logically deducible consequences following by rigidly deductive processes without appeal to intuition4--,,fitch1 ,g4 ,d4 ,the ,fourth ,dimension simply ,explained 7,new ,york1 #aiaj71 p4 #eh4 #abf4 ,the whole of ,mathematics consists in the organization of a series of aids to the imagination in the process of reasoning4 --,,whitehead1 ,a4 ,n4 ,universal ,algebra 7,cambridge1 #ahih71 p4 #ab4 #abg4 ,pure mathematics consists entirely of such asseverations as that1 if such and such a proposition is true of anything1 then such and such another proposition is true of that thing4 ,it is essential not to discuss whether the first proposition is really true1 and not to mention what the anything is of which it is supposed to be true4444 ,if our hypothesis is about anything and not about some one or more particular things1 then our deductions constitute mathematics4 ,thus mathematics may be defined as the subject in which we never know what we are talking about1 nor whether what we are saying is true4--,,russell1 ,,bertrand4 ,recent ,work on the ,principles of ,mathematics1 ,international ,monthly1 ,vol4 #d 7#aija71 p4 #hd4 #abh4 ,pure ,mathematics is the class of all propositions of the form 8p implies q10 where p and q are propositions containing one or more variables1 the same in the two propositions1 and neither p nor q contains any constants except logical constants4 ,and logical constants are all notions definable in terms of the following3 ,implication1 the relation of a term to a class of which it is a member1 the notion of such that1 the notion of relation1 and such further notions as may be involved in the general notion of propositions of the above form4 ,in addition to these1 ,mathematics uses a notion which is not a constituent of the propositions which it considers--namely1 the notion of truth4--,,russell1 ,,bertrand4 ,principles of ,mathematics 7,cambridge1 #aijc71 p4 #a4 #abi4 ,the object of pure ,physic is the unfolding of the laws of the intelligible world2 the object of pure ,mathematic that of unfolding the laws of human intelligence4--,,sylvester1 ,j4 ,j4 ,on a theorem1 connected with ,newton's ,rule1 etc41 ,collected ,mathematical ,papers1 ,vol4 #c1 p4 #dbd4 #acj4 ,first of all1 we ought to observe1 that mathematical propositions1 properly so called1 are always judgments a priori1 and not empirical1 because they carry along with them necessity1 which can never be deduced from experience4 ,if people should object to this1 ,i am quite willing to confine my statements to pure mathematics1 the very concept of which implies that it does not contain empirical1 but only pure knowledge a priori4--,,kant1 ,,immanuel4 ,critique of ,pure ,reason ,mller1 7,new ,york1 #aijj71 p4 #gbj4 #aca4 ,mathematics1 the science of the ideal1 becomes the means of investigating1 understanding and making known the world of the real4 ,the complex is expressed in terms of the simple4 ,from one point of view mathematics may be defined as the science of successive substitutions of simpler concepts for more complex4444 --,,white1 ,,william ,f4 ,a ,scrap-book of ,elementary ,mathematics1 7,chicago1 #aijh71 p4 #bae4 #acb4 ,the critical mathematician has abandoned the search for truth4 ,he no longer flatters himself that his propositions are or can be known to him or to any other human being to be true2 and he contents himself with aiming at the correct1 or the consistent4 ,the distinction is not annulled nor even blurred by the reflection that consistency contains immanently a kind of truth4 ,he is not absolutely certain1 but he believes profoundly that it is possible to find various sets of a few propositions each such that the propositions of each set are compatible1 that the propositions of each such set imply other propositions1 and that the latter can be deduced from the former with certainty4 ,that is to say1 he believes that there are systems of coherent or consistent propositions1 and he regards it his business to discover such systems4 ,any such system is a branch of mathematics4 --,,keyser1 ,c4 ,j4 ,science1 ,new ,series1 ,vol4 #ce1 p4 #ajg4 #acc4 ,mathematics is the study of ideal constructions 7often applicable to real problems71 and the discovery thereby of relations between the parts of these constructions1 before unknown4--,,peirce1 ,c4 ,s4 ,century ,dictionary1 ,article 8,mathematics40 #acd4 ,mathematics is that form of intelligence in which we bring the objects of the phenomenal world under the control of the conception of quantity4 ,provisional definition4--,,howison1 ,g4 ,h4 ,the ,departments of ,mathematics1 and their ,mutual ,relations2 ,journal of ,speculative ,philosophy1 ,vol4 #e1 p4 #afd4 #ace4 ,mathematics is the science of the functional laws and transformations which enable us to convert figured extension and rated motion into number4--,,howison1 ,g4 ,h4 ,the ,departments of ,mathematics1 and their ,mutual ,relations2 ,journal of ,speculative ,philosophy1 ,vol4 #e1 p4 #agj4 ,,chapter ,,ii ,,the ,,nature ,,of ,,mathematics #bja4 ,mathematics1 from the earliest times to which the history of human reason can reach1 has followed1 among that wonderful people of the ,greeks1 the safe way of science4 ,but it must not be supposed that it was as easy for mathematics as for logic1 in which reason is concerned with itself alone1 to find1 or rather to make for itself that royal road4 ,i believe1 on the contrary1 that there was a long period of tentative work 7chiefly still among the ,egyptians71 and that the change is to be ascribed to a revolution1 produced by the happy thought of a single man1 whose experiments pointed unmistakably to the path that had to be followed1 and opened and traced out for the most distant times the safe way of a science4 ,the history of that intellectual revolution1 which was far more important than the passage round the celebrated ,cape of ,good ,hope1 and the name of its fortunate author1 have not been preserved to us4444 ,a new light flashed on the first man who demonstrated the properties of the isosceles triangle 7whether his name was ,thales or any other name71 for he found that he had not to investigate what he saw in the figure1 or the mere concepts of that figure1 and thus to learn its properties2 but that he had to produce 7by construction7 what he had himself1 according to concepts a priori1 placed into that figure and represented in it1 so that1 in order to know anything with certainty a priori1 he must not attribute to that figure anything beyond what necessarily follows from what he has himself placed into it1 in accordance with the concept4 --,,kant1 ,,immanuel4 ,critique of ,pure ,reason1 ,preface to the ,second ,edition ,mller1 7,new ,york1 #aijj71 p4 #fij4 #bjb4 ,when followed in the proper spirit1 there is no study in the world which brings into more harmonious action all the faculties of the mind than the one mathematics of which ,i stand here as the humble representative and advocate4 ,there is none other which prepares so many agreeable surprises for its followers1 more wonderful than the transformation scene of a pantomime1 or1 like this1 seems to raise them1 by successive steps of initiation to higher and higher states of conscious intellectual being4--,,sylvester1 ,j4 ,j4 ,a ,plea for the ,mathematician1 ,nature1 ,vol4 #a1 p4 #bfa4 #bjc4 ,thought-economy is most highly developed in mathematics1 that science which has reached the highest formal development1 and on which natural science so frequently calls for assistance4 ,strange as it may seem1 the strength of mathematics lies in the avoidance of all unnecessary thoughts1 in the utmost economy of thought-operations4 ,the symbols of order1 which we call numbers1 form already a system of wonderful simplicity and economy4 ,when in the multiplication of a number with several digits we employ the multiplication table and thus make use of previously accomplished results rather than to repeat them each time1 when by the use of tables of logarithms we avoid new numerical calculations by replacing them by others long since performed1 when we employ determinants instead of carrying through from the beginning the solution of a system of equations1 when we decompose new integral expressions into others that are familiar1--we see in all this but a faint reflection of the intellectual activity of a ,lagrange or ,cauchy1 who with the keen discernment of a military commander marshalls a whole troop of completed operations in the execution of a new one4--,,mach1 ,e4 ,populr-wissenschafliche ,vorlesungen 7#aijh71 pp4 #bbd-bbe4 #bjd4 ,pure mathematics proves itself a royal science both through its content and form1 which contains within itself the cause of its being and its methods of proof4 ,for in complete independence mathematics creates for itself the object of which it treats1 its magnitudes and laws1 its formulas and symbols4 --,,dillmann1 ,e4 ,die ,mathematik die ,fackeltrgerin einer neuen ,zeit 7,stuttgart1 #ahhi71 p4 #id4 #bje4 ,the essence of mathematics lies in its freedom4 --,,cantor1 ,,george4 ,mathematische ,annalen1 ,bd4 #ba1 p4 #efd4 #bjf4 ,mathematics pursues its own course unrestrained1 not indeed with an unbridled licence which submits to no laws1 but rather with the freedom which is determined by its own nature and in conformity with its own being4--,,hankel1 ,,hermann4 ,die ,entwickelung der ,mathematik in den letzten ,jahrhunderten 7,tbingen1 #ahhd71 p4 #af4 #bjg4 ,mathematics is perfectly free in its development and is subject only to the obvious consideration1 that its concepts must be free from contradictions in themselves1 as well as definitely and orderly related by means of definitions to the previously existing and established concepts4--,,cantor1 ,,george4 ,grundlagen einer allgemeinen ,manigfaltigkeitslehre 7,leipzig1 #ahhc71 ,sect4 #h4 #bjh4 ,mathematicians assume the right to choose1 within the limits of logical contradiction1 what path they please in reaching their results4--,,adams1 ,,henry4 ,a ,letter to ,american ,teachers of ,history 7,washington1 #aiaj71 ,introduction1 p4 v4 #bji4 ,mathematics is the predominant science of our time2 its conquests grow daily1 though without noise2 he who does not employ it for himself1 will some day find it employed against himself4--,,herbart1 ,j4 ,f4 ,werke ,kehrbach 7,langensalza1 #ahij71 ,bd4 #e1 p4 #aje4 #baj4 ,mathematics is not the discoverer of laws1 for it is not induction2 neither is it the framer of theories1 for it is not hypothesis2 but it is the judge over both1 and it is the arbiter to which each must refer its claims2 and neither law can rule nor theory explain without the sanction of mathematics4 --,,peirce1 ,,benjamin4 ,linear ,associative ,algebra1 ,american ,journal of ,mathematics1 ,vol4 #d 7#ahha71 p4 #ig4 #baa4 ,mathematics is a science continually expanding2 and its growth1 unlike some political and industrial events1 is attended by universal acclamation4--,,white1 ,h4 ,s4 ,congress of ,arts and ,sciences 7,boston and ,new ,york1 #aije71 ,vol4 #a1 p4 #dee4 #bab4 ,mathematics accomplishes really nothing outside of the realm of magnitude2 marvellous1 however1 is the skill with which it masters magnitude wherever it finds it4 ,we recall at once the network of lines which it has spun about heavens and earth2 the system of lines to which azimuth and altitude1 declination and right ascension1 longitude and latitude are referred2 those abscissas and ordinates1 tangents and normals1 circles of curvature and evolutes2 those trigonometric and logarithmic functions which have been prepared in advance and await application4 ,a look at this apparatus is sufficient to show that mathematicians are not magicians1 but that everything is accomplished by natural means2 one is rather impressed by the multitude of skilful machines1 numerous witnesses of a manifold and intensely active industry1 admirably fitted for the acquisition of true and lasting treasures4--,,herbart1 ,j4 ,f4 ,werke ,kehrbach 7,langensalza1 #ahij71 ,bd4 #e1 p4 #aja4 #bac4 ,they mathematicians only take those things into consideration1 of which they have clear and distinct ideas1 designating them by proper1 adequate1 and invariable names1 and premising only a few axioms which are most noted and certain to investigate their affections and draw conclusions from them1 and agreeably laying down a very few hypotheses1 such as are in the highest degree consonant with reason and not to be denied by anyone in his right mind4 ,in like manner they assign generations or causes easy to be understood and readily admitted by all1 they preserve a most accurate order1 every proposition immediately following from what is supposed and proved before1 and reject all things howsoever specious and probable which can not be inferred and deduced after the same manner4--,,barrow1 ,,isaac4 ,mathematical ,lectures 7,london1 #agcd71 p4 #ff4 #bad4 ,the dexterous management of terms and being able to fend and prove with them1 ,i know has and does pass in the world for a great part of learning2 but it is learning distinct from knowledge1 for knowledge consists only in perceiving the habitudes and relations of ideas one to another1 which is done without words2 the intervention of sounds helps nothing to it4 ,and hence we see that there is least use of distinction where there is most knowledge3 ,i mean in mathematics1 where men have determined ideas with known names to them2 and so1 there being no room for equivocations1 there is no need of distinctions4 --,,locke1 ,,john4 ,conduct of the ,understanding1 ,sect4 #ca4 #bae4 ,in mathematics it sophistry had no place from the beginning3 ,mathematicians having had the wisdom to define accurately the terms they use1 and to lay down1 as axioms1 the first principles on which their reasoning is grounded4 ,accordingly we find no parties among mathematicians1 and hardly any disputes4 --,,reid1 ,,thomas4 ,essays on the ,intellectual ,powers of ,man1 ,essay #a1 chap4 #a4 #baf4 ,in most sciences one generation tears down what another has built and what one has established another undoes4 ,in ,mathematics alone each generation builds a new story to the old structure4--,,hankel1 ,,hermann4 ,die ,entwickelung der ,mathematik in den letzten ,jahrhunderten 7,tbingen1 #ahhd71 p4 #be4 #bag4 ,mathematics1 the priestess of definiteness and clearness4--,,herbart1 ,j4 ,f4 ,werke ,kehrbach 7,langensalza1 #ahij71 ,bd4 #a1 p4 #aga4 #bah4 444 mathematical analysis is co-extensive with nature itself1 it defines all perceivable relations1 measures times1 spaces1 forces1 temperatures2 it is a difficult science which forms but slowly1 but preserves carefully every principle once acquired2 it increases and becomes stronger incessantly amidst all the changes and errors of the human mind4 ,its chief attribute is clearness2 it has no means for expressing confused ideas4 ,it compares the most diverse phenomena and discovers the secret analogies which unite them4 ,if matter escapes us1 as that of air and light because of its extreme tenuity1 if bodies are placed far from us in the immensity of space1 if man wishes to know the aspect of the heavens at successive periods separated by many centuries1 if gravity and heat act in the interior of the solid earth at depths which will forever be inaccessible1 mathematical analysis is still able to trace the laws of these phenomena4 ,it renders them present and measurable1 and appears to be the faculty of the human mind destined to supplement the brevity of life and the imperfection of the senses1 and what is even more remarkable1 it follows the same course in the study of all phenomena2 it explains them in the same language1 as if in witness to the unity and simplicity of the plan of the universe1 and to make more manifest the unchangeable order which presides over all natural causes4 --,,fourier1 ,j4 ,thorie ,analytique de la ,chaleur1 ,discours ,prliminaire4 #bai4 ,let us now declare the means whereby our understanding can rise to knowledge without fear of error4 ,there are two such means3 intuition and deduction4 ,by intuition ,i mean not the varying testimony of the senses1 nor the deductive judgment of imagination naturally extravagant1 but the conception of an attentive mind so distinct and so clear that no doubt remains to it with regard to that which it comprehends2 or1 what amounts to the same thing1 the self-evidencing conception of a sound and attentive mind1 a conception which springs from the light of reason alone1 and is more certain1 because more simple1 than deduction itself4444 ,it may perhaps be asked why to intuition we add this other mode of knowing1 by deduction1 that is to say1 the process which1 from something of which we have certain knowledge1 draws consequences which necessarily follow therefrom4 ,but we are obliged to admit this second step2 for there are a great many things which1 without being evident of themselves1 nevertheless bear the marks of certainty if only they are deduced from true and incontestable principles by a continuous and uninterrupted movement of thought1 with distinct intuition of each thing2 just as we know that the last link of a long chain holds to the first1 although we can not take in with one glance of the eye the intermediate links1 provided that1 after having run over them in succession1 we can recall them all1 each as being joined to its fellows1 from the first up to the last4 ,thus we distinguish intuition from deduction1 inasmuch as in the latter case there is conceived a certain progress or succession1 while it is not so in the former2444 whence it follows that primary propositions1 derived immediately from principles1 may be said to be known1 according to the way we view them1 now by intuition1 now by deduction2 although the principles themselves can be known only by intuition1 the remote consequences only by deduction4 --,,descartes4 ,rules for the ,direction of the ,mind1 ,philosophy of ,d4 ,torrey 7,new ,york1 #ahib71 pp4 #fd1 #fe4 #bbj4 ,analysis and natural philosophy owe their most important discoveries to this fruitful means1 which is called induction4 ,newton was indebted to it for his theorem of the binomial and the principle of universal gravity4--,,laplace4 ,a ,philosophical ,essay on ,probabilities ,truscott and ,emory 7,new ,york #aijb71 p4 #agf4 #bba4 ,there is in every step of an arithmetical or algebraical calculation a real induction1 a real inference from facts to facts1 and what disguises the induction is simply its comprehensive nature1 and the consequent extreme generality of its language4 --,,mill1 ,j4 ,s4 ,system of ,logic1 ,bk4 #b1 chap4 #f1 #b4 #bbb4 ,it would appear that ,deductive and ,demonstrative ,sciences are all1 without exception1 ,inductive ,sciences3 that their evidence is that of experience1 but that they are also1 in virtue of the peculiar character of one indispensable portion of the general formulae according to which their inductions are made1 ,hypothetical ,sciences4 ,their conclusions are true only upon certain suppositions1 which are1 or ought to be1 approximations to the truth1 but are seldom1 if ever1 exactly true2 and to this hypothetical character is to be ascribed the peculiar certainty1 which is supposed to be inherent in demonstration4--,,mill1 ,j4 ,s4 ,system of ,logic1 ,bk4 #b1 chap4 #f1 #a4 #bbc4 ,the peculiar character of mathematical truth is1 that it is necessarily and inevitably true2 and one of the most important lessons which we learn from our mathematical studies is a knowledge that there are such truths1 and a familiarity with their form and character4 ,this lesson is not only lost1 but read backward1 if the student is taught that there is no such difference1 and that mathematical truths themselves are learned by experience4--,,whewell1 ,w4 ,thoughts on the ,study of ,mathematics4 ,principles of ,english ,university ,education 7,london1 #ahch74 #bbd4 ,these sciences1 ,geometry1 ,theoretical ,arithmetic and ,algebra1 have no principles besides definitions and axioms1 and no process of proof but deduction2 this process1 however1 assuming a most remarkable character2 and exhibiting a combination of simplicity and complexity1 of rigour and generality1 quite unparalleled in other subjects4--,,whewell1 ,w4 ,the ,philosophy of the ,inductive ,sciences1 ,part #a1 ,bk4 #b1 chap4 #a1 sect4 #b 7,london1 #aheh74 #bbe4 ,the apodictic quality of mathematical thought1 the certainty and correctness of its conclusions1 are due1 not to a special mode of ratiocination1 but to the character of the concepts with which it deals4 ,what is that distinctive characteristic8 ,i answer3 precision1 sharpness1 completeness1#a of definition4 ,but how comes your mathematician by such completeness8 ,there is no mysterious trick involved2 some ideas admit of such precision1 others do not2 and the mathematician is one who deals with those that do4--,,keyser1 ,c4 ,j4 ,the ,universe and ,beyond2 ,hibbert ,journal1 ,vol4 #c 7#aijd-aije71 p4 #cji4 #a i4e41 in terms of the absolutely clear and indefinable4 #bbf4 ,the reasoning of mathematicians is founded on certain and infallible principles4 ,every word they use conveys a determinate idea1 and by accurate definitions they excite the same ideas in the mind of the reader that were in the mind of the writer4 ,when they have defined the terms they intend to make use of1 they premise a few axioms1 or self-evident principles1 that every one must assent to as soon as proposed4 ,they then take for granted certain postulates1 that no one can deny them1 such as1 that a right line may be drawn from any given point to another1 and from these plain1 simple principles they have raised most astonishing speculations1 and proved the extent of the human mind to be more spacious and capacious than any other science4--,,adams1 ,,john4 ,diary1 ,works 7,boston1 #ahej71 ,vol4 #b1 p4 #ba4 #bbg4 ,it may be observed of mathematicians that they only meddle with such things as are certain1 passing by those that are doubtful and unknown4 ,they profess not to know all things1 neither do they affect to speak of all things4 ,what they know to be true1 and can make good by invincible arguments1 that they publish and insert among their theorems4 ,of other things they are silent and pass no judgment at all1 choosing rather to acknowledge their ignorance1 than affirm anything rashly4 ,they affirm nothing among their arguments or assertions which is not most manifestly known and examined with utmost rigour1 rejecting all probable conjectures and little witticisms4 ,they submit nothing to authority1 indulge no affection1 detest subterfuges of words1 and declare their sentiments1 as in a court of justice1 without passion1 without apology2 knowing that their reasons1 as ,seneca testifies of them1 are not brought to persuade1 but to compel4--,,barrow1 ,,isaac4 ,mathematical ,lectures 7,london1 #agcd71 p4 #fd4 #bbh4 ,what is exact about mathematics but exactness8 ,and is not this a consequence of the inner sense of truth8--,,goethe4 ,sprche in ,prosa1 ,natur1 #f1 #idh4 #bbi4 444 the three positive characteristics that distinguish mathematical knowledge from other knowledge 444 may be briefly expressed as follows3 first1 mathematical knowledge bears more distinctly the imprint of truth on all its results than any other kind of knowledge2 secondly1 it is always a sure preliminary step to the attainment of other correct knowledge2 thirdly1 it has no need of other knowledge4--,,schubert1 ,h4 ,mathematical ,essays and ,recreations 7,chicago1 #ahih71 p4 #ce4 #bcj4 ,it is now necessary to indicate more definitely the reason why mathematics not only carries conviction in itself1 but also transmits conviction to the objects to which it is applied4 ,the reason is found1 first of all1 in the perfect precision with which the elementary mathematical concepts are determined2 in this respect each science must look to its own salvation4444 ,but this is not all4 ,as soon as human thought attempts long chains of conclusions1 or difficult matters generally1 there arises not only the danger of error but also the suspicion of error1 because since all details cannot be surveyed with clearness at the same instant one must in the end be satisfied with a belief that nothing has been overlooked from the beginning4 ,every one knows how much this is the case even in arithmetic1 the most elementary use of mathematics4 ,no one would imagine that the higher parts of mathematics fare better in this respect2 on the contrary1 in more complicated conclusions the uncertainty and suspicion of hidden errors increases in rapid progression4 ,how does mathematics manage to rid itself of this inconvenience which attaches to it in the highest degree8 ,by making proofs more rigorous8 ,by giving new rules according to which the old rules shall be applied8 ,not in the least4 ,a very great uncertainty continues to attach to the result of each single computation4 ,but there are checks4 ,in the realm of mathematics each point may be reached by a hundred different ways2 and if each of a hundred ways leads to the same point1 one may be sure that the right point has been reached4 ,a calculation without a check is as good as none4 ,just so it is with every isolated proof in any speculative science whatever2 the proof may be ever so ingenious1 and ever so perfectly true and correct1 it will still fail to convince permanently4 ,he will therefore be much deceived1 who1 in metaphysics1 or in psychology which depends on metaphysics1 hopes to see his greatest care in the precise determination of the concepts and in the logical conclusions rewarded by conviction1 much less by success in transmitting conviction to others4 ,not only must the conclusions support each other1 without coercion or suspicion of subreption1 but in all matters originating in experience1 or judging concerning experience1 the results of speculation must be verified by experience1 not only superficially1 but in countless special cases4--,,herbart1 ,j4 ,f4 ,werke ,kehrbach 7,langensalza1 #ahij71 ,bd4 #e1 p4 #aje4 #bca4 ,in mathematics we behold the conscious logical activity of the human mind in its purest and most perfect form4 ,here we learn to realize the laborious nature of the process1 the great care with which it must proceed1 the accuracy which is necessary to determine the exact extent of the general propositions arrived at1 the difficulty of forming and comprehending abstract concepts2 but here we learn also to place confidence in the certainty1 scope and fruitfulness of such intellectual activity4--,,helmholtz1 ,h4 ,ueber das ,verhltniss der ,naturwissenschaften zur ,gesammtheit der ,wissenschaft1 ,vortrge und ,reden1 ,bd4 #a 7#ahif71 p4 #agf4 #bcb4 ,it is true that mathematics1 owing to the fact that its whole content is built up by means of purely logical deduction from a small number of universally comprehended principles1 has not unfittingly been designated as the science of the self-evident ,selbstverstndlichen4 ,experience however1 shows that for the majority of the cultured1 even of scientists1 mathematics remains the science of the incomprehensible ,unverstndlichen4--,,pringsheim1 ,,alfred4 ,ueber ,wert und angeblichen ,unwert der ,mathematik1 ,jahresbericht der ,deutschen ,mathematiker ,vereinigung 7#aijd71 p4 #ceg4 #bcc4 ,mathematical reasoning is deductive in the sense that it is based upon definitions which1 as far as the validity of the reasoning is concerned 7apart from any existential import71 needs only the test of self-consistency4 ,thus no external verification of definitions is required in mathematics1 as long as it is considered merely as mathematics4--,,whitehead1 ,a4 ,n4 ,universal ,algebra 7,cambridge1 #ahih71 ,preface1 p4 vi4 #bcd4 ,the mathematician pays not the least regard either to testimony or conjecture1 but deduces everything by demonstrative reasoning1 from his definitions and axioms4 ,indeed1 whatever is built upon conjecture1 is improperly called science2 for conjecture may beget opinion1 but cannot produce knowledge4 --,,reid1 ,,thomas4 ,essays on the ,intellectual ,powers of ,man1 ,essay #a1 chap4 #c4 #bce4 444 for the saving the long progression of the thoughts to remote and first principles in every case1 the mind should provide itself several stages2 that is to say1 intermediate principles1 which it might have recourse to in the examining those positions that come in its way4 ,these1 though they are not self-evident principles1 yet1 if they have been made out from them by a wary and unquestionable deduction1 may be depended on as certain and infallible truths1 and serve as unquestionable truths to prove other points depending upon them1 by a nearer and shorter view than remote and general maxims4444 ,and thus mathematicians do1 who do not in every new problem run it back to the first axioms through all the whole train of intermediate propositions4 ,certain theorems that they have settled to themselves upon sure demonstration1 serve to resolve to them multitudes of propositions which depend on them1 and are as firmly made out from thence as if the mind went afresh over every link of the whole chain that tie them to first self-evident principles4--,,locke1 ,,john4 ,the ,conduct of the ,understanding1 ,sect4 #ba4 #bcf4 ,those intervening ideas1 which serve to show the agreement of any two others1 are called proofs2 and where the agreement or disagreement is by this means plainly and clearly perceived1 it is called demonstration2 it being shown to the understanding1 and the mind made to see that it is so4 ,a quickness in the mind to find out these intermediate ideas1 7that shall discover the agreement or disagreement of any other7 and to apply them right1 is1 ,i suppose1 that which is called sagacity4--,,locke1 ,,john4 ,an ,essay concerning ,human ,understanding1 ,bk4 #f1 chaps4 #b1 #c4 #bcg4 444 the speculative propositions of mathematics do not relate to facts2 444 all that we are convinced of by any demonstration in the science1 is of a necessary connection subsisting between certain suppositions and certain conclusions4 ,when we find these suppositions actually take place in a particular instance1 the demonstration forces us to apply the conclusion4 ,thus1 if ,i could form a triangle1 the three sides of which were accurately mathematical lines1 ,i might affirm of this individual figure1 that its three angles are equal to two right angles2 but1 as the imperfection of my senses puts it out of my power to be1 in any case1 certain of the exact correspondence of the diagram which ,i delineate1 with the definitions given in the elements of geometry1 ,i never can apply with confidence to a particular figure1 a mathematical theorem4 ,on the other hand1 it appears from the daily testimony of our senses that the speculative truths of geometry may be applied to material objects with a degree of accuracy sufficient for the purposes of life2 and from such applications of them1 advantages of the most important kind have been gained to society4--,,stewart1 ,,dugald4 ,elements of the ,philosophy of the ,human ,mind1 ,part #c1 chap4 #a1 sect4 #c4 #bch4 ,no process of sound reasoning can establish a result not contained in the premises4--,,mellor1 ,j4 ,w4 ,higher ,mathematics for ,students of ,chemistry and ,physics 7,new ,york1 #aijb71 p4 #b4 #bci4 444 we cannot get more out of the mathematical mill than we put into it1 though we may get it in a form infinitely more useful for our purpose4--,,hopkinson1 ,,john4 ,james ,forrest ,lecture1 #ahid4 #bdj4 ,the iron labor of conscious logical reasoning demands great perseverance and great caution2 it moves on but slowly1 and is rarely illuminated by brilliant flashes of genius4 ,it knows little of that facility with which the most varied instances come thronging into the memory of the philologist or historian4 ,rather is it an essential condition of the methodical progress of mathematical reasoning that the mind should remain concentrated on a single point1 undisturbed alike by collateral ideas on the one hand1 and by wishes and hopes on the other1 and moving on steadily in the direction it has deliberately chosen4 --,,helmholtz1 ,h4 ,ueber das ,verhltniss der ,naturwissenschaften zur ,gesammtheit der ,wissenschaft1 ,vortrge und ,reden1 ,bd4 #a 7#ahif71 p4 #agh4 #bda4 ,if it were always necessary to reduce everything to intuitive knowledge1 demonstration would often be insufferably prolix4 ,this is why mathematicians have had the cleverness to divide the difficulties and to demonstrate separately the intervening propositions4 ,and there is art also in this2 for as the mediate truths 7which are called lemmas1 since they appear to be a digression7 may be assigned in many ways1 it is well1 in order to aid the understanding and memory1 to choose of them those which greatly shorten the process1 and appear memorable and worthy in themselves of being demonstrated4 ,but there is another obstacle1 viz43 that it is not easy to demonstrate all the axioms1 and to reduce demonstrations wholly to intuitive knowledge4 ,and if we had chosen to wait for that1 perhaps we should not yet have the science of geometry4--,,leibnitz1 ,g4 ,w4 ,new ,essay on ,human ,understanding ,langley1 ,bk4 #d1 chaps4 #b1 #h4 #bdb4 ,in ,pure ,mathematics1 where all the various truths are necessarily connected with each other1 7being all necessarily connected with those hypotheses which are the principles of the science71 an arrangement is beautiful in proportion as the principles are few2 and what we admire perhaps chiefly in the science1 is the astonishing variety of consequences which may be demonstrably deduced from so small a number of premises4 --,,stewart1 ,,dugald4 ,the ,elements of the ,philosophy of the ,human ,mind1 ,part #c1 chap4 #a1 sect4 #c4 #bdc4 ,whenever 444 a controversy arises in mathematics1 the issue is not whether a thing is true or not1 but whether the proof might not be conducted more simply in some other way1 or whether the proposition demonstrated is sufficiently important for the advancement of the science as to deserve especial enunciation and emphasis1 or finally1 whether the proposition is not a special case of some other and more general truth which is as easily discovered4--,,schubert1 ,h4 ,mathematical ,essays and ,recreations 7,chicago1 #ahih71 p4 #bh4 #bdd4 444 just as the astronomer1 the physicist1 the geologist1 or other student of objective science looks about in the world of sense1 so1 not metaphorically speaking but literally1 the mind of the mathematician goes forth in the universe of logic in quest of the things that are there2 exploring the heights and depths for facts--ideas1 classes1 relationships1 implications1 and the rest2 observing the minute and elusive with the powerful microscope of his ,infinitesimal ,analysis2 observing the elusive and vast with the limitless telescope of his ,calculus of the ,infinite2 making guesses regarding the order and internal harmony of the data observed and collocated2 testing the hypotheses1 not merely by the complete induction peculiar to mathematics1 but1 like his colleagues of the outer world1 resorting also to experimental tests and incomplete induction2 frequently finding it necessary1 in view of unforeseen disclosures1 to abandon one hopeful hypothesis or to transform it by retrenchment or by enlargement3--thus1 in his own domain1 matching1 point for point1 the processes1 methods and experience familiar to the devotee of natural science4 --,,keyser1 ,,cassius ,j4 ,lectures on ,science1 ,philosophy and ,art 7,new ,york1 #aijh71 p4 #bf4 #bde4 ,that mathematics 8do not cultivate the power of generalization10 444 will be admitted by no person of competent knowledge1 except in a very qualified sense4 ,the generalizations of mathematics1 are1 no doubt1 a different thing from the generalizations of physical science2 but in the difficulty of seizing them1 and the mental tension they require1 they are no contemptible preparation for the most arduous efforts of the scientific mind4 ,even the fundamental notions of the higher mathematics1 from those of the differential calculus upwards are products of a very high abstraction4444 ,to perceive the mathematical laws common to the results of many mathematical operations1 even in so simple a case as that of the binomial theorem1 involves a vigorous exercise of the same faculty which gave us ,kepler's laws1 and rose through those laws to the theory of universal gravitation4 ,every process of what has been called ,universal ,geometry--the great creation of ,descartes and his successors1 in which a single train of reasoning solves whole classes of problems at once1 and others common to large groups of them--is a practical lesson in the management of wide generalizations1 and abstraction of the points of agreement from those of difference among objects of great and confusing diversity1 to which the purely inductive sciences cannot furnish many superior4 ,even so elementary an operation as that of abstracting from the particular configuration of the triangles or other figures1 and the relative situation of the particular lines or points1 in the diagram which aids the apprehension of a common geometrical demonstration1 is a very useful1 and far from being always an easy1 exercise of the faculty of generalization so strangely imagined to have no place or part in the processes of mathematics4--,,mill1 ,,john ,,stuart4 ,an ,examination of ,sir ,william ,hamilton's ,philosophy 7,london1 #ahgh71 pp4 #fab1 #fac4 #bdf4 ,when the greatest of ,american logicians1 speaking of the powers that constitute the born geometrician1 had named ,conception1 ,imagination1 and ,generalization1 he paused4 ,thereupon from one of the audience there came the challenge1 8,what of reason80 ,the instant response1 not less just than brilliant1 was3 8,ratiocination--that is but the smooth pavement on which the chariot rolls40--,,keyser1 ,c4 ,j4 ,lectures on ,science1 ,philosophy and ,art 7,new ,york1 #aijh71 p4 #ca4 #bdg4 444 the reasoning process employed in mathematics is not different from that of any other branch of knowledge1 444 but there is required1 and in a great degree1 that attention of mind which is in some part necessary for the acquisition of all knowledge1 and in this branch is indispensably necessary4 ,this must be given in its fullest intensity2 444 the other elements especially characteristic of a mathematical mind are quickness in perceiving logical sequence1 love of order1 methodical arrangement and harmony1 distinctness of conception4--,,price1 ,b4 ,treatise on ,infinitesimal ,calculus 7,oxford1 #ahfh71 ,vol4 #c1 p4 #f4 #bdh4 ,histories make men wise2 poets1 witty2 the mathematics1 subtile2 natural philosophy1 deep2 moral1 grave2 logic and rhetoric1 able to contend4--,,bacon1 ,,francis4 ,essays1 ,of ,studies4 #bdi4 ,the ,mathematician deals with two properties of objects only1 number and extension1 and all the inductions he wants have been formed and finished ages ago4 ,he is now occupied with nothing but deduction and verification4--,,huxley1 ,t4 ,h4 ,on the ,educational ,value of the ,natural ,history ,sciences2 ,lay ,sermons1 ,addresses and ,reviews2 7,new ,york1 #ahgb71 p4 #hg4 #bej4 ,mathematics is that subject which knows nothing of observation1 nothing of experiment1 nothing of induction1 nothing of causation4--,,huxley1 ,t4 ,h4 ,the ,scientific ,aspects of ,positivism1 ,fortnightly ,review 7#ahih72 ,lay ,sermons1 ,addresses and ,reviews1 7,new ,york1 #ahgb71 p4 #afi4 #bea4 ,we are told that 8,mathematics is that study which knows nothing of observation1 nothing of experiment1 nothing of induction1 nothing of causation40 ,i think no statement could have been made more opposite to the facts of the case2 that mathematical analysis is constantly invoking the aid of new principles1 new ideas1 and new methods1 not capable of being defined by any form of words1 but springing direct from the inherent powers and activities of the human mind1 and from continually renewed introspection of that inner world of thought of which the phenomena are as varied and require as close attention to discern as those of the outer physical world 7to which the inner one in each individual man may1 ,i think1 be conceived to stand somewhat in the same relation of correspondence as a shadow to the object from which it is projected1 or as the hollow palm of one hand to the closed fist which it grasps of the other71 that it is unceasingly calling forth the faculties of observation and comparison1 that one of its principal weapons is induction1 that it has frequent recourse to experimental trial and verification1 and that it affords a boundless scope for the exercise of the highest efforts of the imagination and invention4 --,,sylvester1 ,j4 ,j4 ,presidential ,address to ,british ,association1 ,exeter ,british ,association ,report 7#ahfi71 pp4 #a-i42 ,collected ,mathematical ,papers1 ,vol4 #b1 p4 #fed4 #beb4 ,the actual evolution of mathematical theories proceeds by a process of induction strictly analogous to the method of induction employed in building up the physical sciences2 observation1 comparison1 classification1 trial1 and generalisation are essential in both cases4 ,not only are special results1 obtained independently of one another1 frequently seen to be really included in some generalisation1 but branches of the subject which have been developed quite independently of one another are sometimes found to have connections which enable them to be synthesised in one single body of doctrine4 ,the essential nature of mathematical thought manifests itself in the discernment of fundamental identity in the mathematical aspects of what are superficially very different domains4 ,a striking example of this species of immanent identity of mathematical form was exhibited by the discovery of that distinguished mathematician 444 ,major ,mac,mahon1 that all possible ,latin squares are capable of enumeration by the consideration of certain differential operators4 ,here we have a case in which an enumeration1 which appears to be not amenable to direct treatment1 can actually be carried out in a simple manner when the underlying identity of the operation is recognised with that involved in certain operations due to differential operators1 the calculus of which belongs superficially to a wholly different region of thought from that relating to ,latin squares4 --,,hobson1 ,e4 ,w4 ,presidential ,address ,british ,association for the ,advancement of ,science 7#aiaj72 ,nature1 ,vol4 #hd1 p4 #bij4 #bec4 ,it has been asserted 444 that the power of observation is not developed by mathematical studies2 while the truth is1 that2 from the most elementary mathematical notion that arises in the mind of a child to the farthest verge to which mathematical investigation has been pushed and applied1 this power is in constant exercise4 ,by observation1 as here used1 can only be meant the fixing of the attention upon objects 7physical or mental7 so as to note distinctive peculiarities--to recognize resemblances1 differences1 and other relations4 ,now the first mental act of the child recognizing the distinction between one and more than one1 between one and two1 two and three1 etc41 is exactly this4 ,so1 again1 the first geometrical notions are as pure an exercise of this power as can be given4 ,to know a straight line1 to distinguish it from a curve2 to recognize a triangle and distinguish the several forms--what are these1 and all perception of form1 but a series of observations8 ,nor is it alone in securing these fundamental conceptions of number and form that observation plays so important a part4 ,the very genius of the common geometry as a method of reasoning--a system of investigation--is1 that it is but a series of observations4 ,the figure being before the eye in actual representation1 or before the mind in conception1 is so closely scrutinized1 that all its distinctive features are perceived2 auxiliary lines are drawn 7the imagination leading in this71 and a new series of inspections is made2 and thus1 by means of direct1 simple observations1 the investigation proceeds4 ,so characteristic of common geometry is this method of investigation1 that ,comte1 perhaps the ablest of all writers upon the philosophy of mathematics1 is disposed to class geometry1 as to its method1 with the natural sciences1 being based upon observation4 ,moreover1 when we consider applied mathematics1 we need only to notice that the exercise of this faculty is so essential1 that the basis of all such reasoning1 the very material with which we build1 have received the name observations4 ,thus we might proceed to consider the whole range of the human faculties1 and find for the most of them ample scope for exercise in mathematical studies4 ,certainly1 the memory will not be found to be neglected4 ,the very first steps in number--counting1 the multiplication table1 etc41 make heavy demands on this power2 while the higher branches require the memorizing of formulas which are simply appalling to the uninitiated4 ,so the imagination1 the creative faculty of the mind1 has constant exercise in all original mathematical investigations1 from the solution of the simplest problems to the discovery of the most recondite principle2 for it is not by sure1 consecutive steps1 as many suppose1 that we advance from the known to the unknown4 ,the imagination1 not the logical faculty1 leads in this advance4 ,in fact1 practical observation is often in advance of logical exposition4 ,thus1 in the discovery of truth1 the imagination habitually presents hypotheses1 and observation supplies facts1 which it may require ages for the tardy reason to connect logically with the known4 ,of this truth1 mathematics1 as well as all other sciences1 affords abundant illustrations4 ,so remarkably true is this1 that today it is seriously questioned by the majority of thinkers1 whether the sublimest branch of mathematics1--the infinitesimal calculus--has anything more than an empirical foundation1 mathematicians themselves not being agreed as to its logical basis4 ,that the imagination1 and not the logical faculty1 leads in all original investigation1 no one who has ever succeeded in producing an original demonstration of one of the simpler propositions of geometry1 can have any doubt4 ,nor are induction1 analogy1 the scrutinization of premises or the search for them1 or the balancing of probabilities1 spheres of mental operations foreign to mathematics4 ,no one1 indeed1 can claim pre-eminence for mathematical studies in all these departments of intellectual culture1 but it may1 perhaps1 be claimed that scarcely any department of science affords discipline to so great a number of faculties1 and that none presents so complete a gradation in the exercise of these faculties1 from the first principles of the science to the farthest extent of its applications1 as mathematics4--,,olney1 ,,edward4 ,kiddle and ,schem's ,encyclopedia of ,education1 7,new ,york1 #ahgg71 ,article 8,mathematics40 #bed4 ,the opinion appears to be gaining ground that this very general conception of functionality1 born on mathematical ground1 is destined to supersede the narrower notion of causation1 traditional in connection with the natural sciences4 ,as an abstract formulation of the idea of determination in its most general sense1 the notion of functionality includes and transcends the more special notion of causation as a one-sided determination of future phenomena by means of present conditions2 it can be used to express the fact of the subsumption under a general law of past1 present1 and future alike1 in a sequence of phenomena4 ,from this point of view the remark of ,huxley that ,mathematics 8knows nothing of causation0 could only be taken to express the whole truth1 if by the term 8causation0 is understood 8efficient causation40 ,the latter notion has1 however1 in recent times been to an increasing extent regarded as just as irrelevant in the natural sciences as it is in ,mathematics2 the idea of thorough-going determinancy1 in accordance with formal law1 being thought to be alone significant in either domain4--,,hobson1 ,e4 ,w4 ,presidential ,address ,british ,association for the ,advancement of ,science 7#aiaj72 ,nature1 ,vol4 #hd1 p4 #bij4 #bee4 ,most1 if not all1 of the great ideas of modern mathematics have had their origin in observation4 ,take1 for instance1 the arithmetical theory of forms1 of which the foundation was laid in the diophantine theorems of ,fermat1 left without proof by their author1 which resisted all efforts of the myriad-minded ,euler to reduce to demonstration1 and only yielded up their cause of being when turned over in the blow-pipe flame of ,gauss's transcendent genius2 or the doctrine of double periodicity1 which resulted from the observation of ,jacobi of a purely analytical fact of transformation2 or ,legendre's law of reciprocity2 or ,sturm's theorem about the roots of equations1 which1 as he informed me with his own lips1 stared him in the face in the midst of some mechanical investigations connected 7if my memory serves me right7 with the motion of compound pendulums2 or ,huyghen's method of continued fractions1 characterized by ,lagrange as one of the principal discoveries of that great mathematician1 and to which he appears to have been led by the construction of his ,planetary ,automaton2 or the new algebra1 speaking of which one of my predecessors 7,mr4 ,spottiswoode7 has said1 not without just reason and authority1 from this chair1 8that it reaches out and indissolubly connects itself each year with fresh branches of mathematics1 that the theory of equations has become almost new through it1 algebraic geometry transfigured in its light1 that the calculus of variations1 molecular physics1 and mechanics0 7he might1 if speaking at the present moment1 go on to add the theory of elasticity and the development of the integral calculus7 8have all felt its influence40--,,sylvester1 ,j4 ,j4 ,a ,plea for the ,mathematician1 ,nature1 ,vol4 #a1 p4 #bch2 ,collected ,mathematical ,papers1 ,vol4 #b1 pp4 #fee1 #fef4 #bef4 ,the ability to imagine relations is one of the most indispensable conditions of all precise thinking4 ,no subject can be named1 in the investigation of which it is not imperatively needed2 but it can be nowhere else so thoroughly acquired as in the study of mathematics4--,,fiske1 ,,john4 ,darwinism and other ,essays 7,boston1 #ahic71 p4 #bif4 #beg4 ,the great science mathematics occupies itself at least just as much with the power of imagination as with the power of logical conclusion4--,,herbart1 ,f4 ,j4 ,pestalozzi's ,idee eines ,,abc der ,anschauung4 ,werke ,kehrbach 7,langensaltza1 #ahij71 ,bd4 #a1 p4 #agd4 #beh4 ,the moving power of mathematical invention is not reasoning but imagination4--,,de ,,morgan1 ,a4 ,quoted in ,graves' ,life of ,sir ,w4 ,r4 ,hamilton1 ,vol4 #c 7#ahhi71 p4 #bai4 #bei4 ,there is an astonishing imagination1 even in the science of mathematics4444 ,we repeat1 there was far more imagination in the head of ,archimedes than in that of ,homer4--,,voltaire4 ,a ,philosophical ,dictionary 7,boston1 #ahha71 ,vol4 #c1 p4 #dj4 ,article 8,imagination40 #bfj4 ,as the prerogative of ,natural ,science is to cultivate a taste for observation1 so that of ,mathematics is1 almost from the starting point1 to stimulate the faculty of invention4 --,,sylvester1 ,j4 ,j4 ,a ,plea for the ,mathematician1 ,nature1 ,vol4 #a1 p4 #bfa2 ,collected ,mathematical ,papers1 ,vol4 #b 7,cambridge1 #aijh71 p4 #gag4 #bfa4 ,a marveilous newtrality have these things mathematicall1 and also a strange participation between things supernaturall1 immortall1 intellectuall1 simple and indivisible1 and things naturall1 mortall1 sensible1 componded and divisible4--,,dee1 ,,john4 ,euclid 7#aegj71 ,preface4 #bfb4 ,mathematics stands forth as that which unites1 mediates between ,man and ,nature1 inner and outer world1 thought and perception1 as no other subject does4--,,froebel4 ,herford translation 7,london1 #ahic71 ,vol4 #a1 p4 #hd4 #bfc4 ,the intrinsic character of mathematical research and knowledge is based essentially on three properties3 first1 on its conservative attitude towards the old truths and discoveries of mathematics2 secondly1 on its progressive mode of development1 due to the incessant acquisition of new knowledge on the basis of the old2 and thirdly1 on its self-sufficiency and its consequent absolute independence4--,,schubert1 ,h4 ,mathematical ,essays and ,recreations 7,chicago1 #ahih71 p4 #bg4 #bfd4 ,our science1 in contrast with others1 is not founded on a single period of human history1 but has accompanied the development of culture through all its stages4 ,mathematics is as much interwoven with ,greek culture as with the most modern problems in ,engineering4 ,she not only lends a hand to the progressive natural sciences but participates at the same time in the abstract investigations of logicians and philosophers4 --,,klein1 ,f4 ,klein und ,riecke3 ,ueber angewandte ,mathematik und ,physik 7#aijj71 p4 #bbh4 #bfe4 ,there is probably no other science which presents such different appearances to one who cultivates it and to one who does not1 as mathematics4 ,to this person it is ancient1 venerable1 and complete2 a body of dry1 irrefutable1 unambiguous reasoning4 ,to the mathematician1 on the other hand1 his science is yet in the purple bloom of vigorous youth1 everywhere stretching out after the 8attainable but unattained0 and full of the excitement of nascent thoughts2 its logic is beset with ambiguities1 and its analytic processes1 like ,bunyan's road1 have a quagmire on one side and a deep ditch on the other and branch off into innumerable by-paths that end in a wilderness4 --,,chapman1 ,c4 ,h4 ,bulletin ,american ,mathematical ,society1 ,vol4 #b 7,first series71 p4 #fa4 #bff4 ,mathematical science is in my opinion an indivisible whole1 an organism whose vitality is conditioned upon the connection of its parts4 ,for with all the variety of mathematical knowledge1 we are still clearly conscious of the similarity of the logical devices1 the relationship of the ideas in mathematics as a whole and the numerous analogies in its different departments4 ,we also notice that1 the farther a mathematical theory is developed1 the more harmoniously and uniformly does its construction proceed1 and unsuspected relations are disclosed between hitherto separated branches of the science4 ,so it happens that1 with the extension of mathematics1 its organic character is not lost but manifests itself the more clearly4--,,hilbert1 ,d4 ,mathematical ,problems1 ,bulletin ,american ,mathematical ,society1 ,vol4 #h1 p4 #dgh4 #bfg4 ,the mathematics have always been the implacable enemies of scientific romances4--,,arago4 ,oeuvres 7#ahee71 t4 #c1 p4 #dih4 #bfh4 ,those skilled in mathematical analysis know that its object is not simply to calculate numbers1 but that it is also employed to find the relations between magnitudes which cannot be expressed in numbers and between functions whose law is not capable of algebraic expression4--,,cournot1 ,,augustin4 ,mathematical ,theory of the ,principles of ,wealth ,bacon1 ,n4 ,t41 7,new ,york1 #ahig71 p4 #c4 #bfi4 ,coterminous with space and coeval with time is the ,kingdom of ,mathematics2 within this range her dominion is supreme2 otherwise than according to her order nothing can exist2 in contradiction to her laws nothing takes place4 ,on her mysterious scroll is to be found written for those who can read it that which has been1 that which is1 and that which is to come4 ,everything material which is the subject of knowledge has number1 order1 or position2 and these are her first outlines for a sketch of the universe4 ,if our feeble hands cannot follow out the details1 still her part has been drawn with an unerring pen1 and her work cannot be gainsaid4 ,so wide is the range of mathematical sciences1 so indefinitely may it extend beyond our actual powers of manipulation that at some moments we are inclined to fall down with even more than reverence before her majestic presence4 ,but so strictly limited are her promises and powers1 about so much that we might wish to know does she offer no information whatever1 that at other moments we are fain to call her results but a vain thing1 and to reject them as a stone where we had asked for bread4 ,if one aspect of the subject encourages our hopes1 so does the other tend to chasten our desires1 and he is perhaps the wisest1 and in the long run the happiest1 among his fellows1 who has learned not only this science1 but also the larger lesson which it directly teaches1 namely1 to temper our aspirations to that which is possible1 to moderate our desires to that which is attainable1 to restrict our hopes to that of which accomplishment1 if not immediately practicable1 is at least distinctly within the range of conception4--,,spottiswoode1 ,w4 ,quoted in ,sonnenschein's ,encyclopedia of ,education 7,london1 #aijf71 p4 #bjh4 #bgj4 ,but it is precisely mathematics1 and the pure science generally1 from which the general educated public and independent students have been debarred1 and into which they have only rarely attained more than a very meagre insight4 ,the reason of this is twofold4 ,in the first place1 the ascendant and consecutive character of mathematical knowledge renders its results absolutely insusceptible of presentation to persons who are unacquainted with what has gone before1 and so necessitates on the part of its devotees a thorough and patient exploration of the field from the very beginning1 as distinguished from those sciences which may1 so to speak1 be begun at the end1 and which are consequently cultivated with the greatest zeal4 ,the second reason is that1 partly through the exigencies of academic instruction1 but mainly through the martinet traditions of antiquity and the influence of medival logic-mongers1 the great bulk of the elementary text-books of mathematics have unconsciously assumed a very repellant form1--something similar to what is termed in the theory of protective mimicry in biology 8the terrifying form40 ,and it is mainly to this formidableness and touch-me-not character of exterior1 concealing withal a harmless body1 that the undue neglect of typical mathematical studies is to be attributed4--,,mccormack1 ,t4 ,j4 ,preface to ,de ,morgan's ,elementary ,illustrations of the ,differential and ,integral ,calculus 7,chicago1 #ahii74 #bga4 ,mathematics in gross1 it is plain1 are a grievance in natural philosophy1 and with reason3 for mathematical proofs1 like diamonds1 are hard as well as clear1 and will be touched with nothing but strict reasoning4 ,mathematical proofs are out of the reach of topical arguments2 and are not to be attacked by the equivocal use of words or declaration1 that make so great a part of other discourses1--nay1 even of controversies4--,,locke1 ,,john4 ,second ,reply to the ,bishop of ,worcester4 #bgb4 ,the belief that mathematics1 because it is abstract1 because it is static and cold and gray1 is detached from life1 is a mistaken belief4 ,mathematics1 even in its purest and most abstract estate1 is not detached from life4 ,it is just the ideal handling of the problems of life1 as sculpture may idealize a human figure or as poetry or painting may idealize a figure or a scene4 ,mathematics is precisely the ideal handling of the problems of life1 and the central ideas of the science1 the great concepts about which its stately doctrines have been built up1 are precisely the chief ideas with which life must always deal and which1 as it tumbles and rolls about them through time and space1 give it its interests and problems1 and its order and rationality4 ,that such is the case a few indications will suffice to show4 ,the mathematical concepts of constant and variable are represented familiarly in life by the notions of fixedness and change4 ,the concept of equation or that of an equational system1 imposing restriction upon variability1 is matched in life by the concept of natural and spiritual law1 giving order to what were else chaotic change and providing partial freedom in lieu of none at all4 ,what is known in mathematics under the name of limit is everywhere present in life in the guise of some ideal1 some excellence high-dwelling among the rocks1 an 8ever flying perfect0 as ,emerson calls it1 unto which we may approximate nearer and nearer1 but which we can never quite attain1 save in aspiration4 ,the supreme concept of functionality finds its correlate in life in the all-pervasive sense of interdependence and mutual determination among the elements of the world4 ,what is known in mathematics as transformation--that is1 lawful transfer of attention1 serving to match in orderly fashion the things of one system with those of another--is conceived in life as a process of transmutation by which1 in the flux of the world1 the content of the present has come out of the past and in its turn1 in ceasing to be1 gives birth to its successor1 as the boy is father to the man and as things1 in general1 become what they are not4 ,the mathematical concept of invariance and that of infinitude1 especially the imposing doctrines that explain their meanings and bear their names--,what are they but mathematicizations of that which has ever been the chief of life's hopes and dreams1 of that which has ever been the object of its deepest passion and of its dominant enterprise1 ,i mean the finding of the worth that abides1 the finding of permanence in the midst of change1 and the discovery of a presence1 in what has seemed to be a finite world1 of being that is infinite8 ,it is needless further to multiply examples of a correlation that is so abounding and complete as indeed to suggest a doubt whether it be juster to view mathematics as the abstract idealization of life than to regard life as the concrete realization of mathematics4--,,keyser1 ,c4 ,j4 ,the ,humanization of the ,teaching of ,mathematics2 ,science1 ,new ,series1 ,vol4 #ce1 pp4 #fde-fdf4 #bgc4 ,mathematics1 like dialectics1 is an organ of the inner higher sense2 in its execution it is an art like eloquence4 ,both alike care nothing for the content1 to both nothing is of value but the form4 ,it is immaterial to mathematics whether it computes pennies or guineas1 to rhetoric whether it defends truth or error4--,,goethe4 ,wilhelm ,meisters ,wanderjahre1 ,zweites ,buch4 #bgd4 ,the genuine spirit of ,mathesis is devout4 ,no intellectual pursuit more truly leads to profound impressions of the existence and attributes of a ,creator1 and to a deep sense of our filial relations to him1 than the study of these abstract sciences4 ,who can understand so well how feeble are our conceptions of ,almighty ,power1 as he who has calculated the attraction of the sun and the planets1 and weighed in his balance the irresistible force of the lightning8 ,who can so well understand how confused is our estimate of the ,eternal ,wisdom1 as he who has traced out the secret laws which guide the hosts of heaven1 and combine the atoms on earth8 ,who can so well understand that man is made in the image of his ,creator1 as he who has sought to frame new laws and conditions to govern imaginary worlds1 and found his own thoughts similar to those on which his ,creator has acted8 --,,hill1 ,,thomas4 ,the ,imagination in ,mathematics2 ,north ,american ,review1 ,vol4 #he1 p4 #bbf4 #bge4 444 what is physical is subject to the laws of mathematics1 and what is spiritual to the laws of ,god1 and the laws of mathematics are but the expression of the thoughts of ,god4--,,hill1 ,,thomas4 ,the ,uses of ,mathesis2 ,bibliotheca ,sacra1 ,vol4 #cb1 p4 #ebc4 #bgf4 ,it is in the inner world of pure thought1 where all entia dwell1 where is every type of order and manner of correlation and variety of relationship1 it is in this infinite ensemble of eternal verities whence1 if there be one cosmos or many of them1 each derives its character and mode of being1--it is there that the spirit of mathesis has its home and its life4 ,is it a restricted home1 a narrow life1 static and cold and grey with logic1 without artistic interest1 devoid of emotion and mood and sentiment8 ,that world1 it is true1 is not a world of solar light1 not clad in the colours that liven and glorify the things of sense1 but it is an illuminated world1 and over it all and everywhere throughout are hues and tints transcending sense1 painted there by radiant pencils of psychic light1 the light in which it lies4 ,it is a silent world1 and1 nevertheless1 in respect to the highest principle of art--the interpenetration of content and form1 the perfect fusion of mode and meaning--it even surpasses music4 ,in a sense1 it is a static world1 but so1 too1 are the worlds of the sculptor and the architect4 ,the figures1 however1 which reason constructs and the mathematic vision beholds1 transcend the temple and the statue1 alike in simplicity and in intricacy1 in delicacy and in grace1 in symmetry and in poise4 ,not only are this home and this life thus rich in sthetic interests1 really controlled and sustained by motives of a sublimed and supersensuous art1 but the religious aspiration1 too1 finds there1 especially in the beautiful doctrine of invariants1 the most perfect symbols of what it seeks--the changeless in the midst of change1 abiding things in a world of flux1 configurations that remain the same despite the swirl and stress of countless hosts of curious transformations4 ,the domain of mathematics is the sole domain of certainty4 ,there and there alone prevail the standards by which every hypothesis respecting the external universe and all observation and all experiment must be finally judged4 ,it is the realm to which all speculation and all thought must repair for chastening and sanitation--the court of last resort1 ,i say it reverently1 for all intellection whatsoever1 whether of demon or man or deity4 ,it is there that mind as mind attains its highest estate1 and the condition of knowledge there is the ultimate object1 the tantalising goal of the aspiration1 the ,anders-,streben1 of all other knowledge of every kind4--,,keyser1 ,c4 ,j4 ,the ,universe and ,beyond2 ,hibbert ,journal1 ,vol4 #c 7#aijd-aije71 pp4 #cac-cad4 ,,chapter ,,iii ,,estimates ,,of ,,mathematics #cja4 ,the world of ideas which it mathematics discloses or illuminates1 the contemplation of divine beauty and order which it induces1 the harmonious connection of its parts1 the infinite hierarchy and absolute evidence of the truths with which mathematical science is concerned1 these1 and such like1 are the surest grounds of its title of human regard1 and would remain unimpaired were the plan of the universe unrolled like a map at our feet1 and the mind of man qualified to take in the whole scheme of creation at a glance4--,,sylvester1 ,j4 ,j4 ,a ,plea for the ,mathematician1 ,nature1 #a1 p4 #bfb2 ,collected ,mathematical ,papers 7,cambridge1 #aijh71 #b1 p4 #fei4 #cjb4 ,it may well be doubted whether1 in all the range of ,science1 there is any field so fascinating to the explorer--so rich in hidden treasures--so fruitful in delightful surprises--as that of ,pure ,mathematics4 ,the charm lies chiefly 444 in the absolute certainty of its results3 for that is what1 beyond all mental treasures1 the human intellect craves for4 ,let us only be sure of something6 ,more light1 more light6 8,and if our fate be death1 give light and let us die60 ,this is the cry that1 through all the ages1 is going up from perplexed ,humanity1 and ,science has little else to offer1 that will really meet the demands of its votaries1 than the conclusions of ,pure ,mathematics4 --,,dodgson1 ,c4 ,l4 ,a ,new ,theory of ,parallels 7,london1 #ahie71 ,introduction4 #cjc4 ,in every case the awakening touch has been the mathematical spirit1 the attempt to count1 to measure1 or to calculate4 ,what to the poet or the seer may appear to be the very death of all his poetry and all his visions--the cold touch of the calculating mind1--this has proved to be the spell by which knowledge has been born1 by which new sciences have been created1 and hundreds of definite problems put before the minds and into the hands of diligent students4 ,it is the geometrical figure1 the dry algebraical formula1 which transforms the vague reasoning of the philosopher into a tangible and manageable conception2 which represents1 though it does not fully describe1 which corresponds to1 though it does not explain1 the things and processes of nature3 this clothes the fruitful1 but otherwise indefinite1 ideas in such a form that the strict logical methods of thought can be applied1 that the human mind can in its inner chamber evolve a train of reasoning the result of which corresponds to the phenomena of the outer world4--,,merz1 ,j4 ,t4 ,a ,history of ,european ,thought in the ,nineteenth ,century 7,edinburgh and ,london1 #aijd71 ,vol4 #a1 p4 #cad4 #cjd4 ,mathematics 444 the ideal and norm of all careful thinking4--,,hall1 ,g4 ,,stanley4 ,educational ,problems 7,new ,york1 #aiaa71 p4 #cic4 #cje4 ,mathematics is the only true metaphysics4 --,,thomson1 ,w4 7,,lord ,,kelvin74 ,thompson1 ,s4 ,p43 ,life of ,lord ,kelvin 7,london1 #aiaj71 p4 #aj4 #cjf4 ,he who knows not mathematics and the results of recent scientific investigation dies without knowing truth4 --,,schellbach1 ,c4 ,h4 ,quoted in ,young's ,teaching of ,mathematics 7,london1 #aijg71 p4 #dd4 #cjg4 ,the reasoning of mathematics is a type of perfect reasoning4--,,barnett1 ,p4 ,a4 ,common ,sense in ,education and ,teaching 7,new ,york1 #aije71 p4 #bbb4 #cjh4 ,mathematics1 once fairly established on the foundation of a few axioms and definitions1 as upon a rock1 has grown from age to age1 so as to become the most solid fabric that human reason can boast4--,,reid1 ,,thomas4 ,essays on the ,intellectual ,powers of ,man1 #dth4 ,ed41 p4 #dfa4 #cji4 ,the analytical geometry of ,descartes and the calculus of ,newton and ,leibniz have expanded into the marvelous mathematical method--more daring than anything that the history of philosophy records--of ,lobachevsky and ,riemann1 ,gauss and ,sylvester4 ,indeed1 mathematics1 the indispensable tool of the sciences1 defying the senses to follow its splendid flights1 is demonstrating today1 as it never has been demonstrated before1 the supremacy of the pure reason4--,,butler1 ,,nicholas ,,murray4 ,the ,meaning of ,education and other ,essays and ,addresses 7,new ,york1 #aije71 p4 #de4 #caj4 ,mathematics is the gate and key of the sciences4444 ,neglect of mathematics works injury to all knowledge1 since he who is ignorant of it cannot know the other sciences or the things of this world4 ,and what is worse1 men who are thus ignorant are unable to perceive their own ignorance and so do not seek a remedy4--,,bacon1 ,,roger4 ,opus ,majus1 ,part #d1 ,distinctia ,prima1 cap4 #a4 #caa4 ,just as it will never be successfully challenged that the ,french language1 progressively developing and growing more perfect day by day1 has the better claim to serve as a developed court and world language1 so no one will venture to estimate lightly the debt which the world owes to mathematicians1 in that they treat in their own language matters of the utmost importance1 and govern1 determine and decide whatever is subject1 using the word in the highest sense1 to number and measurement4 --,,goethe4 ,sprche in ,prosa1 ,natur1 ,,iii1 #hfh4 #cab4 ,do not imagine that mathematics is hard and crabbed1 and repulsive to common sense4 ,it is merely the etherealization of common sense4--,,thomson1 ,w4 7,,lord ,,kelvin74 ,thompson1 ,s4 ,p43 ,life of ,lord ,kelvin 7,london1 #aiaj71 p4 #aaci4 #cac4 ,the advancement and perfection of mathematics are intimately connected with the prosperity of the ,state4--,,napoleon ,i4 ,correspondance de ,napolon1 t4 #bd 7#ahfh71 p4 #aab4 #cad4 ,the love of mathematics is daily on the increase1 not only with us but in the army4 ,the result of this was unmistakably apparent in our last campaigns4 ,bonaparte himself has a mathematical head1 and though all who study this science may not become geometricians like ,laplace or ,lagrange1 or heroes like ,bonaparte1 there is yet left an influence upon the mind which enables them to accomplish more than they could possibly have achieved without this training4--,,lalande4 ,quoted in ,bruhns' ,alexander von ,humboldt 7#ahgb71 ,bd4 #a1 p4 #bcb4 #cae4 ,in ,pure ,mathematics1 where all the various truths are necessarily connected with each other1 7being all necessarily connected with those hypotheses which are the principles of the science71 an arrangement is beautiful in proportion as the principles are few2 and what we admire perhaps chiefly in the science1 is the astonishing variety of consequences which may be demonstrably deduced from so small a number of premises4 --,,stewart1 ,,dugald4 ,philosophy of the ,human ,mind1 ,part #c1 chap4 #a1 sect4 #c2 ,collected ,works ,hamilton 7,edinburgh1 #ahed71 ,vol4 #d4 #caf4 ,it is curious to observe how differently these great men ,plato and ,bacon estimated the value of every kind of knowledge4 ,take ,arithmetic for example4 ,plato1 after speaking slightly of the convenience of being able to reckon and compute in the ordinary transactions of life1 passes to what he considers as a far more important advantage4 ,the study of the properties of numbers1 he tells us1 habituates the mind to the contemplation of pure truth1 and raises us above the material universe4 ,he would have his disciples apply themselves to this study1 not that they may be able to buy or sell1 not that they may qualify themselves to be shop-keepers or travelling merchants1 but that they may learn to withdraw their minds from the ever-shifting spectacle of this visible and tangible world1 and to fix them on the immutable essences of things4 ,bacon1 on the other hand1 valued this branch of knowledge only on account of its uses with reference to that visible and tangible world which ,plato so much despised4 ,he speaks with scorn of the mystical arithmetic of the later ,platonists1 and laments the propensity of mankind to employ1 on mere matters of curiosity1 powers the whole exertion of which is required for purposes of solid advantage4 ,he advises arithmeticians to leave these trifles1 and employ themselves in framing convenient expressions which may be of use in physical researches4--,,macaulay4 ,lord ,bacon3 ,edinburgh ,review1 ,july1 #ahcg4 ,critical and ,miscellaneous ,essays 7,new ,york1 #ahgi71 ,vol4 #a1 p4 #cig4 #cag4 ,ath4 ,there still remain three studies suitable for freemen4 ,calculation in arithmetic is one of them2 the measurement of length1 surface1 and depth is the second2 and the third has to do with the revolutions of the stars in reference to one another 444 there is in them something that is necessary and cannot be set aside1 444 if ,i am not mistaken1 something of divine necessity2 for as to the human necessities of which men often speak when they talk in this manner1 nothing can be more ridiculous than such an application of the words4 ,cle4 ,and what necessities of knowledge are there1 ,stranger1 which are divine and not human8 ,ath4 ,i conceive them to be those of which he who has no use nor any knowledge at all cannot be a god1 or demi-god1 or hero to mankind1 or able to take any serious thought or charge of them4 --,,plato4 ,republic1 ,bk4 #g4 ,jowett's ,dialogues of ,plato 7,new ,york1 #ahig71 ,vol4 #d1 p4 #ccd4 #cah4 ,those who assert that the mathematical sciences make no affirmation about what is fair or good make a false assertion2 for they do speak of these and frame demonstrations of them in the most eminent sense of the word4 ,for if they do not actually employ these names1 they do not exhibit even the results and the reasons of these1 and therefore can be hardly said to make any assertion about them4 ,of what is fair1 however1 the most important species are order and symmetry1 and that which is definite1 which the mathematical sciences make manifest in a most eminent degree4 ,and since1 at least1 these appear to be the causes of many things--now1 ,i mean1 for example1 order1 and that which is a definite thing1 it is evident that they would assert1 also1 the existence of a cause of this description1 and its subsistence after the same manner as that which is fair subsists in4--,,aristotle4 ,metaphysics ,mac,mahon ,bk4 #ab1 chap4 #c4 #cai4 ,many arts there are which beautify the mind of man2 of all other none do more garnish and beautify it than those arts which are called mathematical4--,,billingsley1 ,h4 ,the ,elements of ,geometrie of the most ancient ,philosopher ,euclide of ,megara 7,london1 #aegj71 ,note to the ,reader4 #cbj4 ,as the sun eclipses the stars by his brilliancy1 so the man of knowledge will eclipse the fame of others in assemblies of the people if he proposes algebraic problems1 and still more if he solves them4--,,brahmagupta4 ,quoted in ,cajori's ,history of ,mathematics 7,new ,york1 #ahig71 p4 #ib4 #cba4 ,so highly did the ancients esteem the power of figures and numbers1 that ,democritus ascribed to the figures of atoms the first principles of the variety of things2 and ,pythagoras asserted that the nature of things consisted of numbers4 --,,bacon1 ,,lord4 ,de ,augmentis1 ,bk4 #c2 ,advancement of ,learning1 ,bk4 #b4 #cbb4 ,there has not been any science so much esteemed and honored as this of mathematics1 nor with so much industry and vigilance become the care of great men1 and labored in by the potentates of the world1 viz4 emperors1 kings1 princes1 etc4 --,,franklin1 ,,benjamin4 ,on the ,usefulness of ,mathematics1 ,works 7,boston1 #ahdj71 ,vol4 #b1 p4 #bh4 #cbc4 ,whatever may have been imputed to some other studies under the notion of insignificancy and loss of time1 yet these mathematics1 ,i believe1 never caused repentance in any1 except it was for their remissness in the prosecution of them4 --,,franklin1 ,,benjamin4 ,on the ,usefulness of ,mathematics1 ,works 7,boston1 #ahdj71 ,vol4 #b1 p4 #fi4 #cbd4 ,what science can there be more noble1 more excellent1 more useful for men1 more admirably high and demonstrative1 than this of the mathematics8--,,franklin1 ,,benjamin4 ,on the ,usefulness of ,mathematics1 ,works 7,boston1 #ahdj71 ,vol4 #b1 p4 #fi4 #cbe4 ,the great truths with which it mathematics deals1 are clothed with austere grandeur1 far above all purposes of immediate convenience or profit4 ,it is in them that our limited understandings approach nearest to the conception of that absolute and infinite1 towards which in most other things they aspire in vain4 ,in the pure mathematics we contemplate absolute truths1 which existed in the divine mind before the morning stars sang together1 and which will continue to exist there1 when the last of their radiant host shall have fallen from heaven4 ,they existed not merely in metaphysical possibility1 but in the actual contemplation of the supreme reason4 ,the pen of inspiration1 ranging all nature and life for imagery to set forth the ,creator's power and wisdom1 finds them best symbolized in the skill of the surveyor4 8,he meted out heaven as with a span20 and an ancient sage1 neither falsely nor irreverently1 ventured to say1 that 8,god is a geometer40--,,everett1 ,,edward4 ,orations and ,speeches 7,boston1 #ahgj71 ,vol4 #c1 p4 #ead4 #cbf4 ,there is no science which teaches the harmonies of nature more clearly than mathematics1 4444--,,carus1 ,,paul4 ,andrews3 ,magic ,squares and ,cubes 7,chicago1 #aijh71 ,introduction4 #cbg4 ,for it being the nature of the mind of man 7to the extreme prejudice of knowledge7 to delight in the spacious liberty of generalities1 as in a champion region1 and not in the enclosures of particularity2 the ,mathematics were the goodliest fields to satisfy that appetite4--,,bacon1 ,,lord4 ,de ,augmentis1 ,bk4 #c2 ,advancement of ,learning1 ,bk4 #b4 #cbh4 ,i would have my son mind and understand business1 read little history1 study the mathematics and cosmography2 these are good1 with subordination to the things of ,god4444 ,these fit for public services for which man is born4--,,cromwell1 ,,oliver4 ,letters and ,speeches of ,oliver ,cromwell 7,new ,york1 #ahii71 ,vol4 #a1 p4 #cga4 #cbi4 ,mathematics is the life supreme4 ,the life of the gods is mathematics4 ,all divine messengers are mathematicians4 ,pure mathematics is religion4 ,its attainment requires a theophany4 --,,novalis4 ,schriften 7,berlin1 #aija71 ,bd4 #b1 p4 #bbc4 #ccj4 ,the ,mathematics which effectually exercises1 not vainly deludes or vexatiously torments studious ,minds with obscure ,subtilties1 perplexed ,difficulties1 or contentious ,disquisitions2 which overcomes without ,opposition1 triumphs without ,pomp1 compels without ,force1 and rules absolutely without ,loss of ,liberty2 which does not privately overreach a weak ,faith1 but openly assaults an armed ,reason1 obtains a total ,victory1 and puts on inevitable ,chains2 whose ,words are so many ,oracles1 and ,works as many ,miracles2 which blabs out nothing rashly1 nor designs anything from the ,purpose1 but plainly demonstrates and readily performs all ,things within its ,verge2 which obtrudes no false ,shadow of ,science1 but the very ,science itself1 the ,mind firmly adheres to it1 as soon as possessed of it1 and can never after desert it of its own ,accord1 or be deprived of it by any ,force of others3 ,lastly the ,mathematics1 which depend upon ,principles clear to the ,mind1 and agreeable to ,experience2 which draws certain ,conclusions1 instructs by profitable ,rules1 unfolds pleasant ,questions2 and produces wonderful ,effects2 which is the fruitful ,parent of1 ,i had almost said all1 ,arts1 the unshaken ,foundation of ,sciences1 and the plentiful ,fountain of ,advantage to human ,affairs4--,,barrow1 ,,isaac4 ,oration before the ,university of ,cambridge on being elected ,lucasian ,professor of ,mathematics1 ,mathematical ,lectures 7,london1 #agcd71 p4 #bh4 #cca4 ,doubtless the reasoning faculty1 the mind1 is the leading and characteristic attribute of the human race4 ,by the exercise of this1 man arrives at the properties of the natural bodies4 ,this is science1 properly and emphatically so called4 ,it is the science of pure mathematics2 and in the high branches of this science lies the truly sublime of human acquisition4 ,if any attainment deserves that epithet1 it is the knowledge1 which1 from the mensuration of the minutest dust of the balance1 proceeds on the rising scale of material bodies1 everywhere weighing1 everywhere measuring1 everywhere detecting and explaining the laws of force and motion1 penetrating into the secret principles which hold the universe of ,god together1 and balancing worlds against worlds1 and system against system4 ,when we seek to accompany those who pursue studies at once so high1 so vast1 and so exact2 when we arrive at the discoveries of ,newton1 which pour in day on the works of ,god1 as if a second fiat had gone forth from his own mouth2 when1 further1 we attempt to follow those who set out where ,newton paused1 making his goal their starting-place1 and1 proceeding with demonstration upon demonstration1 and discovery upon discovery1 bring new worlds and new systems of worlds within the limits of the known universe1 failing to learn all only because all is infinite2 however we may say of man1 in admiration of his physical structure1 that 8in form and moving he is express and admirable10 it is here1 and here without irreverence1 we may exclaim1 8,in apprehension how like a god60 ,the study of the pure mathematics will of course not be extensively pursued in an institution1 which1 like this ,boston ,mechanics' ,institute1 has a direct practical tendency and aim4 ,but it is still to be remembered1 that pure mathematics lie at the foundation of mechanical philosophy1 and that it is ignorance only which can speak or think of that sublime science as useless research or barren speculation4--,,webster1 ,,daniel4 ,works 7,boston1 #ahgb71 ,vol4 #a1 p4 #ahj4 #ccb4 ,the school of ,plato has advanced the interests of the race as much through geometry as through philosophy4 ,the modern engineer1 the navigator1 the astronomer1 built on the truths which those early ,greeks discovered in their purely speculative investigations4 ,and if the poetry1 statesmanship1 oratory1 and philosophy of our day owe much to ,plato's divine ,dialogues1 our commerce1 our manufactures1 and our science are equally indebted to his ,conic ,sections4 ,later instances may be abundantly quoted1 to show that the labors of the mathematician have outlasted those of the statesman1 and wrought mightier changes in the condition of the world4 ,not that we would rank the geometer above the patriot1 but we claim that he is worthy of equal honor4 --,,hill1 ,,thomas4 ,imagination in ,mathematics2 ,north ,american ,review1 ,vol4 #he1 p4 #bbh4 #ccc4 ,the discoveries of ,newton have done more for ,england and for the race1 than has been done by whole dynasties of ,british monarchs2 and we doubt not that in the great mathematical birth of #ahec1 the ,quaternions of ,hamilton1 there is as much real promise of benefit to mankind as in any event of ,victoria's reign4--,,hill1 ,,thomas4 ,imagination in ,mathematics2 ,north ,american ,review1 ,vol4 #he1 p4 #bbh4 #ccd4 ,geometrical and ,mechanical phenomena are the most general1 the most simple1 the most abstract of all1--the most irreducible to others4 ,it follows that the study of them is an indispensable preliminary to that of all others4 ,therefore must ,mathematics hold the first place in the hierarchy of the sciences1 and be the point of departure of all ,education1 whether general or special4--,,comte1 ,a4 ,positive ,philosophy ,martineau1 ,introduction1 chap4 #b4 ,,chapter ,,iv ,,the ,,value ,,of ,,mathematics #dja4 ,mathematics because of its nature and structure is peculiarly fitted for high school instruction ,gymnasiallehrfach4 ,especially the higher mathematics1 even if presented only in its elements1 combines within itself all those qualities which are demanded of a secondary subject4 ,it engages1 it fructifies1 it quickens1 compels attention1 is as circumspect as inventive1 induces courage and self-confidence as well as modesty and submission to truth4 ,it yields the essence and kernel of all things1 is brief in form and overflows with its wealth of content4 ,it discloses the depth and breadth of the law and spiritual element behind the surface of phenomena2 it impels from point to point and carries within itself the incentive toward progress2 it stimulates the artistic perception1 good taste in judgment and execution1 as well as the scientific comprehension of things4 ,mathematics1 therefore1 above all other subjects1 makes the student lust after knowledge1 fills him1 as it were1 with a longing to fathom the cause of things and to employ his own powers independently2 it collects his mental forces and concentrates them on a single point and thus awakens the spirit of individual inquiry1 self-confidence and the joy of doing2 it fascinates because of the view-points which it offers and creates certainty and assurance1 owing to the universal validity of its methods4 ,thus1 both what he receives and what he himself contributes toward the proper conception and solution of a problem1 combine to mature the student and to make him skillful1 to lead him away from the surface of things and to exercise him in the perception of their essence4 ,a student thus prepared thirsts after knowledge and is ready for the university and its sciences4 ,thus it appears1 that higher mathematics is the best guide to philosophy and to the philosophic conception of the world 7considered as a self-contained whole7 and of one's own being4--,,dillmann1 ,e4 ,die ,mathematik die ,fackeltrgerin einer neuen ,zeit 7,stuttgart1 #ahhi71 p4 #dj4 #djb4 ,these ,disciplines mathematics serve to inure and corroborate the ,mind to a constant ,diligence in ,study2 to undergo the ,trouble of an attentive ,meditation1 and cheerfully contend with such ,difficulties as lie in the ,way4 ,they wholly deliver us from a credulous ,simplicity1 most strongly fortify us against the ,vanity of ,scepticism1 effectually restrain from a rash ,presumption1 most easily incline us to a due ,assent1 perfectly subject us to the ,government of right ,reason1 and inspire us with ,resolution to wrestle against the unjust ,tyranny of false ,prejudices4 ,if the ,fancy be unstable and fluctuating1 it is to be poised by this ,ballast1 and steadied by this ,anchor1 if the ,wit be blunt it is sharpened upon this ,whetstone2 if luxuriant it is pared by this ,knife2 if headstrong it is restrained by this ,bridle2 and if dull it is roused by this ,spur4 ,the ,steps are guided by no ,lamp more clearly through the dark ,mazes of ,nature1 by no ,thread more surely through the intricate ,labyrinths of ,philosophy1 nor lastly is the ,bottom of ,truth sounded more happily by any other ,line4 ,i will not mention how plentiful a ,stock of ,knowledge the ,mind is furnished from these1 with what wholesome ,food it is nourished1 and what sincere ,pleasure it enjoys4 ,but if ,i speak farther1 ,i shall neither be the only ,person1 nor the first1 who affirms it2 that while the ,mind is abstracted and elevated from sensible ,matter1 distinctly views pure ,forms1 conceives the ,beauty of ,ideas1 and investigates the ,harmony of ,proportions2 the ,manners themselves are sensibly corrected and improved1 the ,affections composed and rectified1 the ,fancy calmed and settled1 and the ,understanding raised and excited to more divine ,contemplation4 ,all which ,i might defend by ,authority1 and confirm by the ,suffrages of the greatest ,philosophers4--,,barrow1 ,,isaac4 ,prefatory ,oration3 ,mathematical ,lectures 7,london1 #agcd71 p4 #ca4 #djc4 ,no school subject so readily furnishes tasks whose purpose can be made so clear1 so immediate and so appealing to the sober second-thought of the immature learner as the right sort of elementary school mathematics4--,,myers1 ,,george4 ,arithmetic in ,public ,school ,education 7,chicago1 #aiaa71 p4 #h4 #djd4 ,mathematics is a type of thought which seems ingrained in the human mind1 which manifests itself to some extent with even the primitive races1 and which is developed to a high degree with the growth of civilization4444 ,a type of thought1 a body of results1 so essentially characteristic of the human mind1 so little influenced by environment1 so uniformly present in every civilization1 is one of which no well-informed mind today can be ignorant4--,,young1 ,j4 ,w4 ,a4 ,the ,teaching of ,mathematics 7,london1 #aijg71 p4 #ad4 #dje4 ,probably among all the pursuits of the ,university1 mathematics pre-eminently demand self-denial1 patience1 and perseverance from youth1 precisely at that period when they have liberty to act for themselves1 and when on account of obvious temptations1 habits of restraint and application are peculiarly valuable4--,,todhunter1 ,,isaac4 ,the ,conflict of ,studies and other ,essays 7,london1 #ahgc71 p4 #ab4 #djf4 ,mathematics renders its best service through the immediate furthering of rigorous thought and the spirit of invention4 --,,herbart ,j4 ,f4 ,mathematischer ,lehrplan fr ,realschulen3 ,werke ,kehrbach 7,langensalza1 #ahij71 ,bd4 #e1 p4 #agj4 #djg4 ,it seems to me that the older subjects1 classics and mathematics1 are strongly to be recommended on the ground of the accuracy with which we can compare the relative performance of the students4 ,in fact the definiteness of these subjects is obvious1 and is commonly admitted4 ,there is however another advantage1 which ,i think belongs in general to these subjects1 that the examinations can be brought to bear on what is really most valuable in these subjects4--,,todhunter1 ,,isaac4 ,conflict of ,studies and other ,essays 7,london1 #ahgc71 pp4 #f1 #g4 #djh4 ,it is better to teach the child arithmetic and ,latin grammar than rhetoric and moral philosophy1 because they require exactitude of performance it is made certain that the lesson is mastered1 and that power of performance is worth more than knowledge4--,,emerson1 ,r4 ,w4 ,lecture on ,education4 #dji4 ,besides accustoming the student to demand complete proof1 and to know when he has not obtained it1 mathematical studies are of immense benefit to his education by habituating him to precision4 ,it is one of the peculiar excellencies of mathematical discipline1 that the mathematician is never satisfied with peu prs4 ,he requires the exact truth4 ,hardly any of the non-mathematical sciences1 except chemistry1 has this advantage4 ,one of the commonest modes of loose thought1 and sources of error both in opinion and in practice1 is to overlook the importance of quantities4 ,mathematicians and chemists are taught by the whole course of their studies1 that the most fundamental difference of quality depends on some very slight difference in proportional quantity2 and that from the qualities of the influencing elements1 without careful attention to their quantities1 false expectation would constantly be formed as to the very nature and essential character of the result produced4--,,mill1 ,j4 ,s4 ,an ,examination of ,sir ,william ,hamilton's ,philosophy 7,london1 #ahgh71 p4 #faa4 #daj4 ,in mathematics ,i can report no deficience1 except it be that men do not sufficiently understand the excellent use of the ,pure ,mathematics1 in that they do remedy and cure many defects in the wit and faculties intellectual4 ,for if the wit be too dull1 they sharpen it2 if too wandering1 they fix it2 if too inherent in the senses1 they abstract it4 ,so that as tennis is a game of no use in itself1 but of great use in respect it maketh a quick eye and a body ready to put itself into all positions2 so in the ,mathematics1 that use which is collateral and intervenient is no less worthy than that which is principal and intended4--,,bacon1 ,,lord4 ,de ,augmentis1 ,bk4 #c2 ,advancement of ,learning1 ,bk4 #b4 #daa4 ,if a man's wit be wandering1 let him study mathematics2 for in demonstrations1 if his wit be called away never so little1 he must begin again4--,,bacon1 ,,lord4 ,essays3 ,on ,studies4 #dab4 ,if one be bird-witted1 that is easily distracted and unable to keep his attention as long as he should1 mathematics provides a remedy2 for in them if the mind be caught away but a moment1 the demonstration has to be commenced anew4--,,bacon1 ,,lord4 ,de ,augmentis1 ,bk4 #f2 ,advancement of ,learning1 ,bk4 #b4 #dac4 ,the metaphysical philosopher from his point of view recognizes mathematics as an instrument of education1 which strengthens the power of attention1 develops the sense of order and the faculty of construction1 and enables the mind to grasp under the simple formulae the quantitative differences of physical phenomena4--,,jowett1 ,b4 ,dialogues of ,plato 7,new ,york1 #ahig71 ,vol4 #b1 p4 #gh4 #dad4 ,nor do ,i know any study which can compete with mathematics in general in furnishing matter for severe and continued thought4 ,metaphysical problems may be even more difficult2 but then they are far less definite1 and1 as they rarely lead to any precise conclusion1 we miss the power of checking our own operations1 and of discovering whether we are thinking and reasoning or merely fancying and dreaming4--,,todhunter1 ,,isaac4 ,conflict of ,studies 7,london1 #ahgc71 p4 #ac4 #dae4 ,another great and special excellence of mathematics is that it demands earnest voluntary exertion4 ,it is simply impossible for a person to become a good mathematician by the happy accident of having been sent to a good school2 this may give him a preparation and a start1 but by his own individual efforts alone can he reach an eminent position4--,,todhunter1 ,,isaac4 ,conflict of ,studies 7,london1 #ahgc71 p4 #b4 #daf4 ,the faculty of resolution is possibly much invigorated by mathematical study1 and especially by that highest branch of it which1 unjustly1 merely on account of its retrograde operations1 has been called1 as if par excellence1 analysis4--,,poe1 ,e4 ,a4 ,the ,murders in ,rue ,morgue4 #dag4 ,he who gives a portion of his time and talent to the investigation of mathematical truth will come to all other questions with a decided advantage over his opponents4 ,he will be in argument what the ancient ,romans were in the field3 to them the day of battle was a day of comparative recreation1 because they were ever accustomed to exercise with arms much heavier than they fought2 and reviews differed from a real battle in two respects3 they encountered more fatigue1 but the victory was bloodless4--,,colton1 ,c4 ,c4 ,lacon 7,new ,york1 #ahff74 #dah4 ,mathematics is the study which forms the foundation of the course ,west ,point ,military ,academy4 ,this is necessary1 both to impart to the mind that combined strength and versatility1 the peculiar vigor and rapidity of comparison necessary for military action1 and to pave the way for progress in the higher military sciences4 ,congressional ,committee on ,military ,affairs1 #ahcd2 ,u4 ,s4 ,bureau of ,education1 ,bulletin #aiab1 ,no4 #b1 p4 #aj4 #dai4 ,mathematics1 among all school subjects1 is especially adapted to further clearness1 definite brevity and precision in expression1 although it offers no exercise in flights of rhetoric4 ,this is due in the first place to the logical rigour with which it develops thought1 avoiding every departure from the shortest1 most direct way1 never allowing empty phrases to enter4 ,other subjects excel in the development of expression in other respects3 translation from foreign languages into the mother tongue gives exercise in finding the proper word for the given foreign word and gives knowledge of laws of syntax1 the study of poetry and prose furnish fit patterns for connected presentation and elegant form of expression1 composition is to exercise the pupil in a like presentation of his own or borrowed thoughts and their development1 the natural sciences teach description of natural objects1 apparatus and processes1 as well as the statement of laws on the grounds of immediate sense-perception4 ,but all these aids for exercise in the use of the mother tongue1 each in its way valuable and indispensable1 do not guarantee1 in the same manner as mathematical training1 the exclusion of words whose concepts1 if not entirely wanting1 are not sufficiently clear4 ,they do not furnish in the same measure that which the mathematician demands particularly as regards precision of expression4--,,reidt1 ,f4 ,anleitung zum mathematischen ,unterricht in hheren ,schulen 7,berlin1 #aijf71 p4 #ag4 #dbj4 ,one rarely hears of the mathematical recitation as a preparation for public speaking4 ,yet mathematics shares with these studies foreign languages1 drawing and natural science their advantages1 and has another in a higher degree than either of them4 ,most readers will agree that a prime requisite for healthful experience in public speaking is that the attention of the speaker and hearers alike be drawn wholly away from the speaker and concentrated upon the thought4 ,in perhaps no other classroom is this so easy as in the mathematical1 where the close reasoning1 the rigorous demonstration1 the tracing of necessary conclusions from given hypotheses1 commands and secures the entire mental power of the student who is explaining1 