from the ,google ,print project47 ,,science ,,and ,,education ,a ,,series ,,of ,,volumes ,,for ,,the ,,promotion ,,of ,,scientific ,,research ,,and ,,educational ,,progress ,,edited ,,by ,j4 ,,mckeen ,,cattell ,,volume ,i--,,the ,,foundations ,,of ,,science ,,under ,,the ,,same ,,editorship ,,science ,,and ,,education4 ,a series of volumes for the promotion of scientific research and educational progress4 ,volume ,i4 ,the ,foundations of ,science4 ,by ,h4 ,,poincar4 ,containing the authorised ,english translation by ,george ,bruce ,halsted of ,science and ,hypothesis1 ,the ,value of ,science1 and ,science and ,method4 ,volume ,,ii4 ,medical ,research and ,education4 ,by ,richard ,mills ,pearce1 ,william ,h4 ,welch1 ,w4 ,h4 ,howell1 ,franklin ,p4 ,mall1 ,lewellys ,f4 ,barker1 ,charles ,s4 ,minot1 ,w4 ,b4 ,cannon1 ,w4 ,t4 ,councilman ,theobald ,smith1 ,g4 ,n4 ,stewart1 ,c4 ,m4 ,jackson1 ,e4 ,p4 ,lyon1 ,james ,b4 ,herrick1 ,john ,m4 ,dodson1 ,c4 ,r4 ,bardeen1 ,w4 ,ophuls1 ,s4 ,j4 ,meltzer1 ,james ,ewing1 ,w4 ,w4 ,keen1 ,henry ,h4 ,donaldson1 ,christian ,a4 ,herter1 and ,henry ,p4 ,bowditch4 ,volume ,,iii4 ,university ,control4 ,by ,j4 ,,mckeen ,,cattell and other authors4 ,,american ,,men ,,of ,,science4 ,a ,biographical ,directory4 ,,science4 ,a weekly journal devoted to the advancement of science4 ,the official organ of the ,american ,association for the ,advancement of ,science4 ,,the ,,popular ,,science ,,monthly4 ,a monthly magazine devoted to the diffusion of science4 ,,the ,,american ,,naturalist4 ,a monthly journal devoted to the biological sciences1 with special reference to the factors of evolution4 ,,the ,,science ,,press ,,new ,,york ,,garrison1 ,n4 ,y4 ,,the ,,foundations ,,of ,,science ,,science ,,and ,,hypothesis ,,the ,,value ,,of ,,science ,,science ,,and ,,method ,,by ,h4 ,,poincar ,,authorized ,,translation ,,by ,,george ,,bruce ,,halsted ,,with ,a ,,special ,,preface ,,by ,,poincar1 ,,and ,,an ,,introduction ,,by ,,josiah ,,royce1 ,,harvard ,,university ,,the ,,science ,,press ,,new ,,york ,,and ,,garrison1 ,n4 ,y4 #aiac ,copyright1 #aiac ,,by ,,the ,,science ,,press ,,press ,,of ,,the ,,new ,,era ,,printing ,,company ,,lancaster1 ,,pa4 ,,contents ,,page ,henri ,poincar ix ,author's ,preface to the ,translation #c ,,science ,,and ,,hypothesis ,introduction by ,royce #i ,introduction #bg ,,part ,i4 ,number and ,magnitude ,,chapter ,i4--,on the ,nature of ,mathematical ,reasoning #ca ,syllogistic ,deduction #ca ,verification and ,proof #cb ,elements of ,arithmetic #cc ,reasoning by ,recurrence #cg ,induction #dj ,mathematical ,construction #da ,,chapter ,,ii4--,mathematical ,magnitude and ,experience #dc ,definition of ,incommensurables #dd ,the ,physical ,continuum #df ,creation of the ,mathematical ,continuum #df ,measurable ,magnitude #di ,various ,remarks 7,curves without ,tangents7 #ej ,the ,physical ,continuum of ,several ,dimensions #eb ,the ,mathematical ,continuum of ,several ,dimensions #ec ,,part ,,ii4 ,space ,,chapter ,,iii4--,the ,non-,euclidean ,geometries #ee ,the ,bolyai-,lobachevski ,geometry #ef ,riemann's ,geometry #eg ,the ,surfaces of ,constant ,curvature #eh ,interpretation of ,non-,euclidean ,geometries #ei ,the ,implicit ,axioms #fj ,the ,fourth ,geometry #fb ,lie's ,theorem #fb ,riemann's ,geometries #fc ,on the ,nature of ,axioms #fc ,,chapter ,,iv4--,space and ,geometry #ff ,geometric ,space and ,perceptual ,space #ff ,visual ,space #fg ,tactile ,space and ,motor ,space #fh ,characteristics of ,perceptual ,space #fi ,change of ,state and ,change of ,position #gj ,conditions of ,compensation #gb ,solid ,bodies and ,geometry #gb ,law of ,homogeneity #gd ,the ,non-,euclidean ,world #ge ,the ,world of ,four ,dimensions #gh ,conclusions #gi ,,chapter ,v4--,experience and ,geometry #ha ,geometry and ,astronomy #ha ,the ,law of ,relativity #hc ,bearing of ,experiments #hf ,supplement 7,what is a ,point87 #hi ,ancestral ,experience #ia ,,part ,,iii4 ,force ,,chapter ,,vi4--,the ,classic ,mechanics #ib ,the ,principle of ,inertia #ic ,the ,law of ,acceleration #ig ,anthropomorphic ,mechanics #ajc ,the ,school of the ,thread #ajd ,,chapter ,,vii4--,relative ,motion and ,absolute ,motion #ajg ,the ,principle of ,relative ,motion #ajg ,newton's ,argument #ajh ,,chapter ,,viii4--,energy and ,thermodynamics #aae ,energetics #aae ,thermodynamics #aai ,general ,conclusions on ,part ,,iii #abc ,,part ,,iv4 ,nature ,,chapter ,,ix4--,hypotheses in ,physics #abg ,the ,rle of ,experiment and ,generalization #abg ,the ,unity of ,nature #acj ,the ,rle of ,hypothesis #acc ,origin of ,mathematical ,physics #acf ,,chapter ,x4--,the ,theories of ,modern ,physics #adj ,meaning of ,physical ,theories #adj ,physics and ,mechanism #add ,present ,state of the ,science #adh ,,chapter ,,xi4--,the ,calculus of ,probabilities #aee ,classification of the ,problems of ,probability #aeh ,probability in ,mathematics #afa ,probability in the ,physical ,sciences #afd ,rouge et noir #afg ,the ,probability of ,causes #afi ,the ,theory of ,errors #agj ,conclusions #agb ,,chapter ,,xii4--,optics and ,electricity #agd ,fresnel's ,theory #agd ,maxwell's ,theory #age ,the ,mechanical ,explanation of ,physical ,phenomena #agg ,,chapter ,,xiii4--,electrodynamics #ahd ,ampre's ,theory #ahd ,closed ,currents #ahe ,action of a ,closed ,current on a ,portion of ,current #ahf ,continuous ,rotations #ahg ,mutual ,action of ,two ,open ,currents #ahi ,induction #aij ,theory of ,helmholtz #aia ,difficulties ,raised by these ,theories #aic ,maxwell's ,theory #aic ,rowland's ,experiment #aid ,the ,theory of ,lorentz #aif ,,the ,,value ,,of ,,science ,translator's ,introduction #bja ,does the ,scientist ,create ,science8 #bja ,the ,mind ,dispelling ,optical ,illusions #bjb ,euclid not ,necessary #bjb ,without ,hypotheses1 no ,science #bjc ,what ,outcome8 #bjc ,introduction #bje ,,part ,i4 ,the ,mathematical ,sciences ,,chapter ,i4--,intuition and ,logic in ,mathematics #baj ,,chapter ,,ii4--,the ,measure of ,time #bbc ,,chapter ,,iii4--,the ,notion of ,space #bce ,qualitative ,geometry #bch ,the ,physical ,continuum of ,several ,dimensions #bdj ,the ,notion of ,point #bdd ,the ,notion of ,displacement #bdg ,visual ,space #beb ,,chapter ,,iv4--,space and its ,three ,dimensions #bef ,the ,group of ,displacements #bef ,identity of ,two ,points #bei ,tactile ,space #bfd ,identity of the ,different ,spaces #bfh ,space and ,empiricism #bga ,rle of the ,semicircular ,canals #bgf ,,part ,,ii4 ,the ,physical ,sciences ,,chapter ,v4--,analysis and ,physics #bgi ,,chapter ,,vi4--,astronomy #bhi ,,chapter ,,vii4--,the ,history of ,mathematical ,physics #big ,the ,physics of ,central ,forces #big ,the ,physics of the ,principles #bii ,,chapter ,,viii4--,the ,present ,crisis in ,physics #cjc ,the ,new ,crisis #cjc ,carnot's ,principle #cjc ,the ,principle of ,relativity #cje ,newton's ,principle #cjh ,lavoisier's ,principle #caj ,mayer's ,principle #cab ,,chapter ,,ix4--,the ,future of ,mathematical ,physics #cad ,the ,principles and ,experiment #cad ,the ,rle of the ,analyst #cad ,aberration and ,astronomy #cae ,electrons and ,spectra #caf ,conventions preceding ,experiment #cag ,future ,mathematical ,physics #cai ,,part ,,iii4 ,the ,objective ,value of ,science ,,chapter ,x4--,is ,science ,artificial8 #cba ,the ,philosophy of ,le,roy #cba ,science1 ,rule of ,action #cbc ,the ,crude ,fact and the ,scientific ,fact #cbe ,nominalism and the ,universal ,invariant #ccc ,,chapter ,,xi4--,science and ,reality #cdj ,contingence and ,determinism #cdj ,objectivity of ,science #cdg ,the ,rotation of the ,earth #cec ,science for ,its ,own ,sake #ced ,,science ,,and ,,method ,introduction #cei ,,book ,i4 ,science and the ,scientist ,,chapter ,i4--,the ,choice of ,facts #cfb ,,chapter ,,ii4--,the ,future of ,mathematics #cfi ,,chapter ,,iii4--,mathematical ,creation #chc ,,chapter ,,iv4--,chance #cie ,,book ,,ii4 ,mathematical ,reasoning ,,chapter ,i4--,the ,relativity of ,space #dac ,,chapter ,,ii4--,mathematical ,definitions and ,teaching #dcj ,,chapter ,,iii4--,mathematics and ,logic #ddh ,,chapter ,,iv4--,the ,new ,logics #dfj ,,chapter ,v4--,the ,latest ,efforts of the ,logisticians #dgb ,,book ,,iii4 ,the ,new ,mechanics ,,chapter ,i4--,mechanics and ,radium #dhf ,,chapter ,,ii4--,mechanics and ,optics #dif ,,chapter ,,iii4--,the ,new ,mechanics and ,astronomy #eab ,,book ,,iv4 ,astronomic ,science ,,chapter ,i4--,the ,milky ,way and the ,theory of ,gases #ebc ,,chapter ,,ii4--,french ,geodesy #ece ,general ,conclusions #edd ,index #edg ,,henri ,,poincar ,,sir ,,george ,,darwin1 worthy son of an immortal father1 said1 referring to what ,poincar was to him and to his work3 ,he must be regarded as the presiding genius--or1 shall ,i say1 my patron saint8 ,henri ,poincar was born ,april #bi1 #ahed1 at ,nancy1 where his father was a physician highly respected4 ,his schooling was broken into by the war of #ahgj-ga1 to get news of which he learned to read the ,german newspapers4 ,he outclassed the other boys of his age in all subjects and in #ahgc passed highest into the cole ,polytechnique1 where1 like ,john ,bolyai at ,maros ,vsrhely1 he followed the courses in mathematics without taking a note and without the syllabus4 ,he proceeded in #ahge to the ,school of ,mines1 and was ,nomm1 ,march #bf1 #ahgi4 ,but he won his doctorate in the ,university of ,paris1 ,august #a1 #ahgi1 and was appointed to teach in the ,facult des ,sciences de ,caen1 ,december #a1 #ahgi1 whence he was quickly called to the ,university of ,paris1 teaching there from ,october #ba1 #ahha1 until his death1 ,july #ag1 #aiab4 ,so it is an error to say he started as an engineer4 ,at the early age of thirty-two he became a member of l',acadmie des ,sciences1 and1 ,march #e1 #aijh1 was chosen ,membre de l',acadmie ,franaise4 ,july #a1 #aiji1 the number of his writings was #dcf4 ,his earliest publication was in #ahgh1 and was not important4 ,afterward came an essay submitted in competition for the ,grand ,prix offered in #ahhj1 but it did not win4 ,suddenly there came a change1 a striking fire1 a bursting forth1 in ,february1 #ahha1 and ,poincar tells us the very minute it happened4 ,mounting an omnibus1 at the moment when ,i put my foot upon the step1 the idea came to me1 without anything in my previous thoughts seeming to foreshadow it1 that the transformations ,i had used to define the ,fuchsian functions were identical with those of non-,euclidean geometry4 ,thereby was opened a perspective new and immense4 ,moreover1 the magic wand of his whole life-work had been grasped1 the ,aladdin's lamp had been rubbed1 non-,euclidean geometry1 whose necromancy was to open up a new theory of our universe1 whose brilliant exposition was commenced in his book ,science and ,hypothesis1 which has been translated into six languages and has already had a circulation of over #bj1jjj4 ,the non-,euclidean notion is that of the possibility of alternative laws of nature1 which in the ,introduction to the lectricit et ,optique1 #aija1 is thus put3 ,if therefore a phenomenon admits of a complete mechanical explanation1 it will admit of an infinity of ,others which will account equally well for all the peculiarities disclosed by experiment4 ,the scheme of laws of nature so largely due to ,newton is merely one of an infinite number of conceivable rational schemes for helping us master and make experience2 it is commode1 convenient2 but perhaps another may be vastly more advantageous4 ,the old conception of true has been revised4 ,the first expression of the new idea occurs on the title page of ,john ,bolyai's marvelous ,science ,absolute of ,space1 in the phrase haud unquam a priori decidenda4 ,with bearing on the history of the earth and moon system and the origin of double stars1 in formulating the geometric criterion of stability1 ,poincar proved the existence of a previously unknown pear-shaped figure1 with the possibility that the progressive deformation of this figure with increasing angular velocity might result in the breaking up of the rotating body into two detached masses4 ,of his treatise ,les ,mthodes nouvelles de la ,mchanique cleste1 ,sir ,george ,darwin says3 ,it is probable that for half a century to come it will be the mine from which humbler investigators will excavate their materials4 ,brilliant was his appreciation of ,poincar in presenting the gold medal of the ,royal ,astronomical ,society4 ,the three others most akin in genius are linked with him by the ,sylvester medal of the ,royal ,society1 the ,lobachevski medal of the ,physico-,mathematical ,society of ,kazan1 and the ,bolyai prize of the ,hungarian ,academy of ,sciences4 ,his work must be reckoned with the greatest mathematical achievements of mankind4 ,the kernel of ,poincar's power lies in an oracle ,sylvester often quoted to me as from ,hesiod3 ,the whole is less than its part4 ,he penetrates at once the divine simplicity of the perfectly general case1 and thence descends1 as from ,olympus1 to the special concrete earthly particulars4 ,a combination of seemingly extremely simple analytic and geometric concepts gave necessary general conclusions of immense scope from which sprang a disconcerting wilderness of possible deductions4 ,and so he leaves a noble1 fruitful heritage4 ,says ,love3 ,his right is recognized now1 and it is not likely that future generations will revise the judgment1 to rank among the greatest mathematicians of all time4 ,,george ,,bruce ,,halsted4 99 99 99 99 99 ,,science ,,and ,,hypothesis 99 99 99 99 99 ,,author',s ,,preface ,,to ,,the ,,translation ,i am exceedingly grateful to ,dr4 ,halsted1 who has been so good as to present my book to ,american readers in a translation1 clear and faithful4 ,every one knows that this savant has already taken the trouble to translate many ,european treatises and thus has powerfully contributed to make the new continent understand the thought of the old4 ,some people love to repeat that ,anglo-,saxons have not the same way of thinking as the ,latins or as the ,germans2 that they have quite another way of understanding mathematics or of understanding physics2 that this way seems to them superior to all others2 that they feel no need of changing it1 nor even of knowing the ways of other peoples4 ,in that they would beyond question be wrong1 but ,i do not believe that is true1 or1 at least1 that is true no longer4 ,for some time the ,english and ,americans have been devoting themselves much more than formerly to the better understanding of what is thought and said on the continent of ,europe4 ,to be sure1 each people will preserve its characteristic genius1 and it would be a pity if it were otherwise1 supposing such a thing possible4 ,if the ,anglo-,saxons wished to become ,latins1 they would never be more than bad ,latins2 just as the ,french1 in seeking to imitate them1 could turn out only pretty poor ,anglo-,saxons4 ,and then the ,english and ,americans have made scientific conquests they alone could have made2 they will make still more of which others would be incapable4 ,it would therefore be deplorable if there were no longer ,anglo-,saxons4 ,but continentals have on their part done things an ,englishman could not have done1 so that there is no need either for wishing all the world ,anglo-,saxon4 ,each has his characteristic aptitudes1 and these aptitudes should be diverse1 else would the scientific concert resemble a quartet where every one wanted to play the violin4 ,and yet it is not bad for the violin to know what the violon-cello is playing1 and vice versa4 ,this it is that the ,english and ,americans are comprehending more and more2 and from this point of view the translations undertaken by ,dr4 ,halsted are most opportune and timely4 ,consider first what concerns the mathematical sciences4 ,it is frequently said the ,english cultivate them only in view of their applications and even that they despise those who have other aims2 that speculations too abstract repel them as savoring of metaphysic4 ,the ,english1 even in mathematics1 are to proceed always from the particular to the general1 so that they would never have an idea of entering mathematics1 as do many ,germans1 by the gate of the theory of aggregates4 ,they are always to hold1 so to speak1 one foot in the world of the senses1 and never burn the bridges keeping them in communication with reality4 ,they thus are to be incapable of comprehending or at least of appreciating certain theories more interesting than utilitarian1 such as the non-,euclidean geometries4 ,according to that1 the first two parts of this book1 on number and space1 should seem to them void of all substance and would only baffle them4 ,but that is not true4 ,and first of all1 are they such uncompromising realists as has been said8 ,are they absolutely refractory1 ,i do not say to metaphysic1 but at least to everything metaphysical8 ,recall the name of ,berkeley1 born in ,ireland doubtless1 but immediately adopted by the ,english1 who marked a natural and necessary stage in the development of ,english philosophy4 ,is this not enough to show they are capable of making ascensions otherwise than in a captive balloon8 ,and to return to ,america1 is not the ,monist published at ,chicago1 that review which even to us seems bold and yet which finds readers8 ,and in mathematics8 ,do you think ,american geometers are concerned only about applications8 ,far from it4 ,the part of the science they cultivate most devotedly is the theory of groups of substitutions1 and under its most abstract form1 the farthest removed from the practical4 ,moreover1 ,dr4 ,halsted gives regularly each year a review of all productions relative to the non-,euclidean geometry1 and he has about him a public deeply interested in his work4 ,he has initiated this public into the ideas of ,hilbert1 and he has even written an elementary treatise on ',rational ,geometry1' based on the principles of the renowned ,german savant4 ,to introduce this principle into teaching is surely this time to burn all bridges of reliance upon sensory intuition1 and this is1 ,i confess1 a boldness which seems to me almost rashness4 ,the ,american public is therefore much better prepared than has been thought for investigating the origin of the notion of space4 ,moreover1 to analyze this concept is not to sacrifice reality to ,i know not what phantom4 ,the geometric language is after all only a language4 ,space is only a word that we have believed a thing4 ,what is the origin of this word and of other words also8 ,what things do they hide8 ,to ask this is permissible2 to forbid it would be1 on the contrary1 to be a dupe of words2 it would be to adore a metaphysical idol1 like savage peoples who prostrate themselves before a statue of wood without daring to take a look at what is within4 ,in the study of nature1 the contrast between the ,anglo-,saxon spirit and the ,latin spirit is still greater4 ,the ,latins seek in general to put their thought in mathematical form2 the ,english prefer to express it by a material representation4 ,both doubtless rely only on experience for knowing the world2 when they happen to go beyond this1 they consider their foreknowledge as only provisional1 and they hasten to ask its definitive confirmation from nature herself4 ,but experience is not all1 and the savant is not passive2 he does not wait for the truth to come and find him1 or for a chance meeting to bring him face to face with it4 ,he must go to meet it1 and it is for his thinking to reveal to him the way leading thither4 ,for that there is need of an instrument2 well1 just there begins the difference--the instrument the ,latins ordinarily choose is not that preferred by the ,anglo-,saxons4 ,for a ,latin1 truth can be expressed only by equations2 it must obey laws simple1 logical1 symmetric and fitted to satisfy minds in love with mathematical elegance4 ,the ,anglo-,saxon to depict a phenomenon will first be engrossed in making a model1 and he will make it with common materials1 such as our crude1 unaided senses show us them4 ,he also makes a hypothesis1 he assumes implicitly that nature1 in her finest elements1 is the same as in the complicated aggregates which alone are within the reach of our senses4 ,he concludes from the body to the atom4 ,both therefore make hypotheses1 and this indeed is necessary1 since no scientist has ever been able to get on without them4 ,the essential thing is never to make them unconsciously4 ,from this point of view again1 it would be well for these two sorts of physicists to know something of each other2 in studying the work of minds so unlike their own1 they will immediately recognize that in this work there has been an accumulation of hypotheses4 ,doubtless this will not suffice to make them comprehend that they on their part have made just as many2 each sees the mote without seeing the beam2 but by their criticisms they will warn their rivals1 and it may be supposed these will not fail to render them the same service4 ,the ,english procedure often seems to us crude1 the analogies they think they discover to us seem at times superficial2 they are not sufficiently interlocked1 not precise enough2 they sometimes permit incoherences1 contradictions in terms1 which shock a geometric spirit and which the employment of the mathematical method would immediately have put in evidence4 ,but most often it is1 on the other hand1 very fortunate that they have not perceived these contradictions2 else would they have rejected their model and could not have deduced from it the brilliant results they have often made to come out of it4 ,and then these very contradictions1 when they end by perceiving them1 have the advantage of showing them the hypothetical character of their conceptions1 whereas the mathematical method1 by its apparent rigor and inflexible course1 often inspires in us a confidence nothing warrants1 and prevents our looking about us4 ,from another point of view1 however1 the two conceptions are very unlike1 and if all must be said1 they are very unlike because of a common fault4 ,the ,english wish to make the world out of what we see4 ,i mean what we see with the unaided eye1 not the microscope1 nor that still more subtile microscope1 the human head guided by scientific induction4 ,the ,latin wants to make it out of formulas1 but these formulas are still the quintessenced expression of what we see4 ,in a word1 both would make the unknown out of the known1 and their excuse is that there is no way of doing otherwise4 ,and yet is this legitimate1 if the unknown be the simple and the known the complex8 ,shall we not get of the simple a false idea1 if we think it like the complex1 or worse yet if we strive to make it out of elements which are themselves compounds8 ,is not each great advance accomplished precisely the day some one has discovered under the complex aggregate shown by our senses something far more simple1 not even resembling it--as when ,newton replaced ,kepler's three laws by the single law of gravitation1 which was something simpler1 equivalent1 yet unlike8 ,one is justified in asking if we are not on the eve of just such a revolution or one even more important4 ,matter seems on the point of losing its mass1 its solidest attribute1 and resolving itself into electrons4 ,mechanics must then give place to a broader conception which will explain it1 but which it will not explain4 ,so it was in vain the attempt was made in ,england to construct the ether by material models1 or in ,france to apply to it the laws of dynamic4 ,the ether it is1 the unknown1 which explains matter1 the known2 matter is incapable of explaining the ether4 ,,poincar4 ,,introduction ,,by ,,professor ,,josiah ,,royce ,,harvard ,,university ,the treatise of a master needs no commendation through the words of a mere learner4 ,but1 since my friend and former fellow student1 the translator of this volume1 has joined with another of my colleagues1 ,professor ,cattell1 in asking me to undertake the task of calling the attention of my fellow students to the importance and to the scope of ,m4 ,poincar's volume1 ,i accept the office1 not as one competent to pass judgment upon the book1 but simply as a learner1 desirous to increase the number of those amongst us who are already interested in the type of researches to which ,m4 ,poincar has so notably contributed4 ,i ,the branches of inquiry collectively known as the ,philosophy of ,science have undergone great changes since the appearance of ,herbert ,spencer's ,first ,principles1 that volume which a large part of the general public in this country used to regard as the representative compend of all modern wisdom relating to the foundations of scientific knowledge4 ,the summary which ,m4 ,poincar gives1 at the outset of his own introduction to the present work1 where he states the view which the 'superficial observer' takes of scientific truth1 suggests1 not indeed ,spencer's own most characteristic theories1 but something of the spirit in which many disciples of ,spencer interpreting their master's formulas used to conceive the position which science occupies in dealing with experience4 ,it was well known to them1 indeed1 that experience is a constant guide1 and an inexhaustible source both of novel scientific results and of unsolved problems2 but the fundamental ,spencerian principles of science1 such as 'the persistence of force1' the 'rhythm of motion' and the rest1 were treated by ,spencer himself as demonstrably objective1 although indeed 'relative' truths1 capable of being tested once for all by the 'inconceivability of the opposite1' and certain to hold true for the whole 'knowable' universe4 ,thus1 whether one dwelt upon the results of such a mathematical procedure as that to which ,m4 ,poincar refers in his opening paragraphs1 or whether1 like ,spencer himself1 one applied the 'first principles' to regions of less exact science1 this confidence that a certain orthodoxy regarding the principles of science was established forever was characteristic of the followers of the movement in question4 ,experience1 lighted up by reason1 seemed to them to have predetermined for all future time certain great theoretical results regarding the real constitution of the 'knowable' cosmos4 ,whoever doubted this doubted 'the verdict of science4' ,some of us well remember how1 when ,stallo's ',principles and ,theories of ,modern ,physics' first appeared1 this sense of scientific