and of his classmates4 ,in what other circumstances do students feel so instinctively that manner counts for so little and mind for so much8 ,in what other circumstances1 therefore1 is a simple1 unaffected1 easy1 graceful manner so naturally and so healthfully cultivated8 ,mannerisms that are mere affectation or the result of bad literary habit recede to the background and finally disappear1 while those peculiarities that are the expression of personality and are inseparable from its activity continually develop1 where the student frequently presents1 to an audience of his intellectual peers1 a connected train of reasoning4444 ,one would almost wish that our institutions of the science and art of public speaking would put over their doors the motto that ,plato had over the entrance to his school of philosophy3 8,let no one who is unacquainted with geometry enter here40--,,white1 ,w4 ,f4 ,a ,scrap-book of ,elementary ,mathematics 7,chicago1 #aijh71 p4 #baj4 #dba4 ,the training which mathematics gives in working with symbols is an excellent preparation for other sciences2 444 the world's work requires constant mastery of symbols4--,,young1 ,j4 ,w4 ,a4 ,the ,teaching of ,mathematics 7,new ,york1 #aijg71 p4 #db4 #dbb4 ,one striking peculiarity of mathematics is its unlimited power of evolving examples and problems4 ,a student may read a book of ,euclid1 or a few chapters of ,algebra1 and within that limited range of knowledge it is possible to set him exercises as real and as interesting as the propositions themselves which he has studied2 deductions which might have pleased the ,greek geometers1 and algebraic propositions which ,pascal and ,fermat would not have disdained to investigate4--,,todhunter1 ,,isaac4 ,private ,study of ,mathematics3 ,conflict of ,studies and other ,essays 7,london1 #ahgc71 p4 #hb4 #dbc4 ,would you have a man reason well1 you must use him to it betimes2 exercise his mind in observing the connection between ideas1 and following them in train4 ,nothing does this better than mathematics1 which therefore1 ,i think should be taught to all who have the time and opportunity1 not so much to make them mathematicians1 as to make them reasonable creatures2 for though we all call ourselves so1 because we are born to it if we please1 yet we may truly say that nature gives us but the seeds of it1 and we are carried no farther than industry and application have carried us4--,,locke1 ,,john4 ,conduct of the ,understanding1 ,sect4 #f4 #dbd4 ,secondly1 the study of mathematics would show them the necessity there is in reasoning1 to separate all the distinct ideas1 and to see the habitudes that all those concerned in the present inquiry have to one another1 and to lay by those which relate not to the proposition in hand1 and wholly to leave them out of the reckoning4 ,this is that which1 in other respects besides quantity is absolutely requisite to just reasoning1 though in them it is not so easily observed and so carefully practised4 ,in those parts of knowledge where it is thought demonstration has nothing to do1 men reason as it were in a lump2 and if upon a summary and confused view1 or upon a partial consideration1 they can raise the appearance of a probability1 they usually rest content2 especially if it be in a dispute where every little straw is laid hold on1 and everything that can but be drawn in any way to give color to the argument is advanced with ostentation4 ,but that mind is not in a posture to find truth that does not distinctly take all the parts asunder1 and1 omitting what is not at all to the point1 draws a conclusion from the result of all the particulars which in any way influence it4 --,,locke1 ,,john4 ,conduct of the ,understanding1 ,sect4 #g4 #dbe4 ,i have before mentioned mathematics1 wherein algebra gives new helps and views to the understanding4 ,if ,i propose these it is not to make every man a thorough mathematician or deep algebraist2 but yet ,i think the study of them is of infinite use even to grown men2 first by experimentally convincing them1 that to make anyone reason well1 it is not enough to have parts wherewith he is satisfied1 and that serve him well enough in his ordinary course4 ,a man in those studies will see1 that however good he may think his understanding1 yet in many things1 and those very visible1 it may fail him4 ,this would take off that presumption that most men have of themselves in this part2 and they would not be so apt to think their minds wanted no helps to enlarge them1 that there could be nothing added to the acuteness and penetration of their understanding4--,,locke1 ,,john4 ,the ,conduct of the ,understanding1 ,sect4 #g4 #dbf4 ,i have mentioned mathematics as a way to settle in the mind a habit of reasoning closely and in train2 not that ,i think it necessary that all men should be deep mathematicians1 but that1 having got the way of reasoning which that study necessarily brings the mind to1 they might be able to transfer it to other parts of knowledge1 as they shall have occasion4 ,for in all sorts of reasoning1 every single argument should be managed as a mathematical demonstration2 the connection and dependence of ideas should be followed till the mind is brought to the source on which it bottoms1 and observes the coherence all along2 4444--,,locke1 ,,john4 ,the ,conduct of the ,understanding1 ,sect4 #g4 #dbg4 ,as an exercise of the reasoning faculty1 pure mathematics is an admirable exercise1 because it consists of reasoning alone1 and does not encumber the student with an exercise of judgment3 and it is well to begin with learning one thing at a time1 and to defer a combination of mental exercises to a later period4--,,whately1 ,r4 ,annotations to ,bacon's ,essays 7,boston1 #ahgc71 ,essay #a1 p4 #dic4 #dbh4 ,it hath been an old remark1 that ,geometry is an excellent ,logic4 ,and it must be owned that when the definitions are clear2 when the postulata cannot be refused1 nor the axioms denied2 when from the distinct contemplation and comparison of figures1 their properties are derived1 by a perpetual well-connected chain of consequences1 the objects being still kept in view1 and the attention ever fixed upon them2 there is acquired a habit of reasoning1 close and exact and methodical2 which habit strengthens and sharpens the mind1 and being transferred to other subjects is of general use in the inquiry after truth4--,,berkely1 ,,george4 ,the ,analyst1 #b2 ,works 7,london1 #ahih71 ,vol4 #c1 p4 #aj4 #dbi4 ,suppose then ,i want to give myself a little training in the art of reasoning2 suppose ,i want to get out of the region of conjecture and probability1 free myself from the difficult task of weighing evidence1 and putting instances together to arrive at general propositions1 and simply desire to know how to deal with my general propositions when ,i get them1 and how to deduce right inferences from them2 it is clear that ,i shall obtain this sort of discipline best in those departments of thought in which the first principles are unquestionably true4 ,for in all our thinking1 if we come to erroneous conclusions1 we come to them either by accepting false premises to start with--in which case our reasoning1 however good1 will not save us from error2 or by reasoning badly1 in which case the data we start from may be perfectly sound1 and yet our conclusions may be false4 ,but in the mathematical or pure sciences1--geometry1 arithmetic1 algebra1 trigonometry1 the calculus of variations or of curves1--we know at least that there is not1 and cannot be1 error in our first principles1 and we may therefore fasten our whole attention upon the processes4 ,as mere exercises in logic1 therefore1 these sciences1 based as they all are on primary truths relating to space and number1 have always been supposed to furnish the most exact discipline4 ,when ,plato wrote over the portal of his school4 8,let no one ignorant of geometry enter here10 he did not mean that questions relating to lines and surfaces would be discussed by his disciples4 ,on the contrary1 the topics to which he directed their attention were some of the deepest problems1--social1 political1 moral1--on which the mind could exercise itself4 ,plato and his followers tried to think out together conclusions respecting the being1 the duty1 and the destiny of man1 and the relation in which he stood to the gods and to the unseen world4 ,what had geometry to do with these things8 ,simply this3 ,that a man whose mind has not undergone a rigorous training in systematic thinking1 and in the art of drawing legitimate inferences from premises1 was unfitted to enter on the discussion of these high topics2 and that the sort of logical discipline which he needed was most likely to be obtained from geometry--the only mathematical science which in ,plato's time had been formulated and reduced to a system4 ,and we in this country ,england have long acted on the same principle4 ,our future lawyers1 clergy1 and statesmen are expected at the ,university to learn a good deal about curves1 and angles1 and numbers and proportions2 not because these subjects have the smallest relation to the needs of their lives1 but because in the very act of learning them they are likely to acquire that habit of steadfast and accurate thinking1 which is indispensable to success in all the pursuits of life4 --,,fitch1 ,j4 ,c4 ,lectures on ,teaching 7,new ,york1 #aijf71 pp4 #bia-bib4 #dcj4 ,it is admitted by all that a finished or even a competent reasoner is not the work of nature alone2 the experience of every day makes it evident that education develops faculties which would otherwise never have manifested their existence4 ,it is1 therefore1 as necessary to learn to reason before we can expect to be able to reason1 as it is to learn to swim or fence1 in order to attain either of those arts4 ,now1 something must be reasoned upon1 it matters not much what it is1 provided it can be reasoned upon with certainty4 ,the properties of mind or matter1 or the study of languages1 mathematics1 or natural history1 may be chosen for this purpose4 ,now of all these1 it is desirable to choose the one which admits of the reasoning being verified1 that is1 in which we can find out by other means1 such as measurement and ocular demonstration of all sorts1 whether the results are true or not4 ,when the guiding property of the loadstone was first ascertained1 and it was necessary to learn how to use this new discovery1 and to find out how far it might be relied on1 it would have been thought advisable to make many passages between ports that were well known before attempting a voyage of discovery4 ,so it is with our reasoning faculties3 it is desirable that their powers should be exerted upon objects of such a nature1 that we can tell by other means whether the results which we obtain are true or false1 and this before it is safe to trust entirely to reason4 ,now the mathematics are peculiarly well adapted for this purpose1 on the following grounds3 #a4 ,every term is distinctly explained1 and has but one meaning1 and it is rarely that two words are employed to mean the same thing4 #b4 ,the first principles are self-evident1 and1 though derived from observation1 do not require more of it than has been made by children in general4 #c4 ,the demonstration is strictly logical1 taking nothing for granted except self-evident first principles1 resting nothing upon probability1 and entirely independent of authority and opinion4 #d4 ,when the conclusion is obtained by reasoning1 its truth or falsehood can be ascertained1 in geometry by actual measurement1 in algebra by common arithmetical calculation4 ,this gives confidence1 and is absolutely necessary1 if1 as was said before1 reason is not to be the instructor1 but the pupil4 #e4 ,there are no words whose meanings are so much alike that the ideas which they stand for may be confounded4 ,between the meaning of terms there is no distinction1 except a total distinction1 and all adjectives and adverbs expressing difference of degrees are avoided4--,,de ,,morgan1 ,,augustus4 ,on the ,study and ,difficulties of ,mathematics 7,chicago1 #ahih71 chap4 #a4 #dca4 ,the instruction of children should aim gradually to combine knowing and doing ,wissen und ,knnen4 ,among all sciences mathematics seems to be the only one of a kind to satisfy this aim most completely4--,,kant1 ,,immanuel4 ,werke ,rosenkranz und ,schubert1 ,bd4 #i 7,leipzig1 #ahch71 p4 #dji4 #dcb4 ,every discipline must be honored for reason other than its utility1 otherwise it yields no enthusiasm for industry4 ,for both reasons1 ,i consider mathematics the chief subject for the common school4 ,no more highly honored exercise for the mind can be found2 the buoyancy ,spannkraft which it produces is even greater than that produced by the ancient languages1 while its utility is unquestioned4--,,herbart1 ,j4 ,f4 ,mathematischer ,lehrplan fr ,realgymnasien1 ,werke ,kehrbach1 7,langensalza1 #ahij71 ,bd4 #e1 p4 #afg4 #dcc4 ,the motive for the study of mathematics is insight into the nature of the universe4 ,stars and strata1 heat and electricity1 the laws and processes of becoming and being1 incorporate mathematical truths4 ,if language imitates the voice of the ,creator1 revealing ,his heart1 mathematics discloses ,his intellect1 repeating the story of how things came into being4 ,and the value of mathematics1 appealing as it does to our energy and to our honor1 to our desire to know the truth and thereby to live as of right in the household of ,god1 is that it establishes us in larger and larger certainties4 ,as literature develops emotion1 understanding1 and sympathy1 so mathematics develops observation1 imagination1 and reason4--,,chancellor1 ,w4 ,e4 ,a ,theory of ,motives1 ,ideals and ,values in ,education 7,boston and ,new ,york1 #aijg71 p4 #djf4 #dcd4 ,mathematics in its pure form1 as arithmetic1 algebra1 geometry1 and the applications of the analytic method1 as well as mathematics applied to matter and force1 or statics and dynamics1 furnishes the peculiar study that gives to us1 whether as children or as men1 the command of nature in this its quantitative aspect2 mathematics furnishes the instrument1 the tool of thought1 which we wield in this realm4--,,harris1 ,w4 ,t4 ,psychologic ,foundations of ,education 7,new ,york1 #ahih71 p4 #cbe4 #dce4 ,little can be understood of even the simplest phenomena of nature without some knowledge of mathematics1 and the attempt to penetrate deeper into the mysteries of nature compels simultaneous development of the mathematical processes4--,,young1 ,j4 ,w4 ,a4 ,the ,teaching of ,mathematics 7,new ,york1 #aijg71 p4 #af4 #dcf4 ,for many parts of nature can neither be invented with sufficient subtility nor demonstrated with sufficient perspicuity nor accommodated unto use with sufficient dexterity1 without the aid and intervening of mathematics4--,,bacon1 ,,lord4 ,de ,augmentis1 ,bk4 #b2 ,advancement of ,learning1 ,bk4 #c4 #dcg4 ,i confess1 that after ,i began 444 to discern how useful mathematicks may be made to physicks1 ,i have often wished that ,i had employed about the speculative part of geometry1 and the cultivation of the specious ,algebra ,i had been taught very young1 a good part of that time and industry1 that ,i had spent about surveying and fortification 7of which ,i remember ,i once wrote an entire treatise7 and other parts of practick mathematicks4 --,,boyle1 ,,robert4 ,the ,usefulness of ,mathematicks to ,natural ,philosophy2 ,works 7,london1 #aggb71 ,vol4 #c1 p4 #dbf4 #dch4 ,mathematics gives the young man a clear idea of demonstration and habituates him to form long trains of thought and reasoning methodically connected and sustained by the final certainty of the result2 and it has the further advantage1 from a purely moral point of view1 of inspiring an absolute and fanatical respect for truth4 ,in addition to all this1 mathematics1 and chiefly algebra and infinitesimal calculus1 excite to a high degree the conception of the signs and symbols--necessary instruments to extend the power and reach of the human mind by summarizing an aggregate of relations in a condensed form and in a kind of mechanical way4 ,these auxiliaries are of special value in mathematics because they are there adequate to their definitions1 a characteristic which they do not possess to the same degree in the physical and mathematical natural8 sciences4 ,there are1 in fact1 a mass of mental and moral faculties that can be put in full play only by instruction in mathematics2 and they would be made still more available if the teaching was directed so as to leave free play to the personal work of the student4 --,,berthelot1 ,m4 ,p4 ,e4 ,m4 ,science as an ,instrument of ,education2 ,popular ,science ,monthly 7#ahig71 p4 #bec4 #dci4 ,mathematical knowledge1 therefore1 appears to us of value not only in so far as it serves as means to other ends1 but for its own sake as well1 and we behold1 both in its systematic external and internal development1 the most complete and purest logical mind-activity1 the embodiment of the highest intellect-esthetics4--,,pringsheim1 ,,alfred4 ,ueber ,wert und angeblichen ,unwert der ,mathematik2 ,jahresbericht der ,deutschen ,mathematiker ,vereinigung1 ,bd4 #ac1 p4 #cha4 #ddj4 ,the advantages which mathematics derives from the peculiar nature of those relations about which it is conversant1 from its simple and definite phraseology1 and from the severe logic so admirably displayed in the concatenation of its innumerable theorems1 are indeed immense1 and well entitled to separate and ample illustration4--,,stewart1 ,,dugald4 ,philosophy of the ,human ,mind1 ,part #b1 chap4 #b1 sect4 #c4 #dda4 ,i do not intend to go deeply into the question how far mathematical studies1 as the representatives of conscious logical reasoning1 should take a more important place in school education4 ,but it is1 in reality1 one of the questions of the day4 ,in proportion as the range of science extends1 its system and organization must be improved1 and it must inevitably come about that individual students will find themselves compelled to go through a stricter course of training than grammar is in a position to supply4 ,what strikes me in my own experience with students who pass from our classical schools to scientific and medical studies1 is first1 a certain laxity in the application of strictly universal laws4 ,the grammatical rules1 in which they have been exercised1 are for the most part followed by long lists of exceptions2 accordingly they are not in the habit of relying implicitly on the certainty of a legitimate deduction from a strictly universal law4 ,secondly1 ,i find them for the most part too much inclined to trust to authority1 even in cases where they might form an independent judgment4 ,in fact1 in philological studies1 inasmuch as it is seldom possible to take in the whole of the premises at a glance1 and inasmuch as the decision of disputed questions often depends on an sthetic feeling for beauty of expression1 or for the genius of the language1 attainable only by long training1 it must often happen that the student is referred to authorities even by the best teachers4 ,both faults are traceable to certain indolence and vagueness of thought1 the sad effects of which are not confined to subsequent scientific studies4 ,but certainly the best remedy for both is to be found in mathematics1 where there is absolute certainty in the reasoning1 and no authority is recognized but that of one's own intelligence4--,,helmholtz1 ,h4 ,on the ,relation of ,natural ,science to ,science in general2 ,popular ,lectures on ,scientific ,subjects2 ,atkinson 7,new ,york1 #aijj71 pp4 #be-bf4 #ddb4 ,what renders a problem definite1 and what leaves it indefinite1 may best be understood from mathematics4 ,the very important idea of solving a problem within limits of error is an element of rational culture1 coming from the same source4 ,the art of totalizing fluctuations by curves is capable of being carried1 in conception1 far beyond the mathematical domain1 where it is first learned4 ,the distinction between laws and coefficients applies in every department of causation4 ,the theory of ,probable ,evidence is the mathematical contribution to ,logic1 and is of paramount importance4--,,bain1 ,,alexander4 ,education as a ,science 7,new ,york1 #ahih71 pp4 #aea-aeb4 #ddc4 ,we receive it as a fact1 that some minds are so constituted as absolutely to require for their nurture the severe logic of the abstract sciences2 that rigorous sequence of ideas which leads from the premises to the conclusion1 by a path1 arduous and narrow1 it may be1 and which the youthful reason may find it hard to mount1 but where it cannot stray2 and on which1 if it move at all1 it must move onward and upward4444 ,even for intellects of a different character1 whose natural aptitude is for moral evidence and those relations of ideas which are perceived and appreciated by taste1 the study of the exact sciences may be recommended as the best protection against the errors into which they are most likely to fall4 ,although the study of language is in many respects no mean exercise in logic1 yet it must be admitted that an eminently practical mind is hardly to be formed without mathematical training4 --,,everett1 ,,edward4 ,orations and ,speeches 7,boston1 #ahgj71 ,vol4 #b1 p4 #eaj4 #ddd4 ,the value of mathematical instruction as a preparation for those more difficult investigations1 consists in the applicability not of its doctrines but of its methods4 ,mathematics will ever remain the past perfect type of the deductive method in general2 and the applications of mathematics to the simpler branches of physics furnish the only school in which philosophers can effectually learn the most difficult and important of their art1 the employment of the laws of simpler phenomena for explaining and predicting those of the more complex4 ,these grounds are quite sufficient for deeming mathematical training an indispensable basis of real scientific education1 and regarding with ,plato1 one who is 1 as wanting in one of the most essential qualifications for the successful cultivation of the higher branches of philosophy4--,,mill1 ,j4 ,s4 ,system of ,logic1 ,bk4 #c1 chap4 #bd1 sect4 #i4 #dde4 ,this science1 ,geometry1 is one of indispensable use and constant reference1 for every student of the laws of nature2 for the relations of space and number are the alphabet in which those laws are written4 ,but besides the interest and importance of this kind which geometry possesses1 it has a great and peculiar value for all who wish to understand the foundations of human knowledge1 and the methods by which it is acquired4 ,for the student of geometry acquires1 with a degree of insight and clearness which the unmathematical reader can but feebly imagine1 a conviction that there are necessary truths1 many of them of a very complex and striking character2 and that a few of the most simple and self-evident truths which it is possible for the mind of man to apprehend1 may1 by systematic deduction1 lead to the most remote and unexpected results4--,,whewell1 ,,william4 ,the ,philosophy of the ,inductive ,sciences1 ,part #a1 ,bk4 #b1 chap4 #d1 sect4 #h 7,london1 #aheh74 #ddf4 ,mathematics1 while giving no quick remuneration1 like the art of stenography or the craft of bricklaying1 does furnish the power for deliberate thought and accurate statement1 and to speak the truth is one of the most social qualities a person can possess4 ,gossip1 flattery1 slander1 deceit1 all spring from a slovenly mind that has not been trained in the power of truthful statement1 which is one of the highest utilities4--,,dutton1 ,s4 ,t4 ,social ,phases of ,education in the ,school and the ,home 7,london1 #aijj71 p4 #cj4 #ddg4 ,it is from this absolute indifference and tranquility of the mind1 that mathematical speculations derive some of their most considerable advantages2 because there is nothing to interest the imagination2 because the judgment sits free and unbiased to examine the point4 ,all proportions1 every arrangement of quantity1 is alike to the understanding1 because the same truths result to it from all2 from greater from lesser1 from equality and inequality4--,,burke1 ,,edmund4 ,on the ,sublime and ,beautiful1 ,part #c1 sect4 #b4 #ddh4 ,out of the interaction of form and content in mathematics grows an acquaintance with methods which enable the student to produce independently within certain though moderate limits1 and to extend his knowledge through his own reflection4 ,the deepening of the consciousness of the intellectual powers connected with this kind of activity1 and the gradual awakening of the feeling of intellectual self-reliance may well be considered as the most beautiful and highest result of mathematical training4 --,,pringsheim1 ,,alfred4 ,ueber ,wert und angeblichen ,unwert der ,mathematik2 ,jahresbericht der ,deutschen ,mathematiker ,vereinigung 7#aijd71 p4 #cgd4 #ddi4 ,he who would know what geometry is1 must venture boldly into its depths and learn to think and feel as a geometer4 ,i believe that it is impossible to do this1 and to study geometry as it admits of being studied and am conscious it can be taught1 without finding the reason invigorated1 the invention quickened1 the sentiment of the orderly and beautiful awakened and enhanced1 and reverence for truth1 the foundation of all integrity of character1 converted into a fixed principle of the mental and moral constitution1 according to the old and expressive adage 8abeunt studia in mores40--,,sylvester1 ,j4 ,j4 ,a probationary ,lecture on ,geometry2 ,collected ,mathematical ,papers 7,cambridge1 #aijh71 ,vol4 #b1 p4 #i4 #dej4 ,mathematical knowledge adds vigour to the mind1 frees it from prejudice1 credulity1 and superstition4--,,arbuthnot1 ,,john4 ,usefulness of ,mathematical ,learning4 #dea4 ,when the boy begins to understand that the visible point is preceded by an invisible point1 that the shortest distance between two points is conceived as a straight line before it is ever drawn with the pencil on paper1 he experiences a feeling of pride1 of satisfaction4 ,and justly so1 for the fountain of all thought has been opened to him1 the difference between the ideal and the real1 potentia et actu1 has become clear to him2 henceforth the philosopher can reveal him nothing new1 as a geometrician he has discovered the basis of all thought4--,,goethe4 ,sprche in ,prosa1 ,ethisches1 ,,vi1 #dee4 #deb4 ,in mathematics1 444 and in natural philosophy since mathematics was applied to it1 we see the noblest instance of the force of the human mind1 and of the sublime heights to which it may rise by cultivation4 ,an acquaintance with such sciences naturally leads us to think well of our faculties1 and to indulge sanguine expectations concerning the improvement of other parts of knowledge4 ,to this ,i may add1 that1 as mathematical and physical truths are perfectly uninteresting in their consequences1 the understanding readily yields its assent to the evidence which is presented to it2 and in this way may be expected to acquire the habit of trusting to its own conclusions1 which will contribute to fortify it against the weaknesses of scepticism1 in the more interesting inquiries after moral truth in which it may afterwards engage4--,,stewart1 ,,dugald4 ,philosophy of the ,human ,mind1 ,part #c1 chap4 #a1 sect4 #c4 #dec4 ,those that can readily master the difficulties of ,mathematics find a considerable charm in the study1 sometimes amounting to fascination4 ,this is far from universal2 but the subject contains elements of strong interest of a kind that constitutes the pleasures of knowledge4 ,the marvellous devices for solving problems elate the mind with the feeling of intellectual power2 and the innumerable constructions of the science leave us lost in wonder4--,,bain1 ,,alexander4 ,education as a ,science 7,new ,york1 #ahih71 p4 #aec4 #ded4 ,thinking is merely the comparing of ideas1 discerning relations of likeness and of difference between ideas1 and drawing inferences4 ,it is seizing general truths on the basis of clearly apprehended particulars4 ,it is but generalizing and particularizing4 ,who will deny that a child can deal profitably with sequences of ideas like3 ,how many marbles are #b marbles and #c marbles8 #b pencils and #c pencils8 #b balls and #c balls8 #b children and #c children8 #b inches and #c inches8 #b feet and #c feet8 #b and #c8 ,who has not seen the countenance of some little learner light up at the end of such a series of questions with the exclamation1 8,why it's always that way4 ,isn't it80 ,this is the glow of pleasure that the generalizing step always affords him who takes the step himself4 ,this is the genuine life-giving joy which comes from feeling that one can successfully take this step4 ,the reality of such a discovery is as great1 and the lasting effect upon the mind of him that makes it is as sure as was that by which the great ,newton hit upon the generalization of the law of gravitation4 ,it is through these thrills of discovery that love to learn and intellectual pleasure are begotten and fostered4 ,good arithmetic teaching abounds in such opportunities4 --,,myers1 ,,george4 ,arithmetic in ,public ,education 7,chicago71 p4 #ac4 #dee4 ,a general course in mathematics should be required of all officers for its practical value1 but no less for its educational value in training the mind to logical forms of thought1 in developing the sense of absolute truthfulness1 together with a confidence in the accomplishment of definite results by definite means4--,,echols1 ,c4 ,p4 ,mathematics at ,west ,point and ,annapolis2 ,u4 ,s4 ,bureau of ,education1 ,bulletin #aiab1 ,no4 #b1 p4 #aa4 #def4 ,exercise in the most rigorous thinking that is possible will of its own accord strengthen the sense of truth and right1 for each advance in the ability to distinguish between correct and false thoughts1 each habit making for rigour in thought development will increase in the sound pupil the ability and the wish to ascertain what is right in life and to defend it4 --,,reidt1 ,f4 ,anleitung zum mathematischen ,unterricht in den hheren ,schulen 7,berlin1 #aijf71 p4 #bh4 #deg4 ,i do not maintain that the chief value of the study of arithmetic consists in the lessons of morality that arise from this study4 ,i claim only that1 to be impressed from day to day1 that there is something that is right as an answer to the questions with which one is able to grapple1 and that there is a wrong answer--that there are ways in which the right answer can be established as right1 that these ways automatically reject error and slovenliness1 and that the learner is able himself to manipulate these ways and to arrive at the establishment of the true as opposed to the untrue1 this relentless hewing to the line and stopping at the line1 must color distinctly the thought life of the pupil with more than a tinge of morality4444 ,to be neighborly with truth1 to feel one's self somewhat facile in ways of recognizing and establishing what is right1 what is correct1 to find the wrong persistently and unfailingly rejected as of no value1 to feel that one can apply these ways for himself1 that one can think and work independently1 have a real1 a positive1 and a purifying effect upon moral character4 ,they are the quiet1 steady undertones of the work that always appeal to the learner for the sanction of his best judgment1 and these are the really significant matters in school work4 ,it is not the noise and bluster1 not even the dramatics or the polemics from the teacher's desk1 that abide longest and leave the deepest and stablest imprint upon character4 ,it is these still1 small voices that speak unmistakably for the right and against the wrong and the erroneous that really form human character4 ,when the school subjects are arranged on the basis of the degree to which they contribute to the moral upbuilding of human character good arithmetic will be well up the list4--,,myers1 ,,george4 ,arithmetic in ,public ,education 7,chicago71 p4 #ah4 #deh4 ,in destroying the predisposition to anger1 science of all kind is useful2 but the mathematics possess this property in the most eminent degree4--,,dr4 ,,rush4 ,quoted in ,day's ,collacon 7,london1 no date74 #dei4 ,the mathematics are the friends to religion1 inasmuch as they charm the passions1 restrain the impetuosity of the imagination1 and purge the mind from error and prejudice4 ,vice is error1 confusion and false reasoning2 and all truth is more or less opposite to it4 ,besides1 mathematical truth may serve for a pleasant entertainment for those hours which young men are apt to throw away upon their vices2 the delightfulness of them being such as to make solitude not only easy but desirable4 --,,arbuthnot1 ,,john4 ,usefulness of ,mathematical ,learning4 #dfj4 ,there is no prophet which preaches the superpersonal ,god more plainly than mathematics4--,,carus1 ,,paul4 ,reflections on ,magic ,squares2 ,monist 7#aijf71 p4 #adg4 #dfa4 ,mathematics must subdue the flights of our reason2 they are the staff of the blind2 no one can take a step without them2 and to them and experience is due all that is certain in physics4 --,,voltaire4 ,oeuvres ,compltes 7,paris1 #ahhj71 t4 #ce1 p4 #bai4 ,,chapter ,v ,,the ,,teaching ,,of ,,mathematics #eja4 ,in mathematics two ends are constantly kept in view3 ,first1 stimulation of the inventive faculty1 exercise of judgment1 development of logical reasoning1 and the habit of concise statement2 second1 the association of the branches of pure mathematics with each other and with applied science1 that the pupil may see clearly the true relations of principles and things4 ,international ,commission on the ,teaching of ,mathematics1 ,american ,report2 ,u4 ,s4 ,bureau of ,education1 ,bulletin #aiab1 ,no4 #d1 p4 #g4 #ejb4 ,the ends to be attained in the teaching of mathematics in the secondary schools are the knowledge of a body of geometrical truths1 the power to draw correct inferences from given premises1 the power to use algebraic processes as a means of finding results in practical problems1 and the awakening of interest in the science of mathematics4 ,international ,commission on the ,teaching of ,mathematics1 ,american ,report2 ,u4 ,s4 ,bureau of ,education1 ,bulletin #aiab1 ,no4 #d1 p4 #g4 #ejc4 ,general preparatory instruction must continue to be the aim in the instruction at the higher institutions of learning4 ,exclusive selection and treatment of subject matter with reference to specific avocations is disadvantageous4 ,resolution adopted by the ,german ,association for the ,advancement of ,scientific and ,mathematical ,instruction2 ,jahresbericht der ,deutschen ,mathematiker ,vereinigung 7#ahif71 p4 #da4 #ejd4 ,in the secondary schools mathematics should be a part of general culture and not contributory to technical training of any kind2 it should cultivate space intuition1 logical thinking1 the power to rephrase in clear language thoughts recognized as correct1 and ethical and esthetic effects2 so treated1 mathematics is a quite indispensable factor of general education in so far as the latter shows its traces in the comprehension of the development of civilization and the ability to participate in the further tasks of civilization4 ,unterrichtsbltter fr ,mathematik und ,naturwissenschaft 7#aijd71 p4 #abh4 #eje4 ,indeed1 the aim of teaching mathematics should be rather to strengthen his the pupil's faculties1 and to supply a method of reasoning applicable to other subjects1 than to furnish him with an instrument for solving practical problems4--,,magnus1 ,,philip4 ,perry's ,teaching of ,mathematics 7,london1 #aijb71 p4 #hd4 #ejf4 ,the participation in the general development of the mental powers without special reference to his future vocation must be recognized as the essential aim of mathematical instruction4--,,reidt1 ,f4 ,anleitung zum ,mathematischen ,unterricht an hheren ,schulen 7,berlin1 #aijf71 p4 #ab4 #ejg4 ,i am of the decided opinion1 that mathematical instruction must have for its first aim a deep penetration and complete command of abstract mathematical theory together with a clear insight into the structure of the system1 and doubt not that the instruction which accomplishes this is valuable and interesting even if it neglects practical applications4 ,if the instruction sharpens the understanding1 if it arouses the scientific interest1 whether mathematical or philosophical1 if finally it calls into life an esthetic feeling for the beauty of a scientific edifice1 the instruction will take on an ethical value as well1 provided that with the interest it awakens also the impulse toward scientific activity4 ,i contend1 therefore1 that even without reference to its applications mathematics in the high schools has a value equal to that of the other subjects of instruction4--,,goetting1 ,e4 ,ueber das ,lehrziel im mathematischen ,unterricht der hheren ,realanstalten2 ,jahresbericht der ,deutschen ,mathematiker ,vereinigung1 ,bd4 #b1 p4 #aib4 #ejh4 ,mathematics will not be properly esteemed in wider circles until more than the a b c of it is taught in the schools1 and until the unfortunate impression is gotten rid of that mathematics serves no other purpose in instruction than the formal training of the mind4 ,the aim of mathematics is its content1 its form is a secondary consideration and need not necessarily be that historic form which is due to the circumstance that mathematics took permanent shape under the influence of ,greek logic4--,,hankel1 ,h4 ,die ,entwickelung der ,mathematik in den letzten ,jahrhunderten 7,tbingen1 #ahhd71 p4 #f4 #eji4 ,the idea that aptitude for mathematics is rarer than aptitude for other subjects is merely an illusion which is caused by belated or neglected beginners4--,,herbart1 ,j4 ,f4 ,umriss pdagogischer ,vorlesungen2 ,werke ,kehrbach 7,langensalza1 #aijb71 ,bd4 #aj1 p4 #aja4 #eaj4 ,i believe that the useful methods of mathematics are easily to be learned by quite young persons1 just as languages are easily learned in youth4 ,what a wondrous philosophy and history underlie the use of almost every word in every language--yet the child learns to use the word unconsciously4 ,no doubt when such a word was first invented it was studied over and lectured upon1 just as one might lecture now upon the idea of a rate1 or the use of ,cartesian co-ordinates1 and we may depend upon it that children of the future will use the idea of the calculus1 and use squared paper as readily as they now cipher4444 ,when ,egyptian and ,chaldean philosophers spent years in difficult calculations1 which would now be thought easy by young children1 doubtless they had the same notions of the depth of their knowledge that ,sir ,william ,thomson might now have of his4 ,how is it1 then1 that ,thomson gained his immense knowledge in the time taken by a ,chaldean philosopher to acquire a simple knowledge of arithmetic8 ,the reason is plain4 ,thomson1 when a child1 was taught in a few years more than all that was known three thousand years ago of the properties of numbers4 ,when it is found essential to a boy's future that machinery should be given to his brain1 it is given to him2 he is taught to use it1 and his bright memory makes the use of it a second nature to him2 but it is not till after-life that he makes a close investigation of what there actually is in his brain which has enabled him to do so much4 ,it is taken because the child has much faith4 ,in after years he will accept nothing without careful consideration4 ,the machinery given to the brain of children is getting more and more complicated as time goes on2 but there is really no reason why it should not be taken in as early1 and used as readily1 as were the axioms of childish education in ancient ,chaldea4--,,perry1 ,,john4 ,the ,teaching of ,mathematics 7,london1 #aijb71 p4 #ad4 #eaa4 ,the ancients devoted a lifetime to the study of arithmetic2 it required days to extract a square root or to multiply two numbers together4 ,is there any harm in skipping all that1 in letting the school boy learn multiplication sums1 and in starting his more abstract reasoning at a more advanced point8 ,where would be the harm in letting the boy assume the truth of many propositions of the first four books of ,euclid1 letting him assume their truth partly by faith1 partly by trial8 ,giving him the whole fifth book of ,euclid by simple algebra8 ,letting him assume the sixth as axiomatic8 ,letting him1 in fact1 begin his severer studies where he is now in the habit of leaving off8 ,we do much less orthodox things4 ,every here and there in one's mathematical studies one makes exceedingly large assumptions1 because the methodical study would be ridiculous even in the eyes of the most pedantic of teachers4 ,i can imagine a whole year devoted to the philosophical study of many things that a student now takes in his stride without trouble4 ,the present method of training the mind of a mathematical teacher causes it to strain at gnats and to swallow camels4 ,such gnats are most of the propositions of the sixth book of ,euclid2 propositions generally about incommensurables2 the use of arithmetic in geometry2 the parallelogram of forces1 etc41 decimals4--,,perry1 ,,john4 ,the ,teaching of ,mathematics 7,london1 #aijd71 p4 #ab4 #eab4 ,the teaching of elementary mathematics should be conducted so that the way should be prepared for the building upon them of the higher mathematics4 ,the teacher should always bear in mind and look forward to what is to come after4 ,the pupil should not be taught what may be sufficient for the time1 but will lead to difficulties in the future4444 ,i think the fault in teaching arithmetic is that of not attending to general principles and teaching instead of particular rules4444 ,i am inclined to attack the teaching of mathematics on the grounds that it does not dwell sufficiently on a few general axiomatic principles4 --,,hudson1 ,w4 ,h4 ,h4 ,perry's ,teaching of ,mathematics 7,london1 #aijd71 p4 #cc4 #eac4 8,mathematics in ,prussia6 ,ah1 sir1 they teach mathematics in ,prussia as you teach your boys rowing in ,england3 they are trained by men who have been trained by men who have themselves been trained for generations back40--,,langley1 ,e4 ,m4 ,perry's ,teaching of ,mathematics 7,london1 #aijd71 p4 #dc4 #ead4 ,a superficial knowledge of mathematics may lead to the belief that this subject can be taught incidentally1 and that exercises akin to counting the petals of flowers or the legs of a grasshopper are mathematical4 ,such work ignores the fundamental idea out of which quantitative reasoning grows--the equality of magnitudes4 ,it leaves the pupil unaware of that relativity which is the essence of mathematical science4 ,numerical statements are frequently required in the study of natural history1 but to repeat these as a drill upon numbers will scarcely lend charm to these studies1 and certainly will not result in mathematical knowledge4--,,speer1 ,w4 ,w4 ,primary ,arithmetic 7,boston1 #ahig71 pp4 #bf-bg4 #eae4 ,mathematics is no more the art of reckoning and computation than architecture is the art of making bricks or hewing wood1 no more than painting is the art of mixing colors on a palette1 no more than the science of geology is the art of breaking rocks1 or the science of anatomy the art of butchering4 --,,keyser1 ,c4 ,j4 ,lectures on ,science1 ,philosophy and ,art 7,new ,york1 #aijh71 p4 #bi4 #eaf4 ,the study of mathematics--from ordinary reckoning up to the higher processes--must be connected with knowledge of nature1 and at the same time with experience1 that it may enter the pupil's circle of thought4--,,herbart1 ,j4 ,f4 ,letters and ,lectures on ,education ,felkin 7,london1 #aijh71 p4 #aag4 #eag4 ,first1 as concerns the success of teaching mathematics4 ,no instruction in the high schools is as difficult as that of mathematics1 since the large majority of students are at first decidedly disinclined to be harnessed into the rigid framework of logical conclusions4 ,the interest of young people is won much more easily1 if sense-objects are made the starting point and the transition to abstract formulation is brought about gradually4 ,for this reason it is psychologically quite correct to follow this course4 ,not less to be recommended is this course if we inquire into the essential purpose of mathematical instruction4 ,formerly it was too exclusively held that this purpose is to sharpen the understanding4 ,surely another important end is to implant in the student the conviction that correct thinking based on true premises secures mastery over the outer world4 ,to accomplish this the outer world must receive its share of attention from the very beginning4 ,doubtless this is true but there is a danger which needs pointing out4 ,it is as in the case of language teaching where the modern tendency is to secure in addition to grammar also an understanding of the authors4 ,the danger lies in grammar being completely set aside leaving the subject without its indispensable solid basis4 ,just so in the teaching of mathematics it is possible to accumulate interesting applications to such an extent as to stunt the essential logical development4 ,this should in no wise be permitted1 for thus the kernel of the whole matter is lost4 ,therefore3 ,we do want throughout a quickening of mathematical instruction by the introduction of applications1 but we do not want that the pendulum1 which in former decades may have inclined too much toward the abstract side1 should now swing to the other extreme2 we would rather pursue the proper middle course4--,,klein1 ,,felix4 ,ueber den ,mathematischen ,unterricht an den hheren ,schulen2 ,jahresbericht der ,deutschen ,mathematiker ,vereinigung1 ,bd4 #aa1 p4 #aca4 #eah4 ,it is above all the duty of the methodical text-book to adapt itself to the pupil's power of comprehension1 only challenging his higher efforts with the increasing development of his imagination1 his logical power and the ability of abstraction4 ,this indeed constitutes a test of the art of teaching1 it is here where pedagogic tact becomes manifest4 ,in reference to the axioms1 caution is necessary4 ,it should be pointed out comparatively early1 in how far the mathematical body differs from the material body4 ,furthermore1 since mathematical bodies are really portions of space1 this space is to be conceived as mathematical space and to be clearly distinguished from real or physical space4 ,gradually the student will become conscious that the portion of the real space which lies beyond the visible stellar universe is not cognizable through the senses1 that we know nothing of its properties and consequently have no basis for judgments concerning it4 ,mathematical space1 on the other hand1 may be subjected to conditions1 for instance1 we may condition its properties at infinity1 and these conditions constitute the axioms1 say the ,euclidean axioms4 ,but every student will require years before the conviction of the truth of this last statement will force itself upon him4 --,,holzm,,ller1 ,,gustav4 ,methodisches ,lehrbuch der ,elementar-,mathematik 7,leipzig1 #aijd71 ,teil #a1 ,vorwort1 pp4 #d-e4 #eai4 ,like almost every subject of human interest1 this one mathematics is just as easy or as difficult as we choose to make it4 ,a lifetime may be spent by a philosopher in discussing the truth of the simplest axiom4 ,the simplest fact as to our existence may fill us with such wonder that our minds will remain overwhelmed with wonder all the time4 ,a ,scotch ploughman makes a working religion out of a system which appalls a mental philosopher4 ,some boys of ten years of age study the methods of the differential calculus2 other much cleverer boys working at mathematics to the age of nineteen have a difficulty in comprehending the fundamental ideas of the calculus4--,,perry1 ,,john4 ,the ,teaching of ,mathematics 7,london1 #aijb71 pp4 #ai-bj4 #ebj4 ,poor teaching leads to the inevitable idea that the subject mathematics is only adapted to peculiar minds1 when it is the one universal science and the one whose four ground-rules are taught us almost in infancy and reappear in the motions of the universe4--,,safford1 ,t4 ,h4 ,mathematical ,teaching 7,boston1 #aijg71 p4 #ai4 #eba4 ,the number of mathematical students 444 would be much augmented if those who hold the highest rank in science would condescend to give more effective assistance in clearing the elements of the difficulties which they present4--,,de ,,morgan1 ,a4 ,study and ,difficulties of ,mathematics 7,chicago1 #aijb71 ,preface4 #ebb4 ,he that could teach mathematics well1 would not be a bad teacher in any of the rest physics1 chemistry1 biology1 psychology unless by the accident of total inaptitude for experimental illustration2 while the mere experimentalist is likely to fall into the error of missing the essential condition of science as reasoned truth2 not to speak of the danger of making the instruction an affair of sensation1 glitter1 or pyrotechnic show4--,,bain1 ,,alexander4 ,education as a ,science 7,new ,york1 #ahih71 p4 #bih4 #ebc4 ,i should like to draw attention to the inexhaustible variety of the problems and exercises which it mathematics furnishes2 these may be graduated to precisely the amount of attainment which may be possessed1 while yet retaining an interest and value4 ,it seems to me that no other branch of study at all compares with mathematics in this4 ,when we propose a deduction to a beginner we give him an exercise in many cases that would have been admired in the vigorous days of ,greek geometry4 ,although grammatical exercises are well suited to insure the great benefits connected with the study of languages1 yet these exercises seem to me stiff and artificial in comparison with the problems of mathematics4 ,it is not absurd to maintain that ,euclid and ,apollonius would have regarded with interest many of the elegant deductions which are invented for the use of our students in geometry2 but it seems scarcely conceivable that the great masters in any other line of study could condescend to give a moment's attention to the elementary books of the beginner4 --,,todhunter1 ,,isaac4 ,conflict of ,studies 7,london1 #ahgc71 pp4 #aj-aa4 #ebd4 ,the visible figures by which principles are illustrated should1 so far as possible1 have no accessories4 ,they should be magnitudes pure and simple1 so that the thought of the pupil may not be distracted1 and that he may know what features of the thing represented he is to pay attention to4 ,report of the ,committee of ,ten on ,secondary ,school ,subjects1 7,new ,york1 #ahid71 p4 #aji4 #ebe4 ,geometrical reasoning1 and arithmetical process1 have each its own office3 to mix the two in elementary instruction1 is injurious to the proper acquisition of both4--,,de ,,morgan1 ,a4 ,trigonometry and ,double ,algebra 7,london1 #ahdi71 p4 #ib4 #ebf4 ,equations are ,expressions of ,arithmetical ,computation1 and properly have no place in ,geometry1 except as far as ,quantities truly ,geometrical 7that is1 ,lines1 ,surfaces1 ,solids1 and ,proportions7 may be said to be some equal to others4 ,multiplications1 ,divisions1 and such sort of ,computations1 are newly received into ,geometry1 and that unwarily1 and contrary to the first ,design of this ,science4 ,for whosoever considers the ,construction of a ,problem by a right ,line and a ,circle1 found out by the first ,geometricians1 will easily perceive that ,geometry was invented that we might expeditiously avoid1 by drawing ,lines1 the ,tediousness of ,computation4 ,therefore these two ,sciences ought not to be confounded4 ,the ,ancients did so industriously distinguish them from one another1 that they never introduced ,arithmetical ,terms into ,geometry4 ,and the ,moderns1 by confounding both1 have lost the ,simplicity in which all the ,elegance of ,geometry consists4 ,wherefore that is ,arithmetically more simple which is determined by the more simple ,equation1 but that is ,geometrically more simple which is determined by the more simple drawing of ,lines2 and in ,geometry1 that ought to be reckoned best which is geometrically most simple4--,,newton4 ,on the ,linear ,construction of ,equations2 ,universal ,arithmetic 7,london1 #agfi71 ,vol4 #b1 p4 #dgj4 #ebg4 ,as long as algebra and geometry proceeded along separate paths1 their advance was slow and their applications limited4 ,but when these sciences joined company1 they drew from each other fresh vitality and thenceforward marched on at a rapid pace toward perfection4--,,lagrange4 ,leons lmentaires sur les ,mathematiques1 ,leon cinquime4 ,mc,cormack4 #ebh4 ,the greatest enemy to true arithmetic work is found in so-called practical or illustrative problems1 which are freely given to our pupils1 of a degree of difficulty and complexity altogether unsuited to their age and mental development4444 ,i am1 myself1 no bad mathematician1 and all the reasoning powers with which nature endowed me have long been as fully developed as they are ever likely to be2 but ,i have1 not infrequently1 been puzzled1 and at times foiled1 by the subtle logical difficulty running through one of these problems1 given to my own children4 ,the head-master of one of our ,boston high schools confessed to me that he had sometimes been unable to unravel one of these tangled skeins1 in trying to help his own daughter through her evening's work4 ,during this summer1 ,dr4 ,fairbairn1 the distinguished head of one of the colleges of ,oxford1 ,england1 told me that not only had he himself encountered a similar difficulty1 in the case of his own children1 but that1 on one occasion1 having as his guest one of the first mathematicians of ,england1 the two together had been completely puzzled by one of these arithmetical conundrums4 --,,walker1 ,f4 ,a4 ,discussions in ,education 7,new ,york1 #ahii71 pp4 #bec-bed4 #ebi4 ,it is often assumed that because the young child is not competent to study geometry systematically he need be taught nothing geometrical2 that because it would be foolish to present to him physics and mechanics as sciences it is useless to present to him any physical or mechanical principles4 ,an error of like origin1 which has wrought incalculable mischief1 denies to the scholar the use of the symbols and methods of algebra in connection with his early essays in numbers because1 forsooth1 he is not as yet capable of mastering quadratics6444 ,the whole infant generation1 wrestling with arithmetic1 seek for a sign and groan and travail together in pain for the want of it2 but no sign is given them save the sign of the prophet ,jonah1 the withered gourd1 fruitless endeavor1 wasted strength4 --,,walker1 ,f4 ,a4 ,industrial ,education2 ,discussions in ,education 7,new ,york1 #ahii71 p4 #acb4 #ecj4 ,particular and contingent inventions in the solution of problems1 which1 though many times more concise than a general method would allow1 yet1 in my judgment1 are less proper to instruct a learner1 as acrostics1 and such kind of artificial poetry1 though never so excellent1 would be but improper examples to instruct one that aims at ,ovidean poetry4--,,newton1 ,,isaac4 ,letter to ,collins1 #afgj2 ,macclesfield1 ,correspondence of ,scientific ,men 7,oxford1 #ahda71 ,vol4 #b1 p4 #cjg4 #eca4 ,the logic of the subject algebra1 which1 both educationally and scientifically speaking1 is the most important part of it1 is wholly neglected4 ,the whole training consists in example grinding4 ,what should have been merely the help to attain the end has become the end itself4 ,the result is that algebra1 as we teach it1 is neither an art nor a science1 but an ill-digested farrago of rules1 whose object is the solution of examination problems4444 ,the result1 so far as problems worked in examinations go1 is1 after all1 very miserable1 as the reiterated complaints of examiners show2 the effect on the examinee is a well-known enervation of mind1 an almost incurable superficiality1 which might be called ,problematic ,paralysis--a disease which unfits a man to follow an argument extending beyond the length of a printed octavo page4 --,,chrystal1 ,,george4 ,presidential ,address ,british ,association for the ,advancement of ,science1 #ahhe2 ,nature1 ,vol4 #cb1 pp4 #ddg-ddh4 #ecb4 ,it is a serious question whether ,america1 following ,england's lead1 has not gone into problem-solving too extensively4 ,certain it is that we are producing no text-books in which the theory is presented in the delightful style which characterizes many of the ,french works 4441 or those of the recent ,italian school1 or1 indeed1 those of the continental writers in general4 --,,smith1 ,d4 ,e4 ,the ,teaching of ,elementary ,mathematics 7,new ,york1 #aijb71 p4 #bai4 #ecc4 ,the problem for a writer of a text-book has come now1 in fact1 to be this--to write a book so neatly trimmed and compacted that no coach1 on looking through it1 can mark a single passage which the candidate for a minimum pass can safely omit4 ,some of these text-books ,i have seen1 where the scientific matter has been1 like the lady's waist in the nursery song1 compressed 8so gent and sma'10 that the thickness barely1 if at all1 surpasses what is devoted to the publisher's advertisements4 ,we shall return1 ,i verily believe1 to the ,compendium of ,martianus ,capella4 ,the result of all this is that science1 in the hands of specialists1 soars higher and higher into the light of day1 while educators and the educated are left more and more to wander in primeval darkness4--,,chrystal1 ,,george4 ,presidential ,address ,british ,association for the ,advancement of ,science1 #ahhe2 ,nature1 ,vol4 #cb1 p4 #ddh4 #ecd4 ,some persons have contended that mathematics ought to be taught by making the illustrations obvious to the senses4 ,nothing can be more absurd or injurious3 it ought to be our never-ceasing effort to make people think1 not feel4--,,coleridge1 ,s4 ,t4 ,lectures on ,shakespere 7,bohn ,library71 p4 #eb4 #ece4 ,i have come to the conclusion that the exertion1 without which a knowledge of mathematics cannot be acquired1 is not materially increased by logical rigor in the method of instruction4 --,,pringsheim1 ,,alfred4 ,jahresbericht der ,deutschen ,mathematiker ,vereinigung 7#ahih71 p4 #adc4 #ecf4 ,the only way in which to treat the elements of an exact and rigorous science is to apply to them all the rigor and exactness possible4--,d',,alembert4 ,quoted by ,de ,morgan3 ,trigonometry and ,double ,algebra 7,london1 #ahdi71 ,title page4 #ecg4 ,it is an error to believe that rigor in proof is an enemy of simplicity4 ,on the contrary we find it confirmed by numerous examples that the rigorous method is at the same time the simpler and the more easily comprehended4 ,the very effort for rigor forces us to find out simpler methods of proof4--,,hilbert1 ,d4 ,mathematical ,problems2 ,bulletin ,american ,mathematical ,society1 ,vol4 #h1 p4 #dda4 #ech4 ,few will deny that even in the first scientific instruction in mathematics the most rigorous method is to be given preference over all others4 ,especially will every teacher prefer a consistent proof to one which is based on fallacies or proceeds in a vicious circle1 indeed it will be morally impossible for the teacher to present a proof of the latter kind consciously and thus in a sense deceive his pupils4 ,notwithstanding these objectionable so-called proofs1 so far as the foundation and the development of the system is concerned1 predominate in our textbooks to the present time4 ,perhaps it will be answered1 that rigorous proof is found too difficult for the pupil's power of comprehension4 ,should this be anywhere the case1--which would only indicate some defect in the plan or treatment of the whole1--the only remedy would be to merely state the theorem in a historic way1 and forego a proof with the frank confession that no proof has been found which could be comprehended by the pupil2 a remedy which is ever doubtful and should only be applied in the case of extreme necessity4 ,but this remedy is to be preferred to a proof which is no proof1 and is therefore either wholly unintelligible to the pupil1 or deceives him with an appearance of knowledge which opens the door to all superficiality and lack of scientific method4--,,grassmann1 ,,hermann4 ,stcke aus dem ,lehrbuche der ,arithmetik2 ,werke1 ,bd4 #b 7,leipsig1 #aijd71 p4 #bif4 #eci4 ,the average ,english author of mathematical texts leaves one under the impression that he has made a bargain with his reader to put before him the truth1 the greater part of the truth1 and nothing but the truth2 and that if he has put the facts of his subject into his book1 however difficult it may be to unearth them1 he has fulfilled his contract with his reader4 ,this is a very much mistaken view1 because effective teaching requires a great deal more than a bare recitation of facts1 even if these are duly set forth in logical order--as in ,english books they often are not4 ,the probable difficulties which will occur to the student1 the objections which the intelligent student will naturally and necessarily raise to some statement of fact or theory--these things our authors seldom or never notice1 and yet a recognition and anticipation of them by the author would be often of priceless value to the student4 ,again1 a touch of humour 7strange as the contention may seem7 in mathematical works is not only possible with perfect propriety1 but very helpful2 and ,i could give instances of this even from the pure mathematics of ,salmon and the physics of ,clerk ,maxwell4 --,,minchin1 ,g4 ,m4 ,perry's ,teaching of ,mathematics 7,london1 #aijb71 pp4 #ei-fa4 #edj4 ,remember this1 the rule for giving an extempore lecture is--let the mind rest from the subject entirely for an interval preceding the lecture1 after the notes are prepared2 the thoughts will ferment without your knowing it1 and enter into new combinations2 but if you keep the mind active upon the subject up to the moment1 the subject will not ferment but stupefy4 --,,de ,,morgan1 ,a4 ,letter to ,hamilton2 ,graves3 ,life of ,w4 ,r4 ,hamilton 7,new ,york1 #ahhb-ahhi71 ,vol4 #c1 p4 #dhg4 ,,chapter ,,vi ,,study ,,and ,,research ,,in ,,mathematics #fja4 ,the first thing to be attended to in reading any algebraic treatise is the gaining a perfect understanding of the different processes there exhibited1 and of their connection with one another4 ,this cannot be attained by the mere reading of the book1 however great the attention which may be given4 ,it is impossible in a mathematical work to fill up every process in the manner in which it must be filled up in the mind of the student before he can be said to have completely mastered it4 ,many results must be given of which the details are suppressed1 such are the additions1 multiplications1 extractions of square roots1 etc41 with which the investigations abound4 ,these must not be taken on trust by the student1 but must be worked out by his own pen1 which must never be out of his own hand while engaged in any mathematical process4--,,de ,,morgan1 ,a4 ,study and ,difficulties of ,mathematics 7,chicago1 #aijb71 chap4 #ab4 #fjb4 ,the student should not lose any opportunity of exercising himself in numerical calculation and particularly in the use of logarithmic tables4 ,his power of applying mathematics to questions of practical utility is in direct proportion to the facility which he possesses in computation4--,,de ,,morgan1 ,a4 ,study and ,difficulties of ,mathematics 7,chicago1 #aijb71 chap4 #ab4 #fjc4 ,the examples which a beginner should choose for practice should be simple and should not contain very large numbers4 ,the powers of the mind cannot be directed to two things at once2 if the complexity of the numbers used requires all the student's attention1 he cannot observe the principle of the rule which he is following4--,,de ,,morgan1 ,a4 ,study and ,difficulties of ,mathematics 7,chicago1 #aijb71 chap4 #c4 #fjd4 ,euclid and ,archimedes are allowed to be knowing1 and to have demonstrated what they say3 and yet whosoever shall read over their writings without perceiving the connection of their proofs1 and seeing what they show1 though he may understand all their words1 yet he is not the more knowing4 ,he may believe1 indeed1 but does not know what they say1 and so is not advanced one jot in mathematical knowledge by all his reading of those approved mathematicians4--,,locke1 ,,john4 ,conduct of the ,understanding1 sect4 #bd4 #fje4 ,the student should read his author with the most sustained attention1 in order to discover the meaning of every sentence4 ,if the book is well written1 it will endure and repay his close attention3 the text ought to be fairly intelligible1 even without illustrative examples4 ,often1 far too often1 a reader hurries over the text without any sincere and vigorous effort to understand it2 and rushes to some example to clear up what ought not to have been obscure1 if it had been adequately considered4 ,the habit of scrupulously investigating the text seems to me important on several grounds4 ,the close scrutiny of language is a very valuable exercise both for studious and practical life4 ,in the higher departments of mathematics the habit is indispensable3 in the long investigations which occur there it would be impossible to interpose illustrative examples at every stage1 the student must therefore encounter and master1 sentence by sentence1 an extensive and complicated argument4--,,todhunter1 ,,isaac4 ,private ,study of ,mathematics2 ,conflict of ,studies and other ,essays 7,london1 #ahgc71 p4 #fg4 #fjf4 ,it must happen that in some cases the author is not understood1 or is very imperfectly understood2 and the question is what is to be done4 ,after giving a reasonable amount of attention to the passage1 let the student pass on1 reserving the obscurity for future efforts4444 ,the natural tendency of solitary students1 ,i believe1 is not to hurry away prematurely from a hard passage1 but to hang far too long over it2 the just pride that does not like to acknowledge defeat1 and the strong will that cannot endure to be thwarted1 both urge to a continuance of effort even when success seems hopeless4 ,it is only by experience we gain the conviction that when the mind is thoroughly fatigued it has neither the power to continue with advantage its course in an assigned direction1 nor elasticity to strike out a new path2 but that1 on the other hand1 after being withdrawn for a time from the pursuit1 it may return and gain the desired end4--,,todhunter1 ,,isaac4 ,private ,study of ,mathematics2 ,conflict of ,studies and other ,essays 7,london1 #ahgc71 p4 #fh4 #fjg4 ,every mathematical book that is worth reading must be read 8backwards and forwards10 if ,i may use the expression4 ,i would modify ,lagrange's advice a little and say1 8,go on1 but often return to strengthen your faith40 ,when you come on a hard or dreary passage1 pass it over2 and come back to it after you have seen its importance or found the need for it further on4 --,,chrystal1 ,,george4 ,algebra1 ,part #b 7,edinburgh1 #ahhi71 ,preface1 p4 #h4 #fjh4 ,the large collection of problems which our modern ,cambridge books supply will be found to be almost an exclusive peculiarity of these books2 such collections scarcely exist in foreign treatises on mathematics1 nor even in ,english treatises of an earlier date4 ,this fact shows1 ,i think1 that a knowledge of mathematics may be gained without the perpetual working of examples4444 ,do not trouble yourselves with the examples1 make it your main business1 ,i might almost say your exclusive business1 to understand the text of your author4--,,todhunter1 ,,isaac4 ,private ,study of ,mathematics2 ,conflict of ,studies and other ,essays 7,london1 #ahgc71 p4 #gd4 #fji4 ,in my opinion the ,english excel in the art of writing text-books for mathematical teaching2 as regards the clear exposition of theories and the abundance of excellent examples1 carefully selected1 very few books exist in other countries which can compete with those of ,salmon and many other distinguished ,english authors that could be named4--,,cremona1 ,l4 ,projective ,geometry ,leudesdorf 7,oxford1 #ahhe71 ,preface4 #faj4 ,the solution of fallacies1 which give rise to absurdities1 should be to him who is not a first beginner in mathematics an excellent means of testing for a proper intelligible insight into mathematical truth1 of sharpening the wit1 and of confining the judgment and reason within strictly orderly limits4--,,viola1 ,j4 ,mathematische ,sophismen 7,wien1 #ahfd71 ,vorwort4 #faa4 ,success in the solution of a problem generally depends in a great measure on the selection of the most appropriate method of approaching it2 many properties of conic sections 7for instance7 being demonstrable by a few steps of pure geometry which would involve the most laborious operations with trilinear co-ordinates1 while other properties are almost self-evident under the method of trilinear co-ordinates1 which it would perhaps be actually impossible to prove by the old geometry4 --,,whitworth1 ,w4 ,a4 ,modern ,analytic ,geometry 7,cambridge1 #ahff71 p4 #aed4 #fab4 ,the deep study of nature is the most fruitful source of mathematical discoveries4 ,by offering to research a definite end1 this study has the advantage of excluding vague questions and useless calculations2 besides it is a sure means of forming analysis itself and of discovering the elements which it most concerns us to know1 and which natural science ought always to conserve4--- ,,fourier1 ,j4 ,thorie ,analytique de la ,chaleur1 ,discours ,prliminaire4 #fac4 ,it is certainly true that all physical phenomena are subject to strictly mathematical conditions1 and mathematical processes are unassailable in themselves4 ,the trouble arises from the data employed4 ,most phenomena are so highly complex that one can never be quite sure that he is dealing with all the factors until the experiment proves it4 ,so that experiment is rather the criterion of mathematical conclusions and must lead the way4 --,,dolbear1 ,a4 ,e4 ,matter1 ,ether1 ,motion 7,boston1 #ahid71 p4 #hi4 #fad4 ,students should learn to study at an early stage the great works of the great masters instead of making their minds sterile through the everlasting exercises of college1 which are of no use whatever1 except to produce a new ,arcadia where indolence is veiled under the form of useless activity4444 ,hard study on the great models has ever brought out the strong2 and of such must be our new scientific generation if it is to be worthy of the era to which it is born and of the struggles to which it is destined4 --,,beltrami4 ,giornale di matematiche1 ,vol4 #aa1 p4 #aec4 ,young1 ,j4 ,w4 #fae4 ,the history of mathematics may be instructive as well as agreeable2 it may not only remind us of what we have1 but may also teach us to increase our store4 ,says ,de ,morgan1 8,the early history of the mind of men with regards to mathematics leads us to point out our own errors2 and in this respect it is well to pay attention to the history of mathematics40 ,it warns us against hasty conclusions2 it points out the importance of a good notation upon the progress of the science2 it discourages excessive specialization on the part of the investigator1 by showing how apparently distinct branches have been found to possess unexpected connecting links2 it saves the student from wasting time and energy upon problems which were1 perhaps1 solved long since2 it discourages him from attacking an unsolved problem by the same method which has led other mathematicians to failure2 it teaches that fortifications can be taken by other ways than by direct attack1 that when repulsed from a direct assault it is well to reconnoitre and occupy the surrounding ground and to discover the secret paths by which the apparently unconquerable position can be taken4--,,cajori1 ,f4 ,history of ,mathematics 7,new ,york1 #ahig71 pp4 #a-b4 #faf4 ,the history of mathematics is important also as a valuable contribution to the history of civilization4 ,human progress is closely identified with scientific thought4 ,mathematical and physical researches are a reliable record of intellectual progress4--,,cajori1 ,f4 ,history of ,mathematics 7,new ,york1 #ahig71 p4 #d4 #fag4 ,it would be rash to say that nothing remains for discovery or improvement even in elementary mathematics1 but it may be safely asserted that the ground has been so long and so thoroughly explored as to hold out little hope of profitable return for a casual adventurer4--,,todhunter1 ,,isaac4 ,private ,study of ,mathematics2 ,conflict of ,studies and other ,essays 7,london1 #ahgc71 p4 #gc4 #fah4 ,we do not live in a time when knowledge can be extended along a pathway smooth and free from obstacles1 as at the time of the discovery of the infinitesimal calculus1 and in a measure also when in the development of projective geometry obstacles were suddenly removed which1 having hemmed progress for a long time1 permitted a stream of investigators to pour in upon virgin soil4 ,there is no longer any browsing along the beaten paths2 and into the primeval forest only those may venture who are equipped with the sharpest tools4--,,burkhardt1 ,h4 ,mathematisches und wissenschaftliches ,denken2 ,jahresbericht der ,deutschen ,mathematiker ,vereinigung1 ,bd4 #aa1 p4 #ee4 #fai4 ,though we must not without further consideration condemn a body of reasoning merely because it is easy1 nevertheless we must not allow ourselves to be lured on merely by easiness2 and we should take care that every problem which we choose for attack1 whether it be easy or difficult1 shall have a useful purpose1 that it shall contribute in some measure to the up-building of the great edifice4--,,segre1 ,,corradi4 ,some ,recent ,tendencies in ,geometric ,investigation2 ,rivista di ,matematica 7#ahia71 p4 #fc4 ,bulletin ,american ,mathematical ,society1 #aijd1 p4 #dfe4 ,young1 ,j4 ,w44 #fbj4 ,no mathematician now-a-days sets any store on the discovery of isolated theorems1 except as affording hints of an unsuspected new sphere of thought1 like meteorites detached from some undiscovered planetary orb of speculation4--,,sylvester1 ,j4 ,j4 ,notes to the ,exeter ,association ,address2 ,collected ,mathematical ,papers 7,cambridge1 #aijh71 ,vol4 #b1 p4 #gae4 #fba4 ,isolated1 so-called 8pretty theorems0 have even less value in the eyes of a modern mathematician than the discovery of a new 8pretty flower0 has to the scientific botanist1 though the layman finds in these the chief charm of the respective sciences4 --,,hankel1 ,,hermann4 ,die ,entwickelung der ,mathematik in den letzten ,jahrhunderten 7,tbingen1 #ahhd71 p4 #ae4 #fbb4 ,it is1 so to speak1 a scientific tact1 which must guide mathematicians in their investigations1 and guard them from spending their forces on scientifically worthless problems and abstruse realms1 a tact which is closely related to esthetic tact and which is the only thing in our science which cannot be taught or acquired1 and is yet the indispensable endowment of every mathematician4--,,hankel1 ,,hermann4 ,die ,entwickelung der ,mathematik in den letzten ,jahrhunderten 7,tbingen1 #ahhd71 p4 #ba4 #fbc4 ,the mathematician requires tact and good taste at every step of his work1 and he has to learn to trust to his own instinct to distinguish between what is really worthy of his efforts and what is not2 he must take care not to be the slave of his symbols1 but always to have before his mind the realities which they merely serve to express4 ,for these and other reasons it seems to me of the highest importance that a mathematician should be trained in no narrow school2 a wide course of reading in the first few years of his mathematical study cannot fail to influence for good the character of the whole of his subsequent work4--,,glaisher1 ,j4 ,w4 ,l4 ,presidential ,address ,british ,association for the ,advancement of ,science1 ,section ,a1 7#ahij72 ,nature1 ,vol4 #db1 p4 #dfg4 #fbd4 ,as long as a branch of science offers an abundance of problems1 so long it is alive2 a lack of problems foreshadows extinction or the cessation of independent development4 --,,hilbert1 ,d4 ,mathematical ,problems2 ,bulletin ,american ,mathematical ,society1 ,vol4 #h1 p4 #dch4 #fbe4 ,in mathematics as in other fields1 to find one self lost in wonder at some manifestation is frequently the half of a new discovery4--,,dirichlet1 ,p4 ,g4 ,l4 ,werke1 ,bd4 #b 7,berlin1 #ahig71 p4 #bcc4 #fbf4 ,the student of mathematics often finds it hard to throw off the uncomfortable feeling that his science1 in the person of his pencil1 surpasses him in intelligence1--an impression which the great ,euler confessed he often could not get rid of4 ,this feeling finds a sort of justification when we reflect that the majority of the ideas we deal with were conceived by others1 often centuries ago4 ,in a great measure it is really the intelligence of other people that confronts us in science4 --,,mach1 ,,ernst4 ,popular ,scientific ,lectures 7,chicago1 #aiaj71 p4 #aif4 #fbg4 ,it is probably this fact referring to the circumstance that the problems of the parallel axiom1 the squaring of the circle1 the solution of the equation of the fifth degree1 have finally found fully satisfactory and rigorous solutions along with other philosophical reasons that gives rise to the conviction 7which every mathematician shares1 but which no one has yet supported by proof7 that every definite mathematical problem must necessarily be susceptible of an exact settlement1 either in the form of an actual answer to the question asked1 or by the proof of the impossibility of its solution and therewith the necessary failure of all attempts4444 ,this conviction of the solvability of every mathematical problem is a powerful incentive to the worker4 ,we hear within us the perpetual call3 ,there is the problem4 ,seek its solution4 ,you can find it by pure reason1 for in mathematics there is no ignorabimus4--,,hilbert1 ,d4 ,mathematical ,problems2 ,bulletin ,american ,mathematical ,society1 ,vol4 #h1 pp4 #ddd-dde4 #fbh4 ,he who seeks for methods without having a definite problem in mind seeks for the most part in vain4--,,hilbert1 ,d4 ,mathematical ,problems2 ,bulletin ,american ,mathematical ,society1 ,vol4 #h1 p4 #ddd4 #fbi4 ,a mathematical problem should be difficult in order to entice us1 yet not completely inaccessible1 lest it mock at our efforts4 ,it should be to us a guide post on the mazy paths to hidden truths1 and ultimately a reminder of our pleasure in the successful solution4--,,hilbert1 ,d4 ,mathematical ,problems2 ,bulletin ,american ,mathematical ,society1 ,vol4 #h1 p4 #dch4 #fcj4 ,the great mathematicians have acted on the principle 8,divinez avant de demontrer10 and it is certainly true that almost all important discoveries are made in this fashion4 --,,kasner1 ,,edward4 ,the ,present ,problems in ,geometry2 ,bulletin ,american ,mathematical ,society1 ,vol4 #aa1 p4 #bhe4 #fca4 8,divide et impera0 is as true in algebra as in statecraft2 but no less true and even more fertile is the maxim 8auge et impera40 ,the more to do or to prove1 the easier the doing or the proof4--,,sylvester1 ,j4 ,j4 ,proof of the ,fundamental ,theorem of ,invariants2 ,philosophic ,magazine 7#ahgh71 p4 #ahf2 ,collected ,mathematical ,papers1 ,vol4 #c1 p4 #abf4 #fcb4 ,as in the domains of practical life so likewise in science there has come about a division of labor4 ,the individual can no longer control the whole field of mathematics3 it is only possible for him to master separate parts of it in such a manner as to enable him to extend the boundaries of knowledge by creative research4--,,lampe1 ,e4 ,die reine ,mathematik in den ,jahren #ahhd-ahii1 p4 #aj4 #fcc4 ,with the extension of mathematical knowledge will it not finally become impossible for the single investigator to embrace all departments of this knowledge8 ,in answer let me point out how thoroughly it is ingrained in mathematical science that every real advance goes hand in hand with the invention of sharper tools and simpler methods which at the same time assist in understanding earlier theories and to cast aside some more complicated developments4 ,it is therefore possible for the individual investigator1 when he makes these sharper tools and simpler methods his own1 to find his way more easily in the various branches of mathematics than is possible in any other science4--,,hilbert1 ,d4 ,mathematical ,problems2 ,bulletin ,american ,mathematical ,society1 ,vol4 #h1 p4 #dgi4 #fcd4 ,it would seem at first sight as if the rapid expansion of the region of mathematics must be a source of danger to its future progress4 ,not only does the area widen but the subjects of study increase rapidly in number1 and the work of the mathematician tends to become more and more specialized4 ,it is1 of course1 merely a brilliant exaggeration to say that no mathematician is able to understand the work of any other mathematician1 but it is certainly true that it is daily becoming more and more difficult for a mathematician to keep himself acquainted1 even in a general way1 with the progress of any of the branches of mathematics except those which form the field of his own labours4 ,i believe1 however1 that the increasing extent of the territory of mathematics will always be counteracted by increased facilities in the means of communication4 ,additional knowledge opens to us new principles and methods which may conduct us with the greatest ease to results which previously were most difficult of access2 and improvements in notation may exercise the most powerful effects both in the simplification and accessibility of a subject4 ,it rests with the worker in mathematics not only to explore new truths1 but to devise the language by which they may be discovered and expressed2 and the genius of a great mathematician displays itself no less in the notation he invents for deciphering his subject than in the results attained4444 ,i have great faith in the power of well-chosen notation to simplify complicated theories and to bring remote ones near and ,i think it is safe to predict that the increased knowledge of principles and the resulting improvements in the symbolic language of mathematics will always enable us to grapple satisfactorily with the difficulties arising from the mere extent of the subject4--,,glaisher1 ,j4 ,w4 ,l4 ,presidential ,address ,british ,association for the ,advancement of ,science1 ,section ,a41 7#ahij71 ,nature1 ,vol4 #db1 p4 #dff4 #fce4 ,quite distinct from the theoretical question of the manner in which mathematics will rescue itself from the perils to which it is exposed by its own prolific nature is the practical problem of finding means of rendering available for the student the results which have been already accumulated1 and making it possible for the learner to obtain some idea of the present state of the various departments of mathematics4444 ,the great mass of mathematical literature will be always contained in ,journals and ,transactions1 but there is no reason why it should not be rendered far more useful and accessible than at present by means of treatises or higher text-books4 ,the whole science suffers from want of avenues of approach1 and many beautiful branches of mathematics are regarded as difficult and technical merely because they are not easily accessible4444 ,i feel very strongly that any introduction to a new subject written by a competent person confers a real benefit on the whole science4 ,the number of excellent text-books of an elementary kind that are published in this country makes it all the more to be regretted that we have so few that are intended for the advanced student4 ,as an example of the higher kind of text-book1 the want of which is so badly felt in many subjects1 ,i may mention the second part of ,prof4 ,chrystal's 8,algebra0 published last year1 which in a small compass gives a great mass of valuable and fundamental knowledge that has hitherto been beyond the reach of an ordinary student1 though in reality lying so close at hand4 ,i may add that in any treatise or higher text-book it is always desirable that references to the original memoirs should be given1 and1 if possible1 short historic notices also4 ,i am sure that no subject loses more than mathematics by any attempt to dissociate it from its history4--,,glaisher1 ,j4 ,w4 ,l4 ,presidential ,address ,british ,association for the ,advancement of ,science1 ,section ,a 7#ahij72 ,nature1 ,vol4 #db1 p4 #dff4 #fcf4 ,the more a science advances1 the more will it be possible to understand immediately results which formerly could be demonstrated only by means of lengthy intermediate considerations3 a mathematical subject cannot be considered as finally completed until this end has been attained4--,,gordan1 ,,paul4 ,formensystem binrer ,formen 7,leipzig1 #ahge71 p4 #b4 #fcg4 ,an old ,french geometer used to say that a mathematical theory was never to be considered complete till you had made it so clear that you could explain it to the first man you met in the street4--,,smith1 ,h4 ,j4 ,s4 ,nature1 ,vol4 #h 7#ahgc71 p4 #deb4 #fch4 ,in order to comprehend and fully control arithmetical concepts and methods of proof1 a high degree of abstraction is necessary1 and this condition has at times been charged against arithmetic as a fault4 ,i am of the opinion that all other fields of knowledge require at least an equally high degree of abstraction as mathematics1--provided1 that in these fields the foundations are also everywhere examined with the rigour and completeness which is actually necessary4--,,hilbert1 ,d4 ,die ,theorie der algebraischen ,zahlkorper1 ,vorwort2 ,jahresbericht der ,deutschen ,mathematiker ,vereinigung1 ,bd4 #d4 #fci4 ,the anxious precision of modern mathematics is necessary for accuracy1 444 it is necessary for research4 ,it makes for clearness of thought and for fertility in trying new combinations of ideas4 ,when the initial statements are vague and slipshod1 at every subsequent stage of thought1 common sense has to step in to limit applications and to explain meanings4 ,now in creative thought common sense is a bad master4 ,its sole criterion for judgment is that the new ideas shall look like the old ones1 in other words it can only act by suppressing originality4 --,,whitehead1 ,a4 ,n4 ,introduction to ,mathematics 7,new ,york1 #aiaa71 p4 #aeg4 #fdj4 ,mathematicians attach great importance to the elegance of their methods and their results4 ,this is not pure dilettantism4 ,what is it indeed that gives us the feeling of elegance in a solution1 in a demonstration8 ,it is the harmony of the diverse parts1 their symmetry1 their happy balance2 in a word it is all that introduces order1 all that gives unity1 that permits us to see clearly and to comprehend at once both the ensemble and the details4 ,but this is exactly what yields great results1 in fact the more we see this aggregate clearly and at a single glance1 the better we perceive its analogies with other neighboring objects1 consequently the more chances we have of divining the possible generalizations4 ,elegance may produce the feeling of the unforeseen by the unexpected meeting of objects we are not accustomed to bring together2 there again it is fruitful1 since it thus unveils for us kinships before unrecognized4 ,it is fruitful even when it results only from the contrast between the simplicity of the means and the complexity of the problem set2 it makes us then think of the reason for this contrast and very often makes us see that chance is not the reason2 that it is to be found in some unexpected law4 ,in a word1 the feeling of mathematical elegance is only the satisfaction due to any adaptation of the solution to the needs of our mind1 and it is because of this very adaptation that this solution can be for us an instrument4 ,consequently this esthetic satisfaction is bound up with the economy of thought4--,,poincar1 ,h4 ,the ,future of ,mathematics2 ,monist1 ,vol4 #bj1 p4 #hj4 ,halsted4 #fda4 ,the importance of a result is largely relative1 is judged differently by different men1 and changes with the times and circumstances4 ,it has often happened that great importance has been attached to a problem merely on account of the difficulties which it presented2 and indeed if for its solution it has been necessary to invent new methods1 noteworthy artifices1 etc41 the science has gained more perhaps through these than through the final result4 ,in general we may call important all investigations relating to things which in themselves are important2 all those which have a large degree of generality1 or which unite under a single point of view subjects apparently distinct1 simplifying and elucidating them2 all those which lead to results that promise to be the source of numerous consequences2 etc4 --,,segre1 ,,corradi4 ,some ,recent ,tendencies in ,geometric ,investigations4 ,rivista di ,matematica1 ,vol4 #a1 p4 #dd4 ,bulletin ,american ,mathematical ,society1 #aijd1 p4 #ddd4 ,young1 ,j4 ,w44 #fdb4 ,geometric writings are not rare in which one would seek in vain for an idea at all novel1 for a result which sooner or later might be of service1 for anything in fact which might be destined to survive in the science2 and one finds instead treatises on trivial problems or investigations on special forms which have absolutely no use1 no importance1 which have their origin not in the science itself but in the caprice of the author2 or one finds applications of known methods which have already been made thousands of times2 or generalizations from known results which are so easily made that the knowledge of the latter suffices to give at once the former4 ,now such work is not merely useless2 it is actually harmful because it produces a real incumbrance in the science and an embarrassment for the more serious investigators2 and because often it crowds out certain lines of thought which might well have deserved to be studied4 --,,segre1 ,,corradi4 ,on some ,recent ,tendencies in ,geometric ,investigations2 ,rivista di ,matematica1 #ahia1 p4 #dc4 ,bulletin ,american ,mathematical ,society1 #aijd1 p4 #ddc ,young1 ,j4 ,w44 #fdc4 ,a student who wishes now-a-days to study geometry by dividing it sharply from analysis1 without taking account of the progress which the latter has made and is making1 that student no matter how great his genius1 will never be a whole geometer4 ,he will not possess those powerful instruments of research which modern analysis puts into the hands of modern geometry4 ,he will remain ignorant of many geometrical results which are to be found1 perhaps implicitly1 in the writings of the analyst4 ,and not only will he be unable to use them in his own researches1 but he will probably toil to discover them himself1 and1 as happens very often1 he will publish them as new1 when really he has only rediscovered them4--,,segre1 ,,corradi4 ,on some recent ,tendencies in ,geometrical ,investigations2 ,rivista di ,matematica1 #ahia1 p4 #dc4 ,bulletin ,american ,mathematical ,society1 #aijd1 p4 #ddc ,young1 ,j4 ,w44 #fdd4 ,research may start from definite problems whose importance it recognizes and whose solution is sought more or less directly by all forces4 ,but equally legitimate is the other method of research which only selects the field of its activity and1 contrary to the first method1 freely reconnoitres in the search for problems which are capable of solution4 ,different individuals will hold different views as to the relative value of these two methods4 ,if the first method leads to greater penetration it is also easily exposed to the danger of unproductivity4 ,to the second method we owe the acquisition of large and new fields1 in which the details of many things remain to be determined and explored by the first method4--,,clebsch1 ,a4 ,zum ,gedchtniss an ,julius ,plcker2 ,gttinger ,abhandlungen1 #af1 #ahga1 ,mathematische ,classe1 p4 #f4 #fde4 ,during a conversation with the writer in the last weeks of his life1 ,sylvester remarked as curious that notwithstanding he had always considered the bent of his mind to be rather analytical than geometrical1 he found in nearly every case that the solution of an analytical problem turned upon some quite simple geometrical notion1 and that he was never satisfied until he could present the argument in geometrical language4 --,,macmahon1 ,p4 ,a4 ,proceedings ,london ,royal ,society1 ,vol4 #fc1 p4 #ag4 #fdf4 ,the origin of a science is usually to be sought for not in any systematic treatise1 but in the investigation and solution of some particular problem4 ,this is especially the case in the ordinary history of the great improvements in any department of mathematical science4 ,some problem1 mathematical or physical1 is proposed1 which is found to be insoluble by known methods4 ,this condition of insolubility may arise from one of two causes3 ,either there exists no machinery powerful enough to effect the required reduction1 or the workmen are not sufficiently expert to employ their tools in the performance of an entirely new piece of work4 ,the problem proposed is1 however1 finally solved1 and in its solution some new principle1 or new application of old principles1 is necessarily introduced4 ,if a principle is brought to light it is soon found that in its application it is not necessarily limited to the particular question which occasioned its discovery1 and it is then stated in an abstract form and applied to problems of gradually increasing generality4 ,other principles1 similar in their nature1 are added1 and the original principle itself receives such modifications and extensions as are from time to time deemed necessary4 ,the same is true of new applications of old principles2 the application is first thought to be merely confined to a particular problem1 but it is soon recognized that this problem is but one1 and generally a very simple one1 out of a large class1 to which the same process of investigation and solution are applicable4 ,the result in both of these cases is the same4 ,a time comes when these several problems1 solutions1 and principles are grouped together and found to produce an entirely new and consistent method2 a nomenclature and uniform system of notation is adopted1 and the principles of the new method become entitled to rank as a distinct science4--,,craig1 ,,thomas4 ,a ,treatise on ,projection1 ,preface4 ,u4 ,s4 ,coast and ,geodetic ,survey1 ,treasury ,department ,document1 ,no4 #fa4 #fdg4 ,the aim of research is the discovery of the equations which subsist between the elements of phenomena4--,,mach1 ,,ernst4 ,popular ,scientific ,lectures 7,chicago1 #aiaj71 p4 #bje4 #fdh4 ,let him the author be permitted also in all humility to add 444 that in consequence of the large arrears of algebraical and arithmetical speculations waiting in his mind their turn to be called into outward existence1 he is driven to the alternative of leaving the fruits of his meditations to perish 7as has been the fate of too many foregone theories1 the still-born progeny of his brain1 now forever resolved back again into the primordial matter of thought71 or venturing to produce from time to time such imperfect sketches as the present1 calculated to evoke the mental co-operation of his readers1 in whom the algebraical instinct has been to some extent developed1 rather than to satisfy the strict demands of rigorously systematic exposition4 --,,sylvester1 ,j4 ,j4 ,philosophic ,magazine 7#ahfc71 p4 #dfj4 #fdi4 ,in other branches of science1 where quick publication seems to be so much desired1 there may possibly be some excuse for giving to the world slovenly or ill-digested work1 but there is no such excuse in mathematics4 ,the form ought to be as perfect as the substance1 and the demonstrations as rigorous as those of ,euclid4 ,the mathematician has to deal with the most exact facts of ,nature1 and he should spare no effort to render his interpretation worthy of his subject1 and to give to his work its highest degree of perfection4 8,pauca sed matura0 was ,gauss's motto4--,,glaisher1 ,j4 ,w4 ,l4 ,presidential ,address ,british ,association for the ,advancement of ,science1 ,section ,a1 7#ahij72 ,nature1 ,vol4 #db1 p4 #dfg4 #fej4 ,it is the man not the method that solves the problem4 --,,maschke1 ,h4 ,present ,problems of ,algebra and ,analysis2 ,congress of ,arts and ,sciences 7,new ,york and ,boston1 #aije71 ,vol4 #a1 p4 #ecj4 #fea4 ,today it is no longer questioned that the principles of the analysts are the more far-reaching4 ,indeed1 the synthesists lack two things in order to engage in a general theory of algebraic configurations3 these are on the one hand a definition of imaginary elements1 on the other an interpretation of general algebraic concepts4 ,both of these have subsequently been developed in synthetic form1 but to do this the essential principle of synthetic geometry had to be set aside4 ,this principle which manifests itself so brilliantly in the theory of linear forms and the forms of the second degree1 is the possibility of immediate proof by means of visualized constructions4--,,klein1 ,,felix4 ,riemannsche ,flchen 7,leipzig1 #aijf71 ,bd4 #a1 p4 #bcd4 #feb4 ,abstruse mathematical researches 444 are 444 often abused for having no obvious physical application4 ,the fact is that the most useful parts of science have been investigated for the sake of truth1 and not for their usefulness4 ,a new branch of mathematics1 which has sprung up in the last twenty years1 was denounced by the ,astronomer ,royal before the ,university of ,cambridge as doomed to be forgotten1 on account of its uselessness4 ,now it turns out that the reason why we cannot go further in our investigations of molecular action is that we do not know enough of this branch of mathematics4--,,clifford1 ,w4 ,k4 ,conditions of ,mental ,development2 ,lectures and ,essays 7,london1 #aija71 ,vol4 #a1 p4 #aae4 #fec4 ,in geometry1 as in most sciences1 it is very rare that an isolated proposition is of immediate utility4 ,but the theories most powerful in practice are formed of propositions which curiosity alone brought to light1 and which long remained useless without its being able to divine in what way they should one day cease to be so4 ,in this sense it may be said1 that in real science1 no theory1 no research1 is in effect useless4--,,voltaire4 ,a ,philosophical ,dictionary1 ,article 8,geometry02 7,boston1 #ahha71 ,vol4 #a1 p4 #cgd4 #fed4 ,scientific subjects do not progress necessarily on the lines of direct usefulness4 ,very many applications of the theories of pure mathematics have come many years1 sometimes centuries1 after the actual discoveries themselves4 ,the weapons were at hand1 but the men were not able to use them4 --,,forsyth1 ,a4 ,r4 ,perry's ,teaching of ,mathematics 7,london1 #aijb71 p4 #ce4 #fee4 ,it is no paradox to say that in our most theoretical moods we may be nearest to our most practical applications4 --,,whitehead1 ,a4 ,n4 ,introduction to ,mathematics 7,new ,york71 p4 #ajj4 #fef4 ,although with the majority of those who study and practice in these capacities engineers1 builders1 surveyors1 geographers1 navigators1 hydrographers1 astronomers1 second-hand acquirements1 trite formulas1 and appropriate tables are sufficient for ordinary purposes1 yet these trite formulas and familiar rules were originally or gradually deduced from the profound investigations of the most gifted minds1 from the dawn of science to the present day4444 ,the further developments of the science1 with its possible applications to larger purposes of human utility and grander theoretical generalizations1 is an achievement reserved for a few of the choicest spirits1 touched from time to time by ,heaven to these highest issues4 ,the intellectual world is filled with latent and undiscovered truth as the material world is filled with latent electricity4--,,everett1 ,,edward4 ,orations and ,speeches1 ,vol4 #c 7,boston1 #ahgj71 p4 #eac4 #feg4 ,if we view mathematical speculations with reference to their use1 it appears that they should be divided into two classes4 ,to the first belong those which furnish some marked advantage either to common life or to some art1 and the value of such is usually determined by the magnitude of this advantage4 ,the other class embraces those speculations which1 though offering no direct advantage1 are nevertheless valuable in that they extend the boundaries of analysis and increase our resources and skill4 ,now since many investigations1 from which great advantage may be expected1 must be abandoned solely because of the imperfection of analysis1 no small value should be assigned to those speculations which promise to enlarge the field of analysis4--,,euler4 ,novi ,comm4 ,petr41 ,vol4 #d1 ,preface4 #feh4 ,the discovery of the conic sections1 attributed to ,plato1 first threw open the higher species of form to the contemplation of geometers4 ,but for this discovery1 which was probably regarded in ,plato's time and long after him1 as the unprofitable amusement of a speculative brain1 the whole course of practical philosophy of the present day1 of the science of astronomy1 of the theory of projectiles1 of the art of navigation1 might have run in a different channel2 and the greatest discovery that has ever been made in the history of the world1 the law of universal gravitation1 with its innumerable direct and indirect consequences and applications to every department of human research and industry1 might never to this hour have been elicited4 --,,sylvester1 ,j4 ,j4 ,a ,probationary ,lecture on ,geometry2 ,collected ,mathematical ,papers1 ,vol4 #b 7,cambridge1 #aijh71 p4 #g4 #fei4 ,no more impressive warning can be given to those who would confine knowledge and research to what is apparently useful1 than the reflection that conic sections were studied for eighteen hundred years merely as an abstract science1 without regard to any utility other than to satisfy the craving for knowledge on the part of mathematicians1 and that then at the end of this long period of abstract study1 they were found to be the necessary key with which to attain the knowledge of the most important laws of nature4--,,whitehead1 ,a4 ,n4 ,introduction to ,mathematics 7,new ,york1 ,york1 #aiaa71 pp4 #acf-acg4 #ffj4 ,the ,greeks in the first vigour of their pursuit of mathematical truth1 at the time of ,plato and soon after1 had by no means confined themselves to those propositions which had a visible bearing on the phenomena of nature2 but had followed out many beautiful trains of research concerning various kinds of figures1 for the sake of their beauty alone2 as for instance in their doctrine of ,conic ,sections1 of which curves they had discovered all the principal properties4 ,but it is curious to remark1 that these investigations1 thus pursued at first as mere matters of curiosity and intellectual gratification1 were destined1 two thousand years later1 to play a very important part in establishing that system of celestial motions which succeeded the ,platonic scheme of cycles and epicycles4 ,if the properties of conic sections had not been demonstrated by the ,greeks and thus rendered familiar to the mathematicians of succeeding ages1 ,kepler would probably not have been able to discover those laws respecting the orbits and motions of planets which were the occasion of the greatest revolution that ever happened in the history of science4--,,whewell1 ,w4 ,history of ,scientific ,ideas1 ,bk4 #b1 chap4 #ad1 sect4 #c4 #ffa4 ,the greatest mathematicians1 as ,archimedes1 ,newton1 and ,gauss1 always united theory and applications in equal measure4 --,,klein1 ,,felix4 ,elementarmathematik vom hheren ,standpunkte aus 7,leipzig1 #aiji71 ,bd4 #b1 p4 #cib4 #ffb4 ,we may see how unexpectedly recondite parts of pure mathematics may bear upon physical science1 by calling to mind the circumstance that ,fresnel obtained one of the most curious confirmations of the theory 7the laws of ,circular ,polarization by reflection7 through an interpretation of an algebraical expression1 which1 according to the original conventional meaning of the symbols1 involved an impossible quantity4--,,whewell1 ,w4 ,history of ,scientific ,ideas1 ,bk4 #b1 chap4 #ad1 sect4 #h4 #ffc4 ,a great department of thought must have its own inner life1 however transcendent may be the importance of its relations to the outside4 ,no department of science1 least of all one requiring so high a degree of mental concentration as ,mathematics1 can be developed entirely1 or even mainly1 with a view to applications outside its own range4 ,the increased complexity and specialisation of all branches of knowledge makes it true in the present1 however it may have been in former times1 that important advances in such a department as ,mathematics can be expected only from men who are interested in the subject for its own sake1 and who1 whilst keeping an open mind for suggestions from outside1 allow their thought to range freely in those lines of advance which are indicated by the present state of their subject1 untrammelled by any preoccupation as to applications to other departments of science4 ,even with a view to applications1 if ,mathematics is to be adequately equipped for the purpose of coping with the intricate problems which will be presented to it in the future by ,physics1 ,chemistry and other branches of physical science1 many of these problems probably of a character which we cannot at present forecast1 it is essential that ,mathematics should be allowed to develop freely on its own lines4--,,hobson1 ,e4 ,w4 ,presidential ,address ,british ,association for the ,advancement of ,science1 ,section ,a1 7#aiaj72 ,nature1 ,vol4 #hd1 p4 #bhf4 #ffd4 ,to emphasize this opinion that mathematicians would be unwise to accept practical issues as the sole guide or the chief guide in the current of their investigations1 444 let me take one more instance1 by choosing a subject in which the purely mathematical interest is deemed supreme1 the theory of functions of a complex variable4 ,that at least is a theory in pure mathematics1 initiated in that region1 and developed in that region2 it is built up in scores of papers1 and its plan certainly has not been1 and is not now1 dominated or guided by considerations of applicability to natural phenomena4 ,yet what has turned out to be its relation to practical issues8 ,the investigations of ,lagrange and others upon the construction of maps appear as a portion of the general property of conformal representation2 which is merely the general geometrical method of regarding functional relations in that theory4 ,again1 the interesting and important investigations upon discontinuous two-dimensional fluid motion in hydrodynamics1 made in the last twenty years1 can all be1 and now are all1 ,i believe1 deduced from similar considerations by interpreting functional relations between complex variables4 ,in the dynamics of a rotating heavy body1 the only substantial extension of our knowledge since the time of ,lagrange has accrued from associating the general properties of functions with the discussion of the equations of motion4 ,further1 under the title of conjugate functions1 the theory has been applied to various questions in electrostatics1 particularly in connection with condensors and electrometers4 ,and1 lastly1 in the domain of physical astronomy1 some of the most conspicuous advances made in the last few years have been achieved by introducing into the discussion the ideas1 the principles1 the methods1 and the results of the theory of functions 444 the refined and extremely difficult work of ,poincar and others in physical astronomy has been possible only by the use of the most elaborate developments of some purely mathematical subjects1 developments which were made without a thought of such applications4--,,forsyth1 ,a4 ,r4 ,presidential ,address ,british ,association for the ,advancement of ,science1 ,section ,a1 7#ahig72 ,nature1 ,vol4 #ef1 p4 #cgg4 ,,chapter ,,vii ,,modern ,,mathematics #gja4 ,surely this is the golden age of mathematics4 --,,pierpont1 ,,james4 ,history of ,mathematics in the ,nineteenth ,century2 ,congress of ,arts and ,sciences 7,boston and ,new ,york1 #aije71 ,vol4 #a1 p4 #dic4 #gjb4 ,the golden age of mathematics--that was not the age of ,euclid1 it is ours4 ,ours is the age when no less than six international congresses have been held in the course of nine years4 ,it is in our day that more than a dozen mathematical societies contain a growing membership of more than two thousand men representing the centers of scientific light throughout the great culture nations of the world4 ,it is in our time that over five hundred scientific journals are each devoted in part1 while more than two score others are devoted exclusively1 to the publication of mathematics4 ,it is in our time that the ,jahrbuch ber die ,fortschritte der ,mathematik1 though admitting only condensed abstracts with titles1 and not reporting on all the journals1 has1 nevertheless1 grown to nearly forty huge volumes in as many years4 ,it is in our time that as many as two thousand books and memoirs drop from the mathematical press of the world in a single year1 the estimated number mounting up to fifty thousand in the last generation4 ,finally1 to adduce yet another evidence of a similar kind1 it requires not less than seven ponderous tomes of the forthcoming ,encyclopaedie der ,mathematischen ,wissenschaften to contain1 not expositions1 not demonstrations1 but merely compact reports and bibliographic notices sketching developments that have taken place since the beginning of the nineteenth century4--,,keyser1 ,c4 ,j4 ,lectures on ,science1 ,philosophy and ,art 7,new ,york1 #aijh71 p4 #h4 #gjc4 ,i have said that mathematics is the oldest of the sciences2 a glance at its more recent history will show that it has the energy of perpetual youth4 ,the output of contributions to the advance of the science during the last century and more has been so enormous that it is difficult to say whether pride in the greatness of achievement in this subject1 or despair at his inability to cope with the multiplicity of its detailed developments1 should be the dominant feeling of the mathematician4 ,few people outside of the small circle of mathematical specialists have any idea of the vast growth of mathematical literature4 ,the ,royal ,society ,catalogue contains a list of nearly thirty-nine thousand papers on subjects of ,pure ,mathematics alone1 which have appeared in seven hundred serials during the nineteenth century4 ,this represents only a portion of the total output1 the very large number of treatises1 dissertations1 and monographs published during the century being omitted4--,,hobson1 ,e4 ,w4 ,presidential ,address ,british ,association for the ,advancement of ,science1 ,section ,a1 7#aiaj72 ,nature1 ,vol4 #hd1 p4 #bhe4 #gjd4 ,mathematics is one of the oldest of the sciences2 it is also one of the most active1 for its strength is the vigour of perpetual youth4--,,forsyth1 ,a4 ,r4 ,presidential ,address ,british ,association for the ,advancement of ,science1 ,section ,a1 7#ahig72 ,nature1 ,vol4 #ef1 p4 #cgh4 #gje4 ,the nineteenth century which prides itself upon the invention of steam and evolution1 might have derived a more legitimate title to fame from the discovery of pure mathematics4 --,,russell1 ,,bertrand4 ,international ,monthly1 ,vol4 #d 7#aija71 p4 #hc4 #gjf4 ,one of the chiefest triumphs of modern mathematics consists in having discovered what mathematics really is4 --,,russell1 ,,bertrand4 ,international ,monthly1 ,vol4 #d 7#aija71 p4 #hd4 #gjg4 ,modern mathematics1 that most astounding of intellectual creations1 has projected the mind's eye through infinite time and the mind's hand into boundless space4--,,butler1 ,n4 ,m4 ,the ,meaning of ,education and other ,essays and ,addresses 7,new ,york1 #aije71 p4 #dd4 #gjh4 ,the extraordinary development of mathematics in the last century is quite unparalleled in the long history of this most ancient of sciences4 ,not only have those branches of mathematics which were taken over from the eighteenth century steadily grown1 but entirely new ones have sprung up in almost bewildering profusion1 and many of them have promptly assumed proportions of vast extent4--,,pierpont1 ,j4 ,the ,history of ,mathematics in the ,nineteenth ,century2 ,congress of ,arts and ,sciences 7,boston and ,new ,york1 #aije71 ,vol4 #a1 p4 #dgd4 #gji4 ,the ,modern ,theory of ,functions--that stateliest of all the pure creations of the human intellect4--,,keyser1 ,c4 ,j4 ,lectures on ,science1 ,philosophy and ,art 7,new ,york1 #aijh71 p4 #af4 #gaj4 ,if a mathematician of the past1 an ,archimedes or even a ,descartes1 could view the field of geometry in its present condition1 the first feature to impress him would be its lack of concreteness4 ,there are whole classes of geometric theories which proceed not only without models and diagrams1 but without the slightest 7apparent7 use of spatial intuition4 ,in the main this is due1 to the power of the analytic instruments of investigations as compared with the purely geometric4 --,,kasner1 ,,edward4 ,the ,present ,problems in ,geometry2 ,bulletin ,american ,mathematical ,society1 #aije1 p4 #bhe4 #gaa4 ,in ,euclid each proposition stands by itself2 its connection with others is never indicated2 the leading ideas contained in its proof are not stated2 general principles do not exist4 ,in modern methods1 on the other hand1 the greatest importance is attached to the leading thoughts which pervade the whole2 and general principles1 which bring whole groups of theorems under one aspect1 are given rather than separate propositions4 ,the whole tendency is toward generalization4 ,a straight line is considered as given in its entirety1 extending both ways to infinity1 while ,euclid is very careful never to admit anything but finite quantities4 ,the treatment of the infinite is in fact another fundamental difference between the two methods4 ,euclid avoids it1 in modern mathematics it is systematically introduced1 for only thus is generality obtained4 --,,cayley1 ,,arthur4 ,encyclopedia ,britannica 7#ith edition71 ,article 8,geometry40 #gab4 ,this is one of the greatest advantages of modern geometry over the ancient1 to be able1 through the consideration of positive and negative quantities1 to include in a single enunciation the several cases which the same theorem may present by a change in the relative position of the different parts of a figure4 ,thus in our day the nine principal problems and the numerous particular cases1 which form the object of eighty-three theorems in the two books ,de sectione determinata of ,appolonius constitute only one problem which is resolved by a single equation4--,,chasles1 ,m4 ,histoire de la ,gomtrie1 chap4 #a1 sect4 #ce4 #gac4 ,euclid always contemplates a straight line as drawn between two definite points1 and is very careful to mention when it is to be produced beyond this segment4 ,he never thinks of the line as an entity given once for all as a whole4 ,this careful definition and limitation1 so as to exclude an infinity not immediately apparent to the senses1 was very characteristic of the ,greeks in all their many activities4 ,it is enshrined in the difference between ,greek architecture and ,gothic architecture1 and between ,greek religion and modern religion4 ,the spire of a ,gothic cathedral and the importance of the unbounded straight line in modern ,geometry are both emblematic of the transformation of the modern world4--,,whitehead1 ,a4 ,n4 ,introduction to ,mathematics 7,new ,york1 #aiaa71 p4 #aai4 #gad4 ,the geometrical problems and theorems of the ,greeks always refer to definite1 oftentimes to rather complicated figures4 ,now frequently the points and lines of such a figure may assume very many different relative positions2 each of these possible cases is then considered separately4 ,on the contrary1 present day mathematicians generate their figures one from another1 and are accustomed to consider them subject to variation2 in this manner they unite the various cases and combine them as much as possible by employing negative and imaginary magnitudes4 ,for example1 the problems which ,appolonius treats in his two books ,de sectione rationis1 are solved today by means of a single1 universally applicable construction2 ,apollonius1 on the contrary1 separates it into more than eighty different cases varying only in position4 ,thus1 as ,hermann ,hankel has fittingly remarked1 the ancient geometry sacrifices to a seeming simplicity the true simplicity which consists in the unity of principles2 it attained a trivial sensual presentability at the cost of the recognition of the relations of geometric forms in all their changes and in all the variations of their sensually presentable positions4 --,,reye1 ,,theodore4 ,die synthetische ,geometrie im ,altertum und in der ,neuzeit2 ,jahresbericht der ,deutschen ,mathematiker ,vereinigung1 ,bd4 #b1 pp4 #cdf-cdg4 #gae4 ,it is known that the mathematics prescribed for the high school ,gymnasien is essentially ,euclidean1 while it is modern mathematics1 the theory of functions and the infinitesimal calculus1 which has secured for us an insight into the mechanism and laws of nature4 ,euclidean mathematics is indeed1 a prerequisite for the theory of functions1 but just as one1 though he has learned the inflections of ,latin nouns and verbs1 will not thereby be enabled to read a ,latin author much less to appreciate the beauties of a ,horace1 so ,euclidean mathematics1 that is the mathematics of the high school1 is unable to unlock nature and her laws4 ,euclidean mathematics assumes the completeness and invariability of mathematical forms2 these forms it describes with appropriate accuracy and enumerates their inherent and related properties with perfect clearness1 order1 and completeness1 that is1 ,euclidean mathematics operates on forms after the manner that anatomy operates on the dead body and its members4 ,on the other hand1 the mathematics of variable magnitudes--function theory or analysis--considers mathematical forms in their genesis4 ,by writing the equation of the parabola1 we express its law of generation1 the law according to which the variable point moves4 ,the path1 produced before the eyes of the student by a point moving in accordance to this law1 is the parabola4 ,if1 then1 ,euclidean mathematics treats space and number forms after the manner in which anatomy treats the dead body1 modern mathematics deals1 as it were1 with the living body1 with growing and changing forms1 and thus furnishes an insight1 not only into nature as she is and appears1 but also into nature as she generates and creates1--reveals her transition steps and in so doing creates a mind for and understanding of the laws of becoming4 ,thus modern mathematics bears the same relation to ,euclidean mathematics that physiology or biology 444 bears to anatomy4 ,but it is exactly in this respect that our view of nature is so far above that of the ancients2 that we no longer look on nature as a quiescent complete whole1 which compels admiration by its sublimity and wealth of forms1 but that we conceive of her as a vigorous growing organism1 unfolding according to definite1 as delicate as far-reaching1 laws2 that we are able to lay hold of the permanent amidst the transitory1 of law amidst fleeting phenomena1 and to be able to give these their simplest and truest expression through the mathematical formulas4 --,,dillmann1 ,e4 ,die ,mathematik die ,fackeltrgerin einer neuen ,zeit 7,stuttgart1 #ahhi71 p4 #cg4 #gaf4 ,the ,excellence of ,modern ,geometry is in nothing more evident1 than in those full and adequate ,solutions it gives to ,problems2 representing all possible ,cases in one view1 and in one general ,theorem many times comprehending whole ,sciences2 which deduced at length into ,propositions1 and demonstrated after the manner of the ,ancients1 might well become the subjects of large ,treatises3 ,for whatsoever ,theorem solves the most complicated ,problem of the kind1 does with a due ,reduction reach all the subordinate ,cases4--,,halley1 ,e4 ,an ,instance of the ,excellence of ,modern ,algebra1 etc42 ,philosophical ,transactions1 #afid1 p4 #ifj4 #gag4 ,one of the most conspicuous and distinctive features of mathematical thought in the nineteenth century is its critical spirit4 ,beginning with the calculus1 it soon permeates all analysis1 and toward the close of the century it overhauls and recasts the foundations of geometry and aspires to further conquests in mechanics and in the immense domains of mathematical physics4444 ,a searching examination of the foundations of arithmetic and the calculus has brought to light the insufficiency of much of the reasoning formerly considered as conclusive4--,,pierpont1 ,j4 ,history of ,mathematics in the ,nineteenth ,century2 ,congress of ,arts and ,sciences 7,boston and ,new ,york1 #aije71 ,vol4 #a1 p4 #dhb4 #gah4 ,if we compare a mathematical problem with an immense rock1 whose interior we wish to penetrate1 then the work of the ,greek mathematicians appears to us like that of a robust stonecutter1 who1 with indefatigable perseverance1 attempts to demolish the rock gradually from the outside by means of hammer and chisel2 but the modern mathematician resembles an expert miner1 who first constructs a few passages through the rock and then explodes it with a single blast1 bringing to light its inner treasures4 --,,hankel1 ,,hermann4 ,die ,entwickelung der ,mathematik in den letzten ,jahrhunderten 7,tbingen1 #ahhd71 p4 #i4 #gai4 ,all the modern higher mathematics is based on a calculus of operations1 on laws of thought4 ,all mathematics1 from the first1 was so in reality2 but the evolvers of the modern higher calculus have known that it is so4 ,therefore elementary teachers who1 at the present day1 persist in thinking about algebra and arithmetic as dealing with laws of number1 and about geometry as dealing with laws of surface and solid content1 are doing the best that in them lies to put their pupils on the wrong track for reaching in the future any true understanding of the higher algebras4 ,algebras deal not with laws of number1 but with such laws of the human thinking machinery as have been discovered in the course of investigations on numbers4 ,plane geometry deals with such laws of thought as were discovered by men intent on finding out how to measure surface2 and solid geometry with such additional laws of thought as were discovered when men began to extend geometry into three dimensions4--,,boole ,m4 ,e4 ,logic of ,arithmetic 7,oxford1 #aijc71 ,preface1 pp4 #ah-ai4 #gbj4 ,it is not only a decided preference for synthesis and a complete denial of general methods which characterizes the ancient mathematics as against our newer science modern mathematics3 besides this external formal difference there is another real1 more deeply seated1 contrast1 which arises from the different attitudes which the two assumed relative to the use of the concept of variability4 ,for while the ancients1 on account of considerations which had been transmitted to them from the philosophic school of the ,eleatics1 never employed the concept of motion1 the spatial expression for variability1 in their rigorous system1 and made incidental use of it only in the treatment of phonoromically generated curves1 modern geometry dates from the instant that ,descartes left the purely algebraic treatment of equations and proceeded to investigate the variations which an algebraic expression undergoes when one of its variables assumes a continuous succession of values4--,,hankel1 ,,hermann4 ,untersuchungen ber die unendlich oft oszillierenden und unstetigen ,functionen2 ,ostwald's ,klassiker der exacten ,wissenschaften1 ,no4 #aec1 pp4 #dd-de4 #gba4 ,without doubt one of the most characteristic features of mathematics in the last century is the systematic and universal use of the complex variable4 ,most of its great theories received invaluable aid from it1 and many owe their very existence to it4 --,,pierpont1 ,j4 ,history of ,mathematics in the ,nineteenth ,century2 ,congress of ,arts and ,sciences 7,boston and ,new ,york1 #aije71 ,vol4 #a1 p4 #dgd4 #gbb4 ,the notion1 which is really the fundamental one 7and ,i cannot too strongly emphasise the assertion71 underlying and pervading the whole of modern analysis and geometry1 is that of imaginary magnitude in analysis and of imaginary space in geometry4--,,cayley1 ,,arthur4 ,presidential ,address2 ,collected ,works1 ,vol4 #aa1 p4 #dcd4 #gbc4 ,the solution of the difficulties which formerly surrounded the mathematical infinite is probably the greatest achievement of which our age has to boast4--,,russell1 ,,bertrand4 ,the ,study of ,mathematics2 ,philosophical ,essays 7,london1 #aiaj71 p4 #gg4 #gbd4 ,induction and analogy are the special characteristics of modern mathematics1 in which theorems have given place to theories and no truth is regarded otherwise than as a link in an infinite chain4 8,omne exit in infinitum0 is their favorite motto and accepted axiom4--,,sylvester1 ,j4 ,j4 ,a ,plea for the ,mathematician2 ,nature1 ,vol4 #a1 p4 #bfa4 #gbe4 ,the conception of correspondence plays a great part in modern mathematics4 ,it is the fundamental notion in the science of order as distinguished from the science of magnitude4 ,if the older mathematics were mostly dominated by the needs of mensuration1 modern mathematics are dominated by the conception of order and arrangement4 ,it may be that this tendency of thought or direction of reasoning goes hand in hand with the modern discovery in physics1 that the changes in nature depend not only or not so much on the quantity of mass and energy as on their distribution or arrangement4--,,merz1 ,j4 ,t4 ,history of ,european ,thought in the ,nineteenth ,century 7,edinburgh and ,london1 #aijc71 p4 #gcf4 #gbf4 ,now this establishment of correspondence between two aggregates and investigation of the propositions that are carried over by the correspondence may be called the central idea of modern mathematics4--,,clifford1 ,w4 ,k4 ,philosophy of the ,pure ,sciences2 ,lectures and ,essays 7,london1 #aija71 ,vol4 #a1 p4 #djb4 #gbg4 ,in our century the conceptions substitution and substitution group1 transformation and transformation group1 operation and operation group1 invariant1 differential invariant and differential parameter1 appear more and more clearly as the most important conceptions of mathematics4--,,lie1 ,,sophus4 ,leipziger ,berichte1 ,no4 #dg 7#ahie71 p4 #bfa4 #gbh4 ,generality of points of view and of methods1 precision and elegance in presentation1 have become1 since ,lagrange1 the common property of all who would lay claim to the rank of scientific mathematicians4 ,and1 even if this generality leads at times to abstruseness at the expense of intuition and applicability1 so that general theorems are formulated which fail to apply to a single special case1 if furthermore precision at times degenerates into a studied brevity which makes it more difficult to read an article than it was to write it2 if1 finally1 elegance of form has well-nigh become in our day the criterion of the worth or worthlessness of a proposition1--yet are these conditions of the highest importance to a wholesome development1 in that they keep the scientific material within the limits which are necessary both intrinsically and extrinsically if mathematics is not to spend itself in trivialities or smother in profusion4 --,,hankel1 ,,hermann4 ,die ,entwickelung der ,mathematik in den letzten ,jahrhunderten 7,tbingen1 #ahhd71 pp4 #ad-ae4 #gbi4 ,the development of abstract methods during the past few years has given mathematics a new and vital principle which furnishes the most powerful instrument for exhibiting the essential unity of all its branches4--,,young1 ,j4 ,w4 ,fundamental ,concepts of ,algebra and ,geometry 7,new ,york1 #aiaa71 p4 #bbe4 #gcj4 ,everybody praises the incomparable power of the mathematical method1 but so is everybody aware of its incomparable unpopularity4--,,rosanes1 ,j4 ,jahresbericht der ,deutschen ,mathematiker ,vereinigung1 ,bd4 #ac1 p4 #ag4 #gca4 ,indeed the modern developments of mathematics constitute not only one of the most impressive1 but one of the most characteristic1 phenomena of our age4 ,it is a phenomenon1 however1 of which the boasted intelligence of a 8universalized0 daily press seems strangely unaware2 and there is no other great human interest1 whether of science or of art1 regarding which the mind of the educated public is permitted to hold so many fallacious opinions and inferior estimates4--,,keyser1 ,c4 ,j4 ,lectures on ,science1 ,philosophy and ,arts 7,new ,york1 #aijh71 p4 #h4 #gcb4 ,it may be asserted without exaggeration that the domain of mathematical knowledge is the only one of which our otherwise omniscient journalism has not yet possessed itself4 --,,pringsheim1 ,,alfred4 ,ueber ,wert und angeblichen ,unwert der ,mathematik2 ,jahresbericht der ,deutschen ,mathematiker ,vereinigung1 7#aijd7 p4 #ceg4 #gcc4 ,the inaccessibility of special fields of mathematics1 except by the regular way of logically antecedent acquirements1 renders the study discouraging or hateful to weak or indolent minds4--,,lefevre1 ,,arthur4 ,number and its ,algebra 7,boston1 #aijc71 sect4 #bbc4 #gcd4 ,the majority of mathematical truths now possessed by us presuppose the intellectual toil of many centuries4 ,a mathematician1 therefore1 who wishes today to acquire a thorough understanding of modern research in this department1 must think over again in quickened tempo the mathematical labors of several centuries4 ,this constant dependence of new truths on old ones stamps mathematics as a science of uncommon exclusiveness and renders it generally impossible to lay open to uninitiated readers a speedy path to the apprehension of the higher mathematical truths4 ,for this reason1 too1 the theories and results of mathematics are rarely adapted for popular presentation4444 ,this same inaccessibility of mathematics1 although it secures for it a lofty and aristocratic place among the sciences1 also renders it odious to those who have never learned it1 and who dread the great labor involved in acquiring an understanding of the questions of modern mathematics4 ,neither in the languages nor in the natural sciences are the investigations and results so closely interdependent as to make it impossible to acquaint the uninitiated student with single branches or with particular results of these sciences1 without causing him to go through a long course of preliminary study4 --,,schubert1 ,h4 ,mathematical ,essays and ,recreations 7,chicago1 #ahih71 p4 #cb4 #gce4 ,such is the character of mathematics in its profounder depths and in its higher and remoter zones that it is well nigh impossible to convey to one who has not devoted years to its exploration a just impression of the scope and magnitude of the existing body of the science4 ,an imagination formed by other disciplines and accustomed to the interests of another field may scarcely receive suddenly an apocalyptic vision of that infinite interior world4 ,but how amazing and how edifying were such a revelation1 if it only could be made4--,,keyser1 ,c4 ,j4 ,lectures on ,science1 ,philosophy and ,art 7,new ,york1 #aijh71 p4 #f4 #gcf4 ,it is not so long since1 during one of the meetings of the ,association1 one of the leading ,english newspapers briefly described a sitting of this ,section in the words1 8,saturday morning was devoted to pure mathematics1 and so there was nothing of any general interest30 still1 such toleration is better than undisguised and ill-informed hostility4--,,forsyth1 ,a4 ,r4 ,report of the #fgth meeting of the ,british ,association for the ,advancement of ,science4 #gcg4 ,the science of mathematics has grown to such vast proportion that probably no living mathematician can claim to have achieved its mastery as a whole4--,,whitehead1 ,a4 ,n4 ,an ,introduction to ,mathematics 7,new ,york1 #aiaa71 p4 #beb4 #gch4 ,there is perhaps no science of which the development has been carried so far1 which requires greater concentration and will power1 and which by the abstract height of the qualities required tends more to separate one from daily life4 ,provisional ,report of the ,american ,subcommittee of the ,international ,commission on the ,teaching of ,mathematics2 ,bulletin ,american ,society 7#aiaj71 p4 #ig4 #gci4 ,angling may be said to be so like the mathematics1 that it can never be fully learnt4--,,walton1 ,,isaac4 ,the ,complete ,angler1 ,preface4 #gdj4 ,the flights of the imagination which occur to the pure mathematician are in general so much better described in his formul than in words1 that it is not remarkable to find the subject treated by outsiders as something essentially cold and uninteresting--444 the only successful attempt to invest mathematical reasoning with a halo of glory--that made in this section by ,prof4 ,sylvester--is known to a comparative few1 4444 --,,tait1 ,p4 ,g4 ,presidential ,address ,british ,association for the ,advancement of ,science 7#ahga72 ,nature ,vol4 #d1 p4 #bga4 ,,chapter ,,viii ,,the ,,mathematician #hja4 ,the real mathematician is an enthusiast per se4 ,without enthusiasm no mathematics4--,,novalis4 ,schriften 7,berlin1 #aija71 ,zweiter ,teil1 p4 #bbc4 #hjb4 ,it is true that a mathematician1 who is not somewhat of a poet1 will never be a perfect mathematician4--,,weierstrass4 ,quoted by ,mittag-,leffler2 ,compte rendu du deuxime congrs international des mathmaticiens 7,paris1 #aijb71 p4 #adi4 #hjc4 ,the mathematician is perfect only in so far as he is a perfect being1 in so far as he perceives the beauty of truth2 only then will his work be thorough1 transparent1 comprehensive1 pure1 clear1 attractive and even elegant4 ,all this is necessary to resemble ,lagrange4--,,goethe4 ,wilhelm ,meister's ,wanderjahre1 ,zweites ,buch2 ,sprche in ,prosa2 ,natur1 ,,vi1 #iej4 #hjd4 ,a thorough advocate in a just cause1 a penetrating mathematician facing the starry heavens1 both alike bear the semblance of divinity4--,,goethe4 ,wilhelm ,meister's ,wanderjahre1 ,zweites ,buch2 ,sprche in ,prosa2 ,natur1 ,,vi1 #idg4 #hje4 ,mathematicians practice absolute freedom4--,,adams1 ,,henry4 ,a ,letter to ,american ,teachers of ,history 7,washington1 #aiaj71 p4 #afi4 #hjf4 ,the mathematical method is the essence of mathematics4 ,he who fully comprehends the method is a mathematician4--,,novalis4 ,schriften 7,berlin1 #aija71 ,zweiter ,teil1 p4 #aij4 #hjg4 ,he who is unfamiliar with mathematics literally1 he who is a layman in mathematics remains more or less a stranger to our time4--,,dillmann1 ,e4 ,die ,mathematik die ,fackeltrgerin einer neuen ,zeit 7,stuttgart1 #ahhi71 p4 #ci4 #hjh4 ,enlist a great mathematician and a distinguished ,grecian2 your problem will be solved4 ,such men can teach in a dwelling-house as well as in a palace4 ,part of the apparatus they will bring2 part we will furnish4 ,advice given to the ,trustees of ,johns ,hopkins ,university on the choice of a professorial staff4--,,gilman1 ,d4 ,c4 ,report of the ,president of ,johns ,hopkins ,university 7#ahhh71 p4 #bi4 #hji4 ,persons1 who have a decided mathematical talent1 constitute1 as it were1 a favored class4 ,they bear the same relation to the rest of mankind that those who are academically trained bear to those who are not4--,,moebius1 ,p4 ,j4 ,ueber die ,anlage zur ,mathematik 7,leipzig1 #aijj71 p4 #d4 #haj4 ,one may be a mathematician of the first rank without being able to compute4 ,it is possible to be a great computer without having the slightest idea of mathematics4--,,novalis4 ,schriften1 ,zweiter ,teil 7,berlin1 #aija71 p4 #bbc4 #haa4 ,it has long been a complaint against mathematicians that they are hard to convince3 but it is a far greater disqualification both for philosophy1 and for the affairs of life1 to be too easily convinced2 to have too low a standard of proof4 ,the only sound intellects are those which1 in the first instance1 set their standards of proof high4 ,practice in concrete affairs soon teaches them to make the necessary abatement3 but they retain the consciousness1 without which there is no sound practical reasoning1 that in accepting inferior evidence because there is no better to be had1 they do not by that acceptance raise it to completeness4--,,mill1 ,j4 ,s4 ,an ,examination of ,sir ,william ,hamilton's ,philosophy 7,london1 #ahgh71 p4 #faa4 #hab4 ,it is easier to square the circle than to get round a mathematician4--,,de ,,morgan1 ,a4 ,budget of ,paradoxes 7,london1 #ahgb71 p4 #ij4 #hac4 ,mathematicians are like ,frenchmen3 whatever you say to them they translate into their own language and forthwith it is something entirely different4--,,goethe4 ,maximen und ,reflexionen1 ,sechste ,abtheilung4 #had4 ,what ,i chiefly admired1 and thought altogether unaccountable1 was the strong disposition ,i observed in them the mathematicians of ,laputa towards news and politics2 perpetually inquiring into public affairs2 giving their judgments in matters of state2 and passionately disputing every inch of party opinion4 ,i have indeed observed the same disposition among most of the mathematicians ,i have known in ,europe1 although ,i could never discover the least analogy between the two sciences4 --,,swift1 ,,jonathan4 ,gulliver's ,travels1 ,part #c1 chap4 #b4 #hae4 ,the great mathematician1 like the great poet or naturalist or great administrator1 is born4 ,my contention shall be that where the mathematic endowment is found1 there will usually be found associated with it1 as essential implications in it1 other endowments in generous measure1 and that the appeal of the science is to the whole mind1 direct no doubt to the central powers of thought1 but indirectly through sympathy of all1 rousing1 enlarging1 developing1 emancipating all1 so that the faculties of will1 of intellect and feeling learn to respond1 each in its appropriate order and degree1 like the parts of an orchestra to the 8urge and ardor0 of its leader and lord4 --,,keyser1 ,c4 ,j4 ,lectures on ,science1 ,philosophy and ,art 7,new ,york1 #aijh71 p4 #bb4 #haf4 ,whoever limits his exertions to the gratification of others1 whether by personal exhibition1 as in the case of the actor and of the mimic1 or by those kinds of literary composition which are calculated for no end but to please or to entertain1 renders himself1 in some measure1 dependent on their caprices and humours4 ,the diversity among men1 in their judgments concerning the objects of taste1 is incomparably greater than in their speculative conclusions2 and accordingly1 a mathematician will publish to the world a geometrical demonstration1 or a philosopher1 a process of abstract reasoning1 with a confidence very different from what a poet would feel1 in communicating one of his productions even to a friend4--,,stewart1 ,,dugald4 ,elements of the ,philosophy of the ,human ,mind1 ,part #c1 chap4 #a1 sect4 #c4 #hag4 ,considering that1 among all those who up to this time made discoveries in the sciences1 it was the mathematicians alone who had been able to arrive at demonstrations--that is to say1 at proofs certain and evident--,i did not doubt that ,i should begin with the same truths that they have investigated1 although ,i had looked for no other advantage from them than to accustom my mind to nourish itself upon truths and not to be satisfied with false reasons4--,,descartes4 ,discourse upon ,method1 ,part #b2 ,philosophy of ,descartes ,torrey 7,new ,york1 #ahib71 p4 #dh4 #hah4 ,when the late ,sophus ,lie 444 was asked to name the characteristic endowment of the mathematician1 his answer was the following quaternion3 ,phantasie1 ,energie1 ,selbstvertrauen1 ,selbstkritik4--,,keyser1 ,c4 ,j4 ,lectures on ,philosophy1 ,science and ,art 7,new ,york1 #aijh71 p4 #ca4 #hai4 ,the existence of an extensive ,science of ,mathematics1 requiring the highest scientific genius in those who contributed to its creation1 and calling for the most continued and vigorous exertion of intellect in order to appreciate it when created1 etc4--,,mill1 ,j4 ,s4 ,system of ,logic1 ,bk4 #b1 chap4 #d1 sect4 #d4 #hbj4 ,it may be true1 that men1 who are mere mathematicians1 have certain specific shortcomings1 but that is not the fault of mathematics1 for it is equally true of every other exclusive occupation4 ,so there are mere philologists1 mere jurists1 mere soldiers1 mere merchants1 etc4 ,to such idle talk it might further be added3 that whenever a certain exclusive occupation is coupled with specific shortcomings1 it is likewise almost certainly divorced from certain other shortcomings4--,,gauss4 ,gauss-,schumacher ,briefwechsel1 ,bd4 #d1 7,altona1 #ahfb71 p4 #chg4 #hba4 ,mathematical studies 444 when combined1 as they now generally are1 with a taste for physical science1 enlarge infinitely our views of the wisdom and power displayed in the universe4 ,the very intimate connexion indeed1 which1 since the date of the ,newtonian philosophy1 has existed between the different branches of mathematical and physical knowledge1 renders such a character as that of a mere mathematician a very rare and scarcely possible occurrence4--,,stewart1 ,,dugald4 ,elements of the ,philosophy of the ,human ,mind1 part #c1 chap4 #a1 sect4 #c4 #hbb4 ,once when lecturing to a class he ,lord ,kelvin used the word 8mathematician10 and then interrupting himself asked his class3 8,do you know what a mathematician is80 ,stepping to the blackboard he wrote upon it3-- _/ -x e dx _/- ,then putting his finger on what he had written1 he turned to his class and said3 8,a mathematician is one to whom that is as obvious as that twice two makes four is to you4 ,liouville was a mathematician4--,,thompson1 ,s4 ,p4 ,life of ,lord ,kelvin 7,london1 #aiaj71 p4 #aaci4 #hbc4 ,it is not surprising1 in view of the polydynamic constitution of the genuinely mathematical mind1 that many of the major heroes of the science1 men like ,desargues and ,pascal1 ,descartes and ,leibnitz1 ,newton1 ,gauss and ,bolzano1 ,helmholtz and ,clifford1 ,riemann and ,salmon and ,plcker and ,poincar1 have attained to high distinction in other fields not only of science but of philosophy and letters too4 ,and when we reflect that the very greatest mathematical achievements have been due1 not alone to the peering1 microscopic1 histologic vision of men like ,weierstrass1 illuminating the hidden recesses1 the minute and intimate structure of logical reality1 but to the larger vision also of men like ,klein who survey the kingdoms of geometry and analysis for the endless variety of things that nourish there1 as the eye of ,darwin ranged over the flora and fauna of the world1 or as a commercial monarch contemplates its industry1 or as a statesman beholds an empire2 when we reflect not only that the ,calculus of ,probability is a creation of mathematics but that the master mathematician is constantly required to exercise judgment--judgment1 that is1 in matters not admitting of certainty--balancing probabilities not yet reduced nor even reducible perhaps to calculation2 when we reflect that he is called upon to exercise a function analogous to that of the comparative anatomist like ,cuvier1 comparing theories and doctrines of every degree of similarity and dissimilarity of structure2 when1 finally1 we reflect that he seldom deals with a single idea at a time1 but is for the most part engaged in wielding organized hosts of them1 as a general wields at once the division of an army or as a great civil administrator directs from his central office diverse and scattered but related groups of interests and operations2 then1 ,i say1 the current opinion that devotion to mathematics unfits the devotee for practical affairs should be known for false on a priori grounds4 ,and one should be thus prepared to find that as a fact ,gaspard ,monge1 creator of descriptive geometry1 author of the classic 8,applications de l'analyse la gomtrie02 ,lazare ,carnot1 author of the celebrated works1 8,gomtrie de position10 and 8,rflections sur la ,mtaphysique du ,calcul infinitesimal02 ,fourier1 immortal creator of the 8,thorie analytique de la chaleur02 ,arago1 rightful inheritor of ,monge's chair of geometry2 ,poncelet1 creator of pure projective geometry2 one should not be surprised1 ,i say1 to find that these and other mathematicians in a land sagacious enough to invoke their aid1 rendered1 alike in peace and in war1 eminent public service4--,,keyser1 ,c4 ,j4 ,lectures on ,science1 ,philosophy and ,art 7,new ,york1 #aijh71 pp4 #cb-cc4 #hbd4 ,if in ,germany the goddess ,justitia had not the unfortunate habit of depositing the ministerial portfolios only in the cradles of her own progeny1 who knows how many a ,german mathematician might not also have made an excellent minister4 --,,pringsheim1 ,a4 ,jahresbericht der ,deutschen ,mathematiker ,vereinigung1 ,bd4 #ac 7#aijd71 p4 #cgb4 #hbe4 ,we pass with admiration along the great series of mathematicians1 by whom the science of theoretical mechanics has been cultivated1 from the time of ,newton to our own4 ,there is no group of men of science whose fame is higher or brighter4 ,the great discoveries of ,copernicus1 ,galileo1 ,newton1 had fixed all eyes on those portions of human knowledge on which their successors employed their labors4 ,the certainty belonging to this line of speculation seemed to elevate mathematicians above the students of other subjects2 and the beauty of mathematical relations and the subtlety of intellect which may be shown in dealing with them1 were fitted to win unbounded applause4 ,the successors of ,newton and the ,bernoullis1 as ,euler1 ,clairaut1 ,d',alembert1 ,lagrange1 ,laplace1 not to introduce living names1 have been some of the most remarkable men of talent which the world has seen4--,,whewell1 ,w4 ,history of the ,inductive ,sciences1 ,vol4 #a1 ,bk4 #d1 chap4 #f1 sect4 #f4 #hbf4 ,the persons who have been employed on these problems of applying the properties of matter and the laws of motion to the explanation of the phenomena of the world1 and who have brought to them the high and admirable qualities which such an office requires1 have justly excited in a very eminent degree the admiration which mankind feels for great intellectual powers4 ,their names occupy a distinguished place in literary history2 and probably there are no scientific reputations of the last century higher1 and none more merited1 than those earned by great mathematicians who have laboured with such wonderful success in unfolding the mechanism of the heavens2 such for instance as ,d',alembert1 ,clairaut1 ,euler1 ,lagrange1 ,laplace4--,,whewell1 ,w4 ,astronomy and ,general ,physics 7,london1 #ahcc71 ,bk4 #c1 chap4 #d1 p4 #cbg4 #hbg4 ,two extreme views have always been held as to the use of mathematics4 ,to some1 mathematics is only measuring and calculating instruments1 and their interest ceases as soon as discussions arise which cannot benefit those who use the instruments for the purposes of application in mechanics1 astronomy1 physics1 statistics1 and other sciences4 ,at the other extreme we have those who are animated exclusively by the love of pure science4 ,to them pure mathematics1 with the theory of numbers at the head1 is the only real and genuine science1 and the applications have only an interest in so far as they contain or suggest problems in pure mathematics4 ,of the two greatest mathematicians of modern times1 ,newton and ,gauss1 the former can be considered as a representative of the first1 the latter of the second class2 neither of them was exclusively so1 and ,newton's inventions in the science of pure mathematics were probably equal to ,gauss's work in applied mathematics4 ,newton's reluctance to publish the method of fluxions invented and used by him may perhaps be attributed to the fact that he was not satisfied with the logical foundations of the ,calculus2 and ,gauss is known to have abandoned his electro-dynamic speculations1 as he could not find a satisfying physical basis4444 ,newton's greatest work1 the 8,principia01 laid the foundation of mathematical physics2 ,gauss's greatest work1 the 8,disquisitiones ,arithmeticae01 that of higher arithmetic as distinguished from algebra4 ,both works1 written in the synthetic style of the ancients1 are difficult1 if not deterrent1 in their form1 neither of them leading the reader by easy steps to the results4 ,it took twenty or more years before either of these works received due recognition2 neither found favour at once before that great tribunal of mathematical thought1 the ,paris ,academy of ,sciences4444 ,the country of ,newton is still pre-eminent for its culture of mathematical physics1 that of ,gauss for the most abstract work in mathematics4--,,merz1 ,j4 ,t4 ,history of ,european ,thought in the ,nineteenth ,century 7,edinburgh and ,london1 #aijc71 p4 #fcj4 #hbh4 ,as there is no study which may be so advantageously entered upon with a less stock of preparatory knowledge than mathematics1 so there is none in which a greater number of uneducated men have raised themselves1 by their own exertions1 to distinction and eminence4444 ,many of the intellectual defects which1 in such cases1 are commonly placed to the account of mathematical studies1 ought to be ascribed to the want of a liberal education in early youth4--,,stewart1 ,,dugald4 ,elements of the ,philosophy of the ,human ,mind1 ,part #c1 chap4 #a1 sect4 #c4 #hbi4 ,i know1 indeed1 and can conceive of no pursuit so antagonistic to the cultivation of the oratorical faculty 444 as the study of ,mathematics4 ,an eloquent mathematician must1 from the nature of things1 ever remain as rare a phenomenon as a talking fish1 and it is certain that the more anyone gives himself up to the study of oratorical effect the less will he find himself in a fit state to mathematicize4 ,it is the constant aim of the mathematician to reduce all his expressions to their lowest terms1 to retrench every superfluous word and phrase1 and to condense the ,maximum of meaning into the ,minimum of language4 ,he has to turn his eye ever inwards1 to see everything in its dryest light1 to train and inure himself to a habit of internal and impersonal reflection and elaboration of abstract thought1 which makes it most difficult for him to touch or enlarge upon any of those themes which appeal to the emotional nature of his fellow-men4 ,when called upon to speak in public he feels as a man might do who has passed all his life in peering through a microscope1 and is suddenly called upon to take charge of a astronomical observatory4 ,he has to get out of himself1 as it were1 and change the habitual focus of his vision4 --,,sylvester1 ,j4 ,j4 ,baltimore ,address2 ,mathematical ,papers1 ,vol4 #c1 pp4 #gb-gc4 #hcj4 ,an accomplished mathematician1 i4e4 a most wretched orator4--,,barrow1 ,,isaac4 ,mathematical ,lectures 7,london1 #agcd71 p4 #cb4 #hca4 ,nemo mathematicus genium indemnatus habebit4 ,no mathematician#b is esteemed a genius until condemned4 ,juvenal1 ,liberii1 ,satura ,,vi1 #efb4 #b ,used here in the sense of astrologer1 or soothsayer4 #hcb4 ,taking 444 the mathematical faculty1 probably fewer than one in a hundred really possess it1 the great bulk of the population having no natural ability for the study1 or feeling the slightest interest in it4#c ,and if we attempt to measure the amount of variation in the faculty itself between a first-class mathematician and the ordinary run of people who find any kind of calculation confusing and altogether devoid of interest1 it is probable that the former could not be estimated at less than a hundred times the latter1 and perhaps a thousand times would more nearly measure the difference between them4--,,wallace1 ,a4 ,r4 ,darwinism1 chap4 #ae4 #c ,this is the estimate furnished me by two mathematical masters in one of our great public schools of the proportion of boys who have any special taste or capacity for mathematical studies4 ,many more1 of course1 can be drilled into a fair knowledge of elementary mathematics1 but only this small proportion possess the natural faculty which renders it possible for them ever to rank high as mathematicians1 to take any pleasure in it1 or to do any original mathematical work4 #hcc4 444 the present gigantic development of the mathematical faculty is wholly unexplained by the theory of natural selection1 and must be due to some altogether distinct cause4--,,wallace1 ,a4 ,r4 ,darwinism1 chap4 #ae4 #hcd4 ,dr4 ,wallace1 in his 8,darwinism01 declares that he can find no ground for the existence of pure scientists1 especially mathematicians1 on the hypothesis of natural selection4 ,if we put aside the fact that great power in theoretical science is correlated with other developments of increasing brain-activity1 we may1 ,i think1 still account for the existence of pure scientists as ,dr4 ,wallace would himself account for that of worker-bees4 ,their function may not fit them individually to survive in the struggle for existence1 but they are a source of strength and efficiency to the society which produces them4 --,,pearson1 ,,karl4 ,grammar of ,science 7,london1 #aiaa71 ,part #a1 p4 #bba4 #hce4 ,it is only in mathematics1 and to some extent in poetry1 that originality may be attained at an early age1 but even then it is very rare 7,newton and ,keats are examples71 and it is not notable until adolescence is completed4--,,ellis1 ,,havelock4 ,a ,study of ,british ,genius 7,london1 #aijd71 p4 #adb4 #hcf4 ,the ,anglo-,dane appears to possess an aptitude for mathematics which is not shared by the native of any other ,english district as a whole1 and it is in the exact sciences that the ,anglo-,dane triumphs4#d--;,,ellis1 ,,havelock4 ,a ,study of ,british ,genius 7,london1 #aijd71 p4 #fi4 #d ,the mathematical tendencies of ,cambridge are due to the fact that ,cambridge drains the ability of nearly the whole ,anglo-,danish district4 #hcg4 ,in the whole history of the world there was never a race with less liking for abstract reasoning than the ,anglo-,saxon4444 ,common-sense and compromise are believed in1 logical deductions from philosophical principles are looked upon with suspicion1 not only by legislators1 but by all our most learned professional men4--,,perry1 ,,john4 ,the ,teaching of ,mathematics 7,london1 #aijb71 pp4 #bj-ba4 #hch4 ,the degree of exactness of the intuition of space may be different in different individuals1 perhaps even in different races4 ,it would seem as if a strong nave space-intuition were an attribute pre-eminently of the ,teutonic race1 while the critical1 purely logical sense is more fully developed in the ,latin and ,hebrew races4 ,a full investigation of this subject1 somewhat on the lines suggested by ,francis ,galton in his researches on heredity1 might be interesting4--,,klein1 ,,felix4 ,the ,evanston ,colloquium ,lectures 7,new ,york1 #ahid71 p4 #df4 #hci4 ,this the fact that the pursuit of mathematics brings into harmonious action all the faculties of the human mind accounts for the extraordinary longevity of all the greatest masters of the ,analytic art1 the ,dii ,majores of the mathematical ,pantheon4 ,leibnitz lived to the age of #gj2 ,euler to #gf2 ,lagrange to #gg2 ,laplace to #gh2 ,gauss to #gh2 ,plato1 the supposed inventor of the conic sections1 who made mathematics his study and delight1 who called them the handles or aids to philosophy1 the medicine of the soul1 and is said never to have let a day go by without inventing some new theorems1 lived to #hb2 ,newton1 the crown and glory of his race1 to #he2 ,archimedes1 the nearest akin1 probably1 to ,newton in genius1 was #ge1 and might have lived on to be #ajj1 for aught we can guess to the contrary1 when he was slain by the impatient and ill-mannered sergeant1 sent to bring him before the ,roman general1 in the full vigour of his faculties1 and in the very act of working out a problem2 ,pythagoras1 in whose school1 ,i believe1 the word mathematician 7used1 however1 in a somewhat wider than its present sense7 originated1 the second founder of geometry1 the inventor of the matchless theorem which goes by his name1 the pre-cognizer of the undoubtedly mis-called ,copernican theory1 the discoverer of the regular solids and the musical canon who stands at the very apex of this pyramid of fame1 7if we may credit the tradition7 after spending #bb years studying in ,egypt1 and #ab in ,babylon1 opened school when #ef or #eg years old in ,magna ,grcia1 married a young wife when past #fj1 and died1 carrying on his work with energy unspent to the last1 at the age of #ii4 ,the mathematician lives long and lives young2 the wings of his soul do not early drop off1 nor do its pores become clogged with the earthy particles blown from the dusty highways of vulgar life4--,,sylvester1 ,j4 ,j4 ,presidential ,address to the ,british ,association2 ,collected ,mathematical ,papers1 ,vol4 #b 7#aijh71 p4 #feh4 #hdj4 ,the game of chess has always fascinated mathematicians1 and there is reason to suppose that the possession of great powers of playing that game is in many features very much like the possession of great mathematical ability4 ,there are the different pieces to learn1 the pawns1 the knights1 the bishops1 the castles1 and the queen and king4 ,the board possesses certain possible combinations of squares1 as in rows1 diagonals1 etc4 ,the pieces are subject to certain rules by which their motions are governed1 and there are other rules governing the players4444 ,one has only to increase the number of pieces1 to enlarge the field of the board1 and to produce new rules which are to govern either the pieces or the player1 to have a pretty good idea of what mathematics consists4--,,shaw1 ,j4 ,b4 ,what is ,mathematics8 ,bulletin ,american ,mathematical ,society ,vol4 #ah 7#aiab71 pp4 #chf-chg4 #hda4 ,every man is ready to join in the approval or condemnation of a philosopher or a statesman1 a poet or an orator1 an artist or an architect4 ,but who can judge of a mathematician8 ,who will write a review of ,hamilton's ,quaternions1 and show us wherein it is superior to ,newton's ,fluxions8--,,hill1 ,,thomas4 ,imagination in ,mathematics2 ,north ,american ,review1 ,vol4 #he1 p4 #bbd4 #hdb4 ,the pursuit of mathematical science makes its votary appear singularly indifferent to the ordinary interests and cares of men4 ,seeking eternal truths1 and finding his pleasures in the realities of form and number1 he has little interest in the disputes and contentions of the passing hour4 ,his views on social and political questions partake of the grandeur of his favorite contemplations1 and1 while careful to throw his mite of influence on the side of right and truth1 he is content to abide the workings of those general laws by which he doubts not that the fluctuations of human history are as unerringly guided as are the perturbations of the planetary hosts4--,,hill1 ,,thomas4 ,imagination in ,mathematics2 ,north ,american ,review1 ,vol4 #he1 p4 #bbg4 #hdc4 ,there is something sublime in the secrecy in which the really great deeds of the mathematician are done4 ,no popular applause follows the act2 neither contemporary nor succeeding generations of the people understand it4 ,the geometer must be tried by his peers1 and those who truly deserve the title of geometer or analyst have usually been unable to find so many as twelve living peers to form a jury4 ,archimedes so far outstripped his competitors in the race1 that more than a thousand years elapsed before any man appeared1 able to sit in judgment on his work1 and to say how far he had really gone4 ,and in judging of those men whose names are worthy of being mentioned in connection with his1--,galileo1 ,descartes1 ,leibnitz1 ,newton1 and the mathematicians created by ,leibnitz and ,newton's calculus1--we are forced to depend upon their testimony of one another4 ,they are too far above our reach for us to judge of them4--,,hill1 ,,thomas4 ,imagination in ,mathematics2 ,north ,american ,review1 ,vol4 #he1 p4 #bbc4 #hdd4 ,to think the thinkable--that is the mathematician's aim4 --,,keyser1 ,c4 ,j4 ,the ,universe and ,beyond2 ,hibbert ,journal1 ,vol4 #c 7#aijd-aije71 p4 #cab4 #hde4 ,every common mechanic has something to say in his craft about good and evil1 useful and useless1 but these practical considerations never enter into the purview of the mathematician4 --,,aristippus ,,the ,,cyrenaic4 ,quoted in ,hicks1 ,r4 ,d41 ,stoic and ,epicurean1 7,new ,york1 #aiaj7 p4 #baj4 ,,chapter ,,ix ,,persons ,,and ,,anecdotes 7,a-,m7 #ija4 ,alexander is said to have asked ,menchmus to teach him geometry concisely1 but ,menchmus replied3 8,o king1 through the country there are royal roads and roads for common citizens1 but in geometry there is one road for all40 ,stobus 7,edition ,wachsmuth1 ,berlin1 #ahhd71 ,ecl4 #b1 p4 #cj4 #ijb4 ,alexander the king of the ,macedonians1 began like a wretch to learn geometry1 that he might know how little the earth was1 whereof he had possessed very little4 ,thus1 ,i say1 like a wretch for this1 because he was to understand that he did bear a false surname4 ,for who can be great in so small a thing8 ,those things that were delivered were subtile1 and to be learned by diligent attention3 not which that mad man could perceive1 who sent his thoughts beyond the ocean sea4 ,teach me1 saith he1 easy things4 ,to whom his master said3 ,these things be the same1 and alike difficult unto all4 ,think thou that the nature of things saith this4 ,these things whereof thou complainest1 they are the same unto all3 more easy things can be given unto none2 but whosoever will1 shall make those things more easy unto himself4 ,how8 ,with uprightness of mind4--,,seneca4 ,epistle #ia ,thomas ,lodge4 #ijc4 ,archimedes 444 had stated that given the force1 any given weight might be moved1 and even boasted1 we are told1 relying on the strength of demonstration1 that if there were another earth1 by going into it he could remove this4 ,hiero being struck with amazement at this1 and entreating him to make good this problem by actual experiment1 and show some great weight moved by a small engine1 he fixed accordingly upon a ship of burden out of the king's arsenal1 which could not be drawn out of the dock without great labor and many men2 and1 loading her with many passengers and a full freight1 sitting himself the while far off with no great endeavor1 but only holding the head of the pulley in his hand and drawing the cords by degrees1 he drew the ship in a straight line1 as smoothly and evenly1 as if she had been in the sea4 ,the king1 astonished at this1 and convinced of the power of the art1 prevailed upon ,archimedes to make him engines accommodated to all the purposes1 offensive and defensive1 of a siege 444 the apparatus was1 in most opportune time1 ready at hand for the ,syracusans1 and with it also the engineer himself4 --,,plutarch4 ,life of ,marcellus ,dryden4 #ijd4 ,these machines used in the defense of the ,syracusans against the ,romans under ,marcellus he ,archimedes had designed and contrived1 not as matters of any importance1 but as mere amusements in geometry2 in compliance with king ,hiero's desire and request1 some time before1 that he should reduce to practice some part of his admirable speculation in science1 and by accommodating the theoretic truth to sensation and ordinary use1 bring it more within the appreciation of people in general4 ,eudoxus and ,archytas had been the first originators of this far-famed and highly-prized art of mechanics1 which they employed as an elegant illustration of geometrical truths1 and as means of sustaining experimentally1 to the satisfaction of the senses conclusions too intricate for proof by words and diagrams4 ,as1 for example1 to solve the problem1 so often required in constructing geometrical figures1 given the two extremes1 to find the two mean lines of a proportion1 both these mathematicians had recourse to the aid of instruments1 adapting to their purpose certain curves and sections of lines4 ,but what with ,plato's indignation at it1 and his invectives against it as the mere corruption and annihilation of the one good of geometry1--which was thus shamefully turning its back upon the unembodied objects of pure intelligence to recur to sensation1 and to ask help 7not to be obtained without base supervisions and depravation7 from matter2 so it was that mechanics came to be separated from geometry1 and1 repudiated and neglected by philosophers1 took its place as a military art4--,,plutarch4 ,life of ,marcellus ,dryden4 #ije4 ,archimedes was not free from the prevailing notion that geometry was degraded by being employed to produce anything useful4 ,it was with difficulty that he was induced to stoop from speculation to practice4 ,he was half ashamed of those inventions which were the wonder of hostile nations1 and always spoke of them slightingly as mere amusements1 as trifles in which a mathematician might be suffered to relax his mind after intense application to the higher parts of his science4--,,macaulay4 ,lord ,bacon2 ,edinburgh ,review1 ,july #ahcg2 ,critical and ,miscellaneous ,essays 7,new ,york1 #ahgi71 ,vol4 #a1 p4 #chj4 #ijf4 ,call ,archimedes from his buried tomb ,upon the plain of vanished ,syracuse1 ,and feelingly the sage shall make report ,how insecure1 how baseless in itself1 ,is the philosophy1 whose sway depends ,on mere material instruments--how weak ,those arts1 and high inventions1 if unpropped ,by virtue4 --,,wordsworth4 ,the ,excursion4 #ijg4 ,zu ,archimedes kam einst ein wissbegieriger ,jngling4 8,weihe mich10 sprach er zu ihm1 8ein in die gttliche ,kunst1 ,die so herrliche ,frucht dem ,vaterlande getragen1 ,und die ,mauern der ,stadt vor der ,sambuca beschtzt60 8,gttlich nennst du die ,kunst8 ,sie ists10 versetzte der ,weise2 8,aber das war sie1 mein ,sohn1 eh sie dem ,staat noch gedient4 ,willst du nur ,frchte von ihr1 die kann auch die ,sterbliche zeugen2 ,wer um die ,gttin freit1 suche in ihr nicht das ,weib40 --,,schiller4 ,archimedes und der ,schler4 ,to ,archimedes once came a youth intent upon knowledge4 ,said he 8,initiate me into the ,science divine1 ,which to our country has borne glorious fruits in abundance1 ,and which the walls of the town 'gainst the ,sambuca protects40 8,callst thou the science divine8 ,it is so10 the wise man responded2 8,but so it was1 my son1 ere the state by her service was blest4 ,would'st thou have fruit of her only8 ,mortals with that can provide thee1 ,he who the goddess would woo1 seek not the woman in her40 #ijh4 ,archimedes possessed so high a spirit1 so profound a soul1 and such treasures of highly scientific knowledge1 that though these inventions used to defend ,syracuse against the ,romans had now obtained him the renown of more than human sagacity1 he yet would not deign to leave behind him any commentary or writing on such subjects2 but1 repudiating as sordid and ignoble the whole trade of engineering1 and every sort of art that lends itself to mere use and profit1 he placed his whole affection and ambition in those purer speculations where there can be no reference to the vulgar needs of life2 studies1 the superiority of which to all others is unquestioned1 and in which the only doubt can be whether the beauty and grandeur of the subjects examined1 or the precision and cogency of the methods and means of proof1 most deserve our admiration4--,,plutarch4 ,life of ,marcellus ,dryden4 #iji4 ,nothing afflicted ,marcellus so much as the death of ,archimedes1 who was then1 as fate would have it1 intent upon working out some problem by a diagram1 and having fixed his mind alike and his eyes upon the subject of his speculation1 he never noticed the incursion of the ,romans1 nor that the city was taken4 ,in this transport of study and contemplation1 a soldier1 unexpectedly coming up to him1 commanded him to follow to ,marcellus1 which he declined to do before he had worked out his problem to a demonstration2 the soldier1 enraged1 drew his sword and ran him through4 ,others write1 that a ,roman soldier1 running upon him with a drawn sword1 offered to kill him2 and that ,archimedes1 looking back1 earnestly besought him to hold his hand a little while1 that he might not leave what he was at work upon inconclusive and imperfect2 but the soldier1 nothing moved by his entreaty1 instantly killed him4 ,others again relate1 that as ,archimedes was carrying to ,marcellus mathematical instruments1 dials1 spheres1 and angles1 by which the magnitude of the sun might be measured to the sight1 some soldiers seeing him1 and thinking that he carried gold in a vessel1 slew him4 ,certain it is1 that his death was very afflicting to ,marcellus2 and that ,marcellus ever after regarded him that killed him as a murderer2 and that he sought for his kindred and honoured them with signal favours4--,,plutarch4 ,life of ,marcellus ,dryden4 #iaj4 ,archimedes is said to have requested his friends and relations that when he was dead1 they would place over his tomb a sphere containing a cylinder1 inscribing it with the ratio which the containing solid bears to the contained4--,,plutarch4 ,life of ,marcellus ,dryden4 #iaa4 ,archimedes1 who combined a genius for mathematics with a physical insight1 must rank with ,newton1 who lived nearly two thousand years later1 as one of the founders of mathematical physics4444 ,the day 7when having discovered his famous principle of hydrostatics he ran through the streets shouting ,eureka6 ,eureka67 ought to be celebrated as the birthday of mathematical physics2 the science came of age when ,newton sat in his orchard4 --,,whitehead1 ,a4 ,n4 ,an ,introduction to ,mathematics 7,new ,york1 #aiaa71 p4 #ch4 #iab4 ,it is not possible to find in all geometry more difficult and more intricate questions or more simple and lucid explanations than those given by ,archimedes4 ,some ascribe this to his natural genius2 while others think that incredible effort and toil produced these1 to all appearance1 easy and unlaboured results4 ,no amount of investigation of yours would succeed in attaining the proof1 and yet1 once seen1 you immediately believe you would have discovered it2 by so smooth and so rapid a path he leads you to the conclusion required4--,,plutarch4 ,life of ,marcellus ,dryden4 #iac4 ,one feature which will probably most impress the mathematician accustomed to the rapidity and directness secured by the generality of modern methods is the deliberation with which ,archimedes approaches the solution of any one of his main problems4 ,yet this very characteristic1 with its incidental effects1 is calculated to excite the more admiration because the method suggests the tactics of some great strategist who foresees everything1 eliminates everything not immediately conducive to the execution of his plan1 masters every position in its order1 and then suddenly 7when the very elaboration of the scheme has almost obscured1 in the mind of the spectator1 its ultimate object7 strikes the final blow4 ,thus we read in ,archimedes proposition after proposition the bearing of which is not immediately obvious but which we find infallibly used later on2 and we are led by such easy stages that the difficulties of the original problem1 as presented at the outset1 are scarcely appreciated4 ,as ,plutarch says3 8,it is not possible to find in geometry more difficult and troublesome questions1 or more simple and lucid explanations40 ,but it is decidedly a rhetorical exaggeration when ,plutarch goes on to say that we are deceived by the easiness of the successive steps into the belief that anyone could have discovered them for himself4 ,on the contrary1 the studied simplicity and the perfect finish of the treatises involve at the same time an element of mystery4 ,though each step depends on the preceding ones1 we are left in the dark as to how they were suggested to ,archimedes4 ,there is1 in fact1 much truth in a remark by ,wallis to the effect that he seems 8as it were of set purpose to have covered up the traces of his investigation as if he had grudged posterity the secret of his method of inquiry while he wished to extort from them assent to his results40 ,wallis adds with equal reason that not only ,archimedes but nearly all the ancients so hid away from posterity their method of ,analysis 7though it is certain that they had one7 that more modern mathematicians found it easier to invent a new ,analysis than to seek out the old4--,,heath1 ,t4 ,l4 ,the ,works of ,archimedes 7,cambridge1 #ahig71 ,preface4 #iad4 ,it is a great pity ,aristotle had not understood mathematics as well as ,mr4 ,newton1 and made use of it in his natural philosophy with good success3 his example had then authorized the accommodating of it to material things4 --,,locke1 ,,john4 ,second ,reply to the ,bishop of ,worcester4 #iae4 ,the opinion of ,bacon on this subject geometry was diametrically opposed to that of the ancient philosophers4 ,he valued geometry chiefly1 if not solely1 on account of those uses1 which to ,plato appeared so base4 ,and it is remarkable that the longer ,bacon lived the stronger this feeling became4 ,when in #afje he wrote the two books on the ,advancement of ,learning1 he dwelt on the advantages which mankind derived from mixed mathematics2 but he at the same time admitted that the beneficial effect produced by mathematical study on the intellect1 though a collateral advantage1 was 8no less worthy than that which was principal and intended40 ,but it is evident that his views underwent a change4 ,when near twenty years later1 he published the ,de ,augmentis1 which is the ,treatise on the ,advancement of ,learning1 greatly expanded and carefully corrected1 he made important alterations in the part which related to mathematics4 ,he condemned with severity the pretensions of the mathematicians1 8delicias et fastum mathematicorum40 ,assuming the well-being of the human race to be the end of knowledge1 he pronounced that mathematical science could claim no higher rank than that of an appendage or an auxiliary to other sciences4 ,mathematical science1 he says1 is the handmaid of natural philosophy2 she ought to demean herself as such2 and he declares that he cannot conceive by what ill chance it has happened that she presumes to claim precedence over her mistress4--,,macaulay4 ,lord ,bacon3 ,edinburgh ,review1 ,july1 #ahcg2 ,critical and ,miscellaneous ,essays 7,new ,york1 #ahgi71 ,vol4 #a1 p4 #chj4 #iaf4 ,if ,bacon erred here in valuing mathematics only for its uses1 we must acknowledge that we greatly prefer his error to the opposite error of ,plato4 ,we have no patience with a philosophy which1 like those ,roman matrons who swallowed abortives in order to preserve their shapes1 takes pains to be barren for fear of being homely4--,,macaulay4 ,lord ,bacon1 ,edinburgh ,review1 ,july1 #ahcg2 ,critical and ,miscellaneous ,essays 7,new ,york1 #ahgi71 ,vol4 #b1 p4 #cha4 #iag4 ,he ,lord ,bacon appears to have been utterly ignorant of the discoveries which had just been made by ,kepler's calculations 444 he does not say a word about ,napier's ,logarithms1 which had been published only nine years before and reprinted more than once in the interval4 ,he complained that no considerable advance had been made in ,geometry beyond ,euclid1 without taking any notice of what had been done by ,archimedes and ,apollonius4 ,he saw the importance of determining accurately the specific gravities of different substances1 and himself attempted to form a table of them by a rude process of his own1 without knowing of the more scientific though still imperfect methods previously employed by ,archimedes1 ,ghetaldus and ,porta4 ,he speaks of the of ,archimedes in a manner which implies that he did not clearly appreciate either the problem to be solved or the principles upon which the solution depended4 ,in reviewing the progress of ,mechanics1 he makes no mention either of ,archimedes1 or ,stevinus1 ,galileo1 ,guldinus1 or ,ghetaldus4 ,he makes no allusion to the theory of ,equilibrium4 ,he observes that a ball of one pound weight will fall nearly as fast through the air as a ball of two1 without alluding to the theory of acceleration of falling bodies1 which had been made known by ,galileo more than thirty years before4 ,he proposed an inquiry with regard to the lever1--namely1 whether in a balance with arms of different length but equal weight the distance from the fulcrum has any effect upon the inclination--though the theory of the lever was as well understood in his own time as it is now4444 ,he speaks of the poles of the earth as fixed1 in a manner which seems to imply that he was not acquainted with the precession of the equinoxes2 and in another place1 of the north pole being above and the south pole below1 as a reason why in our hemisphere the north winds predominate over the south4--,,spedding1 ,j4 ,works of ,francis ,bacon 7,boston71 ,preface to ,de ,interpretatione ,naturae ,prooemium4 #iah4 ,bacon himself was very ignorant of all that had been done by mathematics2 and1 strange to say1 he especially objected to astronomy being handed over to the mathematicians4 ,leverrier and ,adams1 calculating an unknown planet into a visible existence by enormous heaps of algebra1 furnish the last comment of note on this specimen of the goodness of ,bacon's view4444 ,mathematics was beginning to be the great instrument of exact inquiry3 ,bacon threw the science aside1 from ignorance1 just at the time when his enormous sagacity1 applied to knowledge1 would have made him see the part it was to play4 ,if ,newton had taken ,bacon for his master1 not he1 but somebody else1 would have been ,newton4 --,,de ,,morgan1 ,a4 ,budget of ,paradoxes 7,london1 #ahgb71 pp4 #ec-ed4 #iai4 ,daniel ,bernoulli used to tell two little adventures1 which he said had given him more pleasure than all the other honours he had received4 ,travelling with a learned stranger1 who1 being pleased with his conversation1 asked his name2 8,i am ,daniel ,bernoulli10 answered he with great modesty2 8and ,i10 said the stranger 7who thought he meant to laugh at him7 8am ,isaac ,newton40 ,another time1 having to dine with the celebrated ,koenig1 the mathematician1 who boasted1 with some degree of self-complacency1 of a difficult problem he had solved with much trouble1 ,bernoulli went on doing the honours of his table1 and when they went to drink coffee he presented ,koenig with a solution of the problem more elegant than his own4 --,,hutton1 ,,charles4 ,a ,philosophical and ,mathematical ,dictionary 7,london1 #ahae71 ,vol4 #a1 p4 #bbf4 #ibj4 ,following the example of ,archimedes who wished his tomb decorated with his most beautiful discovery in geometry and ordered it inscribed with a cylinder circumscribed by a sphere1 ,james ,bernoulli requested that his tomb be inscribed with his logarithmic spiral together with the words1 8,eadem mutata resurgo10 a happy allusion to the hope of the ,christians1 which is in a way symbolized by the properties of that curve4 --,,fontenelle4 ,eloge de ,m4 ,bernoulli2 ,oeuvres de ,fontenelle1 t4 #e 7#ageh71 p4 #aab4 #iba4 ,this formula for computing ,bernoulli's numbers was first given by ,james ,bernoulli4 ,he gave no general demonstration2 but was quite aware of the importance of his theorem1 for he boasts that by means of it he calculated intra semi-quadrantem horae6 the sum of the #ajth powers of the first thousand integers1 and found it to be #ia1dji1ibd1bda1dbd1bdc1dbd1bda1ibd1bdb1 ejj4 --,,chrystal1 ,g4 ,algebra1 ,part #b 7,edinburgh1 #ahgi71 p4 #bji4 #ibb4 ,in the year #afib1 ,james ,bernoulli1 discussing the logarithmic spiral or equiangular spiral1 444 shows that it reproduces itself in its evolute1 its involute1 and its caustics of both reflection and refraction1 and then adds3 8,but since this marvellous spiral1 by such a singular and wonderful peculiarity1 pleases me so much that ,i can scarce be satisfied with thinking about it1 ,i have thought that it might not be inelegantly used for a symbolic representation of various matters4 ,for since it always produces a spiral similar to itself1 indeed precisely the same spiral1 however it may be involved or evolved1 or reflected or refracted1 it may be taken as an emblem of a progeny always in all things like the parent1 simillima filia matri4 ,or1 if it is not forbidden to compare a theorem of eternal truth to the mysteries of our faith1 it may be taken as an emblem of the eternal generation of the ,son1 who as an image of the ,father1 emanating from him1 as light from light1 remains with him1 howsoever overshadowed4 ,or1 if you prefer1 since our spira mirabilis remains1 amid all changes1 most persistently itself1 and exactly the same as ever1 it may be used as a symbol1 either of fortitude and constancy in adversity1 or1 of the human body1 which after all its changes1 even after death1 will be restored to its exact and perfect self1 so that1 indeed1 if the fashion of ,archimedes were allowed in these days1 ,i should gladly have my tombstone bear this spiral1 with the motto1 8,though changed1 ,i arise again exactly the same1 ,eadem numero mutata resurgo40--,,hill1 ,,thomas4 ,the ,uses of ,mathesis2 ,bibliotheca ,sacra1 ,vol4 #cb1 pp4 #eae-eaf4 #ibc4 ,babbage was one of the founders of the ,cambridge ,analytical ,society whose purpose he stated was to advocate 8the principles of pure d-ism as opposed to the dot-age of the university40--,,ball1 ,w4 ,w4 ,r4 ,history of ,mathematics 7,london1 #aija71 p4 #dea4 #ibd4 ,bolyai ,janos when in garrison with cavalry officers1 was provoked by thirteen of them and accepted all their challenges on condition that he be permitted after each duel to play a bit on his violin4 ,he came out victor from his thirteen duels1 leaving his thirteen adversaries on the square4--,,halsted1 ,g4 ,b4 ,bolyai's ,science ,absolute of ,space 7,austin1 #ahif71 ,introduction1 p4 #bi4 #ibe4 ,bolyai ,janos projected a universal language for speech as we have it for music and mathematics4--,,halsted1 ,g4 ,b4 ,bolyai's ,science ,absolute of ,space 7,austin1 #ahif71 ,introduction1 p4 #bi4 #ibf4 ,bolyai's ,science ,absolute of ,space--the most extraordinary two dozen pages in the history of thought6 --,,halsted1 ,g4 ,b4 ,bolyai's ,science ,absolute of ,space 7,austin1 #ahif71 ,introduction1 p4 #ah4 #ibg4 ,wolfgang ,bolyai was extremely modest4 ,no monument1 said he1 should stand over his grave1 only an apple-tree1 in memory of the three apples3 the two of ,eve and ,paris1 which made hell out of earth1 and that of ,newton1 which elevated the earth again into the circle of the heavenly bodies4--,,cajori1 ,f4 ,history of ,elementary ,mathematics 7,new ,york1 #aiaj71 p4 #bgc4 #ibh4 ,bernard ,bolzano dispelled the clouds that throughout all the foregone centuries had enveloped the notion of ,infinitude in darkness1 completely sheared the great term of its vagueness without shearing it of its strength1 and thus rendered it forever available for the purposes of logical discourse4--,,keyser1 ,c4 ,j4 ,lectures on ,science1 ,philosophy and ,art 7,new ,york1 #aijh71 p4 #db4 #ibi4 ,let me tell you how at one time the famous mathematician ,euclid became a physician4 ,it was during a vacation1 which ,i spent in ,prague as ,i most always did1 when ,i was attacked by an illness never before experienced1 which manifested itself in chilliness and painful weariness of the whole body4 ,in order to ease my condition ,i took up ,euclid's ,elements and read for the first time his doctrine of ratio1 which ,i found treated there in a manner entirely new to me4 ,the ingenuity displayed in ,euclid's presentation filled me with such vivid pleasure1 that forthwith ,i felt as well as ever4--,,bolzano1 ,,bernard4 ,selbstbiographie 7,wien1 #ahge71 p4 #bj4 #icj4 ,mr4 ,cayley1 of whom it may be so truly said1 whether the matter he takes in hand be great or small1 8nihil tetigit quod non ornavit104444--,,sylvester1 ,j4 ,j4 ,philosophic ,transactions of the ,royal ,society1 ,vol4 #ag 7#ahfd71 p4 #fje4 #ica4 ,it is not ,cayley's way to analyze concepts into their ultimate elements4444 ,but he is master of the empirical utilization of the material3 in the way he combines it to form a single abstract concept which he generalizes and then subjects to computative tests1 in the way the newly acquired data are made to yield at a single stroke the general comprehensive idea to the subsequent numerical verification of which years of labor are devoted4 ,cayley is thus the natural philosopher among mathematicians4--,,noether1 ,m4 ,mathematische ,annalen1 ,bd4 #df 7#ahie71 p4 #dgi4 #icb4 ,when ,cayley had reached his most advanced generalizations he proceeded to establish them directly by some method or other1 though he seldom gave the clue by which they had first been obtained3 a proceeding which does not tend to make his papers easy reading4444 ,his literary style is direct1 simple and clear4 ,his legal training had an influence1 not merely upon his mode of arrangement but also upon his expression2 the result is that his papers are severe and present a curious contrast to the luxuriant enthusiasm which pervades so many of ,sylvester's papers4 ,he used to prepare his work for publication as soon as he carried his investigations in any subject far enough for his immediate purpose4444 ,a paper once written out was promptly sent for publication2 this practice he maintained throughout life4444 ,the consequence is that he has left few arrears of unfinished or unpublished papers2 his work has been given by himself to the world4--,,forsyth1 ,a4 ,r4 ,proceedings of ,london ,royal ,society1 ,vol4 #eh 7#ahie71 pp4 #bc-bd4 #icc4 ,cayley was singularly learned in the work of other men1 and catholic in his range of knowledge4 ,yet he did not read a memoir completely through3 his custom was to read only so much as would enable him to grasp the meaning of the symbols and understand its scope4 ,the main result would then become to him a subject of investigation3 he would establish it 7or test it7 by algebraic analysis and1 not infrequently1 develop it so to obtain other results4 ,this faculty of grasping and testing rapidly the work of others1 together with his great knowledge1 made him an invaluable referee2 his services in this capacity were used through a long series of years by a number of societies to which he was almost in the position of standing mathematical advisor4 --,,forsyth1 ,a4 ,r4 ,proceedings ,london ,royal ,society1 ,vol4 #eh 7#ahie71 pp4 #aa-ab4 #icd4 ,bertrand1 ,darboux1 and ,glaisher have compared ,cayley to ,euler1 alike for his range1 his analytical power1 and1 not least1 for his prolific production of new views and fertile theories4 ,there is hardly a subject in the whole of pure mathematics at which he has not worked4--,,forsyth1 ,a4 ,r4 ,proceedings ,london ,royal ,society1 ,vol4 #eh 7#ahie71 p4 #ba4 #ice4 ,the mathematical talent of ,cayley was characterized by clearness and extreme elegance of analytical form2 it was re-enforced by an incomparable capacity for work which has caused the distinguished scholar to be compared with ,cauchy4 --,,hermite1 ,c4 ,comptes ,rendus1 t4 #abj 7#ahie71 p4 #bcd4 #icf4 ,j4 ,j4 ,sylvester was an enthusiastic supporter of reform in the teaching of geometry4 ,the difference in attitude on this question between the two foremost ,british mathematicians1 ,j4 ,j4 ,sylvester1 the algebraist1 and ,arthur ,cayley1 the algebraist and geometer1 was grotesque4 ,sylvester wished to bury ,euclid 8deeper than e'er plummet sounded0 out of the schoolboy's reach2 ,cayley1 an ardent admirer of ,euclid1 desired the retention of ,simson's ,euclid4 ,when reminded that this treatise was a mixture of ,euclid and ,simson1 ,cayley suggested striking out ,simson's additions and keeping strictly to the original treatise4 --,,cajori1 ,f4 ,history of ,elementary ,mathematics 7,new ,york1 #aiaj71 p4 #bhe4 #icg4 ,tait once urged the advantage of ,quaternions on ,cayley 7who never used them71 saying3 8,you know ,quaternions are just like a pocket-map40 8,that may be10 replied ,cayley1 8but you've got to take it out of your pocket1 and unfold it1 before it's of any use40 ,and he dismissed the subject with a smile4 --,,thompson1 ,s4 ,p4 ,life of ,lord ,kelvin 7,london1 #aiaj71 p4 #aacg4 #ich4 ,as he ,clifford spoke he appeared not to be working out a question1 but simply telling what he saw4 ,without any diagram or symbolic aid he described the geometrical conditions on which the solution depended1 and they seemed to stand out visibly in space4 ,there were no longer consequences to be deduced1 but real and evident facts which only required to be seen4444 ,so whole and complete was his vision that for the time the only strange thing was that anybody should fail to see it in the same way4 ,when one endeavored to call it up again1 and not till then1 it became clear that the magic of genius had been at work1 and that the common sight had been raised to that higher perception by the power that makes and transforms ideas1 the conquering and masterful quality of the human mind which ,goethe called in one word das ,dmonische4--,,pollock1 ,f4 ,clifford's ,lectures and ,essays 7,new ,york1 #aija71 ,vol4 #a1 ,introduction1 pp4 #e-f4 #ici4 ,much of his ,clifford's best work was actually spoken before it was written4 ,he gave most of his public lectures with no visible preparation beyond very short notes1 and the outline seemed to be filled in without effort or hesitation4 ,afterwards he would revise the lecture from a shorthand writer's report1 or sometimes write down from memory almost exactly what he had said4 ,it fell out now and then1 however1 that neither of these things was done2 in such cases there is now no record of the lecture at all4--,,pollock1 ,f4 ,clifford's ,lectures and ,essays 7,new ,york1 #aija71 ,vol4 #a1 ,introduction1 p4 #aj4 #idj4 ,i cannot find anything showing early aptitude for acquiring languages2 but that he ,clifford had it and was fond of exercising it in later life is certain4 ,one practical reason for it was the desire of being able to read mathematical papers in foreign journals2 but this would not account for his taking up ,spanish1 of which he acquired a competent knowledge in the course of a tour to the ,pyrenees4 ,when he was at ,algiers in #ahgf he began ,arabic1 and made progress enough to follow in a general way a course of lessons given in that language4 ,he read modern ,greek fluently1 and at one time he was furious about ,sanskrit4 ,he even spent some time on hieroglyphics4 ,a new language is a riddle before it is conquered1 a power in the hand afterwards3 to ,clifford every riddle was a challenge1 and every chance of new power a divine opportunity to be seized4 ,hence he was likewise interested in the various modes of conveying and expressing language invented for special purposes1 such as the ,morse alphabet and shorthand4444 ,i have forgotten to mention his command of ,french and ,german1 the former of which he knew very well1 and the latter quite sufficiently24444--,,pollock1 ,f4 ,clifford's ,lectures and ,essays 7,new ,york1 #aija71 ,vol4 #a1 ,introduction1 pp4 #aa-ab4 #ida4 ,the most remarkable thing was his ,clifford's great strength as compared with his weight1 as shown in some exercises4 ,at one time he could pull up on the bar with either hand1 which is well known to be one of the greatest feats of strength4 ,his nerve at dangerous heights was extraordinary4 ,i am appalled now to think that he climbed up and sat on the cross bars of the weathercock on a church tower1 and when by way of doing something worse ,i went up and hung by my toes to the bars he did the same4 ,quoted from a letter by one of ,clifford's friends to ,pollock1 ,f43 ,clifford's ,lectures and ,essays 7,new ,york1 #aija71 ,vol4 #a1 ,introduction1 p4 #h4 #idb4 ,comte may truly be said to have created the philosophy of higher mathematics4--,,mill1 ,j4 ,s4 ,system of ,logic 7,new ,york1 #ahdf71 p4 #cfi4 #idc4 ,these specimens1 which ,i could easily multiply1 may suffice to justify a profound distrust of ,auguste ,comte1 wherever he may venture to speak as a mathematician4 ,but his vast general ability1 and that personal intimacy with the great ,fourier1 which ,i most willingly take his own word for having enjoyed1 must always give an interest to his views on any subject of pure or applied mathematics4--,,hamilton1 ,w4 ,r4 ,graves' ,life of ,w4 ,r4 ,hamilton 7,new ,york1 #ahhb-ahhi71 ,vol4 #c1 p4 #dge4 #idd4 ,the manner of ,demoivre's death has a certain interest for psychologists4 ,shortly before it1 he declared that it was necessary for him to sleep some ten minutes or a quarter of an hour longer each day than the preceding one3 the day after he had thus reached a total of something over twenty-three hours he slept up to the limit of twenty-four hours1 and then died in his sleep4--,,ball1 ,w4 ,w4 ,r4 ,history of ,mathematics 7,london1 #aiaa71 p4 #cid4 #ide4 ,de ,morgan was explaining to an actuary what was the chance that a certain proportion of some group of people would at the end of a given time be alive2 and quoted the actuarial formula1 involving 1 which1 in answer to a question1 he explained stood for the ratio of the circumference of a circle to its diameter4 ,his acquaintance1 who had so far listened to the explanation with interest1 interrupted him and exclaimed1 8,my dear friend1 that must be a delusion1 what can a circle have to do with the number of people alive at a given time80 --,,ball1 ,w4 ,w4 ,r4 ,mathematical ,recreations and ,problems 7,london1 #ahif71 p4 #ahj2 ,see also ,de ,morgan's ,budget of ,paradoxes 7,london1 #ahgb71 p4 #agb4 #idf4 ,a few days afterwards1 ,i went to him the same actuary referred to in #ide and very gravely told him that ,i had discovered the law of human mortality in the ,carlisle ,table1 of which he thought very highly4 ,i told him that the law was involved in this circumstance4 ,take the table of the expectation of life1 choose any age1 take its expectation and make the nearest integer a new age1 do the same with that1 and so on2 begin at what age you like1 you are sure to end at the place where the age past is equal1 or most nearly equal1 to the expectation to come4 8,you don't mean that this always happens80--8,try it40 ,he did try1 again and again2 and found it as ,i said4 8,this is1 indeed1 a curious thing2 this is a discovery60 ,i might have sent him about trumpeting the law of life3 but ,i contented myself with informing him that the same thing would happen with any table whatsoever in which the first column goes up and the second goes down24444--,,de ,,morgan1 ,a4 ,budget of ,paradoxes 7,london1 #ahgb71 p4 #agb4 #idg4 ,de ,morgan relates that some person had made up #hjj anagrams on his name1 of which he had seen about #fej4 ,commenting on these he says3 ,two of these ,i have joined in the title-page3 ,ut agendo surgamus arguendo gustamus4 ,a few of the others are personal remarks4 ,great gun6 do us a sum6 is a sneer at my pursuit2 but1 _/ n u ,go6 great sum6 a du _/ is more dignified4444 ,adsum1 nugator1 suge6 is addressed to a student who continues talking after the lecture has commenced3 4444 ,graduatus sum6 nego applies to one who declined to subscribe for an ,m4 ,a4 degree4 --,,de ,,morgan1 ,,augustus4 ,budget of ,paradoxes 7,london1 #ahgb71 p4 #hb4 #idh4 ,descartes is the completest type which history presents of the purely mathematical type of mind--that in which the tendencies produced by mathematical cultivation reign unbalanced and supreme4--,,mill1 ,j4 ,s4 ,an ,examination of ,sir ,w4 ,hamilton's ,philosophy 7,london1 #ahgh71 p4 #fbf4 #idi4 ,to ,descartes1 the great philosopher of the #agth century1 is due the undying credit of having removed the bann which until then rested upon geometry4 ,the analytical geometry1 as ,descartes' method was called1 soon led to an abundance of new theorems and principles1 which far transcended everything that ever could have been reached upon the path pursued by the ancients4--,,hankel1 ,h4 ,die ,entwickelung der ,mathematik in den letzten ,jahrhunderten 7,tbingen1 #ahhd71 p4 #aj4 #iej4 ,the application of algebra has far more than any of his metaphysical speculations1 immortalized the name of ,descartes1 and constitutes the greatest single step ever made in the progress of the exact sciences4--,,mill1 ,j4 ,s4 ,an ,examination of ,sir ,w4 ,hamilton's ,philosophy 7,london1 #ahgh71 p4 #fag4 #iea4 444 1 4 444 they say that ,ptolemy once asked him 7,euclid7 whether there was in geometry no shorter way than that of the elements1 and he replied1 8,there is no royal road to geometry40--,,proclus4 7,edition ,friedlein1 #ahgc71 ,prol4 ,,ii1 #ci4 #ieb4 ,someone who had begun to read geometry with ,euclid1 when he had learned the first proposition1 asked ,euclid1 8,but what shall ,i get by learning these things80 whereupon ,euclid called his slave and said1 8,give him three-pence1 since he must make gain out of what he learns40--,,stob,,us4 7,edition ,wachsmuth1 #ahhd71 ,ecl4 ,,ii4 #iec4 ,the sacred writings excepted1 no ,greek has been so much read and so variously translated as ,euclid4#e--;,,de ,,morgan1 ,a4 ,smith's ,dictionary of ,greek and ,roman ,biology and ,mythology 7,london1 #aijb71 ,article1 8,eucleides40 #e ,riccardi's ,bibliografia ,euclidea 7,bologna1 #ahhg71 lists nearly two thousand editions4 #ied4 ,the thirteen books of ,euclid must have been a tremendous advance1 probably even greater than that contained in the 8,principia0 of ,newton4--,,de ,,morgan1 ,a4 ,smith's ,dictionary of ,greek and ,roman ,biography and ,mythology 7,london1 #aijb71 ,article1 8,eucleides40 #iee4 ,to suppose that so perfect a system as that of ,euclid's ,elements was produced by one man1 without any preceding model or materials1 would be to suppose that ,euclid was more than man4 ,we ascribe to him as much as the weakness of human understanding will permit1 if we suppose that the inventions in geometry1 which had been made in a tract of preceding ages1 were by him not only carried much further1 but digested into so admirable a system1 that his work obscured all that went before it1 and made them be forgot and lost4--,,reid1 ,,thomas4 ,essay on the ,powers of the ,human ,mind 7,edinburgh1 #ahab71 ,vol4 #b1 p4 #cfh4 #ief4 ,it is the invaluable merit of the great ,basle mathematician ,leonhard ,euler1 to have freed the analytical calculus