orthodoxy was shocked amongst many of our ,american readers and teachers of science4 ,i myself can recall to mind some highly authoritative reviews of that work in which the author was more or less sharply taken to task for his ignorant presumption in speaking with the freedom that he there used regarding such sacred possessions of humanity as the fundamental concepts of physics4 ,that very book1 however1 has quite lately been translated into ,german as a valuable contribution to some of the most recent efforts to reconstitute a modern 'philosophy of nature4' ,and whatever may be otherwise thought of ,stallo's critical methods1 or of his results1 there can be no doubt that1 at the present moment1 if his book were to appear for the first time1 nobody would attempt to discredit the work merely on account of its disposition to be agnostic regarding the objective reality of the concepts of the kinetic theory of gases1 or on account of its call for a logical rearrangement of the fundamental concepts of the theory of energy4 ,we are no longer able so easily to know heretics at first sight4 ,for we now appear to stand in this position3 ,the control of natural phenomena1 which through the sciences men have attained1 grows daily vaster and more detailed1 and in its details more assured4 ,phenomena men know and predict better than ever4 ,but regarding the most general theories1 and the most fundamental1 of science1 there is no longer any notable scientific orthodoxy4 ,thus1 as knowledge grows firmer and wider1 conceptual construction becomes less rigid4 ,the field of the theoretical philosophy of nature--yes1 the field of the logic of science--this whole region is to-day an open one4 ,whoever will work there must indeed accept the verdict of experience regarding what happens in the natural world4 ,so far he is indeed bound4 ,but he may undertake without hindrance from mere tradition the task of trying afresh to reduce what happens to conceptual unity4 ,the circle-squarers and the inventors of devices for perpetual motion are indeed still as unwelcome in scientific company as they were in the days when scientific orthodoxy was more rigidly defined2 but that is not because the foundations of geometry are now viewed as completely settled1 beyond controversy1 nor yet because the 'persistence of force' has been finally so defined as to make the 'opposite inconceivable' and the doctrine of energy beyond the reach of novel formulations4 ,no1 the circle-squarers and the inventors of devices for perpetual motion are to-day discredited1 not because of any unorthodoxy of their general philosophy of nature1 but because their views regarding special facts and processes stand in conflict with certain equally special results of science which themselves admit of very various general theoretical interpretations4 ,certain properties of the irrational number pi are known1 in sufficient multitude to justify the mathematician in declining to listen to the arguments of the circle-squarer2 but1 despite great advances1 and despite the assured results of ,dedekind1 of ,cantor1 of ,weierstrass and of various others1 the general theory of the logic of the numbers1 rational and irrational1 still presents several important features of great obscurity2 and the philosophy of the concepts of geometry yet remains1 in several very notable respects1 unconquered territory1 despite the work of ,hilbert and of ,pieri1 and of our author himself4 ,the ordinary inventors of the perpetual motion machines still stand in conflict with accepted generalizations2 but nobody knows as yet what the final form of the theory of energy will be1 nor can any one say precisely what place the phenomena of the radioactive bodies will occupy in that theory4 ,the alchemists would not be welcome workers in modern laboratories2 yet some sorts of transformation and of evolution of the elements are to-day matters which theory can find it convenient1 upon occasion1 to treat as more or less exactly definable possibilities2 while some newly observed phenomena tend to indicate1 not indeed that the ancient hopes of the alchemists were well founded1 but that the ultimate constitution of matter is something more fluent1 less invariant1 than the theoretical orthodoxy of a recent period supposed4 ,again1 regarding the foundations of biology1 a theoretical orthodoxy grows less possible1 less definable1 less conceivable 7even as a hope7 the more knowledge advances4 ,once 'mechanism' and 'vitalism' were mutually contradictory theories regarding the ultimate constitution of living bodies4 ,now they are obviously becoming more and more 'points of view1' diverse but not necessarily conflicting4 ,so far as you find it convenient to limit your study of vital processes to those phenomena which distinguish living matter from all other natural objects1 you may assume1 in the modern 'pragmatic' sense1 the attitude of a 'neo-vitalist4' ,so far1 however1 as you are able to lay stress1 with good results1 upon the many ways in which the life processes can be assimilated to those studied in physics and in chemistry1 you work as if you were a partisan of 'mechanics4' ,in any case1 your special science prospers by reason of the empirical discoveries that you make4 ,and your theories1 whatever they are1 must not run counter to any positive empirical results4 ,but otherwise1 scientific orthodoxy no longer predetermines what alone it is respectable for you to think about the nature of living substance4 ,this gain in the freedom of theory1 coming1 as it does1 side by side with a constant increase of a positive knowledge of nature1 lends itself to various interpretations1 and raises various obvious questions4 ,,ii ,one of the most natural of these interpretations1 one of the most obvious of these questions1 may be readily stated4 ,is not the lesson of all these recent discussions simply this1 that general theories are simply vain1 that a philosophy of nature is an idle dream1 and that the results of science are coextensive with the range of actual empirical observation and of successful prediction8 ,if this is indeed the lesson1 then the decline of theoretical orthodoxy in science is--like the eclipse of dogma in religion--merely a further lesson in pure positivism1 another proof that man does best when he limits himself to thinking about what can be found in human experience1 and in trying to plan what can be done to make human life more controllable and more reasonable4 ,what we are free to do as we please--is it any longer a serious business8 ,what we are free to think as we please--is it of any further interest to one who is in search of truth8 ,if certain general theories are mere conceptual constructions1 which to-day are1 and to-morrow are cast into the oven1 why dignify them by the name of philosophy8 ,has science any place for such theories8 ,why be a 'neo-vitalist1' or an 'evolutionist1' or an 'atomist1' or an ',energetiker'8 ,why not say1 plainly3 ,such and such phenomena1 thus and thus described1 have been observed2 such and such experiences are to be expected1 since the hypotheses by the terms of which we are required to expect them have been verified too often to let us regard the agreement with experience as due merely to chance2 so much then with reasonable assurance we know2 all else is silence--or else is some matter to be tested by another experiment8 ,why not limit our philosophy of science strictly to such a counsel of resignation8 ,why not substitute1 for the old scientific orthodoxy1 simply a confession of ignorance1 and a resolution to devote ourselves to the business of enlarging the bounds of actual empirical knowledge8 ,such comments upon the situation just characterized are frequently made4 ,unfortunately1 they seem not to content the very age whose revolt from the orthodoxy of traditional theory1 whose uncertainty about all theoretical formulations1 and whose vast wealth of empirical discoveries and of rapidly advancing special researches1 would seem most to justify these very comments4 ,never has there been better reason than there is to-day to be content1 if rational man could be content1 with a pure positivism4 ,the splendid triumphs of special research in the most various fields1 the constant increase in our practical control over nature--these1 our positive and growing possessions1 stand in glaring contrast to the failure of the scientific orthodoxy of a former period to fix the outlines of an ultimate creed about the nature of the knowable universe4 ,why not 'take the cash and let the credit go'8 ,why pursue the elusive theoretical 'unification' any further1 when what we daily get from our sciences is an increasing wealth of detailed information and of practical guidance8 ,as a fact1 however1 the known answer of our own age to these very obvious comments is a constant multiplication of new efforts towards large and unifying theories4 ,if theoretical orthodoxy is no longer clearly definable1 theoretical construction was never more rife4 ,the history of the doctrine of evolution1 even in its most recent phases1 when the theoretical uncertainties regarding the 'factors of evolution' are most insisted upon1 is full of illustrations of this remarkable union of scepticism in critical work with courage regarding the use of the scientific imagination4 ,the history of those controversies regarding theoretical physics1 some of whose principal phases ,m4 ,poincar1 in his book1 sketches with the hand of the master1 is another illustration of the consciousness of the time4 ,men have their freedom of thought in these regions2 and they feel the need of making constant and constructive use of this freedom4 ,and the men who most feel this need are by no means in the majority of cases professional metaphysicians--or students who1 like myself1 have to view all these controversies amongst the scientific theoreticians from without as learners4 ,these large theoretical constructions are due1 on the contrary1 in a great many cases to special workers1 who have been driven to the freedom of philosophy by the oppression of experience1 and who have learned in the conflict with special problems the lesson that they now teach in the form of general ideas regarding the philosophical aspects of science4 ,why1 then1 does science actually need general theories1 despite the fact that these theories inevitably alter and pass away8 ,what is the service of a philosophy of science1 when it is certain that the philosophy of science which is best suited to the needs of one generation must be superseded by the advancing insight of the next generation8 ,why must that which endlessly grows1 namely1 man's knowledge of the phenomenal order of nature1 be constantly united in men's minds with that which is certain to decay1 namely1 the theoretical formulation of special knowledge in more or less completely unified systems of doctrine8 ,i understand our author's volume to be in the main an answer to this question4 ,to be sure1 the compact and manifold teachings which this text contains relate to a great many different special issues4 ,a student interested in the problems of the philosophy of mathematics1 or in the theory of probabilities1 or in the nature and office of mathematical physics1 or in still other problems belonging to the wide field here discussed1 may find what he wants here and there in the text1 even in case the general issues which give the volume its unity mean little to him1 or even if he differs from the author's views regarding the principal issues of the book4 ,but in the main1 this volume must be regarded as what its title indicates--a critique of the nature and place of hypothesis in the work of science and a study of the logical relations of theory and fact4 ,the result of the book is a substantial justification of the scientific utility of theoretical construction--an abandonment of dogma1 but a vindication of the rights of the constructive reason4 ,,iii ,the most notable of the results of our author's investigation of the logic of scientific theories relates1 as ,i understand his work1 to a topic which the present state of logical investigation1 just summarized1 makes especially important1 but which has thus far been very inadequately treated in the text-books of inductive logic4 ,the useful hypotheses of science are of two kinds3 #a4 ,the hypotheses which are valuable precisely because they are either verifiable or else refutable through a definite appeal to the tests furnished by experience2 and #b4 ,the hypotheses which1 despite the fact that experience suggests them1 are valuable despite1 or even because1 of the fact that experience can neither confirm nor refute them4 ,the contrast between these two kinds of hypotheses is a prominent topic of our author's discussion4 ,hypotheses of the general type which ,i have here placed first in order are the ones which the text-books of inductive logic and those summaries of scientific method which are customary in the course of the elementary treatises upon physical science are already accustomed to recognize and to characterize4 ,the value of such hypotheses is indeed undoubted4 ,but hypotheses of the type which ,i have here named in the second place are far less frequently recognized in a perfectly explicit way as useful aids in the work of special science4 ,one usually either fails to admit their presence in scientific work1 or else remains silent as to the reasons of their usefulness4 ,our author's treatment of the work of science is therefore especially marked by the fact that he explicitly makes prominent both the existence and the scientific importance of hypotheses of this second type4 ,they occupy in his discussion a place somewhat analogous to each of the two distinct positions occupied by the 'categories' and the 'forms of sensibility1' on the one hand1 and by the 'regulative principles of the reason1' on the other hand1 in the ,kantian theory of our knowledge of nature4 ,that is1 these hypotheses which can neither be confirmed nor refuted by experience appear1 in ,m4 ,poincar's account1 partly 7like the conception of 'continuous quantity'7 as devices of the understanding whereby we give conceptual unity and an invisible connectedness to certain types of phenomenal facts which come to us in a discrete form and in a confused variety2 and partly 7like the larger organizing concepts of science7 as principles regarding the structure of the world in its wholeness2 i4 e41 as principles in the light of which we try to interpret our experience1 so as to give to it a totality and an inclusive unity such as ,euclidean space1 or such as the world of the theory of energy is conceived to possess4 ,thus viewed1 ,m4 ,poincar's logical theory of this second class of hypotheses undertakes to accomplish1 with modern means and in the light of to-day's issues1 a part of what ,kant endeavored to accomplish in his theory of scientific knowledge with the limited means which were at his disposal4 ,those aspects of science which are determined by the use of the hypotheses of this second kind appear in our author's account as constituting an essential human way of viewing nature1 an interpretation rather than a portrayal or a prediction of the objective facts of nature1 an adjustment of our conceptions of things to the internal needs of our intelligence1 rather than a grasping of things as they are in themselves4 ,to be sure1 ,m4 ,poincar's view1 in this portion of his work1 obviously differs1 meanwhile1 from that of ,kant1 as well as this agrees1 in a measure1 with the spirit of the ,kantian epistemology4 ,i do not mean therefore to class our author as a ,kantian4 ,for ,kant1 the interpretations imposed by the 'forms of sensibility1' and by the 'categories of the understanding1' upon our doctrine of nature are rigidly predetermined by the unalterable 'form' of our intellectual powers4 ,we 'must' thus view facts1 whatever the data of sense must be4 ,this1 of course1 is not ,m4 ,poincar's view4 ,a similarly rigid predetermination also limits the ,kantian 'ideas of the reason' to a certain set of principles whose guidance of the course of our theoretical investigations is indeed only 'regulative1' but is 'a priori1' and so unchangeable4 ,for ,m4 ,poincar1 on the contrary1 all this adjustment of our interpretations of experience to the needs of our intellect is something far less rigid and unalterable1 and is constantly subject to the suggestions of experience4 ,we must indeed interpret in our own way2 but our way is itself only relatively determinate2 it is essentially more or less plastic2 other interpretations of experience are conceivable4 ,those that we use are merely the ones found to be most convenient4 ,but this convenience is not absolute necessity4 ,unverifiable and irrefutable hypotheses in science are indeed1 in general1 indispensable aids to the organization and to the guidance of our interpretation of experience4 ,but it is experience itself which points out to us what lines of interpretation will prove most convenient4 ,instead of ,kant's rigid list of a priori 'forms1' we consequently have in ,m4 ,poincar's account a set of conventions1 neither wholly subjective and arbitrary1 nor yet imposed upon us unambiguously by the external compulsion of experience4 ,the organization of science1 so far as this organization is due to hypotheses of the kind here in question1 thus resembles that of a constitutional government--neither absolutely necessary1 nor yet determined apart from the will of the subjects1 nor yet accidental--a free1 yet not a capricious establishment of good order1 in conformity with empirical needs4 ,characteristic remains1 however1 for our author1 as1 in his decidedly contrasting way1 for ,kant1 the thought that without principles which at every stage transcend precise confirmation through such experience as is then accessible the organization of experience is impossible4 ,whether one views these principles as conventions or as a priori 'forms1' they may therefore be described as hypotheses1 but as hypotheses that1 while lying at the basis of our actual physical sciences1 at once refer to experience and help us in dealing with experience1 and are yet neither confirmed nor refuted by the experiences which we possess or which we can hope to attain4 ,three special instances or classes of instances1 according to our author's account1 may be used as illustrations of this general type of hypotheses4 ,they are3 7#a7 ,the hypothesis of the existence of continuous extensive quanta in nature2 7#b7 ,the principles of geometry2 7#c7 ,the principles of mechanics and of the general theory of energy4 ,in case of each of these special types of hypotheses we are at first disposed1 apart from reflection1 to say that we find the world to be thus or thus1 so that1 for instance1 we can confirm the thesis according to which nature contains continuous magnitudes2 or can prove or disprove the physical truth of the postulates of ,euclidean geometry2 or can confirm by definite experience the objective validity of the principles of mechanics4 ,a closer examination reveals1 according to our author1 the incorrectness of all such opinions4 ,hypotheses of these various special types are needed2 and their usefulness can be empirically shown4 ,they are in touch with experience2 and that they are not merely arbitrary conventions is also verifiable4 ,they are not a priori necessities2 and we can easily conceive intelligent beings whose experience could be best interpreted without using these hypotheses4 ,yet these hypotheses are not subject to direct confirmation or refutation by experience4 ,they stand then in sharp contrast to the scientific hypotheses of the other1 and more frequently recognized1 type1 i4 e41 to the hypotheses which can be tested by a definite appeal to experience4 ,to these other hypotheses our author attaches1 of course1 great importance4 ,his treatment of them is full of a living appreciation of the significance of empirical investigation4 ,but the central problem of the logic of science thus becomes the problem of the relation between the two fundamentally distinct types of hypotheses1 i4 e41 between those which can not be verified or refuted through experience1 and those which can be empirically tested4 ,,iv ,the detailed treatment which ,m4 ,poincar gives to the problem thus defined must be learned from his text4 ,it is no part of my purpose to expound1 to defend or to traverse any of his special conclusions regarding this matter4 ,yet ,i can not avoid observing that1 while ,m4 ,poincar strictly confines his illustrations and his expressions of opinion to those regions of science wherein1 as special investigator1 he is himself most at home1 the issues which he thus raises regarding the logic of science are of even more critical importance and of more impressive interest when one applies ,m4 ,poincar's methods to the study of the concepts and presuppositions of the organic and of the historical and social sciences1 than when one confines one's attention1 as our author here does1 to the physical sciences4 ,it belongs to the province of an introduction like the present to point out1 however briefly and inadequately1 that the significance of our author's ideas extends far beyond the scope to which he chooses to confine their discussion4 ,the historical sciences1 and in fact all those sciences such as geology1 and such as the evolutionary sciences in general1 undertake theoretical constructions which relate to past time4 ,hypotheses relating to the more or less remote past stand1 however1 in a position which is very interesting from the point of view of the logic of science4 ,directly speaking1 no such hypothesis is capable of confirmation or of refutation1 because we can not return into the past to verify by our own experience what then happened4 ,yet indirectly1 such hypotheses may lead to predictions of coming experience4 ,these latter will be subject to control4 ,thus1 ,schliemann's confidence that the legend of ,troy had a definite historical foundation led to predictions regarding what certain excavations would reveal4 ,in a sense somewhat different from that which filled ,schliemann's enthusiastic mind1 these predictions proved verifiable4 ,the result has been a considerable change in the attitude of historians toward the legend of ,troy4 ,geological investigation leads to predictions regarding the order of the strata or the course of mineral veins in a district1 regarding the fossils which may be discovered in given formations1 and so on4 ,these hypotheses are subject to the control of experience4 ,the various theories of evolutionary doctrine include many hypotheses capable of confirmation and of refutation by empirical tests4 ,yet1 despite all such empirical control1 it still remains true that whenever a science is mainly concerned with the remote past1 whether this science be archeology1 or geology1 or anthropology1 or ,old ,testament history1 the principal theoretical constructions always include features which no appeal to present or to accessible future experience can ever definitely test4 ,hence the suspicion with which students of experimental science often regard the theoretical constructions of their confrres of the sciences that deal with the past4 ,the origin of the races of men1 of man himself1 of life1 of species1 of the planet2 the hypotheses of anthropologists1 of archeologists1 of students of 'higher criticism'--all these are matters which the men of the laboratory often regard with a general incredulity as belonging not at all to the domain of true science4 ,yet no one can doubt the importance and the inevitableness of endeavoring to apply scientific method to these regions also4 ,science needs theories regarding the past history of the world4 ,and no one who looks closer into the methods of these sciences of past time can doubt that verifiable and unverifiable hypotheses are in all these regions inevitably interwoven2 so that1 while experience is always the guide1 the attitude of the investigator towards experience is determined by interests which have to be partially due to what ,i should call that 'internal meaning1' that human interest in rational theoretical construction which inspires the scientific inquiry2 and the theoretical constructions which prevail in such sciences are neither unbiased reports of the actual constitution of an external reality1 nor yet arbitrary constructions of fancy4 ,these constructions in fact resemble in a measure those which ,m4 ,poincar in this book has analyzed in the case of geometry4 ,they are constructions molded1 but not predetermined in their details1 by experience4 ,we report facts2 we let the facts speak2 but we1 as we investigate1 in the popular phrase1 'talk back' to the facts4 ,we interpret as well as report4 ,man is not merely made for science1 but science is made for man4 ,it expresses his deepest intellectual needs1 as well as his careful observations4 ,it is an effort to bring internal meanings into harmony with external verifications4 ,it attempts therefore to control1 as well as to submit1 to conceive with rational unity1 as well as to accept data4 ,its arts are those directed towards self-possession as well as towards an imitation of the outer reality which we find4 ,it seeks therefore a disciplined freedom of thought4 ,the discipline is as essential as the freedom2 but the latter has also its place4 ,the theories of science are human1 as well as objective1 internally rational1 as well as 7when that is possible7 subject to external tests4 ,in a field very different from that of the historical sciences1 namely1 in a science of observation and of experiment1 which is at the same time an organic science1 ,i have been led in the course of some study of the history of certain researches to notice the existence of a theoretical conception which has proved extremely fruitful in guiding research1 but which apparently resembles in a measure the type of hypotheses of which ,m4 ,poincar speaks when he characterizes the principles of mechanics and of the theory of energy4 ,i venture to call attention here to this conception1 which seems to me to illustrate ,m4 ,poincar's view of the functions of hypothesis in scientific work4 ,the modern science of pathology is usually regarded as dating from the earlier researches of ,virchow1 whose ',cellular ,pathology' was the outcome of a very careful and elaborate