from all geometrical bonds1 and thus to have established analysis as an independent science1 which from his time on has maintained an unchallenged leadership in the field of mathematics4--,,hankel1 ,h4 ,die ,entwickelung der ,mathematik in den letzten ,jahrhunderten 7,tbingen1 #ahhd71 p4 #ab4 #ieg4 ,we may safely say1 that the whole form of modern mathematical thinking was created by ,euler4 ,it is only with the greatest difficulty that one is able to follow the writings of any author immediately preceding ,euler1 because it was not yet known how to let the formulas speak for themselves4 ,this art ,euler was the first one to teach4--,,rudio1 ,f4 ,quoted by ,ahrens ,w43 ,scherz und ,ernst in der ,mathematik 7,leipzig1 #aijd71 p4 #bea4 #ieh4 ,the general knowledge of our author ,leonhard ,euler was more extensive than could well be expected1 in one who had pursued1 with such unremitting ardor1 mathematics and astronomy as his favorite studies4 ,he had made a very considerable progress in medical1 botanical1 and chemical science4 ,what was still more extraordinary1 he was an excellent scholar1 and possessed in a high degree what is generally called erudition4 ,he had attentively read the most eminent writers of ancient ,rome2 the civil and literary history of all ages and all nations was familiar to him2 and foreigners1 who were only acquainted with his works1 were astonished to find in the conversation of a man1 whose long life seemed solely occupied in mathematical and physical researches and discoveries1 such an extensive acquaintance with the most interesting branches of literature4 ,in this respect1 no doubt1 he was much indebted to an uncommon memory1 which seemed to retain every idea that was conveyed to it1 either from reading or from meditation4--,,hutton1 ,,charles4 ,philosophical and ,mathematical ,dictionary 7,london1 #ahae71 pp4 #dic-did4 #iei4 ,euler could repeat the ,aeneid from the beginning to the end1 and he could even tell the first and last lines in every page of the edition which he used4 ,in one of his works there is a learned memoir on a question in mechanics1 of which1 as he himself informs us1 a verse of ,aeneid#f gave him the first idea4 --,,brewster1 ,,david4 ,letters of ,euler 7,new ,york1 #ahgb71 ,vol4 #a1 p4 #bd4 #f ,the line referred to is3 8,the anchor drops1 the rushing keel is staid40 #ifj4 ,most of his ,euler's memoirs are contained in the transactions of the ,academy of ,sciences at ,st4 ,petersburg1 and in those of the ,academy at ,berlin4 ,from #agbh to #aghc a large portion of the ,petropolitan transactions were filled by his writings4 ,he had engaged to furnish the ,petersburg ,academy with memoirs in sufficient number to enrich its acts for twenty years--a promise more than fulfilled1 for down to #ahah ,euler died in #agic the volumes usually contained one or more papers of his4 ,it has been said that an edition of ,euler's complete works would fill #af1jjj quarto pages4--,,cajori1 ,f4 ,history of ,mathematics 7,new ,york1 #ahig71 pp4 #bec-bed4 #ifa4 ,euler who could have been called almost without metaphor1 and certainly without hyperbole1 analysis incarnate4--,,arago4 ,oeuvres1 t4 #b 7#ahed71 p4 #dcc4 #ifb4 ,euler calculated without any apparent effort1 just as men breathe1 as eagles sustain themselves in the air4--,,arago4 ,oeuvres1 t4 #b 7#ahed71 p4 #acc4 #ifc4 ,two of his ,euler's pupils having computed to the #agth term1 a complicated converging series1 their results differed one unit in the fiftieth cipher2 and an appeal being made to ,euler1 he went over the calculation in his mind1 and his decision was found correct4--,,brewster1 ,,david4 ,letters of ,euler 7,new ,york1 #ahgb71 ,vol4 #b1 p4 #bb4 #ifd4 ,in #agce the solving of an astronomical problem1 proposed by the ,academy1 for which several eminent mathematicians had demanded several months' time1 was achieved in three days by ,euler with aid of improved methods of his own4444 ,with still superior methods this same problem was solved by the illustrious ,gauss in one hour4--,,cajori1 ,f4 ,history of ,mathematics 7,new ,york1 #ahig71 p4 #bdh4 #ife4 ,euler's ,tentamen novae theorae musicae had no great success1 as it contained too much geometry for musicians1 and too much music for geometers4--,,fuss1 ,n4 ,quoted by ,brewster3 ,letters of ,euler 7,new ,york1 #ahgb71 ,vol4 #a1 p4 #bf4 #iff4 ,euler was a believer in ,god1 downright and straight-forward4 ,the following story is told by ,thiebault1 in his ,souvenirs de vingt ans de sjour ,berlin1 4444 ,thiebault says that he has no personal knowledge of the truth of the story1 but that it was believed throughout the whole of the north of ,europe4 ,diderot paid a visit to the ,russian ,court at the invitation of the ,empress4 ,he conversed very freely1 and gave the younger members of the ,court circle a good deal of lively atheism4 ,the ,empress was much amused1 but some of her counsellors suggested that it might be desirable to check these expositions of doctrine4 ,the ,empress did not like to put a direct muzzle on her guest's tongue1 so the following plot was contrived4 ,diderot was informed that a learned mathematician was in possession of an algebraical demonstration of the existence of ,god1 and would give it him before all the ,court1 if he desired to hear it4 ,diderot gladly consented3 though the name of the mathematician is not given1 it was ,euler4 ,he advanced toward ,diderot1 and said gravely1 and in a tone of perfect conviction3 a bn ,monsieur1 ------- x1 donc ,dieu existe2 repondez6 n ,diderot1 to whom algebra was ,hebrew1 was embarrassed and disconcerted2 while peals of laughter rose on all sides4 ,he asked permission to return to ,france at once1 which was granted4 --,,de ,,morgan1 ,a4 ,budget of ,paradoxes 7,london1 #ahgb71 p4 #bea4 #ifg4 ,fermat died with the belief that he had found a long-sought-for law of prime numbers in the formula #bbn #a a prime1 but he admitted that he was unable to prove it rigorously4 ,the law is not true1 as was pointed out by ,euler in the example #bbe #a #d1bid1ifg1big #f1gjj1dag times #fda4 ,the ,american lightning calculator ,zerah ,colburn1 when a boy1 readily found the factors but was unable to explain the method by which he made his marvellous mental computation4--,,cajori1 ,f4 ,history of ,mathematics 7,new ,york1 #ahig71 p4 #ahj4 #ifh4 ,i crave the liberty to conceal my name1 not to suppress it4 ,i have composed the letters of it written in ,latin in this sentence-- ,in ,mathesi a sole fundes4#g --,,flamsteed1 ,j4 ,macclesfield3 ,correspondence of ,scientific ,men 7,oxford1 #ahda71 ,vol4 #b1 p4 #ij4 #g ,johannes ,flamsteedius4 #ifi4 ,to the ,memory of ,fourier ,fourier6 with solemn and profound delight1 ,joy born of awe1 but kindling momently ,to an intense and thrilling ecstacy1 ,i gaze upon thy glory and grow bright3 ,as if irradiate with beholden light2 ,as if the immortal that remains of thee ,attuned me to thy spirit's harmony1 ,breathing serene resolve and tranquil might4 ,revealed appear thy silent thoughts of youth1 ,as if to consciousness1 and all that view ,prophetic1 of the heritage of truth ,to thy majestic years of manhood due3 ,darkness and error fleeing far away1 ,and the pure mind enthroned in perfect day4 --,,hamilton1 ,w4 ,r4 ,graves' ,life of ,w4 ,r4 ,hamilton1 7,new ,york1 #ahhb71 ,vol4 #a1 p4 #eif4 #igj4 ,astronomy and ,pure ,mathematics are the magnetic poles toward which the compass of my mind ever turns4--,,gauss ,,to ,,bolyai4 ,briefwechsel 7,schmidt-,stakel71 7#ahii71 p4 #ee4 #iga4 ,gauss calculated the elements of the planet ,ceres and his analysis proved him to be the first of theoretical astronomers no less than the greatest of 8arithmeticians40--,,ball1 ,w4 ,w4 ,r4 ,history of ,mathematics 7,london1 #aija71 p4 #deh4 #igb4 ,the mathematical giant ,gauss1 who from his lofty heights embraces in one view the stars and the abysses4444--,,bolyai1 ,w4 ,kurzer ,grundriss eines ,versuchs 7,maros ,vasarhely1 #ahea71 p4 #dd4 #igc4 ,almost everything1 which the mathematics of our century has brought forth in the way of original scientific ideas1 attaches to the name of ,gauss4--,,kronecker1 ,l4 ,zahlentheorie1 ,teil #a 7,leipzig1 #aija71 p4 #dc4 #igd4 ,i am giving this winter two courses of lectures to three students1 of which one is only moderately prepared1 the other less than moderately1 and the third lacks both preparation and ability4 ,such are the onera of a mathematical profession4 --,,gauss ,,to ,,bessel1 #ahaj4 ,gauss-,bessel ,briefwechsel 7#ahhj71 p4 #ajg4 #ige4 ,gauss once said 8,mathematics is the queen of the sciences and number-theory the queen of mathematics40 ,if this be true we may add that the ,disquisitiones is the ,magna ,charta of number-theory4 ,the advantage which science gained by ,gauss' long-lingering method of publication is this3 ,what he put into print is as true and important today as when first published2 his publications are statutes1 superior to other human statutes in this1 that nowhere and never has a single error been detected in them4 ,this justifies and makes intelligible the pride with which ,gauss said in the evening of his life of the first larger work of his youth3 8,the ,disquisitiones arithmeticae belong to history40 --,,cantor1 ,m4 ,allgemeine ,deutsche ,biographie1 ,bd4 #h 7#ahgh71 p4 #dce4 #igf4 ,here ,i am at the limit which ,god and nature has assigned to my individuality4 ,i am compelled to depend upon word1 language and image in the most precise sense1 and am wholly unable to operate in any manner whatever with symbols and numbers which are easily intelligible to the most highly gifted minds4--,,goethe4 ,letter to ,naumann 7#ahbf72 ,vogel3 ,goethe's ,selbstzeugnisse 7,leipzig1 #aijc71 p4 #ef4 #igg4 ,dirichlet was not satisfied to study ,gauss' 8,disquisitiones arithmeticae0 once or several times1 but continued throughout life to keep in close touch with the wealth of deep mathematical thoughts which it contains by perusing it again and again4 ,for this reason the book was never placed on the shelf but had an abiding place on the table at which he worked4444 ,dirichlet was the first one1 who not only fully understood this work1 but made it also accessible to others4 --,,kummer1 ,e4 ,e4 ,dirichlet3 ,werke1 ,bd4 #b1 p4 #cae4 #igh4 ,the famous attack of ,sir ,william ,hamilton on the tendency of mathematical studies affords the most express evidence of those fatal lacunae in the circle of his knowledge1 which unfitted him for taking a comprehensive or even an accurate view of the processes of the human mind in the establishment of truth4 ,if there is any pre-requisite which all must see to be indispensable in one who attempts to give laws to the human intellect1 it is a thorough acquaintance with the modes by which human intellect has proceeded1 in the case where1 by universal acknowledgment1 grounded on subsequent direct verification1 it has succeeded in ascertaining the greatest number of important and recondite truths4 ,this requisite ,sir ,w4 ,hamilton had not1 in any tolerable degree1 fulfilled4 ,even of pure mathematics he apparently knew little but the rudiments4 ,of mathematics as applied to investigating the laws of physical nature2 of the mode in which the properties of number1 extension1 and figure1 are made instrumental to the ascertainment of truths other than arithmetical or geometrical--it is too much to say that he had even a superficial knowledge3 there is not a line in his works which shows him to have had any knowledge at all4--,,mill1 ,j4 ,s4 ,examination of ,sir ,william ,hamilton's ,philosophy 7,london1 #ahgh71 p4 #fjg4 #igi4 ,helmholtz--the physiologist who learned physics for the sake of his physiology1 and mathematics for the sake of his physics1 and is now in the first rank of all three4 --,,clifford1 ,w4 ,k4 ,aims and ,instruments of ,scientific ,thought2 ,lectures and ,essays1 ,vol4 #a 7,london1 #aija71 p4 #afe4 #ihj4 ,it is said of ,jacobi1 that he attracted the particular attention and friendship of ,bckh1 the director of the philological seminary at ,berlin1 by the great talent he displayed for philology1 and only at the end of two years' study at the ,university1 and after a severe mental struggle1 was able to make his final choice in favor of mathematics4--,,sylvester1 ,j4 ,j4 ,collected ,mathematical ,papers1 ,vol4 #b 7,cambridge1 #aijh71 p4 #fea4 #iha4 ,when ,dr4 ,johnson felt1 or fancied he felt1 his fancy disordered1 his constant recurrence was to the study of arithmetic4--,,boswell1 ,j4 ,life of ,johnson 7,harper's ,edition1 #ahga71 ,vol4 #b1 p4 #bfd4 #ihb4 ,endowed with two qualities1 which seemed incompatible with each other1 a volcanic imagination and a pertinacity of intellect which the most tedious numerical calculations could not daunt1 ,kepler conjectured that the movements of the celestial bodies must be connected together by simple laws1 or1 to use his own expression1 by harmonic laws4 ,these laws he undertook to discover4 ,a thousand fruitless attempts1 errors of calculation inseparable from a colossal undertaking1 did not prevent him a single instant from advancing resolutely toward the goal of which he imagined he had obtained a glimpse4 ,twenty-two years were employed by him in this investigation1 and still he was not weary of it6 ,what1 in reality1 are twenty-two years of labor to him who is about to become the legislator of worlds2 who shall inscribe his name in ineffaceable characters upon the frontispiece of an immortal code2 who shall be able to exclaim in dithyrambic language1 and without incurring the reproach of anyone1 8,the die is cast2 ,i have written my book2 it will be read either in the present age or by posterity1 it matters not which2 it may well await a reader1 since ,god has waited six thousand years for an interpreter of his words40--,,arago4 ,eulogy on ,laplace3 ,baden ,powell ,smithsonian ,report1 #ahgd1 p4 #acb4 #ihc4 ,the great masters of modern analysis are ,lagrange1 ,laplace1 and ,gauss1 who were contemporaries4 ,it is interesting to note the marked contrast in their styles4 ,lagrange is perfect both in form and matter1 he is careful to explain his procedure1 and though his arguments are general they are easy to follow4 ,laplace on the other hand explains nothing1 is indifferent to style1 and1 if satisfied that his results are correct1 is content to leave them either with no proof or with a faulty one4 ,gauss is as exact and elegant as ,lagrange1 but even more difficult to follow than ,laplace1 for he removes every trace of the analysis by which he reached his results1 and studies to give a proof which while rigorous shall be as concise and synthetical as possible4--,,ball1 ,w4 ,w4 ,r4 ,history of ,mathematics 7,london1 #aija71 p4 #dfc4 #ihd4 ,lagrange1 in one of the later years of his life1 imagined that he had overcome the difficulty of the parallel axiom4 ,he went so far as to write a paper1 which he took with him to the ,institute1 and began to read it4 ,but in the first paragraph something struck him which he had not observed3 he muttered ,il faut que j'y songe encore1 and put the paper in his pocket4 --,,de ,,morgan1 ,a4 ,budget of ,paradoxes 7,london1 #ahgb71 p4 #agc4 #ihe4 ,i never come across one of ,laplace's 8,thus it plainly appears0 without feeling sure that ,i have hours of hard work before me to fill up the chasm and find out and show how it plainly appears4--,,bowditch1 ,n4 ,quoted by ,cajori3 ,teaching and ,history of ,mathematics in the ,u4 ,s4 7,washington1 #ahif71 p4 #ajd4 #ihf4 ,biot1 who assisted ,laplace in revising it ,the ,mecnique ,cleste for the press1 says that ,laplace himself was frequently unable to recover the details in the chain of reasoning1 and if satisfied that the conclusions were correct1 he was content to insert the constantly recurring formula1 8,il est is a voir40 --,,ball1 ,w4 ,w4 ,r4 ,history of ,mathematics 7,london1 #aija71 p #dbg4 #ihg4 ,it would be difficult to name a man more remarkable for the greatness and the universality of his intellectual powers than ,leibnitz4--,,mill1 ,j4 ,s4 ,system of ,logic1 ,bk4 #b1 chap4 #e1 sect4 #f4 #ihh4 ,the influence of his ,leibnitz's genius in forming that peculiar taste both in pure and in mixed mathematics which has prevailed in ,france1 as well as in ,germany1 for a century past1 will be found1 upon examination1 to have been incomparably greater than that of any other individual4--,,stewart1 ,,dugald4 ,philosophy of the ,human ,mind1 ,part #c1 chap4 #a1 sect4 #c4 #ihi4 ,leibnitz's discoveries lay in the direction in which all modern progress in science lies1 in establishing order1 symmetry1 and harmony1 i4e41 comprehensiveness and perspicuity1--rather than in dealing with single problems1 in the solution of which followers soon attained greater dexterity than himself4 --,,merz1 ,j4 ,t4 ,leibnitz1 ,chap4 #f4 #iij4 ,it was his ,leibnitz's love of method and order1 and the conviction that such order and harmony existed in the real world1 and that our success in understanding it depended upon the degree and order which we could attain in our own thoughts1 that originally was probably nothing more than a habit which by degrees grew into a formal rule4#h ,this habit was acquired by early occupation with legal and mathematical questions4 ,we have seen how the theory of combinations and arrangements of elements had a special interest for him4 ,we also saw how mathematical calculations served him as a type and model of clear and orderly reasoning1 and how he tried to introduce method and system into logical discussions1 by reducing to a small number of terms the multitude of compound notions he had to deal with4 ,this tendency increased in strength1 and even in those early years he elaborated the idea of a general arithmetic1 with a universal language of symbols1 or a characteristic which would be applicable to all reasoning processes1 and reduce philosophical investigations to that simplicity and certainty which the use of algebraic symbols had introduced into mathematics4 #h ,this sentence has been reworded for the purpose of this quotation4 ,a mental attitude such as this is always highly favorable for mathematical as well as for philosophical investigations4 ,wherever progress depends upon precision and clearness of thought1 and wherever such can be gained by reducing a variety of investigations to a general method1 by bringing a multitude of notions under a common term or symbol1 it proves inestimable4 ,it necessarily imports the special qualities of number--viz41 their continuity1 infinity and infinite divisibility--like mathematical quantities--and destroys the notion that irreconcilable contrasts exist in nature1 or gaps which cannot be bridged over4 ,thus1 in his letter to ,arnaud1 ,leibnitz expresses it as his opinion that geometry1 or the philosophy of space1 forms a step to the philosophy of motion--i4e41 of corporeal things--and the philosophy of motion a step to the philosophy of mind4 --,,merz1 ,j4 ,t4 ,leibnitz 7,philadelphia71 pp4 #dd-de4 #iia4 ,leibnitz believed he saw the image of creation in his binary arithmetic in which he employed only two characters1 unity and zero4 ,since ,god may be represented by unity1 and nothing by zero1 he imagined that the ,supreme ,being might have drawn all things from nothing1 just as in the binary arithmetic all numbers are expressed by unity with zero4 ,this idea was so pleasing to ,leibnitz1 that he communicated it to the ,jesuit ,grimaldi1 ,president of the ,mathematical ,board of ,china1 with the hope that this emblem of the creation might convert to ,christianity the reigning emperor who was particularly attached to the sciences4 --,,laplace4 ,essai ,philosophique sur les ,probabilits2 ,oeuvres 7,paris1 #ahif71 t4 #g1 p4 #aai4 #iib4 ,sophus ,lie1 great comparative anatomist of geometric theories4--,,keyser1 ,c4 ,j4 ,lectures on ,science1 ,philosophy and ,art 7,new ,york1 #aijh71 p4 #ca4 #iic4 ,it has been the final aim of ,lie from the beginning to make progress in the theory of differential equations2 as subsidiary to this may be regarded both his geometrical developments and the theory of continuous groups4--,,klein1 ,f4 ,lectures on ,mathematics 7,new ,york1 #aiaa71 p4 #bd4 #iid4 ,to fully understand the mathematical genius of ,sophus ,lie1 one must not turn to books recently published by him in collaboration with ,dr4 ,engel1 but to his earlier memoirs1 written during the first years of his scientific career4 ,there ,lie shows himself the true geometer that he is1 while in his later publications1 finding that he was but imperfectly understood by the mathematicians accustomed to the analytic point of view1 he adopted a very general analytic form of treatment that is not always easy to follow4--,,klein1 ,f4 ,lectures on ,mathematics 7,new ,york1 #aiaa71 p4 #i4 #iie4 ,it is said that the composing of the ,lilawati was occasioned by the following circumstance4 ,lilawati was the name of the author's ,bhascara daughter1 concerning whom it appeared1 from the qualities of the ascendant at her birth1 that she was destined to pass her life unmarried1 and to remain without children4 ,the father ascertained a lucky hour for contracting her in marriage1 that she might be firmly connected and have children4 ,it is said that when that hour approached1 he brought his daughter and his intended son near him4 ,he left the hour cup on the vessel of water and kept in attendance a time-knowing astrologer1 in order that when the cup should subside in the water1 those two precious jewels should be united4 ,but1 as the intended arrangement was not according to destiny1 it happened that the girl1 from a curiosity natural to children1 looked into the cup1 to observe the water coming in at the hole1 when by chance a pearl separated from her bridal dress1 fell into the cup1 and1 rolling down to the hole1 stopped the influx of water4 ,so the astrologer waited in expectation of the promised hour4 ,when the operation of the cup had thus been delayed beyond all moderate time1 the father was in consternation1 and examining1 he found that a small pearl had stopped the course of the water1 and that the long-expected hour was passed4 ,in short1 the father1 thus disappointed1 said to his unfortunate daughter1 ,i will write a book of your name1 which shall remain to the latest times--for a good name is a second life1 and the ground-work of eternal existence4--,,fizi4 ,preface to the ,lilawati4 ,quoted by ,a4 ,hutton3 ,a ,philosophical and ,mathematical ,dictionary1 ,article 8,algebra0 7,london1 #ahae74 #iif4 ,is there anyone whose name cannot be twisted into either praise or satire8 ,i have had given to me1 ,thomas ,babington ,macaulay ,mouths big3 a ,cantab anomaly4 --,,de ,,morgan1 ,a4 ,budget of ,paradoxes 7,london1 #ahgb71 p4 #hc4 ,,chapter ,x ,,persons ,,and ,,anecdotes 7,n-,z7 #ajja4 ,when he had a few moments for diversion1 he ,napoleon not unfrequently employed them over a book of logarithms1 in which he always found recreation4--,,abbott1 ,j4 ,s4 ,c4 ,napoleon ,bonaparte 7,new ,york1 #aijd71 ,vol4 #a1 chap4 #aj4 #ajjb4 ,the name of ,sir ,isaac ,newton has by general consent been placed at the head of those great men who have been the ornaments of their species4444 ,the philosopher ,laplace1 indeed1 to whom posterity will probably assign a place next to ,newton1 has characterized the ,principia as pre-eminent above all the productions of human intellect4--,,brewster1 ,d4 ,life of ,sir ,isaac ,newton 7,london1 #ahca71 pp4 #a1 #b4 #ajjc4 ,newton and ,laplace need myriads of ages and thick-strewn celestial areas4 ,one may say a gravitating solar system is already prophesied in the nature of ,newton's mind4--,,emerson4 ,essay on ,history4 #ajjd4 ,the law of gravitation is indisputably and incomparably the greatest scientific discovery ever made1 whether we look at the advance which it involved1 the extent of truth disclosed1 or the fundamental and satisfactory nature of this truth4 --,,whewell1 ,w4 ,history of the ,inductive ,sciences1 ,bk4 #g1 chap4 #b1 sect4 #e4 #ajje4 ,newton's theory is the circle of generalization which includes all the others as ,kepler's laws1 ,ptolemy's theory1 etc42--the highest point of the inductive ascent2--the catastrophe of the philosophic drama to which ,plato had prologized2--the point to which men's minds had been journeying for two thousand years4--,,whewell1 ,w4 ,history of the ,inductive ,sciences1 ,bk4 #g1 chap4 #b1 sect4 #e4 #ajjf4 ,the efforts of the great philosopher ,newton were always superhuman2 the questions which he did not solve were incapable of solution in his time4--,,arago4 ,eulogy on ,laplace1 ,baden ,powell ,smithsonian ,report1 #ahgd1 p4 #acc4 #ajjg4 ,nature and ,nature's laws lay hid in night3 ,god said1 8,let ,newton be60 and all was light4 --,,pope1 ,a4 ,epitaph intended for ,sir ,isaac ,newton4 #ajjh4 ,there ,priest of ,nature6 dost thou shine1 ,newton6 a ,king among the ,kings divine4 --,,southey4 ,translation of a ,greek ,ode on ,astronomy4 #ajji4 ,o'er ,nature's laws ,god cast the veil of night1 ,out-blaz'd a ,newton's soul--and all was light4 --,,hill1 ,,aaron4 ,on ,sir ,isaac ,newton4 #ajaj4 ,taking mathematics from the beginning of the world to the time when ,newton lived1 what he had done was much the better half4--,,leibnitz4 ,quoted by ,f4 ,r4 ,moulton3 ,introduction to ,astronomy 7,new ,york1 #aijf71 p4 #aii4 #ajaa4 ,newton was the greatest genius that ever existed1 and the most fortunate1 for we cannot find more than once a system of the world to establish4--,,lagrange4 ,quoted by ,f4 ,r4 ,moulton3 ,introduction to ,astronomy 7,new ,york1 #aijf71 p4 #aii4 #ajab4 ,a monument to ,newton6 a monument to ,shakespeare6 ,look up to ,heaven--look into the ,human ,heart4 ,till the planets and the passions--the affections and the fixed stars are extinguished--their names cannot die4--,,wilson1 ,,john4 ,noctes ,ambrosianae4 #ajac4 ,such men as ,newton and ,linnaeus are incidental1 but august1 teachers of religion4--,,wilson1 ,,john4 ,essays3 ,education of the ,people4 #ajad4 ,sir ,isaac ,newton1 the supreme representative of ,anglo-,saxon genius4--,,ellis1 ,,havelock4 ,study of ,british ,genius 7,london1 #aijd71 p4 #di4 #ajae4 ,throughout his life ,newton must have devoted at least as much attention to chemistry and theology as to mathematics4444 --,,ball1 ,w4 ,w4 ,r4 ,history of ,mathematics 7,london1 #aija71 p4 #cce4 #ajaf4 ,there was a time when he ,newton was possessed with the old fooleries of astrology2 and another when he was so far gone in those of chemistry1 as to be upon the hunt after the philosopher's stone4--,,rev4 ,j4 ,,spence4 ,anecdotes1 ,observations1 and ,characters of ,books and ,men 7,london1 #ahfh71 p4 #ed4 #ajag4 ,for several years this great man ,newton was intensely occupied in endeavoring to discover a way of changing the base metals into gold4444 ,there were periods when his furnace fires were not allowed to go out for six weeks2 he and his secretary sitting up alternate nights to replenish them4--,,parton1 ,,james4 ,sir ,isaac ,newton4 #ajah4 ,on the day of ,cromwell's death1 when ,newton was sixteen1 a great storm raged all over ,england4 ,he used to say1 in his old age1 that on that day he made his first purely scientific experiment4 ,to ascertain the force of the wind1 he first jumped with the wind and then against it2 and1 by comparing these distances with the extent of his own jump on a calm day1 he was enabled to compute the force of the storm4 ,when the wind blew thereafter1 he used to say it was so many feet strong4 --,,parton1 ,,james4 ,sir ,isaac ,newton4 #ajai4 ,newton lectured now and then to the few students who chose to hear him2 and it is recorded that very frequently he came to the lecture-room and found it empty4 ,on such occasions he would remain fifteen minutes1 and then1 if no one came1 return to his apartments4--,,parton1 ,,james4 ,sir ,isaac ,newton4 #ajbj4 ,sir ,isaac ,newton1 though so deep in algebra and fluxions1 could not readily make up a common account3 and1 when he was ,master of the ,mint1 used to get somebody else to make up his accounts for him4--,,rev4 ,j4 ,,spence4 ,anecdotes1 ,observations1 and ,characters of ,books and ,men 7,london1 #aheh71 p4 #acb4 #ajba4 ,we have one of his ,newton's college memorandum-books1 which is highly interesting4 ,the following are some of the entries3 8,drills1 gravers1 a hone1 a hammer1 and a mandril1 #es420 8a magnet1 #afs420 8compasses1 #bs420 8glass bubbles1 #ds420 8at the tavern several other times1 #a20 8spent on my cousin1 #abs420 8on other acquaintances1 #ajs420 8,philosophical ,intelligences1 #is4 #f;d420 8lost at cards twice1 #aes420 8at the tavern twice1 #cs4 #f;d420 8to three prisms1 #c20 8four ounces of putty1 #as4 #d;d420 8,bacon's ,miscellanies1 #as4 #f;d420 8a bible binding1 #cs420 8for oranges to my sister1 #ds4 #b;d420 8for aquafortis1 sublimate1 oyle pink1 fine silver1 antimony1 vinegar1 spirit of wine1 white lead1 salt of tartar1 #b20 8,theatrum chemicum1 #a #hs40--,,parton1 ,,james4 ,sir ,isaac ,newton4 #ajbb4 ,on one occasion1 when he was giving a dinner to some friends at the university1 he left the table to get them a bottle of wine2 but1 on his way to the cellar1 he fell into reflection1 forgot his errand and his company1 went to his chamber1 put on his surplice1 and proceeded to the chapel4 ,sometimes he would go into the street half dressed1 and on discovering his condition1 run back in great haste1 much abashed4 ,often1 while strolling in his garden1 he would suddenly stop1 and then run rapidly to his room1 and begin to write1 standing1 on the first piece of paper that presented itself4 ,intending to dine in the public hall1 he would go out in a brown study1 take the wrong turn1 walk a while1 and then return to his room1 having totally forgotten the dinner4 ,once having dismounted from his horse to lead him up a hill1 the horse slipped his head out of the bridle2 but ,newton1 oblivious1 never discovered it till1 on reaching a tollgate at the top of the hill1 he turned to remount and perceived that the bridle which he held in his hand had no horse attached to it4 ,his secretary records that his forgetfulness of his dinner was an excellent thing for his old housekeeper1 who 8sometimes found both dinner and supper scarcely tasted of1 which the old woman has very pleasantly and mumpingly gone away with40 ,on getting out of bed in the morning1 he has been discovered to sit on his bedside for hours without dressing himself1 utterly absorbed in thought4--,,parton1 ,,james4 ,sir ,isaac ,newton4 #ajbc4 ,i don't know what ,i may seem to the world1 but1 as to myself1 ,i seem to have been only as a boy playing on the seashore1 and diverting myself in now and then finding a smoother pebble or a prettier shell than ordinary1 whilst the great ocean of truth lay all undiscovered before me4--,,newton1 ,i4 ,quoted by ,rev4 ,j4 ,spence3 ,anecdotes1 ,observations1 and ,characters of ,books and ,men 7,london1 #aheh71 p4 #dj4 #ajbd4 ,if ,i have seen farther than ,descartes1 it is by standing on the shoulders of giants4--,,newton1 ,i4 ,quoted by ,james ,parton3 ,sir ,isaac ,newton4 #ajbe4 ,newton could not admit that there was any difference between him and other men1 except in the possession of such habits as 444 perseverance and vigilance4 ,when he was asked how he made his discoveries1 he answered1 8by always thinking about them20 and at another time he declared that if he had done anything1 it was due to nothing but industry and patient thought3 8,i keep the subject of my inquiry constantly before me1 and wait till the first dawning opens gradually1 by little and little1 into a full and clear light40--,,whewell1 ,w4 ,history of the ,inductive ,sciences1 ,bk4 #g1 chap4 #b1 sect4 #e4 #ajbf4 ,newton took no exercise1 indulged in no amusements1 and worked incessantly1 often spending eighteen or nineteen hours out of the twenty-four in writing4--,,ball1 ,w4 ,w4 ,r4 ,history of ,mathematics 7,london1 #aija71 p4 #ceh4 #ajbg4 ,foreshadowings of the principles and even of the language of the infinitesimal calculus can be found in the writings of ,napier1 ,kepler1 ,cavalieri1 ,pascal1 ,fermat1 ,wallis1 and ,barrow4 ,it was ,newton's good luck to come at a time when everything was ripe for the discovery1 and his ability enabled him to construct almost at once a complete calculus4--,,ball1 ,w4 ,w4 ,r4 ,history of ,mathematics 7,london1 #aija71 p4 #cef4 #ajbh4 ,kepler's suggestion of gravitation with the inverse distance1 and ,bouillaud's proposed substitution of the inverse square of the distance1 are things which ,newton knew better than his modern readers4 ,i have discovered two anagrams on his name1 which are quite conclusive3 the notion of gravitation was not new2 but ,newton went on4--,,de ,,morgan1 ,a4 ,budget of ,paradoxes 7,london1 #ahgb71 p4 #hb4 #ajbi4 ,for other great mathematicians or philosophers1 he ,gauss used the epithets magnus1 or clarus1 or clarissimus2 for ,newton alone he kept the prefix summus4--,,ball1 ,w4 ,w4 ,r4 ,history of ,mathematics 7,london1 #aija71 p4 #cfb4 #ajcj4 ,to know him ,sylvester was to know one of the historic figures of all time1 one of the immortals2 and when he was really moved to speak1 his eloquence equalled his genius4--,,halsted1 ,g4 ,b4 ,f4 ,cajori's ,teaching and ,history of ,mathematics in the ,u4 ,s4 7,washington1 #ahij71 p4 #bfe4 #ajca4 ,professor ,sylvester's first high class at the new university ,johns ,hopkins consisted of only one student1 ,g4 ,b4 ,halsted1 who had persisted in urging ,sylvester to lecture on the modern algebra4 ,the attempt to lecture on this subject led him into new investigations in quantics4--,,cajori1 ,f4 ,teaching and ,history of ,mathematics in the ,u4 ,s4 7,washington1 #ahij71 p4 #bfd4 #ajcb4 ,but for the persistence of a student of this university in urging upon me his desire to study with me the modern algebra ,i should never have been led into this investigation2 and the new facts and principles which ,i have discovered in regard to it 7important facts1 ,i believe71 would1 so far as ,i am concerned1 have remained still hidden in the womb of time4 ,in vain ,i represented to this inquisitive student that he would do better to take up some other subject lying less off the beaten track of study1 such as the higher parts of the calculus or elliptic functions1 or the theory of substitutions1 or ,i wot not what besides4 ,he stuck with perfect respectfulness1 but with invincible pertinacity1 to his point4 ,he would have the new algebra 7,heaven knows where he had heard about it1 for it is almost unknown in this continent71 that or nothing4 ,i was obliged to yield1 and what was the consequence8 ,in trying to throw light upon an obscure explanation in our text-book1 my brain took fire1 ,i plunged with re-quickened zeal into a subject which ,i had for years abandoned1 and found food for thoughts which have engaged my attention for a considerable time past1 and will probably occupy all my powers of contemplation advantageously for several months to come4--,,sylvester1 ,j4 ,j4 ,johns ,hopkins ,commemoration ,day ,address2 ,collected ,mathematical ,papers1 ,vol4 #c1 p4 #gf4 #ajcc4 ,sylvester was incapable of reading mathematics in a purely receptive way4 ,apparently a subject either fired in his brain a train of active and restless thought1 or it would not retain his attention at all4 ,to a man of such a temperament1 it would have been peculiarly helpful to live in an atmosphere in which his human associations would have supplied the stimulus which he could not find in mere reading4 ,the great modern work in the theory of functions and in allied disciplines1 he never became acquainted with4444 ,what would have been the effect if1 in the prime of his powers1 he had been surrounded by the influences which prevail in ,berlin or in ,gttingen8 ,it may be confidently taken for granted that he would have done splendid work in those domains of analysis1 which have furnished the laurels of the great mathematicians of ,germany and ,france in the second half of the present century4--,,franklin1 ,f4 ,johns ,hopkins ,university ,circulars #af 7#ahig71 p4 #ed4 #ajcd4 ,if we survey the mathematical works of ,sylvester1 we recognize indeed a considerable abundance1 but in contradistinction to ,cayley--not a versatility toward separate fields1 but1 with few exceptions--a confinement to arithmetic-algebraic branches4444 ,the concept of ,function of a continuous variable1 the fundamental concept of modern mathematics1 plays no role1 is indeed scarcely mentioned in the entire work of ,sylvester--,sylvester was combinatorist combinatoriker4--,,noether1 ,m4 ,mathematische ,annalen1 ,bd4 #ej 7#ahih71 pp4 #acd-ace4 #ajce4 ,sylvester's methods6 ,he had none4 8,three lectures will be delivered on a ,new ,universal ,algebra10 he would say2 then1 8,the course must be extended to twelve40 ,it did last all the rest of that year4 ,the following year the course was to be ,substitutions-,theorie1 by ,netto4 ,we all got the text4 ,he lectured about three times1 following the text closely and stopping sharp at the end of the hour4 ,then he began to think about matrices again4 8,i must give one lecture a week on those10 he said4 ,he could not confine himself to the hour1 nor to the one lecture a week4 ,two weeks were passed1 and ,netto was forgotten entirely and never mentioned again4 ,statements like the following were not unfrequent in his lectures3 8,i haven't proved this1 but ,i am as sure as ,i can be of anything that it must be so4 ,from this it will follow1 etc40 ,at the next lecture it turned out that what he was so sure of was false4 ,never mind1 he kept on forever guessing and trying1 and presently a wonderful discovery followed1 then another and another4 ,afterward he would go back and work it all over again1 and surprise us with all sorts of side lights4 ,he then made another leap in the dark1 more treasures were discovered1 and so on forever4--,,davis1 ,e4 ,w4 ,cajori's ,teaching and ,history of ,mathematics in the ,u4,s4 7,washington1 #ahij71 pp4 #bfe-bff4 #ajcf4 ,i can see him ,sylvester now1 with his white beard and few locks of gray hair1 his forehead wrinkled o'er with thoughts1 writing rapidly his figures and formulae on the board1 sometimes explaining as he wrote1 while we1 his listeners1 caught the reflected sounds from the board4 ,but stop1 something is not right1 he pauses1 his hand goes to his forehead to help his thought1 he goes over the work again1 emphasizes the leading points1 and finally discovers his difficulty4 ,perhaps it is some error in his figures1 perhaps an oversight in the reasoning4 ,sometimes1 however1 the difficulty is not elucidated1 and then there is not much to the rest of the lecture4 ,but at the next lecture we would hear of some new discovery that was the outcome of that difficulty1 and of some article for the ,journal1 which he had begun4 ,if a text-book had been taken up at the beginning1 with the intention of following it1 that text-book was most likely doomed to oblivion for the rest of the term1 or until the class had been made listeners to every new thought and principle that had sprung from the laboratory of his mind1 in consequence of that first difficulty4 ,other difficulties would soon appear1 so that no text-book could last more than half of the term4 ,in this way his class listened to almost all of the work that subsequently appeared in the ,journal4 ,it seemed to be the quality of his mind that he must adhere to one subject4 ,he would think about it1 talk about it to his class1 and finally write about it for the ,journal4 ,the merest accident might start him1 but once started1 every moment1 every thought was given to it1 and1 as much as possible1 he read what others had done in the same direction2 but this last seemed to be his real point2 he could not read without finding difficulties in the way of understanding the author4 ,thus1 often his own work reproduced what had been done by others1 and he did not find it out until too late4 ,a notable example of this is in his theory of cyclotomic functions1 which he had reproduced in several foreign journals1 only to find that he had been greatly anticipated by foreign authors4 ,it was manifest1 one of the critics said1 that the learned professor had not read ,kummer's elementary results in the theory of ideal primes4 ,yet ,professor ,smith's report on the theory of numbers1 which contained a full synopsis of ,kummer's theory1 was ,professor ,sylvester's constant companion4 ,this weakness of ,professor ,sylvester1 in not being able to read what others had done1 is perhaps a concomitant of his peculiar genius4 ,other minds could pass over little difficulties and not be troubled by them1 and so go on to a final understanding of the results of the author4 ,but not so with him4 ,a difficulty1 however small1 worried him1 and he was sure to have difficulties until the subject had been worked over in his own way1 to correspond with his own mode of thought4 ,to read the work of others1 meant therefore to him an almost independent development of it4 ,like the man whose pleasure in life is to pioneer the way for society into the forests1 his rugged mind could derive satisfaction only in hewing out its own paths2 and only when his efforts brought him into the uncleared fields of mathematics did he find his place in the ,universe4--,,hathaway1 ,a4 ,s4 ,f4 ,cajori's ,teaching and ,history of ,mathematics in the ,u4 ,s4 7,washington1 #ahij71 pp4 #bff-bfg4 #ajcg4 ,professor ,cayley has since informed me that the theorem about whose origin ,i was in doubt1 will be found in ,schlfli's 8,de ,eliminatione40 ,this is not the first unconscious plagiarism ,i have been guilty of towards this eminent man whose friendship ,i am proud to claim4 ,a more glaring case occurs in a note by me in the 8,comptes ,rendus10 on the twenty-seven straight lines of cubic surfaces1 where ,i believe ,i have followed 7like one walking in his sleep71 down to the very nomenclature and notation1 the substance of a portion of a paper inserted by ,schlfli in the 8,mathematical ,journal10 which bears my name as one of the editors upon the face4--,,sylvester1 ,j4 ,j4 ,philosophical ,transactions of the ,royal ,society 7#ahfd71 p4 #fdb4 #ajch4 ,he ,sylvester had one remarkable peculiarity4 ,he seldom remembered theorems1 propositions1 etc41 but had always to deduce them when he wished to use them4 ,in this he was the very antithesis of ,cayley1 who was thoroughly conversant with everything that had been done in every branch of mathematics4 ,i remember once submitting to ,sylvester some investigations that ,i had been engaged on1 and he immediately denied my first statement1 saying that such a proposition had never been heard of1 let alone proved4 ,to his astonishment1 ,i showed him a paper of his own in which he had proved the proposition2 in fact1 ,i believe the object of his paper had been the very proof which was so strange to him4--,,durfee1 ,w4 ,p4 ,f4 ,cajori's ,teaching and ,history of ,mathematics in the ,u4 ,s4 7,washington1 #ahij71 p4 #bfh4 #ajci4 ,a short1 broad man of tremendous vitality1 the physical type of ,hereward1 the last of the ,english1 and his brother-in-arms1 ,winter1 ,sylvester's capacious head was ever lost in the highest cloud-lands of pure mathematics4 ,often in the dead of night he would get his favorite pupil1 that he might communicate the very last product of his creative thought4 ,everything he saw suggested to him something new in the higher algebra4 ,this transmutation of everything into new mathematics was a revelation to those who knew him intimately4 ,they began to do it themselves4 ,his ease and fertility of invention proved a constant encouragement1 while his contempt for provincial stupidities1 such as the ,american hieroglyphics for and e1 which have even found their way into ,webster's ,dictionary1 made each young worker apply to himself the strictest tests4--,,halsted1 ,g4 ,b4 ,f4 ,cajori's ,teaching and ,history of ,mathematics in the ,u4 ,s4 7,washington1 #ahij71 p4 #bfe4 #ajdj4 ,sylvester's writings are flowery and eloquent4 ,he was able to make the dullest subject bright1 fresh and interesting4 ,his enthusiasm is evident in every line4 ,he would get quite close up to his subject1 so that everything else looked small in comparison1 and for the time would think and make others think that the world contained no finer matter for contemplation4 ,his handwriting was bad1 and a trouble to his printers4 ,his papers were finished with difficulty4 ,no sooner was the manuscript in the editor's hands than alterations1 corrections1 ameliorations and generalizations would suggest themselves to his mind1 and every post would carry further directions to the editors and printers4--,,macmahon4 ,p4 ,a4 ,nature1 ,vol4 #ee 7#ahig71 p4 #did4 #ajda4 ,the enthusiasm of ,sylvester for his own work1 which manifests itself here as always1 indicates one of his characteristic qualities3 a high degree of subjectivity in his productions and publications4 ,sylvester was so fully possessed by the matter which for the time being engaged his attention1 that it appeared to him and was designated by him as the summit of all that is important1 remarkable and full of future promise4 ,it would excite his phantasy and power of imagination in even a greater measure than his power of reflection1 so much so that he could never marshal the ability to master his subject-matter1 much less to present it in an orderly manner4 ,considering that he was also somewhat of a poet1 it will be easier to overlook the poetic flights which pervade his writing1 often bombastic1 sometimes furnishing apt illustrations2 more damaging is the complete lack of form and orderliness of his publications and their sketchlike character1 444 which must be accredited at least as much to lack of objectivity as to a superfluity of ideas4 ,again1 the text is permeated with associated emotional expressions1 bizarre utterances and paradoxes and is everywhere accompanied by notes1 which constitute an essential part of ,sylvester's method of presentation1 embodying relations1 whether proximate or remote1 which momentarily suggested themselves4 ,these notes1 full of inspiration and occasional flashes of genius1 are the more stimulating owing to their incompleteness4 ,but none of his works manifest a desire to penetrate the subject from all sides and to allow it to mature2 each mere surmise1 conceptions which arose during publication1 immature thoughts and even errors were ushered into publicity at the moment of their inception1 with utmost carelessness1 and always with complete unfamiliarity of the literature of the subject4 ,nowhere is there the least trace of self-criticism4 ,no one can be expected to read the treatises entire1 for in the form in which they are available they fail to give a clear view of the matter under contemplation4 ,sylvester's was not a harmoniously gifted or well-balanced mind1 but rather an instinctively active and creative mind1 free from egotism4 ,his reasoning moved in generalizations1 was frequently influenced by analysis and at times was guided even by mystical numerical relations4 ,his reasoning consists less frequently of pure intelligible conclusions than of inductions1 or rather conjectures incited by individual observations and verifications4 ,in this he was guided by an algebraic sense1 developed through long occupation with processes of forms1 and this led him luckily to general fundamental truths which in some instances remain veiled4 ,his lack of system is here offset by the advantage of freedom from purely mechanical logical activity4 ,the exponents of his essential characteristics are an intuitive talent and a faculty of invention to which we owe a series of ideas of lasting value and bearing the germs of fruitful methods4 ,to no one more fittingly than to ,sylvester can be applied one of the mottos of the ,philosophic ,magazine3 8,admiratio generat quaestionem1 quaestio investigationem investigatio inventionem40--,,noether1 ,m4 ,mathematische ,annalen1 ,bd4 #ej 7#ahih71 pp4 #aee-afj4 #ajdb4 ,perhaps ,i may without immodesty lay claim to the appellation of ,mathematical ,adam1 as ,i believe that ,i have given more names 7passed into general circulation7 of the creatures of the mathematical reason than all the other mathematicians of the age combined4--,,sylvester1 ,j4 ,j4 ,nature1 ,vol4 #cg 7#ahhg-ahhh71 p4 #afb4 #ajdc4 ,tait dubbed ,maxwell dp_/dt1 for according to thermodynamics dp_/dt ,,jcm 7where ,c denotes ,carnot's function7 the initials of 7,j4 ,c47 ,maxwell's name4 ,on the other hand ,maxwell denoted ,thomson by ,t and ,tait by ,t2 so that it became customary to quote ,thomson and ,tait's ,treatise on ,natural ,philosophy as ,t and ,t4--,,macfarlane1 ,a4 ,bibliotheca ,mathematica1 ,bd4 #c 7#aijc71 p4 #ahi4 #ajdd4 ,in future times ,tait will be best known for his work in the quaternion analysis4 ,had it not been for his expositions1 developments and applications1 ,hamilton's invention would be today1 in all probability1 a mathematical curiosity4 --,,macfarlane1 ,a4 ,bibliotheca ,mathematica1 ,bd4 #c 7#aijc71 p4 #ahi4 #ajde4 ,not seldom did he ,sir ,william ,thomson1 in his writings1 set down some mathematical statement with the prefacing remark 8it is obvious that0 to the perplexity of mathematical readers1 to whom the statement was anything but obvious from such mathematics as preceded it on the page4 ,to him it was obvious for physical reasons that might not suggest themselves at all to the mathematician1 however competent4--,,thompson1 ,s4 ,p4 ,life of ,lord ,kelvin 7,london1 #aiaj71 p4 #aacf4 #ajdf4 ,the following is one of the many stories told of 8old ,donald ,mc,farlane0 the faithful assistant of ,sir ,william ,thomson4 ,the father of a new student when bringing him to the ,university1 after calling to see the ,professor ,thomson drew his assistant to one side and besought him to tell him what his son must do that he might stand well with the ,professor4 8,you want your son to stand weel with the ,profeessorr80 asked ,mc,farlane4 8,yes40 8,weel1 then1 he must just have a guid bellyful o' mathematics60 --,,thompson1 ,s4 ,p4 ,life of ,lord ,kelvin 7,london1 #aiaj71 p4 #dbj4 #ajdg4 ,the following story 7here a little softened from the vernacular7 was narrated by ,lord ,kelvin himself when dining at ,trinity ,hall3-- ,a certain rough ,highland lad at the university had done exceedingly well1 and at the close of the session gained prizes both in mathematics and in metaphysics4 ,his old father came up from the farm to see his son receive the prizes1 and visited the ,college4 ,thomson was deputed to show him round the place4 8,weel1 ,mr4 ,thomson10 asked the old man1 8and what may these mathematics be1 for which my son has getten a prize80 8,i told him10 replied ,thomson1 8that mathematics meant reckoning with figures1 and calculating40 8,oo ay10 said the old man1 8he'll ha' getten that fra' me3 ,i were ever a braw hand at the countin'40 ,after a pause he resumed3 8,and what1 ,mr4 ,thomson1 might these metapheesics be80 8,i endeavoured10 replied ,thomson1 8to explain how metaphysics was the attempt to express in language the indefinite40 ,the old ,highlander stood still and scratched his head4 8,oo ay3 may be he'll ha' getten that fra' his mither4 ,she were aye a bletherin' body40--,,thompson1 ,s4 ,p4 ,life of ,lord ,kelvin 7,london1 #aiaj71 p4 #aabd4 #ajdh4 ,lord ,kelvin1 unable to meet his classes one day1 posted the following notice on the door of his lecture room1-- 8,professor ,thomson will not meet his classes today40 ,the disappointed class decided to play a joke on the professor4 ,erasing the 8c0 they left the legend to read1-- 8,professor ,thomson will not meet his lasses today40 ,when the class assembled the next day in anticipation of the effect of their joke1 they were astonished and chagrined to find that the professor had outwitted them4 ,the legend of yesterday was now found to read1-- 8,professor ,thomson will not meet his asses today40#i --,,northrup1 ,,cyrus4 ,university of ,washington ,address1 ,november #b1 #aijh4 #i ,author's note4 ,my colleague1 ,dr4 ,e4 ,t4 ,bell1 informs me that this same anecdote is associated with the name of ,j4 ,s4 ,blackie1 ,professor of ,greek at ,aberdeen and ,edinburgh4 #ajdi4 ,one morning a great noise proceeded from one of the classrooms of the ,braunsberger gymnasium and on investigation it was found that ,weierstrass1 who was to give the recitation1 had not appeared4 ,the director went in person to ,weierstrass' dwelling and on knocking was told to come in4 ,there sat ,weierstrass by a glimmering lamp in a darkened room though it was daylight outside4 ,he had worked the night through and had not noticed the approach of daylight4 ,when the director reminded him of the noisy throng of students who were waiting for him1 his only reply was that he could impossibly interrupt his work2 that he was about to make an important discovery which would attract attention in scientific circles4--,,lampe1 ,e4 ,karl ,weierstrass3 ,jahrbuch der ,deutschen ,mathematiker ,vereinigung1 ,bd4 #f 7#ahig71 pp4 #ch-ci4 #ajej4 ,weierstrass related 444 that he followed ,sylvester's papers on the theory of algebraic forms very attentively until ,sylvester began to employ ,hebrew characters4 ,that was more than he could stand and after that he quit him4--,,lampe1 ,e4 ,naturwissenschaftliche ,rundschau1 ,bd4 #ab 7#ahig71 p4 #cfa4 ,,chapter ,,xi ,,mathematics ,,as ,a ,,fine ,,art #aaja4 ,the world of idea which it discloses or illuminates1 the contemplation of divine beauty and order which it induces1 the harmonious connexion of its parts1 the infinite hierarchy and absolute evidence of the truths with which it is concerned1 these1 and such like1 are the surest grounds of the title of mathematics to human regard1 and would remain unimpeached and unimpaired were the plan of the universe unrolled like a map at our feet1 and the mind of man qualified to take in the whole scheme of creation at a glance4--,,sylvester1 ,j4 ,j4 ,presidential ,address1 ,british ,association ,report 7#ahfi72 ,collected ,mathematical ,papers1 ,vol4 #b1 p4 #fei4 #aajb4 ,mathematics has a triple end4 ,it should furnish an instrument for the study of nature4 ,furthermore it has a philosophic end1 and1 ,i venture to say1 an end esthetic4 ,it ought to incite the philosopher to search into the notions of number1 space1 and time2 and1 above all1 adepts find in mathematics delights analogous to those that painting and music give4 ,they admire the delicate harmony of number and of forms2 they are amazed when a new discovery discloses for them an unlooked for perspective2 and the joy they thus experience1 has it not the esthetic character although the senses take no part in it8 ,only the privileged few are called to enjoy it fully1 it is true2 but is it not the same with all the noblest arts8 ,hence ,i do not hesitate to say that mathematics deserves to be cultivated for its own sake1 and that the theories not admitting of application to physics deserve to be studied as well as others4--,,poincar1 ,,henri4 ,the ,relation of ,analysis and ,mathematical ,physics2 ,bulletin ,american ,mathematical ,society1 ,vol4 #d 7#ahii71 p4 #bdh4 #aajc4 ,i like to look at mathematics almost more as an art than as a science2 for the activity of the mathematician1 constantly creating as he is1 guided though not controlled by the external world of the senses1 bears a resemblance1 not fanciful ,i believe but real1 to the activity of an artist1 of a painter let us say4 ,rigorous deductive reasoning on the part of the mathematician may be likened here to technical skill in drawing on the part of the painter4 ,just as no one can become a good painter without a certain amount of skill1 so no one can become a mathematician without the power to reason accurately up to a certain point4 ,yet these qualities1 fundamental though they are1 do not make a painter or mathematician worthy of the name1 nor indeed are they the most important factors in the case4 ,other qualities of a far more subtle sort1 chief among which in both cases is imagination1 go to the making of a good artist or good mathematician4 --,b,,cher1 ,,maxime4 ,fundamental ,conceptions and ,methods in ,mathematics2 ,bulletin ,american ,mathematical ,society1 ,vol4 #i 7#aijd71 p4 #acc4 #aajd4 ,mathematics1 rightly viewed1 possesses not only truth1 but supreme beauty--a beauty cold and austere1 like that of sculpture1 without appeal to any part of our weaker nature1 without the gorgeous trappings of painting or music1 yet sublimely pure1 and capable of a stern perfection such as only the greatest art can show4 ,the true spirit of delight1 the exaltation1 the sense of being more than man1 which is the touchstone of the highest excellence1 is to be found in mathematics as surely as in poetry4 ,what is best in mathematics deserves not merely to be learned as a task1 but to be assimilated as a part of daily thought1 and brought again and again before the mind with ever-renewed encouragement4 ,real life is1 to most men1 a long second-best1 a perpetual compromise between the real and the possible2 but the world of pure reason knows no compromise1 no practical limitations1 no barrier to the creative activity embodying in splendid edifices the passionate aspiration after the perfect from which all great work springs4 ,remote from human passions1 remote even from the pitiful facts of nature1 the generations have gradually created an ordered cosmos1 where pure thought can dwell as in its natural home1 and where one1 at least1 of our nobler impulses can escape from the dreary exile of the natural world4 --,,russell1 ,,bertrand4 ,the ,study of ,mathematics3 ,philosophical ,essays 7,london1 #aiaj71 p4 #gc4 #aaje4 ,it was not alone the striving for universal culture which attracted the great masters of the ,renaissance1 such as ,brunellesco1 ,leonardo de ,vinci1 ,raphael1 ,michael ,angelo and especially ,albrecht ,drer1 with irresistible power to the mathematical sciences4 ,they were conscious that1 with all the freedom of the individual phantasy1 art is subject to necessary laws1 and conversely1 with all its rigor of logical structure1 mathematics follows esthetic laws4--,,rudio1 ,f4 ,virchow-,holtzendorf3 ,sammlung gemeinverstndliche wissenschaftliche ,vortrge1 ,heft #adb1 p4 #ai4 #aajf4 ,surely the claim of mathematics to take a place among the liberal arts must now be admitted as fully made good4 ,whether we look at the advances made in modern geometry1 in modern integral calculus1 or in modern algebra1 in each of these three a free handling of the material employed is now possible1 and an almost unlimited scope is left to the regulated play of fancy4 ,it seems to me that the whole of aesthetic 7so far as at present revealed7 may be regarded as a scheme having four centres1 which may be treated as the four apices of a tetrahedron1 namely ,epic1 ,music1 ,plastic1 and ,mathematic4 ,there will be found a common plane to every three of these1 outside of which lies the fourth2 and through every two may be drawn a common axis opposite to the axis passing through the other two4 ,so far is certain and demonstrable4 ,i think it also possible that there is a centre of gravity to each set of three1 and that the line joining each such centre with the outside apex will intersect in a common point--the centre of gravity of the whole body of aesthetic2 but what that centre is or must be ,i have not had time to think out4 --,,sylvester1 ,j4 ,j4 ,proof of the hitherto undemonstrated ,fundamental ,theorem of ,invariants3 ,collected ,mathematical ,papers1 ,vol4 #c1 p4 #abc4 #aajg4 ,it is with mathematics not otherwise than it is with music1 painting or poetry4 ,anyone can become a lawyer1 doctor or chemist1 and as such may succeed well1 provided he is clever and industrious1 but not every one can become a painter1 or a musician1 or a mathematician3 general cleverness and industry alone count here for nothing4--,,moebius1 ,p4 ,j4 ,ueber die ,anlage zur ,mathematik 7,leipzig1 #aijj71 p4 #e4 #aajh4 ,the true mathematician is always a good deal of an artist1 an architect1 yes1 of a poet4 ,beyond the real world1 though perceptibly connected with it1 mathematicians have intellectually created an ideal world1 which they attempt to develop into the most perfect of all worlds1 and which is being explored in every direction4 ,none has the faintest conception of this world1 except he who knows it4--,,pringsheim1 ,a4 ,jahresbericht der ,deutschen ,mathematiker ,vereinigung1 ,bd4 #cb1 p4 #cha4 #aaji4 ,who has studied the works of such men as ,euler1 ,lagrange1 ,cauchy1 ,riemann1 ,sophus ,lie1 and ,weierstrass1 can doubt that a great mathematician is a great artist8 ,the faculties possessed by such men1 varying greatly in kind and degree with the individual1 are analogous with those requisite for constructive art4 ,not every mathematician possesses in a specially high degree that critical faculty which finds its employment in the perfection of form1 in conformity with the ideal of logical completeness2 but every great mathematician possesses the rarer faculty of constructive imagination4--,,hobson1 ,e4 ,w4 ,presidential ,address ,british ,association for the ,advancement of ,science 7#aiaj7 ,nature1 ,vol4 #hd1 p4 #bij4 #aaaj4 ,mathematics has beauties of its own--a symmetry and proportion in its results1 a lack of superfluity1 an exact adaptation of means to ends1 which is exceedingly remarkable and to be found elsewhere only in the works of the greatest beauty4 ,it was a felicitous expression of ,goethe's to call a noble cathedral 8frozen music10 but it might even better be called 8petrified mathematics40 ,the beauties of mathematics--of simplicity1 of symmetry1 of completeness--can and should be exemplified even to young children4 ,when this subject is properly and concretely presented1 the mental emotion should be that of enjoyment of beauty1 not that of repulsion from the ugly and the unpleasant4--,,young1 ,j4 ,w4 ,a4 ,the ,teaching of ,mathematics 7,new ,york1 #aijg71 p4 #dd4 #aaaa4 ,a peculiar beauty reigns in the realm of mathematics1 a beauty which resembles not so much the beauty of art as the beauty of nature and which affects the reflective mind1 which has acquired an appreciation of it1 very much like the latter4 --,,kummer1 ,e4 ,e4 ,berliner ,monatsberichte 7#ahfg71 p4 #cie4 #aaab4 ,mathematics make the mind attentive to the objects which it considers4 ,this they do by entertaining it with a great variety of truths1 which are delightful and evident1 but not obvious4 ,truth is the same thing to the understanding as music to the ear and beauty to the eye4 ,the pursuit of it does really as much gratify a natural faculty implanted in us by our wise ,creator as the pleasing of our senses3 only in the former case1 as the object and faculty are more spiritual1 the delight is more pure1 free from regret1 turpitude1 lassitude1 and intemperance that commonly attend sensual pleasures4--,,arbuthnot1 ,,john4 ,usefulness of ,mathematical ,learning4 #aaac4 ,however far the calculating reason of the mathematician may seem separated from the bold flight of the artist's phantasy1 it must be remembered that these expressions are but momentary images snatched arbitrarily from among the activities of both4 ,in the projection of new theories the mathematician needs as bold and creative a phantasy as the productive artist1 and in the execution of the details of a composition the artist too must calculate dispassionately the means which are necessary for the successful consummation of the parts4 ,common to both is the creation1 the generation1 of forms out of mind4--,,lampe1 ,e4 ,die ,entwickelung der ,mathematik1 etc4 7,berlin1 #ahic71 p4 #d4 #aaad4 ,as pure truth is the polar star of our science mathematics1 so it is the great advantage of our science over others that it awakens more easily the love of truth in our pupils4444 ,if ,hegel justly said1 8,whoever does not know the works of the ancients1 has lived without knowing beauty10 ,schellbach responds with equal right1 8,who does not know mathematics1 and the results of recent scientific investigation1 dies without knowing truth40--,,simon1 ,,max4 ,quoted in ,j4 ,w4 ,a4 ,young3 ,teaching of ,mathematics 7,new ,york1 #aijg71 p4 #dd4 #aaae4 ,bchsel in his reminiscences from the life of a country parson relates that he sought his recreation in ,lacroix's ,differential ,calculus and thus found intellectual refreshment for his calling4 ,instances like this make manifest the great advantage which occupation with mathematics affords to one who lives remote from the city and is compelled to forego the pleasures of art4 ,the entrancing charm of mathematics1 which captivates every one who devotes himself to it1 and which is comparable to the fine frenzy under whose ban the poet completes his work1 has ever been incomprehensible to the spectator and has often caused the enthusiastic mathematician to be held in derision4 ,a classic illustration is the example of ,archimedes1 4444--,,lampe1 ,e4 ,die ,entwickelung der ,mathematik1 etc4 7,berlin #ahic71 p4 #bb4 #aaaf4 ,among the memoirs of ,kirchhoff are some of uncommon beauty4 ,beauty1 ,i hear you ask1 do not the ,graces flee where integrals stretch forth their necks8 ,can anything be beautiful1 where the author has no time for the slightest external embellishment8444 ,yet it is this very simplicity1 the indispensableness of each word1 each letter1 each little dash1 that among all artists raises the mathematician nearest to the ,world-creator2 it establishes a sublimity which is equalled in no other art1--something like it exists at most in symphonic music4 ,the ,pythagoreans recognized already the similarity between the most subjective and the most objective of the arts4444 ,ultima se tangunt4 ,how expressive1 how nicely characterizing withal is mathematics6 ,as the musician recognizes ,mozart1 ,beethoven1 ,schubert in the first chords1 so the mathematician would distinguish his ,cauchy1 ,gauss1 ,jacobi1 ,helmholtz in a few pages4 ,extreme external elegance1 sometimes a somewhat weak skeleton of conclusions characterizes the ,french2 the ,english1 above all ,maxwell1 are distinguished by the greatest dramatic bulk4 ,who does not know ,maxwell's dynamic theory of gases8 ,at first there is the majestic development of the variations of velocities1 then enter from one side the equations of condition and from the other the equations of central motions1--higher and higher surges the chaos of formulas1--suddenly four words burst forth3 8,put n #e40 ,the evil demon ,v disappears like the sudden ceasing of the basso parts in music1 which hitherto wildly permeated the piece2 what before seemed beyond control is now ordered as by magic4 ,there is no time to state why this or that substitution was made1 he who cannot feel the reason may as well lay the book aside2 ,maxwell is no program-musician who explains the notes of his composition4 ,forthwith the formulas yield obediently result after result1 until the temperature-equilibrium of a heavy gas is reached as a surprising final climax and the curtain drops4444 ,kirchhoff's whole tendency1 and its true counterpart1 the form of his presentation1 was different4444 ,he is characterized by the extreme precision of his hypotheses1 minute execution1 a quiet rather than epic development with utmost rigor1 never concealing a difficulty1 always dispelling the faintest obscurity4 ,to return once more to my allegory1 he resembled ,beethoven1 the thinker in tones4--,he who doubts that mathematical compositions can be beautiful1 let him read his memoir on ,absorption and ,emission 7,gesammelte ,abhandlungen1 ,leipzig1 #ahhb1 p4 #ega-eih7 or the chapter of his mechanics devoted to ,hydrodynamics4--,,boltzmann1 ,l4 ,gustav ,robert ,kirchhoff 7,leipzig #ahhh71 pp4 #bh-cj4 #aaag4 ,on poetry and geometric truth1 ,and their high privilege of lasting life1 ,from all internal injury exempt1 ,i mused2 upon these chiefly3 and at length1 ,my senses yielding to the sultry air1 ,sleep seized me1 and ,i passed into a dream4 --,,wordsworth4 ,the ,prelude1 ,bk4 #e4 #aaah4 ,geometry seems to stand for all that is practical1 poetry for all that is visionary1 but in the kingdom of the imagination you will find them close akin1 and they should go together as a precious heritage to every youth4--,,milner1 ,,florence4 ,school ,review1 #ahih1 p4 #aad4 #aaai4 ,the beautiful has its place in mathematics as elsewhere4 ,the prose of ordinary intercourse and of business correspondence might be held to be the most practical use to which language is put1 but we should be poor indeed without the literature of imagination4 ,mathematics too has its triumphs of the creative imagination1 its beautiful theorems1 its proofs and processes whose perfection of form has made them classic4 ,he must be a 8practical0 man who can see no poetry in mathematics4 --,,white1 ,w4 ,f4 ,a ,scrap-book of ,elementary ,mathematics 7,chicago1 #aijh71 p4 #bjh4 #aabj4 ,i venture to assert that the feelings one has when the beautiful symbolism of the infinitesimal calculus first gets a meaning1 or when the delicate analysis of ,fourier has been mastered1 or while one follows ,clerk ,maxwell or ,thomson into the strange world of electricity1 now growing so rapidly in form and being1 or can almost feel with ,stokes the pulsations of light that gives nature to our eyes1 or track with ,clausius the courses of molecules we can measure1 even if we know with certainty that we can never see them--,i venture to assert that these feelings are altogether comparable to those aroused in us by an exquisite poem or a lofty thought4--,,workman1 ,w4 ,p4 ,f4 ,spencer3 ,aim and ,practice of ,teaching 7,new ,york1 #ahig71 p4 #aid4 #aaba4 ,it is an open secret to the few who know it1 but a mystery and stumbling block to the many1 that ,science and ,poetry are own sisters2 insomuch that in those branches of scientific inquiry which are most abstract1 most formal1 and most remote from the grasp of the ordinary sensible imagination1 a higher power of imagination akin to the creative insight of the poet is most needed and most fruitful of lasting work4--,,pollock1 ,f4 ,clifford's ,lectures and ,essays 7,new ,york1 #aija71 ,vol4 #a1 ,introduction1 p4 #a4 #aabb4 ,it is as great a mistake to maintain that a high development of the imagination is not essential to progress in mathematical studies as to hold with ,ruskin and others that science and poetry are antagonistic pursuits4--,,hoffman1 ,f4 ,s4 ,sphere of ,science 7,london1 #ahih71 p4 #ajg4 #aabc4 ,we have heard much about the poetry of mathematics1 but very little of it has as yet been sung4 ,the ancients had a juster notion of their poetic value than we4 ,the most distinct and beautiful statements of any truth must take at last the mathematical form4 ,we might so simplify the rules of moral philosophy1 as well as of arithmetic1 that one formula would express them both4--,,thoreau1 ,h4 ,d4 ,a ,week on the ,concord and ,merrimac ,rivers 7,boston1 #ahic71 p4 #dgg4 #aabd4 ,we do not listen with the best regard to the verses of a man who is only a poet1 nor to his problems if he is only an algebraist2 but if a man is at once acquainted with the geometric foundation of things and with their festal splendor1 his poetry is exact and his arithmetic musical4--,,emerson1 ,r4 ,w4 ,society and ,solitude1 ,chap4 #g1 ,works and ,days4 #aabe4 ,mathesis and ,poetry are 444 the utterance of the same power of imagination1 only that in the one case it is addressed to the head1 and in the other1 to the heart4--,,hill1 ,,thomas4 ,north ,american ,review1 ,vol4 #he1 p4 #bcj4 #aabf4 ,the ,mathematics are usually considered as being the very antipodes of ,poesy4 ,yet ,mathesis and ,poesy are of the closest kindred1 for they are both works of the imagination4 ,poesy is a creation1 a making1 a fiction2 and the ,mathematics have been called1 by an admirer of them1 the sublimest and most stupendous of fictions4 ,it is true1 they are not only 1 learning1 but 1 a creation4--,,hill1 ,,thomas4 ,north ,american ,review1 ,vol4 #he1 p4 #bbi4 #aabg4 ,music and poesy used to quicken you3 ,the mathematics1 and the metaphysics1 ,fall to them as you find your stomach serves you4 ,no profit grows1 where is no pleasure ta'en3-- ,in brief1 sir1 study what you most affect4 --,,shakespeare4 ,taming of the ,shrew1 ,act #a1 ,scene #a4 #aabh4 ,music has much resemblance to algebra4--,,novalis4 ,schriften1 ,teil #b 7,berlin1 #aija71 p4 #edi4 #aabi4 ,i do present you with a man of mine1 ,cunning in music and in mathematics1 ,to instruct her fully in those sciences1 ,whereof1 ,i know1 she is not ignorant4 --,,shakespeare4 ,taming of the ,shrew1 ,act #b1 ,scene #a4 #aacj4 ,saturated with that speculative spirit then pervading the ,greek mind1 he ,pythagoras endeavoured to discover some principle of homogeneity in the universe4 ,before him1 the philosophers of the ,ionic school had sought it in the matter of things2 ,pythagoras looked for it in the structure of things4 ,he observed the various numerical relations or analogies between numbers and the phenomena of the universe4 ,being convinced that it was in numbers and their relations that he was to find the foundation to true philosophy1 he proceeded to trace the origin of all things to numbers4 ,thus he observed that musical strings of equal lengths stretched by weights having the proportion of #a_/b1 #b_/c1 #c_/d1 produced intervals which were an octave1 a fifth and a fourth4 ,harmony1 therefore1 depends on musical proportion2 it is nothing but a mysterious numerical relation4 ,where harmony is1 there are numbers4 ,hence the order and beauty of the universe have their origin in numbers4 ,there are seven intervals in the musical scale1 and also seven planets crossing the heavens4 ,the same numerical relations which underlie the former must underlie the latter4 ,but where number is1 there is harmony4 ,hence his spiritual ear discerned in the planetary motions a wonderful 8,harmony of spheres40--,,cajori1 ,f4 ,history of ,mathematics 7,new ,york1 #ahig71 p4 #fg4 #aaca4 ,may not ,music be described as the ,mathematic of sense1 ,mathematic as ,music of the reason8 the soul of each the same6 ,thus the musician feels ,mathematic1 the mathematician thinks ,music1--,music the dream1 ,mathematic the working life--each to receive its consummation from the other when the human intelligence1 elevated to its perfect type1 shall shine forth glorified in some future ,mozart-,dirichlet or ,beethoven-,gauss--a union already not indistinctly foreshadowed in the genius and labours of a ,helmholtz6--,,sylvester1 ,j4 ,j4 ,on ,newton's ,rule for the ,discovery of ,imaginary ,roots2 ,collected ,mathematical ,papers1 ,vol4 #b1 p4 #dai4 #aacb4 ,just as the musician is able to form an acoustic image of a composition which he has never heard played by merely looking at its score1 so the equation of a curve1 which he has never seen1 furnishes the mathematician with a complete picture of its course4 ,yea1 even more3 as the score frequently reveals to the musician niceties which would escape his ear because of the complication and rapid change of the auditory impressions1 so the insight which the mathematician gains from the equation of a curve is much deeper than that which is brought about by a mere inspection of the curve4--,,pringsheim1 ,a4 ,jahresbericht der ,deutschen ,mathematiker ,vereinigung1 ,bd4 #ac1 p4 #cfd4 #aacc4 ,mathematics and music1 the most sharply contrasted fields of scientific activity which can be found1 and yet related1 supporting each other1 as if to show forth the secret connection which ties together all the activities of our mind1 and which leads us to surmise that the manifestations of the artist's genius are but the unconscious expressions of a mysteriously acting rationality4--,,helmholtz1 ,h4 ,vortrge und ,reden1 ,bd4 #a 7,braunschweig1 #ahhd71 p4 #hb4 #aacd4 ,among all highly civilized peoples the golden age of art has always been closely coincident with the golden age of the pure sciences1 particularly with mathematics1 the most ancient among them4 ,this coincidence must not be looked upon as accidental1 but as natural1 due to an inner necessity4 ,just as art can thrive only when the artist1 relieved of the anxieties of existence1 can listen to the inspirations of his spirit and follow in their lead1 so mathematics1 the most ideal of the sciences1 will yield its choicest blossoms only when life's dismal phantom dissolves and fades away1 when the striving after naked truth alone predominates1 conditions which prevail only in nations while in the prime of their development4--,,lampe1 ,e4 ,die ,entwickelung der ,mathematik etc4 7,berlin1 #ahic71 p4 #d4 #aace4 ,till the fifteenth century little progress appears to have been made in the science or practice of music2 but since that era it has advanced with marvelous rapidity1 its progress being curiously parallel with that of mathematics1 inasmuch as great musical geniuses appeared suddenly among different nations1 equal in their possession of this special faculty to any that have since arisen4 ,as with the mathematical so with the musical faculty--it is impossible to trace any connection between its possession and survival in the struggle for existence4 --,,wallace1 ,a4 ,r4 ,darwinism1 ,chap4 #ae4 #aacf4 ,in my opinion1 there is absolutely no trustworthy proof that talents have been improved by their exercise through the course of a long series of generations4 ,the ,bach family shows that musical talent1 and the ,bernoulli family that mathematical power1 can be transmitted from generation to generation1 but this teaches us nothing as to the origin of such talents4 ,in both families the high-watermark of talent lies1 not at the end of the series of generations1 as it should do if the results of practice are transmitted1 but in the middle4 ,again1 talents frequently appear in some member of a family which has not been previously distinguished4 ,gauss was not the son of a mathematician2 ,handel's father was a surgeon1 of whose musical powers nothing is known2 ,titian was the son and also the nephew of a lawyer1 while he and his brother1 ,francesco ,vecellio1 were the first painters in a family which produced a succession of seven other artists with diminishing talents4 ,these facts do not1 however1 prove that the condition of the nerve-tracts and centres of the brain1 which determine the specific talent1 appeared for the first time in these men3 the appropriate condition surely existed previously in their parents1 although it did not achieve expression4 ,they prove1 as it seems to me1 that a high degree of endowment in a special direction1 which we call talent1 cannot have arisen from the experience of previous generations1 that is1 by the exercise of the brain in the same specific direction4--,,weismann1 ,,august4 ,essays upon ,heredity ,a4 ,e4 ,shipley1 7,oxford1 #ahia71 ,vol4 #a1 p4 #ig4 ,,chapter ,,xii ,,mathematics ,,as ,a ,,language #abja4 ,the new mathematics is a sort of supplement to language1 affording a means of thought about form and quantity and a means of expression1 more exact1 compact1 and ready than ordinary language4 ,the great body of physical science1 a great deal of the essential facts of financial science1 and endless social and political problems are only accessible and only thinkable to those who have had a sound training in mathematical analysis1 and the time may not be very remote when it will be understood that for complete initiation as an efficient citizen of one of the new great complex world wide states that are now developing1 it is as necessary to be able to compute1 to think in averages and maxima and minima1 as it is now to be able to read and to write4 --,,wells1 ,h4 ,g4 ,mankind in the ,making 7,london1 #aijd71 pp4 #aia-aib4 #abjb4 ,mathematical language is not only the simplest and most easily understood of any1 but the shortest also4--,,brougham1 ,h4 ,l4 ,works 7,edinburgh1 #ahgb71 ,vol4 #g1 p4 #cag4 #abjc4 ,mathematics is the science of definiteness1 the necessary vocabulary of those who know4--,,white1 ,w4 ,f4 ,a ,scrap-book of ,elementary ,mathematics 7,chicago1 #aijh71 p4 #g4 #abjd4 ,mathematics1 too1 is a language1 and as concerns its structure and content it is the most perfect language which exists1 superior to any vernacular2 indeed1 since it is understood by every people1 mathematics may be called the language of languages4 ,through it1 as it were1 nature herself speaks2 through it the ,creator of the world has spoken1 and through it the ,preserver of the world continues to speak4 --,,dillmann1 ,c4 ,die ,mathematik die ,fackeltrgerin einer neuen ,zeit 7,stuttgart1 #ahhi71 p4 #e4 #abje4 ,would it sound too presumptuous to speak of perception as a quintessence of sensation1 language 7that is1 communicable thought7 of perception1 mathematics of language8 ,we should then have four terms differentiating from inorganic matter and from each other the ,vegetable1 ,animal1 ,rational1 and ,super-sensual modes of existence4--,,sylvester1 ,j4 ,j4 ,presidential ,address1 ,british ,association2 ,collected ,mathematical ,papers1 ,vol4 #b1 p4 #feb4 #abjf4 ,little could ,plato have imagined1 when1 indulging his instinctive love of the true and beautiful for their own sakes1 he entered upon these refined speculations and revelled in a world of his own creation1 that he was writing the grammar of the language in which it would be demonstrated in after ages that the pages of the universe are written4--,,sylvester1 ,j4 ,j4 ,a ,probationary ,lecture on ,geometry2 ,collected ,mathematical ,papers1 ,vol4 #b1 p4 #g4 #abjg4 ,it is the symbolic language of mathematics only which has yet proved sufficiently accurate and comprehensive to demand familiarity with this conception of an inverse process4 --,,venn1 ,,john4 ,symbolic ,logic 7,london and ,new ,york1 #ahid71 p4 #gd4 #abjh4 ,without this language mathematics most of the intimate analogies of things would have remained forever unknown to us2 and we should forever have been ignorant of the internal harmony of the world1 which is the only true objective reality4444 ,this harmony 444 is the sole objective reality1 the only truth we can attain2 and when ,i add that the universal harmony of the world is the source of all beauty1 it will be understood what price we should attach to the slow and difficult progress which little by little enables us to know it better4--,,poincar1 ,h4 ,the ,value of ,science ,halsted ,popular ,science ,monthly1 #aijf1 pp4 #aie-aif4 #abji4 ,the most striking characteristic of the written language of algebra and of the higher forms of the calculus is the sharpness of definition1 by which we are enabled to reason upon the symbols by the mere laws of verbal logic1 discharging our minds entirely of the meaning of the symbols1 until we have reached a stage of the process where we desire to interpret our results4 ,the ability to attend to the symbols1 and to perform the verbal1 visible changes in the position of them permitted by the logical rules of the science1 without allowing the mind to be perplexed with the meaning of the symbols until the result is reached which you wish to interpret1 is a fundamental part of what is called analytical power4 ,many students find themselves perplexed by a perpetual attempt to interpret not only the result1 but each step of the process4 ,they thus lose much of the benefit of the labor-saving machinery of the calculus and are1 indeed1 frequently incapacitated for using it4--,,hill1 ,,thomas4 ,uses of ,mathesis2 ,bibliotheca ,sacra1 ,vol4 #cb1 p4 #eje4 #abaj4 ,the prominent reason why a mathematician can be judged by none but mathematicians1 is that he uses a peculiar language4 ,the language of mathesis is special and untranslatable4 ,in its simplest forms it can be translated1 as1 for instance1 we say a right angle to mean a square corner4 ,but you go a little higher in the science of mathematics1 and it is impossible to dispense with a peculiar language4 ,it would defy all the power of ,mercury himself to explain to a person ignorant of the science what is meant by the single phrase 8functional exponent40 ,how much more impossible1 if we may say so1 would it be to explain a whole treatise like ,hamilton's ,quaternions1 in such a wise as to make it possible to judge of its value6 ,but to one who has learned this language1 it is the most precise and clear of all modes of expression4 ,it discloses the thought exactly as conceived by the writer1 with more or less beauty of form1 but never with obscurity4 ,it may be prolix1 as it often is among ,french writers2 may delight in mere verbal metamorphoses1 as in the ,cambridge ,university of ,england2 or adopt the briefest and clearest forms1 as under the pens of the geometers of our ,cambridge2 but it always reveals to us precisely the writer's thought4 --,,hill1 ,,thomas4 ,north ,american ,review1 ,vol4 #he1 pp4 #bbd-bbe4 #abaa4 ,the domain1 over which the language of analysis extends its sway1 is1 indeed1 relatively limited1 but within this domain it so infinitely excels ordinary language that its attempt to follow the former must be given up after a few steps4 ,the mathematician1 who knows how to think in this marvelously condensed language1 is as different from the mechanical computer as heaven from earth4--,,pringsheim1 ,a4 ,jahresberichte der ,deutschen ,mathematiker ,vereinigung1 ,bd4 #ac1 p4 #cfg4 #abab4 ,the results of systematic symbolical reasoning must always express general truths1 by their nature2 and do not1 for their justification1 require each of the steps of the process to represent some definite operation upon quantity4 ,the absolute universality of the interpretation of symbols is the fundamental principle of their use4--,,whewell1 ,,william4 ,the ,philosophy of the ,inductive ,sciences1 ,part ,i1 ,bk4 #b1 chap4 #ab1 sect4 #b 7,london1 #aheh74 #abac4 ,anyone who understands algebraic notation1 reads at a glance in an equation results reached arithmetically only with great labour and pains4--,,cournot1 ,a4 ,theory of ,wealth ,n4 ,t4 ,bacon1 7,new ,york1 #ahig71 p4 #d4 #abad4 ,as arithmetic and algebra are sciences of great clearness1 certainty1 and extent1 which are immediately conversant about signs1 upon the skilful use whereof they entirely depend1 so a little attention to them may possibly help us to judge of the progress of the mind in other sciences1 which1 though differing in nature1 design1 and object1 may yet agree in the general methods of proof and inquiry4--,,berkeley1 ,,george4 ,alciphron1 or the ,minute ,philosopher1 ,dialogue #g1 sect4 #ab4 #abae4 ,in general the position as regards all such new calculi is this--,that one cannot accomplish by them anything that could not be accomplished without them4 ,however1 the advantage is1 that1 provided such a calculus corresponds to the inmost nature of frequent needs1 anyone who masters it thoroughly is able--without the unconscious inspiration of genius which no one can command--to solve the respective problems1 yea1 to solve them mechanically in complicated cases in which1 without such aid1 even genius becomes powerless4 ,such is the case with the invention of general algebra1 with the differential calculus1 and in a more limited region with ,lagrange's calculus of variations1 with my calculus of congruences1 and with ,mbius's calculus4 ,such conceptions unite1 as it were1 into an organic whole countless problems which otherwise would remain isolated and require for their separate solution more or less application of inventive genius4--,,gauss1 ,c4 ,j4 ,werke1 ,bd4 #h1 p4 #bih4 #abaf4 ,the invention of what we may call primary or fundamental notation has been but little indebted to analogy1 evidently owing to the small extent of ideas in which comparison can be made useful4 ,but at the same time analogy should be attended to1 even if for no other reason than that1 by making the invention of notation an art1 the exertion of individual caprice ceases to be allowable4 ,nothing is more easy than the invention of notation1 and nothing of worse example and consequence than the confusion of mathematical expressions by unknown symbols4 ,if new notation be advisable1 permanently or temporarily1 it should carry with it some mark of distinction from that which is already in use1 unless it be a demonstrable extension of the latter4 --,,de ,,morgan1 ,a4 ,calculus of ,functions2 ,encyclopedia ,metropolitana1 ,addition to ,article #bf4 #abag4 ,before the introduction of the ,arabic notation1 multiplication was difficult1 and the division even of integers called into play the highest mathematical faculties4 ,probably nothing in the modern world could have more astonished a ,greek mathematician than to learn that1 under the influence of compulsory education1 the whole population of ,western ,europe1 from the highest to the lowest1 could perform the operation of division for the largest numbers4 ,this fact would have seemed to him a sheer impossibility4444 ,our modern power of easy reckoning with decimal fractions is the most miraculous result of a perfect notation4--,,whitehead1 ,a4 ,n4 ,introduction to ,mathematics 7,new ,york1 #aiaa71 p4 #ei4 #abah4 ,mathematics is often considered a difficult and mysterious science1 because of the numerous symbols which it employs4 ,of course1 nothing is more incomprehensible than a symbolism which we do not understand4 ,also a symbolism1 which we only partially understand and are unaccustomed to use1 is difficult to follow4 ,in exactly the same way the technical terms of any profession or trade are incomprehensible to those who have never been trained to use them4 ,but this is not because they are difficult in themselves4 ,on the contrary they have invariably been introduced to make things easy4 ,so in mathematics1 granted that we are giving any serious attention to mathematical ideas1 the symbolism is invariably an immense simplification4 --,,whitehead1 ,a4 ,n4 ,introduction to ,mathematics 7,new ,york1 #aiaa71 pp4 #ei-fj4 #abai4 ,symbolism is useful because it makes things difficult4 ,now in the beginning everything is self-evident1 and it is hard to see whether one self-evident proposition follows from another or not4 ,obviousness is always the enemy to correctness4 ,hence we must invent a new and difficult symbolism in which nothing is obvious4444 ,thus the whole of ,arithmetic and ,algebra has been shown to require three indefinable notions and five indemonstrable propositions4--,,russell1 ,,bertrand4 ,international ,monthly1 #aija1 p4 #he4 #abbj4 ,the employment of mathematical symbols is perfectly natural when the relations between magnitudes are under discussion2 and even if they are not rigorously necessary1 it would hardly be reasonable to reject them1 because they are not equally familiar to all readers and because they have sometimes been wrongly used1 if they are able to facilitate the exposition of problems1 to render it more concise1 to open the way to more extended developments1 and to avoid the digressions of vague argumentation4--,,cournot1 ,a4 ,theory of ,wealth ,n4 ,t4 ,bacon1 7,new ,york1 #ahig71 pp4 #c-d4 #abba4 ,an all-inclusive geometrical symbolism1 such as ,hamilton and ,grassmann conceived of1 is impossible4--,,burkhardt1 ,h4 ,jahresbericht der ,deutschen ,mathematiker ,vereinigung1 ,bd4 #e1 p4 #eb4 #abbb4 ,the language of analysis1 most perfect of all1 being in itself a powerful instrument of discoveries1 its notations1 especially when they are necessary and happily conceived1 are so many germs of new calculi4--,,laplace4 ,oeuvres1 t4 #g 7,paris1 #ahif71 p4 xl4 ,,chapter ,,xiii ,,mathematics ,,and ,,logic #acja4 ,mathematics belongs to every inquiry1 moral as well as physical4 ,even the rules of logic1 by which it is rigidly bound1 could not be deduced without its aid4 ,the laws of argument admit of simple statement1 but they must be curiously transposed before they can be applied to the living speech and verified by observation4 ,in its pure and simple form the syllogism cannot be directly compared with all experience1 or it would not have required an ,aristotle to discover it4 ,it must be transmuted into all the possible shapes in which reasoning loves to clothe itself4 ,the transmutation is the mathematical process in the establishment of the law4--,,peirce1 ,,benjamin4 ,linear ,associative ,algebra2 ,american ,journal of ,mathematics1 ,vol4 #d 7#ahha71 p4 #ig4 #acjb4 ,in mathematics we see the conscious logical activity of our mind in its purest and most perfect form2 here is made manifest to us all the labor and the great care with which it progresses1 the precision which is necessary to determine exactly the source of the established general theorems1 and the difficulty with which we form and comprehend abstract conceptions2 but we also learn here to have confidence in the certainty1 breadth1 and fruitfulness of such intellectual labor4 --,,helmholtz1 ,h4 ,vortrge und ,reden1 ,bd4 #a 7,braunschweig1 #ahif71 p4 #agf4 #acjc4 ,mathematical demonstrations are a logic of as much or more use1 than that commonly learned at schools1 serving to a just formation of the mind1 enlarging its capacity1 and strengthening it so as to render the same capable of exact reasoning1 and discerning truth from falsehood in all occurrences1 even in subjects not mathematical4 ,for which reason it is said1 the ,egyptians1 ,persians1 and ,lacedaemonians seldom elected any new kings1 but such as had some knowledge in the mathematics1 imagining those1 who had not1 men of imperfect judgments1 and unfit to rule and govern4--,,franklin1 ,,benjamin4 ,usefulness of ,mathematics2 ,works 7,boston1 #ahdj71 ,vol4 #b1 p4 #fh4 #acjd4 ,the mathematical conception is1 from its very nature1 abstract2 indeed its abstractness is usually of a higher order than the abstractness of the logician4--,,chrystal1 ,,george4 ,encyclopedia ,britannica 7,ninth ,edition71 ,article 8,mathematics40 #acje4 ,mathematics1 that giant pincers of scientific logic4444 --,,halsted1 ,g4 ,b4 ,science 7#aije71 p4 #afa4 #acjf4 ,logic has borrowed the rules of geometry without understanding its power4444 ,i am far from placing logicians by the side of geometers who teach the true way to guide the reason4444 ,the method of avoiding error is sought by every one4 ,the logicians profess to lead the way1 the geometers alone reach it1 and aside from their science there is no true demonstration4 --,,pascal4 ,quoted by ,a4 ,rebire3 ,mathmatiques et ,mathmaticiens 7,paris1 #ahih71 pp4 #afb-afc4 #acjg4 ,mathematics1 like dialectics1 is an organ of the higher sense1 in its execution it is an art like eloquence4 ,to both nothing but the form is of value2 neither cares anything for content4 ,whether mathematics considers pennies or guineas1 whether rhetoric defends truth or error1 is perfectly immaterial to either4--,,goethe4 ,sprche in ,prosa1 ,natur ,,iv1 #idf4 #acjh4 ,confined to its true domain1 mathematical reasoning is admirably adapted to perform the universal office of sound logic3 to induce in order to deduce1 in order to construct4444 ,it contents itself to furnish1 in the most favorable domain1 a model of clearness1 of precision1 and consistency1 the close contemplation of which is alone able to prepare the mind to render other conceptions also as perfect as their nature permits4 ,its general reaction1 more negative than positive1 must consist1 above all1 in inspiring us everywhere with an invincible aversion for vagueness1 inconsistency1 and obscurity1 which may always be really avoided in any reasoning whatsoever1 if we make sufficient effort4--,,comte1 ,a4 ,subjective ,synthesis4 #acji4 ,formal thought1 consciously recognized as such1 is the means of all exact knowledge2 and a correct understanding of the main formal sciences1 ,logic and ,mathematics1 is the proper and only safe foundation for a scientific education4--,,lefevre1 ,,arthur4 ,number and its ,algebra 7,boston1 ,sect4 #bbb47 #acaj4 ,it has come to pass1 ,i know not how1 that ,mathematics and ,logic1 which ought to be but the handmaids of ,physic1 nevertheless presume on the strength of the certainty which they possess to exercise dominion over it4--,,bacon1 ,,francis4 ,de ,augmentis1 ,bk4 #c4 #acaa4 ,we may regard geometry as a practical logic1 for the truths which it considers1 being the most simple and most sensible of all1 are1 for this reason1 the most susceptible to easy and ready application of the rules of reasoning4--,d',,alembert4 ,quoted in ,a4 ,rebire3 ,mathmatiques et ,mathmaticiens 7,paris1 #ahih71 pp4 #aea-aeb4 #acab4 ,there are notable examples enough of demonstration outside of mathematics1 and it may be said that ,aristotle has already given some in his 8,prior ,analytics40 ,in fact logic is as susceptible of demonstration as geometry1 4444 ,archimedes is the first1 whose works we have1 who has practised the art of demonstration upon an occasion where he is treating of physics1 as he has done in his book on ,equilibrium4 ,furthermore1 jurists may be said to have many good demonstrations2 especially the ancient ,roman jurists1 whose fragments have been preserved to us in the ,pandects4--,,leibnitz1 ,g4 ,w4 ,new ,essay on ,human ,understanding ,langley1 ,bk4 #d1 chap4 #b1 sect4 #ab4 #acac4 ,it is commonly considered that mathematics owes its certainty to its reliance on the immutable principles of formal logic4 ,this 444 is only half the truth imperfectly expressed4 ,the other half would be that the principles of formal logic owe such a degree of permanence as they have largely to the fact that they have been tempered by long and varied use by mathematicians4 8,a vicious circle60 you will perhaps say4 ,i should rather describe it as an example of the process known by mathematicians as the method of successive approximation4--,b,,cher1 ,,maxime4 ,bulletin of the ,american ,mathematical ,society1 ,vol4 #aa1 p4 #abj4 #acad4 ,whatever advantage can be attributed to logic in directing and strengthening the action of the understanding is found in a higher degree in mathematical study1 with the immense added advantage of a determinate subject1 distinctly circumscribed1 admitting of the utmost precision1 and free from the danger which is inherent in all abstract logic1--of leading to useless and puerile rules1 or to vain ontological speculations4 ,the positive method1 being everywhere identical1 is as much at home in the art of reasoning as anywhere else3 and this is why no science1 whether biology or any other1 can offer any kind of reasoning1 of which mathematics does not supply a simpler and purer counterpart4 ,thus1 we are enabled to eliminate the only remaining portion of the old philosophy which could even appear to offer any real utility2 the logical part1 the value of which is irrevocably absorbed by mathematical science4--,,comte1 ,a4 ,positive ,philosophy ,martineau1 7,london1 #ahge71 ,vol4 #a1 pp4 #cba-cbb4 #acae4 ,we know that mathematicians care no more for logic than logicians for mathematics4 ,the two eyes of exact science are mathematics and logic3 the mathematical sect puts out the logical eye1 the logical sect puts out the mathematical eye2 each believing that it can see better with one eye than with two4 --,,de ,,morgan1 ,a4 ,quoted in ,f4 ,cajori3 ,history of ,mathematics 7,new ,york1 #ahig71 p4 #caf4 #acaf4 ,the progress of the art of rational discovery depends in a great part upon the art of characteristic 7ars characteristica74 ,the reason why people usually seek demonstrations only in numbers and lines and things represented by these is none other than that there are not1 outside of numbers1 convenient characters corresponding to the notions4--,,leibnitz1 ,g4 ,w4 ,philosophische ,schriften ,gerhardt ,bd4 #h1 p4 #aih4 #acag4 ,the influence of the mathematics of ,leibnitz upon his philosophy appears chiefly in connection with his law of continuity and his prolonged efforts to establish a ,logical ,calculus4444 ,to find a ,logical ,calculus 7implying a universal philosophical language or system of signs7 is an attempt to apply in theological and philosophical investigations an analytic method analogous to that which had proved so successful in ,geometry and ,physics4 ,it seemed to ,leibnitz that if all the complex and apparently disconnected ideas which make up our knowledge could be analysed into their simple elements1 and if these elements could each be represented by a definite sign1 we should have a kind of 8alphabet of human thoughts40 ,by the combination of these signs 7letters of the alphabet of thought7 a system of true knowledge would be built up1 in which reality would be more and more adequately represented or symbolized4444 ,in many cases the analysis may result in an infinite series of elements2 but the principles of the ,infinitesimal ,calculus in mathematics have shown that this does not necessarily render calculation impossible or inaccurate4 ,thus it seemed to ,leibnitz that a synthetic calculus1 based upon a thorough analysis1 would be the most effective instrument of knowledge that could be devised4 8,i feel10 he says1 8that controversies can never be finished1 nor silence imposed upon the ,sects1 unless we give up complicated reasonings in favor of simple calculations1 words of vague and uncertain meaning in favor of fixed symbols40 ,thus it will appear that 8every paralogism is nothing but an error of calculation40 8,when controversies arise1 there will be no more necessity of disputation between two philosophers than between two accountants4 ,nothing will be needed but that they should take pen in hand1 sit down with their counting-tables1 and 7having summoned a friend1 if they like7 say to one another3 ,let us calculate40--,,latta1 ,,robert4 ,leibnitz1 ,the ,monadology1 etc4 7,oxford1 #ahih71 p4 #he4 #acah4 ,pure mathematics was discovered by ,boole in a work which he called 8,the ,laws of ,thought04444 ,his work was concerned with formal logic1 and this is the same thing as mathematics4 --,,russell1 ,,bertrand4 ,international ,monthly1 #aija1 p4 #hc4 #acai4 ,mathematics is but the higher development of ,symbolic ,logic4--,,whetham1 ,w4 ,c4 ,d4 ,recent ,development of ,physical ,science 7,philadelphia1 #aijd71 p4 #cd4 #acbj4 ,symbolic ,logic has been disowned by many logicians on the plea that its interest is mathematical1 and by many mathematicians on the plea that its interest is logical4 --,,whitehead1 ,a4 ,n4 ,universal ,algebra 7,cambridge1 #ahih71 ,preface1 p4 #f4 #acba4 444 the two great components of the critical movement1 though distinct in origin and following separate paths1 are found to converge at last in the thesis3 ,symbolic ,logic is ,mathematics1 ,mathematics is ,symbolic ,logic1 the twain are one4--,,keyser1 ,c4 ,j4 ,lectures on ,science1 ,philosophy and ,art 7,new ,york1 #aijh71 p4 #ai4 #acbb4 ,the emancipation of logic from the yoke of ,aristotle very much resembles the emancipation of geometry from the bondage of ,euclid2 and1 by its subsequent growth and diversification1 logic1 less abundantly perhaps but not less certainly than geometry1 has illustrated the blessings of freedom4--,,keyser1 ,c4 ,j4 ,science1 ,vol4 #ce 7#aiab71 p4 #ajh4 #acbc4 ,i would express it as my personal view1 which is probably not yet shared generally1 that pure mathematics seems to me merely a branch of general logic2 that branch which is based on the concept of numbers1 to whose economic advantages is to be attributed the tremendous development which this particular branch has undergone as compared with the remaining branches of logic1 which until the most recent times have remained almost stationary4--,,schr,,der1 ,e4 ,ueber ,pasigraphie etc42 ,verhandlungen des #a4 ,internationalen ,mathematiker-,kongresses 7,leipzig1 #ahih71 p4 #adi4 #acbd4 ,if logical training is to consist1 not in repeating barbarous scholastic formulas or mechanically tacking together empty majors and minors1 but in acquiring dexterity in the use of trustworthy methods of advancing from the known to the unknown1 then mathematical investigation must ever remain one of its most indispensable instruments4 ,once inured to the habit of accurately imagining abstract relations1 recognizing the true value of symbolic conceptions1 and familiarized with a fixed standard of proof1 the mind is equipped for the consideration of quite other objects than lines and angles4 ,the twin treatises of ,adam ,smith on social science1 wherein1 by deducing all human phenomena first from the unchecked action of selfishness and then from the unchecked action of sympathy1 he arrives at mutually-limiting conclusions of transcendent practical importance1 furnish for all time a brilliant illustration of the value of mathematical methods and mathematical discipline4--,,fiske1 ,,john4 ,darwinism and other ,essays 7,boston1 #ahic71 pp4 #big-bih4 #acbe4 ,no irrational exaggeration of the claims of ,mathematics can ever deprive that part of philosophy of the property of being the natural basis of all logical education1 through its simplicity1 abstractness1 generality1 and freedom from disturbance by human passion4 ,there1 and there alone1 we find in full development the art of reasoning1 all the resources of which1 from the most spontaneous to the most sublime1 are continually applied with far more variety and fruitfulness than elsewhere24444 ,the more abstract portion of mathematics may in fact be regarded as an immense repository of logical resources1 ready for use in scientific deduction and co-ordination4 --,,comte1 ,a4 ,positive ,philosophy ,martineau1 7,london1 #ahge71 ,vol4 #b1 p4 #dci4 #acbf4 ,logic it is called referring to ,whitehead and ,russell's ,principia ,mathematica and logic it is1 the logic of propositions and functions and classes and relations1 by far the greatest 7not merely the biggest7 logic that our planet has produced1 so much that is new in matter and in manner2 but it is also mathematics1 a prolegomenon to the science1 yet itself mathematics in its most genuine sense1 differing from other parts of the science only in the respects that it surpasses these in fundamentality1 generality and precision1 and lacks traditionality4 ,few will read it1 but all will feel its effect1 for behind it is the urgence and push of a magnificent past3 two thousand five hundred years of record and yet longer tradition of human endeavor to think aright4--,,keyser1 ,c4 ,j4 ,science1 ,vol4 #ce 7#aiab71 p4 #aaj4 ,,chapter ,,xiv ,,mathematics ,,and ,,philosophy #adja4 ,socrates is praised by all the centuries for having called philosophy from heaven to men on earth2 but if1 knowing the condition of our science1 he should come again and should look once more to heaven for a means of curing men1 he would there find that to mathematics1 rather than to the philosophy of today1 had been given the crown because of its industry and its most happy and brilliant successes4--,,herbart1 ,j4 ,f4 ,werke ,kehrbach1 7,langensalza1 #ahij71 ,bd4 #e1 p4 #ie4 #adjb4 ,it is the embarrassment of metaphysics that it is able to accomplish so little with the many things that mathematics offers her4--,,kant1 ,e4 ,metaphysische ,anfangsgrnde der ,naturwissenschaft1 ,vorrede4 #adjc4 ,philosophers1 when they have possessed a thorough knowledge of mathematics1 have been among those who have enriched the science with some of its best ideas4 ,on the other hand it must be said that1 with hardly an exception1 all the remarks on mathematics made by those philosophers who have possessed but a slight or hasty or late-acquired knowledge of it are entirely worthless1 being either trivial or wrong4--,,whitehead1 ,a4 ,n4 ,introduction to ,mathematics 7,new ,york1 #aiaa71 p4 #aac4 #adjd4 ,the union of philosophical and mathematical productivity1 which besides in ,plato we find only in ,pythagoras1 ,descartes and ,leibnitz1 has always yielded the choicest fruits to mathematics3 ,to the first we owe scientific mathematics in general1 ,plato discovered the analytic method1 by means of which mathematics was elevated above the view-point of the elements1 ,descartes created the analytical geometry1 our own illustrious countryman discovered the infinitesimal calculus--and just these are the four greatest steps in the development of mathematics4 --,,hankel1 ,,hermann4 ,geschichte der ,mathematik im ,altertum und im ,mittelalter 7,leipzig1 #ahgd71 pp4 #adi-aej4 #adje4 ,without mathematics one cannot fathom the depths of philosophy2 without philosophy one cannot fathom the depths of mathematics2 without the two one cannot fathom anything4 --,,bordas-,,demoulins4 ,quoted in ,a4 ,rebire3 ,mathmatiques et ,mathmaticiens 7,paris1 #ahih71 p4 #adg4 #adjf4 ,in the end mathematics is but simple philosophy1 and philosophy1 higher mathematics in general4--,,novalis4 ,schriften 7,berlin1 #aija71 ,teil #b1 p4 #ddc4 #adjg4 ,it is a safe rule to apply that1 when a mathematical or philosophical author writes with a misty profundity1 he is talking nonsense4--,,whitehead1 ,a4 ,n4 ,introduction to ,mathematics 7,new ,york1 #aiaa71 p4 #bbg4 #adjh4 ,the real finisher of our education is philosophy1 but it is the office of mathematics to ward off the dangers of philosophy4--,,herbart1 ,j4 ,f4 ,pestalozzi's ,idee eines ,,abc der ,anschauung2 ,werke ,kehrbach1 7,langensalza1 #ahij71 ,bd4 #a1 p4 #afh4 #adji4 ,since antiquity mathematics has been regarded as the most indispensable school for philosophic thought and in its highest spheres the research of the mathematician is indeed most closely related to pure speculation4 ,mathematics is the most perfect union between exact knowledge and theoretical thought4--,,curtius1 ,e4 ,berliner ,monatsberichte 7#ahgc71 p4 #eag4 #adaj4 ,geometry has been1 throughout1 of supreme importance in the history of knowledge4--,,russell1 ,,bertrand4 ,foundations of ,geometry 7,cambridge1 #ahig71 p4 #ed4 #adaa4 ,he is unworthy of the name of man who is ignorant of the fact that the diagonal of a square is incommensurable with its side4--,,plato4 ,quoted by ,sophie ,germain3 ,mmoire sur les surfaces lastiques4 #adab4 ,mathematics1 considered as a science1 owes its origin to the idealistic needs of the ,greek philosophers1 and not as fable has it1 to the practical demands of ,egyptian economics4444 ,adam was no zologist when he gave names to the beasts of the field1 nor were the ,egyptian surveyors mathematicians4--,,hankel1 ,h4 ,die ,entwickelung der ,mathematik in den letzten ,jahrhunderten 7,tbingen1 #ahhd71 p4 #g4 #adac4 ,there are only two ways open to man for attaining a certain knowledge of truth3 clear intuition and necessary deduction4--,,descartes4 ,rules for the ,direction of the ,mind2 ,torrey's ,the ,philosophy of ,descartes 7,new ,york1 #ahib71 p4 #ajd4 #adad4 ,mathematicians have1 in many cases1 proved some things to be possible and others to be impossible1 which1 without demonstration1 would not have been believed4444 ,mathematics afford many instances of impossibilities in the nature of things1 which no man would have believed1 if they had not been strictly demonstrated4 ,perhaps1 if we were able to reason demonstratively in other subjects1 to as great extent as in mathematics1 we might find many things to be impossible1 which we conclude1 without hesitation1 to be possible4--,,reid1 ,,thomas4 ,essay on the ,intellectual ,powers of ,man1 ,essay #d1 chap4 #c4 #adae4 ,if philosophers understood mathematics1 they would know that indefinite speech1 which permits each one to think what he pleases and produces a constantly increasing difference of opinion1 is utterly unable1 in spite of all fine words and even in spite of the magnitude of the objects which are under contemplation1 to maintain a balance against a science which instructs and advances through every word which it utters and which at the same time wins for itself endless astonishment1 not through its survey of immense spaces1 but through the exhibition of the most prodigious human ingenuity which surpasses all power of description4--,,herbart1 ,j4 ,f4 ,werke ,kehrbach 7,langensalza1 #ahij71 ,bd4 #e1 p4 #aje4 #adaf4 ,german intellect is an excellent thing1 but when a ,german product is presented it must be analysed4 ,most probably it is a combination of intellect 7,i7 and tobacco-smoke 7,t74 ,certainly ,i,t1 and ,i,t1 occur2 but ,i,t is more common1 and ,i,t and ,i,t occur4 ,in many cases metaphysics 7,m7 occurs and ,i hold that ,ia,tb,mc never occurs without b c #b;a4 ,n4 ,b4--,be careful1 in analysing the compounds of the three1 not to confound ,t and ,m1 which are strongly suspected to be isomorphic4 ,thus1 ,i,t,m may easily be confounded with ,i,t4 ,as far as ,i dare say anything1 those who have placed ,hegel1 ,fichte1 etc41 in the rank of the extenders of ,kant have imagined ,t and ,m to be identical4--,,de ,,morgan1 ,a4 ,graves' ,life of ,w4 ,r4 ,hamilton 7,new ,york1 #ahhb-ahhi71 ,vol4 #ac1 p4 #ddf4 #adag4 ,the discovery of ,ceres was made by ,g4 ,piazzi of ,palermo2 and it was the more interesting as its announcement occurred simultaneously with a publication by ,hegel in which he severely criticized astronomers for not paying more attention to philosophy1 a science1 said he1 which would at once have shown them that there could not possibly be more than seven planets1 and a study of which would therefore have prevented an absurd waste of time in looking for what in the nature of things could never be found4--,,ball1 ,w4 ,w4 ,r4 ,history of ,mathematics 7,london1 #aija71 p4 #deh4 #adah4 ,but who shall parcel out ,his intellect by geometric rules1 ,split like a province into round and square8 --,,wordsworth4 ,the ,prelude1 ,bk4 #b4 #adai4 ,and ,proposition1 gentle maid1 ,who soothly ask'd stern ,demonstration's aid1 4444 --,,coleridge1 ,s4 ,t4 ,a ,mathematical ,problem4 #adbj4 ,mathematics connect themselves on the one side with common life and physical science2 on the other side with philosophy in regard to our notions of space and time1 and in the questions which have arisen as to the universality and necessity of the truths of mathematics and the foundation of our knowledge of them4--,,cayley1 ,,arthur4 ,british ,association ,address 7#ahhh72 ,collected ,mathematical ,papers1 ,vol4 #aa1 p4 #dcj4 #adba4 ,mathematical teaching 444 trains the mind to capacities1 which 444 are of the closest kin to those of the greatest metaphysician and philosopher4 ,there is some color of truth for the opposite doctrine in the case of elementary algebra4 ,the resolution of a common equation can be reduced to almost as mechanical a process as the working of a sum in arithmetic4 ,the reduction of the question to an equation1 however1 is no mechanical operation1 but one which1 according to the degree of its difficulty1 requires nearly every possible grade of ingenuity3 not to speak of the new1 and in the present state of the science insoluble1 equations1 which start up at every fresh step attempted in the application of mathematics to other branches of knowledge4--,,mill1 ,j4 ,s4 ,an ,examination of ,sir ,william ,hamilton's ,philosophy 7,london1 #ahgh71 p4 #fae4 #adbb4 ,the value of mathematical instruction as a preparation for those more difficult investigations1 consists in the applicability not of its doctrines1 but of its methods4 ,mathematics will ever remain the most perfect type of the ,deductive ,method in general2 and the applications of mathematics to the simpler branches of physics1 furnish the only school in which philosophers can effectually learn the most difficult and important portion of their art1 the employment of the laws of the simpler phenomena for explaining and predicting those of the more complex4 ,these grounds are quite sufficient for deeming mathematical training an indispensable basis of real scientific education1 and regarding1 with ,plato1 one who is 1 as wanting in one of the most essential qualifications for the successful cultivation of the higher branches of philosophy4--,,mill1 ,j4 ,s4 ,system of ,logic1 ,bk4 #c1 chap4 #bd1 sect4 #i4 #adbc4 ,in metaphysical reasoning1 the process is always short4 ,the conclusion is but a step or two1 seldom more1 from the first principles or axioms on which it is grounded1 and the different conclusions depend not one upon another4 ,it is otherwise in mathematical reasoning4 ,here the field has no limits4 ,one proposition leads on to another1 that to a third1 and so on without end4 ,if it should be asked1 why demonstrative reasoning has so wide a field in mathematics1 while1 in other abstract subjects1 it is confined within very narrow limits1 ,i conceive this is chiefly owing to the nature of quantity1 444 mathematical quantities being made up of parts without number1 can touch in innumerable points1 and be compared in innumerable different ways4--,,reid1 ,,thomas4 ,essays on the ,powers of the ,human ,mind 7,edinburgh1 #ahab71 ,vol4 #b1 pp4 #dbb-dbc4 #adbd4 ,the power of ,reason 444 is unquestionably the most important by far of those which are comprehended under the general title of ,intellectual4 ,it is on the right use of this power that our success in the pursuit of both knowledge and of happiness depends2 and it is by the exclusive possession of it that man is distinguished1 in the most essential respects1 from the lower animals4 ,it is1 indeed1 from their subserviency to its operations1 that the other faculties 444 derive their chief value4--,,stewart1 ,,dugald4 ,philosophy of the ,human ,mind2 ,collected ,works 7,edinburgh1 #ahed71 ,vol4 #h1 p4 #e4 #adbe4 ,when 444 ,i asked myself why was it then that the earliest philosophers would admit to the study of wisdom only those who had studied mathematics1 as if this science was the easiest of all and the one most necessary for preparing and disciplining the mind to comprehend the more advanced1 ,i suspected that they had knowledge of a mathematical science different from that of our time4444 ,i believe ,i find some traces of these true mathematics in ,pappus and ,diophantus1 who1 although they were not of extreme antiquity1 lived nevertheless in times long preceding ours4 ,but ,i willingly believe that these writers themselves1 by a culpable ruse1 suppressed the knowledge of them2 like some artisans who conceal their secret1 they feared1 perhaps1 that the ease and simplicity of their method1 if become popular1 would diminish its importance1 and they preferred to make themselves admired by leaving to us1 as the product of their art1 certain barren truths deduced with subtlety1 rather than to teach us that art itself1 the knowledge of which would end our admiration4--,,descartes4 ,rules for the ,direction of the ,mind2 ,philosophy of ,descartes ,torrey1 7,new ,york1 #ahib71 pp4 #gj-ga4 #adbf4 ,if we rightly adhere to our rule that is1 that we should occupy ourselves only with those subjects in reference to which the mind is capable of acquiring certain and indubitable knowledge there will remain but few things to the study of which we can devote ourselves4 ,there exists in the sciences hardly a single question upon which men of intellectual ability have not held different opinions4 ,but whenever two men pass contrary judgment on the same thing1 it is certain that one of the two is wrong4 ,more than that1 neither of them has the truth2 for if one of them had a clear and precise insight into it1 he could so exhibit it to his opponent as to end the discussion by compelling his conviction4444 ,it follows from this1 if we reckon rightly1 that among existing sciences there remain only geometry and arithmetic1 to which the observance of our rule would bring us4 --,,descartes4 ,rules for the ,direction of the ,mind2 ,philosophy of ,descartes ,torrey1 7,new ,york1 #ahib71 p4 #fb4 #adbg4 ,the same reason which led ,plato to recommend the study of arithmetic led him to recommend also the study of geometry4 ,the vulgar crowd of geometricians1 he says1 will not understand him4 ,they have practice always in view4 ,they do not know that the real use of the science is to lead men to the knowledge of abstract1 essential1 eternal truth4 7,plato's ,republic1 ,book #g74 ,indeed if we are to believe ,plutarch1 ,plato carried his feeling so far that he considered geometry as degraded by being applied to any purpose of vulgar utility4 ,archytas1 it seems1 had framed machines of extraordinary power on mathematical principles4 7,plutarch1 ,sympos41 ,,viii41 and ,life of ,marcellus4 ,the machines of ,archytas are also mentioned by ,aulus ,gellius and ,diogenes ,laertius74 ,plato remonstrated with his friend1 and declared that this was to degrade a noble intellectual exercise into a low craft1 fit only for carpenters and wheelwrights4 ,the office of geometry1 he said1 was to discipline the mind1 not to minister to the base wants of the body4 ,his interference was successful2 and from that time according to ,plutarch1 the science of mechanics was considered unworthy of the attention of a philosopher4 --,,macaulay4 ,lord ,bacon2 ,edinburgh ,review1 ,july1 #ahcg4 #adbh4 ,the intellectual habits of the ,mathematicians are1 in some respects1 the same with those of the ,metaphysicians we have been now considering2 but1 in other respects1 they differ widely4 ,both are favourable to the improvement of the power of attention1 but not in the same manner1 nor in the same degree4 ,those of the metaphysician give capacity of fixing the attention on the subjects of our consciousness1 without being distracted by things external2 but they afford little or no exercise to that species of attention which enables us to follow long processes of reasoning1 and to keep in view all the various steps of an investigation till we arrive at the conclusion4 ,in mathematics1 such processes are much longer than in any other science2 and hence the study of it is peculiarly calculated to strengthen the power of steady and concatenated thinking1--a power which1 in all the pursuits of life1 whether speculative or active1 is one of the most valuable endowments we can possess4 ,this command of attention1 however1 it may be proper to add1 is to be acquired1 not by the practice of modern methods1 but by the study of ,greek geometry1 more particularly1 by accustoming ourselves to pursue long trains of demonstration1 without availing ourselves of the aid of any sensible diagrams2 the thoughts being directed solely by those ideal delineations which the powers of conception and of memory enable us to form4--,,stewart1 ,,dugald4 ,philosophy of the ,human ,mind1 ,part #c1 chap4 #a1 sect4 #c4 #adbi4 ,they the ,greeks speculated and theorized under a lively persuasion that a ,science of every part of nature was possible1 and was a fit object for the exercise of a man's best faculties2 and they were speedily led to the conviction that such a science must clothe its conclusions in the language of mathematics4 ,this conviction is eminently conspicuous in the writings of ,plato4444 ,probably no succeeding step in the discovery of the ,laws of ,nature was of so much importance as the full adoption of this pervading conviction1 that there must be ,mathematical ,laws of ,nature1 and that it is the business of ,philosophy to discover these ,laws4 ,this conviction continues1 through all the succeeding ages of the history of the science1 to be the animating and supporting principle of scientific investigation and discovery4 --,,whewell1 ,w4 ,history of the ,inductive ,sciences1 ,vol4 #a1 bk4 #b1 chap4 #c4 #adcj4 ,for to pass by those ,ancients1 the wonderful ,pythagoras1 the sagacious ,democritus1 the divine ,plato1 the most subtle and very learned ,aristotle1 ,men whom every ,age has hitherto acknowledged as deservedly honored1 as the greatest ,philosophers1 the ,ring-leaders of ,arts2 in whose ,judgments how much these ,studies mathematics were esteemed1 is abundantly proclaimed in ,history and confirmed by their famous ,monuments1 which are everywhere interspersed and bespangled with ,mathematical ,reasonings and ,examples1 as with so many ,stars2 and consequently anyone not in some ,degree conversant in these ,studies will in vain expect to understand1 or unlock their hidden ,meanings1 without the ,help of a ,mathematical ,key3 ,for who can play well on ,aristotle's ,instrument but with a ,mathematical ,quill2 or not be altogether deaf to the ,lessons of natural ,philosophy1 while ignorant of ,geometry8 ,who void of 7,geometry shall ,i say1 or7 ,arithmetic can comprehend ,plato's ,socrates lisping with ,children concerning ,square ,numbers2 or can conceive ,plato himself treating not only of the ,universe1 but the ,polity of ,commonwealths regulated by the ,laws of ,geometry1 and formed according to a ,mathematical ,plan8--,,barrow1 ,,isaac4 ,mathematical ,lectures 7,london1 #agcd71 pp4 #bf-bg4 #adca4 ,and ,reason now through number1 time1 and space ,darts the keen lustre of her serious eye2 ,and learns from facts compar'd the laws to trace ,whose long procession leads to ,deity --,,beattie1 ,,james4 ,the ,minstrel1 ,bk4 #b1 stanza #dg4 #adcb4 ,that ,egyptian and ,chaldean wisdom mathematical wherewith ,moses and ,daniel were furnished1 4444--,,hooker1 ,,richard4 ,ecclesiastical ,polity1 ,bk4 #c1 sect4 #h4 #adcc4 ,general and certain truths are only founded in the habitudes and relations of abstract ideas4 ,a sagacious and methodical application of our thoughts1 for the finding out of these relations1 is the only way to discover all that can be put with truth and certainty concerning them into general propositions4 ,by what steps we are to proceed in these1 is to be learned in the schools of mathematicians1 who1 from very plain and easy beginnings1 by gentle degrees1 and a continued chain of reasonings1 proceed to the discovery and demonstration of truths that appear at first sight beyond human capacity4 ,the art of finding proofs1 and the admirable method they have invented for the singling out and laying in order those intermediate ideas that demonstratively show the equality or inequality of unapplicable quantities1 is that which has carried them so far and produced such wonderful and unexpected discoveries2 but whether something like this1 in respect of other ideas1 as well as those of magnitude1 may not in time be found out1 ,i will not determine4 ,this1 ,i think1 ,i may say1 that if other ideas that are the real as well as the nominal essences of their species1 were pursued in the way familiar to mathematicians1 they would carry our thoughts further1 and with greater evidence and clearness than possibly we are apt to imagine4--,,locke1 ,,john4 ,an ,essay concerning ,human ,understanding1 ,bk4 #d1 chap4 #ab1 sect4 #g4 #adcd4 ,those long chains of reasoning1 quite simple and easy1 which geometers are wont to employ in the accomplishment of their most difficult demonstrations1 led me to think that everything which might fall under the cognizance of the human mind might be connected together in a similar manner1 and that1 provided only that one should take care not to receive anything as true which was not so1 and if one were always careful to preserve the order necessary for deducing one truth from another1 there would be none so remote at which he might not at last arrive1 nor so concealed which he might not discover4--,,descartes4 ,discourse upon ,method1 part #b2 ,the ,philosophy of ,descartes ,torrey1 7,new ,york1 #ahib71 p4 #dg4 #adce4 ,if anyone wished to write in mathematical fashion in metaphysics or ethics1 nothing would prevent him from so doing with vigor4 ,some have professed to do this1 and we have a promise of mathematical demonstrations outside of mathematics2 but it is very rare that they have been successful4 ,this is1 ,i believe1 because they are disgusted with the trouble it is necessary to take for a small number of readers where they would ask as in ,persius3 ,quis leget haec1 and reply3 ,vel duo vel nemo4 --,,leibnitz4 ,new ,essay concerning ,human ,understanding1 ,langley1 ,bk #b1 chap4 #bi1 sect4 #ab4 #adcf4 ,it is commonly asserted that mathematics and philosophy differ from one another according to their objects1 the former treating of quantity1 the latter of quality4 ,all this is false4 ,the difference between these sciences cannot depend on their object2 for philosophy applies to everything1 hence also to quanta1 and so does mathematics in part1 inasmuch as everything has magnitude4 ,it is only the different kind of rational knowledge or application of reason in mathematics and philosophy which constitutes the specific difference between these two sciences4 ,for philosophy is rational knowledge from mere concepts1 mathematics1 on the contrary1 is rational knowledge from the construction of concepts4 ,we construct concepts when we represent them in intuition a priori1 without experience1 or when we represent in intuition the object which corresponds to our concept of it4--,the mathematician can never apply his reason to mere concepts1 nor the philosopher to the construction of concepts4--,in mathematics the reason is employed in concreto1 however1 the intuition is not empirical1 but the object of contemplation is something a priori4 ,in this1 as we see1 mathematics has an advantage over philosophy1 the knowledge in the former being intuitive1 in the latter1 on the contrary1 only discursive4 ,but the reason why in mathematics we deal more with quantity lies in this1 that magnitudes can be constructed in intuition a priori1 while qualities1 on the contrary1 do not permit of being represented in intuition4--,,kant1 ,e4 ,logik2 ,werke ,hartenstein1 7,leipzig1 #ahfh71 ,bd4 #h1 pp4 #bc-bd4 #adcg4 ,kant has divided human ideas into the two categories of quantity and quality1 which1 if true1 would destroy the universality of ,mathematics2 but ,descartes' fundamental conception of the relation of the concrete to the abstract in ,mathematics abolishes this division1 and proves that all ideas of quality are reducible to ideas of quantity4 ,he had in view geometrical phenomena only2 but his successors have included in this generalization1 first1 mechanical phenomena1 and1 more recently1 those of heat4 ,there are now no geometers who do not consider it of universal application1 and admit that every phenomenon may be as logically capable of being represented by an equation as a curve or a motion1 if only we were always capable 7which we are very far from being7 of first discovering1 and then resolving it4 ,the limitations of ,mathematical science are not1 then1 in its nature4 ,the limitations are in our intelligence3 and by these we find the domain of the science remarkably restricted1 in proportion as phenomena1 in becoming special1 become complex4 --,,comte1 ,a4 ,positive ,philosophy ,martineau1 ,bk4 #a1 chap4 #a4 #adch4 ,the great advantage of the mathematical sciences over the moral consists in this1 that the ideas of the former1 being sensible1 are always clear and determinate1 the smallest distinction between them being immediately perceptible1 and the same terms are still expressive of the same ideas1 without ambiguity or variation4 ,an oval is never mistaken for a circle1 nor an hyperbola for an ellipsis4 ,the isosceles and scalenum are distinguished by boundaries more exact than vice and virtue1 right or wrong4 ,if any term be defined in geometry1 the mind readily1 of itself1 substitutes on all occasions1 the definition for the thing defined3 ,or even when no definition is employed1 the object itself may be represented to the senses1 and by that means be steadily and clearly apprehended4 ,but the finer sentiments of the mind1 the operations of the understanding1 the various agitations of the passions1 though really in themselves distinct1 easily escape us1 when surveyed by reflection2 nor is it in our power to recall the original object1 so often as we have occasion to contemplate it4 ,ambiguity1 by this means1 is gradually introduced into our reasonings3 ,similar objects are readily taken to be the same3 ,and the conclusion becomes at last very wide off the premises4--,,hume1 ,,david4 ,an ,inquiry concerning ,human ,understanding1 sect4 #g1 part #a4 #adci4 ,one part of these disadvantages in moral ideas which has made them be thought not capable of demonstration1 may in a good measure be remedied by definitions1 setting down that collection of simple ideas1 which every term shall stand for2 and then using the terms steadily and constantly for that precise collection4 ,and what methods algebra1 or something of that kind1 may hereafter suggest1 to remove the other difficulties1 it is not easy to foretell4 ,confident1 ,i am1 that if men would in the same method1 and with the same indifferency1 search after moral as they do mathematical truths1 they would find them have a stronger connexion one with another1 and a more necessary consequence from our clear and distinct ideas1 and to come nearer perfect demonstration than is commonly imagined4--,,locke1 ,,john4 ,an ,essay concerning ,human ,understanding1 ,bk4 #d1 chap4 #c1 sect4 #bj4 #addj4 ,that which in this respect has given the advantage to the ideas of quantity1 and made them thought more capable of certainty and demonstration than moral ideas1 is1 ,first1 ,that they can be set down and represented by sensible marks1 which have a greater and nearer correspondence with them than any words or sounds whatsoever4 ,diagrams drawn on paper are copies of the ideas in the mind1 and not liable to the uncertainty that words carry in their signification4 ,an angle1 circle1 or square1 drawn in lines1 lies open to the view1 and cannot be mistaken3 it remains unchangeable1 and may at leisure be considered and examined1 and the demonstration be revised1 and all the parts of it may be gone over more than once1 without any danger of the least change in the ideas4 ,this cannot be done in moral ideas3 we have no sensible marks that resemble them1 whereby we can set them down2 we have nothing but words to express them by2 which1 though when written they remain the same1 yet the ideas they stand for may change in the same man2 and it is seldom that they are not different in different persons4 ,secondly1 ,another thing that makes the greater difficulty in ethics is1 ,that moral ideas are commonly more complex than those of the figures ordinarily considered in mathematics4 ,from whence these two inconveniences follow3--,first1 that their names are of more uncertain signification1 the precise collection of simple ideas they stand for not being so easily agreed on2 and so the sign that is used for them in communication always1 and in thinking often1 does not steadily carry with it the same idea4 ,upon which the same disorder1 confusion1 and error follow1 as would if a man1 going to demonstrate something of an heptagon1 should1 in the diagram he took to do it1 leave out one of the angles1 or by oversight make the figure with an angle more than the name ordinarily imported1 or he intended it should when at first he thought of his demonstration4 ,this often happens1 and is hardly avoidable in very complex moral ideas1 where the same name being retained1 an angle1 i4e4 one simple idea is left out1 or put in the complex one 7still called by the same name7 more at one time than another4 ,secondly1 ,from the complexedness of these moral ideas there follows another inconvenience1 viz41 that the mind cannot easily retain those precise combinations so exactly and perfectly as is necessary in the examination of the habitudes and correspondences1 agreements or disagreements1 of several of them one with another2 especially where it is to be judged of by long deductions and the intervention of several other complex ideas to show the agreement or disagreement of two remote ones4 --,,locke1 ,,john4 ,an ,essay concerning ,human ,understanding1 ,bk4 #d1 chap4 #c1 sect4 #ai4 #adda4 ,it has been generally taken for granted1 that mathematics alone are capable of demonstrative certainty3 but to have such an agreement or disagreement as may be intuitively perceived1 being1 as ,i imagine1 not the privileges of the ideas of number1 extension1 and figure alone1 it may possibly be the want of due method and application in us1 and not of sufficient evidence in things1 that demonstration has been thought to have so little to do in other parts of knowledge1 and been scarce so much as aimed at by any but mathematicians4 ,for whatever ideas we have wherein the mind can perceive the immediate agreement or disagreement that is between them1 there the mind is capable of intuitive knowledge1 and where it can perceive the agreement or disagreement of any two ideas1 by an intuitive perception of the agreement or disagreement they have with any intermediate ideas1 there the mind is capable of demonstration3 which is not limited to the idea of extension1 figure1 number1 and their modes4--,,locke1 ,,john4 ,an ,essay concerning ,human ,understanding1 ,bk4 #d1 chap4 #b1 sect4 #i4 #addb4 ,now ,i shall remark again what ,i have already touched upon more than once1 that it is a common opinion that only mathematical sciences are capable of a demonstrative certainty2 but as the agreement and disagreement which may be known intuitively is not a privilege belonging only to the ideas of numbers and figures1 it is perhaps for want of application on our part that mathematics alone have attained to demonstrations4 --,,leibnitz4 ,new ,essay concerning ,human ,understanding1 ,bk4 #d1 chap4 #b1 sect4 #i ,langley4 ,,chapter ,,xv ,,mathematics ,,and ,,science #aeja4 ,how comes it about that the knowledge of other sciences1 which depend upon this mathematics1 is painfully sought1 and that no one puts himself to the trouble of studying this science itself8 ,i should certainly be surprised1 if ,i did not know that everybody regarded it as being very easy1 and if ,i had not long ago observed that the human mind1 neglecting what it believes to be easy1 is always in haste to run after what is novel and advanced4--,,descartes4 ,rules for the ,direction of the ,mind2 ,philosophy of ,descartes ,torrey1 7,new ,york1 #ahib71 p4 #gb4 #aejb4 ,all quantitative determinations are in the hands of mathematics1 and it at once follows from this that all speculation which is heedless of mathematics1 which does not enter into partnership with it1 which does not seek its aid in distinguishing between the manifold modifications that must of necessity arise by a change of quantitative determinations1 is either an empty play of thoughts1 or at most a fruitless effort4 ,in the field of speculation many things grow which do not start from mathematics nor give it any care1 and ,i am far from asserting that all that thus grow are useless weeds1 among them may be many noble plants1 but without mathematics none will develop to complete maturity4--,,herbart1 ,j4 ,f4 ,werke 7,kehrbach71 7,langensalza1 #ahij71 ,bd4 #e1 p4 #ajf4 #aejc4 ,there are few things which we know1 which are not capable of being reduc'd to a ,mathematical ,reasoning1 and when they cannot1 it's a sign our knowledge of them is very small and confus'd2 and where a mathematical reasoning can be had1 it's as great folly to make use of any other1 as to grope for a thing in the dark1 when you have a candle standing by you4--,,arbuthnot4 ,quoted in ,todhunter's ,history of the ,theory of ,probability 7,cambridge and ,london1 #ahfe71 p4 #ea4 #aejd4 ,mathematical ,analysis is 444 the true rational basis of the whole system of our positive knowledge4--,,comte1 ,a4 ,positive ,philosophy ,martineau1 ,bk4 #a1 chap4 #a4 #aeje4 ,it is only through ,mathematics that we can thoroughly understand what true science is4 ,here alone we can find in the highest degree simplicity and severity of scientific law1 and such abstraction as the human mind can attain4 ,any scientific education setting forth from any other point1 is faulty in its basis4--,,comte1 ,a4 ,positive ,philosophy ,martineau1 ,bk4 #a1 chap4 #a4 #aejf4 ,in the present state of our knowledge we must regard ,mathematics less as a constituent part of natural philosophy than as having been1 since the time of ,descartes and ,newton1 the true basis of the whole of natural philosophy2 though it is1 exactly speaking1 both the one and the other4 ,to us it is of less use for the knowledge of which it consists1 substantial and valuable as that knowledge is1 than as being the most powerful instrument that the human mind can employ in the investigation of the laws of natural phenomena4--,,comte1 ,a4 ,positive ,philosophy ,martineau1 ,introduction1 chap4 #b4 #aejg4 ,the concept of mathematics is the concept of science in general4--,,novalis4 ,schriften 7,berlin1 #aija71 ,teil #b1 p4 #bbb4 #aejh4 ,i contend1 that each natural science is real science only in so far as it is mathematical4444 ,it may be that a pure philosophy of nature in general 7that is1 a philosophy which concerns itself only with the general concepts of nature7 is possible without mathematics1 but a pure science of nature dealing with definite objects 7physics or psychology71 is possible only by means of mathematics1 and since each natural science contains only as much real science as it contains a priori knowledge1 each natural science becomes real science only to the extent that it permits the application of mathematics4 --,,kant1 ,e4 ,metaphysische ,anfangsgrnde der ,naturwissenschaft1 ,vorrede4 #aeji4 ,the theory most prevalent among teachers is that mathematics affords the best training for the reasoning powers2444 ,the modern1 and to my mind true1 theory is that mathematics is the abstract form of the natural sciences2 and that it is valuable as a training of the reasoning powers1 not because it is abstract1 but because it is a representation of actual things4--,,safford1 ,t4 ,h4 ,mathematical ,teaching etc4 7,boston1 #ahhf71 p4 #i4 #aeaj4 ,it seems to me that no one science can so well serve to co-ordinate and1 as it were1 bind together all of the sciences as the queen of them all1 mathematics4--,,davis1 ,e4 ,w4 ,proceedings ,nebraska ,academy of ,sciences for #ahif 7,lincoln1 #ahig71 p4 #bhb4 #aeaa4 ,and as for ,mixed ,mathematics1 ,i may only make this prediction1 that there cannot fail to be more kinds of them1 as nature grows further disclosed4--,,bacon1 ,,francis4 ,advancement of ,learning1 ,bk4 #b2 ,de ,augmentis1 ,bk4 #c4 #aeab4 ,besides the exercise in keen comprehension and the certain discovery of truth1 mathematics has another formative function1 that of equipping the mind for the survey of a scientific system4--,,grassmann1 ,h4 ,stcke aus dem ,lehrbuche der ,arithmetik2 ,werke 7,leipzig1 #aijd71 ,bd4 #b1 p4 #bih4 #aeac4 ,mathematicks may help the naturalists1 both to frame hypotheses1 and to judge of those that are proposed to them1 especially such as relate to mathematical subjects in conjunction with others4--,,boyle1 ,,robert4 ,works 7,london1 #aggb71 ,vol4 #c1 p4 #dbi4 #aead4 ,the more progress physical sciences make1 the more they tend to enter the domain of mathematics1 which is a kind of centre to which they all converge4 ,we may even judge of the degree of perfection to which a science has arrived by the facility with which it may be submitted to calculation4--,,quetelet4 ,quoted in ,e4 ,mailly's ,eulogy on ,quetelet2 ,smithsonian ,report1 #ahgd1 p4 #agc4 #aeae4 ,the mathematical formula is the point through which all the light gained by science passes in order to be of use to practice2 it is also the point in which all knowledge gained by practice1 experiment1 and observation must be concentrated before it can be scientifically grasped4 ,the more distant and marked the point1 the more concentrated will be the light coming from it1 the more unmistakable the insight conveyed4 ,all scientific thought1 from the simple gravitation formula of ,newton1 through the more complicated formulae of physics and chemistry1 the vaguer so called laws of organic and animated nature1 down to the uncertain statements of psychology and the data of our social and historical knowledge1 alike partakes of this characteristic1 that it is an attempt to gather up the scattered rays of light1 the different parts of knowledge1 in a focus1 from whence it can be again spread out and analyzed1 according to the abstract processes of the thinking mind4 ,but only when this can be done with a mathematical precision and accuracy is the image sharp and well-defined1 and the deductions clear and unmistakable4 ,as we descend from the mechanical1 through the physical1 chemical1 and biological1 to the mental1 moral1 and social sciences1 the process of focalization becomes less and less perfect1--the sharp point1 the focus1 is replaced by a larger or smaller circle1 the contours of the image become less and less distinct1 and with the possible light which we gain there is mingled much darkness1 the sources of many mistakes and errors4 ,but the tendency of all scientific thought is toward clearer and clearer definition2 it lies in the direction of a more and more extended use of mathematical measurements1 of mathematical formulae4--,,merz1 ,j4 ,t4 ,history of ,european ,thought in the #aith ,century 7,edinburgh and ,london1 #aijd71 ,vol4 #a1 p4 #ccc4 #aeaf4 ,from the very outset of his investigations the physicist has to rely constantly on the aid of the mathematician1 for even in the simplest cases1 the direct results of his measuring operations are entirely without meaning until they have been submitted to more or less of mathematical discussion4 ,and when in this way some interpretation of the experimental results has been arrived at1 and it has been proved that two or more physical quantities stand in a definite relation to each other1 the mathematician is very often able to infer1 from the existence of this relation1 that the quantities in question also fulfill some other relation1 that was previously unsuspected4 ,thus when ,coulomb1 combining the functions of experimentalist and mathematician1 had discovered the law of the force exerted between two particles of electricity1 it became a purely mathematical problem1 not requiring any further experiment1 to ascertain how electricity is distributed upon a charged conductor and this problem has been solved by mathematicians in several cases4--,,foster1 ,g4 ,c4 ,presidential ,address ,british ,association for the ,advancement of ,science1 ,section ,a 7#ahgg72 ,nature1 ,vol4 #af1 p4 #cab-cac4 #aeag4 ,without consummate mathematical skill1 on the part of some investigators at any rate1 all the higher physical problems would be sealed to us2 and without competent skill on the part of the ordinary student no idea can be formed of the nature and cogency of the evidence on which the solutions rest4 ,mathematics are not merely a gate through which we may approach if we please1 but they are the only mode of approach to large and important districts of thought4--,,venn1 ,,john4 ,symbolic ,logic 7,london and ,new ,york1 #ahid71 ,introduction1 p4 xix4 #aeah4 ,much of the skill of the true mathematical physicist and of the mathematical astronomer consists in the power of adapting methods and results carried out on an exact mathematical basis to obtain approximations sufficient for the purposes of physical measurements4 ,it might perhaps be thought that a scheme of ,mathematics on a frankly approximative basis would be sufficient for all the practical purposes of application in ,physics1 ,engineering ,science1 and ,astronomy1 and no doubt it would be possible to develop1 to some extent at least1 a species of ,mathematics on these lines4 ,such a system would1 however1 involve an intolerable awkwardness and prolixity in the statements of results1 especially in view of the fact that the degree of approximation necessary for various purposes is very different1 and thus that unassigned grades of approximation would have to be provided for4 ,moreover1 the mathematician working on these lines would be cut off from the chief sources of inspiration1 the ideals of exactitude and logical rigour1 as well as from one of his most indispensable guides to discovery1 symmetry1 and permanence of mathematical form4 ,the history of the actual movements of mathematical thought through the centuries shows that these ideals are the very life-blood of the science1 and warrants the conclusion that a constant striving toward their attainment is an absolutely essential condition of vigorous growth4 ,these ideals have their roots in irresistible impulses and deep-seated needs of the human mind1 manifested in its efforts to introduce intelligibility in certain great domains of the world of thought4--,,hobson1 ,e4 ,w4 ,presidential ,address ,british ,association for the ,advancement of ,science1 ,section ,a 7#aiaj72 ,nature1 ,vol4 #hd1 pp4 #bhe-bhf4 #aeai4 ,the immense part which those laws laws of number and extension take in giving a deductive character to the other departments of physical science1 is well known2 and is not surprising1 when we consider that all causes operate according to mathematical laws4 ,the effect is always dependent upon1 or in mathematical language1 is a function of1 the quantity of the agent2 and generally of its position also4 ,we cannot1 therefore1 reason respecting causation1 without introducing considerations of quantity and extension at every step2 and if the nature of the phenomena admits of our obtaining numerical data of sufficient accuracy1 the laws of quantity become the grand instruments for calculating forward to an effect1 or backward to a cause4 --,,mill1 ,j4 ,s4 ,system of ,logic1 ,bk4 #c1 chap4 #bd1 sect4 #i4 #aebj4 ,the ordinary mathematical treatment of any applied science substitutes exact axioms for the approximate results of experience1 and deduces from these axioms the rigid mathematical conclusions4 ,in applying this method it must not be forgotten that the mathematical developments transcending the limits of exactness of the science are of no practical value4 ,it follows that a large portion of abstract mathematics remains without finding any practical application1 the amount of mathematics that can be usefully employed in any science being in proportion to the degree of accuracy attained in the science4 ,thus1 while the astronomer can put to use a wide range of mathematical theory1 the chemist is only just beginning to apply the first derivative1 i4e4 the rate of change at which certain processes are going on2 for second derivatives he does not seem to have found any use as yet4--,,klein1 ,f4 ,lectures on ,mathematics 7,new ,york1 #aiaa71 p4 #dg4 #aeba4 ,the bond of union among the physical sciences is the mathematical spirit and the mathematical method which pervades them4444 ,our knowledge of nature1 as it advances1 continuously resolves differences of quality into differences of quantity4 ,all exact reasoning--indeed all reasoning--about quantity is mathematical reasoning2 and thus as our knowledge increases1 that portion of it which becomes mathematical increases at a still more rapid rate4--,,smith1 ,h4 ,j4 ,s4 ,presidential ,address ,british ,association for the ,advancement of ,science1 ,section ,a 7#ahgc72 ,nature1 ,vol4 #h1 p4 #ddi4 #aebb4 ,another way of convincing ourselves how largely this process of assimilation of mathematics by physics has gone on would be to try to conceive the effect of some intellectual catastrophe1 supposing such a thing possible1 whereby all knowledge of mathematics should be swept away from men's minds4 ,would it not be that the departure of mathematics would be the destruction of physics8 ,objective physical phenomena would1 indeed1 remain as they are now1 but physical science would cease to exist4 ,we should no doubt see the same colours on looking into a spectroscope or polariscope1 vibrating strings would produce the same sounds1 electrical machines would give sparks1 and galvanometer needles would be deflected2 but all these things would have lost their meaning2 they would be but as the dry bones--the disjecta membra--of what is now a living and growing science4 ,to follow this conception further1 and to try to image to ourselves in some detail what would be the kind of knowledge of physics which would remain possible1 supposing all mathematical ideas to be blotted out1 would be extremely interesting1 but it would lead us directly into a dim and entangled region where the subjective seems to be always passing itself off for the objective1 and where ,i at least could not attempt to lead the way1 gladly as ,i would follow any one who could show where a firm footing is to be found4 ,but without venturing to do more than to look from a safe distance over this puzzling ground1 we may see clearly enough that mathematics is the connective tissue of physics1 binding what would else be merely a list of detached observations into an organized body of science4--,,foster1 ,g4 ,c4 ,presidential ,address ,british ,association for the ,advancement of ,science1 ,section ,a 7#ahgg72 ,nature1 ,vol4 #af1 p4 #cac4 #aebc4 ,in ,plato's time mathematics was purely a play of the free intellect2 the mathematic-mystical reveries of a ,pythagoras foreshadowed a far-reaching significance1 but such a significance 7except in the case of music7 was as yet entirely a matter of fancy2 yet even in that time mathematics was the prerequisite to all other studies6 ,but today1 when mathematics furnishes the only language by means of which we may formulate the most comprehensive laws of nature1 laws which the ancients scarcely dreamed of1 when moreover mathematics is the only means by which these laws may be understood1--how few learn today anything of the real essence of our mathematics6444 ,in the schools of today mathematics serves only as a disciplinary study1 a mental gymnastic2 that it includes the highest ideal value for the comprehension of the universe1 one dares scarcely to think of in view of our present day instruction4--,,lindeman1 ,f4 ,lehren und ,lernen in der ,mathematik 7,mnchen1 #aijd71 p4 #ad4 #aebd4 ,all applications of mathematics consist in extending the empirical knowledge which we possess of a limited number or region of accessible phenomena into the region of the unknown and inaccessible2 and much of the progress of pure analysis consists in inventing definite conceptions1 marked by symbols1 of complicated operations2 in ascertaining their properties as independent objects of research2 and in extending their meaning beyond the limits they were originally invented for1--thus opening out new and larger regions of thought4--,,merz1 ,j4 ,t4 ,history of ,european ,thought in the #aith ,century 7,edinburgh and ,london1 #aijc71 ,vol4 #a1 p4 #fih4 #aebe4 ,all the effects of nature are only mathematical results of a small number of immutable laws4--,,laplace4 ,a ,philosophical ,essay on ,probabilities ,truscott and ,emory 7,new ,york1 #aijb71 p4 #agg2 ,oeuvres1 t4 #g1 p4 #aci4 #aebf4 ,what logarithms are to mathematics that mathematics are to the other sciences4--,,novalis4 ,schriften 7,berlin1 #aija71 ,teil #b1 p4 #bbb4 #aebg4 ,any intelligent man may now1 by resolutely applying himself for a few years to mathematics1 learn more than the great ,newton knew after half a century of study and meditation4 --,,macaulay4 ,milton2 ,critical and ,miscellaneous ,essays 7,new ,york1 #ahgi71 ,vol4 #a1 p4 #ac4 #aebh4 ,in questions of science the authority of a thousand is not worth the humble reasoning of a single individual4--,,galileo4 ,quoted in ,arago's ,eulogy on ,laplace2 ,smithsonian ,report1 #ahgd1 p4 #afd4 #aebi4 ,behind the artisan is the chemist1 behind the chemist a physicist1 behind the physicist a mathematician4--,,white1 ,w4 ,f4 ,scrap-book of ,elementary ,mathematics 7,chicago1 #aijh71 p4 #bag4 #aecj4 ,the advance in our knowledge of physics is largely due to the application to it of mathematics1 and every year it becomes more difficult for an experimenter to make any mark in the subject unless he is also a mathematician4--,,ball1 ,w4 ,w4 ,r4 ,history of ,mathematics 7,london1 #aija71 p4 #ejc4 #aeca4 ,in very many cases the most obvious and direct experimental method of investigating a given problem is extremely difficult1 or for some reason or other untrustworthy4 ,in such cases the mathematician can often point out some other problem more accessible to experimental treatment1 the solution of which involves the solution of the former one4 ,for example1 if we try to deduce from direct experiments the law according to which one pole of a magnet attracts or repels a pole of another magnet1 the observed action is so much complicated with the effects of the mutual induction of the magnets and of the forces due to the second pole of each magnet1 that it is next to impossible to obtain results of any great accuracy4 ,gauss1 however1 showed how the law which applied in the case mentioned can be deduced from the deflections undergone by a small suspended magnetic needle when it is acted upon by a small fixed magnet placed successively in two determinate positions relatively to the needle2 and being an experimentalist as well as a mathematician1 he showed likewise how these deflections can be measured very easily and with great precision4--,,foster1 ,g4 ,c4 ,presidential ,address ,british ,association for the ,advancement of ,science1 ,section ,a 7#ahgg72 ,nature1 ,vol4 #af1 p4 #cac4 #aecb4 ,give me to learn each secret cause2 ,let ,number's1 ,figure's1 ,motion's laws ,reveal'd before me stand2 ,these to great ,nature's scenes apply1 ,and round the globe1 and through the sky1 ,disclose her working hand4 --,,akenside1 ,m4 ,hymn to ,science4 #aecc4 ,now there are several scores1 upon which skill in mathematicks may be useful to the experimental philosopher4 ,for there are some general advantages1 which mathematicks may bring to the minds of men1 to whatever study they apply themselves1 and consequently to the student of natural philosophy2 namely1 that these disciplines are wont to make men accurate1 and very attentive to the employment that they are about1 keeping their thoughts from wandering1 and inuring them to patience in going through with tedious and intricate demonstrations2 besides1 that they much improve reason1 by accustoming the mind to deduce successive consequences1 and judge of them without easily acquiescing in anything but demonstration4--,,boyle1 ,,robert4 ,works 7,london1 #aggb71 ,vol4 #c1 p4 #dbf4 #aecd4 ,it is not easy to anatomize the constitution and the operations of a mind like ,newton's which makes such an advance in knowledge4 ,yet we may observe that there must exist in it1 in an eminent degree1 the elements which compose the mathematical talent4 ,it must possess distinctness of intuition1 tenacity and facility in tracing logical connection1 fertility of invention1 and a strong tendency to generalization4--,,whewell1 ,w4 ,history of the ,inductive ,sciences 7,new ,york1 #ahid71 ,vol4 #a1 p4 #daf4 #aece4 ,the domain of physics is no proper field for mathematical pastimes4 ,the best security would be in giving a geometrical training to physicists1 who need not then have recourse to mathematicians1 whose tendency is to despise experimental science4 ,by this method will that union between the abstract and the concrete be effected which will perfect the uses of mathematical1 while extending the positive value of physical science4 ,meantime1 the use of analysis in physics is clear enough4 ,without it we should have no precision1 and no co-ordination2 and what account could we give of our study of heat1 weight1 light1 etc48 ,we should have merely series of unconnected facts1 in which we could foresee nothing but by constant recourse to experiment2 whereas1 they now have a character of rationality which fits them for purposes of prevision4--,,comte1 ,a4 ,positive ,philosophy ,martineau1 ,bk4 #c1 chap4 #a4 #aecf4 ,it must ever be remembered that the true positive spirit first came forth from the pure sources of mathematical science2 and it is only the mind that has imbibed it there1 and which has been face to face with the lucid truths of geometry and mechanics1 that can bring into full action its natural positivity1 and apply it in bringing the most complex studies into the reality of demonstration4 ,no other discipline can fitly prepare the intellectual organ4--,,comte1 ,a4 ,positive ,philosophy ,martineau1 ,bk4 #c1 chap4 #a4 #aecg4 ,during the last two centuries and a half1 physical knowledge has been gradually made to rest upon a basis which it had not before4 ,it has become mathematical4 ,the question now is1 not whether this or that hypothesis is better or worse to the pure thought1 but whether it accords with observed phenomena in those consequences which can be shown necessarily to follow from it1 if it be true4 ,even in those sciences which are not yet under the dominion of mathematics1 and perhaps never will be1 a working copy of the mathematical process has been made4 ,this is not known to the followers of those sciences who are not themselves mathematicians1 and who very often exalt their horns against the mathematics in consequence4 ,they might as well be squaring the circle1 for any sense they show in this particular4--,,de ,,morgan1 ,a4 ,a ,budget of ,paradoxes 7,london1 #ahgb71 p4 #b4 #aech4 ,among the mere talkers so far as mathematics are concerned1 are to be ranked three out of four of those who apply mathematics to physics1 who1 wanting a tool only1 are very impatient of everything which is not of direct aid to the actual methods which are in their hands4--,,de ,,morgan1 ,a4 ,graves' ,life of ,sir ,william ,rowan ,hamilton 7,new ,york1 #ahhb-ahhi71 ,vol4 #c1 p4 #cdh4 #aeci4 ,something has been said about the use of mathematics in physical science1 the mathematics being regarded as a weapon forged by others1 and the study of the weapon being completely set aside4 ,i can only say that there is danger of obtaining untrustworthy results in physical science1 if only the results of mathematics are used2 for the person so using the weapon can remain unacquainted with the conditions under which it can be rightly applied4444 ,the results are often correct1 sometimes are incorrect2 the consequence of the latter class of cases is to throw doubt upon all the applications of such a worker until a result has been otherwise tested4 ,moreover1 such a practice in the use of mathematics leads a worker to a mere repetition in the use of familiar weapons2 he is unable to adapt them with any confidence when some new set of conditions arise with a demand for a new method3 for want of adequate instruction in the forging of the weapon1 he may find himself1 sooner or later in the progress of his subject1 without any weapon worth having4 --,,forsyth1 ,a4 ,r4 ,perry's ,teaching of ,mathematics 7,london1 #aijb71 p4 #cf4 #aedj4 ,if in the range of human endeavor after sound knowledge there is one subject that needs to be practical1 it surely is ,medicine4 ,yet in the field of ,medicine it has been found that branches such as biology and pathology must be studied for themselves and be developed by themselves with the single aim of increasing knowledge2 and it is then that they can be best applied to the conduct of living processes4 ,so also in the pursuit of mathematics1 the path of practical utility is too narrow and irregular1 not always leading far4 ,the witness of history shows that1 in the field of natural philosophy1 mathematics will furnish the more effective assistance if1 in its systematic development1 its course can freely pass beyond the ever-shifting domain of use and application4--,,forsyth1 ,a4 ,r4 ,presidential ,address ,british ,association for the ,advancement of ,science1 ,section ,a2 ,nature1 ,vol4 #ef 7#ahig71 p4 #cgg4 #aeda4 ,if the ,greeks had not cultivated ,conic ,sections1 ,kepler could not have superseded ,ptolemy2 if the ,greeks had cultivated ,dynamics1 ,kepler might have anticipated ,newton4--,,whewell1 ,w4 ,history of the ,inductive ,science 7,new ,york1 #ahid71 ,vol4 #a1 p4 #caa4 #aedb4 ,if we may use the great names of ,kepler and ,newton to signify stages in the progress of human discovery1 it is not too much to say that without the treatises of the ,greek geometers on the conic sections there could have been no ,kepler1 without ,kepler no ,newton1 and without ,newton no science in the modern sense of the term1 or at least no such conception of nature as now lies at the basis of all our science1 of nature as subject in the smallest as well as in its greatest phenomena1 to exact quantitative relations1 and to definite numerical laws4 --,,smith1 ,h4 ,j4 ,s4 ,presidential ,address ,british ,association for the ,advancement of ,science1 ,section ,a2 ,nature1 ,vol4 #h 7#ahgc71 p4 #dej4 #aedc4 ,the silent work of the great ,regiomontanus in his chamber at ,nuremberg computed the ,ephemerides which made possible the discovery of ,america by ,columbus4--,,rudio1 ,f4 ,quoted in ,max ,simon's ,geschichte der ,mathematik im ,altertum 7,berlin1 #aiji71 ,einleitung1 p4 xi4 #aedd4 ,the calculation of the eclipses of ,jupiter's satellites1 many a man might have been disposed1 originally1 to regard as a most unprofitable study4 ,but the utility of it to navigation 7in the determination of longitudes7 is now well known4--,,whately1 ,r4 ,annotations to ,bacon's ,essays 7,boston1 #aghc71 p4 #dib4 #aede4 ,who could have imagined1 when ,galvani observed the twitching of the frog muscles as he brought various metals in contact with them1 that eighty years later ,europe would be overspun with wires which transmit messages from ,madrid to ,st4 ,petersburg with the rapidity of lightning1 by means of the same principle whose first manifestations this anatomist then observed6444 ,he who seeks for immediate practical use in the pursuit of science1 may be reasonably sure1 that he will seek in vain4 ,complete knowledge and complete understanding of the action of forces of nature and of the mind1 is the only thing that science can aim at4 ,the individual investigator must find his reward in the joy of new discoveries1 as new victories of thought over resisting matter1 in the esthetic beauty which a well-ordered domain of knowledge affords1 where all parts are intellectually related1 where one thing evolves from another1 and all show the marks of the mind's supremacy2 he must find his reward in the consciousness of having contributed to the growing capital of knowledge on which depends the supremacy of man over the forces hostile to the spirit4--,,helmholtz1 ,h4 ,vortrge und ,reden 7,braunschweig1 #ahhd71 ,bd4 #a1 p4 #adb4 #aedf4 ,when the time comes that knowledge will not be sought for its own sake1 and men will not press forward simply in a desire of achievement1 without hope of gain1 to extend the limits of human knowledge and information1 then1 indeed1 will the race enter upon its decadence4--,,hughes1 ,c4 ,e4 ,quoted in ,d4 ,e4 ,smith's ,teaching of ,geometry 7,boston1 #aiaa71 p4 #i4 #aedg4 ,in the ,opus ,majus of ,roger ,bacon there is a chapter1 in which it is proved by reason1 that all sciences require mathematics4 ,and the arguments which are used to establish this doctrine1 show a most just appreciation of the office of mathematics in science4 ,they are such as follows3 ,that other sciences use examples taken from mathematics as the most evident3--,that mathematical knowledge is1 as it were1 innate to us1 on which point he refers to the well-known dialogue of ,plato1 as quoted by ,cicero3--,that this science1 being the easiest1 offers the best introduction to the more difficult3--,that in mathematics1 things as known to us are identical with things as known to nature3--,that we can here entirely avoid doubt and error1 and obtain certainty and truth3--,that mathematics is prior to other sciences in nature1 because it takes cognizance of quantity1 which is apprehended by intuition 7intuitu intellectus74 8,moreover10 he adds1 8there have been found famous men1 as ,robert1 bishop of ,lincoln1 and ,brother ,adam ,marshman 7de ,marisco71 and many others1 who by the power of mathematics have been able to explain the causes of things2 as may be seen in the writings of these men1 for instance1 concerning the ,rainbow and ,comets1 and the generation of heat1 and climates1 and the celestial bodies40--,,whewell1 ,w4 ,history of the ,inductive ,sciences 7,new ,york1 #ahid71 ,vol4 #a1 p4 #eai4 ,bacon1 ,roger3 ,opus ,majus1 ,part #d1 ,distinctia ,prima1 cap4 #c4 #aedh4 ,the analysis which is based upon the conception of function discloses to the astronomer and physicist not merely the formulae for the computation of whatever desired distances1 times1 velocities1 physical constants2 it moreover gives him insight into the laws of the processes of motion1 teaches him to predict future occurrences from past experiences and supplies him with means to a scientific knowledge of nature1 i4e4 it enables him to trace back whole groups of various1 sometimes extremely heterogeneous1 phenomena to a minimum of simple fundamental laws4 --,,pringsheim1 ,a4 ,jahresbericht der ,deutschen ,mathematiker ,vereinigung1 ,bd4 #ac1 p4 #cff4 #aedi4 8,as is known1 scientific physics dates its existence from the discovery of the differential calculus4 ,only when it was learned how to follow continuously the course of natural events1 attempts1 to construct by means of abstract conceptions the connection between phenomena1 met with success4 ,to do this two things are necessary3 ,first1 simple fundamental concepts with which to construct2 second1 some method by which to deduce1 from the simple fundamental laws of the construction which relate to instants of time and points in space1 laws for finite intervals and distances1 which alone are accessible to observation 7can be compared with experience740 ,riemann4 ,the first of the two problems here indicated by ,riemann consists in setting up the differential equation1 based upon physical facts and hypotheses4 ,the second is the integration of this differential equation and its application to each separate concrete case1 this is the task of mathematics4--,,weber1 ,,heinrich4 ,die partiellen ,differentialgleichungen der mathematischen ,physik 7,braunschweig1 #ahhb71 ,bd4 #a1 ,vorrede4 #aeej4 ,mathematics is the most powerful instrument which we possess for this purpose to trace into their farthest results those general laws which an inductive philosophy has supplied3 in many sciences a profound knowledge of mathematics is indispensable for a successful investigation4 ,in the most delicate researches into the theories of light1 heat1 and sound it is the only instrument2 they have properties which no other language can express2 and their argumentative processes are beyond the reach of other symbols4--,,price1 ,b4 ,treatise on ,infinitesimal ,calculus 7,oxford1 #aheh71 ,vol4 #c1 p4 #e4 #aeea4 ,notwithstanding the eminent difficulties of the mathematical theory of sonorous vibrations1 we owe to it such progress as has yet been made in acoustics4 ,the formation of the differential equations proper to the phenomena is1 independent of their integration1 a very important acquisition1 on account of the approximations which mathematical analysis allows between questions1 otherwise heterogeneous1 which lead to similar equations4 ,this fundamental property1 whose value we have so often to recognize1 applies remarkably in the present case2 and especially since the creation of mathematical thermology1 whose principal equations are strongly analogous to those of vibratory motion4--,this means of investigation is all the more valuable on account of the difficulties in the way of direct inquiry into the phenomena of sound4 ,we may decide the necessity of the atmospheric medium for the transmission of sonorous vibrations2 and we may conceive of the possibility of determining by experiment the duration of the propagation1 in the air1 and then through other media2 but the general laws of the vibrations of sonorous bodies escape immediate observation4 ,we should know almost nothing of the whole case if the mathematical theory did not come in to connect the different phenomena of sound1 enabling us to substitute for direct observation an equivalent examination of more favorable cases subjected to the same law4 ,for instance1 when the analysis of the problem of vibrating chords has shown us that1 other things being equal1 the number of oscillations is in inverse proportion to the length of the chord1 we see that the most rapid vibrations of a very short chord may be counted1 since the law enables us to direct our attention to very slow vibrations4 ,the same substitution is at our command in many cases in which it is less direct4--,,comte1 ,a4 ,positive ,philosophy ,martineau1 ,bk4 #c1 chap4 #d4 #aeeb4 ,problems relative to the uniform propagation1 or to the varied movements of heat in the interior of solids1 are reduced 444 to problems of pure analysis1 and the progress of this part of physics will depend in consequence upon the advance which may be made in the art of analysis4 ,the differential equations 444 contain the chief results of the theory2 they express1 in the most general and concise manner1 the necessary relations of numerical analysis to a very extensive class of phenomena2 and they connect forever with mathematical science one of the most important branches of natural philosophy4--,,fourier1 ,j4 ,theory of ,heat ,freeman1 7,cambridge1 #ahgh71 ,chap4 #c1 p4 #aca4 #aeec4 ,the effects of heat are subject to constant laws which cannot be discovered without the aid of mathematical analysis4 ,the object of the theory is to demonstrate these laws2 it reduces all physical researches on the propagation of heat1 to problems of the integral calculus1 whose elements are given by experiment4 ,no subject has more extensive relations with the progress of industry and the natural sciences2 for the action of heat is always present1 it influences the processes of the arts1 and occurs in all the phenomena of the universe4--,,fourier1 ,j4 ,theory of ,heat ,freeman1 7,cambridge1 #ahgh71 ,chap4 #a1 p4 #ab4 #aeed4 ,dealing with any and every amount of static electricity1 the mathematical mind has balanced and adjusted them with wonderful advantage1 and has foretold results which the experimentalist can do no more than verify4444 ,so in respect of the force of gravitation1 it has calculated the results of the power in such a wonderful manner as to trace the known planets through their courses and perturbations1 and in so doing has discovered a planet before unknown4--,,faraday4 ,some ,thoughts on the ,conservation of ,force4 #aeee4 ,certain branches of natural philosophy 7such as physical astronomy and optics71 444 are1 in a great measure1 inaccessible to those who have not received a regular mathematical education4444 --,,stewart1 ,,dugald4 ,philosophy of the ,human ,mind1 ,part #c1 chap4 #a1 sect4 #c4 #aeef4 ,so intimate is the union between mathematics and physics that probably by far the larger part of the accessions to our mathematical knowledge have been obtained by the efforts of mathematicians to solve the problems set to them by experiment1 and to create 8for each successive class of phenomena1 a new calculus or a new geometry1 as the case might be1 which might prove not wholly inadequate to the subtlety of nature40 ,sometimes1 indeed1 the mathematician has been before the physicist1 and it has happened that when some great and new question has occurred to the experimentalist or the observer1 he has found in the armoury of the mathematician the weapons which he has needed ready made to his hand4 ,but1 much oftener1 the questions proposed by the physicist have transcended the utmost powers of the mathematics of the time1 and a fresh mathematical creation has been needed to supply the logical instrument requisite to interpret the new enigma4--,,smith1 ,h4 ,j4 ,s4 ,presidential ,address ,british ,association for the ,advancement of ,science1 ,section ,a2 ,nature1 ,vol4 #h 7#ahgc71 p4 #dej4 #aeeg4 ,of all the great subjects which belong to the province of his section1 take that which at first sight is the least within the domain of mathematics--,i mean meteorology4 ,yet the part which mathematics plays in meteorology increases every year1 and seems destined to increase4 ,not only is the theory of the simplest instruments essentially mathematical1 but the discussions of the observations--upon which1 be it remembered1 depend the hopes which are already entertained with increasing confidence1 of reducing the most variable and complex of all known phenomena to exact laws--is a problem which not only belongs wholly to mathematics1 but which taxes to the utmost the resources of the mathematics which we now possess4--,,smith1 ,h4 ,j4 ,s4 ,presidential ,address ,british ,association for the ,advancement of ,science1 ,section ,a2 ,nature1 ,vol4 #h 7#ahgc71 p4 #ddi4 #aeeh4 ,you know that if you make a dot on a piece of paper1 and then hold a piece of ,iceland spar over it1 you will see not one dot but two4 ,a mineralogist1 by measuring the angles of a crystal1 can tell you whether or no it possesses this property without looking through it4 ,he requires no scientific thought to do that4 ,but ,sir ,william ,roman ,hamilton 444 knowing these facts and also the explanation of them which ,fresnel had given1 thought about the subject1 and he predicted that by looking through certain crystals in a particular direction we should see not two dots but a continuous circle4 ,mr4 ,lloyd made the experiment1 and saw the circle1 a result which had never been even suspected4 ,this has always been considered one of the most signal instances of scientific thought in the domain of physics4 --,,clifford1 ,w4 ,k4 ,lectures and ,essays 7,new ,york1 #aija71 ,vol4 #a1 p4 #add4 #aeei4 ,the discovery of this planet ,neptune is justly reckoned as the greatest triumph of mathematical astronomy4 ,uranus failed to move precisely in the path which the computers predicted for it1 and was misguided by some unknown influence to an extent which a keen eye might almost see without telescopic aid4444 ,these minute discrepancies constituted the data which were found sufficient for calculating the position of a hitherto unknown planet1 and bringing it to light4 ,leverrier wrote to ,galle1 in substance3 8,direct your telescope to a point on the ecliptic in the constellation of ,aquarius1 in longitude #cbf1 and you will find within a degree of that place a new planet1 looking like a star of about the ninth magnitude1 and having a perceptible disc40 ,the planet was found at ,berlin on the night of ,sept4 #bf1 #ahdf1 in exact accordance with this prediction1 within half an hour after the astronomers began looking for it1 and only about #eb distant from the precise point that ,leverrier had indicated4 --,,young1 ,c4 ,a4 ,general ,astronomy 7,boston1 #ahia71 ,art4 #fec4 #aefj4 ,i am convinced that the future progress of chemistry as an exact science depends very much indeed upon the alliance with mathematics4--,,frankland1 ,a4 ,american ,journal of ,mathematics1 ,vol4 #a1 p4 #cdi4 #aefa4 ,it is almost impossible to follow the later developments of physical or general chemistry without a working knowledge of higher mathematics4--,,mellor1 ,j4 ,w4 ,higher ,mathematics 7,new ,york1 #aijb71 ,preface4 #aefb4 444 ,mount where science guides2 ,go measure earth1 weigh air1 and state the tides2 ,instruct the planets in what orb to run1 ,correct old time1 and regulate the sun4 --,,thomson1 ,w4 ,on the ,figure of the ,earth1 ,title page4 #aefc4 ,admission to its sanctuary referring to astronomy and to the privileges and feelings of a votary1 is only to be gained by one means1--sound and sufficient knowledge of mathematics1 the great instrument of all exact inquiry1 without which no man can ever make such advances in this or any other of the higher departments of science as can entitle him to form an independent opinion on any subject of discussion within their range4 --,,herschel1 ,j4 ,outlines of ,astronomy1 ,introduction1 sect4 #g4 #aefd4 ,the long series of connected truths which compose the science of astronomy1 have been evolved from the appearances and observations by calculation1 and a process of reasoning entirely geometrical4 ,it was not without reason that ,plato called geometry and arithmetic the wings of astronomy2 for it is only by means of these two sciences that we can give a rational account of any of the appearances1 or connect any fact with theory1 or even render a single observation available to the most common astronomical purpose4 ,it is by geometry that we are enabled to reason our way up through the apparent motions to the real orbits of the planets1 and to assign their positions1 magnitudes and eccentricities4 ,and it is by application of geometry--a sublime geometry1 indeed1 invented for the purpose--to the general laws of mechanics1 that we demonstrate the law of gravitation1 trace it through its remotest effects on the different planets1 and1 comparing these effects with what we observe1 determine the densities and weights of the minutest bodies belonging to the system4 ,the whole science of astronomy is in fact a tissue of geometrical reasoning1 applied to the data of observation2 and it is from this circumstance that it derives its peculiar character of precision and certainty4 ,to disconnect it from geometry1 therefore1 and to substitute familiar illustrations and vague description for close and logical reasoning1 is to deprive it of its principal advantages1 and to reduce it to the condition of an ordinary province of natural history4 ,edinburgh ,review1 ,vol4 #eh 7#ahcc-ahcd71 p4 #afh4 #aefe4 ,but geometry is not only the instrument of astronomical investigation1 and the bond by which the truths are enchained together1--it is also the instrument of explanation1 affording1 by the peculiar brevity and perspicuity of its technical processes1 not only aid to the learner1 but also such facilities to the teacher as he will find it very difficult to supply1 if he voluntarily undertakes to forego its assistance4 ,few undertakings1 indeed1 are attended with greater difficulty than that of attempting to exhibit the connecting links of a chain of mathematical reasoning1 when we lay aside the technical symbols and notation which relieve the memory1 and speak at once to the eyes and the understanding34444 ,edinburgh ,review1 ,vol4 #eh 7#ahcc-ahcd71 p4 #afi4 #aeff4 ,with an ordinary acquaintance of trigonometry1 and the simplest elements of algebra1 one may take up any well-written treatise on plane astronomy1 and work his way through it1 from beginning to end1 with perfect ease2 and he will acquire1 in the course of his progress1 from the mere examples put before him1 an infinitely more correct and precise idea of astronomical methods and theories1 than he could obtain in a lifetime from the most eloquent general descriptions that ever were written4 ,at the same time he will be strengthening himself for farther advances1 and accustoming his mind to habits of close comparison and rigid demonstration1 which are of infinitely more importance than the acquisition of stores of undigested facts4 ,edinburgh ,review1 ,vol4 #eh 7#ahcc-ahcd71 p4 #agj4 #aefg4 ,while the telescope serves as a means of penetrating space1 and of bringing its remotest regions nearer us1 mathematics1 by inductive reasoning1 have led us onwards to the remotest regions of heaven1 and brought a portion of them within the range of our possibilities2 nay1 in our own times--so propitious to the extension of knowledge--the application of all the elements yielded by the present conditions of astronomy has even revealed to the intellectual eyes a heavenly body1 and assigned to it its place1 orbit1 mass1 before a single telescope has been directed towards it4--,,humboldt1 ,a4 ,cosmos ,otte1 ,vol4 #b1 part #b1 sect4 #c4 #aefh4 ,mighty are numbers1 joined with art resistless4--,,euripides4 ,hecuba1 ,line #hhd4 #aefi4 ,no single instrument of youthful education has such mighty power1 both as regards domestic economy and politics1 and in the arts1 as the study of arithmetic4 ,above all1 arithmetic stirs up him who is by nature sleepy and dull1 and makes him quick to learn1 retentive1 shrewd1 and aided by art divine he makes progress quite beyond his natural powers4--,,plato4 ,laws ,jowett1 ,bk4 #e1 p4 #gdg4 #aegj4 ,for all the higher arts of construction some acquaintance with mathematics is indispensable4 ,the village carpenter1 who1 lacking rational instruction1 lays out his work by empirical rules learned in his apprenticeship1 equally with the builder of a ,britannia ,bridge1 makes hourly reference to the laws of quantitative relations4 ,the surveyor on whose survey the land is purchased2 the architect in designing a mansion to be built on it2 the builder in preparing his estimates2 his foreman in laying out the foundations2 the masons in cutting the stones2 and the various artisans who put up the fittings2 are all guided by geometrical truths4 ,railway-making is regulated from beginning to end by mathematics3 alike in the preparation of plans and sections2 in staking out the lines2 in the mensuration of cuttings and embankments2 in the designing1 estimating1 and building of bridges1 culverts1 viaducts1 tunnels1 stations4 ,and similarly with the harbors1 docks1 piers1 and various engineering and architectural works that fringe the coasts and overspread the face of the country1 as well as the mines that run underneath it4 ,out of geometry1 too1 as applied to astronomy1 the art of navigation has grown2 and so1 by this science1 has been made possible that enormous foreign commerce which supports a large part of our population1 and supplies us with many necessaries and most of our luxuries4 ,and nowadays even the farmer1 for the correct laying out of his drains1 has recourse to the level--that is1 to geometrical principles4--,,spencer1 ,,herbert4 ,education1 chap4 #a4 #aega4 ,arithmetic is another of the great master-keys of life4 ,with it the astronomer opens the depths of the heavens2 the engineer1 the gates of the mountains2 the navigator1 the pathways of the deep4 ,the skillful arrangement1 the rapid handling of figures1 is a perfect magician's wand4 ,the mighty commerce of the ,united ,states1 foreign and domestic1 passes through the books kept by some thousands of diligent and faithful clerks4 ,eight hundred bookkeepers1 in the ,bank of ,england1 strike the monetary balance of half the civilized world4 ,their skill and accuracy in applying the common rules of arithmetic are as important as the enterprise and capital of the merchant1 or the industry and courage of the navigator4 ,i look upon a well-kept ledger with something of the pleasure with which ,i gaze on a picture or a statue4 ,it is a beautiful work of art4--,,everett1 ,,edward4 ,orations and ,speeches 7,boston1 #ahgj71 ,vol4 #c1 p4 #dg4 #aegb4 ,mathematics is the fruitful ,parent of1 ,i had almost said all1 ,arts1 the unshaken ,foundation of ,sciences1 and the plentiful ,fountain of ,advantage to ,human ,affairs4 ,in which last ,respect1 we may be said to receive from the ,mathematics1 the principal ,delights of ,life1 ,securities of ,health1 ,increase of ,fortune1 and ,conveniences of ,labour3 ,that we dwell elegantly and commodiously1 build decent ,houses for ourselves1 erect stately ,temples to ,god1 and leave wonderful ,monuments to ,posterity3 ,that we are protected by those ,rampires from the ,incursions of the ,enemy2 rightly use ,arms1 skillfully range an ,army1 and manage ,war by ,art1 and not by the ,madness of wild ,beasts3 ,that we have safe ,traffick through the deceitful ,billows1 pass in a direct ,road through the tractless ,ways of the ,sea1 and come to the designed ,ports by the uncertain ,impulse of the ,winds3 ,that we rightly cast up our ,accounts1 do ,business expeditiously1 dispose1 tabulate1 and calculate scattered ,ranks of ,numbers1 and easily compute them1 though expressive of huge ,heaps of ,sand1 nay immense ,hills of ,atoms3 ,that we make pacifick ,separations of the ,bounds of ,lands1 examine the ,moments of ,weights in an equal ,balance1 and distribute every one his own by a just ,measure3 ,that with a light ,touch we thrust forward vast ,bodies which way we will1 and stop a huge ,resistance with a very small ,force3 ,that we accurately delineate the ,face of this ,earthly ,orb1 and subject the ,oeconomy of the ,universe to our ,sight3 ,that we aptly digest the flowing ,series of ,time1 distinguish what is acted by due ,intervals1 rightly account and discern the various ,returns of the ,seasons1 the stated ,periods of ,years and ,months1 the alternate ,increments of ,days and ,nights1 the doubtful ,limits of ,light and ,shadow1 and the exact ,differences of ,hours and ,minutes3 ,that we derive the subtle ,virtue of the ,solar ,rays to our ,uses1 infinitely extend the ,sphere of ,sight1 enlarge the near ,appearances of ,things1 bring to ,hand ,things remote1 discover ,things hidden1 search ,nature out of her ,concealments1 and unfold her dark ,mysteries3 ,that we delight our ,eyes with beautiful ,images1 cunningly imitate the ,devices and portray the ,works of ,nature2 imitate did ,i say8 nay excel1 while we form to ourselves ,things not in being1 exhibit ,things absent1 and represent ,things past3 ,that we recreate our ,minds and delight our ,ears with melodious ,sounds1 attemperate the inconstant ,undulations of the ,air to musical ,tunes1 add a pleasant ,voice to a sapless ,log and draw a sweet ,eloquence from a rigid ,metal2 celebrate our ,maker with an harmonious ,praise1 and not unaptly imitate the blessed ,choirs of ,heaven3 ,that we approach and examine the inaccessible ,seats of the ,clouds1 the distant ,tracts of ,land1 unfrequented ,paths of the ,sea2 lofty ,tops of the ,mountains1 low ,bottoms of the ,valleys1 and deep ,gulphs of the ,ocean3 ,that in ,heart we advance to the ,saints themselves above1 yea draw them to us1 scale the etherial ,towers1 freely range through the celestial ,fields1 measure the ,magnitudes1 and determine the ,interstices of the ,stars1 prescribe inviolable ,laws to the ,heavens themselves1 and confine the wandering ,circuits of the ,stars within fixed ,bounds3 ,lastly1 that we comprehend the vast ,fabrick of the ,universe1 admire and contemplate the wonderful ,beauty of the ,divine ,workmanship1 and to learn the incredible ,force and ,sagacity of our own ,minds1 by certain ,experiments1 and to acknowledge the ,blessings of ,heaven with pious ,affection4--,,barrow1 ,,isaac4 ,mathematical ,lectures 7,london1 #agcd71 pp4 #bg-cj4 #aegc4 ,analytical and graphical treatment of statistics is employed by the economist1 the philanthropist1 the business expert1 the actuary1 and even the physician1 with the most surprisingly valuable results2 while symbolic language involving mathematical methods has become a part of wellnigh every large business4 ,the handling of pig-iron does not seem to offer any opportunity for mathematical application4 ,yet graphical and analytical treatment of the data from long-continued experiments with this material at ,bethlehem1 ,pennsylvania1 resulted in the discovery of the law that fatigue varied in proportion to a certain relation between the load and the periods of rest4 ,practical application of this law increased the amount handled by each man from twelve and a half to forty-seven tons per day4 ,such study would have been impossible without preliminary acquaintance with the simple invariable elements of mathematics4--,,karpinsky1 ,l4 ,high ,school ,education 7,new ,york1 #aiab71 chap4 #f1 p4 #acd4 #aegd4 ,they computation and arithmetic belong then1 it seems1 to the branches of learning which we are now investigating2--for a military man must necessarily learn them with a view to the marshalling of his troops1 and so must a philosopher with the view of understanding real being1 after having emerged from the unstable condition of becoming1 or else he can never become an apt reasoner4 ,that is the fact he replied4 ,but the guardian of ours happens to be both a military man and a philosopher4 ,unquestionably so4 ,it would be proper then1 ,glaucon1 to lay down laws for this branch of science and persuade those about to engage in the most important state-matters to apply themselves to computation1 and study it1 not in the common vulgar fashion1 but with the view of arriving at the contemplation of the nature of numbers by the intellect itself1--not for the sake of buying and selling as anxious merchants and retailers1 but for war also1 and that the soul may acquire a facility in turning itself from what is in the course of generation to truth and real being4--,,plato4 ,republic ,davis1 ,bk4 #g1 p4 #ebe4 #aege4 ,the scientific part of ,arithmetic and ,geometry would be of more use for regulating the thoughts and opinions of men than all the great advantage which ,society receives from the general application of them3 and this use cannot be spread through the ,society by the practice2 for the ,practitioners1 however dextrous1 have no more knowledge of the ,science than the very instruments with which they work4 ,they have taken up the ,rules as they found them delivered down to them by scientific men1 without the least inquiry after the ,principles from which they are derived3 and the more accurate the ,rules1 the less occasion there is for inquiring after the ,principles1 and consequently1 the more difficult it is to make them turn their attention to the ,first ,principles2 and1 therefore1 a ,nation ought to have both ,scientific and ,practical ,mathematicians4--,,williamson1 ,,james4 ,elements of ,euclid with ,dissertations 7,oxford1 #agha74 #aegf4 ,where there is nothing to measure there is nothing to calculate1 hence it is impossible to employ mathematics in psychological investigations4 ,thus runs the syllogism compounded of an adherence to usage and an apparent truth4 ,as to the latter1 it is wholly untrue that we may calculate only where we have measured4 ,exactly the opposite is true4 ,every hypothetically assumed law of quantitative combination1 even such as is recognized as invalid1 is subject to calculation2 and in case of deeply hidden but important matters it is imperative to try on hypotheses and to subject the consequences which flow from them to precise computation until it is found which one of the various hypotheses coincides with experience4 ,thus the ancient astronomers tried eccentric circles1 and ,kepler tried the ellipse to account for the motion of the planets1 the latter also compared the squares of the times of revolution with the cubes of the mean distances before he discovered their agreement4 ,in like manner ,newton tried whether a gravitation1 varying inversely as the square of the distance1 sufficed to keep the moon in its orbit about the earth2 if this supposition had failed him1 he would have tried some other power of the distance1 as the fourth or fifth1 and deduced the corresponding consequences to compare them with the observations4 ,just this is the greatest benefit of mathematics1 that it enables us to survey the possibilities whose range includes the actual1 long before we have adequate definite experience2 this makes it possible to employ very incomplete indications of experience to avoid at least the crudest errors4 ,long before the transit of ,venus was employed in the determination of the sun's parallax1 it was attempted to determine the instant at which the sun illumines exactly one-half of the moon's disk1 in order to compute the sun's distance from the known distance of the moon from the earth4 ,this was not possible1 for1 owing to psychological reasons1 our method of measuring time is too crude to give us the desired instant with sufficient accuracy2 yet the attempt gave us the knowledge that the sun's distance from us is at least several hundred times as great as that of the moon4 ,this illustration shows clearly that even a very imperfect estimate of a magnitude in a case where no precise observation is possible1 may become very instructive1 if we know how to exploit it4 ,was it necessary to know the scale of our solar system in order to learn of its order in general8 ,or1 taking an illustration from another field1 was it impossible to investigate the laws of motion until it was known exactly how far a body falls in a second at some definite place8 ,not at all4 ,such determinations of fundamental measures are in themselves exceedingly difficult1 but fortunately1 such investigations form a class of their own2 our knowledge of fundamental laws does not need to wait on these4 ,to be sure1 computation invites measurement1 and every easily observed regularity of certain magnitudes is an incentive to mathematical investigation4 --,,herbart1 ,j4 ,f4 ,werke ,kehrbach1 7,langensalza1 #ahij71 ,bd4 #e1 p4 #ig4 #aegg4 ,those who pass for naturalists1 have1 for the most part1 been very little1 or not at all1 versed in mathematicks1 if not also jealous of them4--,,boyle1 ,,robert4 ,works 7,london1 #aggb71 ,vol4 #c1 p4 #dbf4 #aegh4 ,however hurtful may have been the incursions of the geometers1 direct and indirect1 into a domain which it is not for them to cultivate1 the physiologists are not the less wrong in turning away from mathematics altogether4 ,it is not only that without mathematics they could not receive their due preliminary training in the intervening sciences3 it is further necessary for them to have geometrical and mechanical knowledge1 to understand the structure and the play of the complex apparatus of the living1 and especially the animal organism4 ,animal mechanics1 statical and dynamical1 must be unintelligible to those who are ignorant of the general laws of rational mechanics4 ,the laws of equilibrium and motion are 444 absolutely universal in their action1 depending wholly on the energy1 and not at all on the nature of the forces considered3 and the only difficulty is in their numerical application in cases of complexity4 ,thus1 discarding all idea of a numerical application in biology1 we perceive that the general theorems of statics and dynamics must be steadily verified in the mechanism of living bodies1 on the rational study of which they cast an indispensable light4 ,the highest orders of animals act in repose and motion1 like any other mechanical apparatus of a similar complexity1 with the one difference of the mover1 which has no power to alter the laws of motion and equilibrium4 ,the participation of rational mechanics in positive biology is thus evident4 ,mechanics cannot dispense with geometry2 and beside1 we see how anatomical and physiological speculations involve considerations of form and position1 and require a familiar knowledge of the principal