induction4 ,virchow1 himself1 felt a strong aversion to mere speculation4 ,he endeavored to keep close to observation1 and to relieve medical science from the control of fantastic theories1 such as those of the ,naturphilosophen had been4 ,yet ,virchow's researches were1 as early as #ahdg1 or still earlier1 already under the guidance of a theoretical presupposition which he himself states as follows3 ,we have learned to recognize1 he says1 that diseases are not autonomous organisms1 that they are no entities that have entered into the body1 that they are no parasites which take root in the body1 but that they merely show us the course of the vital processes under altered conditions 7'dasz sie nur ,ablauf der ,lebenserscheinungen unter vernderten ,bedingungen darstellen'74 ,the enormous importance of this theoretical presupposition for all the early successes of modern pathological investigation is generally recognized by the experts4 ,i do not doubt this opinion4 ,it appears to be a commonplace of the history of this science4 ,but in ,virchow's later years this very presupposition seemed to some of his contemporaries to be called in question by the successes of recent bacteriology4 ,the question arose whether the theoretical foundations of ,virchow's pathology had not been set aside4 ,and in fact the theory of the parasitical origin of a vast number of diseased conditions has indeed come upon an empirical basis to be generally recognized4 ,yet to the end of his own career ,virchow stoutly maintained that in all its essential significance his own fundamental principle remained quite untouched by the newer discoveries4 ,and1 as a fact1 this view could indeed be maintained4 ,for if diseases proved to be the consequences of the presence of parasites1 the diseases themselves1 so far as they belonged to the diseased organism1 were still not the parasites1 but were1 as before1 the reaction of the organism to the vernderte ,bedingungen which the presence of the parasites entailed4 ,so ,virchow could well insist4 ,and if the famous principle in question is only stated with sufficient generality1 it amounts simply to saying that if a disease involves a change in an organism1 and if this change is subject to law at all1 then the nature of the organism and the reaction of the organism to whatever it is which causes the disease must be understood in case the disease is to be understood4 ,for this very reason1 however1 ,virchow's theoretical principle in its most general form could be neither confirmed nor refuted by experience4 ,it would remain empirically irrefutable1 so far as ,i can see1 even if we should learn that the devil was the true cause of all diseases4 ,for the devil himself would then simply predetermine the vernderte ,bedingungen to which the diseased organism would be reacting4 ,let bullets or bacteria1 poisons or compressed air1 or the devil be the ,bedingungen to which a diseased organism reacts1 the postulate that ,virchow states in the passage just quoted will remain irrefutable1 if only this postulate be interpreted to meet the case4 ,for the principle in question merely says that whatever entity it may be1 bullet1 or poison1 or devil1 that affects the organism1 the disease is not that entity1 but is the resulting alteration in the process of the organism4 ,i insist1 then1 that this principle of ,virchow's is no trial supposition1 no scientific hypothesis in the narrower sense--capable of being submitted to precise empirical tests4 ,it is1 on the contrary1 a very precious leading idea1 a theoretical interpretation of phenomena1 in the light of which observations are to be made--'a regulative principle' of research4 ,it is equivalent to a resolution to search for those detailed connections which link the processes of disease to the normal process of the organism4 ,such a search undertakes to find the true unity1 whatever that may prove to be1 wherein the pathological and the normal processes are linked4 ,now without some such leading idea1 the cellular pathology itself could never have been reached2 because the empirical facts in question would never have been observed4 ,hence this principle of ,virchow's was indispensable to the growth of his science4 ,yet it was not a verifiable and not a refutable hypothesis4 ,one value of unverifiable and irrefutable hypotheses of this type lies1 then1 in the sort of empirical inquiries which they initiate1 inspire1 organize and guide4 ,in these inquiries hypotheses in the narrower sense1 that is1 trial propositions which are to be submitted to definite empirical control1 are indeed everywhere present4 ,and the use of the other sort of principles lies wholly in their application to experience4 ,yet without what ,i have just proposed to call the 'leading ideas' of a science1 that is1 its principles of an unverifiable and irrefutable character1 suggested1 but not to be finally tested1 by experience1 the hypotheses in the narrower sense would lack that guidance which1 as ,m4 ,poincar has shown1 the larger ideas of science give to empirical investigation4 ,v ,i have dwelt1 no doubt1 at too great length upon one aspect only of our author's varied and well-balanced discussion of the problems and concepts of scientific theory4 ,of the hypotheses in the narrower sense and of the value of direct empirical control1 he has also spoken with the authority and the originality which belong to his position4 ,and in dealing with the foundations of mathematics he has raised one or two questions of great philosophical import into which ,i have no time1 even if ,i had the right1 to enter here4 ,in particular1 in speaking of the essence of mathematical reasoning1 and of the difficult problem of what makes possible novel results in the field of pure mathematics1 ,m4 ,poincar defends a thesis regarding the office of 'demonstration by recurrence'--a thesis which is indeed disputable1 which has been disputed and which ,i myself should be disposed1 so far as ,i at present understand the matter1 to modify in some respects1 even in accepting the spirit of our author's assertion4 ,yet there can be no doubt of the importance of this thesis1 and of the fact that it defines a characteristic that is indeed fundamental in a wide range of mathematical research4 ,the philosophical problems that lie at the basis of recurrent proofs and processes are1 as ,i have elsewhere argued1 of the most fundamental importance4 ,these1 then1 are a few hints relating to the significance of our author's discussion1 and a few reasons for hoping that our own students will profit by the reading of the book as those of other nations have already done4 ,of the person and of the life-work of our author a few words are here1 in conclusion1 still in place1 addressed1 not to the students of his own science1 to whom his position is well known1 but to the general reader who may seek guidance in these pages4 ,jules ,henri ,poincar was born at ,nancy1 in #ahed1 the son of a professor in the ,faculty of ,medicine at ,nancy4 ,he studied at the cole ,polytechnique and at the cole des ,mines1 and later received his doctorate in mathematics in #ahgi4 ,in #ahhc he began courses of instruction in mathematics at the cole ,polytechnique2 in #ahhf received a professorship of mathematical physics in the ,faculty of ,sciences at ,paris2 then became member of the ,academy of ,sciences at ,paris1 in #ahhg1 and devoted his life to instruction and investigation in the regions of pure mathematics1 of mathematical physics and of celestial mechanics4 ,his list of published treatises relating to various branches of his chosen sciences is long2 and his original memoirs have included several momentous investigations1 which have gone far to transform more than one branch of research4 ,his presence at the ,international ,congress of ,arts and ,science in ,st4 ,louis was one of the most noticeable features of that remarkable gathering of distinguished foreign guests4 ,in ,poincar the reader meets1 then1 not one who is primarily a speculative student of general problems for their own sake1 but an original investigator of the highest rank in several distinct1 although interrelated1 branches of modern research4 ,the theory of functions--a highly recondite region of pure mathematics--owes to him advances of the first importance1 for instance1 the definition of a new type of functions4 ,the 'problem of the three bodies1' a famous and fundamental problem of celestial mechanics1 has received from his studies a treatment whose significance has been recognized by the highest authorities4 ,his international reputation has been confirmed by the conferring of more than one important prize for his researches4 ,his membership in the most eminent learned societies of various nations is widely extended2 his volumes bearing upon various branches of mathematics and of mathematical physics are used by special students in all parts of the learned world2 in brief1 he is1 as geometer1 as analyst and as a theoretical physicist1 a leader of his age4 ,meanwhile1 as contributor to the philosophical discussion of the bases and methods of science1 ,m4 ,poincar has long been active4 ,when1 in #ahic1 the admirable ,revue de ,mtaphysique et de ,morale began to appear1 ,m4 ,poincar was soon found amongst the most satisfactory of the contributors to the work of that journal1 whose office it has especially been to bring philosophy and the various special sciences 7both natural and moral7 into a closer mutual understanding4 ,the discussions brought together in the present volume are in large part the outcome of ,m4 ,poincar's contributions to the ,revue de ,mtaphysique et de ,morale4 ,the reader of ,m4 ,poincar's book is in presence1 then1 of a great special investigator who is also a philosopher4 ,,science ,,and ,,hypothesis ,,introduction ,for a superficial observer1 scientific truth is beyond the possibility of doubt2 the logic of science is infallible1 and if the scientists are sometimes mistaken1 this is only from their mistaking its rules4 ,the mathematical verities flow from a small number of self-evident propositions by a chain of impeccable reasonings2 they impose themselves not only on us1 but on nature itself4 ,they fetter1 so to speak1 the ,creator and only permit him to choose between some relatively few solutions4 ,a few experiments then will suffice to let us know what choice he has made4 ,from each experiment a crowd of consequences will follow by a series of mathematical deductions1 and thus each experiment will make known to us a corner of the universe4 ,behold what is for many people in the world1 for scholars getting their first notions of physics1 the origin of scientific certitude4 ,this is what they suppose to be the rle of experimentation and mathematics4 ,this same conception1 a hundred years ago1 was held by many savants who dreamed of constructing the world with as little as possible taken from experiment4 ,on a little more reflection it was perceived how great a place hypothesis occupies2 that the mathematician can not do without it1 still less the experimenter4 ,and then it was doubted if all these constructions were really solid1 and believed that a breath would overthrow them4 ,to be skeptical in this fashion is still to be superficial4 ,to doubt everything and to believe everything are two equally convenient solutions2 each saves us from thinking4 ,instead of pronouncing a summary condemnation1 we ought therefore to examine with care the rle of hypothesis2 we shall then recognize1 not only that it is necessary1 but that usually it is legitimate4 ,we shall also see that there are several sorts of hypotheses2 that some are verifiable1 and once confirmed by experiment become fruitful truths2 that others1 powerless to lead us astray1 may be useful to us in fixing our ideas2 that others1 finally1 are hypotheses only in appearance and are reducible to disguised definitions or conventions4 ,these last are met with above all in mathematics and the related sciences4 ,thence precisely it is that these sciences get their rigor2 these conventions are the work of the free activity of our mind1 which1 in this domain1 recognizes no obstacle4 ,here our mind can affirm1 since it decrees2 but let us understand that while these decrees are imposed upon our science1 which1 without them1 would be impossible1 they are not imposed upon nature4 ,are they then arbitrary8 ,no1 else were they sterile4 ,experiment leaves us our freedom of choice1 but it guides us by aiding us to discern the easiest way4 ,our decrees are therefore like those of a prince1 absolute but wise1 who consults his council of state4 ,some people have been struck by this character of free convention recognizable in certain fundamental principles of the sciences4 ,they have wished to generalize beyond measure1 and1 at the same time1 they have forgotten that liberty is not license4 ,thus they have reached what is called nominalism1 and have asked themselves if the savant is not the dupe of his own definitions and if the world he thinks he discovers is not simply created by his own caprice4#a ,under these conditions science would be certain1 but deprived of significance4 #a ,see ,le ,roy1 ',science et ,philosophie1' ,revue de ,mtaphysique et de ,morale1 #aija4 ,if this were so1 science would be powerless4 ,now every day we see it work under our very eyes4 ,that could not be if it taught us nothing of reality4 ,still1 the things themselves are not what it can reach1 as the nave dogmatists think1 but only the relations between things4 ,outside of these relations there is no knowable reality4 ,such is the conclusion to which we shall come1 but for that we must review the series of sciences from arithmetic and geometry to mechanics and experimental physics4 ,what is the nature of mathematical reasoning8 ,is is really deductive1 as is commonly supposed8 ,a deeper analysis shows us that it is not1 that it partakes in a certain measure of the nature of inductive reasoning1 and just because of this is it so fruitful4 ,none the less does it retain its character of rigor absolute2 this is the first thing that had to be shown4 ,knowing better now one of the instruments which mathematics puts into the hands of the investigator1 we had to analyze another fundamental notion1 that of mathematical magnitude4 ,do we find it in nature1 or do we ourselves introduce it there8 ,and1 in this latter case1 do we not risk marring everything8 ,comparing the rough data of our senses with that extremely complex and subtile concept which mathematicians call magnitude1 we are forced to recognize a difference2 this frame into which we wish to force everything is of our own construction2 but we have not made it at random4 ,we have made it1 so to speak1 by measure and therefore we can make the facts fit into it without changing what is essential in them4 ,another frame which we impose on the world is space4 ,whence come the first principles of geometry8 ,are they imposed on us by logic8 ,lobachevski has proved not1 by creating non-,euclidean geometry4 ,is space revealed to us by our senses8 ,still no1 for the space our senses could show us differs absolutely from that of the geometer4 ,is experience the source of geometry8 ,a deeper discussion will show us it is not4 ,we therefore conclude that the first principles of geometry are only conventions2 but these conventions are not arbitrary and if transported into another world 7that ,i call the non-,euclidean world and seek to imagine71 then we should have been led to adopt others4 ,in mechanics we should be led to analogous conclusions1 and should see that the principles of this science1 though more directly based on experiment1 still partake of the conventional character of the geometric postulates4 ,thus far nominalism triumphs2 but now we arrive at the physical sciences1 properly so called4 ,here the scene changes2 we meet another sort of hypotheses and we see their fertility4 ,without doubt1 at first blush1 the theories seem to us fragile1 and the history of science proves to us how ephemeral they are2 yet they do not entirely perish1 and of each of them something remains4 ,it is this something we must seek to disentangle1 since there and there alone is the veritable reality4 ,the method of the physical sciences rests on the induction which makes us expect the repetition of a phenomenon when the circumstances under which it first happened are reproduced4 ,if all these circumstances could be reproduced at once1 this principle could be applied without fear2 but that will never happen2 some of these circumstances will always be lacking4 ,are we absolutely sure they are unimportant8 ,evidently not4 ,that may be probable1 it can not be rigorously certain4 ,hence the important rle the notion of probability plays in the physical sciences4 ,the calculus of probabilities is therefore not merely a recreation or a guide to players of baccarat1 and we must seek to go deeper with its foundations4 ,under this head ,i have been able to give only very incomplete results1 so strongly does this vague instinct which lets us discern probability defy analysis4 ,after a study of the conditions under which the physicist works1 ,i have thought proper to show him at work4 ,for that ,i have taken instances from the history of optics and of electricity4 ,we shall see whence have sprung the ideas of ,fresnel1 of ,maxwell1 and what unconscious hypotheses were made by ,ampre and the other founders of electrodynamics4 ,,part ,i ,,number ,,and ,,magnitude ,,chapter ,i ,,on ,,the ,,nature ,,of ,,mathematical ,,reasoning ,i ,the very possibility of the science of mathematics seems an insoluble contradiction4 ,if this science is deductive only in appearance1 whence does it derive that perfect rigor no one dreams of doubting8 ,if1 on the contrary1 all the propositions it enunciates can be deduced one from another by the rules of formal logic1 why is not mathematics reduced to an immense tautology8 ,the syllogism can teach us nothing essentially new1 and1 if everything is to spring from the principle of identity1 everything should be capable of being reduced to it4 ,shall we then admit that the enunciations of all those theorems which fill so many volumes are nothing but devious ways of saying ,a is ,a8 ,without doubt1 we can go back to the axioms1 which are at the source of all these reasonings4 ,if we decide that these can not be reduced to the principle of contradiction1 if still less we see in them experimental facts which could not partake of mathematical necessity1 we have yet the resource of classing them among synthetic a priori judgments4 ,this is not to solve the difficulty1 but only to baptize it2 and even if the nature of synthetic judgments were for us no mystery1 the contradiction would not have disappeared1 it would only have moved back2 syllogistic reasoning remains incapable of adding anything to the data given it3 these data reduce themselves to a few axioms1 and we should find nothing else in the conclusions4 ,no theorem could be new if no new axiom intervened in its demonstration2 reasoning could give us only the immediately evident verities borrowed from direct intuition2 it would be only an intermediary parasite1 and therefore should we not have good reason to ask whether the whole syllogistic apparatus did not serve solely to disguise our borrowing8 ,the contradiction will strike us the more if we open any book on mathematics2 on every page the author will announce his intention of generalizing some proposition already known4 ,does the mathematical method proceed from the particular to the general1 and1 if so1 how then can it be called deductive8 ,if finally the science of number were purely analytic1 or could be analytically derived from a small number of synthetic judgments1 it seems that a mind sufficiently powerful could at a glance perceive all its truths2 nay more1 we might even hope that some day one would invent to express them a language sufficiently simple to have them appear self-evident to an ordinary intelligence4 ,if we refuse to admit these consequences1 it must be conceded that mathematical reasoning has of itself a sort of creative virtue and consequently differs from the syllogism4 ,the difference must even be profound4 ,we shall not1 for example1 find the key to the mystery in the frequent use of that rule according to which one and the same uniform operation applied to two equal numbers will give identical results4 ,all these modes of reasoning1 whether or not they be reducible to the syllogism properly so called1 retain the analytic character1 and just because of that are powerless4 ,,ii ,the discussion is old2 ,leibnitz tried to prove #b and #b make #d2 let us look a moment at his demonstration4 ,i will suppose the number #a defined and also the operation x #a which consists in adding unity to a given number x4 ,these definitions1 whatever they be1 do not enter into the course of the reasoning4 ,i define then the numbers #b1 #c and #d by the equalities 7#a7 #a #a #b2 7#b7 #b #a #c2 7#c7 #c #a #d4 ,in the same way1 ,i define the operation x #b by the relation3 7#d7 x #b 7x #a7 #a4 ,that presupposed1 we have #b #a #a #c #a 7,definition #b71 #c #a #d 7,definition #c71 #b #b 7#b #a7 #a 7,definition #d71 whence #b #b #d ,q4,e4,d4 ,it can not be denied that this reasoning is purely analytic4 ,but ask any mathematician3 ',that is not a demonstration properly so called1' he will say to you3 'that is a verification4' ,we have confined ourselves to comparing two purely conventional definitions and have ascertained their identity2 we have learned nothing new4 ,verification differs from true demonstration precisely because it is purely analytic and because it is sterile4 ,it is sterile because the conclusion is nothing but the premises translated into another language4 ,on the contrary1 true demonstration is fruitful because the conclusion here is in a sense more general than the premises4 ,the equality #b #b #d is thus susceptible of a verification only because it is particular4 ,every particular enunciation in mathematics can always be verified in this same way4 ,but if mathematics could be reduced to a series of such verifications1 it would not be a science4 ,so a chess-player1 for example1 does not create a science in winning a game4 ,there is no science apart from the general4 ,it may even be said the very object of the exact sciences is to spare us these direct verifications4 ,,iii ,let us1 therefore1 see the geometer at work and seek to catch his process4 ,the task is not without difficulty2 it does not suffice to open a work at random and analyze any demonstration in it4 ,we must first exclude geometry1 where the question is complicated by arduous problems relative to the rle of the postulates1 to the nature and the origin of the notion of space4 ,for analogous reasons we can not turn to the infinitesimal analysis4 ,we must seek mathematical thought where it has remained pure1 that is1 in arithmetic4 ,a choice still is necessary2 in the higher parts of the theory of numbers1 the primitive mathematical notions have already undergone an elaboration so profound that it becomes difficult to analyze them4 ,it is1 therefore1 at the beginning of arithmetic that we must expect to find the explanation we seek1 but it happens that precisely in the demonstration of the most elementary theorems the authors of the classic treatises have shown the least precision and rigor4 ,we must not impute this to them as a crime2 they have yielded to a necessity2 beginners are not prepared for real mathematical rigor2 they would see in it only useless and irksome subtleties2 it would be a waste of time to try prematurely to make them more exacting2 they must pass over rapidly1 but without skipping stations1 the road traversed slowly by the founders of the science4 ,why is so long a preparation necessary to become habituated to this perfect rigor1 which1 it would seem1 should naturally impress itself upon all good minds8 ,this is a logical and psychological problem well worthy of study4 ,but we shall not take it up2 it is foreign to our purpose2 all ,i wish to insist on is that1 not to fail of our purpose1 we must recast the demonstrations of the most elementary theorems and give them1 not the crude form in which they are left1 so as not to harass beginners1 but the form that will satisfy a skilled geometer4 ,,definition ,,of ,,addition4--,i suppose already defined the operation x #a1 which consists in adding the number #a to a given number x4 ,this definition1 whatever it be1 does not enter into our subsequent reasoning4 ,we now have to define the operation x a1 which consists in adding the number a to a given number x4 ,supposing we have defined the operation x 7a - #a71 the operation x a will be defined by the equality 7#a7 x a x 7a - #a7 #a4 ,we shall know then what x a is when we know what x 7a - #a7 is1 and as ,i have supposed that to start with we knew what x #a is1 we can define successively and 'by recurrence' the operations x #b1 x #c1 etc4 ,this definition deserves a moment's attention2 it is of a particular nature which already distinguishes it from the purely logical definition2 the equality 7#a7 contains an infinity of distinct definitions1 each having a meaning only when one knows the preceding4 ,,properties ,,of ,,addition4--,associativity4--,i say that a 7b c7 7a b7 c4 ,in fact the theorem is true for c #a2 it is then written a 7b #a7 7a b7 #a1 which1 apart from the difference of notation1 is nothing but the equality 7#a71 by which ,i have just defined addition4 ,supposing the theorem true for c gamma1 ,i say it will be true for c gamma #a4 ,in fact1 supposing 7a b7 gamma a 7b gamma71 it follows that 7a b7 gamma #a a 7b gamma7 #a or by definition 7#a7 7a b7 7gamma #a7 a 7b gamma #a7 a b 7gamma #a71 which shows1 by a series of purely analytic deductions1 that the theorem is true for gamma #a4 ,being true for c #a1 we thus see successively that so it is for c #b1 for c #c1 etc4 ,commutativity4--#a ,i say that a #a #a a4 ,the theorem is evidently true for a #a2 we can verify by purely analytic reasoning that if it is true for a gamma it will be true for a gamma #a2 for then 7gamma #a7 #a 7#a gamma7 #a #a 7gamma #a72 now it is true for a #a1 therefore it will be true for a #b1 for a #c1 etc41 which is expressed by saying that the enunciated proposition is demonstrated by recurrence4 #b ,i say that a b b a4 ,the theorem