geometrical laws which may cast light upon these complex relations4--,,comte1 ,a4 ,positive ,philosophy ,martineau1 ,bk4 #e1 chap4 #a4 #aegi4 ,in mathematics we find the primitive source of rationality2 and to mathematics must the biologists resort for means to carry on their researches4--,,comte1 ,a4 ,positive ,philosophy ,martineau1 ,bk4 #e1 chap4 #a4 #aehj4 ,in this school of mathematics must they biologists learn familiarly the real characters and conditions of scientific evidence1 in order to transfer it afterwards to the province of their own theories4 ,the study of it here1 in the most simple and perfect cases1 is the only sound preparation for its recognition in the most complex4 ,the study is equally necessary for the formation of intellectual habits2 for obtaining an aptitude in forming and sustaining positive abstractions1 without which the comparative method cannot be used in either anatomy or physiology4 ,the abstraction which is to be the standard of comparison must be first clearly formed1 and then steadily maintained in its integrity1 or the analysis becomes abortive3 and this is so completely in the spirit of mathematical combinations1 that practice in them is the best preparation for it4 ,a student who cannot accomplish the process in the more simple case may be assured that he is not qualified for the higher order of biological researches1 and must be satisfied with the humbler office of collecting materials for the use of minds of another order4 ,hence arises another use of mathematical training2--that of testing and classifying minds1 as well as preparing and guiding them4 ,probably as much good would be done by excluding the students who only encumber the science by aimless and desultory inquiries1 as by fitly instituting those who can better fulfill its conditions4--,,comte1 ,a4 ,positive ,philosophy ,martineau1 ,bk4 #e1 chap4 #a4 #aeha4 ,there seems no sufficient reason why the use of scientific fictions1 so common in the hands of geometers1 should not be introduced into biology1 if systematically employed1 and adopted with sufficient sobriety4 ,in mathematical studies1 great advantages have arisen from imagining a series of hypothetical cases1 the consideration of which1 though artificial1 may aid the clearing up of the real subject1 or its fundamental elaboration4 ,this art is usually confounded with that of hypotheses2 but it is entirely different2 inasmuch as in the latter case the solution alone is imaginary2 whereas in the former1 the problem itself is radically ideal4 ,its use can never be in biology comparable to what it is in mathematics3 but it seems to me that the abstract character of the higher conceptions of comparative biology renders them susceptible of such treatment4 ,the process will be to intercalate1 among different known organisms1 certain purely fictitious organisms1 so imagined as to facilitate their comparison1 by rendering the biological series more homogeneous and continuous3 and it might be that several might hereafter meet with more or less of a realization among organisms hitherto unexplored4 ,it may be possible1 in the present state of our knowledge of living bodies1 to conceive of a new organism capable of fulfilling certain given conditions of existence4 ,however that may be1 the collocation of real cases with well-imagined ones1 after the manner of geometers1 will doubtless be practised hereafter1 to complete the general laws of comparative anatomy and physiology1 and possibly to anticipate occasionally the direct exploration4 ,even now1 the rational use of such an artifice might greatly simplify and clear up the ordinary system of biological instruction4 ,but it is only the highest order of investigators who can be trusted with it4 ,whenever it is adopted1 it will constitute another ground of relation between biology and mathematics4--,,comte1 ,a4 ,positive ,philosophy ,martineau1 ,bk4 #e1 chap4 #a4 #aehb4 ,i think it may safely enough be affirmed1 that he1 that is not so much as indifferently skilled in mathematicks1 can hardly be more than indifferently skilled in the fundamental principles of physiology4--,,boyle1 ,,robert4 ,works 7,london1 #aggb71 ,vol4 #c1 p4 #dcj4 #aehc4 ,it is not only possible but necessary that mathematics be applied to psychology2 the reason for this necessity lies briefly in this3 that by no other means can be reached that which is the ultimate aim of all speculation1 namely conviction4--,,herbart1 ,j4 ,f4 ,werke ,kehrbach1 7,langensalza1 #ahij71 ,bd4 #e1 p4 #ajd4 #aehd4 ,all more definite knowledge must start with computation2 and this is of most important consequences not only for the theory of memory1 of imagination1 of understanding1 but as well for the doctrine of sensations1 of desires1 and affections4 --,,herbart1 ,j4 ,f4 ,werke ,kehrbach1 7,langensalza1 #ahij71 ,bd4 #e1 p4 #ajc4 #aehe4 ,in the near future mathematics will play an important part in medicine3 already there are increasing indications that physiology1 descriptive anatomy1 pathology and therapeutics cannot escape mathematical legitimation4--,,dessoir1 ,,max4 ,westermann's ,monatsberichte1 ,bd4 #gg1 p4 #chj2 ,ahrens3 ,scherz und ,ernst in der ,mathematik 7,leipzig1 #aijd71 p4 #cie4 #aehf4 ,the social sciences mathematically developed are to be the controlling factors in civilization4--,,white1 ,w4 ,f4 ,a ,scrap-book of ,elementary ,mathematics 7,chicago1 #aijh71 p4 #bjh4 #aehg4 ,it is clear that this education referring to education preparatory to the science of sociology must rest on a basis of mathematical philosophy1 even apart from the necessity of mathematics to the study of inorganic philosophy4 ,it is only in the region of mathematics that sociologists1 or anybody else1 can obtain a true sense of scientific evidence1 and form the habit of rational and decisive argumentation2 can1 in short1 learn to fulfill the logical conditions of all positive speculation1 by studying universal positivism at its source4 ,this training1 obtained and employed with the more care on account of the eminent difficulty of social science1 is what sociologists have to seek in mathematics4--,,comte1 ,a4 ,positive ,philosophy ,martineau1 ,bk4 #f1 chap4 #d4 #aehh4 ,it is clear that the individual as a social unit and the state as a social aggregate require a certain modicum of mathematics1 some arithmetic and algebra1 to conduct their affairs4 ,under this head would fall the theory of interest1 simple and compound1 matters of discount and amortization1 and1 if lotteries hold a prominent place in raising moneys1 as in some states1 questions of probability must be added4 ,as the state becomes more highly organized and more interested in the scientific analysis of its life1 there appears an urgent necessity for various statistical information1 and this can be properly obtained1 reduced1 correlated1 and interpreted only when the guiding spirit in the work have the necessary mathematical training in the theory of statistics4 7,figures may not lie1 but statistics compiled unscientifically and analyzed incompetently are almost sure to be misleading1 and when this condition is unnecessarily chronic the so-called statisticians may well be called liars47 ,the dependence of insurance of various kinds on statistical information and the very great place which insurance occupies in the modern state1 albeit often controlled by private corporations instead of by the government1 makes the theories of paramount importance to our social life4--,,wilson1 ,e4 ,b4 ,bulletin ,american ,mathematical ,society1 ,vol4 #ah 7#aiab71 p4 #dfc4 #aehi4 ,the theory of probabilities and the theory of errors now constitute a formidable body of knowledge of great mathematical interest and of great practical importance4 ,though developed largely through the applications to the more precise sciences of astronomy1 geodesy1 and physics1 their range of applicability extends to all the sciences2 and they are plainly destined to play an increasingly important rle in the development and in the applications of the sciences of the future4 ,hence their study is not only a commendable element in a liberal education1 but some knowledge of them is essential to a correct understanding of daily events4--,,woodward1 ,r4 ,s4 ,probability and ,theory of ,errors 7,new ,york1 #aijf71 ,preface4 #aeij4 ,it was not to be anticipated that a new science the science of probabilities which took its rise in games of chance1 and which had long to encounter an obloquy1 hardly yet extinct1 due to the prevailing idea that its only end was to facilitate and encourage the calculations of gamblers1 could ever have attained its present status--that its aid should be called for in every department of natural science1 both to assist in discovery1 which it has repeatedly done 7even in pure mathematics71 to minimize the unavoidable errors of observation1 and to detect the presence of causes as revealed by observed events4 ,nor are commercial and other practical interests of life less indebted to it3 wherever the future has to be forecasted1 risk to be provided against1 or the true lessons to be deduced from statistics1 it corrects for us the rough conjectures of common sense1 and decides which course is really1 according to the lights of which we are in possession1 the wisest for us to pursue4--,,crofton1 ,m4 ,w4 ,encyclopedia ,britannica1 #ith ,edition2 ,article 8,probability40 #aeia4 ,the calculus of probabilities1 when confined within just limits1 ought to interest1 in an equal degree1 the mathematician1 the experimentalist1 and the statesman4 ,from the time when ,pascal and ,fermat established its first principles1 it has rendered1 and continues daily to render1 services of the most eminent kind4 ,it is the calculus of probabilities1 which1 after having suggested the best arrangements of the tables of population and mortality1 teaches us to deduce from those numbers1 in general so erroneously interpreted1 conclusions of a precise and useful character2 it is the calculus of probabilities which alone can regulate justly the premiums to be paid for assurances2 the reserve funds for the disbursements of pensions1 annuities1 discounts1 etc4 ,it is under its influence that lotteries and other shameful snares cunningly laid for avarice and ignorance have definitely disappeared4--,,arago4 ,eulogy on ,laplace ,baden-,powell1 ,smithsonian ,report1 #ahgd1 p4 #afd4 #aeib4 ,men were surprised to hear that not only births1 deaths1 and marriages1 but the decisions of tribunals1 the results of popular elections1 the influence of punishments in checking crime1 the comparative values of medical remedies1 the probable limits of error in numerical results in every department of physical inquiry1 the detection of causes1 physical1 social1 and moral1 nay1 even the weight of evidence and the validity of logical argument1 might come to be surveyed with the lynx-eyed scrutiny of a dispassionate analysis4--,,herschel1 ,j4 ,quoted in ,encyclopedia ,britannica1 #ith ,edition2 ,article 8,probability40 #aeic4 ,if economists expect of the application of the mathematical method any extensive concrete numerical results1 and it is to be feared that like other non-mathematicians all too many of them think of mathematics as merely an arithmetical science1 they are bound to be disappointed and to find a paucity of results in the works of the few of their colleagues who use that method4 ,but they should rather learn1 as the mathematicians among them know full well1 that mathematics is much broader1 that it has an abstract quantitative 7or even qualitative7 side1 that it deals with relations as well as numbers1 4444--,,wilson1 ,e4 ,b4 ,bulletin ,american ,mathematical ,society1 ,vol4 #ah 7#aiab71 p4 #dfd4 #aeid4 ,the effort of the economist is to see1 to picture the inter-play of economic elements4 ,the more clearly cut these elements appear in his vision1 the better2 the more elements he can grasp and hold in his mind at once1 the better4 ,the economic world is a misty region4 ,the first explorers used unaided vision4 ,mathematics is the lantern by which what before was dimly visible now looms up in firm1 bold outlines4 ,the old phantasmagoria disappear4 ,we see better4 ,we also see further4--,,fisher1 ,,irving4 ,transactions of ,connecticut ,academy1 ,vol4 #i 7#ahib71 p4 #aai4 #aeie4 ,in the great inquiries of the moral and social sciences 444 mathematics 7,i always mean ,applied ,mathematics7 affords the only sufficient type of deductive art4 ,up to this time1 ,i may venture to say that no one ever knew what deduction is1 as a means of investigating the laws of nature1 who had not learned it from mathematics1 nor can any one hope to understand it thoroughly1 who has not1 at some time in his life1 known enough of mathematics to be familiar with the instrument at work4 --,,mill1 ,j4 ,s4 ,an ,examination of ,sir ,william ,hamilton's ,philosophy 7,london1 #ahgh71 p4 #fbb4 #aeif4 ,let me pass on to say a word or two about the teaching of mathematics as an academic training for general professional life4 ,it has immense capabilities in that respect4 ,if you consider how much of the effectiveness of an administrator depends upon the capacity for co-ordinating appropriately a number of different ideas1 precise accuracy of definition1 rigidity of proof1 and sustained reasoning1 strict in every step1 and when you consider what substitutes for these things nine men out of every ten without special training have to put up with1 it is clear that a man with a mathematical training has incalculable advantages4--,,shaw1 ,w4 ,h4 ,perry's ,teaching of ,mathematics 7,london1 #aijb71 p4 #gc4 #aeig4 ,before you enter on the study of law a sufficient ground work must be laid4444 ,mathematics and natural philosophy are so useful in the most familiar occurrences of life and are so peculiarly engaging and delightful as would induce everyone to wish an acquaintance with them4 ,besides this1 the faculties of the mind1 like the members of a body1 are strengthened and improved by exercise4 ,mathematical reasoning and deductions are1 therefore1 a fine preparation for investigating the abstruse speculations of the law4--,,jefferson1 ,,thomas4 ,quoted in ,cajori's ,teaching and ,history of ,mathematics in the ,u4 ,s4 7,washington1 #ahij71 p4 #ce4 #aeih4 ,it has been observed in ,england of the study of law1--though the acquisition of the most difficult parts of its learning1 the interpretation of laws1 the comparison of authorities1 and the construction of instruments1 would seem to require philological and critical training2 though the weighing of evidence and the investigation of probable truth belong to the province of the moral sciences1 and the peculiar duties of the advocate require rhetorical skill1--yet that a large proportion of the most distinguished members of the profession has proceeded from the university 7that of ,cambridge7 most celebrated for the cultivation of mathematical studies4--,,everett1 ,,edward4 ,orations and ,speeches 7,boston1 #ahgj71 ,vol4 #b1 p4 #eaa4 #aeii4 ,all historic science tends to become mathematical4 ,mathematical power is classifying power4--,,novalis4 ,schriften 7,berlin1 #aija71 ,teil #b1 p4 #aib4 #aeii;a4 ,history has never regarded itself as a science of statistics4 ,it was the ,science of ,vital ,energy in relation with time2 and of late this radiating centre of its life has been steadily tending1--together with every form of physical and mechanical energy1--toward mathematical expression4--,,adam1 ,,henry4 ,a ,letter to ,american ,teachers of ,history 7,washington1 #aiaj71 p4 #aae4 #aeii;b4 ,mathematics can be shown to sustain a certain relation to rhetoric and may aid in determining its laws4--,,sherman ,l4 ,a4 ,university of ,nebraska ,studies1 ,vol4 #a1 p4 #acj4 ,,chapter ,,xvi ,,arithmetic #afja4 ,there is no problem in all mathematics that cannot be solved by direct counting4 ,but with the present implements of mathematics many operations can be performed in a few minutes which without mathematical methods would take a lifetime4 --,,mach1 ,,ernst4 ,popular ,scientific ,lectures ,mc,cormack 7,chicago1 #ahih71 p4 #aig4 #afjb4 ,there is no inquiry which is not finally reducible to a question of ,numbers2 for there is none which may not be conceived of as consisting in the determination of quantities by each other1 according to certain relations4--,,comte1 ,a4 ,positive ,philosophy ,martineau1 ,bk4 #a1 chap4 #a4 #afjc4 ,pythagoras says that number is the origin of all things1 and certainly the law of number is the key that unlocks the secrets of the universe4 ,but the law of number possesses an immanent order1 which is at first sight mystifying1 but on a more intimate acquaintance we easily understand it to be intrinsically necessary2 and this law of number explains the wondrous consistency of the laws of nature4--,,carus1 ,,paul4 ,reflections on ,magic ,squares2 ,monist1 ,vol4 #af 7#aijf71 p4 #aci4 #afjd4 ,an ancient writer said that arithmetic and geometry are the wings of mathematics2 ,i believe one can say without speaking metaphorically that these two sciences are the foundation and essence of all the sciences which deal with quantity4 ,not only are they the foundation1 they are also1 as it were1 the capstones2 for1 whenever a result has been arrived at1 in order to use that result1 it is necessary to translate it into numbers or into lines2 to translate it into numbers requires the aid of arithmetic1 to translate it into lines necessitates the use of geometry4--,,lagrange4 ,leons ,elmentaires sur les ,mathmatiques1 ,leon seconde4 #afje4 ,it is number which regulates everything and it is measure which establishes universal order4444 ,a quiet peace1 an inviolable order1 an inflexible security amidst all change and turmoil characterize the world which mathematics discloses and whose depths it unlocks4--,,dillmann1 ,e4 ,die ,mathematik die ,fackeltrgerin einer neuen ,zeit 7,stuttgart1 #ahhi71 p4 #ab4 #afjf4 ,number1 the inducer of philosophies1 ,the synthesis of letters1 4444 --,,aeschylus4 ,quoted in1 ,thomson1 ,j4 ,a41 ,introduction to ,science1 chap4 #a 7,london74 #afjg4 ,amongst all the ideas we have1 as there is none suggested to the mind by more ways1 so there is none more simple1 than that of unity1 or one3 it has no shadow of variety or composition in it2 every object our senses are employed about2 every idea in our understanding2 every thought of our minds1 brings this idea along with it4 ,and therefore it is the most intimate to our thoughts1 as well as it is1 in its agreement to all other things1 the most universal idea we have4--,,locke1 ,,john4 ,an ,essay concerning ,human ,understanding1 ,bk4 #b1 chap4 #af1 sect4 #a4 #afjh4 ,the simple modes of number are of all other the most distinct2 every the least variation1 which is an unit1 making each combination as clearly different from that which approacheth nearest to it1 as the most remote2 two being as distinct from one1 as two hundred2 and the idea of two as distinct from the idea of three1 as the magnitude of the whole earth is from that of a mite4--,,locke1 ,,john4 ,an ,essay concerning ,human ,understanding1 ,bk4 #b1 chap4 #af1 sect4 #c4 #afji4 ,the number of a class is the class of all classes similar to the given class4--,,russell1 ,,bertrand4 ,principles of ,mathematics 7,cambridge1 #aijc71 p4 #aae4 #afaj4 ,number is that property of a group of distinct things which remains unchanged during any change to which the group may be subjected which does not destroy the distinctness of the individual things4--,,fine1 ,h4 ,b4 ,number-system of ,algebra 7,boston and ,new ,york1 #ahij71 p4 #c4 #afaa4 ,the science of arithmetic may be called the science of exact limitation of matter and things in space1 force1 and time4 --,,parker1 ,f4 ,w4 ,talks on ,pedagogics 7,new ,york1 #ahid71 p4 #fd4 #afab4 ,arithmetic is the science of the ,evaluation of ,functions1 ,algebra is the science of the ,transformation of ,functions4 --,,howison1 ,g4 ,h4 ,journal of ,speculative ,philosophy1 ,vol4 #e1 p4 #age4 #afac4 ,that arithmetic rests on pure intuition of time is not so obvious as that geometry is based on pure intuition of space1 but it may be readily proved as follows4 ,all counting consists in the repeated positing of unity2 only in order to know how often it has been posited1 we mark it each time with a different word3 these are the numerals4 ,now repetition is possible only through succession3 but succession rests on the immediate intuition of time1 it is intelligible only by means of this latter concept3 hence counting is possible only by means of time4--,this dependence of counting on time is evidenced by the fact that in all languages multiplication is expressed by 8times0 mal1 that is1 by a concept of time2 sexies1 1 six fois1 six times4 --,,schopenhauer1 ,a4 ,die ,welt als ,vorstellung und ,wille2 ,werke 7,frauenstaedt7 7,leipzig1 #ahgg71 ,bd4 #c1 p4 #ci4 #afad4 ,the miraculous powers of modern calculation are due to three inventions3 the ,arabic ,notation1 ,decimal ,fractions and ,logarithms4--,,cajori1 ,f4 ,history of ,mathematics 7,new ,york1 #ahig71 p4 #afa4 #afae4 ,the grandest achievement of the ,hindoos and the one which1 of all mathematical investigations1 has contributed most to the general progress of intelligence1 is the invention of the principle of position in writing numbers4--,,cajori1 ,f4 ,history of ,mathematics 7,new ,york1 #ahig71 p4 #hg4 #afaf4 ,the invention of logarithms and the calculation of the earlier tables form a very striking episode in the history of exact science1 and1 with the exception of the ,principia of ,newton1 there is no mathematical work published in the country which has produced such important consequences1 or to which so much interest attaches as to ,napier's ,descriptio4 --,,glaisher1 ,j4 ,w4 ,l4 ,encyclopedia ,britannica1 #ith ,edition2 ,article 8,logarithms40 #afag4 ,all minds are equally capable of attaining the science of numbers3 yet we find a prodigious difference in the powers of different men1 in that respect1 after they have grown up1 because their minds have been more or less exercised in it4 --,,johnson1 ,,samuel4 ,boswell's ,life of ,johnson1 ,harper's ,edition 7#ahga71 ,vol4 #b1 p4 #cc4 #afah4 ,the method of arithmetical teaching is perhaps the best understood of any of the methods concerned with elementary studies4--,,bain1 ,,alexander4 ,education as a ,science 7,new ,york1 #ahih71 p4 #bhh4 #afai4 ,what a benefite that onely thyng is1 to haue the witte whetted and sharpened1 ,i neade not trauell to declare1 sith all men confesse it to be as greate as maie be4 ,excepte any witlesse persone thinke he maie bee to wise4 ,but he that most feareth that1 is leaste in daunger of it4 ,wherefore to conclude1 ,i see moare menne to acknowledge the benefite of nomber1 than ,i can espie willying to studie1 to attaine the benefites of it4 ,many praise it1 but fewe dooe greatly practise it3 onlesse it bee for the vulgare practice1 concernying ,merchaundes trade4 ,wherein the desire and hope of gain1 maketh many willying to sustaine some trauell4 ,for aide of whom1 ,i did sette forth the first parte of ,arithmetike4 ,but if thei knewe how faree this seconde parte1 doeeth excell the firste parte1 thei would not accoumpte any tyme loste1 that were emploied in it4 ,yea thei would not thinke any tyme well bestowed till thei had gotten soche habilitie by it1 that it might be their aide in al other studies4 --,,recorde1 ,,robert4 ,whetstone of ,witte 7,london1 #aeeg74 #afbj4 ,you see then1 my friend1 ,i observed1 that our real need of this branch of science arithmetic is probably because it seems to compel the soul to use our intelligence in the search after pure truth4 ,aye1 remarked he1 it does this to a remarkable extent4 ,have you ever noticed that those who have a turn for arithmetic are1 with scarcely an exception1 naturally quick in all sciences2 and that men of slow intellect1 if they be trained and exercised in this study 444 become invariably quicker than they were before8 ,exactly so1 he replied4 ,and1 moreover1 ,i think you will not easily find that many things give the learner and student more trouble than this4 ,of course not4 ,on all these accounts1 then1 we must not omit this branch of science1 but those with the best of talents should be instructed therein4--,,plato4 ,republic ,davis1 ,bk4 #g1 chap4 #h4 #afba4 ,arithmetic has a very great and elevating effect1 compelling the soul to reason about abstract number1 and if visible or tangible objects are obtruding upon the argument1 refusing to be satisfied4--,,plato4 ,republic ,jowett1 ,bk4 #g1 p4 #ebe4 #afbb4 ,good arithmetic contributes powerfully to purposive effort1 to concentration1 to tenacity of purpose1 to generalship1 to faith in right1 and to the joy of achievement1 which are the elements that make up efficient citizenship4444 ,good arithmetic exalts thinking1 furnishes intellectual pleasure1 adds appreciably to love of right1 and subordinates pure memory4--,,myers1 ,,george4 ,monograph on ,arithmetic in ,public ,education 7,chicago71 p4 #ba4 #afbc4 ,on the one side we may say that the purpose of number work is to put a child in possession of the machinery of calculation2 on the other side it is to give him a better mastery of the world through a clear 7mathematical7 insight into the varied physical objects and activities4 ,the whole world1 from one point of view1 can be definitely interpreted and appreciated by mathematical measurements and estimates4 ,arithmetic in the common school should give a child this point of view1 the ability to see and estimate things with a mathematical eye4--,,mcmurray1 ,c4 ,a4 ,special ,method in ,arithmetic 7,new ,york1 #aijf71 p4 #ah4 #afbd4 ,we are so accustomed to hear arithmetic spoken of as one of the three fundamental ingredients in all schemes of instruction1 that it seems like inquiring too curiously to ask why this should be4 ,reading1 ,writing1 and ,arithmetic--these three are assumed to be of co-ordinate rank4 ,are they indeed co-ordinate1 and if so on what grounds8 ,in this modern 8trivium0 the art of reading is put first4 ,well1 there is no doubt as to its right to the foremost place4 ,for reading is the instrument of all our acquisition4 ,it is indispensable4 ,there is not an hour in our lives in which it does not make a great difference to us whether we can read or not4 ,and the art of ,writing1 too2 that is the instrument of all communication1 and it becomes1 in one form or other1 useful to us every day4 ,but ,counting--doing sums1--how often in life does this accomplishment come into exercise8 ,beyond the simplest additions1 and the power to check the items of a bill1 the arithmetical knowledge required of any well-informed person in private life is very limited4 ,for all practical purposes1 whatever ,i may have learned at school of fractions1 or proportion1 or decimals1 is1 unless ,i happen to be in business1 far less available to me in life than a knowledge1 say1 of history of my own country1 or the elementary truths of physics4 ,the truth is1 that regarded as practical arts1 reading1 writing1 and arithmetic have no right to be classed together as co-ordinate elements of education2 for the last of these is considerably less useful to the average man or woman not only than the other two1 but than many others that might be named4 ,but reading1 writing1 and such mathematical or logical exercise as may be gained in connection with the manifestation of numbers1 have a right to constitute the primary elements of instruction4 ,and ,i believe that arithmetic1 if it deserves the high place that it conventionally holds in our educational system1 deserves it mainly on the ground that it is to be treated as a logical exercise4 ,it is the only branch of mathematics which has found its way into primary and early education2 other departments of pure science being reserved for what is called higher or university instruction4 ,but all the arguments in favor of teaching algebra and trigonometry to advanced students1 apply equally to the teaching of the principles or theory of arithmetic to schoolboys4 ,it is calculated to do for them exactly the same kind of service1 to educate one side of their minds1 to bring into play one set of faculties which cannot be so severely or properly exercised in any other department of learning4 ,in short1 relatively to the needs of a beginner1 ,arithmetic1 as a science1 is just as valuable--it is certainly quite as intelligible--as the higher mathematics to a university student4--,,fitch1 ,j4 ,g4 ,lectures on ,teaching 7,new ,york1 #aijf71 pp4 #bfg-bfh4 #afbe4 ,what mathematics1 therefore are expected to do for the advanced student at the university1 ,arithmetic1 if taught demonstratively1 is capable of doing for the children even of the humblest school4 ,it furnishes training in reasoning1 and particularly in deductive reasoning4 ,it is a discipline in closeness and continuity of thought4 ,it reveals the nature of fallacies1 and refuses to avail itself of unverified assumptions4 ,it is the one department of school-study in which the sceptical and inquisitive spirit has the most legitimate scope2 in which authority goes for nothing4 ,in other departments of instruction you have a right to ask for the scholar's confidence1 and to expect many things to be received on your testimony with the understanding that they will be explained and verified afterwards4 ,but here you are justified in saying to your pupil 8,believe nothing which you cannot understand4 ,take nothing for granted40 ,in short1 the proper office of arithmetic is to serve as elementary training in logic4 ,all through your work as teachers you will bear in mind the fundamental difference between knowing and thinking2 and will feel how much more important relatively to the health of the intellectual life the habit of thinking is than the power of knowing1 or even facility of achieving visible results4 ,but here this principle has special significance4 ,it is by ,arithmetic more than by any other subject in the school course that the art of thinking--consecutively1 closely1 logically--can be effectually taught4--,,fitch1 ,j4 ,g4 ,lectures on ,teaching 7,new ,york1 #aijf71 pp4 #bib-bic4 #afbf4 ,arithmetic and geometry1 those wings on which the astronomer soars as high as heaven4--,,boyle1 ,,robert4 ,usefulness of ,mathematics to ,natural ,philosophy2 ,works 7,london1 #aggb71 ,vol4 #c1 p4 #dbi4 #afbg4 ,arithmetical symbols are written diagrams and geometrical figures are graphic formulas4--,,hilbert1 ,d4 ,mathematical ,problems2 ,bulletin ,american ,mathematical ,society1 ,vol4 #h 7#aijb71 p4 #ddc4 #afbh4 ,arithmetic and geometry are much more certain than the other sciences1 because the objects of them are in themselves so simple and so clear that they need not suppose anything which experience can call in question1 and both proceed by a chain of consequences which reason deduces one from another4 ,they are also the easiest and clearest of all the sciences1 and their object is such as we desire2 for1 except for want of attention1 it is hardly supposable that a man should go astray in them4 ,we must not be surprised1 however1 that many minds apply themselves by preference to other studies1 or to philosophy4 ,indeed everyone allows himself more freely the right to make his guess if the matter be dark than if it be clear1 and it is much easier to have on any question some vague ideas than to arrive at the truth itself on the simplest of all4--,,descartes4 ,rules for the ,direction of the ,mind2 ,torrey's ,philosophy of ,descartes 7,new ,york1 #ahib71 p4 #fc4 #afbi4 ,why are wise few1 fools numerous in the excesse8 ',cause1 wanting number1 they are numberlesse4 --,,lovelace4 ,noah ,bridges3 ,vulgar ,arithmetike 7,london1 #afei71 p4 #abg4 #afcj4 ,the clearness and distinctness of each mode of number from all others1 even those that approach nearest1 makes me apt to think that demonstrations in numbers1 if they are not more evident and exact than in extension1 yet they are more general in their use1 and more determinate in their application4 ,because the ideas of numbers are more precise and distinguishable than in extension2 where every equality and excess are not so easy to be observed or measured2 because our thoughts cannot in space arrive at any determined smallness beyond which it cannot go1 as an unit2 and therefore the quantity or proportion of any the least excess cannot be discovered4--,,locke1 ,,john4 ,an ,essay concerning ,human ,understanding1 ,bk4 #b1 chap4 #af1 sect4 #d4 #afca4 ,battalions of figures are like battalions of men1 not always as strong as is supposed4--,,sage1 ,m4 ,mrs4 ,piper and the ,society for ,psychical ,research ,robertson 7,new ,york1 #aiji71 p4 #aea4 #afcb4 ,number was born in superstition and reared in mystery1444 numbers were once made the foundation of religion and philosophy1 and the tricks of figures have had a marvellous effect on a credulous people4--,,parker1 ,f4 ,w4 ,talks on ,pedagogics 7,new ,york1 #ahid71 ,p4 #fd4 #afcc4 ,a rule to trick th' arithmetic4--,,kipling1 ,r4 ,to the ,true ,romance4 #afcd4 ,god made integers1 all else is the work of man4 --,,kronecker1 ,l4 ,jahresberichte der ,deutschen ,mathematiker ,vereinigung1 ,bd4 #b1 p4 #ai4 #afce4 ,plato said 8 40 ,jacobi changed this to 8 40 ,then came ,kronecker and created the memorable expression 8,die ganzen ,zahlen hat ,gott gemacht1 alles andere ist ,menschenwerk40--,,klein1 ,f4 ,jahresbericht der ,deutschen ,mathematiker ,vereinigung1 ,bd4 #f1 p4 #acf4 #afcf4 ,integral numbers are the fountainhead of all mathematics4 --,,minkowski1 ,h4 ,diophantische ,approximationen 7,leipzig1 #aijg71 ,vorrede4 #afcg4 ,the 8,disquisitiones ,arithmeticae0 that great book with seven seals4--,,merz1 ,j4 ,t4 ,a ,history of ,european ,thought in the ,nineteenth ,century 7,edinburgh and ,london1 #aijh71 p4 #gba4 #afch4 ,it may fairly be said that the germs of the modern algebra of linear substitutions and concomitants are to be found in the fifth section of the ,disquisitiones ,arithmeticae2 and inversely1 every advance in the algebraic theory of forms is an acquisition to the arithmetical theory4--,,mathews1 ,g4 ,b4 ,theory of ,numbers 7,cambridge1 #ahib71 ,part #a1 sect4 #dh4 #afci4 ,strictly speaking1 the theory of numbers has nothing to do with negative1 or fractional1 or irrational quantities1 as such4 ,no theorem which cannot be expressed without reference to these notions is purely arithmetical3 and no proof of an arithmetical theorem1 can be considered finally satisfactory if it intrinsically depends upon extraneous analytical theories4 --,,mathews1 ,g4 ,b4 ,theory of ,numbers 7,cambridge1 #ahib71 ,part #a1 sect4 #a4 #afdj4 ,many of the greatest masters of the mathematical sciences were first attracted to mathematical inquiry by problems relating to numbers1 and no one can glance at the periodicals of the present day which contain questions for solution without noticing how singular a charm such problems still continue to exert4 ,the interest in numbers seems implanted in the human mind1 and it is a pity that it should not have freer scope in this country4 ,the methods of the theory of numbers are peculiar to itself1 and are not readily acquired by a student whose mind has for years been familiarized with the very different treatment which is appropriate to the theory of continuous magnitude2 it is therefore extremely desirable that some portion of the theory should be included in the ordinary course of mathematical instruction at our ,university4 ,from the moment that ,gauss1 in his wonderful treatise of #ahja1 laid down the true lines of the theory1 it entered upon a new day1 and no one is likely to be able to do useful work in any part of the subject who is unacquainted with the principles and conceptions with which he endowed it4--,,glaisher1 ,j4 ,w4 ,l4 ,presidential ,address ,british ,association for the ,advancement of ,science 7#ahij72 ,nature1 ,vol4 #db1 p4 #dfg4 #afda4 ,let us look for a moment at the general significance of the fact that calculating machines actually exist1 which relieve mathematicians of the purely mechanical part of numerical computations1 and which accomplish the work more quickly and with a greater degree of accuracy2 for the machine is not subject to the slips of the human calculator4 ,the existence of such a machine proves that computation is not concerned with the significance of numbers1 but that it is concerned essentially only with the formal laws of operation2 for it is only these that the machine can obey--having been thus constructed--an intuitive perception of the significance of numbers being out of the question4--,,klein1 ,f4 ,elementarmathematik vom hheren ,standpunkte aus4 7,leipzig1 #aijh71 ,bd4 #a1 p4 #ec4 #afdb4 ,mathematics is the queen of the sciences and arithmetic the queen of mathematics4 ,she often condescends to render service to astronomy and other natural sciences1 but in all relations she is entitled to the first rank4--,,gauss4 ,sartorius von ,waltershausen3 ,gauss zum ,gedchtniss4 7,leipzig1 #ahff71 p4 #gi4 #afdc4 ,zu ,archimedes kam ein wissbegieriger ,jngling1 ,weihe mich1 sprach er zu ihm1 ein in die gttliche ,kunst1 ,die so herrliche ,dienste der ,sternenkunde geleistet1 ,hinter dem ,uranos noch einen ,planeten entdeckt4 ,gttlich nennst ,du die ,kunst1 sie ist's1 versetzte der ,weise1 ,aber sie war es1 bevor noch sie den ,kosmos erforscht1 ,ehe sie herrliche ,dienste der ,sternenkunde geleistet1 ,hinter dem ,uranos noch einen ,planeten entdeckt4 ,was ,du im ,kosmos erblickst1 ist nur der ,gttlichen ,abglanz1 ,in der ,olympier ,schaar thronet die ewige ,zahl4 --,,jacobi1 ,c4 ,g4 ,j4 ,journal fr ,mathematik1 ,bd4 #aja 7#ahhg71 p4 #cch4 ,to ,archimedes came a youth intent upon knowledge1 ,quoth he1 8,initiate me into the science divine ,which to astronomy1 lo6 such excellent service has rendered1 ,and beyond ,uranus' orb a hidden planet revealed40 8,call'st thou the science divine8 ,so it is10 the wise man responded1 8,but so it was long before its light on the ,cosmos it shed1 ,ere in astronomy's realm such excellent service it rendered1 ,and beyond ,uranus' orb a hidden planet revealed4 ,only reflection divine is that which ,cosmos discloses1 ,number herself sits enthroned among ,olympia's hosts40 #afdd4 ,the higher arithmetic presents us with an inexhaustible store of interesting truths1--of truths too1 which are not isolated1 but stand in a close internal connexion1 and between which1 as our knowledge increases1 we are continually discovering new and sometimes wholly unexpected ties4 ,a great part of its theories derives an additional charm from the peculiarity that important propositions1 with the impress of simplicity upon them1 are often easily discoverable by induction1 and yet are of so profound a character that we cannot find their demonstration till after many vain attempts2 and even then1 when we do succeed1 it is often by some tedious and artificial process1 while the simpler methods may long remain concealed4--,,gauss1 ,c4 ,f4 ,preface to ,eisenstein's ,mathematische ,abhandlungen 7,berlin1 #ahdg71 ,h4 ,j4 ,s4 ,smith4 #afde4 ,the ,theory of ,numbers has acquired a great and increasing claim to the attention of mathematicians4 ,it is equally remarkable for the number and importance of its results1 for the precision and rigorousness of its demonstrations1 for the variety of its methods1 for the intimate relations between truths apparently isolated which it sometimes discloses1 and for the numerous applications of which it is susceptible in other parts of analysis4--,,smith1 ,h4 ,j4 ,s4 ,report on the ,theory of ,numbers1 ,british ,association1 #ahei2 ,collected ,mathematical ,papers1 ,vol4 #a1 p4 #ch4 #afdf4 ,the invention of the symbol by ,gauss affords a striking example of the advantage which may be derived from an appropriate notation1 and marks an epoch in the development of the science of arithmetic4--,,mathews1 ,g4 ,b4 ,theory of ,numbers 7,cambridge1 #ahib71 ,part #a1 sect4 #bi4 #afdg4 ,as ,gauss first pointed out1 the problem of cyclotomy1 or division of the circle into a number of equal parts1 depends in a very remarkable way upon arithmetical considerations4 ,we have here the earliest and simplest example of those relations of the theory of numbers to transcendental analysis1 and even to pure geometry1 which so often unexpectedly present themselves1 and which1 at first sight1 are so mysterious4--,,mathews1 ,g4 ,b4 ,theory of ,numbers 7,cambridge1 #ahib71 ,part #a1 sect4 #afg4 #afdh4 ,i have sometimes thought that the profound mystery which envelops our conceptions relative to prime numbers depends upon the limitations of our faculties in regard to time1 which like space may be in its essence poly-dimensional1 and that this and such sort of truths would become self-evident to a being whose mode of perception is according to superficially as distinguished from our own limitation to linearly extended time4--,,sylvester1 ,j4 ,j4 ,collected ,mathematical ,papers1 ,vol4 #d1 p4 #fjj1 footnote4 ,,chapter ,,xvii ,,algebra #agja4 ,the science of algebra1 independently of any of its uses1 has all the advantages which belong to mathematics in general as an object of study1 and which it is not necessary to enumerate4 ,viewed either as a science of quantity1 or as a language of symbols1 it may be made of the greatest service to those who are sufficiently acquainted with arithmetic1 and who have sufficient power of comprehension to enter fairly upon its difficulties4 --,,de ,,morgan1 ,a4 ,elements of ,algebra 7,london1 #ahcg71 ,preface4 #agjb4 ,algebra is generous1 she often gives more than is asked of her4--,d',,alembert4 ,quoted in ,bulletin ,american ,mathematical ,society1 ,vol4 #b 7#aije71 p4 #bhe4 #agjc4 ,the operations of symbolic arithmetick seem to me to afford men one of the clearest exercises of reason that ,i ever yet met with1 nothing being there to be performed without strict and watchful ratiocination1 and the whole method and progress of that appearing at once upon the paper1 when the operation is finished1 and affording the analyst a lasting1 and1 as it were1 visible ratiocination4--,,boyle1 ,,robert4 ,works 7,london1 #aggb71 ,vol4 #c1 p4 #dbf4 #agjd4 ,the human mind has never invented a labor-saving machine equal to algebra4-- ,the ,nation1 ,vol4 #cc1 p4 #bcg4 #agje4 ,they that are ignorant of ,algebra cannot imagine the wonders in this kind are to be done by it3 and what further improvements and helps advantageous to other parts of knowledge the sagacious mind of man may yet find out1 it is not easy to determine4 ,this at least ,i believe1 that the ideas of quantity are not those alone that are capable of demonstration and knowledge2 and that other1 and perhaps more useful1 parts of contemplation1 would afford us certainty1 if vices1 passions1 and domineering interest did not oppose and menace such endeavours4 --,,locke1 ,,john4 ,an ,essay concerning ,human ,understanding1 ,bk4 #d1 chap4 #c1 sect4 #ah4 #agjf4 ,algebra is but written geometry and geometry is but figured algebra4--,,germain1 ,,sophie4 ,mmoire sur la surfaces lastiques4 #agjg4 ,so long as algebra and geometry proceeded separately their progress was slow and their application limited1 but when these two sciences were united1 they mutually strengthened each other1 and marched together at a rapid pace toward perfection4 --,,lagrange4 ,leons lmentaires sur les ,mathmatiques1 ,leon ,cinquime4 #agjh4 ,the laws of algebra1 though suggested by arithmetic1 do not depend on it4 ,they depend entirely on the conventions by which it is stated that certain modes of grouping the symbols are to be considered as identical4 ,this assigns certain properties to the marks which form the symbols of algebra4 ,the laws regulating the manipulation of algebraic symbols are identical with those of arithmetic4 ,it follows that no algebraic theorem can ever contradict any result which could be arrived at by arithmetic2 for the reasoning in both cases merely applies the same general laws to different classes of things4 ,if an algebraic theorem can be interpreted in arithmetic1 the corresponding arithmetical theorem is therefore true4--,,whitehead1 ,a4 ,n4 ,universal ,algebra 7,cambridge1 #ahih71 p4 #b4 #agji4 ,that a formal science like algebra1 the creation of our abstract thought1 should thus1 in a sense1 dictate the laws of its own being1 is very remarkable4 ,it has required the experience of centuries for us to realize the full force of this appeal4 --,,mathews1 ,g4 ,b4 ,f4 ,spencer3 ,chapters on ,aims and ,practice of ,teaching 7,london1 #ahii71 p4 #ahd4 #agaj4 ,the rules of algebra may be investigated by its own principles1 without any aid from geometry2 and although in many cases the two sciences may serve to illustrate each other1 there is not now the least necessity in the more elementary parts to call in the aid of the latter in expounding the former4 --,,chrystal1 ,,george4 ,encyclopedia ,britannica1 #ith ,edition2 ,article 8,algebra40 #agaa4 ,algebra1 as an art1 can be of no use to any one in the business of life2 certainly not as taught in the schools4 ,i appeal to every man who has been through the school routine whether this be not the case4 ,taught as an art it is of little use in the higher mathematics1 as those are made to feel who attempt to study the differential calculus without knowing more of the principles than is contained in books of rules4 --,,de ,,morgan1 ,a4 ,elements of ,algebra 7,london1 #ahcg71 ,preface4 #agab4 ,we may always depend upon it that algebra1 which cannot be translated into good ,english and sound common sense1 is bad algebra4--,,clifford1 ,w4 ,k4 ,common ,sense in the ,exact ,sciences 7,london1 #ahhe71 chap4 #a1 sect4 #g4 #agac4 ,the best review of arithmetic consists in the study of algebra4--,,cajori1 ,f4 ,teaching and ,history of ,mathematics in ,u4 ,s4 7,washington1 #ahif71 p4 #aaj4 #agad4 ,algebra has for its object the resolution of equations2 taking this expression in its full logical meaning1 which signifies the transformation of implicit functions into equivalent explicit ones4 ,in the same way arithmetic may be defined as destined to the determination of the values of functions4444 ,we will briefly say that ,algebra is the ,calculus of ,functions1 and ,arithmetic the ,calculus of ,values4--,,comte1 ,a4 ,philosophy of ,mathematics ,gillespie 7,new ,york1 #ahea71 p4 #ee4 #agae4 444 the subject matter of algebraic science is the abstract notion of time2 divested of1 or not yet clothed with1 any actual knowledge which we may possess of the real ,events of ,history1 or any conception which we may frame of ,cause and ,effect in ,nature2 but involving1 what indeed it cannot be divested of1 the thought of possible ,succession1 or of pure1 ideal ,progression4--,,hamilton1 ,w4 ,r4 ,graves' ,life of ,hamilton 7,new ,york1 #ahhb-ahhi71 ,vol4 #c1 p4 #fcc4 #agaf4 444 instead of seeking to attain consistency and uniformity of system1 as some modern writers have attempted1 by banishing this thought of time from the higher ,algebra1 ,i seek to attain the same object1 by systematically introducing it into the lower or earlier parts of the science4--,,hamilton1 ,w4 ,r4 ,graves' ,life of ,hamilton 7,new ,york1 #ahhb-ahhi71 ,vol4 #c1 p4 #fcd4 #agag4 ,the circumstances that algebra has its origin in arithmetic1 however widely it may in the end differ from that science1 led ,sir ,isaac ,newton to designate it 8,universal ,arithmetic10 a designation which1 vague as it is1 indicates its character better than any other by which it has been attempted to express its functions--better certainly1 to ordinary minds1 than the designation which has been applied to it by ,sir ,william ,rowan ,hamilton1 one of the greatest mathematicians the world has seen since the days of ,newton--8the ,science of ,pure ,time20 or even than the title by which ,de ,morgan would paraphrase ,hamilton's words--8the ,calculus of ,succession40--,,chrystal1 ,,george4 ,encyclopedia ,britannica1 #ith ,edition2 ,article 8,algebra40 #agah4 ,time is said to have only one dimension1 and space to have three dimensions4444 ,the mathematical quaternion partakes of both these elements2 in technical language it may be said to be 8time plus space10 or 8space plus time30 and in this sense it has1 or at least involves a reference to1 four dimensions4444 ,and how the ,one of ,time1 of ,space the ,three1 ,might in the ,chain of ,symbols girdled be4 --,,hamilton1 ,w4 ,r4 ,graves' ,life of ,hamilton 7,new ,york1 #ahhb-ahhi71 ,vol4 #c1 p4 #fce4 #agai4 ,it is confidently predicted1 by those best qualified to judge1 that in the coming centuries ,hamilton's ,quaternions will stand out as the great discovery of our nineteenth century4 ,yet how silently has the book taken its place upon the shelves of the mathematician's library6 ,perhaps not fifty men on this side of the ,atlantic have seen it1 certainly not five have read it4 --,,hill1 ,,thomas4 ,north ,american ,review1 ,vol4 #he1 p4 #bbc4 #agbj4 ,i think the time may come when double algebra will be the beginner's tool2 and quaternions will be where double algebra is now4 ,the ,lord only knows what will come above the quaternions4 --,,de ,,morgan1 ,a4 ,graves' ,life of ,hamilton 7,new ,york1 #ahhb-ahhi71 ,vol4 #c1 p4 #dic4 #agba4 ,quaternions came from ,hamilton after his really good work had been done2 and though beautifully ingenious1 have been an unmixed evil to those who have touched them in any way1 including ,clerk ,maxwell4--,,thomson1 ,,william4 ,thompson1 ,s4 ,p43 ,life of ,lord ,kelvin 7,london1 #aiaj71 p4 #aach4 #agbb4 ,the whole affair quaternions has in respect to mathematics a value not inferior to that of 8,volapuk0 in respect to language4--,,thomson1 ,,william4 ,thompson1 ,s4 ,p43 ,life of ,lord ,kelvin 7,london1 #aiaj71 p4 #aach4 #agbc4 ,a quaternion of maladies6 ,do send me some formula by help of which ,i may so doctor them that they may all become imaginary or positively equal to nothing4--,,sedgwick4 ,graves' ,life of ,hamilton 7,new ,york1 #ahhb-ahhi71 ,vol4 #c1 p4 #b4 #agbd4 ,if nothing more could be said of ,quaternions than that they enable us to exhibit in a singularly compact and elegant form1 whose meaning is obvious at a glance on account of the utter inartificiality of the method1 results which in the ordinary ,cartesian co-ordinates are of the utmost complexity1 a very powerful argument for their use would be furnished4 ,but it would be unjust to ,quaternions to be content with such a statement2 for we are fully entitled to say that in all cases1 even in those to which the ,cartesian methods seem specially adapted1 they give as simple an expression as any other method2 while in the great majority of cases they give a vastly simpler one4 ,in the common methods a judicious choice of co-ordinates is often of immense importance in simplifying an investigation2 in ,quaternions there is usually no choice1 for 7except when they degrade to mere scalars7 they are in general utterly independent of any particular directions in space1 and select of themselves the most natural reference lines for each particular problem4 --,,tait1 ,p4 ,g4 ,presidential ,address ,british ,association for the ,advancement of ,science 7#ahga72 ,nature1 ,vol4 #d1 p4 #bgj4 #agbe4 ,comparing a ,quaternion investigation1 no matter in what department1 with the equivalent ,cartesian one1 even when the latter has availed itself to the utmost of the improvements suggested by ,higher ,algebra1 one can hardly help making the remark that they contrast even more strongly than the decimal notation with the binary scale1 or with the old ,greek arithmetic--or than the well-ordered subdivisions of the metrical system with the preposterous no-systems of ,great ,britain1 a mere fragment of which 7in the form of ,table of ,weights and ,measures7 form1 perhaps the most effective1 if not the most ingenious1 of the many instruments of torture employed in our elementary teaching4--,,tait1 ,p4 ,g4 ,presidential ,address ,british ,association for the ,advancement of ,science 7#ahga72 ,nature1 ,vol4 #d1 p4 #bga4 #agbf4 ,it is true that1 in the eyes of the pure mathematician1 ,quaternions have one grand and fatal defect4 ,they cannot be applied to space of n dimensions1 they are contented to deal with those poor three dimensions in which mere mortals are doomed to dwell1 but which cannot bound the limitless aspirations of a ,cayley or a ,sylvester4 ,from the physical point of view this1 instead of a defect1 is to be regarded as the greatest possible recommendation4 ,it shows1 in fact1 ,quaternions to be the special instrument so constructed for application to the ,actual as to have thrown overboard everything which is not absolutely necessary1 without the slightest consideration whether or no it was thereby being rendered useless for application to the ,inconceivable4--,,tait1 ,p4 ,g4 ,presidential ,address ,british ,association for the ,advancement of ,science 7#ahga72 ,nature1 ,vol4 #d1 p4 #bga4 #agbg4 ,there is an old epigram which assigns the empire of the sea to the ,english1 of the land to the ,french1 and of the clouds to the ,germans4 ,surely it was from the clouds that the ,germans fetched and 2 the ideas which these symbols have generated are much too important to the welfare of humanity to have come from the sea or from the land4--,,whitehead1 ,a4 ,n4 ,an ,introduction to ,mathematics 7,new ,york1 #aiaa71 p4 #hf4 #agbh4 ,now as to what pertains to these ,surd numbers 7which1 as it were by way of reproach and calumny1 having no merit of their own are also styled ,irrational1 ,irregular1 and ,inexplicable7 they are by many denied to be numbers properly speaking1 and are wont to be banished from arithmetic to another ,science1 7which yet is no science7 viz4 algebra4--,,barrow1 ,,isaac4 ,mathematical ,lectures 7,london1 #agcd71 p4 #dd4 #agbi4 ,if it is true as ,whewell says1 that the essence of the triumphs of science and its progress consists in that it enables us to consider evident and necessary1 views which our ancestors held to be unintelligible and were unable to comprehend1 then the extension of the number concept to include the irrational1 and we will at once add1 the imaginary1 is the greatest forward step which pure mathematics has ever taken4--,,hankel1 ,,hermann4 ,theorie der ,complexen ,zahlen 7,leipzig1 #ahfg71 p4 #fj4 #agcj4 ,that this subject of imaginary magnitudes has hitherto been considered from the wrong point of view and surrounded by a mysterious obscurity1 is to be attributed largely to an ill-adapted notation4 ,if for instance1 #a1a1 #a had been called direct1 inverse1 and lateral units1 instead of positive1 negative1 and imaginary 7or even impossible7 such an obscurity would have been out of question4--,,gauss1 ,c4 ,f4 ,theoria residiorum biquadraticorum1 ,commentatio secunda2 ,werke1 ,bd4 #b 7,goettingen1 #ahfc71 p4 #agg4 #agca4 444 the imaginary1 this bosom-child of complex mysticism4 --,d,,hring1 ,,eugen4 ,kritische ,geschichte der allgemeinen ,principien der ,mechanik 7,leipzig1 #ahgg71 p4 #eag4 #agcb4 ,judged by the only standards which are admissible in a pure doctrine of numbers i is imaginary in the same sense as the negative1 the fraction1 and the irrational1 but in no other sense2 all are alike mere symbols devised for the sake of representing the results of operations even when these results are not numbers 7positive integers74--,,fine1 ,h4 ,b4 ,the ,number-,system of ,algebra 7,boston1 #ahij71 p4 #cf4 #agcc4 ,this symbol -#a is restricted to a precise signification as the representative of perpendicularity in quaternions1 and this wonderful algebra of space is intimately dependent upon the special use of the symbol for its symmetry1 elegance1 and power4 ,the immortal author of quaternions has shown that there are other significations which may attach to the symbol in other cases4 ,but the strongest use of the symbol is to be found in its magical power of doubling the actual universe1 and placing by its side an ideal universe1 its exact counterpart1 with which it can be compared and contrasted1 and1 by means of curiously connecting fibres1 form with it an organic whole1 from which modern analysis has developed her surpassing geometry4--,,peirce1 ,,benjamin4 ,on the ,uses and ,transformations of ,linear ,algebras2 ,american ,journal of ,mathematics1 ,vol4 #d 7#ahha71 p4 #baf4 #agcd4 ,the conception of the inconceivable imaginary1 this measurement of what not only does not1 but cannot exist1 is one of the finest achievements of the human intellect4 ,no one can deny that such imaginings are indeed imaginary4 ,but they lead to results grander than any which flow from the imagination of the poet4 ,the imaginary calculus is one of the masterkeys to physical science4 ,these realms of the inconceivable afford in many places our only mode of passage to the domains of positive knowledge4 ,light itself lay in darkness until this imaginary calculus threw light upon light4 ,and in all modern researches into electricity1 magnetism1 and heat1 and other subtile physical inquiries1 these are the most powerful instruments4--,,hill1 ,,thomas4 ,north ,american ,review1 ,vol4 #he1 p4 #bce4 #agce4 ,all the fruitful uses of imaginaries1 in ,geometry1 are those which begin and end with real quantities1 and use imaginaries only for the intermediate steps4 ,now in all such cases1 we have a real spatial interpretation at the beginning and end of our argument1 where alone the spatial interpretation is important2 in the intermediate links1 we are dealing in purely algebraic manner with purely algebraic quantities1 and may perform any operations which are algebraically permissible4 ,if the quantities with which we end are capable of spatial interpretation1 then1 and only then1 our results may be regarded as geometrical4 ,to use geometrical language1 in any other case1 is only a convenient help to the imagination4 ,to speak1 for example1 of projective properties which refer to the circular points1 is a mere memoria technica for purely algebraical properties2 the circular points are not to be found in space1 but only in the auxiliary quantities by which geometrical equations are transformed4 ,that no contradictions arise from the geometrical interpretation of imaginaries is not wonderful2 for they are interpreted solely by the rules of ,algebra1 which we may admit as valid in their interpretation to imaginaries4 ,the perception of space being wholly absent1 ,algebra rules supreme1 and no inconsistency can arise4--,,russell1 ,,bertrand4 ,foundations of ,geometry 7,cambridge1 #ahig71 p4 #de4 #agcf4 ,indeed1 if one understands by algebra the application of arithmetic operations to composite magnitudes of all kinds1 whether they be rational or irrational number or space magnitudes1 then the learned ,brahmins of ,hindostan are the true inventors of algebra4--,,hankel1 ,,hermann4 ,geschichte der ,mathematik im ,altertum und ,mittelalter 7,leipzig1 #ahgd71 p4 #aie4 #agcg4 ,it is remarkable to what extent ,indian mathematics enters into the science of our time4 ,both the form and the spirit of the arithmetic and algebra of modern times are essentially ,indian and not ,grecian4--,,cajori1 ,f4 ,history of ,mathematics 7,new ,york1 #ahig71 p4 #ajj4 #agch4 ,there are many questions in this science algebra which learned men have to this time in vain attempted to solve2 and they have stated some of these questions in their writings1 to prove that this science contains difficulties1 to silence those who pretend they find nothing in it above their ability1 to warn mathematicians against undertaking to answer every question that may be proposed1 and to excite men of genius to attempt their solution4 ,of these ,i have selected seven4 #a4 ,to divide #aj into two parts1 such1 that when each part is added to its square-root and the sums multiplied together1 the product is equal to the supposed number4 #b4 ,what square is that1 which being increased or diminished by #aj1 the sum and remainder are both square numbers8 #c4 ,a person said he owed to ,zaid #aj all but the square-root of what he owed to ,amir1 and that he owed ,amir #e all but the square-root of what he owed ,zaid4 #d4 ,to divide a cube number into two cube numbers4 #e4 ,to divide #aj into two parts such1 that if each is divided by the other1 and the two quotients are added together1 the sum is equal to one of the parts4 #f4 ,there are three square numbers in continued geometric proportion1 such1 that the sum of the three is a square number4 #g4 ,there is a square1 such1 that when it is increased and diminished by its root and #b1 the sum and the difference are squares4--,,khulasat-,,al-,,hisab4 ,algebra2 quoted in ,hutton3 ,a ,philosophical and ,mathematical ,dictionary 7,london1 #ahae71 ,vol4 #a1 p4 #gj4 #agci4 ,the solution of such questions as these referring to the solution of cubic equations depends on correct judgment1 aided by the assistance of ,god4--,,bija ,,ganita4 ,quoted in ,hutton3 ,a ,philosophical and ,mathematical ,dictionary 7,london1 #ahae71 ,vol4 #a1 p4 #fe4 #agdj4 ,for what is the theory of determinants8 ,it is an algebra upon algebra2 a calculus which enables us to combine and foretell the results of algebraical operations1 in the same way as algebra itself enables us to dispense with the performance of the special operations of arithmetic4 ,all analysis must ultimately clothe itself under this form4--,,sylvester1 ,j4 ,j4 ,philosophical ,magazine1 ,vol4 #a1 7#ahea71 p4 #cjj2 ,collected ,mathematical ,papers1 ,vol4 #a1 p4 #bdg4 #agda4 ,fuchs4 ,fast mcht' ich nun moderne ,algebra studieren4 ,meph4 ,ich wnschte nicht euch irre zu fhren4 ,was diese ,wissenschaft betrifft1 ,es ist so schwer1 die leere ,form zu meiden1 ,und wenn ihr es nicht recht begrifft1 ,vermgt die ,indices ihr kaum zu unterscheiden4 ,am ,besten ist's1 wenn ihr nur ,einem traut ,und auf des ,meister's ,formeln baut4 ,im ,ganzen--haltet euch an die ,symbole4 ,dann geht ihr zu der ,forschung ,wohle ,ins sichre ,reich der ,formeln ein4 ,fuchs4 ,ein ,resultat muss beim ,symbole sein8 ,meph4 ,schon gut6 ,nur muss man sich nicht alzu ngstlich qulen4 ,denn eben1 wo die ,resultate fehlen1 ,stellt ein ,symbol zur rechten ,zeit sich ein4 ,symbolisch lsst sich alles schreiben1 ,msst nur im ,allgemeinen bleiben4 ,wenn man der ,gleichung ,lsung nicht erkannte1 ,schreibt man sie als ,determinante4 ,schreib' was du willst1 nur rechne nie was aus4 ,symbole lassen trefflich sich traktieren1 ,mit einem ,strich ist alles auszufhren1 ,und mit ,symbolen kommt man immer aus4 --,,lasswitz1 ,,kurd4 ,der ,faust-,tragdie 7-n7ter ,teil2 ,zeitschrift fr mathematischen und naturwissenschaftlichen ,unterricht1 ,bd4 #ad1 p4 #cag4 ,fuchs4 ,to study modern algebra ,i'm most persuaded4 ,meph4 ',twas not my wish to lead thee astray4 ,but as concerns this science1 truly ',tis difficult to avoid the empty form1 ,and should'st thou lack clear comprehension1 ,scarcely the indices thou'll know apart4 ',tis safest far to trust but one ,and built upon your master's formulas4 ,on the whole--cling closely to your symbols4 ,then1 for the weal of research you may gain ,an entrance to the formula's sure domain4 ,fuchs4 ,the symbol1 it must lead to some result8 ,meph4 ,granted4 ,but never worry about results1 ,for1 mind you1 just where the results are wanting ,a symbol at the nick of time appears4 ,to symbolic treatment all things yield1 ,provided we stay in the general field4 ,should a solution prove elusive1 ,write the equation in determinant form4 ,write what you please1 but never calculate4 ,symbols are patient and long suffering1 ,a single stroke completes the whole affair4 ,symbols for every purpose do suffice4 #agdb4 ,as all roads are said to lead to ,rome1 so ,i find1 in my own case at least1 that all algebraic inquiries sooner or later end at the ,capitol of ,modern ,algebra over whose shining portal is inscribed 8,theory of ,invariants40--,,sylvester1 ,j4 ,j4 ,on ,newton's ,rule for the ,discovery of ,imaginary ,roots2 ,collected ,mathematical ,papers1 ,vol4 #b1 p4 #chj4 #agdc4 ,if we consider the beauty of the theorem ,sylvester's ,theorem on ,newton's ,rule for the ,discovery of ,imaginary ,roots which has now been expounded1 the interest which belongs to the rule associated with the great name of ,newton1 and the long lapse of years during which the reason and extent of that rule remained undiscovered by mathematicians1 among whom ,maclaurin1 ,waring and ,euler are explicitly included1 we must regard ,professor ,sylvester's investigations made to the ,theory of ,equations in modern times1 justly to be ranked with those of ,fourier1 ,sturm and ,cauchy4--,,todhunter1 ,i4 ,theory of ,equations 7,london1 #aijd71 p4 #bej4 #agdd4 ,considering the remarkable elegance1 generality1 and simplicity of the method ,homer's ,method of finding the numerical values of the roots of an equation1 it is not a little surprising that it has not taken a more prominent place in current mathematical textbooks4444 ,as a matter of fact1 its spirit is purely arithmetical2 and its beauty1 which can only be appreciated after one has used it in particular cases1 is of that indescribably simple kind1 which distinguishes the use of position in the decimal notation and the arrangement of the simple rules of arithmetic4 ,it is1 in short1 one of those things whose invention was the creation of a commonplace4 --,,chrystal1 ,,george4 ,algebra 7,london and ,edinburgh1 #ahic71 ,vol4 #a1 chap4 #ae1 sect4 #be4 #agde4 ,to a missing member of a family group of terms in an algebraical formula4 ,lone and discarded one6 divorced by fate1 ,far from thy wished-for fellows--whither art flown8 ,where lingerest thou in thy bereaved estate1 ,like some lost star1 or buried meteor stone8 ,thou mindst me much of that presumptuous one ,who loth1 aught less than greatest1 to be great1 ,from ,heaven's immensity fell headlong down ,to live forlorn1 self-centred1 desolate3 ,or who1 like ,heraclid1 hard exile bore1 ,now buoyed by hope1 now stretched on rack of fear1 ,till throned ,asta1 wafting to his ear ,words of dim portent through the ,atlantic roar1 ,bade him 8the sanctuary of the ,muse revere ,and strew with flame the dust of ,isis' shore40 --,,sylvester1 ,j4 ,j4 ,inaugural ,lecture1 ,oxford1 #ahhe2 ,nature1 ,vol4 #cc1 p4 #bbh4 #agdf4 ,in every subject of inquiry there are certain entities1 the mutual relations of which1 under various conditions1 it is desirable to ascertain4 ,a certain combination of these entities are submitted to certain processes or are made the subjects of certain operations4 ,the theory of invariants in its widest scientific meaning determines these combinations1 elucidates their properties1 and expresses results when possible in terms of them4 ,many of the general principles of political science and economics can be represented by means of invariantive relations connecting the factors which enter as entities into the special problems4 ,the great principle of chemical science which asserts that when elementary or compound bodies combine with one another the total weight of the materials is unchanged1 is another case in point4 ,again1 in physics1 a given mass of gas under the operation of varying pressure and temperature has the well-known invariant1 pressure multiplied by volume and divided by absolute temperature4444 ,in mathematics the entities under examination may be arithmetical1 algebraical1 or geometrical2 the processes to which they are subjected may be any of those which are met with in mathematical work4444 ,it is the principle which is so valuable4 ,it is the idea of invariance that pervades today all branches of mathematics4--,,macmahon1 ,p4 ,a4 ,presidential ,address ,british ,association for the ,advancement of ,science 7#aija72 ,nature1 ,vol4 #fd1 p4 #dha4 #agdg4 ,the theory of invariants has invaded the domain of geometry1 and has almost re-created the analytical theory2 but it has done more than this for the investigations of ,cayley have required a full reconsideration of the very foundations of geometry4 ,it has exercised a profound influence upon the theory of algebraic equations2 it has made its way into the theory of differential equations2 and the generalisation of its ideas is opening out new regions of most advanced and profound functional analysis4 ,and so far from its course being completed1 its questions fully answered1 or its interest extinct1 there is no reason to suppose that a term can be assigned to its growth and its influence4--,,forsyth1 ,a4 ,r4 ,presidential ,address ,british ,association for the ,advancement of ,science 7#ahig72 ,nature1 ,vol4 #ef1 p4 #cgh4 #agdh4 444 the doctrine of ,invariants1 a theory filling the heavens like a light-bearing ether1 penetrating all the branches of geometry and analysis1 revealing everywhere abiding configurations in the midst of change1 everywhere disclosing the eternal reign of the law of form4--,,keyser1 ,c4 ,j4 ,lectures on ,science1 ,philosophy and ,art 7,new ,york1 #aijh71 p4 #bh4 #agdi4 ,it is in the mathematical doctrine of ,invariance1 the realm wherein are sought and found configurations and types of being that1 amidst the swirl and stress of countless hosts of transformations remain immutable1 and the spirit dwells in contemplation of the serene and eternal reign of the subtile laws of ,form1 it is there that ,theology may find1 if she will1 the clearest conceptions1 the noblest symbols1 the most inspiring intimations1 the most illuminating illustrations1 and the surest guarantees of the object of her teaching and her quest1 an ,eternal ,being1 unchanging in the midst of the universal flux4 --,,keyser1 ,c4 ,j4 ,lectures on ,science1 ,philosophy and ,art 7,new ,york1 #aijh71 p4 #db4 #agej4 ,i think that young chemists desirous of raising their science to its proper rank would act wisely in making themselves master betimes of the theory of algebraic forms4 ,what mechanics is to physics1 that ,i think is algebraic morphology1 founded at option on the theory of partitions or ideal elements1 or both1 is destined to be to the chemistry of the future 444 invariants and isomerism are sister theories4--,,sylvester1 ,j4 ,j4 ,american ,journal of ,mathematics1 ,vol4 #a 7#ahgh71 p4 #abf4 #agea4 ,the great notion of ,group1 444 though it had barely merged into consciousness a hundred years ago1 has meanwhile become a concept of fundamental importance and prodigious fertility1 not only affording the basis of an imposing doctrine--the ,theory of ,groups--but therewith serving also as a bond of union1 a kind of connective tissue1 or rather as an immense cerebro-spinal system1 uniting together a large number of widely dissimilar doctrines as organs of a single body4--,,keyser1 ,c4 ,j4 ,lectures on ,science1 ,philosophy and ,art 7,new ,york1 #aijh71 p4 #ab4 #ageb4 ,in recent times the view becomes more and more prevalent that many branches of mathematics are nothing but the theory of invariants of special groups4--,,lie1 ,,sophus4 ,continuierliche ,gruppen--,scheffers 7,leipzig1 #ahic71 p4 #ffe4 #agec4 ,universal ,algebra has been looked on with some suspicion by many mathematicians1 as being without intrinsic mathematical interest and as being comparatively useless as an engine of investigation4444 ,but it may be shown that ,universal ,algebra has the same claim to be a serious subject of mathematical study as any other branch of mathematics4--,,whitehead1 ,a4 ,n4 ,universal ,algebra 7,cambridge1 #ahih71 ,preface1 p4 vi4 #aged4 ,function theory was1 in effect1 founded by ,cauchy2 but1 outside his own investigations1 it at first made slow and hesitating progress4 ,at the present day1 its fundamental ideas may be said almost to govern most departments of the analysis of continuous quantity4 ,on many of them1 it has shed a completely new light2 it has educed relations between them before unknown4 ,it may be doubted whether any subject is at the present day so richly endowed with variety of method and fertility of resource2 its activity is prodigious1 and no less remarkable than its activity is its freshness4--,,forsyth1 ,a4 ,r4 ,presidential ,address ,british ,association for the ,advancement of ,science 7#ahig72 ,nature1 ,vol4 #ef1 p4 #cgh4 #agee4 ,let me mention one other contribution which this theory ,theory of functions of a complex variable has made to knowledge lying somewhat outside our track4 ,during the rigorous revision to which the foundations of the theory have been subjected in its re-establishment by ,weierstrass1 new ideas as regards number and continuity have been introduced4 ,with him and with others influenced by him1 there has thence sprung a new theory of higher arithmetic2 and with its growth1 much has concurrently been effected in the elucidation of the general notions of number and quantity4444 ,it thus appears to be the fact that1 as with ,plato1 or ,descartes1 or ,leibnitz1 or ,kant1 the activity of pure mathematics is again lending some assistance to the better comprehension of those notions of time1 space1 number1 quantity1 which underlie a philosophical conception of the universe4 --,,forsyth1 ,a4 ,r4 ,presidential ,address ,british ,association for the ,advancement of ,science 7#ahig72 ,nature1 ,vol4 #ef1 p4 #cgh4 ,,chapter ,,xviii ,,geometry #ahja4 ,the science of figures is most glorious and beautiful4 ,but how inaptly it has received the name geometry6--,,frischlinus1 ,n4 ,dialog #a4 #ahjb4 ,plato said that ,god geometrizes continually4--,,plutarch4 ,convivialium disputationum1 liber #h1 #b4 #ahjc4 4 ,let no one ignorant of geometry enter my door4--,,plato4 ,tzetzes1 ,chiliad1 #h1 #igb4 #ahjd4 ,all the authorities agree that he ,plato made a study of geometry or some exact science an indispensable preliminary to that of philosophy4 ,the inscription over the entrance to his school ran 8,let none ignorant of geometry enter my door10 and on one occasion an applicant who knew no geometry is said to have been refused admission as a student4--,,ball1 ,w4 ,w4 ,r4 ,history of ,mathematics 7,london1 #aija71 p4 #de4 #ahje4 ,form and size constitute the foundation of all search for truth4--,,parker1 ,f4 ,w4 ,talks on ,pedagogics 7,new ,york1 #ahid71 p4 #gb4 #ahjf4 ,at present the science of geometry is in flat contradiction to the language which geometricians use1 as will hardly be denied by those who have any acquaintance with the study3 for they speak of finding the side of a square1 and applying and adding1 and so on1 as if they were engaged in some business1 and as if all their propositions had a practical end in view3 whereas in reality the science is pursued wholly for the sake of knowledge4 ,certainly1 he said4 ,then must not a further admission be made8 ,what admission8 ,the admission that this knowledge at which geometry aims is of the eternal1 and not of the perishing and transient4 ,that may be easily allowed4 ,geometry1 no doubt1 is the knowledge of what eternally exists4 ,then1 my noble friend1 geometry will draw the soul towards truth1 and create the mind of philosophy1 and raise up that which is now unhappily allowed to fall down4--,,plato4 ,republic ,jowett-,davies1 ,bk4 #g1 p4 #ebg4 #ahjg4 ,among them the ,greeks geometry was held in highest honor3 nothing was more glorious than mathematics4 ,but we have limited the usefulness of this art to measuring and calculating4 --,,cicero4 ,tusculanae ,disputationes1 #a1 #b1 #e4 #ahjh4 ,geometria1 ,through which a man hath the sleight ,of length1 and brede1 of depth1 of height4 --,,gower1 ,,john4 ,confessio ,amantis1 ,bk4 #g4 #ahji4 ,geometrical truths are in a way asymptotes to physical truths1 that is to say1 the latter approach the former indefinitely near without ever reaching them exactly4--,d',,alembert4 ,quoted in ,rebire3 ,mathmatiques et ,mathmaticiens 7,paris1 #ahih71 p4 #aj4 #ahaj4 ,geometry exhibits the most perfect example of logical stratagem4--,,buckle1 ,h4 ,t4 ,history of ,civilization in ,england 7,new ,york1 #ahia71 ,vol4 #b1 p4 #cdb4 #ahaa4 ,it is the glory of geometry that from so few principles1 fetched from without1 it is able to accomplish so much4--,,newton4 ,philosophiae ,naturalis ,principia ,mathematica1 ,praefat4 #ahab4 ,geometry is the application of strict logic to those properties of space and figure which are self-evident1 and which therefore cannot be disputed4 ,but the rigor of this science is carried one step further2 for no property1 however evident it may be1 is allowed to pass without demonstration1 if that can be given4 ,the question is therefore to demonstrate all geometrical truths with the smallest possible number of assumptions4 --,,de ,,morgan1 ,a4 ,on the ,study and ,difficulties of ,mathematics 7,chicago1 #aijb71 p4 #bca4 #ahac4 ,geometry is a true natural science3--only more simple1 and therefore more perfect than any other4 ,we must not suppose that1 because it admits the application of mathematical analysis1 it is therefore a purely logical science1 independent of observation4 ,every body studied by geometers presents some primitive phenomena which1 not being discoverable by reasoning1 must be due to observation alone4--,,comte1 ,a4 ,positive ,philosophy ,martineau1 ,bk4 #a1 chap4 #c4 #ahad4 ,geometry in every proposition speaks a language which experience never dares to utter2 and indeed of which she but half comprehends the meaning4 ,experience sees that the assertions are true1 but she sees not how profound and absolute is their truth4 ,she unhesitatingly assents to the laws which geometry delivers1 but she does not pretend to see the origin of their obligation4 ,she is always ready to acknowledge the sway of pure scientific principles as a matter of fact1 but she does not dream of offering her opinion on their authority as a matter of right2 still less can she justly claim to herself the source of that authority4--,,whewell1 ,,william4 ,the ,philosophy of the ,inductive ,sciences1 ,part #a1 ,bk4 #a1 chap4 #f1 sect4 #a 7,london1 #aheh74 #ahae4 ,geometry is the science created to give understanding and mastery of the external relations of things2 to make easy the explanation and description of such relations and the transmission of this mastery4--,,halsted1 ,g4 ,b4 ,proceedings of the ,american ,association for the ,advancement of ,science 7#aijd71 p4 #cei4 #ahaf4 ,a mathematical point is the most indivisible and unique thing which art can present4--,,donne1 ,,john4 ,letters1 #ba4 #ahag4 ,it is certain that from its completeness1 uniformity and faultlessness1 from its arrangement and progressive character1 and from the universal adoption of the completest and best line of argument1 ,euclid's 8,elements0 stand pre-eminently at the head of all human productions4 ,in no science1 in no department of knowledge1 has anything appeared like this work3 for upward of #bjjj years it has commanded the admiration of mankind1 and that period has suggested little toward its improvement4--,,kelland1 ,p4 ,lectures on the ,principles of ,demonstrative ,mathematics 7,london1 #ahdc71 p4 #ag4 #ahah4 ,in comparing the performance in ,euclid with that in ,arithmetic and ,algebra there could be no doubt that ,euclid had made the deepest and most beneficial impression3 in fact it might be asserted that this constituted by far the most valuable part of the whole training to which such persons students1 the majority of which were not distinguished for mathematical taste and power were subjected4--,,todhunter1 ,i4 ,essay on ,elementary ,geometry2 ,conflict of ,studies and other ,essays 7,london1 #ahgc71 p4 #afg4 #ahai4 ,in ,england the geometry studied is that of ,euclid1 and ,i hope it never will be any other2 for this reason1 that so much has been written on ,euclid1 and all the difficulties of geometry have so uniformly been considered with reference to the form in which they appear in ,euclid1 that the study of that author is a better key to a great quantity of useful reading than any other4 --,,de ,,morgan1 ,a4 ,elements of ,algebra 7,london1 #ahcg71 ,introduction4 #ahbj4 ,this book ,euclid has been for nearly twenty-two centuries the encouragement and guide of that scientific thought which is one thing with the progress of man from a worse to a better state4 ,the encouragement2 for it contained a body of knowledge that was really known and could be relied on1 and that moreover was growing in extent and application4 ,for even at the time this book was written--shortly after the foundation of the ,alexandrian ,museum--,mathematics was no longer the merely ideal science of the ,platonic school1 but had started on her career of conquest over the whole world of ,phenomena4 ,the guide2 for the aim of every scientific student of every subject was to bring his knowledge of that subject into a form as perfect as that which geometry had attained4 ,far up on the great mountain of ,truth1 which all the sciences hope to scale1 the foremost of that sacred sisterhood was seen1 beckoning for the rest to follow her4 ,and hence she was called1 in the dialect of the ,phythagoreans1 8the purifier of the reasonable soul40--,,clifford1 ,w4 ,k4 ,lectures and ,essays 7,london1 #aija71 ,vol4 #a1 p4 #ced4 #ahba4 ,euclid at once the inspiration and aspiration of scientific thought4--,,clifford1 ,w4 ,k4 ,lectures and ,essays 7,london1 #aija71 ,vol #a1 p4 #cee4 #ahbb4 ,the 8elements0 of the ,great ,alexandrian remain for all time the first1 and one may venture to assert1 the only perfect model of logical exactness of principles1 and of rigorous development of theorems4 ,if one would see how a science can be constructed and developed to its minutest details from a very small number of intuitively perceived axioms1 postulates1 and plain definitions1 by means of rigorous1 one would almost say chaste1 syllogism1 which nowhere makes use of surreptitious or foreign aids1 if one would see how a science may thus be constructed one must turn to the elements of ,euclid4--,,hankel1 ,h4 ,die ,entwickelung der ,mathematik in den letzten ,jahrhunderten 7,tbingen1 #ahhd71 p4 #g4 #ahbc4 ,if we consider him ,euclid as meaning to be what his commentators have taken him to be1 a model of the most unscrupulous formal rigour1 we can deny that he has altogether succeeded1 though we admit that he made the nearest approach4 --,,de ,,morgan1 ,a4 ,smith's ,dictionary of ,greek and ,roman ,biography and ,mythology 7,london1 #aijb72 ,article 8,eucleides40 #ahbd4 ,the ,elements of ,euclid is as small a part of mathematics as the ,iliad is of literature2 or as the sculpture of ,phidias is of the world's total art4--,,keyser1 ,c4 ,j4 ,lectures on ,science1 ,philosophy and ,art 7,new ,york1 #aijh71 p4 #h4 #ahbe4 ,i should rejoice to see 444 ,euclid honourably shelved or buried 8deeper than did ever plummet sound0 out of the schoolboys' reach2 morphology introduced into the elements of algebra2 projection1 correlation1 and motion accepted as aids to geometry2 the mind of the student quickened and elevated and his faith awakened by early initiation into the ruling ideas of polarity1 continuity1 infinity1 and familiarization with the doctrines of the imaginary and inconceivable4--,,sylvester1 ,j4 ,j4 ,a ,plea for the ,mathematician2 ,nature1 ,vol4 #a1 p4 #bfa4 #ahbf4 ,the early study of ,euclid made me a hater of geometry1 444 and yet1 in spite of this repugnance1 which had become a second nature in me1 whenever ,i went far enough into any mathematical question1 ,i found ,i touched1 at last1 a geometrical bottom4--,,sylvester1 ,j4 ,j4 ,a ,plea for the ,mathematician2 ,nature1 ,vol4 #a1 p4 #bfb4 #ahbg4 ,newton had so remarkable a talent for mathematics that ,euclid's ,geometry seemed to him 8a trifling book10 and he wondered that any man should have taken the trouble to demonstrate propositions1 the truth of which was so obvious to him at the first glance4 ,but1 on attempting to read the more abstruse geometry of ,descartes1 without having mastered the elements of the science1 he was baffled1 and was glad to come back again to his ,euclid4--,,parton1 ,,james4 ,sir ,isaac ,newton4 #ahbh4 ,as to the need of improvement there can be no question whilst the reign of ,euclid continues4 ,my own idea of a useful course is to begin with arithmetic1 and then not ,euclid but algebra4 ,next1 not ,euclid1 but practical geometry1 solid as well as plane2 not demonstration1 but to make acquaintance4 ,then not ,euclid1 but elementary vectors1 conjoined with algebra1 and applied to geometry4 ,addition first2 then the scalar product4 ,elementary calculus should go on simultaneously1 and come into the vector algebraic geometry after a bit4 ,euclid might be an extra course for learned men1 like ,homer4 ,but ,euclid for children is barbarous4--,,heaviside1 ,,oliver4 ,electro-,magnetic ,theory 7,london1 #ahic71 ,vol4 #a1 p4 #adh4 #ahbi4 ,geometry is nothing if it be not rigorous1 and the whole educational value of the study is lost1 if strictness of demonstration be trifled with4 ,the methods of ,euclid are1 by almost universal consent1 unexceptionable in point of rigour4 --,,smith1 ,h4 ,j4 ,s4 ,nature1 ,vol4 #h1 p4 #dej4 #ahcj4 ,to seek for proof of geometrical propositions by an appeal to observation proves nothing in reality1 except that the person who has recourse to such grounds has no due apprehension of the nature of geometrical demonstration4 ,we have heard of persons who convince themselves by measurement that the geometrical rule respecting the squares on the sides of a right-angles triangle was true3 but these were persons whose minds had been engrossed by practical habits1 and in whom speculative development of the idea of space had been stifled by other employments4--,,whewell1 ,,william4 ,the ,philosophy of the ,inductive ,sciences1 7,london1 #aheh71 ,part #a1 ,bk4 #b1 chap4 #a1 sect4 #d4 #ahca4 ,no one has ever given so easy and natural a chain of geometrical consequences as ,euclid4 ,there is a never-erring truth in the results4--,,de ,,morgan1 ,a4 ,smith's ,dictionary of ,greek and ,roman ,biography and ,mythology 7,london1 #aijb72 ,article 8,eucleides40 #ahcb4 ,beyond question1 ,egyptian geometry1 such as it was1 was eagerly studied by the early ,greek philosophers1 and was the germ from which in their hands grew that magnificent science to which every ,englishman is indebted for his first lessons in right seeing and thinking4--,,gow1 ,,james4 ,a ,short ,history of ,greek ,mathematics 7,cambridge1 #ahhd71 p4 #aca4 #ahcc4 ,a figure and a step onward3 ,not a figure and a florin4 --,,motto ,,of ,,the ,,pythagorean ,,brotherhood4 ,w4 ,b4 ,frankland3 ,story of ,euclid 7,london1 #aijb71 p4 #cc4 #ahcd4 ,the doctrine of proportion1 as laid down in the fifth book of ,euclid1 is1 probably1 still unsurpassed as a masterpiece of exact reasoning2 although the cumbrousness of the forms of expression which were adopted in the old geometry has led to the total exclusion of this part of the elements from the ordinary course of geometrical education4 ,a zealous defender of ,euclid might add with truth that the gap thus created in the elementary teaching of mathematics has never been adequately supplied4 --,,smith1 ,h4 ,j4 ,s4 ,presidential ,address ,british ,association for the ,advancement of ,science 7#ahgc72 ,nature1 ,vol4 #h1 p4 #dea4 #ahce4 ,the ,definition in the ,elements1 according to ,clavius1 is this3 ,magnitudes are said to be in the same ,reason ratio1 a first to a second1 and a third to a fourth1 when the ,equimultiples of the first and third according to any ,multiplication whatsoever are both together either short of1 equal to1 or exceed the ,equimultiples of the second and fourth1 if those be taken1 which answer one another4444 ,such is ,euclid's ,definition of ,proportions2 that scare-,crow at which the over modest or slothful ,dispositions of ,men are generally affrighted3 they are modest1 who distrust their own ,ability1 as soon as a ,difficulty appears1 but they are slothful that will not give some ,attention for the learning of ,sciences2 as if while we are involved in ,obscurity we could clear ourselves without ,labour4 ,both of which ,sorts of ,persons are to be admonished1 that the former be not discouraged1 nor the latter refuse a little ,care and ,diligence when a ,thing requires some ,study4--,,barrow1 ,,isaac4 ,mathematical ,lectures 7,london1 #agcd71 p4 #chh4 #ahcf4 ,of all branches of human knowledge1 there is none which1 like it geometry has sprung a completely armed ,minerva from the head of ,jupiter2 none before whose death-dealing ,aegis doubt and inconsistency have so little dared to raise their eyes4 ,it escapes the tedious and troublesome task of collecting experimental facts1 which is the province of the natural sciences in the strict sense of the word3 the sole form of its scientific method is deduction4 ,conclusion is deduced from conclusion1 and yet no one of common sense doubts but that these geometrical principles must find their practical application in the real world about us4 ,land surveying1 as well as architecture1 the construction of machinery no less than mathematical physics1 are continually calculating relations of space of the most varied kinds by geometrical principles2 they expect that the success of their constructions and experiments shall agree with their calculations2 and no case is known in which this expectation has been falsified1 provided the calculations were made correctly and with sufficient data4--,,helmholtz1 ,h4 ,the ,origin and ,significance of ,geometrical ,axioms2 ,popular ,scientific ,lectures ,atkinson1 ,second ,series 7,new ,york1 #ahha71 p4 #bg4 #ahcg4 ,the amazing triumphs of this branch of mathematics geometry show how powerful a weapon that form of deduction is which proceeds by an artificial reparation of facts1 in themselves inseparable4--,,buckle1 ,h4 ,t4 ,history of ,civilization in ,england 7,new ,york1 #ahia71 ,vol4 #b1 p4 #cdc4 #ahch4 ,every theorem in geometry is a law of external nature1 and might have been ascertained by generalizing from observation and experiment1 which in this case resolve themselves into comparisons and measurements4 ,but it was found practicable1 and being practicable was desirable1 to deduce these truths by ratiocination from a small number of general laws of nature1 the certainty and universality of which was obvious to the most careless observer1 and which compose the first principles and ultimate premises of the science4--,,mill1 ,j4 ,s4 ,system of ,logic1 ,bk4 #c1 chap4 #bd1 sect4 #g4 #ahci4 ,all such reasonings natural philosophy1 chemistry1 agriculture1 political economy1 etc4 are1 in comparison with mathematics1 very complex2 requiring so much more than that does1 beyond the process of merely deducing the conclusion logically from the premises3 so that it is no wonder that the longest mathematical demonstration should be much more easily constructed and understood1 than a much shorter train of just reasoning concerning real facts4 ,the former has been aptly compared to a long and steep1 but even and regular1 flight of steps1 which tries the breath1 and the strength1 and the perseverance only2 while the latter resembles a short1 but rugged and uneven1 ascent up a precipice1 which requires a quick eye1 agile limbs1 and a firm step2 and in which we have to tread now on this side1 now on that--ever considering as we proceed1 whether this or that projection will afford room for our foot1 or whether some loose stone may not slide from under us4 ,there are probably as many steps of pure reasoning in one of the longer of ,euclid's demonstrations1 as in the whole of an argumentative treatise on some other subject1 occupying perhaps a considerable volume4--,,whately1 ,r4 ,elements of ,logic1 ,bk4 #d1 chap4 #b1 sect4 #e4 #ahdj4 ,geometry that held acquaintance with the stars1 ,and wedded soul to soul in purest bond ,of reason1 undisturbed by space or time4 --,,wordsworth4 ,the ,prelude1 ,bk4 #e4 #ahda4 ,the statement that a given individual has received a sound geometrical training implies that he has segregated from the whole of his sense impressions a certain set of these impressions1 that he has eliminated from their consideration all irrelevant impressions 7in other words1 acquired a subjective command of these impressions71 that he has developed on the basis of these impressions an ordered and continuous system of logical deduction1 and finally that he is capable of expressing the nature of these impressions and his deductions therefrom in terms simple and free from ambiguity4 ,now the slightest consideration will convince any one not already conversant with the idea1 that the same sequence of mental processes underlies the whole career of any individual in any walk of life if only he is not concerned entirely with manual labor2 consequently a full training in the performance of such sequences must be regarded as forming an essential part of any education worthy of the name4 ,moreover the full appreciation of such processes has a higher value than is contained in the mental training involved1 great though this be1 for it induces an appreciation of intellectual unity and beauty which plays for the mind that part which the appreciation of schemes of shape and color plays for the artistic faculties2 or1 again1 that part which the appreciation of a body of religious doctrine plays for the ethical aspirations4 ,now geometry is not the sole possible basis for inculcating this appreciation4 ,logic is an alternative for adults1 provided that the individual is possessed of sufficient wide1 though rough1 experience on which to base his reasoning4 ,geometry is1 however1 highly desirable in that the objective bases are so simple and precise that they can be grasped at an early age1 that the amount of training for the imagination is very large1 that the deductive processes are not beyond the scope of ordinary boys1 and finally that it affords a better basis for exercise in the art of simple and exact expression than any other possible subject of a school course4 --,,carson1 ,g4 ,w4 ,l4 ,the ,functions of ,geometry as a ,subject of ,education 7,tonbridge1 #aiaj71 p4 #c4 #ahdb4 ,it seems to me that the thing that is wanting in the education of women is not the acquaintance with any facts1 but accurate and scientific habits of thought1 and the courage to think that true which appears unlikely4 ,and for supplying this want there is a special advantage in geometry1 namely that it does not require study of a physically laborious kind1 but rather that rapid intuition which women certainly possess2 so that it is fit to become a scientific pursuit for them4--,,clifford1 ,w4 ,k4 ,quoted by ,pollock in ,clifford's ,lectures and ,essays 7,london1 #aija71 ,vol4 #a1 ,introduction1 p4 #dc4 #ahdc4 ,on the lecture slate ,the circle rounded under female hands ,with flawless demonstration4 --,,tennyson4 ,the ,princess1 ,,ii1 l4 #dic4 #ahdd4 ,it is plain that that part of geometry which bears upon strategy does concern us4 ,for in pitching camps1 or in occupying positions1 or in closing or extending the lines of an army1 and in all the other manuvres of an army whether in battle or on the march1 it will make a great difference to a general1 whether he is a geometrician or not4--,,plato4 ,republic1 ,bk4 #g1 p4 #ebf4 #ahde4 ,then nothing should be more effectually enacted1 than that the inhabitants of your fair city should learn geometry4 ,moreover the science has indirect effects1 which are not small4 ,of what kind are they8 he said4 ,there are the military advantages of which you spoke1 ,i said2 and in all departments of study1 as experience proves1 any one who has studied geometry is infinitely quicker of apprehension4--,,plato4 ,republic ,jowett1 ,bk4 #g1 p4 #ebg4 #ahdf4 ,it is doubtful if we have any other subject that does so much to bring to the front the danger of carelessness1 of slovenly reasoning1 of inaccuracy1 and of forgetfulness as this science of geometry1 which has been so polished and perfected as the centuries have gone on4--,,smith1 ,d4 ,e4 ,the ,teaching of ,geometry 7,boston1 #aiaa71 p4 #ab4 #ahdg4 ,the culture of the geometric imagination1 tending to produce precision in remembrance and invention of visible forms will1 therefore1 tend directly to increase the appreciation of works of belles-letters4--,,hill1 ,,thomas4 ,the ,uses of ,mathesis2 ,bibliotheca ,sacra1 ,vol4 #cb1 p4 #ejd4 #ahdh4 ,yet may we not entirely overlook ,the pleasures gathered from the rudiments ,of geometric science4 ,though advanced ,in these inquiries1 with regret ,i speak1 ,no farther than the threshold1 there ,i found ,both elevation and composed delight3 ,with ,indian awe and wonder1 ignorance pleased ,with its own struggles1 did ,i meditate ,on the relations those abstractions bear ,to ,nature's laws4 99 99 99 99 99 ,more frequently from the same source ,i drew ,a pleasure quiet and profound1 a sense ,of permanent and universal sway1 ,and paramount belief2 there1 recognized ,a type1 for finite natures1 of the one ,supreme ,existence1 the surpassing life ,which to the boundaries of space and time1 ,of melancholy space and doleful time1 ,superior and incapable of change1 ,nor touched by welterings of passion--is1 ,and hath the name of ,god4 ,transcendent peace ,and silence did wait upon these thoughts ,that were a frequent comfort to my youth4 99 99 99 99 99 ,mighty is the charm ,of those abstractions to a mind beset ,with images and haunted by himself1 ,and specially delightful unto me ,was that clear synthesis built up aloft ,so gracefully2 even then when it appeared ,not more than a mere plaything1 or a toy ,to sense embodied3 not the thing it is ,in verity1 an independent world1 ,created out of pure intelligence4 --,,wordsworth4 ,the ,prelude1 ,bk4 #f4 #ahdi4 ',tis told by one whom stormy waters threw1 ,with fellow-sufferers by the shipwreck spared1 ,upon a desert coast1 that having brought ,to land a single volume1 saved by chance1 ,a treatise of ,geometry1 he wont1 ,although of food and clothing destitute1 ,and beyond common wretchedness depressed1 ,to part from company1 and take this book 7,then first a self taught pupil in its truths7 ,to spots remote1 and draw his diagrams ,with a long staff upon the sand1 and thus ,did oft beguile his sorrow1 and almost ,forget his feeling3 --,,wordsworth4 ,the ,prelude1 ,bk4 #f4 #ahej4 ,we study art because we receive pleasure from the great works of the masters1 and probably we appreciate them the more because we have dabbled a little in pigments or in clay4 ,we do not expect to be composers1 or poets1 or sculptors1 but we wish to appreciate music and letters and the fine arts1 and to derive pleasure from them and be uplifted by them4444 ,so it is with geometry4 ,we study it because we derive pleasure from contact with a great and ancient body of learning that has occupied the attention of master minds during the thousands of years in which it has been perfected1 and we are uplifted by it4 ,to deny that our pupils derive this pleasure from the study is to confess ourselves poor teachers1 for most pupils do have positive enjoyment in the pursuit of geometry1 in spite of the tradition that leads them to proclaim a general dislike for all study4 ,this enjoyment is partly that of the game1--the playing of a game that can always be won1 but that cannot be won too easily4 ,it is partly that of the aesthetic1 the pleasure of symmetry of form1 the delight of fitting things together4 ,but probably it lies chiefly in the mental uplift that geometry brings1 the contact with absolute truth1 and the approach that one makes to the ,infinite4 ,we are not quite sure of any one thing in biology2 our knowledge of geology is relatively very slight1 and the economic laws of society are uncertain to every one except some individual who attempts to set them forth2 but before the world was fashioned the square on the hypotenuse was equal to the sum of the squares on the other two sides of a right triangle1 and it will be so after this world is dead2 and the inhabitant of ,mars1 if he exists1 probably knows its truth as we know it4 ,the uplift of this contact with absolute truth1 with truth eternal1 gives pleasure to humanity to a greater or less degree1 depending upon the mental equipment of the particular individual2 but it probably gives an appreciable amount of pleasure to every student of geometry who has a teacher worthy of the name4--,,smith1 ,d4 ,e4 ,the ,teaching of ,geometry 7,boston1 #aiaa71 p4 #af4 #ahea4 ,no other person can judge better of either the merits of a writer and the merits of his works than himself2 for none have had access to a closer or more deliberate examination of them4 ,it is for this reason1 that in proportion that the value of a work is intrinsic1 and independent of opinion1 the less eagerness will the author feel to conciliate the suffrages of the public4 ,hence that inward satisfaction1 so pure and so complete1 which the study of geometry yields4 ,the progress which an individual makes in this science1 the degree of eminence which he attains in it1 all this may be measured with the same rigorous accuracy as the methods about which his thoughts are employed4 ,it is only when we entertain some doubts about the justness of our own standard1 that we become anxious to relieve ourselves from our uncertainty1 by comparing it with the standard of another4 ,now1 in all matters which fall under the cognizance of taste1 this standard is necessarily somewhat variable2 depending on a sort of gross estimate1 always a little arbitrary1 either in whole or in part2 and liable to continual alteration in its dimensions1 from negligence1 temper1 or caprice4 ,in consequence of these circumstances ,i have no doubt1 that if men lived separate from each other1 and could in such a situation occupy themselves about anything but self-preservation1 they would prefer the study of the exact sciences to the cultivation of the agreeable arts4 ,it is chiefly on account of others1 that a man aims at excellence in the latter1 it is on his own account that he devotes himself to the former4 ,in a desert island1 accordingly1 ,i should think that a poet could scarcely be vain2 whereas a geometrician might still enjoy the pride of discovery4--,d',,alembert4 ,essai sur les ,gens ,lettres2 ,melages 7,amsterdam #agfd71 t4 #a1 p4 #ccd4 #aheb4 ,if it were required to determine inclined planes of varying inclinations of such lengths that a free rolling body would descend on them in equal times1 any one who understands the mechanical laws involved would admit that this would necessitate sundry preparations4 ,but in the circle the proper arrangement takes place of its own accord for an infinite variety of positions yet with the greatest accuracy in each individual case4 ,for all chords which meet the vertical diameter whether at its highest or lowest point1 and whatever their inclinations1 have this in common3 that the free descent along them takes place in equal times4 ,i remember1 one bright pupil1 who1 after ,i had stated and demonstrated this theorem to him1 and he had caught the full import of it1 was moved as by a miracle4 ,and1 indeed1 there is just cause for astonishment and wonder when one beholds such a strange union of manifold things in accordance with such fruitful rules in so plain and simple an object as the circle4 ,moreover1 there is no miracle in nature1 which because of its pervading beauty or order1 gives greater cause for astonishment1 unless it be1 for the reason that its causes are not so clearly comprehended1 marvel being a daughter of ignorance4--,,kant4 ,der einzig mgliche ,beweisgrund zu einer ,demonstration des ,daseins ,gottes2 ,werke 7,hartenstein71 ,bd4 #b1 p4 #acg4 #ahec4 ,these examples taken from the geometry of the circle indicate what a countless number of other such harmonic relations obtain in the properties of space1 many of which are manifested in the relations of the various classes of curves in higher geometry1 all of which1 besides exercising the understanding through intellectual insight1 affect the emotion in a similar or even greater degree than the occasional beauties of nature4--,,kant4 ,der einzig mgliche ,beweisgrund zu einer ,demonstration des ,daseins ,gottes2 ,werke 7,hartenstein71 ,bd4 #b1 p4 #ach4 #ahed4 ,but neither thirty years1 nor thirty centuries1 affect the clearness1 or the charm1 of ,geometrical truths4 ,such a theorem as 8the square of the hypotenuse of a right-angled triangle is equal to the sum of the squares of the sides0 is as dazzlingly beautiful now as it was in the day when ,pythagoras first discovered it1 and celebrated its advent1 it is said1 by sacrificing a hecatomb of oxen--a method of doing honor to ,science that has always seemed to me slightly exaggerated and uncalled-for4 ,one can imagine oneself1 even in these degenerate days1 marking the epoch of some brilliant scientific discovery by inviting a convivial friend or two1 to join one in a beefsteak and a bottle of wine4 ,but a hecatomb of oxen6 ,it would produce a quite inconvenient supply of beef4--,,dodgson1 ,c4 ,l4 ,a ,new ,theory of ,parallels 7,london1 #ahie71 ,introduction1 p4 #af4 #ahee4 ,after ,pythagoras discovered his fundamental theorem he sacrificed a hecatomb of oxen4 ,since that time all dunces#aj ,ochsen tremble whenever a new truth is discovered4--,,boerne4 ,quoted in ,moszkowski3 ,die unsterbliche ,kiste 7,berlin1 #aijh71 p4 #ah4 #aj ,in the ,german vernacular a dunce or blockhead is called an ox4 #ahef4 ,vom ,pythagorieschen ,lehrsatz4 ,die ,wahrheit1 sie besteht in ,ewigkeit1 ,wenn erst die blde ,welt ihr ,licht erkannt3 ,der ,lehrsatz1 nach ,pythagoras benannt1 ,gilt heute1 wie er galt in seiner ,zeit4 ,ein ,opfer hat ,pythagoras geweiht ,den ,gttern1 die den ,lichtstrahl ihm gesandt2 ,es thaten kund1 geschlachtet und verbrannt1 ,ein hundert ,ochsen seine ,dankbarkeit4 ,die ,ochsen seit den ,tage1 wenn sie wittern1 ,dass eine neue ,wahrheit sich enthlle1 ,erheben ein unmenschliches ,gebrlle2 ,pythagoras erfllt sie mit ,entsetzen2 ,und machtlos1 sich dem ,licht zu wiedersetzen1 ,verschiessen sie die ,augen und erzittern4 --,,chamisso1 ,,adelbert ,,von4 ,gedichte1 #ahce 7,haushenbusch71 7,berlin1 #ahhi71 p4 #cjb4 ,truth lasts throughout eternity1 ,when once the stupid world its light discerns3 ,the theorem1 coupled with ,pythagoras' name1 ,holds true today1 as't did in olden times4 ,a splendid sacrifice ,pythagoras brought ,the gods1 who blessed him with this ray divine2 ,a great burnt offering of a hundred kine1 ,proclaimed afar the sage's gratitude4 ,now since that day1 all cattle blockheads when they scent ,new truth about to see the light of day1 ,in frightful bellowings manifest their dismay2 ,pythagoras fills them all with terror2 ,and powerless to shut out light by error1 ,in sheer despair they shut their eyes and tremble4 #aheg4 ,to the question 8,which is the signally most beautiful of geometrical truths80 ,frankland replies3 8,one star excels another in brightness1 but the very sun will be1 by common consent1 a property of the circle ,euclid1 ,book #c1 ,proposition #ca selected for particular mention by ,dante1 that greatest of all exponents of the beautiful40--,,frankland1 ,w4 ,b4 ,the ,story of ,euclid 7,london1 #aijb71 p4 #gj4 #aheh4 ,as one ,who vers'd in geometric lore1 would fain ,measure the circle2 and1 though pondering long ,and deeply1 that beginning1 which he needs1 ,finds not2 e'en such was ,i1 intent to scan ,the novel wonder1 and trace out the form1 ,how to the circle fitted1 and therein ,how plac'd3 but the flight was not for my wing2 --,,dante4 ,paradise ,carey ,canto #cc1 lines #abb-abi4 #ahei4 ,if geometry were as much opposed to our passions and present interests as is ethics1 we should contest it and violate it but little less1 notwithstanding all the demonstrations of ,euclid and of ,archimedes1 which you would call dreams and believe full of paralogisms2 and ,joseph ,scaliger1 ,hobbes1 and others1 who have written against ,euclid and ,archimedes1 would not find themselves in such a small company as at present4--,,leibnitz4 ,new ,essays concerning ,human ,understanding ,langley1 ,bk4 #a1 chap4 #b1 sect4 #ab4 #ahfj4 ,i have no fault to find with those who teach geometry4 ,that science is the only one which has not produced sects2 it is founded on analysis and on synthesis and on the calculus2 it does not occupy itself with probable truth2 moreover it has the same method in every country4--,,frederick ,,the ,,great4 ,oeuvres 7,decker71 t4 #g1 p4 #ajj4 #ahfa4 ,there are1 undoubtedly1 the most ample reasons for stating both the principles and theorems of geometry in their general form1 4444 ,but1 that an unpractised learner1 even in making use of one theorem to demonstrate another1 reasons rather from particular to particular than from the general proposition1 is manifest from the difficulty he finds in applying a theorem to a case in which the configuration of the diagram is extremely unlike that of the diagram by which the original theorem was demonstrated4 ,a difficulty which1 except in cases of unusual mental powers1 long practice can alone remove1 and removes chiefly by rendering us familiar with all the configurations consistent with the general conditions of the theorem4--,,mill1 ,j4 ,s4 ,system of ,logic1 ,bk4 #b1 chap4 #c1 sect4 #c4 #ahfb4 ,the reason why ,i impute any defect to geometry1 is1 because its original and fundamental principles are deriv'd merely from appearances2 and it may perhaps be imagin'd1 that this defect must always attend it1 and keep it from ever reaching a greater exactness in the comparison of objects or ideas1 than what our eye or imagination alone is able to attain4 ,i own that this defect so far attends it1 as to keep it from ever aspiring to a full certainty4 ,but since these fundamental principles depend on the easiest and least deceitful appearances1 they bestow on their consequences a degree of exactness1 of which these consequences are singly incapable4--,,hume1 ,d4 ,a ,treatise of ,human ,nature1 ,part #c1 sect4 #a4 #ahfc4 ,i have already observed1 that geometry1 or the art1 by which we fix the proportions of figures1 tho' it much excels both in universality and exactness1 the loose judgments of the senses and imagination2 yet never attains a perfect precision and exactness4 ,its first principles are still drawn from the general appearance of the objects2 and that appearance can never afford us any security1 when we examine the prodigious minuteness of which nature is susceptible4444 ,there remain1 therefore1 algebra and arithmetic as the only sciences1 in which we can carry on a chain of reasoning to any degree of intricacy1 and yet preserve a perfect exactness and certainty4--,,hume1 ,d4 ,a ,treatise of ,human ,nature1 ,part #c1 sect4 #a4 #ahfd4 ,all geometrical reasoning is1 in the last resort1 circular3 if we start by assuming points1 they can only be defined by the lines or planes which relate them2 and if we start by assuming lines or planes1 they can only be defined by the points through which they pass4--,,russell1 ,,bertrand4 ,foundations of ,geometry 7,cambridge1 #ahig71 p4 #abj4 #ahfe4 ,the description of right lines and circles1 upon which ,geometry is founded1 belongs to ,mechanics4 ,geometry does not teach us to draw these lines1 but requires them to be drawn4444 it requires that the learner should first be taught to describe these accurately1 before he enters upon ,geometry2 then it shows how by these operations problems may be solved4 ,to describe right lines and circles are problems1 but not geometrical problems4 ,the solution of these problems is required from ,mechanics2 by ,geometry the use of them1 when solved1 is shown4444 ,therefore ,geometry is founded in mechanical practice1 and is nothing but that part of universal ,mechanics which accurately proposes and demonstrates the art of measuring4 ,but since the manual arts are chiefly conversant in the moving of bodies1 it comes to pass that ,geometry is commonly referred to their magnitudes1 and ,mechanics to their motion4--,,newton4 ,philosophiae ,naturalis ,principia ,mathematica1 ,praefat4 #ahff4 ,we must1 then1 admit 444 that there is an independent science of geometry just as there is an independent science of physics1 and that either of these may be treated by mathematical methods4 ,thus geometry becomes the simplest of the natural sciences1 and its axioms are of the nature of physical laws1 to be tested by experience and to be regarded as true only within the limits of error of observation--,b,,cher1 ,,maxime4 ,bulletin ,american ,mathematical ,society1 ,vol4 #b 7#aijd71 p4 #abd4 #ahfg4 ,geometry is not an experimental science2 experience forms merely the occasion for our reflecting upon the geometrical ideas which pre-exist in us4 ,but the occasion is necessary1 if it did not exist we should not reflect1 and if our experiences were different1 doubtless our reflections would also be different4 ,space is not a form of sensibility2 it is an instrument which serves us not to represent things to ourselves1 but to reason upon things4--,,poincar1 ,h4 ,on the ,foundations of ,geometry2 ,monist1 ,vol4 #i 7#ahih-ahii71 p4 #da4 #ahfh4 ,it has been said that geometry is an instrument4 ,the comparison may be admitted1 provided it is granted at the same time that this instrument1 like ,proteus in the fable1 ought constantly to change its form4--,,arago4 ,oeuvres1 t4 #b 7#ahed71 p4 #fid4 #ahfi4 ,it is essential that the treatment of geometry should be rid of everything superfluous1 for the superfluous is an obstacle to the acquisition of knowledge2 it should select everything that embraces the subject and brings it to a focus1 for this is of the highest service to science2 it must have great regard both to clearness and to conciseness1 for their opposites trouble our understanding2 it must aim to generalize its theorems1 for the division of knowledge into small elements renders it difficult of comprehension4--,,proclus4 ,quoted in ,d4 ,e4 ,smith3 ,the ,teaching of ,geometry 7,boston1 #aiaa71 p4 #ga4 #ahgj4 ,many are acquainted with mathematics1 but mathesis few know4 ,for it is one thing to know a number of propositions and to make some obvious deductions from them1 by accident rather than by any sure method of procedure1 another thing to know clearly the nature and character of the science itself1 to penetrate into its inmost recesses1 and to be instructed by its universal principles1 by which facility in working out countless problems and their proofs is secured4 ,for as the majority of artists1 by copying the same model again and again1 gain certain technical skill in painting1 but no other knowledge of the art of painting than what their eyes suggest1 so many1 having read the books of ,euclid and other geometricians1 are wont to devise1 in imitation of them and to prove some propositions1 but the most profound method of solving more difficult demonstrations and problems they are utterly ignorant of4--,,lafaille1 ,j4 ,c4 ,theoremata de ,centro ,gravitatis 7,anvers1 #afcb71 ,praefat4 #ahga4 ,the elements of plane geometry should precede algebra for every reason known to sound educational theory4 ,it is more fundamental1 more concrete1 and it deals with things and their relations rather than with symbols4--,,butler1 ,n4 ,m4 ,the ,meaning of ,education etc4 7,new ,york1 #aije71 p4 #aga4 #ahgb4 ,the reason why geometry is not so difficult as algebra1 is to be found in the less general nature of the symbols employed4 ,in algebra a general proposition respecting numbers is to be proved4 ,letters are taken which may represent any of the numbers in question1 and the course of the demonstration1 far from making use of a particular case1 does not even allow that any reasoning1 however general in its nature1 is conclusive1 unless the symbols are as general as the arguments4444 ,in geometry on the contrary1 at least in the elementary parts1 any proposition may be safely demonstrated on reasonings on any one particular example4444 ,it also affords some facility that the results of elementary geometry are in many cases sufficiently evident of themselves to the eye2 for instance1 that two sides of a triangle are greater than the third1 whereas in algebra many rudimentary propositions derive no evidence from the senses2 for example1 that ab is always divisible without a remainder by ab4--,,de ,,morgan1 ,a4 ,on the ,study and ,difficulties of ,mathematics 7,chicago1 #aijb71 chap4 #ac4 #ahgc4 ,the principal characteristics of the ancient geometry are3-- 7#a7 ,a wonderful clearness and definiteness of its concepts and an almost perfect logical rigour of its conclusions4 7#b7 ,a complete want of general principles and methods4444 ,in the demonstration of a theorem1 there were1 for the ancient geometers1 as many different cases requiring separate proof as there were different positions of the lines4 ,the greatest geometers considered it necessary to treat all possible cases independently of each other1 and to prove each with equal fulness4 ,to devise methods by which all the various cases could all be disposed of with one stroke1 was beyond the power of the ancients4--,,cajori1 ,f4 ,history of ,mathematics 7,new ,york1 #ahig71 p4 #fb4 #ahgd4 ,it has been observed that the ancient geometers made use of a kind of analysis1 which they employed in the solution of problems1 although they begrudged to posterity the knowledge of it4--,,descartes4 ,rules for the ,direction of the ,mind2 ,the ,philosophy of ,descartes ,torrey 7,new ,york1 #ahib71 p4 #fh4 #ahge4 ,the ancients studied geometry with reference to the bodies under notice1 or specially3 the moderns study it with reference to the phenomena to be considered1 or generally4 ,the ancients extracted all they could out of one line or surface1 before passing to another2 and each inquiry gave little or no assistance in the next4 ,the moderns1 since ,descartes1 employ themselves on questions which relate to any figure whatever4 ,they abstract1 to treat by itself1 every question relating to the same geometrical phenomenon1 in whatever bodies it may be considered4 ,geometers can thus rise to the study of new geometrical conceptions1 which1 applied to the curves investigated by the ancients1 have brought out new properties never suspected by them4--,,comte4 ,positive ,philosophy ,martineau1 ,bk4 #a1 chap4 #c4 #ahgf4 ,it is astonishing that this subject projective geometry should be so generally ignored1 for mathematics offers nothing more attractive4 ,it possesses the concreteness of the ancient geometry without the tedious particularity1 and the power of the analytical geometry without the reckoning1 and by the beauty of its ideas and methods illustrates the esthetic generality which is the charm of higher mathematics1 but which the elementary mathematics generally lacks4 ,report of the ,committee of ,ten on ,secondary ,school ,studies 7,chicago1 #ahid71 p4 #aaf4 #ahgg4 ,there exist a small number of very simple fundamental relations which contain the scheme1 according to which the remaining mass of theorems in projective geometry permit of orderly and easy development4 ,by a proper appropriation of a few fundamental relations one becomes master of the whole subject2 order takes the place of chaos1 one beholds how all parts fit naturally into each other1 and arrange themselves serially in the most beautiful order1 and how related parts combine into well-defined groups4 ,in this manner one arrives1 as it were1 at the elements1 which nature herself employs in order to endow figures with numberless properties with the utmost economy and simplicity4--,,steiner1 ,j4 ,werke1 ,bd4 #a 7#ahha71 p4 #bcc4 #ahgh4 ,euclid once said to his king ,ptolemy1 who1 as is easily understood1 found the painstaking study of the 8,elements0 repellant1 8,there exists no royal road to mathematics40 ,but we may add3 ,modern geometry is a royal road4 ,it has disclosed 8the organism1 by means of which the most heterogeneous phenomena in the world of space are united one with another0 7,steiner71 and has1 as we may say without exaggeration1 almost attained to the scientific ideal4--,,hankel1 ,h4 ,die ,entwickelung der ,mathematik in den letzten ,jahrhunderten 7,tbingen1 #ahfi74 #ahgi4 ,the two mathematically fundamental things in projective geometry are anharmonic ratio1 and the quadrilateral construction4 ,everything else follows mathematically from these two4 --,,russell1 ,,bertrand4 ,foundations of ,geometry 7,cambridge1 #ahig71 p4 #abb4 #ahhj4 444 ,projective ,geometry3 a boundless domain of countless fields where reals and imaginaries1 finites and infinites1 enter on equal terms1 where the spirit delights in the artistic balance and symmetric interplay of a kind of conceptual and logical counterpoint1--an enchanted realm where thought is double and flows throughout in parallel streams4--,,keyser1 ,c4 ,j4 ,lectures on ,science1 ,philosophy and ,arts 7,new ,york1 #aijh71 p4 #b4 #ahha4 ,the ancients1 in the early days of the science1 made great use of the graphic method1 even in the form of construction2 as when ,aristarchus of ,samos estimated the distance of the sun and moon from the earth on a triangle constructed as nearly as possible in resemblance to the right-angled triangle formed by the three bodies at the instant when the moon is in quadrature1 and when therefore an observation of the angle at the earth would define the triangle4 ,archimedes himself1 though he was the first to introduce calculated determinations into geometry1 frequently used the same means4 ,the introduction of trigonometry lessened the practice2 but did not abolish it4 ,the ,greeks and ,arabians employed it still for a great number of investigations for which we now consider the use of the ,calculus indispensable4--,,comte1 ,a4 ,positive ,philosophy ,martineau1 ,bk4 #a1 chap4 #c4 #ahhb4 ,a mathematical problem may usually be attacked by what is termed in military parlance the method of 8systematic approach20 that is to say1 its solution may be gradually felt for1 even though the successive steps leading to that solution cannot be clearly foreseen4 ,but a ,descriptive ,geometry problem must be seen through and through before it can be attempted4 ,the entire scope of its conditions1 as well as each step toward its solution1 must be grasped by the imagination4 ,it must be 8taken by assault40 --,,clarke1 ,g4 ,s4 ,quoted in ,w4 ,s4 ,hall3 ,descriptive ,geometry 7,new ,york1 #aijb71 chap4 #a4 #ahhc4 ,the grand use of ,descriptive ,geometry is in its application to the industrial arts2--its few abstract problems1 capable of invariable solution1 relating essentially to the contacts and intersections of surfaces2 so that all the geometrical questions which may arise in any of the various arts of construction1--as stone-cutting1 carpentry1 perspective1 dialing1 fortification1 etc41--can always be treated as simple individual cases of a single theory1 the solution being certainly obtainable through the particular circumstances of each case4 ,this creation must be very important in the eyes of philosophers who think that all human achievement1 thus far1 is only a first step toward a philosophical renovation of the labours of mankind2 towards that precision and logical character which can alone ensure the future progression of all arts4444 ,of ,descriptive ,geometry1 it may further be said that it usefully exercises the student's faculty of ,imagination1--of conceiving of complicated geometrical combinations in space2 and that1 while it belongs to the geometry of the ancients by the character of its solutions1 it approaches to the geometry of the moderns by the nature of the questions which compose it4--,,comte1 ,a4 ,positive ,philosophy ,martineau1 ,bk4 #a1 chap4 #c4 #ahhd4 ,there is perhaps nothing which so occupies1 as it were1 the middle position of mathematics1 as trigonometry4 --,,herbart1 ,j4 ,f4 ,idee eines ,,abc der ,anschauung2 ,werke 7,kehrbach7 7,langensalza1 #ahij71 ,bd4 #a1 p4 #agd4 #ahhe4 ,trigonometry contains the science of continually undulating magnitude3 meaning magnitude which becomes alternately greater and less1 without any termination to succession of increase and decrease4444 ,all trigonometric functions are not undulating3 but it may be stated that in common algebra nothing but infinite series undulate3 in trigonometry nothing but infinite series do not undulate4--,,de ,,morgan1 ,a4 ,trigonometry and ,double ,algebra 7,london1 #ahdi71 ,bk4 #a1 chap4 #a4 #ahhf4 ,sin is odious to me1 even though ,laplace made use of it2 should it be feared that sin might become ambiguous1 which would perhaps never occur1 or at most very rarely when speaking of sin 771 well then1 let us write 7sin71 but not sin1 which by analogy should signify sin7sin74--,,gauss4 ,gauss-,schumacher ,briefwechsel1 ,bd4 #c1 p4 #bib2 ,bd4 #d1 p4 #fc4 #ahhg4 ,perhaps to the student there is no part of elementary mathematics so repulsive as is spherical trigonometry4--,,tait1 ,p4 ,g4 ,encyclopedia ,britannica1 #ith ,edition2 ,article 8,quaternions40 #ahhh4 8,napier's ,rule of circular parts0 is perhaps the happiest example of artificial memory that is known4--,,cajori1 ,f4 ,history of ,mathematics 7,new ,york1 #ahig71 p4 #afe4 #ahhi4 ,the analytical equations1 unknown to the ancients1 which ,descartes first introduced into the study of curves and surfaces1 are not restricted to the properties of figures1 and to those properties which are the object of rational mechanics2 they apply to all phenomena in general4 ,there cannot be a language more universal and more simple1 more free from errors and obscurities1 that is to say1 better adapted to express the invariable relations of nature4--,,fourier4 ,thorie ,analytique de la ,chaleur1 ,discours ,prliminaire4 #ahij4 ,it is impossible not to feel stirred at the thought of the emotions of men at certain historic moments of adventure and discovery--,columbus when he first saw the ,western shore1 ,pizarro when he stared at the ,pacific ,ocean1 ,franklin when the electric spark came from the string of his kite1 ,galileo when he first turned his telescope to the heavens4 ,such moments are also granted to students in the abstract regions of thought1 and high among them must be placed the morning when ,descartes lay in bed and invented the method of co-ordinate geometry4--,,whitehead1 ,a4 ,n4 ,an ,introduction to ,mathematics 7,new ,york1 #aiaa71 p4 #abb4 #ahia4 ,it is often said that an equation contains only what has been put into it4 ,it is easy to reply that the new form under which things are found often constitutes by itself an important discovery4 ,but there is something more3 analysis1 by the simple play of its symbols1 may suggest generalizations far beyond the original limits4--,,picard1 ,e4 ,bulletin ,american ,mathematical ,society1 ,vol4 #b 7#aije71 p4 #dji4 #ahib4 ,it is not the ,simplicity of the ,equation1 but the ,easiness of the ,description1 which is to determine the ,choice of our ,lines for the ,constructions of ,problems4 ,for the ,equation that expresses a ,parabola is more simple than that that expresses the ,circle1 and yet the ,circle1 by its more simple ,construction1 is admitted before it4--,,newton4 ,the ,linear ,constructions of ,equations2 ,universal ,arithmetic 7,london1 #agfi71 ,vol4 #b1 p4 #dfh4 #ahic4 ,the pursuit of mathematics unfolds its formative power completely only with the transition from the elementary subjects to analytical geometry4 ,unquestionably the simplest geometry and algebra already accustom the mind to sharp quantitative thinking1 as also to assume as true only axioms and what has been proven4 ,but the representation of functions by curves or surfaces reveals a new world of concepts and teaches the use of one of the most fruitful methods1 which the human mind ever employed to increase its own effectiveness4 ,what the discovery of this method by ,vieta and ,descartes brought to humanity1 that it brings today to every one who is in any measure endowed for such things3 a life-epoch-making beam of light ,lichtblick4 ,this method has its roots in the farthest depths of human cognition and so has an entirely different significance1 than the most ingenious artifice which serves a special purpose4--,,bois-,,reymond1 ,,emil ,,du4 ,reden1 ,bd4 #a 7,leipzig1 #ahhe71 p4 #bhg4 #ahid4 ,song of the ,screw4 ,a moving form or rigid mass1 ,under whate'er conditions ,along successive screws must pass ,between each two positions4 ,it turns around and slides along-- ,this is the burden of my song4 ,the pitch of screw1 if multiplied ,by angle of rotation1 ,will give the distance it must glide ,in motion of translation4 ,infinite pitch means pure translation1 ,and zero pitch means pure rotation4 ,two motions on two given screws1 ,with amplitudes at pleasure1 ,into a third screw-motion fuse1 ,whose amplitude we measure ,by parallelogram construction 7,a very obvious deduction74 ,its axis cuts the nodal line ,which to both screws is normal1 ,and generates a form divine1 ,whose name1 in language formal1 ,is 8surface-ruled of third degree40 ,cylindroid is the name for me4 ,rotation round a given line ,is like a force along1 ,if to say couple you decline1 ,you're clearly in the wrong2-- ',tis obvious1 upon reflection1 ,a line is not a mere direction4 ,so couples with translations too ,in all respects agree2 ,and thus there centres in the screw ,a wondrous harmony ,of ,kinematics and of ,statics1-- ,the sweetest thing in mathematics4 ,the forces on one given screw1 ,with motion on a second1 ,in general some work will do1 ,whose magnitude is reckoned ,by angle1 force1 and what we call ,the coefficient virtual4 ,rotation now to force convert1 ,and force into rotation2 ,unchanged the work1 we can assert1 ,in spite of transformation4 ,and if two screws no work can claim1 ,reciprocal will be their name4 ,five numbers will a screw define1 ,a screwing motion1 six2 ,for four will give the axial line1 ,one more the pitch will fix2 ,and hence we always can contrive ,one screw reciprocal to five4 ,screws--two1 three1 four or five1 combined 7,no question here of six71 ,yield other screws which are confined ,within one screw complex4 ,thus we obtain the clearest notion ,of freedom and constraint of motion4 ,in complex ,,iii1 three several screws ,at every point you find1 ,or if you one direction choose1 ,one screw is to your mind2 ,and complexes of order ,,iii4 ,their own reciprocals may be4 ,in ,,iv1 wherever you arrive1 ,you find of screws a cone1 ,on every line of complex ,v4 ,there is precisely one2 ,at each point of this complex rich1 ,a plane of screws have given pitch4 ,but time would fail me to discourse ,of ,order and ,degree2 ,of ,impulse1 ,energy and ,force1 ,and ,reciprocity4 ,all these and more1 for motions small1 ,have been discussed by ,dr4 ,ball4 --,,anonymous4 ,,chapter ,,xix ,,the ,,calculus ,,and ,,allied ,,topics #aija4 ,it may be said that the conceptions of differential quotient and integral1 which in their origin certainly go back to ,archimedes1 were introduced into science by the investigations of ,kepler1 ,descartes1 ,cavalieri1 ,fermat and ,wallis4444 ,the capital discovery that differentiation and integration are inverse operations belongs to ,newton and ,leibnitz4--,,lie1 ,,sophus4 ,leipziger ,berichte1 #dg 7#ahie71 ,math4-phys4 ,classe1 p4 #ec4 #aijb4 ,it appears that ,fermat1 the true inventor of the differential calculus1 considered that calculus as derived from the calculus of finite differences by neglecting infinitesimals of higher orders as compared with those of a lower order4444 ,newton1 through his method of fluxions1 has since rendered the calculus more analytical1 he also simplified and generalized the method by the invention of his binomial theorem4 ,leibnitz has enriched the differential calculus by a very happy notation4 --,,laplace4 ,ls ,intgrales ,dfinies1 etc42 ,oeuvres1 t4 #ab 7,paris1 #ahih71 p4 #cei4 #aijc4 ,professor ,peacock's ,algebra1 and ,mr4 ,whewell's ,doctrine of ,limits should be studied by every one who desires to comprehend the evidence of mathematical truths1 and the meaning of the obscure processes of the calculus2 while1 even after mastering these treatises1 the student will have much to learn on the subject from ,m4 ,comte1 of whose admirable work one of the most admirable portions is that in which he may truly be said to have created the philosophy of the higher mathematics4 --,,mill1 ,j4 ,s4 ,system of ,logic1 ,bk4 #c1 chap4 #bd1 sect4 #f4 #aijd4 ,if we must confine ourselves to one system of notation then there can be no doubt that that which was invented by ,leibnitz is better fitted for most of the purposes to which the infinitesimal calculus is applied than that of fluxions1 and for some 7such as the calculus of variations7 it is indeed almost essential4--,,ball1 ,w4 ,w4 ,r4 ,history of ,mathematics 7,london1 #aija71 p4 #cga4 #aije4 ,the difference between the method of infinitesimals and that of limits 7when exclusively adopted7 is1 that in the latter it is usual to retain evanescent quantities of higher orders until the end of the calculation and then neglect them4 ,on the other hand1 such quantities are neglected from the commencement in the infinitesimal method1 from the conviction that they cannot affect the final result1 as they must disappear when we proceed to the limit4--,,williamson1 ,b4 ,encyclopedia ,britannica1 #ith ,edition2 ,article 8,infinitesimal ,calculus10 sect4 #ad4 #aijf4 ,when we have grasped the spirit of the infinitesimal method1 and have verified the exactness of its results either by the geometrical method of prime and ultimate ratios1 or by the analytical method of derived functions1 we may employ infinitely small quantities as a sure and valuable means of shortening and simplifying our proofs4--,,lagrange4 ,mchanique ,analytique1 ,preface2 ,oeuvres1 t4 #b 7,paris1 #ahhh71 p4 #ad4 #aijg4 ,the essential merit1 the sublimity1 of the infinitesimal method lies in the fact that it is as easily performed as the simplest method of approximation1 and that it is as accurate as the results of an ordinary calculation4 ,this advantage would be lost1 or at least greatly impaired1 if1 under the pretense of securing greater accuracy throughout the whole process1 we were to substitute for the simpler method given by ,leibnitz1 one less convenient and less in harmony with the probable course of natural events4444 ,the objections which have been raised against the infinitesimal method are based on the false supposition that the errors due to neglecting infinitely small quantities during the actual calculation will continue to exist in the result of the calculation4--,,carnot1 ,l4 ,rflections sur la ,mtaphysique du ,calcul ,infinitsimal 7,paris1 #ahac71 p4 #bae4 #aijh4 ,a limiting ratio is neither more nor less difficult to define than an infinitely small quantity4--,,carnot1 ,l4 ,rflections sur la ,mtaphysique du ,calcul ,infinitsimal 7,paris1 #ahac71 p4 #baj4 #aiji4 ,a limit is a peculiar and fundamental conception1 the use of which in proving the propositions of ,higher ,geometry cannot be superseded by any combination of other hypotheses and definitions4 ,the axiom just noted that what is true up to the limit is true at the limit1 is involved in the very conception of a limit3 and this principle1 with its consequences1 leads to all the results which form the subject of the higher mathematics1 whether proved by the consideration of evanescent triangles1 by the processes of the ,differential ,calculus1 or in any other way4 --,,whewell1 ,w4 ,the ,philosophy of the ,inductive ,sciences1 ,part #a1 bk4 #b1 chap4 #ab1 sect4 #a1 7,london1 #aheh74 #aiaj4 ,the differential calculus has all the exactitude of other algebraic operations4--,,laplace4 ,thorie ,analytique des ,probabilits1 ,introduction2 ,oeuvres1 t4 #g 7,paris1 #ahhf71 p4 #cg4 #aiaa4 ,the method of fluxions is probably one of the greatest1 most subtle1 and sublime discoveries of any age3 it opens a new world to our view1 and extends our knowledge1 as it were1 to infinity2 carrying us beyond the bounds that seemed to have been prescribed to the human mind1 at least infinitely beyond those to which the ancient geometry was confined4--,,hutton1 ,,charles4 ,a ,philosophical and ,mathematical ,dictionary 7,london1 #ahae71 ,vol4 #a1 p4 #ebe4 #aiab4 ,the states and conditions of matter1 as they occur in nature1 are in a state of perpetual flux1 and these qualities may be effectively studied by the ,newtonian method 7,methodus fluxionem7 whenever they can be referred to number or subjected to measurement 7real or imaginary74 ,by the aid of ,newton's calculus the mode of action of natural changes from moment to moment can be portrayed as faithfully as these words represent the thoughts at present in my mind4 ,from this1 the law which controls the whole process can be determined with unmistakable certainty by pure calculation4--,,mellor1 ,j4 ,w4 ,higher ,mathematics for ,students of ,chemistry and ,physics 7,london1 #aijb71 ,prologue4 #aiac4 ,the calculus is the greatest aid we have to the appreciation of physical truth in the broadest sense of the word4 --,,osgood1 ,w4 ,f4 ,bulletin ,american ,mathematical ,society1 ,vol4 #ac 7#aijg71 p4 #dfg4 #aiad4 ,infinitesimal analysis is the most powerful weapon of thought yet devised by the wit of man4--,,smith1 ,w4 ,b4 ,infinitesimal ,analysis 7,new ,york1 #ahih71 ,preface1 p4 vii4 #aiae4 ,the method of ,fluxions is the general key by help whereof the modern mathematicians unlock the secrets of ,geometry1 and consequently of ,nature4 ,and1 as it is that which hath enabled them so remarkably to outgo the ancients in discovering theorems and solving problems1 the exercise and application thereof is become the main if not sole employment of all those who in this age pass for profound geometers4--,,berkeley1 ,,george4 ,the ,analyst1 sect4 #c4 #aiaf4 ,i have at last become fully satisfied that the language and idea of infinitesimals should be used in the most elementary instruction--under all safeguards of course4--,,de ,,morgan1 ,a4 ,graves' ,life of ,w4 ,r4 ,hamilton 7,new ,york1 #ahhb-ahhi71 ,vol4 #c1 p4 #dgi4 #aiag4 ,pupils should be taught how to differentiate and how to integrate simple algebraic expressions before we attempt to teach them geometry and these other complicated things4 ,the dreadful fear of the symbols is entirely broken down in those cases where at the beginning the teaching of the calculus is adopted4 ,then after the pupil has mastered those symbols you may begin geometry or anything you please4 ,i would also abolish out of the school that thing called geometrical conics4 ,there is a great deal of superstition about conic sections4 ,the student should be taught the symbols of the calculus and the simplest use of these symbols at the earliest age1 instead of these being left over until he has gone to the ,college or ,university4 --,,thompson1 ,s4 ,p4 ,perry's ,teaching of ,mathematics 7,london1 #aijb71 p4 #di4 #aiah4 ,every one versed in the matter will agree that even the elements of a scientific study of nature can be understood only by those who have a knowledge of at least the elements of the differential and integral calculus1 as well as of analytical geometry--i4e4 the so-called lower part of the higher mathematics4444 ,we should raise the question1 whether sufficient time could not be reserved in the curricula of at least the science high schools ,realanstalten to make room for these subjects4444 ,the first consideration would be to entirely relieve from the mathematical requirements of the university ,hochschule certain classes of students who can get along without extended mathematical knowledge1 or to make the necessary mathematical knowledge accessible to them in a manner which1 for various reasons1 has not yet been adopted by the university4 ,among such students ,i would count architects1 also the chemists and in general the students of the so-called descriptive natural sciences4 ,i am moreover of the opinion--and this has been for long a favorite idea of mine--1 that it would be very useful to medical students to acquire such mathematical knowledge as is indicated by the above described modest limits2 for it seems impossible to understand far-reaching physiological investigations1 if one is terrified as soon as a differential or integration symbol appears4--,,klein1 ,f4 ,jahresbericht der ,deutschen ,mathematiker ,vereinigung1 ,bd4 #b 7#aijb71 p4 #aca4 #aiai4 ,common integration is only the memory of differentiation 444 the different artifices by which integration is effected1 are changes1 not from the known to the unknown1 but from forms in which memory will not serve us to those in which it will4--,,de ,,morgan1 ,a4 ,transactions ,cambridge ,philosophical ,society1 ,vol4 #h 7#ahdd71 p4 #ahh4 #aibj4 ,given for one instant an intelligence which could comprehend all the forces by which nature is animated and the respective positions of the beings which compose it1 if moreover this intelligence were vast enough to submit these data to analysis1 it would embrace in the same formula both the movements of the largest bodies in the universe and those of the lightest atom3 to it nothing would be uncertain1 and the future as the past would be present to its eyes4 ,the human mind offers a feeble outline of that intelligence1 in the perfection which it has given to astronomy4 ,its discoveries in mechanics and in geometry1 joined to that of universal gravity1 have enabled it to comprehend in the same analytical expressions the past and future states of the world system4--,,laplace4 ,thorie ,analytique des ,probabilits1 ,introduction2 ,oeuvres1 t4 #g 7,paris1 #ahhf71 p4 #f4 #aiba4 ,there is perhaps the same relation between the action of natural selection during one generation and the accumulated result of a hundred thousand generations1 that there exists between differential and integral4 ,how seldom are we able to follow completely this latter relation although we subject it to calculation4 ,do we on that account doubt the correctness of our integrations8--,,bois-,,reymond1 ,,emil ,,du4 ,reden1 ,bd4 #a 7,leipzig1 #ahhe71 p4 #bbh4 #aibb4 ,it seems to be expected of every pilgrim up the slopes of the mathematical ,parnassus1 that he will at some point or other of his journey sit down and invent a definite integral or two towards the increase of the common stock4--,,sylvester1 ,j4 ,j4 ,notes to the ,meditation on ,poncelet's ,theorem2 ,mathematical ,papers1 ,vol4 #b1 p4 #bad4 #aibc4 ,the experimental verification of a theory concerning any natural phenomenon generally rests on the result of an integration4--,,mellor1 ,j4 ,w4 ,higher ,mathematics for ,students of ,chemistry and ,physics 7,new ,york1 #aijb71 p4 #aej4 #aibd4 ,among all the mathematical disciplines the theory of differential equations is the most important4444 ,it furnishes the explanation of all those elementary manifestations of nature which involve time4444--,,lie1 ,,sophus4 ,leipziger ,berichte1 #dg 7#ahie72 ,math4-phys4 ,classe1 p4 #bfb4 #aibe4 ,if the mathematical expression of our ideas leads to equations which cannot be integrated1 the working hypothesis will either have to be verified some other way1 or else relegated to the great repository of unverified speculations4--,,mellor1 ,j4 ,w4 ,higher ,mathematics for ,students of ,chemistry and ,physics 7,new ,york1 #aijb71 p4 #aeg4 #aibf4 ,it is well known that the central problem of the whole of modern mathematics is the study of the transcendental functions defined by differential equations4--,,klein1 ,f4 ,lectures on ,mathematics 7,new ,york1 #aiaa71 p4 #h4 #aibg4 ,every one knows what a curve is1 until he has studied enough mathematics to become confused through the countless number of possible exceptions4444 ,a curve is the totality of points1 whose co-ordinates are functions of a parameter which may be differentiated as often as may be required4--,,klein1 ,f4 ,elementar ,mathematik vom hheren ,standpunkte aus4 7,leipzig4 #aiji7 ,vol4 #b1 p4 #ced4 #aibh4 ,fourier's theorem is not only one of the most beautiful results of modern analysis1 but it may be said to furnish an indispensable instrument in the treatment of nearly every recondite question in modern physics4 ,to mention only sonorous vibrations1 the propagation of electric signals along telegraph wires1 and the conduction of heat by the earth's crust1 as subjects in their generality intractable without it1 is to give but a feeble idea of its importance4--,,thomson ,,and ,,tait4 ,elements of ,natural ,philosophy1 chap4 #a4 #aibi4 ,the principal advantage arising from the use of hyperbolic functions is that they bring to light some curious analogies between the integrals of certain irrational functions4 --,,byerly1 ,w4 ,e4 ,integral ,calculus 7,boston1 #ahij71 p4 #cj4 #aicj4 ,hyperbolic functions are extremely useful in every branch of pure physics and in the applications of physics whether to observational and experimental sciences or to technology4 ,thus whenever an entity 7such as light1 velocity1 electricity1 or radio-activity7 is subject to gradual absorption or extinction1 the decay is represented by some form of hyperbolic functions4 ,mercator's projection is likewise computed by hyperbolic functions4 ,whenever mechanical strains are regarded great enough to be measured they are most simply expressed in terms of hyperbolic functions4 ,hence geological deformations invariably lead to such expressions4444--,,walcott1 ,c4 ,d4 ,smithsonian ,mathematical ,tables1 ,hyperbolic ,functions 7,washington1 #aiji71 ,advertisement4 #aica4 ,geometry may sometimes appear to take the lead over analysis1 but in fact precedes it only as a servant goes before his master to clear the path and light him on the way4 ,the interval between the two is as wide as between empiricism and science1 as between the understanding and the reason1 or as between the finite and the infinite4--,,sylvester1 ,j4 ,j4 ,philosophic ,magazine1 ,vol4 #ca 7#ahff71 p4 #eba4 #aicb4 ,nature herself exhibits to us measurable and observable quantities in definite mathematical dependence2 the conception of a function is suggested by all the processes of nature where we observe natural phenomena varying according to distance or to time4 ,nearly all the 8known0 functions have presented themselves in the attempt to solve geometrical1 mechanical1 or physical problems4--,,merz1 ,j4 ,t4 ,a ,history of ,european ,thought in the ,nineteenth ,century 7,edinburgh and ,london1 #aijc71 p4 #fif4 #aicc4 ,that flower of modern mathematical thought--the notion of a function4--,,mccormack1 ,,thomas ,j4 ,on the ,nature of ,scientific ,law and ,scientific ,explanation1 ,monist1 ,vol4 #aj 7#ahii-aijj71 p4 #eee4 #aicd4 ,fuchs4 ,ich bin von alledem so consterniert1 ,als wrde mir ein ,kreis im ,kopfe quadriert4 ,meph4 ,nachher vor alien andern ,sachen ,msst ihe euch an die ,funktionen-,theorie machen4 ,da seht1 dass ihr tiefsinnig fasst1 ,was sich zu integrieren nicht passt4 ,an ,theoremen wird's euch nicht fehlen1 ,msst nur die ,verschwindungspunkte zhlen1 ,umkehren1 abbilden1 auf der ,eb'ne 'rumfahren ,und mit den ,theta-,produkten nicht sparen4 --,,lasswitz1 ,,kurd4 ,der ,faust-,tragdie 7-n7ter ,tiel2 ,zeitschrift fr den math4-natur4 ,unterricht1 ,bd4 #ad 7#ahhc71 p4 #caf4 ,fuchs4 ,your words fill me with an awful dread1 ,seems like a circle were squared in my head4 ,meph4 ,next in order you certainly ought ,on function-theory bestow your thought1 ,and penetrate with contemplation ,what resists your attempts at integration4 ,you'll find no dearth of theorems there-- ,to vanishing-points give proper care-- ,enumerate1 reciprocate1 ,nor forget to delineate1 ,traverse the plane from end to end1 ,and theta-functions freely spend4 #aice4 ,the student should avoid founding results upon divergent series1 as the question of their legitimacy is disputed upon grounds to which no answer commanding anything like general assent has yet been given4 ,but they may be used as means of discovery1 provided that their results be verified by other means before they are considered as established4--,,de ,,morgan1 ,a4 ,trigonometry and ,double ,algebra 7,london1 #ahdi71 p4 #ee4 #aicf4 ,there is nothing now which ever gives me any thought or care in algebra except divergent series1 which ,i cannot follow the ,french in rejecting4--,,de ,,morgan1 ,a4 ,graves' ,life of ,w4 ,r4 ,hamilton 7,new ,york1 #ahhb-ahhi71 ,vol4 #c1 p4 #bdi4 #aicg4 ,it is a strange vicissitude of our science that these divergent series which early in the century were supposed to be banished once and for all from rigorous mathematics should at its close be knocking at the door for readmission4--,,pierpont1 ,j4 ,congress of ,arts and ,sciences 7,boston and ,new ,york1 #aije71 ,vol4 #a1 p4 #dgf4 #aich4 ,zeno was concerned with three problems4444 ,these are the problem of the infinitesimal1 the infinite1 and continuity4444 ,from him to our own day1 the finest intellects of each generation in turn attacked these problems1 but achieved broadly speaking nothing4444 ,weierstrass1 ,dedekind1 and ,cantor1 444 have completely solved them4 ,their solutions 444 are so clear as to leave no longer the slightest doubt of difficulty4 ,this achievement is probably the greatest of which the age can boast4444 ,the problem of the infinitesimal was solved by ,weierstrass1 the solution of the other two was begun by ,dedekind and definitely accomplished by ,cantor4--,,russell1 ,,bertrand4 ,international ,monthly1 ,vol4 #d 7#aija71 p4 #hi4 #aici4 ,it was not till ,leibnitz and ,newton1 by the discovery of the differential calculus1 had dispelled the ancient darkness which enveloped the conception of the infinite1 and had clearly established the conception of the continuous and continuous change1 that a full and productive application of the newly-found mechanical conceptions made any progress4--,,helmholtz1 ,h4 ,aim and ,progress of ,physical ,science2 ,popular ,lectures ,flight 7,new ,york1 #aijj71 p4 #cgb4 #aidj4 ,the idea of an infinitesimal involves no contradiction4444 ,as a mathematician1 ,i prefer the method of infinitesimals to that of limits1 as far easier and less infested with snares4--,,pierce1 ,c4 ,f4 ,the ,law of ,mind2 ,monist1 ,vol4 #b 7#ahia-ahib71 pp4 #edc1 #ede4 #aida4 ,the chief objection against all abstract reasonings is derived from the ideas of space and time2 ideas1 which1 in common life and to a careless view1 are very clear and intelligible1 but when they pass through the scrutiny of the profound sciences 7and they are the chief object of these sciences7 afford principles1 which seem full of obscurity and contradiction4 ,no priestly dogmas1 invented on purpose to tame and subdue the rebellious reason of mankind1 ever shocked common sense more than the doctrine of the infinite divisibility of extension1 with its consequences2 as they are pompously displayed by all geometricians and metaphysicians1 with a kind of triumph and exultation4 ,a real quantity1 infinitely less than any finite quantity1 containing quantities infinitely less than itself1 and so on in infinitum2 this is an edifice so bold and prodigious1 that it is too weighty for any pretended demonstration to support1 because it shocks the clearest and most natural principles of human reason4 ,but what renders the matter more extraordinary1 is1 that these seemingly absurd opinions are supported by a chain of reasoning1 the clearest and most natural2 nor is it possible for us to allow the premises without admitting the consequences4 ,nothing can be more convincing and satisfactory than all the conclusions concerning the properties of circles and triangles2 and yet1 when these are once received1 how can we deny1 that the angle of contact between a circle and its tangent is infinitely less than any rectilineal angle1 that as you may increase the diameter of the circle in infinitum1 this angle of contact becomes still less1 even in infinitum1 and that the angle of contact between other curves and their tangents may be infinitely less than those between any circle and its tangent1 and so on1 in infinitum8 ,the demonstration of these principles seems as unexceptionable as that which proves the three angles of a triangle to be equal to two right ones1 though the latter opinion be natural and easy1 and the former big with contradiction and absurdity4 ,reason here seems to be thrown into a kind of amazement and suspense1 which1 without the suggestion of any sceptic1 gives her a diffidence of herself1 and of the ground on which she treads4 ,she sees a full light1 which illuminates certain places2 but that light borders upon the most profound darkness4 ,and between these she is so dazzled and confounded1 that she scarcely can pronounce with certainty and assurance concerning any one object4--,,hume1 ,,david4 ,an ,inquiry concerning ,human ,understanding1 ,sect4 #ab1 part #b4 #aidb4 ,he who can digest a second or third fluxion1 a second or third difference1 need not1 methinks1 be squeamish about any point in ,divinity4--,,berkeley1 ,g4 ,the ,analyst1 sect4 #g4 #aidc4 ,and what are these fluxions8 ,the velocities of evanescent increments4 ,and what are these same evanescent increments8 ,they are neither finite quantities1 nor quantities infinitely small1 nor yet nothing4 ,may we not call them ghosts of departed quantities8--,,berkeley1 ,g4 ,the ,analyst1 sect4 #ce4 #aidd4 ,it is said that the minutest errors are not to be neglected in mathematics2 that the fluxions are celerities1 not proportional to the finite increments1 though ever so small2 but only to the moments or nascent increments1 whereof the proportion alone1 and not the magnitude1 is considered4 ,and of the aforesaid fluxions there be other fluxions1 which fluxions of fluxions are called second fluxions4 ,and the fluxions of these second fluxions are called third fluxions3 and so on1 fourth1 fifth1 sixth1 etc41 ad infinitum4 ,now1 as our ,sense is strained and puzzled with the perception of objects extremely minute1 even so the ,imagination1 which faculty derives from sense1 is very much strained and puzzled to frame clear ideas of the least particle of time1 or the least increment generated therein3 and much more to comprehend the moments1 or those increments of the flowing quantities in status nascenti1 in their first origin or beginning to exist1 before they become finite particles4 ,and it seems still more difficult to conceive the abstracted velocities of such nascent imperfect entities4 ,but the velocities of the velocities1 the second1 third1 fourth1 and fifth velocities1 etc41 exceed1 if ,i mistake not1 all human understanding4 ,the further the mind analyseth and pursueth these fugitive ideas the more it is lost and bewildered2 the objects1 at first fleeting and minute1 soon vanishing out of sight4 ,certainly1 in any sense1 a second or third fluxion seems an obscure ,mystery4 ,the incipient celerity of an incipient celerity1 the nascent augment of a nascent augment1 i4e4 of a thing which hath no magnitude2 take it in what light you please1 the clear conception of it will1 if ,i mistake not1 be found impossible2 whether it be so or no ,i appeal to the trial of every thinking reader4 ,and if a second fluxion be inconceivable1 what are we to think of third1 fourth1 fifth fluxions1 and so on without end4--,,berkeley1 ,g4 ,the ,analyst1 sect1 #d4 #aide4 ,the infinite divisibility of finite extension1 though it is not expressly laid down either as an axiom or theorem in the elements of that science1 yet it is throughout the same everywhere supposed and thought to have so inseparable and essential a connection with the principles and demonstrations in ,geometry1 that mathematicians never admit it into doubt1 or make the least question of it4 ,and1 as this notion is the source whence do spring all those amusing geometrical paradoxes which have such a direct repugnancy to the plain common sense of mankind1 and are admitted with so much reluctance into a mind not yet debauched by learning2 so it is the principal occasion of all that nice and extreme subtility which renders the study of ,mathematics so difficult and tedious4--,,berkeley1 ,g4 ,on the ,principles of ,human ,knowledge1 ,sect4 #abc4 #aidf4 ,to avoid misconception1 it should be borne in mind that infinitesimals are not regarded as being actual quantities in the ordinary acceptation of the words1 or as capable of exact representation4 ,they are introduced for the purpose of abridgment and simplification of our reasonings1 and are an ultimate phase of magnitude when it is conceived by the mind as capable of diminution below any assigned quantity1 however small4444 ,moreover such quantities are neglected1 not1 as ,leibnitz stated1 because they are infinitely small in comparison with those that are retained1 which would produce an infinitely small error1 but because they must be neglected to obtain a rigorous result2 since such result must be definite and determinate1 and consequently independent of these variable indefinitely small quantities4 ,it may be added that the precise principles of the infinitesimal calculus1 like those of any other science1 cannot be thoroughly apprehended except by those who have already studied the science1 and made some progress in the application of its principles4 --,,williamson1 ,b4 ,encyclopedia ,britannica1 #ith ,edition2 ,article 8,infinitesimal ,calculus10 ,sect4 #ab1 #ad4 #aidg4 ,we admit1 in geometry1 not only infinite magnitudes1 that is to say1 magnitudes greater than any assignable magnitude1 but infinite magnitudes infinitely greater1 the one than the other4 ,this astonishes our dimension of brains1 which is only about six inches long1 five broad1 and six in depth1 in the largest heads4 --,,voltaire4 ,a ,philosophical ,dictionary2 ,article 8,infinity40 7,boston1 #ahha74 #aidh4 ,infinity is the land of mathematical hocus pocus4 ,there ,zero the magician is king4 ,when ,zero divides any number he changes it without regard to its magnitude into the infinitely small great81 and inversely1 when divided by any number he begets the infinitely great small84 ,in this domain the circumference of the circle becomes a straight line1 and then the circle can be squared4 ,here all ranks are abolished1 for ,zero reduces everything to the same level one way or another4 ,happy is the kingdom where ,zero rules6--,,carus1 ,,paul4 ,logical and ,mathematical ,thought2 ,monist1 ,vol4 #bj 7#aiji-aiaj71 p4 #fi4 #aidi4 ,great fleas have little fleas upon their backs to bite 'em1 ,and little fleas have lesser fleas1 and so ad infinitum4 ,and the great fleas themselves1 in turn1 have greater fleas to go on2 ,while these again have greater still1 and greater still1 and so on4 --,,de ,,morgan1 ,a4 ,budget of ,paradoxes 7,london1 #ahgb71 p4 #cgg4 #aiej4 ,we have adroitly defined the infinite in arithmetic by a loveknot1 in this manner 2 but we possess not therefore the clearer notion of it4--,,voltaire4 ,a ,philosophical ,dictionary2 ,article 8,infinity40 7,boston1 #ahha74 #aiea4 ,i protest against the use of infinite magnitude as something completed1 which in mathematics is never permissible4 ,infinity is merely a facon de parler1 the real meaning being a limit which certain ratios approach indefinitely near1 while others are permitted to increase without restriction4--,,gauss4 ,brief an ,schumacher 7#ahca72 ,werke1 ,bd4 #h p4 #baf4 #aieb4 ,in spite of the essential difference between the conceptions of the potential and the actual infinite1 the former signifying a variable finite magnitude increasing beyond all finite limits1 while the latter is a fixed1 constant quantity lying beyond all finite magnitudes1 it happens only too often that the one is mistaken for the other4444 ,owing to a justifiable aversion to such illegitimate actual infinities and the influence of the modern epicuric-materialistic tendency1 a certain horror infiniti has grown up in extended scientific circles1 which finds its classic expression and support in the letter of ,gauss see #aiea1 yet it seems to me that the consequent uncritical rejection of the legitimate actual infinite is no lesser violation of the nature of things1 which must be taken as they are4--,,cantor1 ,g4 ,zum ,problem des actualen ,unendlichen2 ,natur und ,offenbarung1 ,bd4 #cb 7#ahhf71 p4 #bbf4 #aiec4 ,the ,infinite is often confounded with the ,indefinite1 but the two conceptions are diametrically opposed4 ,instead of being a quantity with unassigned yet assignable limits1 the ,infinite is not a quantity at all1 since it neither admits of augmentation nor diminution1 having no assignable limits2 it is the operation of continuously withdrawing any limits that may have been assigned3 the endless addition of new quantities to the old3 the flux of continuity4 ,the ,infinite is no more a quantity than ,zero is a quantity4 ,if ,zero is the sign of a vanished quantity1 the ,infinite is a sign of that continuity of ,existence which has been ideally divided into discrete parts in the affixing of limits4 --,,lewes1 ,g4 ,h4 ,problems of ,life and ,mind 7,boston1 #ahge71 ,vol4 #b1 p4 #chd4 #aied4 ,a great deal of misunderstanding is avoided if it be remembered that the terms infinity1 infinite1 zero1 infinitesimal must be interpreted in connexion with their context1 and admit a variety of meanings according to the way in which they are defined4--,,mathews1 ,g4 ,b4 ,theory of ,numbers 7,cambridge1 #ahib71 ,part #a1 sect4 #ajd4 #aiee4 ,this further is observable in number1 that it is that which the mind makes use of in measuring all things that by us are measurable1 which principally are expansion and duration2 and our idea of infinity1 even when applied to those1 seems to be nothing but the infinity of number4 ,for what else are our ideas of ,eternity and ,immensity1 but the repeated additions of certain ideas of imagined parts of duration and expansion1 with the infinity of number2 in which we can come to no end of addition8 --,,locke1 ,,john4 ,an ,essay concerning ,human ,understanding1 ,bk4 #b1 chap4 #af1 sect4 #h4 #aief4 ,but of all other ideas1 it is number1 which ,i think furnishes us with the clearest and most distinct idea of infinity we are capable of4--,,locke1 ,,john4 ,an ,essay concerning ,human ,understanding1 ,bk4 #b1 chap4 #ag1 sect4 #i4 #aieg4 ,willst du ins ,unendliche schreiten8 ,geh nur im ,endlichen nach allen ,seiten6 ,willst du dich am ,ganzen erquicken1 ,so musst du das ,ganze im ,kleinsten erblicken4 --,,goethe4 ,gott1 ,gemt und ,welt 7#ahae74 ,would'st thou the infinite essay8 ,the finite but traverse in every way4 ,would'st in the whole delight thy heart8 ,learn to discern the whole in its minutest part4 #aieh4 ,ich hufe ungeheure ,zahlen1 ,gebrge ,millionen auf1 ,ich setze ,zeit auf ,zeit und ,welt auf ,welt zu ,hauf1 ,und wenn ich von der grausen ,hh' ,mit ,schwindeln wieder nach dir seh1' ,ist alle ,macht der ,zahl1 vermehrt zu tausendmalen1 ,noch nicht ein ,theil von dir4 ,ich zieh' sie ab1 und du liegst ganz vor mir4 --,,haller1 ,,albr4 ,,von4 ,quoted in ,hegel3 ,wissenschaft der ,logik1 ,buch #a1 ,abschnitt #b1 ,kap4 #b1 ,c1 b4 ,numbers upon numbers pile1 ,mountains millions high1 ,time on time and world on world amass1 ,then1 if from the dreadful hight1 alas6 ,dizzy-brained1 ,i turn thee to behold1 ,all the power of number1 increased thousandfold1 ,not yet may match thy part4 ,subtract what ,i will1 wholly whole thou art4 #aiei4 ,a collection of terms is infinite when it contains as parts other collections which have just as many terms in it as it has4 ,if you can take away some of the terms of a collection1 without diminishing the number of terms1 then there is an infinite number of terms in the collection4--,,russell1 ,,bertrand4 ,international ,monthly1 ,vol4 #d 7#aija71 p4 #ic4 #aifj4 ,an assemblage 7ensemble1 collection1 group1 manifold7 of elements 7things1 no matter what7 is infinite or finite according as it has or has not a part to which the whole is just equivalent in the sense that between the elements composing that part and those composing the whole there subsists a unique and reciprocal 7one-to-one7 correspondence4--,,keyser1 ,c4 ,j4 ,the ,axioms of ,infinity2 ,hibbert ,journal1 ,vol4 #b 7#aijc-aijd71 p4 #eci4 #aifa4 ,whereas in former times the ,infinite betrayed its presence not indeed to the faculties of ,logic but only to the spiritual ,imagination and ,sensibility1 mathematics has shown 444 that the structure of ,transfinite ,being is open to exploration by the organon of ,thought4--,,keyser1 ,c4 ,j4 ,lectures on ,science1 ,philosophy and ,art 7,new ,york1 #aijh71 p4 #db4 #aifb4 ,the mathematical theory of probability is a science which aims at reducing to calculation1 where possible1 the amount of credence due to propositions or statements1 or to the occurrence of events1 future or past1 more especially as contingent or dependent upon other propositions or events the probability of which is known4--,,crofton1 ,m4 ,w4 ,encyclopedia ,britannica1 #ith ,edition2 ,article1 8,probability40 #aifc4 ,the theory of probabilities is at bottom nothing but common sense reduced to calculus2 it enables us to appreciate with exactness that which accurate minds feel with a sort of instinct for which ofttimes they are unable to account4 ,if we consider the analytical methods to which this theory has given birth1 the truth of the principles on which it is based1 the fine and delicate logic which their employment in the solution of problems requires1 the public utilities whose establishment rests upon it1 the extension which it has received and which it may still receive through its application to the most important problems of natural philosophy and the moral sciences2 if again we observe that1 even in matters which cannot be submitted to the calculus1 it gives us the surest suggestions for the guidance of our judgments1 and that it teaches us to avoid the illusions which often mislead us1 then we shall see that there is no science more worthy of our contemplations nor a more useful one for admission to our system of public education4--,,laplace4 ,thorie ,analytique des ,probabilitis1 ,introduction2 ,oeuvres1 t4 #g 7,paris1 #ahhf71 p4 #aec4 #aifd4 ,it is a truth very certain that1 when it is not in our power to determine what is true1 we ought to follow what is most probable4--,,descartes4 ,discourse on ,method1 ,part #c4 #aife4 ,as demonstration is the showing the agreement or disagreement of two ideas1 by the intervention of one or more proofs1 which have a constant1 immutable1 and visible connexion one with another2 so probability is nothing but the appearance of such an agreement or disagreement1 by the intervention of proofs1 whose connexion is not constant and immutable1 or at least is not perceived to be so1 and it is enough to induce the mind to judge the proposition to be true or false1 rather than contrary4--,,locke1 ,,john4 ,an ,essay concerning ,human ,understanding1 ,bk4 #d1 chap4 #ae1 sect4 #a4 #aiff4 ,the difference between necessary and contingent truths is indeed the same as that between commensurable and incommensurable numbers4 ,for the reduction of commensurable numbers to a common measure is analogous to the demonstration of necessary truths1 or their reduction to such as are identical4 ,but as1 in the case of surd ratios1 the reduction involves an infinite process1 and yet approaches a common measure1 so that a definite but unending series is obtained1 so also contingent truths require an infinite analysis1 which ,god alone can accomplish4--,,leibnitz4 ,philosophische ,schriften ,gerhardt ,bd4 #g 7,berlin1 #ahij71 p4 #bjj4 #aifg4 ,the theory in question theory of probability affords an excellent illustration of the application of the theory of permutation and combinations which is the fundamental part of the algebra of discrete quantity2 it forms in the elementary parts an excellent logical exercise in the accurate use of terms and in the nice discrimination of shades of meaning2 and1 above all1 it enters into the regulation of some of the most important practical concerns of modern life4--,,chrystal1 ,,george4 ,algebra1 ,vol4 #b 7,edinburgh1 #ahhi71 chap4 #cf1 sect4 #a4 #aifh4 ,there is possibly no branch of mathematics at once so interesting1 so bewildering1 and of so great practical importance as the theory of probabilities4 ,its history reveals both the wonders that can be accomplished and the bounds that cannot be transcended by mathematical science4 ,it is the link between rigid deduction and the vast field of inductive science4 ,a complete theory of probabilities would be the complete theory of the formation of belief4 ,it is certainly a pity then1 that1 to quote ,m4 ,bertrand1 8one cannot well understand the calculus of probabilities without having read ,laplace's work10 and that 8one cannot read ,laplace's work without having prepared oneself for it by the most profound mathematical studies40--,,davis1 ,e4 ,w4 ,bulletin ,american ,mathematical ,society1 ,vol4 #a 7#ahid-ahie71 p4 #af4 #aifi4 ,the most important questions of life are1 for the most part1 really only problems of probability4 ,strictly speaking one may even say that nearly all our knowledge is problematical2 and in the small number of things which we are able to know with certainty1 even in the mathematical sciences themselves1 induction and analogy1 the principal means for discovering truth1 are based on probabilities1 so that the entire system of human knowledge is connected with this theory4--,,laplace4 ,thorie ,analytique des ,probabilitis1 ,introduction2 ,oeuvres1 t4 #g 7,paris1 #ahhf71 p4 #e4 #aigj4 ,there is no more remarkable feature in the mathematical theory of probability than the manner in which it has been found to harmonize with1 and justify1 the conclusions to which mankind have been led1 not by reasoning1 but by instinct and experience1 both of the individual and of the race4 ,at the same time it has corrected1 extended1 and invested them with a definiteness and precision of which these crude1 though sound1 appreciations of common sense were till then devoid4--,,crofton1 ,m4 ,w4 ,encyclopedia ,britannica1 #ith ,edition2 ,article 8,probability40 #aiga4 ,it is remarkable that a science probabilities which began with the consideration of games of chance1 should have become the most important object of human knowledge4--,,laplace4 ,thorie ,analytique des ,probabilitis1 ,introduction2 ,oeuvres1 t4 #g 7,paris1 #ahhf71 p4 #aeb4 #aigb4 ,not much has been added to the subject of probability since the close of ,laplace's career4 ,the history of science records more than one parallel to this abatement of activity4 ,when such a genius has departed1 the field of his labours seems exhausted for the time1 and little left to be gleaned by his successors4 ,it is to be regretted that so little remains to us of the inner workings of such gifted minds1 and of the clue by which each of their discoveries was reached4 ,the didactic and synthetic form in which these are presented to the world retains but faint traces of the skilful inductions1 the keen and delicate perception of fitness and analogy1 and the power of imagination 444 which have doubtless guided such a master as ,laplace or ,newton in shaping out such great designs--only the minor details of which have remained over1 to be supplied by the less cunning hand of commentator and disciple4--,,crofton1 ,m4 ,w4 ,encyclopedia ,britannica1 #ith ,edition2 ,article 8,probability40 #aigc4 ,the theory of errors may be defined as that branch of mathematics which is concerned1 first1 with the expression of the resultant effect of one or more sources of error to which computed and observed quantities are subject2 and1 secondly1 with the determination of the relation between the magnitude of an error and the probability of its occurrence4--,,woodward1 ,r4 ,s4 ,probability and ,theory of ,errors 7,new ,york1 #aijf71 p4 #cj4 #aigd4 ,of all the applications of the doctrine of probability none is of greater utility than the theory of errors4 ,in astronomy1 geodesy1 physics1 and chemistry1 as in every science which attains precision in measuring1 weighing1 and computing1 a knowledge of the theory of errors is indispensable4 ,by the aid of this theory the exact sciences have made great progress during the nineteenth century1 not only in the actual determinations of the constants of nature1 but also in the fixation of clear ideas as to the possibilities of future conquests in the same direction4 ,nothing1 for example1 is more satisfactory and instructive in the history of science than the success with which the unique method of least squares has been applied to the problems presented by the earth and the other members of the solar system4 ,so great1 in fact1 are the practical value and theoretical importance of least squares1 that it is frequently mistaken for the whole theory of errors1 and is sometimes regarded as embodying the major part of the doctrine of probability itself4--,,woodward1 ,r4 ,s4 ,probability and ,theory of ,errors 7,new ,york1 #aijf71 pp4 #i-aj4 #aige4 ,direct and inverse ratios have been applied by an ingenious author to measure human affections1 and the moral worth of actions4 ,an eminent ,mathematician attempted to ascertain by calculation1 the ratio in which the evidence of facts must decrease in the course of time1 and fixed the period when the evidence of the facts on which ,christianity is founded shall become evanescent1 and when in consequence no faith shall be found on the earth4--,,reid1 ,,thomas4 ,essays on the ,powers of the ,human ,mind 7,edinburgh1 #ahab71 ,vol4 #b1 p4 #djh4 ,,chapter ,,xx ,,the ,,fundamental ,,concepts1 ,,time ,,and ,,space #bjja4 ,kant's ,doctrine of ,time4 ,i4 ,time is not an empirical concept deduced from any experience1 for neither co-existence nor succession would enter into our perception1 if the representation of time were not given a priori4 ,only when this representation a priori is given1 can we imagine that certain things happen at the same time 7simultaneously7 or at different times 7successively74 ,,ii4 ,time is a necessary representation on which all intuitions depend4 ,we cannot take away time from phenomena in general1 though we can well take away phenomena out of time4 ,in time alone is reality of phenomena possible4 ,all phenomena may vanish1 but time itself 7as the general condition of their possibility7 cannot be done away with4 ,,iii4 ,on this a priori necessity depends also the possibility of apodictic principles of the relations of time1 or of axioms of time in general4 ,time has one dimension only2 different times are not simultaneous1 but successive1 while different spaces are never successive1 but simultaneous4 ,such principles cannot be derived from experience1 because experience could not impart to them absolute universality nor apodictic certainty4444 ,,iv4 ,time is not a discursive1 or what is called a general concept1 but a pure form of sensuous intuition4 ,different times are parts only of one and the same time4444 ,v4 ,to say that time is infinite means no more than that every definite quantity of time is possible only by limitations of one time which forms the foundation of all times4 ,the original representation of time must therefore be given as unlimited4 ,but when the parts themselves and every quantity of an object can be represented as determined by limitation only1 the whole representation cannot be given by concepts 7for in that case the partial representation comes first71 but must be founded on immediate intuition4--,,kant1 ,i4 ,critique of ,pure ,reason ,max ,mller 7,new ,york1 #aijj71 pp4 #bd-be4 #bjjb4 ,kant's ,doctrine of ,space4 ,i4 ,space is not an empirical concept which has been derived from external experience4 ,for in order that certain sensations should be referred to something outside myself1 i4e4 to something in a different part of space from that where ,i am2 again1 in order that ,i may be able to represent them as side by side1 that is1 not only as different1 but as in different places1 the representation of space must already be there4444 ,,ii4 ,space is a necessary representation a priori1 forming the very foundation of all external intuitions4 ,it is impossible to imagine that there should be no space1 though one might very well imagine that there should be space without objects to fill it4 ,space is therefore regarded as a condition of the possibility of phenomena1 not as a determination produced by them2 it is a representation a priori which necessarily precedes all external phenomena4 ,,iii4 ,on this necessity of an a priori representation of space rests the apodictic certainty of all geometrical principles1 and the possibility of their construction a priori4 ,for if the intuition of space were a concept gained a posteriori1 borrowed from general external experience1 the first principles of mathematical definition would be nothing but perceptions4 ,they would be exposed to all the accidents of perception1 and there being but one straight line between two points would not be a necessity1 but only something taught in each case by experience4 ,whatever is derived from experience possesses a relative generality only1 based on induction4 ,we should therefore not be able to say more than that1 so far as hitherto observed1 no space has yet been found having more than three dimensions4 ,,iv4 ,space is not a discursive or so-called general concept of the relations of things in general1 but a pure intuition4 ,for1 first of all1 we can imagine one space only1 and if we speak of many spaces1 we mean parts only of one and the same space4 ,nor can these parts be considered as antecedent to the one and all-embracing space and1 as it were1 its component parts out of which an aggregate is formed1 but they can be thought of as existing within it only4 ,space is essentially one2 its multiplicity1 and therefore the general concept of spaces in general1 arises entirely from limitations4 ,hence it follows that1 with respect to space1 an intuition a priori1 which is not empirical1 must form the foundation of all conceptions of space4444 ,v4 ,space is represented as an infinite given quantity4 ,now it is quite true that every concept is to be thought as a representation1 which is contained in an infinite number of different possible representations 7as their common characteristic71 and therefore comprehends them3 but no concept1 as such1 can be thought as if it contained in itself an infinite number of representations4 ,nevertheless1 space is so thought 7for all parts of infinite space exist simultaneously74 ,consequently1 the original representation of space is an intuition a priori1 and not a concept4--,,kant1 ,i4 ,critique of ,pure ,reason ,max ,mller 7,new ,york1 #aijj71 pp4 #ah-bj and ,supplement #h4 #bjjc4 ,schopenhauer's ,predicabilia a priori4#aa ,,of ,,time ,,of ,,space #a4 ,there is but one time1 all #a4 ,there is but one space1 different times are parts of all different spaces are it4 parts of it4 #b4 ,different times are not #b4 ,different spaces are not simultaneous but successive4 successive but simultaneous4 #c4 ,everything in time may be #c4 ,everything in space may be thought of as non-existent1 thought of as non-existent1 but not time4 but not space4 #d4 ,time has three divisions3 #d4 ,space has three dimensions3 past1 present and future1 height1 breadth1 and which form two directions length4 with a point of indifference4 #e4 ,time is infinitely #e4 ,space is infinitely divisible4 divisible4 #f4 ,time is homogeneous and a #f4 ,space is homogeneous and a continuum3 i4e4 no part is continuum3 i4e4 no part different from another1 nor is different from another1 separated by something nor separated by something which is not time4 which is not space4 #g4 ,time has no beginning nor #g4 ,space has no limits end1 but all beginning and ,grnzen1 but all limits end is in time4 are in space4 #h4 ,time makes counting #h4 ,space makes measurement possible4 possible4 #i4 ,rhythm exists only in time4 #i4 ,symmetry exists only in space4 #aj4 ,the laws of time are a #aj4 ,the laws of space are a priori conceptions4 priori conceptions4 #aa4 ,time is perceptible a #aa4 ,space is immediately priori1 but only by a perceptible a priori4 means of a line-image4 #ab4 ,time has no permanence but #ab4 ,space never passes but is passes the moment it is permanent throughout present4 all time4 #ac4 ,time never rests4 #ac4 ,space never moves4 #ad4 ,everything in time has #ad4 ,everything in space has duration4 position4 #ae4 ,time has no duration1 but #ae4 ,space has no motion1 but all duration is in time2 all motion is in space2 time is the persistence of space is the change in what is permanent in position of that which contrast with its restless moves in contrast to its course4 imperturbable rest4 #af4 ,motion is only possible in #af4 ,motion is only possible in time4 space4 #ag4 ,velocity1 the space being #ag4 ,velocity1 the time being the same1 is in the inverse the same1 is in the direct ratio of the time4 ratio of the space4 #ah4 ,time is not directly #ah4 ,space is measurable directly measurable by means of through itself and itself but only by means of indirectly through motion motion which takes place in which takes place in both both space and time4444 time and space4444 #ai4 ,time is omnipresent3 each #ai4 ,space is eternal3 each part of it is everywhere4 part of it exists always4 #bj4 ,in time alone all things #bj4 ,in space alone all things are successive4 are simultaneous4 #ba4 ,time makes possible the #ba4 ,space makes possible the change of accidents4 endurance of substance4 #bb4 ,each part of time contains #bb4 ,no part of space contains all substance4 the same substance as another4 #bc4 ,time is the principium #bc4 ,space is the principium individuationis4 individuationis4 #bd4 ,the now is without #bd4 ,the point is without duration4 extension4 #be4 ,time of itself is empty and #be4 ,space is of itself empty indeterminate4 and indeterminate4 #bf4 ,each moment is conditioned #bf4 ,the relation of each by the one which precedes boundary in space to every it1 and only so far as this other is determined by its one has ceased to exist4 relation to any one4 7,principle of sufficient 7,principle of sufficient reason of being in time47 reason of being in space47 #bg4 ,time makes ,arithmetic #bg4 ,space makes ,geometry possible4 possible4 #bh4 ,the simple element of #bh4 ,the element of ,geometry ,arithmetic is unity4 is the point4 --,,schopenhauer1 ,a4 ,die ,welt als ,vorstellung und ,wille2 ,werke 7,frauenstdt7 7,leipzig1 #ahgg71 ,bd4 #b1 p4 #ee4 #aa ,schopenhauer's table contains a third column headed 8of matter0 which has here been omitted4 #bjjd4 ,the clear possession of the ,idea of ,space is the first requisite for all geometrical reasoning2 and this clearness of idea may be tested by examining whether the axioms offer themselves to the mind as evident4--,,whewell1 ,,william4 ,the ,philosophy of the ,inductive ,sciences1 ,part #a1 ,bk4 #b1 chap4 #d1 sect4 #d 7,london1 #aheh74 #bjje4 ,geometrical axioms are neither synthetic a priori conclusions nor experimental facts4 ,they are conventions3 our choice1 amongst all possible conventions1 is guided by experimental facts2 but it remains free1 and is only limited by the necessity of avoiding all contradiction4444 ,in other words1 axioms of geometry are only definitions in disguise4 ,that being so what ought one to think of this question3 ,is the ,euclidean ,geometry true8 ,the question is nonsense4 ,one might as well ask whether the metric system is true and the old measures false2 whether ,cartesian co-ordinates are true and polar co-ordinates false4 --,,poincar1 ,h4 ,non-,euclidean ,geometry2 ,nature1 ,vol #de 7#ahia-ahib71 p4 #djg4 #bjjf4 ,i do in no wise share this view that the axioms are arbitrary propositions which we assume wholly at will1 and that in like manner the fundamental conceptions are in the end only arbitrary symbols with which we operate but consider it the death of all science3 in my judgment the axioms of geometry are not arbitrary1 but reasonable propositions which generally have the origin in space intuition and whose separate content and sequence is controlled by reasons of expediency4--,,klein1 ,f4 ,elementarmathematik vom hheren ,standpunkte aus 7,leipzig1 #aiji71 ,bd4 #b1 p4 #chd4 #bjjg4 ,euclid's ,postulate #e ,the ,parallel ,axiom4 ,that1 if a straight line falling on two straight lines make the interior angles on the same side less than two right angles1 the two straight lines1 if produced indefinitely1 meet on that side on which are the angles less than the two right angles4--,,euclid4 ,the ,thirteen ,books of ,euclid's ,elements ,t4 ,l4 ,heath ,vol4 #a 7,cambridge1 #aijh71 p4 #bjb4 #bjjh4 ,it must be admitted that ,euclid's ,parallel ,axiom is unsatisfactory as the basis of a theory of parallel straight lines4 ,it cannot be regarded as either simple or self-evident1 and it therefore falls short of the essential characteristics of an axiom4444--,,hall1 ,h4 ,s4 and ,,stevens1 ,f4 ,h4 ,euclid's ,elements 7,london1 #ahib71 p4 #ee4 #bjji4 ,we may still well declare the parallel axiom the simplest assumption which permits us to represent spatial relations1 and so it will be true generally1 that concepts and axioms are not immediate facts of intuition1 but rather the idealizations of these facts chosen for reasons of expediency4--,,klein1 ,f4 ,elementarmathematik vom1 hheren ,standpunkte aus 7,leipzig1 #aiji71 ,bd4 #b1 p4 #chb4 #bjaj4 ,the characteristic features of our space are not necessities of thought1 and the truth of ,euclid's axioms1 in so far as they specially differentiate our space from other conceivable spaces1 must be established by experience and by experience only4--,,ball1 ,r4 ,s4 ,encyclopedia ,britannica1 #ith ,edition2 ,article 8,measurement40 #bjaa4 ,mathematical and physiological researches have shown that the space of experience is simply an actual case of many conceivable cases1 about whose peculiar properties experience alone can instruct us4--,,mach1 ,,ernst4 ,popular ,scientific ,lectures 7,chicago1 #aiaj71 p4 #bje4 #bjab4 ,the familiar definition3 ,an axiom is a self-evident truth1 means if it means anything1 that the proposition which we call an axiom has been approved by us in the light of our experience and intuition4 ,in this sense mathematics has no axioms1 for mathematics is a formal subject over which formal and not material implication reigns4--,,wilson1 ,e4 ,b4 ,bulletin ,american ,mathematical ,society1 ,vol4 #b 7#aijd-aije71 p4 #ha4 #bjac4 ,the proof of self-evident propositions may seem1 to the uninitiated1 a somewhat frivolous occupation4 ,to this we might reply that it is often by no means self-evident that one obvious proposition follows from another obvious proposition2 so that we are really discovering new truths when we prove what is evident by a method which is not evident4 ,but a more interesting retort is1 that since people have tried to prove obvious propositions1 they have found that many of them are false4 ,self-evidence is often a mere will-o'-the-wisp1 which is sure to lead us astray if we take it as our guide4--,,russell1 ,,bertrand4 ,recent ,work on the ,principles of ,mathematics2 ,international ,monthly1 ,vol4 #d 7#aija71 p4 #hf4 #bjad4 ,the problem of ,euclid's ,parallel ,axiom is now at a par with the squaring of the circle and the trisection of an angle by means of ruler and compass4 ,so far as the mathematical public is concerned1 the famous problem of the parallel is settled for all time4--,,young1 ,,john ,,wesley4 ,fundamental ,concepts of ,algebra and ,geometry 7,new ,york1 #aiaa71 p4 #cb4 #bjae4 ,if the ,euclidean assumptions are true1 the constitution of those parts of space which are at an infinite distance from us1 8geometry upon the plane at infinity10 is just as well known as the geometry of any portion of this room4 ,in this infinite and thoroughly well-known space the ,universe is situated during at least some portion of an infinite and thoroughly well-known time4 ,so that here we have real knowledge of something at least that concerns the ,cosmos2 something that is true throughout the ,immensities and the ,eternities4 ,that something ,lobatchewsky and his successors have taken away4 ,the geometer of to-day knows nothing about the nature of the actually existing space at an infinite distance2 he knows nothing about the properties of this present space in a past or future eternity4 ,he knows1 indeed1 that the laws assumed by ,euclid are true with an accuracy that no direct experiment can approach1 not only in this place where we are1 but in places at a distance from us that no astronomer has conceived2 but he knows this as of ,here and ,now2 beyond this range is a ,there and ,then of which he knows nothing at present1 but may ultimately come to know more4--,,clifford1 ,w4 ,k4 ,lectures and ,essays 7,new ,york1 #aija71 ,vol4 #a1 pp4 #ceh-cei4 #bjaf4 ,the truth is that other systems of geometry are possible1 yet after all1 these other systems are not spaces but other methods of space measurements4 ,there is one space only1 though we may conceive of many different manifolds1 which are contrivances or ideal constructions invented for the purpose of determining space4--,,carus1 ,,paul4 ,science1 ,vol4 #ah 7#aijc71 p4 #ajf4 #bjag4 ,as ,i have formerly stated that from the philosophic side ,non-,euclidean ,geometry has as yet not frequently met with full understanding1 so ,i must now emphasize that it is universally recognized in the science of mathematics2 indeed1 for many purposes1 as for instance in the modern theory of functions1 it is used as an extremely convenient means for the visual representation of highly complicated arithmetical relations4 --,,klein1 ,f4 ,elementarmathematik vom hheren ,standpunkte aus 7,leipzig1 #aiji71 ,bd4 #b1 p4 #cgg4 #bjah4 ,everything in physical science1 from the law of gravitation to the building of bridges1 from the spectroscope to the art of navigation1 would be profoundly modified by any considerable inaccuracy in the hypothesis that our actual space is ,euclidean4 ,the observed truth of physical science1 therefore1 constitutes overwhelming empirical evidence that this hypothesis is very approximately correct1 even if not rigidly true4 --,,russell1 ,,bertrand4 ,foundations of ,geometry 7,cambridge1 #ahig71 p4 #f4 #bjai4 ,the most suggestive and notable achievement of the last century is the discovery of ,non-,euclidean geometry4--,,hilbert1 ,d4 ,quoted by ,g4 ,d4 ,fitch in ,manning's 8,the ,fourth ,dimension ,simply ,explained10 7,new ,york1 #aiaj71 p4 #eh4 #bjbj4 ,non-,euclidean geometry--primate among the emancipators of the human intellect4444--,,keyser1 ,c4 ,j4 ,the ,foundations of ,mathematics2 ,science ,history of the ,universe1 ,vol4 #h 7,new ,york1 #aiji71 p4 #aib4 #bjba4 ,every high school teacher ,gymnasial-lehrer must of necessity know something about non-euclidean geometry1 because it is one of the few branches of mathematics which1 by means of certain catch-phrases1 has become known in wider circles1 and concerning which any teacher is consequently liable to be asked at any time4 ,in physics there are many such matters--almost every new discovery is of this kind--which1 through certain catch-words have become topics of common conversation1 and about which therefore every teacher must of course be informed4 ,think of a teacher of physics who knows nothing of ,roentgen rays or of radium2 no better impression would be made by a mathematician who is unable to give information concerning non-euclidean geometry4 --,,klein1 ,f4 ,elementarmathematik vom hheren ,standpunkte aus 7,leipzig1 #aiji71 ,bd4 #b1 p4 #cgh4 #bjbb4 ,what ,vesalius was to ,galen1 what ,copernicus was to ,ptolemy1 that was ,lobatchewsky to ,euclid4 ,there is1 indeed1 a somewhat instructive parallel between the last two cases4 ,copernicus and ,lobatchewsky were both of ,slavic origin4 ,each of them has brought about a revolution in scientific ideas so great that it can only be compared with that wrought by the other4 ,and the reason of the transcendent importance of these two changes is that they are changes in the conception of the ,cosmos4444 ,and in virtue of these two revolutions the idea of the ,universe1 the ,macrocosm1 the ,all1 as subject of human knowledge1 and therefore of human interest1 has fallen to pieces4--,,clifford1 ,w4 ,k4 ,lectures and ,essays 7,new ,york1 #aija71 ,vol4 #a1 pp4 #cef1 #ceh4 #bjbc4 ,i am exceedingly sorry that ,i have failed to avail myself of our former greater proximity to learn more of your work on the foundations of geometry2 it surely would have saved me much useless effort and given me more peace1 than one of my disposition can enjoy so long as so much is left to consider in a matter of this kind4 ,i have myself made much progress in this matter 7though my other heterogeneous occupations have left me but little time for this purpose72 though the course which ,i have pursued does not lead as much to the desired end1 which you assure me you have reached1 as to the questioning of the truth of geometry4 ,it is true that ,i have found much which many would accept as proof1 but which in my estimation proves nothing1 for instance1 if it could be shown that a rectilinear triangle is possible1 whose area is greater than that of any given surface1 then ,i could rigorously establish the whole of geometry4 ,now most people1 no doubt1 would grant this as an axiom1 but not ,i2 it is conceivable that1 however distant apart the vertices of the triangle might be chosen1 its area might yet always be below a certain limit4 ,i have found several other such theorems1 but none of them satisfies me4--,,gauss4 ,letter to ,bolyai 7#agii72 ,werke1 ,bd4 #h 7,gttingen1 #aijj71 p4 #aei4 #bjbd4 ,on the supposition that ,euclidean geometry is not valid1 it is easy to show that similar figures do not exist2 in that case the angles of an equilateral triangle vary with the side in which ,i see no absurdity at all4 ,the angle is a function of the side and the sides are functions of the angle1 a function which1 of course1 at the same time involves a constant length4 ,it seems somewhat of a paradox to say that a constant length could be given a priori as it were1 but in this again ,i see nothing inconsistent4 ,indeed1 it would be desirable that ,euclidean geometry were not valid1 for then we should possess a general a priori standard of measure4--,,gauss4 ,letter to ,gerling 7#ahaf72 ,werke1 ,bd4 #h 7,gttingen1 #aijj71 p4 #afi4 #bjbe4 ,i am convinced more and more that the necessary truth of our geometry cannot be demonstrated1 at least not by the human intellect to the human understanding4 ,perhaps in another world we may gain other insights into the nature of space which at present are unattainable to us4 ,until then we must consider geometry as of equal rank not with arithmetic1 which is purely a priori1 but with mechanics4--,,gauss4 ,letter to ,olbers 7#ahag72 ,werke1 ,bd4 #h 7,gttingen1 #aijj71 p4 #agg4 #bjbf4 ,there is no doubt that it can be rigorously established that the sum of the angles of a rectilinear triangle cannot exceed #ahj4 ,but it is otherwise with the statement that the sum of the angles cannot be less than #ahj2 this is the real ,gordian knot1 the rocks which cause the wreck of all4444 ,i have been occupied with the problem over thirty years and ,i doubt if anyone has given it more serious attention1 though ,i have never published anything concerning it4 ,the assumption that the angle sum is less than #ahj leads to a peculiar geometry1 entirely different from the ,euclidean1 but throughout consistent with itself4 ,i have developed this geometry to my own satisfaction so that ,i can solve every problem that arises in it with the exception of the determination of a certain constant which cannot be determined a priori4 ,the larger one assumes this constant the more nearly one approaches the ,euclidean geometry1 an infinitely large value makes the two coincide4 ,the theorems of this geometry seem in part paradoxical1 and to the unpracticed absurd2 but on a closer and calm reflection it is found that in themselves they contain nothing impossible4444 ,all my efforts to discover some contradiction1 some inconsistency in this ,non-,euclidean geometry have been fruitless1 the one thing in it that seems contrary to reason is that space would have to contain a definitely determinate 7though to us unknown7 linear magnitude4 ,however1 it seems to me that notwithstanding the meaningless word-wisdom of the metaphysicians we know really too little1 or nothing1 concerning the true nature of space to confound what appears unnatural with the absolutely impossible4 ,should ,non-,euclidean geometry be true1 and this constant bear some relation to magnitudes which come within the domain of terrestrial or celestial measurement1 it could be determined a posteriori4 --,,gauss4 ,letter to ,taurinus 7#ahbd72 ,werke1 ,bd4 #h 7,gttingen1 #aijj71 p4 #ahg4 #bjbg4 ,there is also another subject1 which with me is nearly forty years old1 to which ,i have again given some thought during leisure hours1 ,i mean the foundations of geometry4444 ,here1 too1 ,i have consolidated many things1 and my conviction has1 if possible become more firm that geometry cannot be completely established on a priori grounds4 ,in the mean time ,i shall probably not for a long time yet put my very extended investigations concerning this matter in shape for publication1 possibly not while ,i live1 for ,i fear the cry of the ,botians which would arise should ,i express my whole view on this matter4--,it is curious too1 that besides the known gap in ,euclid's geometry1 to fill which all efforts till now have been in vain1 and which will never be filled1 there exists another defect1 which to my knowledge no one thus far has criticised and which 7though possible7 it is by no means easy to remove4 ,this is the definition of a plane as a surface which wholly contains the line joining any two points4 ,this definition contains more than is necessary to the determination of the surface1 and tacitly involves a theorem which demands proof4--,,gauss4 ,letter to ,bessel 7#ahbi72 ,werke1 ,bd4 #h 7,gttingen1 #aijj71 p4 #bjj4 #bjbh4 ,i will add that ,i have recently received from ,hungary a little paper on ,non-,euclidean geometry1 in which ,i rediscover all my own ideas and results worked out with great elegance1 4444 ,the writer is a very young ,austrian officer1 the son of one of my early friends1 with whom ,i often discussed the subject in #agih1 although my ideas were at that time far removed from the development and