has just been demonstrated for b #a2 it can be verified analytically that if it is true for b beta1 it will be true for b beta #a4 ,the proposition is therefore established by recurrence4 ,,definition ,,of ,,multiplication4--,we shall define multiplication by the equalities4 7#a7 a #a a4 7#b7 a b a 7b - #a7 a4 ,like equality 7#a71 equality 7#b7 contains an infinity of definitions2 having defined a #a1 it enables us to define successively3 a #b1 a #c1 etc4 ,,properties ,,of ,,multiplication4--,distributivity4--,i say that 7a b7 c 7a c7 7b c74 ,we verify analytically that the equality is true for c #a2 then that if the theorem is true for c gamma1 it will be true for c gamma #a4 ,the proposition is1 therefore1 demonstrated by recurrence4 ,commutativity4--#a ,i say that a #a #a a4 ,the theorem is evident for a #a4 ,we verify analytically that if it is true for a alpha1 it will be true for a alpha #a4 #b ,i say that a b b a4 ,the theorem has just been proven for b #a4 ,we could verify analytically that if it is true for b beta1 it will be true for b beta #a4 ,,iv ,here ,i stop this monotonous series of reasonings4 ,but this very monotony has the better brought out the procedure which is uniform and is met again at each step4 ,this procedure is the demonstration by recurrence4 ,we first establish a theorem for n #a2 then we show that if it is true of n - #a1 it is true of n1 and thence conclude that it is true for all the whole numbers4 ,we have just seen how it may be used to demonstrate the rules of addition and multiplication1 that is to say1 the rules of the algebraic calculus2 this calculus is an instrument of transformation1 which lends itself to many more differing combinations than does the simple syllogism2 but it is still an instrument purely analytic1 and incapable of teaching us anything new4 ,if mathematics had no other instrument1 it would therefore be forthwith arrested in its development2 but it has recourse anew to the same procedure1 that is1 to reasoning by recurrence1 and it is able to continue its forward march4 ,if we look closely1 at every step we meet again this mode of reasoning1 either in the simple form we have just given it1 or under a form more or less modified4 ,here then we have the mathematical reasoning par excellence1 and we must examine it more closely4 ,v ,the essential characteristic of reasoning by recurrence is that it contains1 condensed1 so to speak1 in a single formula1 an infinity of syllogisms4 ,that this may the better be seen1 ,i will state one after another these syllogisms which are1 if you will allow me the expression1 arranged in 'cascade4' ,these are of course hypothetical syllogisms4 ,the theorem is true of the number #a4 ,now1 if it is true of #a1 it is true of #b4 ,therefore it is true of #b4 ,now1 if it is true of #b1 it is true of #c4 ,therefore it is true of #c1 and so on4 ,we see that the conclusion of each syllogism serves as minor to the following4 ,furthermore the majors of all our syllogisms can be reduced to a single formula4 ,if the theorem is true of n - #a1 so it is of n4 ,we see1 then1 that in reasoning by recurrence we confine ourselves to stating the minor of the first syllogism1 and the general formula which contains as particular cases all the majors4 ,this never-ending series of syllogisms is thus reduced to a phrase of a few lines4 ,it is now easy to comprehend why every particular consequence of a theorem can1 as ,i have explained above1 be verified by purely analytic procedures4 ,if instead of showing that our theorem is true of all numbers1 we only wish to show it true of the number #f1 for example1 it will suffice for us to establish the first #e syllogisms of our cascade2 #i would be necessary if we wished to prove the theorem for the number #aj2 more would be needed for a larger number2 but1 however great this number might be1 we should always end by reaching it1 and the analytic verification would be possible4 ,and yet1 however far we thus might go1 we could never rise to the general theorem1 applicable to all numbers1 which alone can be the object of science4 ,to reach this1 an infinity of syllogisms would be necessary2 it would be necessary to overleap an abyss that the patience of the analyst1 restricted to the resources of formal logic alone1 never could fill up4 ,i asked at the outset why one could not conceive of a mind sufficiently powerful to perceive at a glance the whole body of mathematical truths4 ,the answer is now easy2 a chess-player is able to combine four moves1 five moves1 in advance1 but1 however extraordinary he may be1 he will never prepare more than a finite number of them2 if he applies his faculties to arithmetic1 he will not be able to perceive its general truths by a single direct intuition2 to arrive at the smallest theorem he can not dispense with the aid of reasoning by recurrence1 for this is an instrument which enables us to pass from the finite to the infinite4 ,this instrument is always useful1 for1 allowing us to overleap at a bound as many stages as we wish1 it spares us verifications1 long1 irksome and monotonous1 which would quickly become impracticable4 ,but it becomes indispensable as soon as we aim at the general theorem1 to which analytic verification would bring us continually nearer without ever enabling us to reach it4 ,in this domain of arithmetic1 we may think ourselves very far from the infinitesimal analysis1 and yet1 as we have just seen1 the idea of the mathematical infinite already plays a preponderant rle1 and without it there would be no science1 because there would be nothing general4 ,,vi ,the judgment on which reasoning by recurrence rests can be put under other forms2 we may say1 for example1 that in an infinite collection of different whole numbers there is always one which is less than all the others4 ,we can easily pass from one enunciation to the other and thus get the illusion of having demonstrated the legitimacy of reasoning by recurrence4 ,but we shall always be arrested1 we shall always arrive at an undemonstrable axiom which will be in reality only the proposition to be proved translated into another language4 ,we can not therefore escape the conclusion that the rule of reasoning by recurrence is irreducible to the principle of contradiction4 ,neither can this rule come to us from experience2 experience could teach us that the rule is true for the first ten or hundred numbers2 for example1 it can not attain to the indefinite series of numbers1 but only to a portion of this series1 more or less long but always limited4 ,now if it were only a question of that1 the principle of contradiction would suffice2 it would always allow of our developing as many syllogisms as we wished2 it is only when it is a question of including an infinity of them in a single formula1 it is only before the infinite that this principle fails1 and there too1 experience becomes powerless4 ,this rule1 inaccessible to analytic demonstration and to experience1 is the veritable type of the synthetic a priori judgment4 ,on the other hand1 we can not think of seeing in it a convention1 as in some of the postulates of geometry4 ,why then does this judgment force itself upon us with an irresistible evidence8 ,it is because it is only the affirmation of the power of the mind which knows itself capable of conceiving the indefinite repetition of the same act when once this act is possible4 ,the mind has a direct intuition of this power1 and experience can only give occasion for using it and thereby becoming conscious of it4 ,but1 one will say1 if raw experience can not legitimatize reasoning by recurrence1 is it so of experiment aided by induction8 ,we see successively that a theorem is true of the number #a1 of the number #b1 of the number #c and so on2 the law is evident1 we say1 and it has the same warranty as every physical law based on observations1 whose number is very great but limited4 ,here is1 it must be admitted1 a striking analogy with the usual procedures of induction4 ,but there is an essential difference4 ,induction applied to the physical sciences is always uncertain1 because it rests on the belief in a general order of the universe1 an order outside of us4 ,mathematical induction1 that is1 demonstration by recurrence1 on the contrary1 imposes itself necessarily because it is only the affirmation of a property of the mind itself4 ,,vii ,mathematicians1 as ,i have said before1 always endeavor to generalize the propositions they have obtained1 and1 to seek no other example1 we have just proved the equality3 a #a #a a and afterwards used it to establish the equality a b b a which is manifestly more general4 ,mathematics can1 therefore1 like the other sciences1 proceed from the particular to the general4 ,this is a fact which would have appeared incomprehensible to us at the outset of this study1 but which is no longer mysterious to us1 since we have ascertained the analogies between demonstration by recurrence and ordinary induction4 ,without doubt recurrent reasoning in mathematics and inductive reasoning in physics rest on different foundations1 but their march is parallel1 they advance in the same sense1 that is to say1 from the particular to the general4 ,let us examine the case a little more closely4 ,to demonstrate the equality a #b #b a it suffices to twice apply the rule 7#a7 a #a #a a and write 7#b7 a #b a #a #a #a a #a #a #a a #b a4 ,the equality 7#b7 thus deduced in purely analytic way from the equality 7#a7 is1 however1 not simply a particular ease of it2 it is something quite different4 ,we can not therefore even say that in the really analytic and deductive part of mathematical reasoning we proceed from the general to the particular in the ordinary sense of the word4 ,the two members of the equality 7#b7 are simply combinations more complicated than the two members of the equality 7#a71 and analysis only serves to separate the elements which enter into these combinations and to study their relations4 ,mathematicians proceed therefore 'by construction1' they 'construct' combinations more and more complicated4 ,coming back then by the analysis of these combinations1 of these aggregates1 so to speak1 to their primitive elements1 they perceive the relations of these elements and from them deduce the relations of the aggregates themselves4 ,this is a purely analytical proceeding1 but it is not1 however1 a proceeding from the general to the particular1 because evidently the aggregates can not be regarded as more particular than their elements4 ,great importance1 and justly1 has been attached to this procedure of 'construction1' and some have tried to see in it the necessary and sufficient condition for the progress of the exact sciences4 ,necessary1 without doubt2 but sufficient1 no4 ,for a construction to be useful and not a vain toil for the mind1 that it may serve as stepping-stone to one wishing to mount1 it must first of all possess a sort of unity enabling us to see in it something besides the juxtaposition of its elements4 ,or1 more exactly1 there must be some advantage in considering the construction rather than its elements themselves4 ,what can this advantage be8 ,why reason on a polygon1 for instance1 which is always decomposable into triangles1 and not on the elementary triangles8 ,it is because there are properties appertaining to polygons of any number of sides and that may be immediately applied to any particular polygon4 ,usually1 on the contrary1 it is only at the cost of the most prolonged exertions that they could be found by studying directly the relations of the elementary triangles4 ,the knowledge of the general theorem spares us these efforts4 ,a construction1 therefore1 becomes interesting only when it can be ranged beside other analogous constructions1 forming species of the same genus4 ,if the quadrilateral is something besides the juxtaposition of two triangles1 this is because it belongs to the genus polygon4 ,moreover1 one must be able to demonstrate the properties of the genus without being forced to establish them successively for each of the species4 ,to attain that1 we must necessarily mount from the particular to the general1 ascending one or more steps4 ,the analytic procedure 'by construction' does not oblige us to descend1 but it leaves us at the same level4 ,we can ascend only by mathematical induction1 which alone can teach us something new4 ,without the aid of this induction1 different in certain respects from physical induction1 but quite as fertile1 construction would be powerless to create science4 ,observe finally that this induction is possible only if the same operation can be repeated indefinitely4 ,that is why the theory of chess can never become a science1 for the different moves of the same game do not resemble one another4 ,,chapter ,,ii ,,mathematical ,,magnitude ,,and ,,experience ,to learn what mathematicians understand by a continuum1 one should not inquire of geometry4 ,the geometer always seeks to represent to himself more or less the figures he studies1 but his representations are for him only instruments2 in making geometry he uses space just as he does chalk2 so too much weight should not be attached to non-essentials1 often of no more importance than the whiteness of the chalk4 ,the pure analyst has not this rock to fear4 ,he has disengaged the science of mathematics from all foreign elements1 and can answer our question3 ',what exactly is this continuum about which mathematicians reason8' ,many analysts who reflect on their art have answered already2 ,monsieur ,tannery1 for example1 in his ,introduction la thorie des fonctions d'une variable4 ,let us start from the scale of whole numbers2 between two consecutive steps1 intercalate one or more intermediary steps1 then between these new steps still others1 and so on indefinitely4 ,thus we shall have an unlimited number of terms2 these will be the numbers called fractional1 rational or commensurable4 ,but this is not yet enough2 between these terms1 which1 however1 are already infinite in number1 it is still necessary to intercalate others called irrational or incommensurable4 ,a remark before going further4 ,the continuum so conceived is only a collection of individuals ranged in a certain order1 infinite in number1 it is true1 but exterior to one another4 ,this is not the ordinary conception1 wherein is supposed between the elements of the continuum a sort of intimate bond which makes of them a whole1 where the point does not exist before the line1 but the line before the point4 ,of the celebrated formula1 'the continuum is unity in multiplicity1' only the multiplicity remains1 the unity has disappeared4 ,the analysts are none the less right in defining their continuum as they do1 for they always reason on just this as soon as they pique themselves on their rigor4 ,but this is enough to apprise us that the veritable mathematical continuum is a very different thing from that of the physicists and that of the metaphysicians4 ,it may also be said perhaps that the mathematicians who are content with this definition are dupes of words1 that it is necessary to say precisely what each of these intermediary steps is1 to explain how they are to be intercalated and to demonstrate that it is possible to do it4 ,but that would be wrong2 the only property of these steps which is used in their reasonings#b is that of being before or after such and such steps2 therefore also this alone should occur in the definition4 #b ,with those contained in the special conventions which serve to define addition and of which we shall speak later4 ,so how the intermediary terms should be intercalated need not concern us2 on the other hand1 no one will doubt the possibility of this operation1 unless from forgetting that possible1 in the language of geometers1 simply means free from contradiction4 ,our definition1 however1 is not yet complete1 and ,i return to it after this over-long digression4 ,,definition ,,of ,,incommensurables4--,the mathematicians of the ,berlin school1 ,kronecker in particular1 have devoted themselves to constructing this continuous scale of fractional and irrational numbers without using any material other than the whole number4 ,the mathematical continuum would be1 in this view1 a pure creation of the mind1 where experience would have no part4 ,the notion of the rational number seeming to them to present no difficulty1 they have chiefly striven to define the incommensurable number4 ,but before producing here their definition1 ,i must make a remark to forestall the astonishment it is sure to arouse in readers unfamiliar with the customs of geometers4 ,mathematicians study not objects1 but relations between objects2 the replacement of these objects by others is therefore indifferent to them1 provided the relations do not change4 ,the matter is for them unimportant1 the form alone interests them4 ,without recalling this1 it would scarcely be comprehensible that ,dedekind should designate by the name incommensurable number a mere symbol1 that is to say1 something very different from the ordinary idea of a quantity1 which should be measurable and almost tangible4 ,let us see now what ,dedekind's definition is3 ,the commensurable numbers can in an infinity of ways be partitioned into two classes1 such that any number of the first class is greater than any number of the second class4 ,it may happen that among the numbers of the first class there is one smaller than all the others2 if1 for example1 we range in the first class all numbers greater than #b1 and #b itself1 and in the second class all numbers less than #b1 it is clear that #b will be the least of all numbers of the first class4 ,the number #b may be chosen as symbol of this partition4 ,it may happen1 on the contrary1 that among the numbers of the second class is one greater than all the others2 this is the case1 for example1 if the first class comprehends all numbers greater than #b1 and the second all numbers less than #b1 and #b itself4 ,here again the number #b may be chosen as symbol of this partition4 ,but it may equally well happen that neither is there in the first class a number less than all the others1 nor in the second class a number greater than all the others4 ,suppose1 for example1 we put in the first class all commensurable numbers whose squares are greater than #b and in the second all whose squares are less than #b4 ,there is none whose square is precisely #b4 ,evidently there is not in the first class a number less than all the others1 for1 however near the square of a number may be to #b1 we can always find a commensurable number whose square is still closer to #b4 ,in ,dedekind's view1 the incommensurable number sqrt7#b7 or 7#b7a_/b is nothing but the symbol of this particular mode of partition of commensurable numbers2 and to each mode of partition corresponds thus a number1 commensurable or not1 which serves as its symbol4 ,but to be content with this would be to forget too far the origin of these symbols2 it remains to explain how we have been led to attribute to them a sort of concrete existence1 and1 besides1 does not the difficulty begin even for the fractional numbers themselves8 ,should we have the notion of these numbers if we had not previously known a matter that we conceive as infinitely divisible1 that is to say1 a continuum8 ,,the ,,physical ,,continuum4--,we ask ourselves then if the notion of the mathematical continuum is not simply drawn from experience4 ,if it were1 the raw data of experience1 which are our sensations1 would be susceptible of measurement4 ,we might be tempted to believe they really are so1 since in these latter days the attempt has been made to measure them and a law has even been formulated1 known as ,fechner's law1 according to which sensation is proportional to the logarithm of the stimulus4 ,but if we examine more closely the experiments by which it has been sought to establish this law1 we shall be led to a diametrically opposite conclusion4 ,it has been observed1 for example1 that a weight ,a of #aj grams and a weight ,b of #aa grams produce identical sensations1 that the weight ,b is just as indistinguishable from a weight ,c of #ab grams1 but that the weight ,a is easily distinguished from the weight ,c4 ,thus the raw results of experience may be expressed by the following relations3 ,a ,b1 ,b ,c1 ,a ,c1 which may be regarded as the formula of the physical continuum4 ,but here is an intolerable discord with the principle of contradiction1 and the need of stopping this has compelled us to invent the mathematical continuum4 ,we are1 therefore1 forced to conclude that this notion has been created entirely by the mind1 but that experience has given the occasion4 ,we can not believe that two quantities equal to a third are not equal to one another1 and so we are led to suppose that ,a is different from ,b and ,b from ,c1 but that the imperfection of our senses has not permitted of our distinguishing them4 ,,creation ,,of ,,the ,,mathematical ,,continuum4--,first ,stage4 ,so far it would suffice1 in accounting for the facts1 to intercalate between ,a and ,b a few terms1 which would remain discrete4 ,what happens now if we have recourse to some instrument to supplement the feebleness of our senses1 if1 for example1 we make use of a microscope8 ,terms such as ,a and ,b1 before indistinguishable1 appear now distinct2 but between ,a and ,b1 now become distinct1 will be intercalated a new term1 ,d1 that we can distinguish neither from ,a nor from ,b4 ,despite the employment of the most highly perfected methods1 the raw results of our experience will always present the characteristics of the physical continuum with the contradiction which is inherent in it4 ,we shall escape it only by incessantly intercalating new terms between the terms already distinguished1 and this operation must be continued indefinitely4 ,we might conceive the stopping of this operation if we could imagine some instrument sufficiently powerful to decompose the physical continuum into discrete elements1 as the telescope resolves the milky way into stars4 ,but this we can not imagine2 in fact1 it is with the eye we observe the image magnified by the microscope1 and consequently this image must always retain the characteristics of visual sensation and consequently those of the physical continuum4 ,nothing distinguishes a length observed directly from the half of this length doubled by the microscope4 ,the whole is homogeneous with the part2 this is a new contradiction1 or rather it would be if the number of terms were supposed finite2 in fact1 it is clear that the part containing fewer terms than the whole could not be similar to the whole4 ,the contradiction ceases when the number of terms is regarded as infinite2 nothing hinders1 for example1 considering the aggregate of whole numbers as similar to the aggregate of even numbers1 which1 however1 is only a part of it2 and1 in fact1 to each whole number corresponds an even number1 its double4 ,but it is not only to escape this contradiction contained in the empirical data that the mind is led to create the concept of a continuum1 formed of an indefinite number of terms4 ,all happens as in the sequence of whole numbers4 ,we have the faculty of conceiving that a unit can be added to a collection of units2 thanks to experience1 we have occasion to exercise this faculty and we become conscious of it2 but from this moment we feel that our power has no limit and that we can count indefinitely1 though we have never had to count more than a finite number of objects4 ,just so1 as soon as we have been led to intercalate means between two consecutive terms of a series1 we feel that this operation can be continued beyond all limit1 and that there is1 so to speak1 no intrinsic reason for stopping4 ,as an abbreviation1 let me call a mathematical continuum of the first order every aggregate of terms formed according to the same law as the scale of commensurable numbers4 ,if we afterwards intercalate new steps according to the law of formation of incommensurable numbers1 we shall obtain what we will call a continuum of the second order4 ,second ,stage4--,we have made hitherto only the first stride2 we have explained the origin of continua of the first order2 but it is necessary to see why even they are not sufficient and why the incommensurable numbers had to be invented4 ,if we try to imagine a line1 it must have the characteristics of the physical continuum1 that is to say1 we shall not be able to represent it except with a certain breadth4 ,two lines then will appear to us under the form of two narrow bands1 and1 if we are content with this rough image1 it is evident that if the two lines cross they will have a common part4 ,but the pure geometer makes a further effort2 without entirely renouncing the aid of the senses1 he tries to reach the concept of the line without breadth1 of the point without extension4 ,this he can only attain to by regarding the line as the limit toward which tends an ever narrowing band1 and the point as the limit toward which tends an ever lessening area4 ,and then1 our two bands1 however narrow they may be1 will always have a common area1 the smaller as they are the narrower1 and whose limit will be what the pure geometer calls a point4 ,this is why it is said two lines which cross have a point in common1 and this truth seems intuitive4 ,but it would imply contradiction if lines were conceived as continua of the first order1 that is to say1 if on the lines traced by the geometer should be found only points having for coordinates rational numbers4 ,the contradiction would be manifest as soon as one affirmed1 for example1 the existence of straights and circles4 ,it is clear1 in fact1 that if the points whose coordinates are commensurable were alone regarded as real1 the circle inscribed in a square and the diagonal of this square would not intersect1 since the coordinates of the point of intersection are incommensurable4 ,that would not yet be sufficient1 because we should get in this way only certain incommensurable numbers and not all those numbers4 ,but conceive of a straight line divided into two rays4 ,each of these rays will appear to our imagination as a band of a certain breadth2 these bands moreover will encroach one on the other1 since there must be no interval between them4 ,the common part will appear to us as a point which will always remain when we try to imagine our bands narrower and narrower1 so that we admit as an intuitive truth that if a straight is cut into two rays their common frontier is a point2 we recognize here the conception of ,dedekind1 in which an incommensurable number was regarded as the common frontier of two classes of rational numbers4 ,such is the origin of the continuum of the second order1 which is the mathematical continuum properly so called4 ,rsum4--,in recapitulation1 the mind has the faculty of creating symbols1 and it is thus that it has constructed the mathematical continuum1 which is only a particular system of symbols4 ,its power is limited only by the necessity of avoiding all contradiction2 but the mind only makes use of this faculty if experience furnishes it a stimulus thereto4 ,in the case considered1 this stimulus was the notion of the physical continuum1 drawn from the rough data of the senses4 ,but this notion leads to a series of contradictions from which it is necessary successively to free ourselves4 ,so we are forced to imagine a more and more complicated system of symbols4 ,that at which we stop is not only exempt from internal contradiction 7it was so already at all the stages we have traversed71 but neither is it in contradiction with various propositions called intuitive1 which are derived from empirical notions more or less elaborated4 ,,measurable ,,magnitude4--,the magnitudes we have studied hitherto are not measurable2 we can indeed say whether a given one of these magnitudes is greater than another1 but not whether it is twice or thrice as great4 ,so far1 ,i have only considered the order in which our terms are ranged4 ,but for most applications that does not suffice4 ,we must learn to compare the interval which separates any two terms4 ,only on this condition does the continuum become a measurable magnitude and the operations of arithmetic applicable4 ,this can only be done by the aid of a new and special convention4 ,we will agree that in such and such a case the interval comprised between the terms ,a and ,b is equal to the interval which separates ,c and ,d4 ,for example1 at the beginning of our work we have set out from the scale of the whole numbers and we have supposed intercalated between two consecutive steps n intermediary steps2 well1 these new steps will be by convention regarded as equidistant4 ,this is a way of defining the addition of two magnitudes1 because if the interval ,,ab is by definition equal to the interval ,,cd1 the interval ,,ad will be by definition the sum of the intervals ,,ab and ,,ac4 ,this definition is arbitrary in a very large measure4 ,it is not completely so1 however4 ,it is subjected to certain conditions and1 for example1 to the rules of commutativity and associativity of addition4 ,but provided the definition chosen satisfies these rules1 the choice is indifferent1 and it is useless to particularize it4 ,,various ,,remarks4--,we can now discuss several important questions3 #a ,is the creative power of the mind exhausted by the creation of the mathematical continuum8 ,no3 the works of ,du ,bois-,reymond demonstrate it in a striking way4 ,we know that mathematicians distinguish between infinitesimals of different orders and that those of the second order are infinitesimal1 not only in an absolute way1 but also in relation to those of the first order4 ,it is not difficult to imagine infinitesimals of fractional or even of irrational order1 and thus we find again that scale of the mathematical continuum which has been dealt with in the preceding pages4 ,further1 there are infinitesimals which are infinitely small in relation to those of the first order1 and1 on the contrary1 infinitely great in relation to those of order #a epsilon1 and that however small epsilon may be4 ,here1 then1 are new terms intercalated in our series1 and if ,i may be permitted to revert to the phraseology lately employed which is very convenient though not consecrated by usage1 ,i shall say that thus has been created a sort of continuum of the third order4 ,it would be easy to go further1 but that would be idle2 one would only be imagining symbols without possible application1 and no one will think of doing that4 ,the continuum of the third order1 to which the consideration of the different orders of infinitesimals leads1 is itself not useful enough to have won citizenship1 and geometers regard it only as a mere curiosity4 ,the mind uses its creative faculty only when experience requires it4 #b ,once in possession of the concept of the mathematical continuum1 is one safe from contradictions analogous to those which gave birth to it8 ,no1 and ,i will give an example4 ,one must be very wise not to regard it as evident that every curve has a tangent2 and in fact if we picture this curve and a straight as two narrow bands we can always so dispose them that they have a part in common without crossing4 ,if we imagine then the breadth of these two bands to diminish indefinitely1 this common part will always subsist and1 at the limit1 so to speak1 the two lines will have a point in common without crossing1 that is to say1 they will be tangent4 ,the geometer who reasons in this way1 consciously or not1 is only doing what we have done above to prove two lines which cut have a point in common1 and his intuition might seem just as legitimate4 ,it would deceive him however4 ,we can demonstrate that there are curves which have no tangent1 if such a curve is defined as an analytic continuum of the second order4 ,without doubt some artifice analogous to those we have discussed above would have sufficed to remove the contradiction2 but1 as this is met with only in very exceptional cases1 it has received no further attention4 ,instead of seeking to reconcile intuition with analysis1 we have been content to sacrifice one of the two1 and as analysis must remain impeccable1 we have decided against intuition4 ,,the ,,physical ,,continuum ,,of ,,several ,,dimensions4--,we have discussed above the physical continuum as derived from the immediate data of our senses1 or1 if you wish1 from the rough results of ,fechner's experiments2 ,i have shown that these results are summed up in the contradictory formulas ,a ,b1 ,b ,c1 ,a ,c4 ,let us now see how this notion has been generalized and how from it has come the concept of many-dimensional continua4 ,consider any two aggregates of sensations4 ,either we can discriminate them one from another1 or we can not1 just as in ,fechner's experiments a weight of #aj grams can be distinguished from a weight of #ab grams1 but not from a weight of #aa grams4 ,this is all that is required to construct the continuum of several dimensions4 ,let us call one of these aggregates of sensations an element4 ,that will be something analogous to the point of the mathematicians2 it will not be altogether the same thing however4 ,we can not say our element is without extension1 since we can not distinguish it from neighboring elements and it is thus surrounded by a sort of haze4 ,if the astronomical comparison may be allowed1 our 'elements' would be like nebulae1 whereas the mathematical points would be like stars4 ,that being granted1 a system of elements will form a continuum if we can pass from any one of them to any other1 by a series of consecutive elements such that each is indistinguishable from the preceding4 ,this linear series is to the line of the mathematician what an isolated element was to the point4 ,before going farther1 ,i must explain what is meant by a cut4 ,consider a continuum ,c and remove from it certain of its elements which for an instant we shall regard as no longer belonging to this continuum4 ,the aggregate of the elements so removed will be called a cut4 ,it may happen that1 thanks to this cut1 ,c may be subdivided into several distinct continua1 the aggregate of the remaining elements ceasing to form a unique continuum4 ,there will then be on ,c two elements1 ,a and ,b1 that must be regarded as belonging to two distinct continua1 and this will be recognized because it will be impossible to find a linear series of consecutive elements of ,c1 each of these elements indistinguishable from the preceding1 the first being ,a and the last ,b1 without one of the elements of this series being indistinguishable from one of the elements of the cut4 ,on the contrary1 it may happen that the cut made is insufficient to subdivide the continuum ,c4 ,to classify the physical continua1 we will examine precisely what are the cuts which must be made to subdivide them4 ,if a physical continuum ,c can be subdivided by a cut reducing to a finite number of elements all distinguishable from one another 7and consequently forming neither a continuum1 nor several continua71 we shall say ,c is a one-dimensional continuum4 ,if1 on the contrary1 ,c can be subdivided only by cuts which are themselves continua1 we shall say ,c has several dimensions4 ,if cuts which are continua of one dimension suffice1 we shall say ,c has two dimensions2 if cuts of two dimensions suffice1 we shall say ,c has three dimensions1 and so on4 ,thus is defined the notion of the physical continuum of several dimensions1 thanks to this very simple fact that two aggregates of sensations are distinguishable or indistinguishable4 ,,the ,,mathematical ,,continuum ,,of ,,several ,,dimensions4--,thence the notion of the mathematical continuum of n dimensions has sprung quite naturally by a process very like that we discussed at the beginning of this chapter4 ,a point of such a continuum1 you know1 appears to us as defined by a system of n distinct magnitudes called its coordinates4 ,these magnitudes need not always be measurable2 there is1 for instance1 a branch of geometry independent of the measurement of these magnitudes1 in which it is only a question of knowing1 for example1 whether on a curve ,,abc1 the point ,b is between the points ,a and ,c1 and not of knowing whether the arc ,,ab is equal to the arc ,,bc or twice as great4 ,this is what is called ,analysis ,situs4 ,this is a whole body of doctrine which has attracted the attention of the greatest geometers and where we see flow one from another a series of remarkable theorems4 ,what distinguishes these theorems from those of ordinary geometry is that they are purely qualitative and that they would remain true if the figures were copied by a draughtsman so awkward as to grossly distort the proportions and replace straights by strokes more or less curved4 ,through the wish to introduce measure next into the continuum just defined this continuum becomes space1 and geometry is born4 ,but the discussion of this is reserved for ,part ,second4 ,,part ,,ii ,,space ,,chapter ,,iii ,,the ,,non-,,euclidean ,,geometries ,every conclusion supposes premises2 these premises themselves either are self-evident and need no demonstration1 or can be established only by relying upon other propositions1 and since we can not go back thus to infinity1 every deductive science1 and in particular geometry1 must rest on a certain number of undemonstrable axioms4 ,all treatises on geometry begin1 therefore1 by the enunciation of these axioms4 ,but among these there is a distinction to be made3 ,some1 for example1 ',things which are equal to the same thing are equal to one another1' are not propositions of geometry1 but propositions of analysis4 ,i regard them as analytic judgments a priori1 and shall not concern myself with them4 ,but ,i must lay stress upon other axioms which are peculiar to geometry4 ,most treatises enunciate three of these explicitly3 #a ,through two points can pass only one straight2 #b ,the straight line is the shortest path from one point to another2 #c ,through a given point there is not more than one parallel to a given straight4 ,although generally a proof of the second of these axioms is omitted1 it would be possible to deduce it from the other two and from those1 much more numerous1 which are implicitly admitted without enunciating them1 as ,i shall explain further on4 ,it was long sought in vain to demonstrate likewise the third axiom1 known as ,euclid's ,postulate4 ,what vast effort has been wasted in this chimeric hope is truly unimaginable4 ,finally1 in the first quarter of the nineteenth century1 and almost at the same time1 a ,hungarian and a ,russian1 ,bolyai and ,lobachevski1 established irrefutably that this demonstration is impossible2 they have almost rid us of inventors of geometries 'sans postulatum'2 since then the ,acadmie des ,sciences receives only about one or two new demonstrations a year4 ,the question was not exhausted2 it soon made a great stride by the publication of ,riemann's celebrated memoir entitled3 ,ueber die ,hypothesen welche der ,geometrie zu ,grunde liegen4 ,this paper has inspired most of the recent works of which ,i shall speak further on1 and among which it is proper to cite those of ,beltrami and of ,helmholtz4 ,,the ,,bolyai-,,lobachevski ,,geometry4--,if it were possible to deduce ,euclid's postulate from the other axioms1 it is evident that in denying the postulate and admitting the other axioms1 we should be led to contradictory consequences2 it would therefore be impossible to base on such premises a coherent geometry4 ,now this is precisely what ,lobachevski did4 ,he assumes at the start that3 ,through a given point can be drawn two parallels to a given straight4 ,and he retains besides all ,euclid's other axioms4 ,from these hypotheses he deduces a series of theorems among which it is impossible to find any contradiction1 and he constructs a geometry whose faultless logic is inferior in nothing to that of the ,euclidean geometry4 ,the theorems are1 of course1 very different from those to which we are accustomed1 and they can not fail to be at first a little disconcerting4 ,thus the sum of the angles of a triangle is always less than two right angles1 and the difference between this sum and two right angles is proportional to the surface of the triangle4 ,it is impossible to construct a figure similar to a given figure but of different dimensions4 ,if we divide a circumference into n equal parts1 and draw tangents at the points of division1 these n tangents will form a polygon if the radius of the circle is small enough2 but if this radius is sufficiently great they will not meet4 ,it is useless to multiply these examples2 ,lobachevski's propositions have no relation to those of ,euclid1 but they are not less logically bound one to another4 ,,riemann',s ,,geometry4--,imagine a world uniquely peopled by beings of no thickness 7height72 and suppose these 'infinitely flat' animals are all in the same plane and can not get out4 ,admit besides that this world is sufficiently far from others to be free from their influence4 ,while we are making hypotheses1 it costs us no more to endow these beings with reason and believe them capable of creating a geometry4 ,in that case1 they will certainly attribute to space only two dimensions4 ,but suppose now that these imaginary animals1 while remaining without thickness1 have the form of a spherical1 and not of a plane1 figure1 and are all on the same sphere without power to get off4 ,what geometry will they construct8 ,first it is clear they will attribute to space only two dimensions2 what will play for them the rle of the straight line will be the shortest path from one point to another on the sphere1 that is to say1 an arc of a great circle2 in a word1 their geometry will be the spherical geometry4 ,what they will call space will be this sphere on which they must stay1 and on which happen all the phenomena they can know4 ,their space will therefore be unbounded since on a sphere one can always go forward without ever being stopped1 and yet it will be finite2 one can never find the end of it1 but one can make a tour of it4 ,well1 ,riemann's geometry is spherical geometry extended to three dimensions4 ,to construct it1 the ,german mathematician had to throw overboard1 not only ,euclid's postulate1 but also the first axiom3 ,only one straight can pass through two points4 ,on a sphere1 through two given points we can draw in general only one great circle 7which1 as we have just seen1 would play the rle of the straight for our imaginary beings72 but there is an exception3 if the two given points are diametrically opposite1 an infinity of great circles can be drawn through them4 ,in the same way1 in ,riemann's geometry 7at least in one of its forms71 through two points will pass in general only a single straight2 but there are exceptional cases where through two points an infinity of straights can pass4 ,there is a sort of opposition between ,riemann's geometry and that of ,lobachevski4 ,thus the sum of the angles of a triangle is3 ,equal to two right angles in ,euclid's geometry2 ,less than two right angles in that of ,lobachevski2 ,greater than two right angles in that of ,riemann4 ,the number of straights through a given point that can be drawn coplanar to a given straight1 but nowhere meeting it1 is equal3 ,to one in ,euclid's geometry2 ,to zero in that of ,riemann2 ,to infinity in that of ,lobachevski4 ,add that ,riemann's space is finite1 although unbounded1 in the sense given above to these two words4 ,,the ,,surfaces ,,of ,,constant ,,curvature4--,one objection still remained possible4 ,the theorems of ,lobachevski and of ,riemann present no contradiction2 but however numerous the consequences these two geometers have drawn from their hypotheses1 they must have stopped before exhausting them1 since their number would be infinite2 who can say then that if they had pushed their deductions farther they would not have eventually reached some contradiction8 ,this difficulty does not exist for ,riemann's geometry1 provided it is limited to two dimensions2 in fact1 as we have seen1 two-dimensional ,riemannian geometry does not differ from spherical geometry1 which is only a branch of ordinary geometry1 and consequently is beyond all discussion4 ,beltrami1 in correlating likewise ,lobachevski's two-dimensional geometry with a branch of ordinary geometry1 has equally refuted the objection so far as it is concerned4 ,here is how he accomplished it4 ,consider any figure on a surface4 ,imagine this figure traced on a flexible and inextensible canvas applied over this surface in such a way that when the canvas is displaced and deformed1 the various lines of this figure can change their form without changing their length4 ,in general1 this flexible and inextensible figure can not be displaced without leaving the surface2 but there are certain particular surfaces for which such a movement would be possible2 these are the surfaces of constant curvature4 ,if we resume the comparison made above and imagine beings without thickness living on one of these surfaces1 they will regard as possible the motion of a figure all of whose lines remain constant in length4 ,on the contrary1 such a movement would appear absurd to animals without thickness living on a surface of variable curvature4 ,these surfaces of constant curvature are of two sorts3 ,some are of positive curvature1 and can be deformed so as to be applied over a sphere4 ,the geometry of these surfaces reduces itself therefore to the spherical geometry1 which is that of ,riemann4 ,the others are of negative curvature4 ,beltrami has shown that the geometry of these surfaces is none other than that of ,lobachevski4 ,the two-dimensional geometries of ,riemann and ,lobachevski are thus correlated to the ,euclidean geometry4 ,,interpretation ,,of ,,non-,,euclidean ,,geometries4--,so vanishes the objection so far as two-dimensional geometries are concerned4 ,it would be easy to extend ,beltrami's reasoning to three-dimensional geometries4 ,the minds that space of four dimensions does not repel will see no difficulty in it1 but they are few4 ,i prefer therefore to proceed otherwise4 ,consider a certain plane1 which ,i shall call the fundamental plane1 and construct a sort of dictionary1 by making correspond each to each a double series of terms written in two columns1 just as correspond in the ordinary dictionaries the words of two languages whose significance is the same3 ,space3 ,portion of space situated above the fundamental plane4 ,plane3 ,sphere cutting the fundamental plane orthogonally4 ,straight3 ,circle cutting the fundamental plane orthogonally4 ,sphere3 ,sphere4 ,circle3 ,circle4 ,angle3 ,angle4 ,distance between two points3 ,logarithm of the cross ratio of these two points and the intersections of the fundamental plane with a circle passing through these two points and cutting it orthogonally4 ,etc41 ,etc4 ,now take ,lobachevski's theorems and translate them with the aid of this dictionary as we translate a ,german text with the aid of a ,german-,english dictionary4 ,we shall thus obtain theorems of the ordinary geometry4 ,for example1 that theorem of ,lobachevski3 'the sum of the angles of a triangle is less than two right angles' is translated thus3 ,if a curvilinear triangle has for sides circle-arcs which prolonged would cut orthogonally the fundamental plane1 the sum of the angles of this curvilinear triangle will be less than two right angles4 ,thus1 however far the consequences of ,lobachevski's hypotheses are pushed1 they will never lead to a contradiction4 ,in fact1 if two of ,lobachevski's theorems were contradictory1 it would be the same with the translations of these two theorems1 made by the aid of our dictionary1 but these translations are theorems of ordinary geometry and no one doubts that the ordinary geometry is free from contradiction4 ,whence comes this certainty and is it justified8 ,that is a question ,i can not treat here because it would require to be enlarged upon1 but which is very interesting and ,i think not insoluble4 ,nothing remains then of the objection above formulated4 ,this is not all4 ,lobachevski's geometry1 susceptible of a concrete interpretation1 ceases to be a vain logical exercise and is capable of applications2 ,i have not the time to speak here of these applications1 nor of the aid that ,klein and ,i have gotten from them for the integration of linear differential equations4 ,this interpretation moreover is not unique1 and several dictionaries analogous to the preceding could be constructed1 which would enable us by a simple 'translation' to transform ,lobachevski's theorems into theorems of ordinary geometry4 ,,the ,,implicit ,,axioms4--,are the axioms explicitly enunciated in our treatises the sole foundations of geometry8 ,we may be assured of the contrary by noticing that after they are successively abandoned there are still left over some propositions common to the theories of ,euclid1 ,lobachevski and ,riemann4 ,these propositions must rest on premises the geometers admit without enunciation4 ,it is interesting to try to disentangle them from the classic demonstrations4 ,stuart ,mill has claimed that every definition contains an axiom1 because in defining one affirms implicitly the existence of the object defined4 ,this is going much too far2 it is rare that in mathematics a definition is given without its being followed by the demonstration of the existence of the object defined1 and when this is dispensed with it is generally because the reader can easily supply it4 ,it must not be forgotten that the word existence has not the same sense when it refers to a mathematical entity and when it is a question of a material object4 ,a mathematical entity exists1 provided its definition implies no contradiction1 either in itself1 or with the propositions already admitted4 ,but if ,stuart ,mill's observation can not be applied to all definitions1 it is none the less just for some of them4 ,the plane is sometimes defined as follows3 ,the plane is a surface such that the straight which joins any two of its points is wholly on this surface4 ,this definition manifestly hides a new axiom2 it is true we might change it1 and that would be preferable1 but then we should have to enunciate the axiom explicitly4 ,other definitions would suggest reflections not less important4 ,such1 for example1 is that of the equality of two figures2 two figures are equal when they can be superposed2 to superpose them one must be displaced until it coincides with the other2 but how shall it be displaced8 ,if we should ask this1 no doubt we should be told that it must be done without altering the shape and as a rigid solid4 ,the vicious circle would then be evident4 ,in fact this definition defines nothing2 it would have no meaning for a being living in a world where there were only fluids4 ,if it seems clear to us1 that is because we are used to the properties of natural solids which do not differ much from those of the ideal solids1 all of whose dimensions are invariable4 ,yet1 imperfect as it may be1 this definition implies an axiom4 ,the possibility of the motion of a rigid figure is not a self-evident truth1 or at least it is so only in the fashion of ,euclid's postulate and not as an analytic judgment a priori would be4 ,moreover1 in studying the definitions and the demonstrations of geometry1 we see that one is obliged to admit without proof not only the possibility of this motion1 but some of its properties besides4 ,this is at once seen from the definition of the straight line4 ,many defective definitions have been given1 but the true one is that which is implied in all the demonstrations where the straight line enters3 ,it may happen that the motion of a rigid figure is such that all the points of a line belonging to this figure remain motionless while all the points situated outside of this line move4 ,such a line will be called a straight line4 ,we have designedly1 in this enunciation1 separated the definition from the axiom it implies4 ,many demonstrations1 such as those of the cases of the equality of triangles1 of the possibility of dropping a perpendicular from a point to a straight1 presume propositions which are not enunciated1 for they require the admission that it is possible to transport a figure in a certain way in space4 ,,the ,,fourth ,,geometry4--,among these implicit axioms1 there is one which seems to me to merit some attention1 because when it is abandoned a fourth geometry can be constructed as coherent as those of ,euclid1 ,lobachevski and ,riemann4 ,to prove that a perpendicular may always be erected at a point ,a to a straight ,,ab1 we consider a straight ,,ac movable around the point ,a and initially