maturity which they have received through the original reflections of this young man4 ,i consider the young geometer v4 ,bolyai a genius of the first rank4--,,gauss4 ,letter to ,gerling 7#ahcb72 ,werke1 ,bd4 #h 7,gttingen1 #aijj71 p4 #bba4 #bjbi4 ,think of the image of the world in a convex mirror4444 ,a well-made convex mirror of moderate aperture represents the objects in front of it as apparently solid and in fixed positions behind its surface4 ,but the images of the distant horizon and of the sun in the sky lie behind the mirror at a limited distance1 equal to its focal length4 ,between these and the surface of the mirror are found the images of all the other objects before it1 but the images are diminished and flattened in proportion to the distance of their objects from the mirror4444 ,yet every straight line or plane in the outer world is represented by a straight 8 line or plane in the image4 ,the image of a man measuring with a rule a straight line from the mirror1 would contract more and more the farther he went1 but with his shrunken rule the man in the image would count out exactly the same number of centimeters as the real man4 ,and1 in general1 all geometrical measurements of lines and angles made with regularly varying images of real instruments would yield exactly the same results as in the outer world1 all lines of sight in the mirror would be represented by straight lines of sight in the mirror4 ,in short1 ,i do not see how men in the mirror are to discover that their bodies are not rigid solids and their experiences good examples of the correctness of ,euclidean axioms4 ,but if they could look out upon our world as we look into theirs without overstepping the boundary1 they must declare it to be a picture in a spherical mirror1 and would speak of us just as we speak of them2 and if two inhabitants of the different worlds could communicate with one another1 neither1 as far as ,i can see1 would be able to convince the other that he had the true1 the other the distorted1 relation4 ,indeed ,i cannot see that such a question would have any meaning at all1 so long as mechanical considerations are not mixed up with it4--,,helmholtz1 ,h4 ,on the ,origin and ,significance of ,geometrical ,axioms2 ,popular ,scientific ,lectures1 second series 7,new ,york1 #ahha71 pp4 #eg-ei4 #bjcj4 ,that space conceived of as a locus of points has but three dimensions needs no argument from the mathematical point of view2 but just as little can we from this point of view prevent the assertion that space has really four or an infinite number of dimensions though we perceive only three4 ,the theory of multiply-extended manifolds1 which enters more and more into the foreground of mathematical research1 is from its very nature perfectly independent of such an assertion4 ,but the form of expression1 which this theory employs1 has indeed grown out of this conception4 ,instead of referring to the individuals of a manifold1 we speak of the points of a higher space1 etc4 ,in itself this form of expression has many advantages1 in that it facilitates comprehension by calling up geometrical intuition4 ,but it has this disadvantage1 that in extended circles1 investigations concerning manifolds of any number of dimensions are considered singular alongside the above-mentioned conception of space4 ,this view is without the least foundation4 ,the investigations in question would indeed find immediate geometric applications if the conception were valid but its value and purpose1 being independent of this conception1 rests upon its essential mathematical content4--,,klein1 ,f4 ,mathematische ,annalen1 ,bd4 #dc 7#ahic71 p4 #ie4 #bjca4 ,we are led naturally to extend the language of geometry to the case of any number of variables1 still using the word point to designate any system of values of n variables 7the cordinates of the point71 the word space 7of n dimensions7 to designate the totality of all these points or systems of values1 curves or surface to designate the spread composed of points whose cordinates are given functions 7with the proper restrictions7 of one or two parameters 7the straight line or plane1 when they are linear fractional functions with the same denominator71 etc4 ,such an extension has come to be a necessity in a large number of investigations1 in order as well to give them the greatest generality as to preserve in them the intuitive character of geometry4 ,but it has been noted that in such use of geometric language we are no longer constructing truly a geometry1 for the forms that we have been considering are essentially analytic1 and that1 for example1 the general projective geometry constructed in this way is in substance nothing more than the algebra of linear transformations4 --,,segre1 ,,corradi4 ,rivista di ,matematica1 ,vol4 ,i 7#ahia71 p4 #ei4 ,j4 ,w4 ,young4 #bjcb4 ,those who can1 in common algebra1 find a square root of #a1 will be at no loss to find a fourth dimension in space in which ,,abc may become ,,abcd3 or1 if they cannot find it1 they have but to imagine it1 and call it an impossible dimension1 subject to all the laws of the three we find possible4 ,and just as #a in common algebra1 gives all its significant combinations true1 so would it be with any number of dimensions of space which the speculator might choose to call into impossible existence4 --,,de ,,morgan1 ,a4 ,trigonometry and ,double ,algebra 7,london1 #ahdi71 ,part #b1 chap4 #c4 #bjcc4 ,the doctrine of non-,euclidean spaces and of hyperspaces in general possesses the highest intellectual interest1 and it requires a far-sighted man to foretell that it can never have any practical importance4--,,smith1 ,w4 ,b4 ,introductory ,modern ,geometry 7,new ,york1 #ahic71 p4 #bgd4 #bjcd4 ,according to his frequently expressed view1 ,gauss considered the three dimensions of space as specific peculiarities of the human soul2 people1 which are unable to comprehend this1 he designated in his humorous mood by the name ,botians4 ,we could imagine ourselves1 he said1 as beings which are conscious of but two dimensions2 higher beings might look at us in a like manner1 and continuing jokingly1 he said that he had laid aside certain problems which1 when in a higher state of being1 he hoped to investigate geometrically4--,,sartorius1 ,w4 ,v4 ,,waltershausen4 ,gauss zum ,gedchtniss 7,leipzig1 #ahef71 p4 #ha4 #bjce4 ,there is many a rational logos1 and the mathematician has high delight in the contemplation of inconsistent systems of consistent relationships4 ,there are1 for example1 a ,euclidean geometry and more than one species of non-,euclidean4 ,as theories of a given space1 these are not compatible4 ,if our universe be1 as ,plato thought1 and nature-science takes for granted1 a space-conditioned1 geometrised affair1 one of these geometries may be1 none of them may be1 not all of them can be1 valid in it4 ,but in the vaster world of thought1 all of them are valid1 there they co-exist1 and interlace among themselves and others1 as differing component strains of a higher1 strictly supernatural1 hypercosmic1 harmony4--,,keyser1 ,c4 ,j4 ,the ,universe and ,beyond2 ,hibbert ,journal1 ,vol4 #c 7#aijd-aije71 p4 #cac4 #bjcf4 ,the introduction into geometrical work of conceptions such as the infinite1 the imaginary1 and the relations of hyperspace1 none of which can be directly imagined1 has a psychological significance well worthy of examination4 ,it gives a deep insight into the resources and working of the human mind4 ,we arrive at the borderland of mathematics and psychology4 --,,merz1 ,j4 ,t4 ,history of ,european ,thought in the ,nineteenth ,century 7,edinburgh and ,london1 #aijc71 p4 #gaf4 #bjcg4 ,among the splendid generalizations effected by modern mathematics1 there is none more brilliant or more inspiring or more fruitful1 and none more commensurate with the limitless immensity of being itself1 than that which produced the great concept designated 444 hyperspace or multidimensional space4 --,,keyser1 ,c4 ,j4 ,mathematical ,emancipations2 ,monist1 ,vol4 #af 7#aijf71 p4 #fe4 #bjch4 ,the great generalization of hyperspace has made it possible to enrich1 quicken and beautify analysis with the terse1 sensuous1 artistic1 stimulating language of geometry4 ,on the other hand1 the hyperspaces are in themselves immeasurably interesting and inexhaustibly rich fields of research4 ,not only does the geometrician find light in them for the illumination of otherwise dark and undiscovered properties of ordinary spaces of intuition1 but he also discovers there wondrous structures quite unknown to ordinary space4444 ,it is by creation of hyperspaces that the rational spirit secures release from limitation4 ,in them it lives ever joyously1 sustained by an unfailing sense of infinite freedom4--,,keyser1 ,c4 ,j4 ,mathematical ,emancipations2 ,monist1 ,vol4 #af 7#aijf71 p4 #hc4 #bjci4 ,mathematicians who busy themselves a great deal with the formal theory of four-dimensional space1 seem to acquire a capacity for imagining this form as easily as the three-dimensional form with which we are all familiar4--,,ostwald1 ,w4 ,natural ,philosophy ,seltzer1 7,new ,york1 #aiaj71 p4 #gg4 #bjdj4 ,fuchs4 ,was soll ich nun aber denn studieren8 ,meph4 ,ihr knnt es mit analytischer ,geometrie probieren4 ,da wird der ,raum euch wohl dressiert1 ,in ,coordinaten eingeschnrt1 ,dass ihr nicht etwa auf gut ,glck ,von der ,figur gewinnt ein ,stck4 ,dann lehret man euch manchen ,tag1 ,dass1 was ihr sonst auf einen ,schlag ,construiertet im ,raume frei1 ,eine ,gleichung dazu ntig sei4 ,zwar war dem ,menschen zu seiner ,erbauung ,die dreidimensionale ,raumanschauung1 ,dass er sieht1 was um ihn passiert1 ,und die ,figuren sich construiert-- ,der ,analytiker tritt herein ,und beweist1 das knnte auch anders sein4 ,gleichungen1 die auf dem ,papiere stehn1 ,die msst' man auch knnen im ,raume sehn2 ,und knnte man's nicht construieren1 ,da msste man's anders definieren4 ,denn was man formt nach ,zahlengesetzen ,msst' uns auch geometrisch erletzen4 ,drum in den unendlich fernen beiden ,imaginren ,punkten mssen sich schneiden ,alle ,kreise fein suberlich1 ,auch ,parallelen1 die treffen sich1 ,und im ,raume kann man daneben ,allerlei ,krmmungsmasse erleben4 ,die ,formeln sind alle wahr und schn1 ,warum sollen sie nicht zu deuten gehn8 ,da preisen's die ,schler aller ,orten1 ,dass das ,gerade ist krumm geworden4 ,nicht-,euklidisch nennt's die ,geometrie1 ,spotted ihrer selbst1 und weiss nicht wie4 ,fuchs4 ,kann euch nicht eben ganz verstehn4 ,meph4 ,das soll den ,philosophen auch so gehn4 ,doch wenn ihr lernt alles reducieren ,und gehrig transformieren1 ,bis die ,formeln den ,sinn verlieren1 ,dann versteht ihr mathematish zu spekulieren4 --,,lasswitz1 ,,kurd4 ,der ,faust-,tragdie 7-n7ter ,teil2 ,zeitschrift fr den math-naturw4 ,unterricht1 ,bd4 #ad 7#ahhh71 p4 #caf4 ,fuchs4 ,to what study then should ,i myself apply8 ,meph4 ,begin with analytical geometry4 ,there all space is properly trained1 ,by cordinates well restrained1 ,that no one by some lucky assay ,carry some part of the figure away4 ,next thou'll be taught to realize1 ,constructions won't help thee to geometrize1 ,and the result of a free construction ,requires an equation for proper deduction4 ,three-dimensional space relation ,exists for human edification1 ,that he may see what about him transpires1 ,and construct such figures as he requires4 ,enters the analyst4 ,forthwith you see ,that all this might otherwise be4 ,equations1 written with pencil or pen1 ,must be visible in space1 and when ,difficulties in construction arise1 ,we need only define it otherwise4 ,for1 what is formed after laws arithmetic ,must also yield some delight geometric4 ,therefore we must not object ,that all circles intersect ,in the circular points at infinity4 ,and all parallels1 they declare1 ,if produced must meet somewhere4 ,so in space1 it can't be denied1 ,any old curvature may abide4 ,the formulas are all fine and true1 ,then why should they not have a meaning too8 ,pupils everywhere praise their fate ,that that now is crooked which once was straight4 ,non-,euclidean1 in fine derision1 ,is what it's called by the geometrician4 ,fuchs4 ,i do not fully follow thee4 ,meph4 ,no better does philosophy4 ,to master mathematical speculation1 ,carefully learn to reduce your equation ,by an adequate transformation ,till the formulas are devoid of interpretation4 ,,chapter ,,xxi ,,paradoxes ,,and ,,curiosities #baja4 ,the pseudomath is a person who handles mathematics as a monkey handles the razor4 ,the creature tried to shave himself as he had seen his master do2 but1 not having any notion of the angle at which the razor was to be held1 he cut his own throat4 ,he never tried it a second time1 poor animal6 but the pseudomath keeps on in his work1 proclaims himself clean shaved1 and all the rest of the world hairy4 ,the graphomath is a person who1 having no mathematics1 attempts to describe a mathematician4 ,novelists perform in this way3 even ,walter ,scott now and then burns his fingers4 ,his dreaming calculator1 ,davy ,ramsay1 swears 8by the bones of the immortal ,napier40 ,scott thought that the philomaths worshipped relics3 so they do in one sense4--,,de ,,morgan1 ,a4 ,budget of ,paradoxes 7,london1 #ahgb71 p4 #dgc4 #bajb4 ,proof requires a person who can give and a person who can receive4444 ,a blind man said1 ,as to the ,sun1 ,i'll take my ,bible oath there's none2 ,for if there had been one to show ,they would have shown it long ago4 ,how came he such a goose to be8 ,did he not know he couldn't see8 ,not he4 --,,de ,,morgan1 ,a4 ,budget of ,paradoxes 7,london1 #ahgb71 p4 #bfb4 #bajc4 ,mathematical research1 with all its wealth of hidden treasure1 is all too apt to yield nothing to our research3 for it is haunted by certain ignes fatui--delusive phantoms1 that float before us1 and seem so fair1 and are all but in our grasp1 so nearly that it never seems to need more than one step further1 and the prize shall be ours6 ,alas for him who has been turned aside from real research by one of these spectres--who has found a music in its mocking laughter--and who wastes his life and energy in the desperate chase6--,,dodgson1 ,c4 ,l4 ,a new ,theory of ,parallels 7,london1 #ahie71 ,introduction4 #bajd4 ,as lightning clears the air of impalpable vapours1 so an incisive paradox frees the human intelligence from the lethargic influence of latent and unsuspected assumptions4 ,paradox is the slayer of ,prejudice4--,,sylvester1 ,j4 ,j4 ,on a ,lady's ,fan etc4 ,collected ,mathematical ,papers1 ,vol4 #c1 p4 #cf4 #baje4 ,when a paradoxer parades capital letters and diagrams which are as good as ,newton's to all who know nothing about it1 some persons wonder why science does not rise and triturate the whole thing4 ,this is why3 all who are fit to read the refutation are satisfied already1 and can1 if they please1 detect the paradoxer for themselves4 ,those who are not fit to do this would not know the difference between the true answer and the new capitals and diagrams on which the delighted paradoxer would declare that he had crumbled the philosophers1 and not they him4 --,,de ,,morgan1 ,a4 ,a ,budget of ,paradoxes 7,london1 #ahgb71 p4 #dhd4 #bajf4 ,demonstrative reason never raises the cry of ,church in ,danger6 and it cannot have any ,dictionary of heresies except a ,budget of ,paradoxes4 ,mistaken claimants are left to ,time and his extinguisher1 with the approbation of all non-claimants3 there is no need of a succession of exposures4 ,time gets through the job in his own workmanlike manner4--,,de ,,morgan1 ,a4 ,a ,budget of ,paradoxes 7,london1 #ahgb71 p4 #dhe4 #bajg4 ,d',israeli speaks of the 8six follies of science10--the quadrature1 the duplication1 the perpetual motion1 the philosopher's stone1 magic1 and astrology4 ,he might as well have added the trisection1 to make the mystic number seven2 but had he done so1 he would still have been very lenient2 only seven follies in all science1 from mathematics to chemistry6 ,science might have said to such a judge--as convicts used to say who got seven years1 expecting it for life1 8,thank you1 my ,lord1 and may you sit there until they are over10--may the ,curiosities of ,literature outlive the ,follies of ,science6--,,de ,,morgan1 ,a4 ,a ,budget of ,paradoxes 7,london1 #ahgb71 p4 #ga4 #bajh4 ,montucla says1 speaking of ,france1 that he finds three notions prevalent among cyclometers3 #a4 ,that there is a large reward offered for success2 #b4 ,that the longitude problem depends on that success2 #c4 ,that the solution is the great end and object of geometry4 ,the same three notions are equally prevalent among the same class in ,england4 ,no reward has ever been offered by the government of either country4 ,the longitude problem in no way depends upon perfect solution2 existing approximations are sufficient to a point of accuracy far beyond what can be wanted4 ,and geometry1 content with what exists1 has long passed on to other matters4 ,sometimes a cyclometer persuades a skipper who has made land in the wrong place that the astronomers are at fault1 for using a wrong measure of the circle2 and the skipper thinks it a very comfortable solution6 ,and this is the utmost that the problem has to do with longitude4--,,de ,,morgan1 ,a4 ,a ,budget of ,paradoxes 7,london1 #ahgb71 p4 #if4 #baji4 ,gregory ,st4 ,vincent is the greatest of circle-squarers1 and his investigations led him into many truths3 he found the property of the arc of the hyperbola which led to ,napier's logarithms being called hyperbolic4 ,montucla says of him1 with sly truth1 that no one ever squared the circle with so much genius1 or1 excepting his principal object1 with so much success4 --,,de ,,morgan1 ,a4 ,a ,budget of ,paradoxes 7,london1 #ahgb71 p4 #gj4 #baaj4 ,when ,i reached geometry1 and became acquainted with the proposition the proof of which has been sought for centuries1 ,i felt irresistibly impelled to try my powers at its discovery4 ,you will consider me foolish if ,i confess that ,i am still earnestly of the opinion to have succeeded in my attempt4--,,bolzano1 ,,bernard4 ,selbstbiographie 7,wien1 #ahge71 p4 #ai4 #baaa4 ,the ,theory of ,parallels4 ,it is known that to complete the theory it is only necessary to demonstrate the following proposition1 which ,euclid assumed as an axiom3 ,prop4 ,if the sum of the interior angles ,,ecf and ,,dbc which two straight lines ,,ec and ,,db make with a third line ,,cp is less than two right angles1 the lines1 if sufficiently produced1 will intersect4 ,illustration3 ,a geometrical drawing of parallel lines and intersecting lines to accompany proof4 ,proof4 ,construct ,,pca equal to the supplement ,,pbd of ,,cbd1 and ,,ecf1 ,,fcg1 etc4 each equal to ,,ace1 so that ,,acf #b4;,,ace1 ,,acg #c4;,,ace1 etc4 ,then however small the angle ,,ace may be1 there exists some number n such that n4,,ace ,,ach will be equal to or greater than ,,acp4 ,again1 take ,,bi1 ,,il1 etc4 each equal to ,,cb1 and draw ,,ik1 ,,lm1 etc4 parallel to ,,bd1 then the figures ,,acbd1 ,,dbik1 ,,kilm1 etc4 are congruent1 and ,,acik #b4;,,abcd1 ,,aclm #c4;,,acbd1 etc4 ,take ,,acno n4,,acbd1 n having the same value as in the expression ,,ach n4,,ace1 then ,,acno is certainly less than ,,acp1 since ,,acno must be increased by ,,onp to be equal to ,,acp4 ,it follows that ,,acno is also less than ,,ach1 and by taking the nth part of each of these1 that ,,acbd is less than ,,ace4 ,but if ,,ace is greater than ,,acbd1 ,,ce and ,,bd must intersect1 for otherwise ,,ace would be a part of ,,acbd4 ,journal fr ,mathematik1 ,bd4 #b 7#ahcd71 p4 #aih4 #baab4 ,are you sure that it is impossible to trisect the angle by ,euclid8 ,i have not to lament a single hour thrown away on the attempt1 but fancy that it is rather a tact1 a feeling1 than a proof1 which makes us think that the thing cannot be done4 ,but would ,gauss's inscription of the regular polygon of seventeen sides have seemed1 a century ago1 much less an impossible thing1 by line and circle8--,,hamilton1 ,w4 ,r4 ,letter to ,de ,morgan 7#aheb74 #baac4 ,one of the most curious of these cases geometrical paradoxers was that of a student1 ,i am not sure but a graduate1 of the ,university of ,virginia1 who claimed that geometers were in error in assuming that a line had no thickness4 ,he published a school geometry based on his views1 which received the endorsement of a well-known ,new ,york school official and1 on the basis of this1 was actually endorsed1 or came very near being endorsed1 as a text-book in the public schools of ,new ,york4 --,,newcomb1 ,,simon4 ,the ,reminiscences of an ,astronomer 7,boston and ,new ,york1 #aijc71 p4 #chh4 #baad4 ,what distinguishes the straight line and circle more than anything else1 and properly separates them for the purpose of elementary geometry8 ,their self-similarity4 ,every inch of a straight line coincides with every other inch1 and off a circle with every other off the same circle4 ,where1 then1 did ,euclid fail8 ,in not introducing the third curve1 which has the same property--the screw4 ,the right line1 the circle1 the screw--the representations of translation1 rotation1 and the two combined--ought to have been the instruments of geometry4 ,with a screw we should never have heard of the impossibility of trisecting an angle1 squaring the circle1 etc4--,,de ,,morgan1 ,a4 ,quoted in ,graves' ,life of ,sir ,w4 ,r4 ,hamilton1 ,vol4 #c 7,new ,york1 #ahhi71 p4 #cdb4 #baae4 ,mad ,mathesis alone was unconfined1 ,too mad for mere material chains to bind1 ,now to pure space lifts her ecstatic stare1 ,now1 running round the circle1 finds it square4 --,,pope1 ,,alexander4 ,the ,dunciad1 ,bk4 #d1 lines #ca-cd4 #baaf4 ,or is't a tart idea1 to procure ,an edge1 and keep the practic soul in ure1 ,like that dear ,chymic dust1 or puzzling quadrature8 --,,quarles1 ,,philip4 ,quoted by ,de ,morgan3 ,budget of ,paradoxes 7,london1 #ahgb71 p4 #dcf4 #baag4 ,quale 'l geometra che tutto s' affige ,per misurar lo cerchio1 e non ritruova1 ,pensando qual principio ond' egli indige4 --,,dante4 ,paradise1 canto #cc1 lines #abb-abe4 ,as doth the expert geometer appear ,who seeks to square the circle1 and whose skill ,finds not the law with which his course to steer4#ab ,quoted in ,frankland's ,story of ,euclid 7,london1 #aijb71 p4 #aja4 #ab ,for another rendition of these same lines see #aheh4 #baah4 ,in ,mathematicks he was greater ,than ,tycho ,brahe1 or ,erra ,pater3 ,for he1 by ,geometrick scale1 ,could take the size of ,pots of ,ale2 ,resolve by ,signs and ,tangents streight1 ,if ,bread or ,butter wanted weight2 ,and wisely tell what hour o' th' day ,the ,clock doth strike1 by ,algebra4 --,,butler1 ,,samuel4 ,hudibras1 ,part #a1 canto #a1 lines #aai-abf4 #baai4 ,i have often been surprised that ,mathematics1 the quintessence of truth1 should have found admirers so few and so languid4 ,frequent considerations and minute scrutiny have at length unravelled the cause2 viz4 that though ,reason is feasted1 ,imagination is starved2 whilst ,reason is luxuriating in its proper ,paradise1 ,imagination is wearily travelling on a dreary desert4--,,coleridge1 ,,samuel4 ,a ,mathematical ,problem4 #babj4 ,at last we entered the palace1 and proceeded into the chamber of presence where ,i saw the king seated on his throne1 attended on each side by persons of prime quality4 ,before the throne1 was a large table filled with globes and spheres1 and mathematical instruments of all kinds4 ,his majesty took not the least notice of us1 although our entrance was not without sufficient noise1 by the concourse of all persons belonging to the court4 ,but he was then deep in a problem1 and we attended an hour1 before he could solve it4 ,there stood by him1 on each side1 a young page with flaps in their hands1 and when they saw he was at leisure1 one of them gently struck his mouth1 and the other his right ear2 at which he started like one awaked on the sudden1 and looking toward me and the company ,i was in1 recollected the occasion of our coming1 whereof he had been informed before4 ,he spake some words1 whereupon immediately a young man with a flap came to my side1 and flapt me gently on the right ear1 but ,i made signs1 as well as ,i could1 that ,i had no occasion for such an instrument2 which1 as ,i afterwards found1 gave his majesty1 and the whole court1 a very mean opinion of my understanding4 ,the king1 as far as ,i could conjecture1 asked me several questions1 and ,i addressed myself to him in all the languages ,i had4 ,when it was found1 that ,i could neither understand nor be understood1 ,i was conducted by his order to an apartment in his palace1 7this prince being distinguished above all his predecessors1 for his hospitality to strangers7 where two servants were appointed to attend me4 ,my dinner was brought1 and four persons of quality1 did me the honour to dine with me4 ,we had two courses of three dishes each4 ,in the first course1 there was a shoulder of mutton cut into an equilateral triangle1 a piece of beef into a rhomboides1 and a pudding into a cycloid4 ,the second course1 was1 two ducks trussed up in the form of fiddles2 sausages and puddings1 resembling flutes and haut-boys1 and a breast of veal in the shape of a harp4 ,the servants cut our bread into cones1 cylinders1 parallelograms1 and several other mathematical figures4--,,swift1 ,,jonathan4 ,gulliver's ,travels2 ,a ,voyage to ,laputa2 ,chap4 #b4 #baba4 ,those to whom the king had entrusted me1 observing how ill ,i was clad1 ordered a taylor to come next morning1 and take measure for a suit of cloaths4 ,this operator did his office after a different manner1 from those of his trade in ,europe4 ,he first took my altitude by a quadrant1 and then1 with rule and compasses1 described the dimensions and outlines of my whole body1 all which he entered upon paper2 and in six days1 brought my cloaths very ill made1 and quite out of shape1 by happening to mistake a figure in the calculation4 ,but my comfort was1 that ,i observed such accidents very frequent1 and little regarded4 --,,swift1 ,,jonathan4 ,gulliver's ,travels2 ,a ,voyage to ,laputa1 ,chap4 #b4 #babb4 ,the knowledge ,i had in mathematics1 gave me great assistance in acquiring their phraseology1 which depended much upon that science1 and music2 and in the latter ,i was not unskilled4 ,their ideas are perpetually conversant in lines and figures4 ,if they would1 for example1 praise the beauty of a woman1 or any other animal1 they describe it by rhombs1 circles1 parallelograms1 ellipses1 and other geometrical terms1 or by words of art drawn from music1 needless here to repeat4 ,i observed in the king's kitchen all sorts of mathematical and musical instruments1 after the figures of which1 they cut up the joints that were served to his majesty's table4--,,swift1 ,,jonathan4 ,gulliver's ,travels2 ,a ,voyage to ,laputa1 ,chap4 #b4 #babc4 ,i was at the mathematical school1 where the master taught his pupils1 after a method1 scarce imaginable to us in ,europe4 ,the propositions1 and demonstrations1 were fairly written on a thin wafer1 with ink composed of a cephalic tincture4 ,this1 the student was to swallow upon a fasting stomach1 and for three days following1 eat nothing but bread and water4 ,as the wafer digested1 the tincture mounted to his brain1 bearing the proposition along with it4 ,but the success has not hitherto been answerable1 partly by some error in the quantum or composition1 and partly by the perverseness of lads2 to whom this bolus is so nauseous1 that they generally steal aside1 and discharge it upwards1 before it can operate2 neither have they been yet persuaded to use so long an abstinence as the prescription requires4--,,swift1 ,,jonathan4 ,gulliver's ,travels2 ,a ,voyage to ,laputa1 ,chap4 #e4 #babd4 ,it is worth observing that some of those who disparage some branch of study in which they are deficient1 will often affect more contempt for it than they really feel4 ,and not unfrequently they will take pains to have it thought that they are themselves well versed in it1 or that they easily might be1 if they thought it worth while2--in short1 that it is not from hanging too high that the grapes are called sour4 ,thus1 ,swift1 in the person of ,gulliver1 represents himself1 while deriding the extravagant passion for ,mathematics among the ,laputians1 as being a good mathematician4 ,yet he betrays his utter ignorance1 by speaking 8of a pudding in the form of a cycloid30 evidently taking the cycloid for a figure1 instead of a line4 ,this may help to explain the difficulty he is said to have had in obtaining his ,degree4--,,whately1 ,r4 ,annotations to ,bacon's ,essays1 ,essay ,l4 #babe4 ,it is natural to think that an abstract science cannot be of much importance in the affairs of human life1 because it has omitted from its consideration everything of real interest4 ,it will be remembered that ,swift1 in his description of ,gulliver's voyage to ,laputa1 is of two minds on this point4 ,he describes the mathematicians of that country as silly and useless dreamers1 whose attention has to be awakened by flappers4 ,also1 the mathematical tailor measures his height by a quadrant1 and deduces his other dimensions by a rule and compasses1 producing a suit of very ill-fitting clothes4 ,on the other hand1 the mathematicians of ,laputa1 by their marvellous invention of the magnetic island floating in the air1 ruled the country and maintained their ascendency over their subjects4 ,swift1 indeed1 lived at a time peculiarly unsuited for gibes at contemporary mathematicians4 ,newton's ,principia had just been written1 one of the great forces which have transformed the modern world4 ,swift might just as well have laughed at an earthquake4 --,,whitehead1 ,a4 ,n4 ,an ,introduction to ,mathematics 7,new ,york1 #aiaa71 p4 #aj4 #babf4 ,illustration3 ,a geometrical drawing including square and four triangles to demonstrate a graphical proof of the theorem of ,pythagoras as described in the poem4 ,here ,i am as you may see a b - ab ,when two ,triangles on me stand ,square of hypothene is plann'd ,but if ,i stand on them instead1 ,the squares of both the sides are read4 --,,airy1 ,g4 ,b4 ,quoted in ,graves' ,life of ,sir ,w4 ,r4 ,hamilton1 ,vol4 #c 7,new ,york1 #ahhi71 p4 #ejb4 #babg4 #c4ada #eib #fec #ehi #gic #bch #dfb #fdc #chc #bgi 444 #c #a #d #a #e #i ,now ,i1 even ,i1 would celebrate #b #f #e #c #e ,in rhymes inapt1 the great #h #i #g #i ,immortal ,syracusan1 rivaled nevermore1 #c #b #c #h #d ,who in his wondrous lore1 #f #b #f ,passed on before1 #d #c #c #h #c #b #g #i ,left men his guidance how to circles mensurate4 --,,orr1 ,a4 ,c4 ,literary ,digest1 ,vol4 #cb 7#aijf71 p4 #hd4 #babh4 ,i take from a biographical dictionary the first five names of poets1 with their ages at death4 ,they are ,aagard1 died at #dh4 ,abeille1 8 8 #gf4 ,abulola1 8 8 #hd4 ,abunowas1 8 8 #dh4 ,accords1 8 8 #de4 ,these five ages have the following characters in common3-- #a4 ,the difference of the two digits composing the number divided by three1 leaves a remainder of one4 #b4 ,the first digit raised to the power indicated by the second1 and then divided by three1 leaves a remainder of one4 #c4 ,the sum of the prime factors of each age1 including one as a prime factor1 is divisible by three4--,,peirce1 ,c4 ,s4 ,a ,theory of ,probable ,inference2 ,studies in ,logic 7,boston1 #ahhc71 p4 #afc4 #babi4 ,in view of the fact that the offered prize for the solution of the problem of ,fermat's ,greater ,theorem is about #be1jjj and that lack of marginal space in his copy of ,diophantus was the reason given by ,fermat for not communicating his proof1 one might be tempted to wish that one could send credit for a dime back through the ages to ,fermat and thus secure this coveted prize1 if it actually existed4 ,this might1 however1 result more seriously than one would at first suppose2 for if ,fermat had bought on credit a dime's worth of paper even during the year of his death1 #affe1 and if this bill had been drawing compound interest at the rate of six per cent1 since that time1 the bill would now amount to more than seven times as much as the prize4 --,,miller1 ,g4 ,a4 ,some ,thoughts on ,modern ,mathematical ,research2 ,science1 ,vol4 #ce 7#aiab71 p4 #hha4 #bacj4 ,if the ,indians hadn't spent the #bd4 ,in #afbf ,peter ,minuit1 first governor of ,new ,netherland1 purchased ,manhattan ,island from the ,indians for about #bd4 ,the rate of interest on money is higher in new countries1 and gradually decreases as wealth accumulates4 ,within the present generation the legal rate in the state has fallen from #g to #f4 ,assume for simplicity a uniform rate of #g from #afbf to the present1 and suppose that the ,indians had put their #bd at interest at that rate 7banking facilities in ,new ,york being always taken for granted67 and had added the interest to the principal yearly4 ,what would be the amount now1 after #bhj years8 #bd 7#a4jg7bhj more than #d1jdb1jjj1jjj4 ,the latest tax assessment available at the time of writing gives the realty for the borough of ,manhattan as #c1hbj1ged4aha4 ,this is estimated to be #gh of the actual value1 making the actual value a little more than #d1hih1djj1jjj4 ,the amount of the ,indians' money would therefore be more than the present assessed valuation but less than the actual valuation4 --,,white1 ,w4 ,f4 ,a ,scrap-book of ,elementary ,mathematics 7,chicago1 #aijh71 pp4 #dg-dh4 #baca4 ,see ,mystery to ,mathematics fly6--,,pope1 ,,alexander4 ,the ,dunciad1 ,bk4 #d1 line #fdg4 #bacb4 ,the ,pythagoreans and ,platonists were carried further by this love of simplicity4 ,pythagoras1 by his skill in mathematics1 discovered that there can be no more than five regular solid figures1 terminated by plane surfaces which are all similar and equal2 to wit1 the tetrahedron1 the cube1 the octahedron1 the dodecahedron1 and the eicosihedron4 ,as nature works in the most simple and regular way1 he thought that all elementary bodies must have one or other of those regular figures2 and that the discovery of the properties and relations of the regular solids must be a key to open the mysteries of nature4 ,this notion of the ,pythagoreans and ,platonists has undoubtedly great beauty and simplicity4 ,accordingly it prevailed1 at least to the time of ,euclid4 ,he was a ,platonic philosopher1 and is said to have wrote all the books of his ,elements1 in order to discover the properties and relations of the five regular solids4 ,the ancient tradition of the intention of ,euclid in writing his elements1 is countenanced by the work itself4 ,for the last book of the elements treats of the regular solids1 and all the preceding are subservient to the last4--,,reid1 ,,thomas4 ,essays on the ,powers of the ,human ,mind 7,edinburgh1 #ahab71 ,vol4 #b1 p4 #djj4 #bacc4 ,in the ,timus of ,plato it is asserted that the particles of the various elements have the forms of these the regular solids4 ,fire has the ,pyramid2 ,earth has the ,cube2 ,water the ,octahedron2 ,air the ,icosahedron2 and the ,dodecahedron is the plan of the ,universe itself4 ,it was natural that when ,plato had learnt that other mathematical properties had a bearing upon the constitution of the ,universe1 he should suppose that the singular property of space1 which the existence of this limited and varied class of solids implied1 should have some corresponding property in the ,universe1 which exists in space4 --,,whewell1 ,w4 ,history of the ,inductive ,sciences1 #crd ,edition1 ,additions to ,bk4 #b4 #bacd4 ,the orbit of the earth is a circle3 round the sphere to which this circle belongs1 describe a dodecahedron2 the sphere including this will give the orbit of ,mars4 ,round ,mars describe a tetrahedron2 the circle including this will be the orbit of ,jupiter4 ,describe a cube round ,jupiter's orbit2 the circle including this will be the orbit of ,saturn4 ,now inscribe in the earth's orbit an icosahedron2 the circle inscribed in it will be the orbit of ,venus4 ,inscribe an octahedron in the orbit of ,venus2 the circle inscribed in it will be ,mercury's orbit4 ,this is the reason of the number of the planets4--,,kepler4 ,mysterium ,cosmographicum ,whewell4 #bace4 ,it will not be thought surprising that ,plato expected that ,astronomy1 when further advanced1 would be able to render an account of many things for which she has not accounted even to this day4 ,thus1 in the passage in the seventh ,book of the ,republic1 he says that the philosopher requires a reason for the proportion of the day to the month1 and the month to the year1 deeper and more substantial than mere observation can give4 ,yet ,astronomy has not yet shown us any reason why the proportion of the times of the earth's rotation on its axis1 the moon's revolution round the earth1 and the earth's revolution round the sun1 might not have been made by the ,creator quite different from what they are4 ,but in asking ,mathematical ,astronomy for reasons which she cannot give1 ,plato was only doing what a great astronomical discoverer1 ,kepler1 did at a later period4 ,one of the questions which ,kepler especially wished to have answered was1 why there are five planets1 and why at such particular distances from the sun8 ,and it is still more curious that he thought he had found the reason of these things1 in the relation of those five regular solids which ,plato was desirous of introducing into the philosophy of the universe4444 ,kepler regards the law which thus determines the number and magnitude of the planetary orbits by means of the five regular solids as a discovery no less remarkable and certain than the ,three ,laws which give his name its imperishable place in the history of astronomy4--,,whewell1 ,w4 ,history of the ,inductive ,sciences1 #crd ,edition1 ,additions to ,bk4 #c4 #bacf4 ,pythagorean philosophers 444 maintained that of two combatants1 he would conquer1 the sum of the numbers expressed by the characters of whose names exceeded the sum of those expressed by the other4 ,it was upon this principle that they explained the relative prowess and fate of the heroes in ,homer1 1 and 1 the sum of the numbers in whose names are #hfa1 #abbe1 and #abgf respectively4--,,peacock1 ,,george4 ,encyclopedia of ,pure ,mathematics 7,london1 #ahdg72 ,article 8,arithmetic10 sect4 #ch4 #bacg4 ,round numbers are always false4--,,johnson1 ,,samuel4 ,johnsoniana2 ,apothegms1 ,sentiment1 etc4 #bach4 ,numero deus impare gaudet ,god in number odd rejoices4 --,,virgil4 ,eclogue1 #h1 #gg4 #baci4 ,why is it that we entertain the belief that for every purpose odd numbers are the most effectual8--,,pliny4 ,natural ,history1 ,bk4 #bh1 chap4 #e4 #badj4 8,then here goes another10 says he1 8to make sure1 ,fore there's luck in odd numbers10 says ,rory ,o',moore4 --,,lover1 ,s4 ,rory ,o',moore4 #bada4 ,this is the third time2 ,i hope1 good luck lies in odd numbers4444 ,they say1 there is divinity in odd numbers1 either in nativity1 chance1 or death4--,,shakespeare4 ,the ,merry ,wives of ,windsor1 ,act #e1 scene #a4 #badb4 ,to add to golden numbers1 golden numbers4--,,decker1 ,,thomas4 ,patient ,grissell1 ,act #a1 scene #a4 #badc4 ,i've read that things inanimate have moved1 ,and1 as with living souls1 have been inform'd1 ,by magic numbers and persuasive sound4 --,,congreve1 ,,richard4 ,the ,morning ,bride1 ,act #a1 scene #a4 #badd4 444 the ,yancos on the ,amazon1 whose name for three is ,poettarrarorincoaroac1 of a length sufficiently formidable to justify the remark of ,la ,condamine3 ,heureusement pour ceux qui ont faire avec eux1 leur ,arithmetique ne va pas plus loin4--,,peacock1 ,,george4 ,encyclopedia of ,pure ,mathematics 7,london1 #ahdg72 ,article 8,arithmetic10 sect4 #cb4 #bade4 ,there are three principal sins1 avarice1 luxury1 and pride2 three sorts of satisfaction for sin1 fasting1 almsgiving1 and prayer2 three persons offended by sin1 ,god1 the sinner himself1 and his neighbour2 three witnesses in heaven1 ,pater1 verbum1 and spiritus sanctus2 three degrees of penitence1 contrition1 confession1 and satisfaction1 which ,dante has represented as the three steps of the ladder that lead to purgatory1 the first marble1 the second black and rugged stone1 and the third red porphyry4 ,there are three sacred orders in the church militant1 sub-diaconati1 diaconiti1 and presbyterati2 there are three parts1 not without mystery1 of the most sacred body made by the priest in the mass2 and three times he says ,agnus ,dei1 and three times1 ,sanctus2 and if we well consider all the devout acts of ,christian worship1 they are found in a ternary combination2 if we wish rightly to partake of the holy communion1 we must three times express our contrition1 ,domine non sum dignus2 but who can say more of the ternary number in a shorter compass1 than what the prophet says1 tu signaculum sanctae trinitatis4 ,there are three ,furies in the infernal regions2 three ,fates1 ,atropos1 ,lachesis1 and ,clotho4 ,there are three theological virtues3 ,fides1 spes1 and charitas4 ,tria sunt pericula mundi3 ,equum currere2 navigare1 et sub tyranno vivere4 ,there are three enemies of the soul3 the ,devil1 the world1 and the flesh4 ,there are three things which are of no esteem3 the strength of a porter1 the advice of a poor man1 and the beauty of a beautiful woman4 ,there are three vows of the ,minorite ,friars3 poverty1 obedience1 and chastity4 ,there are three terms in a continued proportion4 ,there are three ways in which we may commit sin3 corde1 ore1 ope4 ,three principal things in ,paradise3 glory1 riches1 and justice4 ,there are three things which are especially displeasing to ,god3 an avaricious rich man1 a proud poor man1 and a luxurious old man4 ,and all things1 in short1 are founded in three2 that is1 in number1 in weight1 and in measure4 --,,pacioli1 ,author of the first printed treatise on arithmetic4 ,quoted in ,encyclopedia of ,pure ,mathematics 7,london1 #ahdg72 ,article 8,arithmetic10 sect4 #ij4 #badf4 ,ah6 why1 ye ,gods1 should two and two make four8 --,,pope1 ,,alexander4 ,the ,dunciad1 ,bk4 #b1 line #bhe4 #badg4 ,by him who stampt ,the ,four upon the mind1-- ,the ,four1 the fount of nature's endless stream4 --,ascribed to ,,pythagoras4 ,quoted in ,whewell's ,history of the ,inductive ,sciences1 ,bk4 #d1 chap4 #c4 #badh4 ,along the skiey arch the goddess trode1 ,and sought ,harmonia's august abode2 ,the universal plan1 the mystic ,four1 ,defines the figure of the palace floor4 ,solid and square the ancient fabric stands1 ,raised by the labors of unnumbered hands4 --,,nonnus4 ,dionysiac1 #da1 #bge-bhj4 ,whewell4 #badi4 ,the number seventy-seven figures the abolition of all sins by baptism4444 ,the number ten signifies justice and beatitude1 resulting from the creature1 which makes seven with the ,trinity1 which is three3 therefore it is that ,god's commandments are ten in number4 ,the number eleven denotes sin1 because it transgresses ten4444 ,this number seventy-seven is the product of eleven1 figuring sin1 multiplied by seven1 and not by ten1 for seven is the number of the creature4 ,three represents the soul1 which is in some sort an image of ,divinity2 and four represents the body1 on account of its four qualities4444 --,,st4 ,,augustine4 ,sermon #da1 art4 #bc4 #baej4 ,heliodorus says that the ,nile is nothing else than the year1 founding his opinion on the fact that the numbers expressed by the letters 1 ,nile1 are in ,greek arithmetic1 #ej2 #e2 #aj2 #cj2 #gj2 #bjj2 and these figures make up together #cfe1 the number of days in the year4 ,littell's ,living ,age1 ,vol4 #aag1 p4 #chj4 #baea4 ,in treating #fff1 ,bungus ,petri ,bungi ,bergomatis ,numerorum mysteria1 ,bergamo1 #aeia a good ,catholic1 could not compliment the ,pope with it1 but he fixes it on ,martin ,luther with a little forcing4 ,if from ,a to ,i represent #a-i1 from ,k to ,s #aj-ij1 and from ,t to ,z #ajj-ejj1 we see-- ,m ,a ,r ,t ,i ,n ,l ,u ,t ,e ,r ,a #cj #a #hj #ajj #i #dj #bj #bjj #ajj #e #hj #a which gives #fff4 ,again in ,hebrew1 ,lulter ,hebraized form of ,luther does the same3-- #bjj #djj #cj #f #cj --,,de ,,morgan1 ,a4 ,budget of ,paradoxes 7,london1 #ahgb71 p4 #cg4 #baeb4 ,stifel1 the most acute and original of the early mathematicians of ,germany1 444 relates 444 that whilst a monk at ,esslingen in #aebj1 and when infected by the writings of ,luther1 he was reading in the library of his convent the #acth ,chapter of ,revelations1 it struck his mind that the ,beast must signify the ,pope1 ,leo ,x42 ,he then proceeded in pious hope to make the calculation of the sum of the numeral letters in ,leo decimus1 which he found to be ,m1 ,d1 ,c1 ,l1 ,v1 ,i2 the sum which these formed was too great by ,m1 and too little by ,x2 but he bethought him again1 that he has seen the name written ,leo ,x41 and that there were ten letters in ,leo decimus1 from either of which he could obtain the deficient number1 and by interpreting the ,m to mean mysterium1 he found the number required1 a discovery which gave him such unspeakable comfort1 that he believed that his interpretation must have been an immediate inspiration of ,god4 --,,peacock1 ,,george4 ,encyclopedia of ,pure ,mathematics 7,london1 #ahdg72 ,article 8,arithmetic10 sect4 #hi4 #baec4 ,perhaps the best anagram ever made is that by ,dr4 ,burney on ,horatio ,nelson1 so happily transformed into the ,latin sentence so truthful of the great admiral1 ,honor est a ,nilo4 ,reading this1 one is almost persuaded that the hit contained in it has a meaning provided by providence or fate4 ,this is also amusingly illustrated in the case of the ,frenchman ,andr ,pujom1 who1 using j as i1 found in his name the anagram1 ,pendu ,riom4 ,riom being the seat of justice for the province of ,auvergne1 the poor fellow1 impelled by a sort of infatuation1 actually committed a capital offence in that province1 and was hanged at ,riom1 that the anagram might be fulfilled4 ,new ,american ,cyclopedia1 ,vol4 #a2 ,article 8,anagram40 #baed4 ,the most remarkable pseudonym of transposed names adopted by authors is the name of 8,voltaire10 which the celebrated philosopher assumed instead of his family name1 8,franois ,marie ,arouet10 and which is now generally allowed to be an anagram of 8,arouet1 l4 j410 that is1 ,arouet the younger4 ,encyclopedia ,britannica1 #aath ,edition2 ,article 8,anagram40 #baee4 ,perhaps the most beautiful anagram that has ever been composed is by ,jablonsky1 a former rector of the school at ,lissa4 ,the occasion was the following3 ,when while a young man king ,stanislaus of ,poland returned from a journey1 the whole house of ,lescinsky assembled to welcome the family heir4 ,on this occasion ,jablonsky arranged for a school program1 the closing number of which consisted of a ballet by thirteen pupils impersonating youthful heroes4 ,each of them carried a shield on which appeared in gold one of the letters of the words ,domus ,lescinia4 ,at the end of the first dance the children were so arranged that the letters on their shields spelled the words ,domus ,lescinia4 ,at the end of the second dance they read3 ades incolumis 7sound thou art here74 ,after the third3 omnis es lucida 7wholly brilliant art thou72 after the fourth3 lucida sis omen 7bright be the omen74 ,then3 mane sidus loci 7remain our country's star72 and again3 sis columna ,dei 7be a column of ,god72 and finally3 ,i6 scande solium 7,proceed1 ascend the throne74 ,this last was the more beautiful since it proved a true prophecy4 ,even more artificial are the anagrams which transform one verse into another4 ,thus an ,italian scholar beheld in a dream the line from ,horace3 ,grata superveniet1 quae non sperabitur1 hora4 ,this a friend changed to the anagram3 ,est ventura ,rhosina parataque nubere pigro4 ,this induced the scholar1 though an old man1 to marry an unknown lady by the name of ,rosina4--,,heis1 ,,eduard4 ,algebraische ,aufgaben 7,kln1 #ahih71 p4 #cca4 #baef4 ,the following verses read the same whether read forward or backward3-- ,aspice6 nam raro mittit timor arma1 nec ipsa ,si se mente reget1 non tegeret ,nemesis2#ac also1 ,sator ,arepo tenet opera rotas4 --,,heis1 ,,eduard4 ,algebraische ,aufgaben 7,kln1 #ahih71 p4 #cbh4 #ac ,the beginning of a poem which ,johannes a ,lasco wrote on the count ,karl von ,sdermanland4 #baeg4 ,there is a certain spiral of a peculiar form on which a point may have been approaching for centuries the center1 and have nearly reached it1 before we discover that its rate of approach is accelerated4 ,the first thought of the observer1 on seeing the acceleration1 would be to say that it would reach the center sooner than he had before supposed4 ,but as the point comes near the center it suddenly1 although still moving under the same simple law as from the beginning1 makes a very short turn upon its path and flies off rapidly almost in a straight line1 out to an infinite distance4 ,this illustrates that apparent breach of continuity which we sometimes find in a natural law2 that apparently sudden change of character which we sometimes see in man4--,,hill1 ,,thomas4 ,uses of ,mathesis2 ,bibliotheca ,sacra1 ,vol4 #cb1 p4 #eba4 #baeh4 ,one of the most remarkable of ,babbage's illustrations of miracles has never had the consideration in the popular mind which it deserves2 the illustration drawn from the existence of isolated points fulfilling the equation of a curve4444 ,there are definitions of curves which describe not only the positions of every point in a certain curve1 but also of one or more perfectly isolated points2 and if we should attempt to get by induction the definition1 from the observation of the points on the curve1 we might fail altogether to include these isolated points2 which1 nevertheless1 although standing alone1 as miracles to the observer of the course of the points in the curve1 are nevertheless rigorously included in the law of the curve4--,,hill1 ,,thomas4 ,uses of ,mathesis2 ,bibliotheca ,sacra1 ,vol4 #cb1 p4 #eaf4 #baei4 ,pure mathematics is the magician's real wand4--,,novalis4 ,schriften1 ,zweiter ,teil 7,berlin1 #aija71 p4 #bbc4 #bafj4 ,miracles1 considered as antinatural facts1 are amathematical1 but there are no miracles in this sense1 and those so called may be comprehended by means of mathematics1 for to mathematics nothing is miraculous4--,,novalis4 ,schriften1 ,zweiter ,teil 7,berlin1 #aiaa71 p4 #bbb4 ,,index ,black-faced numbers refer to authors ,abbreviations3--m4 mathematics1 math4 mathematical1 math'n4 mathematician4 ,abbott1 #ajja4 ,abstract method1 ,development of1 #gbi4 ,abstract nature of m41 ,reason for1 #fch4 ,abstract reasoning1 ,objection to1 #aida4 ,abstractness1 math41 ,compared with logical1 #acjd4 ,adams1 ,henry1 ,m4 and history1 #aeii4 ,math'ns practice freedom1 #bjh1 #hje4 ,adams1 ,john1 ,method in m41 #bbf4 ,aeneid1 ,euler's knowledge of1 #hei4 ,aeschylus4 ,on number1 #afjf4 ,aim in teaching m41 #eja-ejh1 #eag1 #hdd4 ,airy1 ,pythagorean theorem1 #babf4 ,akenside1 #aecb4 ,alexander1 #ija1 #ijb4 ,algebra1 ,chapter ,,xvii4 ,definitions of1 #aaj1 #agad1 #agae4 ,problems in1 #cbj1 #ecj1 #agch4 ,of use to grown men1 #dbe4 ,and geometry1 #ebe-ebg1 #afaj1 #agjg4 ,advantages of1 #agja1 #agjc1 #agje4 ,laws of1 #agjh-agaj4 ,as an art1 #agaa4 ,review of1 #agac4 ,designations of1 #agag4 ,origin of1 #agcf4 ,burlesque on modern1 #agda4 ,hume on1 #ahfc4 ,algebraic notation1 value of1 #abac1 #abad4 ,algebraic treatises1 ,how to read1 #fja4 ,amusements in m41 #ijd1 #ije4 ,anagrams1 ,on ,de ,morgan1 #idg4 ,on ,domus ,lescinia1 #baee4 ,on ,flamsteed1 #ifh4 ,on ,macaulay1 #iif4 ,on ,nelson1 #baec4 ,on ,newton1 #ajbh4 ,on ,voltaire1 #baed4 ,analysis1 ,invigorates the faculty of resolution1 #daf4 ,relation of geometry to1 #aica4 ,analytical geometry1 #ahhi1 #ahij1 #ahic4 ,method of1 #caj4 ,importance of1 #idi4 ,burlesque on1 #bjdj4 ,ancient geometry1 ,characteristics of1 #gab1 #gad4 ,compared with modern1 #agaa-agaf4 ,method of1 #adbe1 #ahgc-ahge4 ,ancients1 ,m4 among the1 #cba4 ,anecdotes1 ,chapters1 ,,ix1 ,x4 ,anger1 ,m4 destroys predisposition to1 #deh4 ,angling like m41 #gci4 ,anglo-,danes1 ,aptitude for m41 #hcf4 ,anglo-,saxons1 ,aptitude for m41 #hcg4 ,newton as representative of1 #ajad4 ,anonymous1 ,song of the screw1 #ahid4 ,appolonius1 #gab1 #gad4 ,approximate m41 ,why not sufficient1 #aeah4 ,aptitude for m41 #eji1 #eaj1 #ebj1 #hcf-hch1 #igf1 #afag4 ,arabic notation1 #afad4 ,arago1 ,m4 the enemy of scientific romances1 #bfg4 ,euler1 8analysis incarnate10 #ifa4 ,euler as a computer1 #ifb4 ,on ,kepler's discovery1 #ihb4 ,newton's efforts superhuman1 #ajjf4 ,on probabilities1 #aeia4 ,geometry as an instrument1 #ahfh4 ,arbuthnot1 ,m4 frees from prejudice1 credulity and superstition1 #ddi4 ,m4 the friend of religion1 #deh4 ,m4 compared to music1 #aaab4 ,on math1 reasoning1 #aejc4 ,archimedes1 ,his machines1 #ijc1 #ijd4 ,estimate of math1 appliances1 #ijd-ijf1 #ijh4 ,wordsworth on1 #ijf4 ,schiller on1 #ijg4 ,and engineering1 #ijh4 ,death of1 #iji4 ,his tomb1 #iaj4 ,compared with ,newton1 #iaa4 ,character of his work1 #iab1 #iac4 ,applied m41 #acab4 ,architecture and m41 #bgf4 ,archytas1 #ijd4 ,and ,plato1 #adbg4 ,aristippus the ,cyrenaic1 #hde4 ,aristotle1 #iad4 ,on relation of m4 to esthetics1 #cah4 ,arithmetic1 ,chapter ,,xvi4 ,definitions of1 #ajf1 #aaj1 #afaa1 #afab1 #agad4 ,emerson on advantage of study of1 #djh4 ,problems in1 #ebh4 ,a master-key1 #aega4 ,based on concept of time1 #afac4 ,method of teaching1 #afah4 ,purpose of teaching1 #ded1 #afbd4 ,as logic1 #afbd1 #afbe4 ,the queen of m41 #afdb4 ,higher1 #agee4 ,hume on1 #ahfc4 ,arithmetical theorems1 #afci4 ,art1 ,m4 as a fine1 ,chapter ,,xi ,arts1 ,m4 and the1 #aefh-aegj1 #aegc4 ,astronomy and m41 #aeed1 #aeei1 #aefb-aefg4 8,auge et impera410 #fca4 ,authority in science1 #aebh4 ,axioms1 #eah1 #bjae4 ,in geometry1 #ahab1 #bjjd1 #bjjf4 ,def4 in disguise1 #bjje4 ,euclid's1 #bjjg-bjaj1 #bjad4 ,nature of1 #bjab4 ,proofs of1 #bjac4 ,and the idea of space1 #bjjd4 ,babbage1 #ibc4 ,bacon1 ,lord1 ,classification of m41 #ajf4 ,m4 makes men subtile1 #bdh4 ,view of m41 #caf1 #iae1 #iaf4 ,m4 held in high esteem by ancients1 #cba4 ,on the generalizing power of m41 #cbg4 ,on the value of math1 studies1 #daj4 ,m4 develops concentration of mind1 #daa4 ,m4 cures distraction of mind1 #dab4 ,m4 essential to study of nature1 #dcf4 ,his view of m41 #iae1 #iaf4 ,his knowledge of m41 #iag1 #iah4 ,m4 and logic1 #acaj4 ,growth of m41 #aeaa4 ,bacon1 ,roger1 ,neglect of m4 works injury to all science1 #caj4 ,on the value of m41 #aedg4 ,bain1 ,importance of m4 in education1 #ddb4 ,on the charm of the study of m41 #dec4 ,m4 and science teaching1 #ebb4 ,teaching of arithmetic1 #afah4 ,ball1 ,r4 ,s41 #bjaj4 ,ball1 ,w4 ,w4 ,r41 ,on ,babbage1 #ibc4 ,on ,demoivre's death1 #idd4 ,de ,morgan and the actuary1 #ide4 ,gauss as astronomer1 #iga4 ,laplace's 8,it is easy to see40 #ihf4 ,lagrange1 ,laplace and ,gauss contrasted1 #iic4 ,newton's interest in chemistry and theology1 #ajae4 ,on ,newton's method of work1 #ajbf4 ,on ,newton's discovery of the calculus1 #ajbg4 ,gauss's estimate of ,newton1 #ajbi4 ,m4 and philosophy1 #adag4 ,advance in physics1 #aecj4 ,plato on geometry1 #ahjd4 ,notation of the calculus1 #aijd4 ,barnett1 ,m4 the type of perfect reasoning1 #cjg4 ,barrow1 ,on the method of m41 #bac1 #bbg4 ,eulogy of m41 #ccj4 ,m4 as a discipline of the mind1 #djb4 ,m4 and eloquence1 #hcj4 ,philosophy and m41 #adcj4 ,uses of m41 #aegb4 ,on surd numbers1 #agbh4 ,euclid's definition of proportion1 #ahce4 ,beattie1 #adca4 ,beauty of m41 #dec1 #hbd1 #abjh4 ,consists in simplicity1 #bdb1 #cae4 ,sylvester on1 #aaja4 ,russell on1 #aajd4 ,young on1 #aaaj4 ,kummer on1 #aaaa4 ,white on1 #aaai4 ,and truth1 #aaad4 ,boltzmann on1 #aaaf4 ,beltrami1 ,on reading of the masters1 #fad4 ,berkeley1 ,on geometry as logic1 #dbh4 ,on math4 symbols1 #abad4 ,on fluxions1 #aiae1 #aidb-aidd4 ,on infinite divisibility1 #aide4 ,bernoulli1 ,daniel1 #iai4 ,bernoulli1 ,james1 ,legend for his tomb1 #ibj1 #ibb4 ,computation of sum of tenth powers of numbers1 #iba4 ,discussion of logarithmic spiral1 #ibb4 ,berthelot1 ,m4 inspires respect for truth1 #dch4 ,bija ,ganita1 ,solution of problems1 #agci4 ,billingsley1 ,m4 beautifies the mind1 #cai4 ,binary arithmetic1 #iia4 ,biology and m41 #aegi-aeha4 ,biot1 ,laplace's 8,it is easy to see10 #ihf4 ,bcher1 ,m4 likened to painting1 #aajc4 ,interrelation of m4 and logic1 #acac4 ,geometry as a natural science1 #ahff4 ,boerne1 ,on ,pythagoras1 #ahee4 ,bois-,reymond1 ,on the analytic method1 #ahic4 ,natural selection and the calculus1 #aiba4 ,boltzmann1 ,on beauty in m41 #aaaf4 ,bolyai1 ,janos1 ,duel with officers1 #ibd4 ,universal language1 #ibe4 ,science absolute of space1 #ibf4 ,bolyai1 ,wolfgang1 #ibg4 ,on ,gauss1 #igb4 ,bolzano1 #ibh4 ,cured by ,euclid1 #ibi4 ,parallel axiom1 #baaj4 ,book-keeping1 ,importance of the art of1 #aega4 ,boole1 ,m4 ,e4 #gai4 ,boole's ,laws of ,thought1 #acah4 ,borda-,demoulins1 ,philosophy and m41 #adje4 ,boswell1 #iha4 ,bowditch1 ,on ,laplace's 8,thus it plainly appears10 #ihe4 ,boyle1 ,usefulness of m4 to physics1 #dcg4 ,m4 and science1 #aeac1 #aecc4 ,ignorance of m41 #aegg4 ,m4 and physiology1 #aehb4 ,wings of m41 #afbf4 ,advantages of algebra1 #agjc4 ,brahmagupta1 ,estimate of m41 #cbj4 ,brewster1 ,on ,euler's knowledge of the ,aeneid1 #iei4 ,on ,euler as a computer1 #ifc4 ,on ,newton's fame1 #ajjb4 ,brougham1 #abjb4 ,buckle1 ,on geometry1 #ahaj1 #ahcg4 ,burke1 ,on the value of m41 #ddg4 ,burkhardt1 ,on discovery in m41 #fah4 ,on universal symbolism1 #abba4 ,butler1 ,n4 ,m41 ,m4 demonstrates the supremacy of the human reason1 #cji4 ,m4 the most astounding intellectual creation1 #gjg4 ,geometry before algebra1 #ahga4 ,butler1 ,samuel1 #baah4 ,byerly1 ,on hyperbolic functions1 #aibi4 ,cajori1 ,on the value of the history of m41 #fae4 ,on ,bolyai1 #ibg4 ,cayley's view of ,euclid1 #icf4 ,on the extent of ,euler's work1 #ifj4 ,on ,euler's math4 power1 #ifd4 ,on the ,darmstaetter prize1 #ifg4 ,on ,sylvester's first class at ,johns ,hopkins1 #ajca4 ,on music and m4 among the ,pythagoreans1 #aacj4 ,on the greatest achievement of the ,hindoos1 #afae4 ,on modern calculation1 #afad4 ,on review in arithmetic1 #agac4 ,on ,indian m41 #agcg4 ,on the characteristics of ancient geometry1 #ahgc4 ,on ,napier's rule1 #ahhh4 ,calculating machines1 #afda4 ,calculation1 ,importance of1 #fjb4 ,not the sole object of m41 #bfh4 ,calculus1 ,chapter ,,xix4 ,foundation of #bec4 ,as a method1 #cji4 ,may be taught at an early age1 #eai1 #aiag1 #aiah4 ,cambridge m41 #hcf1 #abaj4 ,cantor1 ,on freedom in m41 #bje1 #bjg4 ,on the character of ,gauss's writing1 #ige4 ,zeno's problem1 #aich4 ,on the infinite1 #aieb4 ,carlisle life tables1 #idf4 ,carnot1 ,on limiting ratios1 #aijh4 ,on the infinitesimal method1 #aijg4 ,carson1 ,value of geometrical training1 #ahda4 ,cartesian method1 #ahhi1 #ahij4 ,carus1 ,estimate of m41 #cbf4 ,m4 reveals supernatural ,god1 #dfj4 ,number and nature1 #afjc4 ,zero and infinity1 #aidh4 ,non-euclidean geometry1 #bjaf4 ,cathedral1 8,petrified mathematics10 #aaaj4 ,causation in m41 #bea1 #bed4 ,cayley1 ,advantage of modern geometry over ancient1 #gaa4 ,on the imaginary1 #gbb4 ,sylvester on1 #icj4 ,noether on1 #ica4 ,his style1 #icb4 ,forsyth on1 #icb-icd4 ,his method1 #icc4 ,compared with ,euler1 #icd4 ,hermite on1 #ice4 ,his view of ,euclid1 #icf4 ,his estimate of quaternions1 #icg4 ,m4 and philosophy1 #adbj4 ,certainty of m41 #bbb1 #addj-addb1 #afbh1 #ahfc4 ,chamisso1 ,pythagorean theorem1 #ahef4 ,chancellor1 ,m4 develops observation1 imagination and reason1 #dcc4 ,chapman1 ,different aspects of m41 #bfe4 ,characteristics of m41 #bbe1 #bbi1 #bdg1 #bfc4 ,characteristics of modern m41 #gbj1 #gbd-gbi4 ,charm in m41 #aaae1 #afdj1 #ahdh4 ,chasles1 ,advantage of modern geometry over ancient1 #gab4 ,checks in m41 #bcj4 ,chemistry and m41 #aebj1 #aefj1 #aefa1 #agej4 ,chess1 ,m4 like1 #hdj4 ,chrystal1 ,definition of m41 #aac4 ,definition of quantity1 #aae4 ,on problem solving1 #eca4 ,on modern text-books1 #ecc4 ,how to read m41 #fjg4 ,his algebra1 #fce4 ,on ,bernoulli's numbers1 #iba4 ,on math4 versus logical abstractness1 #acjd4 ,rules of algebra1 #agaj4 ,on universal arithmetic1 #agag4 ,on ,horner's method1 #agdd4 ,on probabilities1 #aifg4 ,cicero1 ,decadence of geometry among ,romans1 #ahjg4 ,circle1 ,properties of1 #aheb1 #aheg4 ,circle-squarers1 #bajh1 #baji4 ,clarke1 ,descriptive geometry1 #ahhb4 ,classic problems1 ,hilbert on1 #fbg4 ,clebsch1 ,on math4 research1 #fdd4 ,clifford1 ,on direct usefulness of math4 results1 #feb4 ,correspondence the central idea of modern m41 #gbf4 ,his vision1 #ich4 ,his method1 #ici4 ,his knowledge of languages1 #idj4 ,his physical strength1 #ida4 ,on ,helmholtz1 #igi4 ,on m4 and mineralogy1 #aeeh4 ,on algebra and good ,english1 #agab4 ,euclid the encouragement and guide of scientific thought1 #ahbj4 ,euclid the inspiration and aspiration of scientific thought1 #ahba4 ,on geometry for girls1 #ahdb4 ,on ,euclid's axioms1 #bjae4 ,on non-,euclidean geometry1 #bjbb ,colburn1 #ifg4 ,coleridge1 ,on problems in m41 #ecd4 ,proposition1 gentle maid1 #adai4 ,m4 the quintessence of truth1 #bjai4 ,colton1 ,on the effect of math4 training1 #dag4 ,commensurable numbers1 #aiff4 ,commerce and m41 #aega4 ,committee of ,ten1 ,on figures in geometry1 #ebd4 ,on projective geometry1 #ahgf4 ,common sense1 ,m4 the etherealization of1 #cab4 ,computation1 ,not m41 #eae4 ,and m41 #haj4 ,not concerned with significance of numbers1 #afda4 ,comte1 ,on the object of m41 #ajc4 ,on the business of concrete m41 #ajd4 ,m4 the indispensable basis of all education1 #ccd4 ,mill on1 #idb4 ,hamilton on1 #idc4 ,m4 and logic1 #acjh1 #acad1 #acbe4 ,on ,kant's view of m41 #adcg4 ,estimate of m41 #aejd4 ,m4 essential to scientific education1 #aeje4 ,m4 and natural philosophy1 #aejf4 ,m4 and physics1 #aece1 #aeea4 ,m4 and science1 #aecf4 ,m4 and biology1 #aegh1 #aehj1 #aeha4 ,m4 and social science1 #aehg4 ,every inquiry reducible to a question of number1 #afjb4 ,definition of algebra and arithmetic1 #agad4 ,geometry a natural science1 #ahac4 ,ancient and modern methods1 #ahge4 ,on the graphic method1 #ahha4 ,on descriptive geometry1 #ahhc4 ,mill's estimate of1 #aijc4 ,congreve1 #badc4 ,congruence1 ,symbol of1 #afdf4 ,conic sections1 #feh1 #ffj1 #aeda1 #aedb4 ,conjecture1 ,m4 free from1 #bcd4 ,contingent truths1 #aiff4 ,controversies in m41 #bae1 #bdc1 #ahei4 ,correlation in m41 #ebe-ebg1 #agjg1 #agaj4 ,correspondence1 ,concept of1 #gbe1 #gbf4 ,coulomb1 #aeaf4 ,counting1 ,every problem can be solved by1 #afja4 ,cournot1 ,on the object of m41 #bfh4 ,on algebraic notation1 #abac4 ,advantage of math1 notation1 #abbj4 ,craig1 ,on the origin of a new science1 #fdf4 ,credulity1 ,m4 frees mind from1 #dej4 ,cremona1 ,on ,english text-books1 #fji4 ,crofton1 ,on value of probabilities1 #aeij4 ,on probabilities1 #aieb1 #aigj1aigb4 ,cromwell1 ,on m4 and public service1 #cbh4 ,curiosities1 ,chapter ,,xxi4 ,curtius1 ,m4 and philosophy1 #adji4 ,curve1 ,definition of1 #aibg4 ,cyclometers1 ,notions of1 #bajh4 ,cyclotomy depends on number theory1 #afdg4 ,d',alembert1 ,on rigor in m41 #ecf4 ,geometry as logic1 #acaa4 ,algebra is generous1 #agjb4 ,geometrical versus physical truths1 #ahji4 ,standards in m41 #ahea4 ,dante1 #aheh1 #baag4 ,darmstaetter prize1 #babi4 ,davis1 ,on ,sylvester's method1 #ajce4 ,m4 and science1 #aeaj4 ,on probability1 #aifh4 ,decimal fractions1 #abag1 #afad4 ,decker1 #badb4 ,dedekind1 ,zeno's ,problem1 #aich4 ,deduction1 ,why necessary1 #bai4 ,m4 based on1 #bbd4 ,and ,intuition1 #adac4 ,dee1 ,on the nature of m41 #bfa4 ,definitions of m41 ,chapter ,i4 ,also #bjje4 ,democritus1 #cba4 ,demoivre1 ,his death1 #idd4 ,demonstrations1 ,locke on1 #bcf4 ,outside of m41 #acab4 ,in m41 #adbc4 ,de ,morgan1 ,imagination in m41 #beh4 ,m4 as an exercise in reasoning1 #dcj4 ,on difficulties in m41 #eba4 ,on correlation in m41 #ebe4 ,on extempore lectures1 #edj4 ,on reading algebraic works1 #fja4 ,on numerical calculations1 #fjb4 ,on practice problems1 #fjc4 ,on the value of the history of m41 #fae1 #faf4 ,on math'ns41 #hab4 ,on ,bacon's knowledge of m41 #iah4 ,and the actuary1 #ide4 ,on life tables1 #idf4 ,anagrams' on 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,kant1 ,on the a priori nature of m41 #acj4 ,m4 follows the safe way of science1 #bja4 ,on the origin of scientific m41 #bja4 ,on m4 in primary education1 #dca4 ,m4 the embarrassment of metaphysics1 #adjb4 ,his view of m41 #adcf1 #adcg4 ,on the difference between m4 and philosophy1 #adcf4 ,on m4 and science1 #aejh4 ,esthetic elements in m41 #aheb1 #ahec4 ,doctrine of time1 #bjja4 ,doctrine of space1 #bjjc4 ,karpinsky1 ,m4 and efficiency1 #afgc4 ,kasner1 8,divinez avant de demontrer10 #fcj4 ,on modern geometry1 #gaj4 ,kelland1 ,on ,euclid's elements1 #ahag4 ,kelvin1 ,lord1 ,see ,william ,thomson4 ,kepler1 ,his method1 #ihb4 ,planetary orbits and the regular solids1 #bacd4 ,keyser1 ,definition of m41 #acb4 ,three characteristics of m41 #bbe4 ,on the method of m41 #bdd4 ,on ratiocination1 #bdf4 ,m4 not detached from life1 #bgc4 ,on the spirit of mathesis1 #bgf4 ,computation not m41 #eae4 ,math1 output of present day1 #gjb4 ,modern theory of functions1 #gji4 ,m4 and journalism1 #gca4 ,difficulty of m41 #gce4 ,m4 appeals to whole mind1 #hae4 ,endowment of math'ns1 #hah4 ,math'ns in public service1 #hbc4 ,the aim of the math'n1 #hdd4 ,on ,bolzano1 #ibi4 ,on ,lie1 #iib4 ,on symbolic logic1 #acba4 ,on the emancipation of logic1 #acbb4 ,on the ,principia ,mathematica1 #acbf4 ,on invariants1 #agbh4 ,on invariance1 #agbi4 ,on the notion of group1 #agea4 ,on the elements of ,euclid1 #ahbd4 ,on protective geometry1 #ahhj4 ,definition of infinite assemblage1 #aifj4 ,on the infinite1 #aifa4 ,on non-euclidean geometry1 #bjce4 ,on hyper-space1 #bjcg1 #bjch4 ,khulasat-al-,hisab1 ,problems1 #agch4 ,kipling1 #afcc4 ,kirchhoff1 ,artistic nature of his works1 #aaaf4 ,klein1 ,definition of m41 #abc4 ,m4 a versatile science1 #bfd4 ,aim in teaching1 #ejg1 #eag4 ,analysts versus synthesists1 #fea4 ,on theory and practice1 #ffa4 ,math1 aptitudes of various races1 #hch4 ,lie's final aim1 #iic4 ,lie's genius1 #iid4 ,on m4 and science1 #aebj4 ,famous aphorisms1 #afce4 ,calculating machines1 #afda4 ,calculus for high schools1 #aiah4 ,on differential equations1 #aibf4 ,definition of a curve1 #aibg4 ,on axioms of geometry1 #bjjf4 ,on the parallel axiom1 #bjji4 ,on non-euclidean geometry1 #bjag1 #bjba4 ,on hyper-space1 #bjcj4 ,kronecker1 ,on the greatness of ,gauss1 #igc4 ,god made integers etc41 #afcd4 ,kummer1 ,on ,dirichlet1 #igg4 ,on beauty in m41 #aaaa4 ,la,faille1 ,mathesis few know1 #ahgj4 ,lagrange1 ,on correlation of algebra and geometry1 #ebg4 ,his style and method1 #ihc4 ,and the parallel axiom1 #ihd4 ,on ,newton1 #ajaa4 ,wings of m41 #afjd4 ,union of algebra and geometry1 #agjg4 ,on the infinitesimal method1 #aijf4 ,lalande1 ,m4 in ,french army1 #cad4 ,langley1 ,m4 in ,prussia1 #eac4 ,lampe1 ,on division of labor in m41 #fcb4 ,on ,weierstrass1 #ajdi4 ,weierstrass and ,sylvester1 #ajej4 ,qualities common to math'ns and artists1 #aaac4 ,charm of m41 #aaae4 ,golden age of art and m4 coincident1 #aacd4 ,language1 ,chapter ,,xii4 ,see also #caa1 #dai1 #ddc1 #aebc1 #ahjd1 #ahhi4 ,laplace1 ,on instruction in m41 #bbj4 ,his style and method1 #ihc4 8,thus it plainly appears10 #ihe1 #ihf4 ,emerson on1 #ajjc4 ,on ,leibnitz1 #iia4 ,on the language of analysis1 #abbb4 ,on m4 and nature1 #aebe4 ,on the origin of the calculus1 #aijb4 ,on the exactitude of the differential calculus1 #aiaj4 ,the universe in a single formula1 #aibj4 ,on probability1 #aifc1 #aifi1 #aiga4 ,laputa1 ,math'ns of1 #babj-babb1 ,math4 school of1 #babc4 ,lasswitz1 ,on modern algebra1 #agda4 ,on function theory1 #aicd4 ,on non-euclidean geometry1 #bjdj4 ,latin squares1 #beb4 ,latta1 ,on ,leibnitz's logical calculus1 #acag4 ,law and m41 #aeig1 #aeih4 ,laws of thought1 #gai1 #acah4 ,leadership1 ,m4 as training for1 #cag4 ,lecture1 ,preparation of1 #edj4 ,lefevre1 ,m4 hateful to weak minds1 #gcc4 ,logic and m41 #acji4 ,leibnitz1 ,on difficulties in m41 #bda4 ,his greatness1 #ihg4 ,his influence1 #ihh4 ,the nature of his work1 #ihi4 ,his math4 tendencies1 #iij4 ,his binary arithmetic1 #iia4 ,on ,newton1 #ajaj4 ,on demonstrations outside of m41 #acab4 ,ars characteristica1 #acaf4 ,his logical calculus1 #acag4 ,union of philosophical and m4 productivity1 #adjd4 ,m4 and philosophy1 #adce4 ,on the certainty of math4 knowledge1 #addb4 ,on controversy in geometry1 #ahei4 ,his differential calculus1 #aijb4 ,his notation of the calculus1 #aijd4 ,on necessary and contingent truth1 #aiff4 ,leverrier1 ,discovery of ,neptune1 #aeei4 ,lewes1 ,on the infinite1 #aiec4 ,lie1 ,on central conceptions in modern m41 #gbg4 ,endowment of math'ns1 #hah4 ,the comparative anatomist1 #iib4 ,aim of his work1 #iic4 ,his genius1 #iid4 ,on groups1 #ageb4 ,on the origin of the calculus1 #aija4 ,on differential equations1 #aibd4 ,liliwati1 ,origin of1 #iie4 ,limitations of math4 science1 #adcg4 ,limits1 ,method of1 #aije1 #aijh1 #aiji1 #aidj4 ,lindeman1 ,on m4 and science1 #aebc4 ,liouville1 #hbb4 ,lobatchewsky1 #bjbb4 ,locke1 ,on the method of m41 #bad1 #bce4 ,on proofs and demonstrations1 #bcf4 ,on the unpopularity of m41 #bga4 ,on m4 as a logical exercise1 #dbc1 #dbd4 ,m4 cures presumption1 #dbe4 ,math1 reasoning of universal application1 #dbf4 ,on reading of classic authors1 #fjd4 ,on ,aristotle1 #iad4 ,on m4 and philosophy1 #adcc4 ,on m4 and moral science1 #adci1 #addj4 ,on the certainty of math4 knowledge1 #addj1 #adda4 ,on unity1 #afjg4 ,on number1 #afjh4 ,on demonstrations in numbers1 #afcj4 ,on the advantages of algebra1 #agje4 ,on infinity1 #aiee1 #aieg4 ,on probability1 #aife4 ,logarithmic spiral1 #ibb4 ,logarithmic tables1 #fjb4 ,logarithms1 #aebf1 #afad1 #afaf4 ,logic and m41 ,chapter ,,xiii4 ,see also #dbc-dcj1 #ddb4 ,logical calculus1 #acaf1 #acag4 ,longevity of math'ns1 #hci4 ,lovelace1 ,why are wise few etc41 #afbi4 ,lover1 #badj4 ,macaulay1 ,plato and ,bacon1 #caf4 ,on ,archimedes1 #ije4 ,bacon's view of m41 #iae1 #iaf4 ,anagram on his name1 #iif4 ,plato and ,archytas1 #adbg4 ,on the power of m41 #aebg4 ,macfarlane1 ,on ,tait1 ,maxwell1 ,thomson1 #ajdb4 ,on ,tait and ,hamilton's quaternions1 #ajdd4 ,mach1 ,on thought-economy in m41 #bjc4 ,m4 seems possessed of intelligence1 #fbf4 ,on aim of research1 #fdg4 ,on m4 and counting1 #afja4 ,on the space of experience1 #bjaa4 ,mac,mahon1 ,latin squares1 #beb4 ,on ,sylvester's bend of mind1 #fde4 ,on ,sylvester's style1 #ajdj4 ,on the idea of invariance1 #agdf4 ,magnitude1 ,grassmann's definition1 #aje4 ,magnus1 ,on the aim in teaching m41 #eje4 ,manhattan ,island1 ,cost of1 #bacj4 ,marcellus1 ,estimate of ,archimedes1 #iji4 ,maschke1 ,man above method1 #fej4 ,masters1 ,on the reading of the1 #fad4 ,mathematic1 ,sylvester on use of term1 #aja4 ,bacon's use of term1 #ajf4 ,mathematical faculty1 ,frequency of1 #hcb4 ,mathematical mill1 ,the1 #bci1 #ahia4 ,mathematical productions1 #fdh1 #fdi4 ,mathematical theory1 ,when complete1 #fcf1 #fcg4 ,mathematical training1 #ddc1 #ddd4 ,maxims of math'ns1 #fcj1 #fca1 #fdi4 ,not a computer1 #abaa4 ,intellectual habits of math'ns1 #adbh4 ,the place of the1 #aebi4 ,characteristics of the mind of a1 #aecd4 ,mathematician1 ,the1 ,chapter ,,viii4 ,mathematics1 ,definitions of1 ,chapter ,i4 ,objects of1 ,chapter ,i4 ,nature of1 ,chapter ,,ii4 ,estimates of1 ,chapter ,,iii4 ,value of1 ,chapter ,,iv4 ,teaching of1 ,chapter ,v4 ,study of1 ,chapter ,,vi4 ,research in1 ,chapter ,,vi4 ,modern1 ,chapter ,,vii4 ,as a fine art1 ,chapter ,,xi4 ,as a language1 ,chapter ,,xii4 ,also #dde1 #ahad4 ,and logic1 ,chapter ,,xiii4 ,and philosophy1 ,chapter ,,xiv4 ,and science1 ,chapter ,,xv4 ,and applications1 ,chapter ,,xv4 ,knowledge most in1 #bad4 ,suppl4 brevity of life1 #bah4 ,the range of1 #bfi4 ,compared to ,french language1 #caa4 ,the care of great men1 #cbb4 ,and professional education1 #dbi4 ,and science teaching1 #ebb4 ,the queen of the sciences1 #ige4 ,advantage over philosophy1 #adcf1 #adch4 ,as an instrument1 #aejf4 ,for its own sake1 #aedj1 #aeda1 #aede1 #aedf4 ,the wings of1 #afjd4 ,mathesis1 #bgd1 #bgf1 #ahgj1 #bjae4 ,mathews1 ,on ,disqu4 ,arith4 #afch4 ,on number theory1 #afci4 ,the symbol 1 #afdf4 ,on ,cyclotomy1 #afdg4 ,laws of algebra1 #agji4 ,on infinite1 zero1 infinitesimal1 #aied4 ,maxims of great math'ns1 #fcj1 #fca1 #fdi4 ,maxwell1 #ajdc1 #aaaf4 ,mc,cormack1 ,on the unpopularity of m41 #bgj4 ,on function1 #aicc4 ,mchanique cleste1 #ihe1 #ihf4 ,medicine1 ,m4 and the study of1 #aehe1 #aiah4 ,mellor1 ,definition of higher m41 #ajh4 ,conclusions involved in premises1 #bch4 ,on m4 and science1 #aefa4 ,on the calculus1 #aiab4 ,on integration1 #aibc1 #aibe4 ,memory in m41 #bec4 ,menchmus1 #ija4 ,mere math'ns1 #hbj1 #hba4 ,merz1 ,on the transforming power of m41 #cjc4 ,on the dominant ideas in m41 #gbe4 ,on extreme views in m41 #hbg4 ,on ,leibnitz's work1 #ihi4 ,on the math4 tendency of ,leibnitz1 #iij4 ,on m4 as a lens1 #aeae4 ,m4 extends knowledge1 #aebd4 ,disquisitiones ,arithmeticae1 #afcg4 ,on functions1 #aicb4 ,on hyper-space1 #bjcf4 ,metaphysics1 ,m4 the only true1 #cje4 ,meteorology and m41 #aeeg4 ,method of m4 #bab-bae1 #bbf1 #bbg1 #bcj1 #bce1 #bdd1 #hjf1 #aegf4 ,metric system1 #agbe4 ,military training1 ,m4 in1 #cad1 #dah1 #aegd4 ,mill1 ,on induction in m41 #bba1 #bbb4 ,on generalization in m41 #bde4 ,on math4 studies1 #dji4 ,on m4 in a scientific education1 #ddd4 ,math'ns hard to convince1 #haa4 ,math'ns require genius1 #hai4 ,on ,comte1 #idb4 ,on ,descartes1 #idb1 #idh4 ,on ,sir ,william ,hamilton's ignorance of m41 #igh4 ,on ,leibnitz1 #ihg4 ,on m4 and philosophy1 #adba4 ,on m4 as training for philosophers1 #adbb4 ,m4 indispensable to science1 #aeai4 ,m4 and social science1 #aeie4 ,on the nature of geometry1 #ahch4 ,on geometrical method1 #ahfa4 ,on the calculus1 #aijc4 ,miller1 ,on the ,darmstaetter prize1 #babi4 ,milner1 ,geometry and poetry1 #aaah4 ,minchin1 ,on ,english text-books1 #eci4 ,mineralogy and m41 #aeeh4 ,minkowski1 ,on integral numbers1 #afcf4 ,miracles and m41 #baeg1 #baeh1 #bafj4 ,mixed m41 ,bacon's definition of1 #ajf4 ,whewell's definition of1 #ajg4 ,modern algebra1 #ajca1 #ajcb1 #afch1 #agda4 ,modern geometry1 #agaj-agac1 #gae1 #gaf1 #ahgh4 ,modern m41 ,chapter ,,vii4 ,moebius1 ,math'ns constitute a favorite class1 #hji4 ,m4 a fine art1 #aajg4 ,moral science and m41 #adch-addj4 ,moral value of m41 ,see ethical value4 ,mottoes1 ,of math'ns1 #fcj1 #fca1 #fdi4 ,of ,pythagoreans1 #ahcc4 ,murray1 ,definition of m41 #aaf4 ,music and m41 #aja1 #bgf1 #ife1 #aajg1 #aaab1 #aaaf1 #aabg1 #aabh1 #aacj-aacc1 #aace1 #aacf4 ,myers1 ,on m4 as a school subject1 #djc4 ,on pleasure in m41 #ded4 ,on the ethical value of m41 #deg4 ,on the value of arithmetic1 #afbb4 ,mysticism and numbers1 #bacf-bada1 #badc4 ,napier's rule1 #ahhh4 ,napoleon1 ,m4 and the welfare of the state1 #cac4 ,his interest in m41 #cad1 #ajja4 ,natural science and m41 ,chapter ,,xv4 ,also #bdd1 #ddd1 #dde1 #eja4 ,natural selection1 #aiba4 ,nature of m41 ,chapter ,,ii4 ,see also #hae1 #abae1 #acjh1 #adbf1 #aebe1 #afbh4 ,nature1 ,study of1 #dcc-dcf1 #ead1 #eaf1 #fab4 ,navigation and m41 #aedc1 #aedd4 ,nelson1 ,anagram on1 #baec4 ,neptune1 ,discovery of1 #aeed1 #aeei4 ,newcomb1 ,on geometrical paradoxers1 #baac4 ,newton1 ,importance of his work1 #ccc4 ,on correlation in m41 #ebf4 ,on problems in algebra1 #ecj4 ,and ,gauss compared1 #hbg4 ,his fame1 #ajjb4 ,emerson on1 #ajjc4 ,whewell on1 #ajjd1 #ajje4 ,arago on1 #ajjf4 ,pope on1 #ajjg4 ,southey on1 #ajjh4 ,hill on1 #ajji4 ,leibnitz on1 #ajaj4 ,lagrange on1 #ajaa4 ,no monument to1 #ajab4 ,wilson on1 #ajab1 #ajac4 ,his genius1 #ajad4 ,his interest in chemistry and theology1 #ajae4 ,and alchemy1 #ajaf1 #ajag4 ,his first experiment1 #ajah4 ,as a lecturer1 #ajai4 ,as an accountant1 #ajbj4 ,his memorandum-book1 #ajba4 ,his absent-mindedness1 #ajbb4 ,estimate of himself1 #ajbc-ajbe4 ,his method of work1 #ajbf4 ,discovery of the calculus1 #ajbg4 ,anagrams on1 #ajbh4 ,gauss's estimate of1 #ajbi4 ,on geometry1 #ahaa4 ,compared with ,euclid1 #ahbg4 ,geometry a mechanical science1 #ahfe4 ,test of simplicity1 #ahib4 ,method of fluxions1 #aijb4 ,newton's rule1 #agdc4 ,nile1 ,origin of name1 #baej4 ,noether1 ,on ,cayley1 #ica4 ,on ,sylvester1 #ajcd1 #ajda4 ,non-euclidean geometry1 #acbb1 #bjaf-bjbi1 #bjcc1 #bjce1 #bjdj4 ,nonnus1 ,on the mystic four1 #badh4 ,northrup1 ,on ,lord ,kelvin1 #ajdh4 ,notation1 ,importance of1 #fcd1 #abbb1 #afdf4 ,value of algebraic1 #abac1 #abad4 ,criterion of good1 #abaf4 ,on ,arabic1 #abag1 #afad4 ,advantage of math41 #abbj4 ,see also symbolism4 ,notions1 ,cardinal of m41 #aaj4 ,indefinable1 #abai4 ,novalis1 ,definition of pure m41 #aab4 ,m4 the life supreme1 #cbi4 ,without enthusiasm no m41 #hja4 ,method is the essence of m41 #hjf4 ,math'ns not good computers1 #haj4 ,music and algebra1 #aabh4 ,philosophy and m41 #adjf4 ,m4 and science1 #aejg1 #aebf4 ,m4 and historic science1 #aeii4 ,m4 and magic1 #baei4 ,m4 and miracles1 #bafj4 ,number1 ,every inquiry reducible to a question of1 #afjb4 ,and nature1 #afjc4 ,regulates all things1 #afje4 ,aeschylus on1 #afjf4 ,definition of1 #afji1 #afaj4 ,and superstition1 #afcb4 ,distinctness of1 #agjg4 ,of the beast1 #baea1 #baeb4 ,number-theory1 ,the queen of m41 #ige4 ,nature of1 #afci4 ,gauss on1 #afdd4 ,smith on1 #afde4 ,notation in1 #afdf4 ,aid to geometry1 #afdg4 ,mystery in1 #afdh4 ,number-work1 ,purpose of1 #afbc4 ,numbers1 ,pythagoras' view of1 #cba4 ,mighty are1 #aefh4 ,aptitude for1 #afag4 ,demonstrations in1 #afcj4 ,prime1 #afdh4 ,necessary truths like1 #aiff4 ,round1 #bacg4 ,odd1 #bach-bada4 ,golden1 #badb4 ,magic1 #badc4 ,obscurity in m4 and philosophy1 #adjg4 ,observation in m41 #bea-bec1 #bee1 #dcc1 #ahcj4 ,obviousness in m41 #ihe1 #ihf1 #ajde4 ,olney1 ,on the nature of m41 #bec4 ,oratory and m41 #hbi1 #hcj4 ,order and arrangement1 #gbe4 ,origin of m41 #adab4 ,orr1 ,memory verse for 1 #babg4 ,osgood1 ,on the calculus1 #aiac4 ,ostwald1 ,on four-dimensional space1 #bjci4 4 ,in actuarial formula1 #ide4 ,memory verse for1 #babg4 ,pacioli1 ,on the number three1 #bade4 ,painting and m41 #aajc1 #aajg4 ,papperitz1 ,on the object of pure m41 #aaa4 ,paradoxes1 ,chapter ,,xxi4 ,parallel axiom1 ,proof of1 #ihd1 #baaj1 #baaa4 ,see also non-euclidean geometry4 ,parker1 ,definition of arithmetic1 #afaa4 ,number born in superstition1 #afcb4 ,on geometry1 #ahje4 ,parton1 ,on ,newton1 #aiag-aiai1 #ajba1 #ajbb1 #ahbg4 ,pascal1 ,logic and m41 #acjf4 ,peacock1 ,on the mysticism of ,greek philosophers1 #bacf4 ,the ,yankos word for three1 #badd4 ,the number of the beast1 #baeb4 ,pearson1 ,m4 and natural selection1 #hcd4 ,peirce1 ,benjamin1 ,definition of m41 #abj4 ,m4 as an arbiter1 #baj4 ,logic dependent on m41 #acja4 ,on the symbol -#a1 #agcc4 ,peirce1 ,c4 ,s4 ,definition of m41 #acc4 ,on accidental relations1 #babh4 ,perry1 ,on the teaching of m41 #eaj1 #eaa1 #fai1 #hcg4 ,persons and anecdotes1 ,chapters ,,ix and ,x4 ,philosophy and m41 ,chapter ,,xiv4 ,also #ccb1 #dja1 #dad1 #ddd1 #dde1 #deb4 ,physics and m41 #abi1 #dcg1 #aeaf1 #aecj1 #aece1 #aech1 #aeci1 #aedh1 #aedi1 #aeej1 #aeee1 #aeef4 ,physiology and m41 #aegh1 #aeha1 #aehb4 ,picard1 ,on the use of equations1 #ahia4 ,pierce1 ,on infinitesimals1 #aidj4 ,pierpont1 ,golden age of m41 #gja4 ,on the progress of m41 #gjh4 ,characteristics of modern m41 #gag4 ,on variability1 #gba4 ,on divergent series1 #aicg4 ,plato1 ,his view of m41 #caf1 #dbi4 ,m4 a study suitable for freemen1 #cag4 ,his conic sections1 #ccb4 ,and ,archimedes1 #ijd4 ,union of math4 and philosophical productivity1 #adjd4 ,diagonal of square1 #adaa4 ,and ,archytas1 #adbg4 ,m4 and the arts1 #aefg4 ,on the value of m41 #aegd4 ,on arithmetic1 #afbj1 #afba4 ,god geometrizes1 #afce1 #afcf4 #agjb4 ,on geometry1 #dbi1 #ahjc1 #ahjd1 #ahjf1 #ahdd1 #ahde4 ,pleasure1 ,element of in m41 #afbb1 #afbi1 #ahdh1 #ahej1 #ahea4 ,pliny1 #bjci4 ,plus and minus signs1 #agbg4 ,plutarch1 ,on ,archimedes1 #ijc1 #ijd1 #ijh-iaj1 #iab4 ,god geometrizes1 #ahjb4 ,poe1 #dag4 ,poetry and m41 ,weierstrass on1 #hjb4 ,pringsheim on1 #aajh4 ,wordsworth on1 #aaag4 ,milner on1 #aaah4 ,workman on1 #aabj4 ,pollock on1 #aaba4 ,hoffman on1 #aabb4 ,thoreau on1 #aabc4 ,emerson on1 #aabd4 ,hill on1 #aabe1 #aabf4 ,shakespeare on1 #aabg4 ,poincar1 ,on elegance in m41 #fdj4 ,m4 has a triple end1 #aajb4 ,m4 as a language1 #abjh4 ,geometry not an experimental science1 #ahfg4 ,on geometrical axioms1 #bjje4 ,point1 #ahaf4 ,political science1 ,m4 and1 #abja1 #acbd4 ,politics1 ,math'ns and1 #had4 ,pollock1 ,on ,clifford1 #ich-ida1 #aaba4 ,pope1 #ijg1 #bjae1 #bjca1 #bjdf4 ,precision in m41 #bbh1 #fci1 #gbh4 ,precocity in m41 #hce4 ,predicabilia a priori1 #bjjc4 ,press1 ,m4 ignored by daily1 #gca1 #gcb4 ,price1 ,characteristics of m41 #bdg4 ,on m4 and physics1 #aeej4 ,prime numbers1 ,sylvester on1 #afdh4 ,principia ,mathematica1 #acbf4 ,pringsheim1 ,m4 the science of the self-evident1 #bcb4 ,m4 should be studied for its own sake1 #dci4 ,on the indirect value of m41 #ddh4 ,on rigor in m41 #ece4 ,on m4 and journalism1 #gcb4 ,on math'ns in public service1 #hbd4 ,math'n somewhat of a poet1 #aajh4 ,on music and m41 #aacb4 ,on the language of m41 #abaa4 ,on m4 and physics1 #aedh4 ,probabilities1 #ddb1 #hbc1 #aehi1 #aeij-aeib1 #aifb-aigb1 #aige4 ,problem solving1 #eca1 #ecb4 ,problems1 ,in m41 #ebc1 #ecd4 ,in arithmetic1 #ebh4 ,in algebra1 #ecj4 ,should be simple1 #fjc4 ,in ,cambridge texts1 #fjh4 ,on solution of1 #faa4 ,on importance of1 #fbd1 #fbh4 ,what constitutes good1 #fbi4 ,aid to research1 #fdd4 ,of modern m41 #aibf4 ,proclus1 ,ptolemy and ,euclid1 #iea4 ,on characteristics of geometry1 #ahfi4 ,progress in m41 #bji1 #baa1 #bab1 #baf1 #bah1 #gjb-gje1 #gjh4 ,projective geometry1 #ahgf1 #ahgg1 #ahgi1 #ahhj4 ,proportion1 ,euclid's doctrine of1 #ahcd4 ,euclid's definition of1 #ahce4 ,proposition1 #abai1 #adai4 ,prussia1 ,m4 in1 #eac4 ,pseudomath1 ,defined1 #baja4 ,psychology and m41 #aegf1 #aehc1 #aehd4 ,ptolemy and ,euclid1 #iea4 ,public service1 ,m4 and1 #hbc1 #hbd1 #acjc1 #aegd4 ,public speaking1 ,m4 and1 #dbj1 #hbi1 #hcj4 ,publications1 ,math4 of present day1 #gjb1 #gjc4 ,pure ,m41 ,bacon's definition of1 #ajf4 ,whewell's definition of1 #ajg4 ,on the object of1 #aaa1 #abi4 ,novalis' conception of1 #aab4 ,hobson's definition of1 #aah4 ,russell's definition of1 #abg1 #abh4 ,pursuit of m41 #hdb4 ,pythagoras1 ,number the nature of things1 #cba4 ,union of math1 and philosophical productivity1 #adjd4 ,the number four1 #badg4 ,pythagorean brotherhood1 ,motto of1 #ahcc4 ,pythagorean theorem1 #ahed-ahef1 #bjbf4 ,pythagoreans1 ,music and ,m41 #aacj4 ,quadrature1 ,see ,squaring of the circle4 ,quantity1 ,chrystal's definition of1 #aae4 ,quarles1 ,on quadrature1 #baaf4 ,quaternions1 #ccc1 #hda1 #icg1 #ajdd1 #abaj1 #agah-agbf4 ,quetelet1 ,growth of m41 #aead4 ,railway-making1 #aegj4 ,reading of m41 #fja1 #fjd-fjf4 ,reason1 ,m4 most solid fabric of human1 #cjh4 ,m4 demonstrates supremacy of human1 #cji4 ,reasoning1 ,m4 a type of perfect1 #cjg4 ,m4 as an exercise in1 #dbc-dbg1 #dbi1 #dcj1 #aejc4 ,recorde1 ,value of arithmetic1 #afai4 ,regiomontanus1 #aedc4 ,regular solids1 #bacb-bace4 ,reid1 ,m4 frees from sophistry1 #bae4 ,conjecture has no place in m41 #bcd4 ,m4 the most solid fabric1 #cjh4 ,on ,euclid's elements1 #iee4 ,m4 manifests what is impossible #adad4 ,on m4 and philosophy1 #adbc4 ,probability and ,christianity1 #aige4 ,on ,pythagoras and the regular solids1 #bacb4 ,reidt1 ,m1 as an exercise in language1 #dai4 ,on the ethical value of m41 #def4 ,on aim in math4 instruction1 #ejf4 ,religion and m41 #bgd-bgf1 #dei1 #dfj1 #ajac4 ,research in m41 ,chapter ,,vi4 ,reversible verses1 #baef4 ,reye1 ,advantages of modern over ancient geometry1 #gad4 ,rhetoric and m41 #aeii4 ,riemann1 ,on m4 and physics1 #aedi4 ,rigor in m41 #ece-ech4 ,rosanes1 ,on the unpopularity of m41 #gcj4 ,royal road1 #bja1 #ija1 #iea1 #aggd4 ,royal science1 ,m4 a1 #bjd4 ,rudio1 ,on ,euler1 #ieg4 ,m4 and great artists1 #aaje4 ,on m4 and navigation1 #aedc4 ,rush1 ,m4 cures predisposition to anger1 #deh4 ,russell1 ,definition of m41 #abg1 #abh4 ,on nineteenth century m41 #gje4 ,chief triumph of modern m41 #gjf4 ,on the infinite1 #gbc4 ,on beauty in m41 #aajd4 ,on the value of symbols1 #abai4 ,on ,boole's ,laws of ,thought1 #acah4 ,principia ,mathematica1 #acbf4 ,on geometry and philosophy1 #adaj4 ,definition of number1 #afji4 ,fruitful uses of imaginaries1 #agce4 ,geometrical reasoning circular1 #ahfd4 ,on projective geometry1 #ahgi4 ,zeno's problems1 #aich4 ,definition of infinite collection1 #aiei4 ,on proofs of axioms1 #bjac4 ,on non-euclidean geometry1 #bjah4 ,safford1 ,on aptitude for m41 #ebj4 ,on m4 and science1 #aeji4 ,sage1 ,battalions of figures1 #afca4 ,sartorius1 ,gauss on the nature of space1 #bjcd4 ,scepticism1 #deb1 #haa4 ,schellbach1 ,estimate of m41 #cjf4 ,on truth1 #aaad4 ,schiller1 ,archimedes and the youth1 #ijg4 ,schopenhauer1 ,arithmetic rests on the concept of time1 #afac4 ,predicabilia a priori1 #bjjc4 ,schrder1 ,m4 as a branch of logic1 #acbc4 ,schubert1 ,three characteristics of m41 #bbi4 ,on controversies in m41 #bdc4 ,characteristics of m41 #bfc4 ,m4 an exclusive science1 #gcd4 ,science and m41 ,chapter ,,xv4 ,m4 an indispensible tool of1 #cji4 ,neglect of m4 works injury to1 #caj4 ,craig on origin of new1 #fdf4 ,greek view of1 #adbi4 ,six follies of1 #bajg4 ,see also #dcc1 #dcf1 #dcg1 #dfa1 #gbe4 ,scientific education1 ,math4 training indispensable basis of1 #ddd4 ,screw1 ,the song of the1 #ahid4 ,as an instrument in geometry1 #baad4 ,sedgwick1 ,quaternion of maladies1 #agbc4 ,segre1 ,on research in m41 #fai4 ,what kind of investigations are important1 #fda4 ,on the worthlessness of certain investigations1 #fdb1 #fdc4 ,on hyper-space1 #bjca4 ,seneca1 ,alexander and geometry1 #ijb4 ,seventy-seven1 ,the number1 #badi4 ,shakespeare1 #aabg1 #aabi1 #bada4 ,shaw1 ,j4 ,b41 ,m4 like game of chess1 #hdj4 ,shaw1 ,w4 ,h41 ,m4 and professional life1 #aeif4 ,sherman1 ,m4 and rhetoric1 #aeii4 ,smith1 ,adam1 #acbd4 ,smith1 ,d4 ,e41 ,on problem solving1 #ecb4 ,value of geometrical training1 #ahdf4 ,reason for studying geometry1 #ahej4 ,smith1 ,h4 ,j4 ,s41 ,when a math4 theory is completed1 #fcg4 ,on the growth of m41 #aeba4 ,on m4 and science1 #aedb4 ,on m4 and physics1 #aeef4 ,on m4 and meteorology1 #aeeg4 ,on number theory1 #afde4 ,rigor in ,euclid1 #ahbi4 ,on ,euclid's doctrine of proportion1 #ahcd4 ,smith1 ,w4 ,b41 ,definition of m41 #aba4 ,on infinitesimal analysis1 #aiad4 ,on non-euclidean and hyperspaces1 #bjcc4 ,simon1 ,on beauty and truth1 #aaad4 ,simplicity in m41 #cae1 #ebf4 ,sin1 ,on the notation of1 #ahhf4 ,six hundred sixty-six1 ,the number1 #baea1 #baeb4 ,social science and m41 #abja1 #aehf1 #aehg4 ,social service1 ,m4 as an aid to1 #cac1 #cad1 #cbh4 ,social value of m41 #def1 #aehh4 ,solitude and m41 #ahdi1 #ahea4 ,sophistry1 ,m4 free from1 #bae4 ,sound1 ,m4 and the theory of1 #aeea4 ,southey1 ,on ,newton1 #ajjh4 ,space1 ,of experience1 #bjaa4 ,kant's doctrine of1 #bjjc4 ,schopenhauer's predicabilia1 #bjjd4 ,whewell1 ,on the idea of1 #bjjd4 ,non-euclidean1 #bjae1 #bjaf1 #bjah4 ,hyper-1 #bjcj1 #bjca1 #bjcc1 #bjcf-bjch4 ,spedding1 ,on ,bacon's knowledge of m41 #iag4 ,speer1 ,on m4 and nature-study1 #ead4 ,spence1 ,on ,newton1 #ajaf1 #ajbj4 ,spencer1 ,on m4 in the arts1 #aegj4 ,spherical trigonometry1 #ahhg4 ,spira mirabilis1 #ibb4 ,spottiswoode1 ,on the kingdom of m41 #bfi4 ,squaring the circle1 #aecg1 #aheh1 #aicd1 #aidh1 #baae-baag4 ,st4 ,augustine1 ,the number seventy seven1 #badi4 ,st4 ,vincent1 ,as a circle-squarer1 #baji4 ,steiner1 ,on projective geometry1 #ahgg4 ,stewart1 ,m4 and facts1 #bcg4 ,on beauty in m41 #bdb4 ,what we most admire in m41 #cae4 ,m4 for its own sake1 #ddj4 ,m4 the noblest instance of force of the human mind1 #deb4 ,math'ns and applause1 #haf4 ,mere math'ns1 #hba4 ,shortcomings of math'ns1 #hbh4 ,on the influence of ,leibnitz1 #ihh4 ,reason supreme1 #adbd4 ,m4 and philosophy compared1 #adbh4 ,m4 and natural philosophy1 #aeee4 ,stifel1 ,the number of the beast1 #baeb4 ,stobus1 ,alexander and ,menchmus1 #ija4 ,euclid and the student1 #ieb4 ,study of m41 ,chapter ,,vi4 ,substitution1 ,concept of1 #gbg4 ,superstition1 ,m4 frees mind from1 #dej4 ,number was born in1 #afcb4 ,surd numbers1 #agbh4 ,surprises1 ,m4 rich in1 #bjb4 ,swift1 ,on m4 and politics1 #had4 ,the math'ns of ,laputa1 #babj-babb4 ,the math4 school of ,laputa1 #babc4 ,his ignorance of m41 #babd1 #babe4 ,sylvester1 ,on the use of the terms mathematic and mathematics1 #aja4 ,order and arrangement the basic ideas of m41 #aji1 #aaj4 ,definition of algebra1 #aaj4 ,definition of arithmetic1 #aaj4 ,definition of geometry1 #aaj4 ,on the object of pure m41 #abi4 ,m4 requires harmonious action of all the faculties1 #bjb4 ,answer to ,huxley1 #bea4 ,on the nature of m41 #bea4 ,on observation in m41 #bee4 ,invention in m41 #bfj4 ,m4 entitled to human regard1 #cja4 ,on the ethical value of m41 #ddi4 ,on isolated theorems1 #fbj4 8,auge et impera40 #fca4 ,his bent of mind1 #fde4 ,apology for imperfections1 #fdh4 ,on theoretical investigations1 #feh4 ,characteristics of modern m41 #gbd4 ,invested m4 with halo of glory1 #gdj4 ,m4 and eloquence1 #hbi4 ,on longevity of math'ns1 #hci4 ,on ,cayley1 #icj4 ,his view of ,euclid1 #icf4 ,jacobi's talent for philology1 #ihj4 ,his eloquence1 #ajcj4 ,researches in quantics1 #ajcb4 ,his weakness1 #ajcc1 #ajcf1 #ajcg4 ,one-sided character of his work1 #ajcd4 ,his method1 #ajce1 #ajcf1 #ajda4 ,his forgetfulness1 #ajcg1 #ajch4 ,relations with students1 #ajci4 ,his style1 #ajdj1 #ajda4 ,his characteristics1 #ajda4 ,his enthusiasm1 #ajda4 ,the math4 ,adam1 #ajdb4 ,and ,weierstrass1 #ajej4 ,on divine beauty and order in m41 #aaja4 ,m4 among the fine arts1 #aajf4 ,on music and m41 #aaca4 ,m4 the quintessence of language1 #abje4 ,m4 the language of the universe1 #abjf4 ,on prime numbers1 #afdh4 ,on determinants1 #agdj4 ,on invariants1 #agdb4 ,contribution to theory of equations1 #agdc4 ,to a missing member etc41 #agde4 ,invariants and isomerism1 #agej4 ,his dislike for ,euclid1 #ahbf4 ,on the invention of integrals1 #aibb4 ,on geometry and analysis1 #aica4 ,on paradoxes1 #bajd4 ,symbolic language1 ,m4 as a1 #abjg1 #abab4 ,use of1 #aegc4 ,symbolic logic1 #acaf-acba4 ,symbolism1 ,on the nature of math41 #abaj4 ,difficulty of math41 #abah4 ,universal impossible1 #abba4 ,see also notation4 ,symbols1 ,burlesque on1 #agda4 ,symbols1 ,m4 leads to mastery of1 #dba4 ,value of math41 #abji1 #abab1 #abai4 ,essential to demonstration1 #acaf4 ,arithmetical1 #afbg4 ,tact in m41 #fbb1 #fbc4 ,tait1 ,on the unpopularity of m41 #gdj4 ,and ,thomson1 #ajdc4 ,and ,hamilton1 #ajdd4 ,on quaternions1 #agbd-agbf4 ,on spherical trigonometry1 #ahhg4 ,talent1 ,math'ns men of1 #hbe4 ,teaching of m41 ,chapter ,v4 ,tennyson1 #ahdc4 ,teutonic race1 ,aptitude for m41 #hch4 ,text-books1 ,chrystal on1 #ecc4 ,minchin on1 #eci4 ,cremona on ,english1 #fji4 ,glaisher on need of1 #fce4 ,thales1 #bja4 ,theoretical investigations1 #feb-ffd4 ,theory and practice1 #ffa4 ,thompson1 ,sylvanus1 ,lord ,kelvin's definition of a math'n1 #hbb4 ,cayley's estimate of quaternions1 #icg4 ,thomson's 8,it is obvious that10 #ajde4 ,anecdote of ,lord ,kelvin1 #ajdf1 #ajdg4 ,on the calculus for beginners1 #aiag4 ,thomson1 ,sir ,william1 ,m4 the only true metaphysics1 #cje4 ,m4 not repulsive to common sense1 #cab4 ,what is a math'n8 #hbb4 ,and ,tait1 #ajdc4 8,it is obvious that10 #ajde4 ,anecdotes concerning1 #ajdf1 #ajdg1 #ajdh4 ,on m4 and astronomy1 #aefb4 ,on quaternions1 #agba1 #agbb4 ,thomson and ,tait1 #ajdc4 ,on ,fourier's theorem1 #aibh4 ,thoreau1 ,on poetry and m41 #aabc4 ,thought-economy in m41 #bjc1 #abji1 #agjd4 ,three1 ,the ,yankos word for1 #badd4 ,pacioli on the number1 #bade4 ,time1 ,arithmetic rests on notion of #afac4 ,as a concept in algebra1 #agae1 #agaf1 #agag4 ,kant's doctrine of1 #bjja4 ,schopenhauer's predicabilia1 #bjjc4 ,todhunter1 ,on m4 as a university subject1 #dje4 ,on m4 as a test of performance1 #djh4 ,on m4 as an instrument in education1 #dad4 ,m4 requires voluntary exertion1 #dae4 ,on exercises1 #dbb4 ,on problems1 #ebc1 #fjh4 ,how to read m41 #fje1 #fjf4 ,on discovery in elementary m41 #fag4 ,on ,sylvester's theorem1 #agdc4 ,on performance in ,euclid1 #ahah4 ,transformation1 ,concept of1 #gbg4 ,trigonometry1 #ahha1 #ahhd-ahhi4 ,trilinear co-ordinates1 #faa4 ,trisection of angle1 #baab4 ,truth1 and m41 #cjf4 ,math'ns must perceive beauty of1 #hjc4 ,and beauty1 #aaad4 ,tzetzes1 ,plato on geom41 #ahjc4 ,unity1 ,locke on the idea of1 #afjg4 ,universal algebra1 #agec4 ,universal arithmetic1 #agag4 ,universal language1 #ibe4 ,unpopularity of m41 #bgj1 #bga1 #gcj-gcf1 #gch1 #gdj1 #aeja1 #afbh4 ,usefulness1 ,as a principle in research1 #feb-fee1 #fei1 #ffd4 ,uses of m41 ,see value of m4 ,value of m41 ,chapter ,,iv4 ,see also #ccj1 #ccc1 #adad1 #adbb1 #aeje1 #aejf1 #aeab1 #aebc1 #aebf1 #aebg1 #aecc1 #aeda1 #aedb1 #aedc1 #aedg-aegf1 #afai-afbf1 #ahda1 #ahdd-ahea4 ,variability1 ,the central idea of modern m41 #gbj1 #gba4 ,venn1 ,on m4 as a symbolic language1 #abjg4 ,m4 the only gate1 #aeag4 ,viola1 ,on the use of fallacies1 #faj4 ,virgil1 #bach4 ,voltaire1 ,archimedes more imaginative than ,homer1 #bei4 ,m4 the staff of the blind1 #dfa4 ,on direct usefulness of results1 #fec4 ,on infinite magnitudes1 #aidg4 ,on the symbol1 #aiej4 ,anagram on1 #baed4 ,walcott1 ,on hyperbolic functions1 #aicj4 ,walker1 ,on problems in arithmetic1 #ebh4 ,on the teaching of geometry1 #ebi4 ,wallace1 ,on the frequency of the math4 faculty1 #hcb4 ,on m4 and natural selection1 #hcc1 #hcd4 ,parallel growth of m4 and music1 #aace4 ,walton1 ,angling like m41 #gci4 ,weber1 ,on m4 and physics1 #aedi4 ,webster1 ,estimate of m41 #cca4 ,weierstrass1 ,math'ns are poets1 #hjb4 ,anecdote concerning1 #ajdi4 ,and ,sylvester1 #ajej4 ,problem of infinitesimals1 #aich4 ,weismann1 ,on the origin of the math4 faculty1 #aacf4 ,wells1 ,on m4 as a world language1 #abja4 ,whately1 ,on m4 as an exercise1 #dbg4 ,on m4 and navigation1 #aedd4 ,on geometrical demonstrations1 #ahci4 ,on ,swift's ignorance of m41 #babd4 ,whetham1 ,on symbolic logic1 #acai4 ,whewell1 ,on mixed and pure math41 #ajg4 ,m4 not an inductive science1 #bbc4 ,nature of m41 #bbd4 ,value of geometry1 #dde4 ,on theoretical investigations1 #ffj1 #ffb4 ,math'ns men of talent1 #hbe4 ,fame of math'ns1 #hbf4 ,on ,newton's greatness1 #ajjd4 ,on ,newton's theory1 #ajje4 ,on ,newton's humility1 #ajbe4 ,on symbols1 #abab4 ,on philosophy and m41 #adbi4 ,on m4 and science1 #aecd4 ,quotation from ,r4 ,bacon1 #aedg4 ,on m4 and applications1 #aeda4 ,geometry and experience1 #ahad4 ,geometry not an inductive science1 #ahcj4 ,on limits1 #aiji4 ,on the idea of space1 #bjjd4 ,on ,plato and the regular solids1 #bacc1 #bace4 ,white1 ,h4 ,s41 ,on the growth of m41 #baa4 ,white1 ,w4 ,f41 ,definition of m41 #aca1 #abjc4 ,m4 as a prerequisite for public speaking1 #dbj4 ,on beauty in m41 #aaai4 ,the place of the math'n1 #aebi4 ,on m4 and social science1 #aehf4 ,the cost of ,manhattan island1 #bacj4 ,whitehead1 ,on the ideal of m41 #aai4 ,definition of m41 #abb4 ,on the scope of m41 #abf4 ,on the nature of m41 #bcc4 ,precision necessary in m41 #fci4 ,on practical applications1 #fee4 ,on theoretical investigations1 #fei4 ,characteristics of ancient geometry1 #gac4 ,on the extent of m41 #gcg4 ,archimedes compared with ,newton1 #iaa4 ,on the ,arabic notation1 #abag4 ,difficulty of math4 notation1 #abah4 ,on symbolic logic1 #acbj4 ,principia ,mathematica1 #acbf4 ,on philosophy and m41 #adjc4 ,on obscurity in m4 and philosophy1 #adjg4 ,on the laws of algebra1 #agjh4 ,on and signs1 #agbg4 ,on universal algebra1 #agec4 ,on the ,cartesian method1 #ahij4 ,on ,swift's ignorance of m41 #babe4 ,whitworth1 ,on the solution of problems1 #faa4 ,williamson1 ,on the value of m41 #aege4 ,infinitesimals and limits1 #aije4 ,on infinitesimals1 #aidf4 ,wilson1 ,e4 ,b41 ,on the social value of m41 #aehh4 ,on m4 and economics1 #aeic4 ,on the nature of axioms1 #bjab4 ,wilson1 ,john1 ,on ,newton and ,shakespeare1 #ajab4 ,newton and ,linnus1 #ajac4 ,woodward1 ,on probabilities1 #aehi4 ,on the theory of errors1 #aigc1 #aigd4 ,wordsworth1 ,w41 ,on ,archimedes1 #ijf4 ,on poetry and geometric truth1 #aaag4 ,on geometric rules1 #adah4 ,on geometry1 #ahdj1 #ahdh4 ,m4 and solitude1 #ahei4 ,workman1 ,on the poetic nature of m41 #aabj4 ,young1 ,c4 ,a41 ,on the discovery of ,neptune1 #aeei4 ,young1 ,c4 ,w41 ,definition of m41 #abd4 ,young1 ,j4 ,w4 ,a41 ,on m4 as type a of thought1 #djd4 ,m4 as preparation for science study1 #dba4 ,m4 essential to comprehension of nature1 #dce4 ,development of abstract methods1 #gbi4 ,beauty in m41 #aaaj4 ,on ,euclid's axiom1 #bjad4 ,zeno1 ,his problems1 #aich4 ,zero1 #aidh1 #aied4 99 99 99 99 99 ,transcriber's ,notes ,punctuation has been standardised4 ,characters in small caps have been replaced by all caps4 ,italic text has been denoted by underscores and bold text by equal signs4 ,em-dash added before all attribution names for consistency4 ,the two omitted illustrations have been identified by an ,illustration3 tag with a short description4 ,mis-alphabetized entries in the ,index have been corrected ,non-printable superscripts are represented by a caret followed by the character1 i4e4 xn4 ,non-printable subscripts have been represented by an underscore followed by the subscript in braces i4e4 a4 ,book was written in a period when many words had not become standarized in their spelling4 ,numerous words have multiple spelling variations in the text4 ,these have been left unchanged unless noted below3 #bcj - 8elmenetary0 corrected to 8elementary0 7the most elementary use of7 #dcg - 8,mathematiks0 corrected to 8,mathematicks0 7,the ,usefulness of ,mathematicks7 as in the quoted text4 #eaa - ,block number shown as #eag #eag - 8hoheren0 corrected to 8hheren0 7hheren ,schulen7 for consistency #edj - duplicate word 8the0 removed 7let the mind7 #feg - 8anaylsis0 corrected to 8analysis0 7field of analysis47 #gbi - 8,geomtry0 corrected to 8,geometry0 7,algebra and ,geometry7 #hbb - end of quote not identified2 placement unclear4 #hbc - 8heros0 corrected to 8heroes0 7many of the major heroes7 #iag - 80 corrected to 80 7speaks of the 7 #ihf - added missing end quote #aacb - 8,vereiningung0 corrected to 8,vereinigung0 7 ,deutschen ,mathematiker ,vereinigung7 #acbe - 8,philosphy0 corrected to 8,philosophy0 7,positive ,philosophy7 #adba - 8#adbg0 corrected to block 8#adba0 7#adba47 #aejc - 8,todhunder's0 corrected to 8,todhunter's0 7,todhunter's ,history of7 #aece - 8uses0 corrected to 8use0 7the use of analysis7 #ahjc - 80 corrected to 80 7 7 #ahgd - 8anaylsis0 corrected to 8analysis0 7a kind of analysis7 #aicj - 8,hyberbolic0 corrected to 8,hyperbolic0 7,mathematical ,tables1 ,hyperbolic ,functions7 #bjji - 8,stanfpunkte0 corrected to 8,standpunkte0 7hheren ,standpunkte aus7 #babf - ,omitted block number added #bace - 8astromomy0 corrected to 8astronomy0 7history of astronomy7 #baea - 8#aj0 corrected to 8#i0 7,a to ,i represent #a-i7 ,appolonius - also spelled 8,apollonius0 but not referenced at #ebc and #iag ,bcher - 8,bocher0 corrected to 8,bcher0 as given in text ,halsted - 8,slyvester0 corrected to 8,sylvester0 7,and ,sylvester7 ,jefferson - 8,om0 corrected to 8,on0 7,on m4 and law7 ,peacock - 8philosphers0 corrected to 8philosophers0 7,greek philosophers7