coincident with the fixed straight ,,ab2 and we make it turn about the point ,a until it comes into the prolongation of ,,ab4 ,thus two propositions are presupposed3 ,first1 that such a rotation is possible1 and next that it may be continued until the two straights come into the prolongation one of the other4 ,if the first point is admitted and the second rejected1 we are led to a series of theorems even stranger than those of ,lobachevski and ,riemann1 but equally exempt from contradiction4 ,i shall cite only one of these theorems and that not the most singular3 ,a real straight may be perpendicular to itself4 ,,lie',s ,,theorem4--,the number of axioms implicitly introduced in the classic demonstrations is greater than necessary1 and it would be interesting to reduce it to a minimum4 ,it may first be asked whether this reduction is possible1 whether the number of necessary axioms and that of imaginable geometries are not infinite4 ,a theorem of ,sophus ,lie dominates this whole discussion4 ,it may be thus enunciated3 ,suppose the following premises are admitted3 #a ,space has n dimensions2 #b ,the motion of a rigid figure is possible2 #c ,it requires p conditions to determine the position of this figure in space4 ,the number of geometries compatible with these premises will be limited4 ,i may even add that if n is given1 a superior limit can be assigned to p4 ,if therefore the possibility of motion is admitted1 there can be invented only a finite 7and even a rather small7 number of three-dimensional geometries4 ,,riemann',s ,,geometries4--,yet this result seems contradicted by ,riemann1 for this savant constructs an infinity of different geometries1 and that to which his name is ordinarily given is only a particular case4 ,all depends1 he says1 on how the length of a curve is defined4 ,now1 there is an infinity of ways of defining this length1 and each of them may be the starting point of a new geometry4 ,that is perfectly true1 but most of these definitions are incompatible with the motion of a rigid figure1 which in the theorem of ,lie is supposed possible4 ,these geometries of ,riemann1 in many ways so interesting1 could never therefore be other than purely analytic and would not lend themselves to demonstrations analogous to those of ,euclid4 ,,on ,,the ,,nature ,,of ,,axioms4--,most mathematicians regard ,lobachevski's geometry only as a mere logical curiosity2 some of them1 however1 have gone farther4 ,since several geometries are possible1 is it certain ours is the true one8 ,experience no doubt teaches us that the sum of the angles of a triangle is equal to two right angles2 but this is because the triangles we deal with are too little2 the difference1 according to ,lobachevski1 is proportional to the surface of the triangle2 will it not perhaps become sensible when we shall operate on larger triangles or when our measurements shall become more precise8 ,the ,euclidean geometry would thus be only a provisional geometry4 ,to discuss this opinion1 we should first ask ourselves what is the nature of the geometric axioms4 ,are they synthetic a priori judgments1 as ,kant said8 ,they would then impose themselves upon us with such force that we could not conceive the contrary proposition1 nor build upon it a theoretic edifice4 ,there would be no non-,euclidean geometry4 ,to be convinced of it take a veritable synthetic a priori judgment1 the following1 for instance1 of which we have seen the preponderant rle in the first chapter3 ,if a theorem is true for the number #a1 and if it has been proved that it is true of n #a provided it is true of n1 it will be true of all the positive whole numbers4 ,then try to escape from that and1 denying this proposition1 try to found a false arithmetic analogous to non-,euclidean geometry--it can not be done2 one would even be tempted at first blush to regard these judgments as analytic4 ,moreover1 resuming our fiction of animals without thickness1 we can hardly admit that these beings1 if their minds are like ours1 would adopt the ,euclidean geometry which would be contradicted by all their experience4 ,should we therefore conclude that the axioms of geometry are experimental verities8 ,but we do not experiment on ideal straights or circles2 it can only be done on material objects4 ,on what then could be based experiments which should serve as foundation for geometry8 ,the answer is easy4 ,we have seen above that we constantly reason as if the geometric figures behaved like solids4 ,what geometry would borrow from experience would therefore be the properties of these bodies4 ,the properties of light and its rectilinear propagation have also given rise to some of the propositions of geometry1 and in particular those of projective geometry1 so that from this point of view one would be tempted to say that metric geometry is the study of solids1 and projective1 that of light4 ,but a difficulty remains1 and it is insurmountable4 ,if geometry were an experimental science1 it would not be an exact science1 it would be subject to a continual revision4 ,nay1 it would from this very day be convicted of error1 since we know that there is no rigorously rigid solid4 ,the axioms of geometry therefore are neither synthetic a priori judgments nor experimental facts4 ,they are conventions2 our choice among all possible conventions is guided by experimental facts2 but it remains free and is limited only by the necessity of avoiding all contradiction4 ,thus it is that the postulates can remain rigorously true even though the experimental laws which have determined their adoption are only approximative4 ,in other words1 the axioms of geometry 7,i do not speak of those of arithmetic7 are merely disguised definitions4 ,then what are we to think of that question3 ,is the ,euclidean geometry true8 ,it has no meaning4 ,as well ask whether the metric system is true and the old measures false2 whether ,cartesian coordinates are true and polar coordinates false4 ,one geometry can not be more true than another2 it can only be more convenient4 ,now1 ,euclidean geometry is1 and will remain1 the most convenient3 #a ,because it is the simplest2 and it is so not only in consequence of our mental habits1 or of ,i know not what direct intuition that we may have of ,euclidean space2 it is the simplest in itself1 just as a polynomial of the first degree is simpler than one of the second2 the formulas of spherical trigonometry are more complicated than those of plane trigonometry1 and they would still appear so to an analyst ignorant of their geometric signification4 #b ,because it accords sufficiently well with the properties of natural solids1 those bodies which our hands and our eyes compare and with which we make our instruments of measure4 ,,chapter ,,iv ,,space ,,and ,,geometry ,let us begin by a little paradox4 ,beings with minds like ours1 and having the same senses as we1 but without previous education1 would receive from a suitably chosen external world impressions such that they would be led to construct a geometry other than that of ,euclid and to localize the phenomena of that external world in a non-,euclidean space1 or even in a space of four dimensions4 ,as for us1 whose education has been accomplished by our actual world1 if we were suddenly transported into this new world1 we should have no difficulty in referring its phenomena to our ,euclidean space4 ,conversely1 if these beings were transported into our environment1 they would be led to relate our phenomena to non-,euclidean space4 ,nay more2 with a little effort we likewise could do it4 ,a person who should devote his existence to it might perhaps attain to a realization of the fourth dimension4 ,,geometric ,,space ,,and ,,perceptual ,,space4--,it is often said the images of external objects are localized in space1 even that they can not be formed except on this condition4 ,it is also said that this space1 which serves thus as a ready prepared frame for our sensations and our representations1 is identical with that of the geometers1 of which it possesses all the properties4 ,to all the good minds who think thus1 the preceding statement must have appeared quite extraordinary4 ,but let us see whether they are not subject to an illusion that a more profound analysis would dissipate4 ,what1 first of all1 are the properties of space1 properly so called8 ,i mean of that space which is the object of geometry and which ,i shall call geometric space4 ,the following are some of the most essential3 #a ,it is continuous2 #b ,it is infinite2 #c ,it has three dimensions2 #d ,it is homogeneous1 that is to say1 all its points are identical one with another2 #e ,it is isotropic1 that is to say1 all the straights which pass through the same point are identical one with another4 ,compare it now to the frame of our representations and our sensations1 which ,i may call perceptual space4 ,,visual ,,space4--,consider first a purely visual impression1 due to an image formed on the bottom of the retina4 ,a cursory analysis shows us this image as continuous1 but as possessing only two dimensions2 this already distinguishes from geometric space what we may call pure visual space4 ,besides1 this image is enclosed in a limited frame4 ,finally1 there is another difference not less important3 this pure visual space is not homogeneous4 ,all the points of the retina1 aside from the images which may there be formed1 do not play the same rle4 ,the yellow spot can in no way be regarded as identical with a point on the border of the retina4 ,in fact1 not only does the same object produce there much more vivid impressions1 but in every limited frame the point occupying the center of the frame will never appear as equivalent to a point near one of the borders4 ,no doubt a more profound analysis would show us that this continuity of visual space and its two dimensions are only an illusion2 it would separate it therefore still more from geometric space1 but we shall not dwell on this remark4 ,sight1 however1 enables us to judge of distances and consequently to perceive a third dimension4 ,but every one knows that this perception of the third dimension reduces itself to the sensation of the effort at accommodation it is necessary to make1 and to that of the convergence which must be given to the two eyes1 to perceive an object distinctly4 ,these are muscular sensations altogether different from the visual sensations which have given us the notion of the first two dimensions4 ,the third dimension therefore will not appear to us as playing the same rle as the other two4 ,what may be called complete visual space is therefore not an isotropic space4 ,it has1 it is true1 precisely three dimensions1 which means that the elements of our visual sensations 7those at least which combine to form the notion of extension7 will be completely defined when three of them are known2 to use the language of mathematics1 they will be functions of three independent variables4 ,but examine the matter a little more closely4 ,the third dimension is revealed to us in two different ways3 by the effort of accommodation and by the convergence of the eyes4 ,no doubt these two indications are always concordant1 there is a constant relation between them1 or1 in mathematical terms1 the two variables which measure these two muscular sensations do not appear to us as independent2 or again1 to avoid an appeal to mathematical notions already rather refined1 we may go back to the language of the preceding chapter and enunciate the same fact as follows3 ,if two sensations of convergence1 ,a and ,b1 are indistinguishable1 the two sensations of accommodation1 ,a' and ,b'1 which respectively accompany them1 will be equally indistinguishable4 ,but here we have1 so to speak1 an experimental fact2 a priori nothing prevents our supposing the contrary1 and if the contrary takes place1 if these two muscular sensations vary independently of one another1 we shall have to take account of one more independent variable1 and 'complete visual space' will appear to us as a physical continuum of four dimensions4 ,we have here even1 ,i will add1 a fact of external experience4 ,nothing prevents our supposing that a being with a mind like ours1 having the same sense organs that we have1 may be placed in a world where light would only reach him after having traversed reflecting media of complicated form4 ,the two indications which serve us in judging distances would cease to be connected by a constant relation4 ,a being who should achieve in such a world the education of his senses would no doubt attribute four dimensions to complete visual space4 ,,tactile ,,space ,,and ,,motor ,,space4--',tactile space' is still more complicated than visual space and farther removed from geometric space4 ,it is superfluous to repeat for touch the discussion ,i have given for sight4 ,but apart from the data of sight and touch1 there are other sensations which contribute as much and more than they to the genesis of the notion of space4 ,these are known to every one2 they accompany all our movements1 and are usually called muscular sensations4 ,the corresponding frame constitutes what may be called motor space4 ,each muscle gives rise to a special sensation capable of augmenting or of diminishing1 so that the totality of our muscular sensations will depend upon as many variables as we have muscles4 ,from this point of view1 motor space would have as many dimensions as we have muscles4 ,i know it will be said that if the muscular sensations contribute to form the notion of space1 it is because we have the sense of the direction of each movement and that it makes an integrant part of the sensation4 ,if this were so1 if a muscular sensation could not arise except accompanied by this geometric sense of direction1 geometric space would indeed be a form imposed upon our sensibility4 ,but ,i perceive nothing at all of this when ,i analyze my sensations4 ,what ,i do see is that the sensations which correspond to movements in the same direction are connected in my mind by a mere association of ideas4 ,it is to this association that what we call 'the sense of direction' is reducible4 ,this feeling therefore can not be found in a single sensation4 ,this association is extremely complex1 for the contraction of the same muscle may correspond1 according to the position of the limbs1 to movements of very different direction4 ,besides1 it is evidently acquired2 it is1 like all associations of ideas1 the result of a habit2 this habit itself results from very numerous experiences2 without any doubt1 if the education of our senses had been accomplished in a different environment1 where we should have been subjected to different impressions1 contrary habits would have arisen and our muscular sensations would have been associated according to other laws4 ,,characteristics ,,of ,,perceptual ,,space4--,thus perceptual space1 under its triple form1 visual1 tactile and motor1 is essentially different from geometric space4 ,it is neither homogeneous1 nor isotropic2 one can not even say that it has three dimensions4 ,it is often said that we 'project' into geometric space the objects of our external perception2 that we 'localize' them4 ,has this a meaning1 and if so what8 ,does it mean that we represent to ourselves external objects in geometric space8 ,our representations are only the reproduction of our sensations2 they can therefore be ranged only in the same frame as these1 that is to say1 in perceptual space4 ,it is as impossible for us to represent to ourselves external bodies in geometric space1 as it is for a painter to paint on a plane canvas objects with their three dimensions4 ,perceptual space is only an image of geometric space1 an image altered in shape by a sort of perspective1 and we can represent to ourselves objects only by bringing them under the laws of this perspective4 ,therefore we do not represent to ourselves external bodies in geometric space1 but we reason on these bodies as if they were situated in geometric space4 ,when it is said then that we 'localize' such and such an object at such and such a point of space1 what does it mean8 ,it simply means that we represent to ourselves the movements it would be necessary to make to reach that object2 and one may not say that to represent to oneself these movements1 it is necessary to project the movements themselves in space and that the notion of space must1 consequently1 pre-exist4 ,when ,i say that we represent to ourselves these movements1 ,i mean only that we represent to ourselves the muscular sensations which accompany them and which have no geometric character whatever1 which consequently do not at all imply the preexistence of the notion of space4 ,,change ,,of ,,state ,,and ,,change ,,of ,,position4--,but1 it will be said1 if the idea of geometric space is not imposed upon our mind1 and if1 on the other hand1 none of our sensations can furnish it1 how could it have come into existence8 ,this is what we have now to examine1 and it will take some time1 but ,i can summarize in a few words the attempt at explanation that ,i am about to develop4 ,none of our sensations1 isolated1 could have conducted us to the idea of space2 we are led to it only in studying the laws1 according to which these sensations succeed each other4 ,we see first that our impressions are subject to change2 but among the changes we ascertain we are soon led to make a distinction4 ,at one time we say that the objects which cause these impressions have changed state1 at another time that they have changed position1 that they have only been displaced4 ,whether an object changes its state or merely its position1 this is always translated for us in the same manner3 by a modification in an aggregate of impressions4 ,how then could we have been led to distinguish between the two8 ,it is easy to account for4 ,if there has only been a change of position1 we can restore the primitive aggregate of impressions by making movements which replace us opposite the mobile object in the same relative situation4 ,we thus correct the modification that happened and we reestablish the initial state by an inverse modification4 ,if it is a question of sight1 for example1 and if an object changes its place before our eye1 we can 'follow it with the eye' and maintain its image on the same point of the retina by appropriate movements of the eyeball4 ,these movements we are conscious of because they are voluntary and because they are accompanied by muscular sensations1 but that does not mean that we represent them to ourselves in geometric space4 ,so what characterizes change of position1 what distinguishes it from change of state1 is that it can always be corrected in this way4 ,it may therefore happen that we pass from the totality of impressions ,a to the totality ,b in two different ways3 #a ,involuntarily and without experiencing muscular sensations2 this happens when it is the object which changes place2 #b ,voluntarily and with muscular sensations2 this happens when the object is motionless1 but we move so that the object has relative motion with reference to us4 ,if this be so1 the passage from the totality ,a to the totality ,b is only a change of position4 ,it follows from this that sight and touch could not have given us the notion of space without the aid of the 'muscular sense4' ,not only could this notion not be derived from a single sensation or even from a series of sensations1 but what is more1 an immobile being could never have acquired it1 since1 not being able to correct by his movements the effects of the changes of position of exterior objects1 he would have had no reason whatever to distinguish them from changes of state4 ,just as little could he have acquired it if his motions had not been voluntary or were unaccompanied by any sensations4 ,,conditions ,,of ,,compensation4--,how is a like compensation possible1 of such sort that two changes1 otherwise independent of each other1 reciprocally correct each other8 ,a mind already familiar with geometry would reason as follows3 ,evidently1 if there is to be compensation1 the various parts of the external object1 on the one hand1 and the various sense organs1 on the other hand1 must be in the same relative position after the double change4 ,and1 for that to be the case1 the various parts of the external object must likewise have retained in reference to each other the same relative position1 and the same must be true of the various parts of our body in regard to each other4 ,in other words1 the external object1 in the first change1 must be displaced as is a rigid solid1 and so must it be with the whole of our body in the second change which corrects the first4 ,under these conditions1 compensation may take place4 ,but we who as yet know nothing of geometry1 since for us the notion of space is not yet formed1 we can not reason thus1 we can not foresee a priori whether compensation is possible4 ,but experience teaches us that it sometimes happens1 and it is from this experimental fact that we start to distinguish changes of state from changes of position4 ,,solid ,,bodies ,,and ,,geometry4--,among surrounding objects there are some which frequently undergo displacements susceptible of being thus corrected by a correlative movement of our own body2 these are the solid bodies4 ,the other objects1 whose form is variable1 only exceptionally undergo like displacements 7change of position without change of form74 ,when a body changes its place and its shape1 we can no longer1 by appropriate movements1 bring back our sense-organs into the same relative situation with regard to this body2 consequently we can no longer reestablish the primitive totality of impressions4 ,it is only later1 and as a consequence of new experiences1 that we learn how to decompose the bodies of variable form into smaller elements1 such that each is displaced almost in accordance with the same laws as solid bodies4 ,thus we distinguish 'deformations' from other changes of state2 in these deformations1 each element undergoes a mere change of position1 which can be corrected1 but the modification undergone by the aggregate is more profound and is no longer susceptible of correction by a correlative movement4 ,such a notion is already very complex and must have been relatively late in appearing2 moreover it could not have arisen if the observation of solid bodies had not already taught us to distinguish changes of position4 ,therefore1 if there were no solid bodies in nature1 there would be no geometry4 ,another remark also deserves a moment's attention4 ,suppose a solid body to occupy successively the positions alpha and beta2 in its first position1 it will produce on us the totality of impressions ,a1 and in its second position the totality of impressions ,b4 ,let there be now a second solid body1 having qualities entirely different from the first1 for example1 a different color4 ,suppose it to pass from the position alpha1 where it gives us the totality of impressions ,a'1 to the position beta1 where it gives the totality of impressions ,b'4 ,in general1 the totality ,a will have nothing in common with the totality ,a'1 nor the totality ,b with the totality ,b'4 ,the transition from the totality ,a to the totality ,b and that from the totality ,a' to the totality ,b' are therefore two changes which in themselves have in general nothing in common4 ,and yet we regard these two changes both as displacements and1 furthermore1 we consider them as the same displacement4 ,how can that be8 ,it is simply because they can both be corrected by the same correlative movement of our body4 ',correlative movement' therefore constitutes the sole connection between two phenomena which otherwise we never should have dreamt of likening4 ,on the other hand1 our body1 thanks to the number of its articulations and muscles1 may make a multitude of different movements2 but all are not capable of 'correcting' a modification of external objects2 only those will be capable of it in which our whole body1 or at least all those of our sense-organs which come into play1 are displaced as a whole1 that is1 without their relative positions varying1 or in the fashion of a solid body4 ,to summarize3 #a ,we are led at first to distinguish two categories of phenomena3 ,some1 involuntary1 unaccompanied by muscular sensations1 are attributed by us to external objects2 these are external changes2 ,others1 opposite in character and attributed by us to the movements of our own body1 are internal changes2 #b ,we notice that certain changes of each of these categories may be corrected by a correlative change of the other category2 #c ,we distinguish among external changes those which have thus a correlative in the other category2 these we call displacements2 and just so among the internal changes1 we distinguish those which have a correlative in the first category4 ,thus are defined1 thanks to this reciprocity1 a particular class of phenomena which we call displacements4 ,the laws of these phenomena constitute the object of geometry4 ,,law ,,of ,,homogeneity4--,the first of these laws is the law of homogeneity4 ,suppose that1 by an external change alpha1 we pass from the totality of impressions ,a to the totality ,b1 then that this change alpha is corrected by a correlative voluntary movement beta1 so that we are brought back to the totality ,a4 ,suppose now that another external change alpha' makes us pass anew from the totality ,a to the totality ,b4 ,experience teaches us that this change alpha' is1 like alpha1 susceptible of being corrected by a correlative voluntary movement beta' and that this movement beta' corresponds to the same muscular sensations as the movement beta which corrected alpha4 ,this fact is usually enunciated by saying that space is homogeneous and isotropic4 ,it may also be said that a movement which has once been produced may be repeated a second and a third time1 and so on1 without its properties varying4 ,in the first chapter1 where we discussed the nature of mathematical reasoning1 we saw the importance which must be attributed to the possibility of repeating indefinitely the same operation4 ,it is from this repetition that mathematical reasoning gets its power2 it is1 therefore1 thanks to the law of homogeneity1 that it has a hold on the geometric facts4 ,for completeness1 to the law of homogeneity should be added a multitude of other analogous laws1 into the details of which ,i do not wish to enter1 but which mathematicians sum up in a word by saying that displacements form 'a group4' ,,the ,,non-,,euclidean ,,world4--,if geometric space were a frame imposed on each of our representations1 considered individually1 it would be impossible to represent to ourselves an image stripped of this frame1 and we could change nothing of our geometry4 ,but this is not the case2 geometry is only the rsum of the laws according to which these images succeed each other4 ,nothing then prevents us from imagining a series of representations1 similar in all points to our ordinary representations1 but succeeding one another according to laws different from those to which we are accustomed4 ,we can conceive then that beings who received their education in an environment where these laws were thus upset might have a geometry very different from ours4 ,suppose1 for example1 a world enclosed in a great sphere and subject to the following laws3 ,the temperature is not uniform2 it is greatest at the center1 and diminishes in proportion to the distance from the center1 to sink to absolute zero when the sphere is reached in which this world is enclosed4 ,to specify still more precisely the law in accordance with which this temperature varies3 ,let ,r be the radius of the limiting sphere2 let r be the distance of the point considered from the center of this sphere4 ,the absolute temperature shall be proportional to ,r#b - r#b4 ,i shall further suppose that1 in this world1 all bodies have the same coefficient of dilatation1 so that the length of any rule is proportional to its absolute temperature4 ,finally1 ,i shall suppose that a body transported from one point to another of different temperature is put immediately into thermal equilibrium with its new environment4 ,nothing in these hypotheses is contradictory or unimaginable4 ,a movable object will then become smaller and smaller in proportion as it approaches the limit-sphere4 ,note first that1 though this world is limited from the point of view of our ordinary geometry1 it will appear infinite to its inhabitants4 ,in fact1 when these try to approach the limit-sphere1 they cool off and become smaller and smaller4 ,therefore the steps they take are also smaller and smaller1 so that they can never reach the limiting sphere4 ,if1 for us1 geometry is only the study of the laws according to which rigid solids move1 for these imaginary beings it will be the study of the laws of motion of solids distorted by the differences of temperature just spoken of4 ,no doubt1 in our world1 natural solids likewise undergo variations of form and volume due to warming or cooling4 ,but we neglect these variations in laying the foundations of geometry1 because1 besides their being very slight1 they are irregular and consequently seem to us accidental4 ,in our hypothetical world1 this would no longer be the case1 and these variations would follow regular and very simple laws4 ,moreover1 the various solid pieces of which the bodies of its inhabitants would be composed would undergo the same variations of form and volume4 ,i will make still another hypothesis2 ,i will suppose light traverses media diversely refractive and such that the index of refraction is inversely proportional to ,r#b - r#b4 ,it is easy to see that1 under these conditions1 the rays of light would not be rectilinear1 but circular4 ,to justify what precedes1 it remains for me to show that certain changes in the position of external objects can be corrected by correlative movements of the sentient beings inhabiting this imaginary world1 and that in such a way as to restore the primitive aggregate of impressions experienced by these sentient beings4 ,suppose in fact that an object is displaced1 undergoing deformation1 not as a rigid solid1 but as a solid subjected to unequal dilatations in exact conformity to the law of temperature above supposed4 ,permit me for brevity to call such a movement a non-,euclidean displacement4 ,if a sentient being happens to be in the neighborhood1 his impressions will be modified by the displacement of the object1 but he can reestablish them by moving in a suitable manner4 ,it suffices if finally the aggregate of the object and the sentient being1 considered as forming a single body1 has undergone one of those particular displacements ,i have just called non-,euclidean4 ,this is possible if it be supposed that the limbs of these beings dilate according to the same law as the other bodies of the world they inhabit4 ,although from the point of view of our ordinary geometry there is a deformation of the bodies in this displacement and their various parts are no longer in the same relative position1 nevertheless we shall see that the impressions of the sentient being have once more become the same4 ,in fact1 though the mutual distances of the various parts may have varied1 yet the parts originally in contact are again in contact4 ,therefore the tactile impressions have not changed4 ,on the other hand1 taking into account the hypothesis made above in regard to the refraction and the curvature of the rays of light1 the visual impressions will also have remained the same4 ,these imaginary beings will therefore like ourselves be led to classify the phenomena they witness and to distinguish among them the 'changes of position' susceptible of correction by a correlative voluntary movement4 ,if they construct a geometry1 it will not be1 as ours is1 the study of the movements of our rigid solids2 it will be the study of the changes of position which they will thus have distinguished and which are none other than the 'non-,euclidean displacements'2 it will be non-,euclidean geometry4 ,thus beings like ourselves1 educated in such a world1 would not have the same geometry as ours4 ,,the ,,world ,,of ,,four ,,dimensions4--,we can represent to ourselves a four-dimensional world just as well as a non-,euclidean4 ,the sense of sight1 even with a single eye1 together with the muscular sensations relative to the movements of the eyeball1 would suffice to teach us space of three dimensions4 ,the images of external objects are painted on the retina1 which is a two-dimensional canvas2 they are perspectives4 ,but1 as eye and objects are movable1 we see in succession various perspectives of the same body1 taken from different points of view4 ,at the same time1 we find that the transition from one perspective to another is often accompanied by muscular sensations4 ,if the transition from the perspective ,a to the perspective ,b1 and that from the perspective ,a' to the perspective ,b' are accompanied by the same muscular sensations1 we liken them one to the other as operations of the same nature4 ,studying then the laws according to which these operations combine1 we recognize that they form a group1 which has the same structure as that of the movements of rigid solids4 ,now1 we have seen that it is from the properties of this group we have derived the notion of geometric space and that of three dimensions4 ,we understand thus how the idea of a space of three dimensions could take birth from the pageant of these perspectives1 though each of them is of only two dimensions1 since they follow one another according to certain laws4 ,well1 just as the perspective of a three-dimensional figure can be made on a plane1 we can make that of a four-dimensional figure on a picture of three 7or of two7 dimensions4 ,to a geometer this is only child's play4 ,we can even take of the same figure several perspectives from several different points of view4 ,we can easily represent to ourselves these perspectives1 since they are of only three dimensions4 ,imagine that the various perspectives of the same object succeed one another1 and that the transition from one to the other is accompanied by muscular sensations4 ,we shall of course consider two of these transitions as two operations of the same nature when they are associated with the same muscular sensations4 ,nothing then prevents us from imagining that these operations combine according to any law we choose1 for example1 so as to form a group with the same structure as that of the movements of a rigid solid of four dimensions4 ,here there is nothing unpicturable1 and yet these sensations are precisely those which would be felt by a being possessed of a two-dimensional retina who could move in space of four dimensions4 ,in this sense we may say the fourth dimension is imaginable4 ,,conclusions4--,we see that experience plays an indispensable rle in the genesis of geometry2 but it would be an error thence to conclude that geometry is1 even in part1 an experimental science4 ,if it were experimental1 it would be only approximative and provisional4 ,and what rough approximation6 ,geometry would be only the study of the movements of solids2 but in reality it is not occupied with natural solids1 it has for object certain ideal solids1 absolutely rigid1 which are only a simplified and very remote image of natural solids4 ,the notion of these ideal solids is drawn from all parts of our mind1 and experience is only an occasion which induces us to bring it forth from them4 ,the object of geometry is the study of a particular 'group'2 but the general group concept pre-exists1 at least potentially1 in our minds4 ,it is imposed on us1 not as form of our sense1 but as form of our understanding4 ,only1 from among all the possible groups1 that must be chosen which will be1 so to speak1 the standard to which we shall refer natural phenomena4 ,experience guides us in this choice without forcing it upon us2 it tells us not which is the truest geometry1 but which is the most convenient4 ,notice that ,i have been able to describe the fantastic worlds above imagined without ceasing to employ the language of ordinary geometry4 ,and1 in fact1 we should not have to change it if transported thither4 ,beings educated there would doubtless find it more convenient to create a geometry different from ours1 and better adapted to their impressions4 ,as for us1 in face of the same impressions1 it is certain we should find it more convenient not to change our habits4 ,,chapter ,v ,,experience ,,and ,,geometry #a4 ,already in the preceding pages ,i have several times tried to show that the principles of geometry are not experimental facts and that in particular ,euclid's postulate can not be proven experimentally4 ,however decisive appear to me the reasons already given1 ,i believe ,i should emphasize this point because here a false idea is profoundly rooted in many minds4 #b4 ,if we construct a material circle1 measure its radius and circumference1 and see if the ratio of these two lengths is equal to pi1 what shall we have done8 ,we shall have made an experiment on the properties of the matter with which we constructed this round thing1 and of that of which the measure used was made4 #c4 ,,geometry ,,and ,,astronomy4--,the question has also been put in another way4 ,if ,lobachevski's geometry is true1 the parallax of a very distant star will be finite2 if ,riemann's is true1 it will be negative4 ,these are results which seem within the reach of experiment1 and there have been hopes that astronomical observations might enable us to decide between the three geometries4 ,but in astronomy 'straight line' means simply 'path of a ray of light4' ,if therefore negative parallaxes were found1 or if it were demonstrated that all parallaxes are superior to a certain limit1 two courses would be open to us2 we might either renounce ,euclidean geometry1 or else modify the laws of optics and suppose that light does not travel rigorously in a straight line4 ,it is needless to add that all the world would regard the latter solution as the more advantageous4 ,the ,euclidean geometry has1 therefore1 nothing to fear from fresh experiments4 #d4 ,is the position tenable1 that certain phenomena1 possible in ,euclidean space1 would be impossible in non-,euclidean space1 so that experience1 in establishing these phenomena1 would directly contradict the non-,euclidean hypothesis8 ,for my part ,i think no such question can be put4 ,to my mind it is precisely equivalent to the following1 whose absurdity is patent to all eyes3 are there lengths expressible in meters and centimeters1 but which can not be measured in fathoms1 feet and inches1 so that experience1 in ascertaining the existence of these lengths1 would directly contradict the hypothesis that there are fathoms divided into six feet8 ,examine the question more closely4 ,i suppose that the straight line possesses in ,euclidean space any two properties which ,i shall call ,a and ,b2 that in non-,euclidean space it still possesses the property ,a1 but no longer has the property ,b2 finally ,i suppose that in both ,euclidean and non-,euclidean space the straight line is the only line having the property ,a4 ,if this were so1 experience would be capable of deciding between the hypothesis of ,euclid and that of ,lobachevski4 ,it would be ascertained that a definite concrete object1 accessible to experiment1 for example1 a pencil of rays of light1 possesses the property ,a2 we should conclude that it is rectilinear1 and then investigate whether or not it has the property ,b4 ,but this is not so2 no property exists which1 like this property ,a1 can be an absolute criterion enabling us to recognize the straight line and to distinguish it from every other line4 ,shall we say1 for instance3 the following is such a property3 the straight line is a line such that a figure of which this line forms a part can be moved without the mutual distances of its points varying and so that all points of this line remain fixed8 ,this1 in fact1 is a property which1 in ,euclidean or non-,euclidean space1 belongs to the straight and belongs only to it4 ,but how shall we ascertain experimentally whether it belongs to this or that concrete object8 ,it will be necessary to measure distances1 and how shall one know that any concrete magnitude which ,i have measured with my material instrument really represents the abstract distance8 ,we have only pushed back the difficulty4 ,in reality the property just enunciated is not a property of the straight line alone1 it is a property of the straight line and distance4 ,for it to serve as absolute criterion1 we should have to be able to establish not only that it does not also belong to a line other than the straight and to distance1 but in addition that it does not belong to a line other than the straight and to a magnitude other than distance4 ,now this is not true4 ,it is therefore impossible to imagine a concrete experiment which can be interpreted in the ,euclidean system and not in the ,lobachevskian system1 so that ,i may conclude3 ,no experience will ever be in contradiction to ,euclid's postulate2 nor1 on the other hand1 will any experience ever contradict the postulate of ,lobachevski4 #e4 ,but it is not enough that the ,euclidean 7or non-,euclidean7 geometry can never be directly contradicted by experience4 ,might it not happen that it can accord with experience only by violating the principle of sufficient reason or that of the relativity of space8 ,i will explain myself3 consider any material system2 we shall have to regard1 on the one hand1 'the state' of the various bodies of this system 7for instance1 their temperature1 their electric potential1 etc471 and1 on the other hand1 their position in space2 and among the data which enable us to define this position we shall1 moreover1 distinguish the mutual distances of these bodies1 which define their relative positions1 from the conditions which define the absolute position of the system and its absolute orientation in space4 ,the laws of the phenomena which will happen in this system will depend on the state of these bodies and their mutual distances2 but1 because of the relativity and passivity of space1 they will not depend on the absolute position and orientation of the system4 ,in other words1 the state of the bodies and their mutual distances at any instant will depend solely on the state of these same bodies and on their mutual distances at the initial instant1 but will not at all depend on the absolute initial position of the system or on its absolute initial orientation4 ,this is what for brevity ,i shall call the law of relativity4 ,hitherto ,i have spoken as a ,euclidean geometer4 ,as ,i have said1 an experience1 whatever it be1 admits of an interpretation on the ,euclidean hypothesis2 but it admits of one equally on the non-,euclidean hypothesis4 ,well1 we have made a series of experiments2 we have interpreted them on the ,euclidean hypothesis1 and we have recognized that these experiments thus interpreted do not violate this 'law of relativity4' ,we now interpret them on the non-,euclidean hypothesis3 this is always possible2 only the non-,euclidean distances of our different bodies in this new interpretation will not generally be the same as the ,euclidean distances in the primitive interpretation4 ,will our experiments1 interpreted in this new manner1 still be in accord with our 'law of relativity'8 ,and if there were not this accord1 should we not have also the right to say experience had proven the falsity of the non-,euclidean geometry8 ,it is easy to see that this is an idle fear2 in fact1 to apply the law of relativity in all rigor1 it must be applied to the entire universe4 ,for if only a part of this universe were considered1 and if the absolute position of this part happened to vary1 the distances to the other bodies of the universe would likewise vary1 their influence on the part of the universe considered would consequently augment or diminish1 which might modify the laws of the phenomena happening there4 ,but if our system is the entire universe1 experience is powerless to give information about its absolute position and orientation in space4 ,all that our instruments1 however perfected they may be1 can tell us will be the state of the various parts of the universe and their mutual distances4 ,so our law of relativity may be thus enunciated3 ,the readings we shall be able to make on our instruments at any instant will depend only on the readings we could have made on these same instruments at the initial instant4 ,now such an enunciation is independent of every interpretation of experimental facts4 ,if the law is true in the ,euclidean interpretation1 it will also be true in the non-,euclidean interpretation4 ,allow me here a short digression4 ,i have spoken above of the data which define the position of the various bodies of the system2 ,i should likewise have spoken of those which define their velocities2 ,i should then have had to distinguish the velocities with which the mutual distances of the different bodies vary2 and1 on the other hand1 the velocities of translation and rotation of the system1 that is to say1 the velocities with which its absolute position and orientation vary4 ,to fully satisfy the mind1 the law of relativity should be expressible thus3 ,the state of bodies and their mutual distances at any instant1 as well as the velocities with which these distances vary at this same instant1 will depend only on the state of those bodies and their mutual distances at the initial instant1 and the velocities with which these distances vary at this initial instant1 but they will not depend either upon the absolute initial position of the system1 or upon its absolute orientation1 or upon the velocities with which this absolute position and orientation varied at the initial instant4 ,unhappily the law thus enunciated is not in accord with experiments1 at least as they are ordinarily interpreted4 ,suppose a man be transported to a planet whose heavens were always covered with a thick curtain of clouds1 so that he could never see the other stars2 on that planet he would live as if it were isolated in space4 ,yet this man could become aware that it turned1 either by measuring its oblateness 7done ordinarily by the aid of astronomic observations1 but capable of being done by purely geodetic means71 or by repeating the experiment of ,foucault's pendulum4 ,the absolute rotation of this planet could therefore be made evident4 ,that is a fact which shocks the philosopher1 but which the physicist is compelled to accept4 ,we know that from this fact ,newton inferred the existence of absolute space2 ,i myself am quite unable to adopt this view4 ,i shall explain why in ,part ,,iii4 ,for the moment it is not my intention to enter upon this difficulty4 ,therefore ,i must resign myself1 in the enunciation of the law of relativity1 to including velocities of every kind among the data which define the state of the bodies4 ,however that may be1 this difficulty is the same for ,euclid's geometry as for ,lobachevski's2 ,i therefore need not trouble myself with it1 and have only mentioned it incidentally4 ,what is important is the conclusion3 experiment can not decide between ,euclid and ,lobachevski4 ,to sum up1 whichever way we look at it1 it is impossible to discover in geometric empiricism a rational meaning4 #f4 ,experiments only teach us the relations of bodies to one another2 none of them bears or can bear on the relations of bodies with space1 or on the mutual relations of different parts of space4 ,yes1 you reply1 a single experiment is insufficient1 because it gives me only a single equation with several unknowns2 but when ,i shall have made enough experiments ,i shall have equations enough to calculate all my unknowns4 ,to know the height of the mainmast does not suffice for calculating the age of the captain4 ,when you have measured every bit of wood in the ship you will have many equations1 but you will know his age no better4 ,all your measurements bearing only on your bits of wood can reveal to you nothing except concerning these bits of wood4 ,just so your experiments1 however numerous they may be1 bearing only on the relations of bodies to one another1 will reveal to us nothing about the mutual relations of the various parts of space4 #g4 ,will you say that if the experiments bear on the bodies1 they bear at least upon the geometric properties of the bodies8 ,but1 first1 what do you understand by geometric properties of the bodies8 ,i assume that it is a question of the relations of the bodies with space2 these properties are therefore inaccessible to experiments which bear only on the relations of the bodies to one another4 ,this alone would suffice to show that there can be no question of these properties4 ,still let us begin by coming to an understanding about the sense of the phrase3 geometric properties of bodies4 ,when ,i say a body is composed of several parts1 ,i assume that ,i do not enunciate therein a geometric property1 and this would remain true even if ,i agreed to give the improper name of points to the smallest parts ,i consider4 ,when ,i say that such a part of such a body is in contact with such a part of such another body1 ,i enunciate a proposition which concerns the mutual relations of these two bodies and not their relations with space4 ,i suppose you will grant me these are not geometric properties2 at least ,i am sure you will grant me these properties are independent of all knowledge of metric geometry4 ,this presupposed1 ,i imagine that we have a solid body formed of eight slender iron rods1 ,,oa1 ,,ob1 ,,oc1 ,,od1 ,,oe1 ,,of1 ,,og1 ,,oh1 united at one of their extremities ,o4 ,let us besides have a second solid body1 for example a bit of wood1 to be marked with three little flecks of ink which ,i shall call alpha1 beta1 gamma4 ,i further suppose it ascertained that alphabetagamma may be brought into contact with ,,ago 7,i mean alpha with ,a1 and at the same time beta with ,g and gamma with ,o71 then that we may bring successively into contact alphabetagamma with ,,bgo1 ,,cgo1 ,,dgo1 ,,ego1 ,,fgo1 then with ,,aho1 ,,bho1 ,,cho1 ,,dho1 ,,eho1 ,,fho1 then alphagamma successively with ,,ab1 ,,bc1 ,,cd1 ,,de1 ,,ef1 ,,fa4 ,these are determinations we may make without having in advance any notion about form or about the metric properties of space4 ,they in no wise bear on the 'geometric properties of bodies4' ,and these determinations will not be possible if the bodies experimented upon move in accordance with a group having the same structure as the ,lobachevskian group 7,i mean according to the same laws as solid bodies in ,lobachevski's geometry74 ,they suffice therefore to prove that these bodies move in accordance with the ,euclidean group1 or at least that they do not move according to the ,lobachevskian group4 ,that they are compatible with the ,euclidean group is easy to see4 ,for they could be made if the body alphabetagamma was a rigid solid of our ordinary geometry presenting the form of a right-angled triangle1 and if the points ,,abcdefgh were the summits of a polyhedron formed of two regular hexagonal pyramids of our ordinary geometry1 having for common base ,,abcdef and for apices the one ,g and the other ,h4 ,suppose now that in place of the preceding determination it is observed that as above alphabetagamma can be successively applied to ,,ago1 ,,bgo1 ,,cgo1 ,,dgo1 ,,ego1 ,,aho1 ,,bho1 ,,cho1 ,,dho1 ,,eho1 ,,fho1 then that alphabeta 7and no longer alphagamma7 can be successively applied to ,,ab1 ,,bc1 ,,cd1 ,,de1 ,,ef and ,,fa4 ,these are determinations which could be made if non-,euclidean geometry were true1 if the bodies alphabetagamma and ,,oabcdefgh were rigid solids1 and if the first were a right-angled triangle and the second a double regular hexagonal pyramid of suitable dimensions4 ,therefore these new determinations are not possible if the bodies move according to the ,euclidean group2 but they become so if it be supposed that the bodies move according to the ,lobachevskian group4 ,they would suffice1 therefore 7if one made them71 to prove that the bodies in question do not move according to the ,euclidean group4 ,thus1 without making any hypothesis about form1 about the nature of space1 about the relations of bodies to space1 and without attributing to bodies any geometric property1 ,i have made observations which have enabled me to show in one case that the bodies experimented upon move according to a group whose structure is ,euclidean1 in the other case that they move according to a group whose structure is ,lobachevskian4 ,and one may not say that the first aggregate of determinations would constitute an experiment proving that space is ,euclidean1 and the second an experiment proving that space is non-,euclidean4 ,in fact one could imagine 7,i say imagine7 bodies moving so as to render possible the second series of determinations4 ,and the proof is that the first mechanician met could construct such bodies if he cared to take the pains and make the outlay4 ,you will not conclude from that1 however1 that space is non-,euclidean4 ,nay1 since the ordinary solid bodies would continue to exist when the mechanician had constructed the strange bodies of which ,i have just spoken1 it would be necessary to conclude that space is at the same time ,euclidean and non-,euclidean4 ,suppose1 for example1 that we have a great sphere of radius ,r and that the temperature decreases from the center to the surface of this sphere according to the law of which ,i have spoken in describing the non-,euclidean world4 ,we might have bodies whose expansion would be negligible and which would act like ordinary rigid solids2 and1 on the other hand1 bodies very dilatable and which would act like non-,euclidean solids4 ,we might have two double pyramids ,,oabcdefgh and ,o',a',b',c',d',e',f',g',h' and two triangles alphabetagamma and alpha'beta'gamma'4 ,the first double pyramid might be rectilinear and the second curvilinear2 the triangle alphabetagamma might be made of inexpansible matter and the other of a very dilatable matter4 ,it would then be possible to make the first observations with the double pyramid ,,oah and the triangle alphabetagamma1 and the second with the double pyramid ,o',a',h' and the triangle alpha'beta'gamma'4 ,and then experiment would seem to prove first that the ,euclidean geometry is true and then that it is false4 ,experiments therefore have a bearing1 not on space1 but on bodies4 ,,supplement #h4 ,to complete the matter1 ,i ought to speak of a very delicate question1 which would require long development2 ,i shall confine myself to summarizing here what ,i have expounded in the ,revue de ,mtaphysique et de ,morale and in ,the ,monist4 ,when we say space has three dimensions1 what do we mean8 ,we have seen the importance of those 'internal changes' revealed to us by our muscular sensations4 ,they may serve to characterize the various attitudes of our body4 ,take arbitrarily as origin one of these attitudes ,a4 ,when we pass from this initial attitude to any other attitude ,b1 we feel a series of muscular sensations1 and this series ,s will define ,b4 ,observe1 however1 that we shall often regard two series ,s and ,s' as defining the same attitude ,b 7since the initial and final attitudes ,a and ,b remaining the same1 the intermediary attitudes and the corresponding sensations may differ74 ,how then shall we recognize the equivalence of these two series8 ,because they may serve to compensate the same external change1 or more generally because1 when it is a question of compensating an external change1 one of the series can be replaced by the other4 ,among these series1 we have distinguished those which of themselves alone can compensate an external change1 and which we have called 'displacements4' ,as we can not discriminate between two displacements which are too close together1 the totality of these displacements presents the characteristics of a physical continuum2 experience teaches us that they are those of a physical continuum of six dimensions2 but we do not yet know how many dimensions space itself has1 we must first solve another question4 ,what is a point of space8 ,everybody thinks he knows1 but that is an illusion4 ,what we see when we try to represent to ourselves a point of space is a black speck on white paper1 a speck of chalk on a blackboard1 always an object4 ,the question should therefore be understood as follows3 ,what do ,i mean when ,i say the object ,b is at the same point that the object ,a occupied just now8 ,or further1 what criterion will enable me to apprehend this8 ,i mean that1 although ,i have not budged 7which my muscular sense tells me71 my first finger which just now touched the object ,a touches at present the object ,b4 ,i could have used other criteria2 for instance another finger or the sense of sight4 ,but the first criterion is sufficient2 ,i know that if it answers yes1 all the other criteria will give the same response4 ,i know it by experience1 ,i can not know it a priori4 ,for the same reason ,i say that touch can not be exercised at a distance2 this is another way of enunciating the same experimental fact4 ,and if1 on the contrary1 ,i say that sight acts at a distance1 it means that the criterion furnished by sight may respond yes while the others reply no4 ,and in fact1 the object1 although moved away1 may form its image at the same point of the retina4 ,sight responds yes1 the object has remained at the same point and touch answers no1 because my finger which just now touched the object1 at present touches it no longer4 ,if experience had shown us that one finger may respond no when the other says yes1 we should likewise say that touch acts at a distance4 ,in short1 for each attitude of my body1 my first finger determines a point1 and this it is1 and this alone1 which defines a point of space4 ,to each attitude corresponds thus a point2 but it often happens that the same point corresponds to several different attitudes 7in this case we say our finger has not budged1 but the rest of the body has moved74 ,we distinguish1 therefore1 among the changes of attitude those where the finger does not budge4 ,how are we led thereto8 ,it is because often we notice that in these changes the object which is in contact with the finger remains in contact with it4 ,range1 therefore1 in the same class all the attitudes obtainable from each other by one of the changes we have thus distinguished4 ,to all the attitudes of the class will correspond the same point of space4 ,therefore to each class will correspond a point and to each point a class4 ,but one may say that what experience arrives at is not the point1 it is this class of changes or1 better1 the corresponding class of muscular sensations4 ,and when we say space has three dimensions1 we simply mean that the totality of these classes appears to us with the characteristics of a physical continuum of three dimensions4 ,one might be tempted to conclude that it is experience which has taught us how many dimensions space has4 ,but in reality here also our experiences have bearing1 not on space1 but on our body and its relations with the neighboring objects4 ,moreover they are excessively crude4 ,in our mind pre-existed the latent idea of a certain number of groups--those whose theory ,lie has developed4 ,which group shall we choose1 to make of it a sort of standard with which to compare natural phenomena8 ,and1 this group chosen1 which of its sub-groups shall we take to characterize a point of space8 ,experience has guided us by showing us which choice best adapts itself to the properties of our body4 ,but its rle is limited to that4 ,,ancestral ,,experience ,it has often been said that if individual experience could not create geometry the same is not true of ancestral experience4 ,but what does that mean8 ,is it meant that we could not experimentally demonstrate ,euclid's postulate1 but that our ancestors have been able to do it8 ,not in the least4 ,it is meant that by natural selection our mind has adapted itself to the conditions of the external world1 that it has adopted the geometry most advantageous to the species3 or in other words the most convenient4 ,this is entirely in conformity with our conclusions2 geometry is not true1 it is advantageous4 ,,part ,,iii ,,force ,,chapter ,,vi ,,the ,,classic ,,mechanics ,the ,english teach mechanics as an experimental science2 on the continent it is always expounded as more or less a deductive and a priori science4 ,the ,english are right1 that goes without saying2 but how could the other method have been persisted in so long8 ,why have the continental savants who have sought to get out of the ruts of their predecessors been usually unable to free themselves completely8 ,on the other hand1 if the principles of mechanics are only of experimental origin1 are they not therefore only approximate and provisional8 ,might not new experiments some day lead us to modify or even to abandon them8 ,such are the questions which naturally obtrude themselves1 and the difficulty of solution comes principally from the fact that the treatises on mechanics do not clearly distinguish between what is experiment1 what is mathematical reasoning1 what is convention1 what is hypothesis4 ,that is not all3 #a ,there is no absolute space and we can conceive only of relative motions2 yet usually the mechanical facts are enunciated as if there were an absolute space to which to refer them4 #b ,there is no absolute time2 to say two durations are equal is an assertion which has by itself no meaning and which can acquire one only by convention4 #c ,not only have we no direct intuition of the equality of two durations1 but we have not even direct intuition of the simultaneity of two events occurring in different places3 this ,i have explained in an article entitled ,la mesure du temps4#c #c ,revue de ,mtaphysique et de ,morale1 t4 ,,vi41 pp4 #a-ac 7,january1 #ahih74 #d ,finally1 our ,euclidean geometry is itself only a sort of convention of language2 mechanical facts might be enunciated with reference to a non-,euclidean space which would be a guide less convenient than1 but just as legitimate as1 our ordinary space2 the enunciation would thus become much more complicated1 but it would remain possible4 ,thus absolute space1 absolute time1 geometry itself1 are not conditions which impose themselves on mechanics2 all these things are no more antecedent to mechanics than the ,french language is logically antecedent to the verities one expresses in ,french4 ,we might try to enunciate the fundamental laws of mechanics in a language independent of all these conventions2 we should thus without doubt get a better idea of what these laws are in themselves2 this is what ,m4 ,andrade has attempted to do1 at least in part1 in his ,leons de mcanique physique4 ,the enunciation of these laws would become of course much more complicated1 because all these conventions have been devised expressly to abridge and simplify this enunciation4 ,as for me1 save in what concerns absolute space1 ,i shall ignore all these difficulties2 not that ,i fail to appreciate them1 far from that2 but we have sufficiently examined them in the first two parts of the book4 ,i shall therefore admit1 provisionally1 absolute time and ,euclidean geometry4 ,,the ,,principle ,,of ,,inertia4--,a body acted on by no force can only move uniformly in a straight line4 ,is this a truth imposed a priori upon the mind8 ,if it were so1 how should the ,greeks have failed to recognize it8 ,how could they have believed that motion stops when the cause which gave birth to it ceases8 ,or again that every body if nothing prevents1 will move in a circle1 the noblest of motions8 ,if it is said that the velocity of a body can not change if there is no reason for it to change1 could it not be maintained just as well that the position of this body can not change1 or that the curvature of its trajectory can not change1 if no external cause intervenes to modify them8 ,is the principle of inertia1 which is not an a priori truth1 therefore an experimental fact8 ,but has any one ever experimented on bodies withdrawn from the action of every force8 and1 if so1 how was it known that these bodies were subjected to no force8 ,the example ordinarily cited is that of a ball rolling a very long time on a marble table2 but why do we say it is subjected to no force8 ,is this because it is too remote from all other bodies to experience any appreciable action from them8 ,yet it is not farther from the earth than if it were thrown freely into the air2 and every one knows that in this case it would experience the influence of gravity due to the attraction of the earth4 ,teachers of mechanics usually pass rapidly over the example of the ball2 but they add that the principle of inertia is verified indirectly by its consequences4 ,they express themselves badly2 they evidently mean it is possible to verify various consequences of a more general principle1 of which that of inertia is only a particular case4 ,i shall propose for this general principle the following enunciation3 ,the acceleration of a body depends only upon the position of this body and of the neighboring bodies and upon their velocities4 ,mathematicians would say the movements of all the material molecules of the universe depend on differential equations of the second order4 ,to make it clear that this is really the natural generalization of the law of inertia1 ,i shall beg you to permit me a bit of fiction4 ,the law of inertia1 as ,i have said above1 is not imposed upon us a priori2 other laws would be quite as compatible with the principle of sufficient reason4 ,if a body is subjected to no force1 in lieu of supposing its velocity not to change1 it might be supposed that it is its position or else its acceleration which is not to change4 ,well1 imagine for an instant that one of these two hypothetical laws is a law of nature and replaces our law of inertia4 ,what would be its natural generalization8 ,a moment's thought will show us4 ,in the first case1 we must suppose that the velocity of a body depends only upon its position and upon that of the neighboring bodies2 in the second case that the change of acceleration of a body depends only upon the position of this body and of the neighboring bodies1 upon their velocities and upon their accelerations4 ,or to speak the language of mathematics1 the differential equations of motion would be of the first order in the first case1 and of the third order in the second case4 ,let us slightly modify our fiction4 ,suppose a world analogous to our solar system1 but where1 by a strange chance1 the orbits of all the planets are without eccentricity and without inclination4 ,suppose further that the masses of these planets are too slight for their mutual perturbations to be sensible4 ,astronomers inhabiting one of these planets could not fail to conclude that the orbit of a star can only be circular and parallel to a certain plane2 the position of a star at a given instant would then suffice to determine its velocity and its whole path4 ,the law of inertia which they would adopt would be the first of the two hypothetical laws ,i have mentioned4 ,imagine now that this system is some day traversed with great velocity by a body of vast mass1 coming from distant constellations4 ,all the orbits would be profoundly disturbed4 ,still our astronomers would not be too greatly astonished2 they would very well divine that this new star was alone to blame for all the mischief4 ,but1 they would say1 when it is gone1 order will of itself be reestablished2 no doubt the distances of the planets from the sun will not revert to what they were before the cataclysm1 but when the perturbing star is gone1 the orbits will again become circular4 ,it would only be when the disturbing body was gone and when nevertheless the orbits1 in lieu of again becoming circular1 became elliptic1 that these astronomers would become conscious of their error and the necessity of remaking all their mechanics4 ,i have dwelt somewhat upon these hypotheses because it seems to me one can clearly comprehend what our generalized law of inertia really is only in contrasting it with a contrary hypothesis4 ,well1 now1 has this generalized law of inertia been verified by experiment1 or can it be8 ,when ,newton wrote the ,principia he quite regarded this truth as experimentally acquired and demonstrated4 ,it was so in his eyes1 not only through the anthropomorphism of which we shall speak further on1 but through the work of ,galileo4 ,it was so even from ,kepler's laws themselves2 in accordance with these laws1 in fact1 the path of a planet is completely determined by its initial position and initial velocity2 this is just what our generalized law of inertia requires4 ,for this principle to be only in appearance true1 for one to have cause to dread having some day to replace it by one of the analogous principles ,i have just now contrasted with it1 would be necessary our having been misled by some amazing chance1 like that which1 in the fiction above developed1 led into error our imaginary astronomers4 ,such a hypothesis is too unlikely to delay over4 ,no one will believe that such coincidences can happen2 no doubt the probability of two eccentricities being both precisely null1 to within errors of observation1 is not less than the probability of one being precisely equal to #j4a1 for instance1 and the other to #j4b1 to within errors of observation4 ,the probability of a simple event is not less than that of a complicated event2 and yet1 if the first happens1 we shall not consent to attribute it to chance2 we should not believe that nature had acted expressly to deceive us4 ,the hypothesis of an error of this sort being discarded1 it may therefore be admitted that in so far as astronomy is concerned1 our law has been verified by experiment4 ,but astronomy is not the whole of physics4 ,may we not fear lest some day a new experiment should come to falsify the law in some domain of physics8 ,an experimental law is always subject to revision2 one should always expect to see it replaced by a more precise law4 ,yet no one seriously thinks that the law we are speaking of will ever be abandoned or amended4 ,why8 ,precisely because it can never be subjected to a decisive test4 ,first of all1 in order that this trial should be complete1 it would be necessary that after a certain time all the bodies in the universe should revert to their initial positions with their initial velocities4 ,it might then be seen whether1 starting from this moment1 they would resume their original paths4 ,but this test is impossible1 it can be only partially applied1 and1 however well it is made1 there will always be some bodies which will not revert to their initial positions2 thus every derogation of the law will easily find its explanation4 ,this is not all2 in astronomy we see the bodies whose motions we study and we usually assume that they are not subjected to the action of other invisible bodies4 ,under these conditions our law must indeed be either verified or not verified4 ,but it is not the same in physics2 if the physical phenomena are due to motions1 it is to the motions of molecules which we do not see4 ,if then the acceleration of one of the bodies we see appears to us to depend on something else besides the positions or velocities of other visible bodies or of invisible molecules whose existence we have been previously led to admit1 nothing prevents our supposing that this something else is the position or the velocity of other molecules whose presence we have not before suspected4 ,the law will find itself safeguarded4 ,permit me to employ mathematical language a moment to express the same thought under another form4 ,suppose we observe n molecules and ascertain that their #cn coordinates satisfy a system of #cn differential equations of the fourth order 7and not of the second order as the law of inertia would require74 ,we know that by introducing #cn auxiliary variables1 a system of #cn equations of the fourth order can be reduced to a system of #fn equations of the second order4 ,if then we suppose these #cn auxiliary variables represent the coordinates of n invisible molecules1 the result is again in conformity with the law of inertia4 ,to sum up1 this law1 verified experimentally in some particular cases1 may unhesitatingly be extended to the most general cases1 since we know that in these general cases experiment no longer is able either to confirm or to contradict it4 ,,the ,,law ,,of ,,acceleration4--,the acceleration of a body is equal to the force acting on it divided by its mass4 ,can this law be verified by experiment8 ,for that it would be necessary to measure the three magnitudes which figure in the enunciation3 acceleration1 force and mass4 ,i assume that acceleration can be measured1 for ,i pass over the difficulty arising from the measurement of time4 ,but how measure force1 or mass8 ,we do not even know what they are4 ,what is mass8 ,according to ,newton1 it is the product of the volume by the density4 ,according to ,thomson and ,tait1 it would be better to say that density is the quotient of the mass by the volume4 ,what is force8 ,it is1 replies ,lagrange1 that which moves or tends to move a body4 ,it is1 ,kirchhoff will say1 the product of the mass by the acceleration4 ,but then1 why not say the mass is the quotient of the force by the acceleration8 ,these difficulties are inextricable4 ,when we say force is the cause of motion1 we talk metaphysics1 and this definition1 if one were content with it1 would be absolutely sterile4 ,for a definition to be of any use1 it must teach us to measure force2 moreover that suffices2 it is not at all necessary that it teach us what force is in itself1 nor whether it is the cause or the effect of motion4 ,we must therefore first define the equality of two forces4 ,when shall we say two forces are equal8 ,it is1 we are told1 when1 applied to the same mass1 they impress upon it the same acceleration1 or when1 opposed directly one to the other1 they produce equilibrium4 ,this definition is only a sham4 ,a force applied to a body can not be uncoupled to hook it up to another body1 as one uncouples a locomotive to attach it to another train4 ,it is therefore impossible to know what acceleration such a force1 applied to such a body1 would impress upon such another body1 if it were applied to it4 ,it is impossible to know how two forces which are not directly opposed would act1 if they were directly opposed4 ,it is this definition we try to materialize1 so to speak1 when we measure a force with a dynamometer1 or in balancing it with a weight4 ,two forces ,f and ,f'1 which for simplicity ,i will suppose vertical and directed upward1 are applied respectively to two bodies ,c and ,c'2 ,i suspend the same heavy body ,p first to the body ,c1 then to the body ,c'2 if equilibrium is produced in both cases1 ,i shall conclude that the two forces ,f and ,f' are equal to one another1 since they are each equal to the weight of the body ,p4 ,but am ,i sure the body ,p has retained the same weight when ,i have transported it from the first body to the second8 ,far from it2 ,i am sure of the contrary2 ,i know the intensity of gravity varies from one point to another1 and that it is stronger1 for instance1 at the pole than at the equator4 ,no doubt the difference is very slight and1 in practise1 ,i shall take no account of it2 but a properly constructed definition should have mathematical rigor2 this rigor is lacking4 ,what ,i say of weight would evidently apply to the force of the resiliency of a dynamometer1 which the temperature and a multitude of circumstances may cause to vary4 ,this is not all2 we can not say the weight of the body ,p may be applied to the body ,c and directly balance the force ,f4 ,what is applied to the body ,c is the action ,a of the body ,p on the body ,c2 the body ,p is submitted on its part1 on the one hand1 to its weight2 on the other hand1 to the reaction ,r of the body ,c on ,p4 ,finally1 the force ,f is equal to the force ,a1 since it balances it2 the force ,a is equal to ,r1 in virtue of the principle of the equality of action and reaction2 lastly1 the force ,r is equal to the weight of ,p1 since it balances it4 ,it is from these three equalities we deduce as consequence the equality of ,f and the weight of ,p4 ,we are therefore obliged in the definition of the equality of the two forces to bring in the principle of the equality of action and reaction2 on this account1 this principle must no longer be regarded as an experimental law1 but as a definition4 ,for recognizing the equality of two forces here1 we are then in possession of two rules3 equality of two forces which balance2 equality of action and reaction4 ,but1 as we have seen above1 these two rules are insufficient2 we are obliged to have recourse to a third rule and to assume that certain forces1 as1 for instance1 the weight of a body1 are constant in magnitude and direction4 ,but this third rule1 as ,i have said1 is an experimental law2 it is only approximately true2 it is a bad definition4 ,we are therefore reduced to ,kirchhoff's definition2 force is equal to the mass multiplied by the acceleration4 ,this 'law of ,newton' in its turn ceases to be regarded as an experimental law1 it is now only a definition4 ,but this definition is still insufficient1 for we do not know what mass is4 ,it enables us doubtless to calculate the relation of two forces applied to the same body at different instants2 it teaches us nothing about the relation of two forces applied to two different bodies4 ,to complete it1 it is necessary to go back anew to ,newton's third law 7equality of action and reaction71 regarded again1 not as an experimental law1 but as a definition4 ,two bodies ,a and ,b act one upon the other2 the acceleration of ,a multiplied by the mass of ,a is equal to the action of ,b upon ,a2 in the same way1 the product of the acceleration of ,b by its mass is equal to the reaction of ,a upon ,b4 ,as1 by definition1 action is equal to reaction1 the masses of ,a and ,b are in the inverse ratio of their accelerations4 ,here we have the ratio of these two masses defined1 and it is for experiment to verify that this ratio is constant4 ,that would be all very well if the two bodies ,a and ,b alone were present and removed from the action of the rest of the world4 ,this is not at all the case2 the acceleration of ,a is not due merely to the action of ,b1 but to that of a multitude of other bodies ,c1 ,d1444 ,to apply the preceding rule1 it is therefore necessary to separate the acceleration of ,a into many components1 and discern which of these components is due to the action of ,b4 ,this separation would still be possible1 if we should assume that the action of ,c upon ,a is simply adjoined to that of ,b upon ,a1 without the presence of the body ,c modifying the action of ,b upon ,a2 or the presence of ,b modifying the action of ,c upon ,a2 if we should assume1 consequently1 that any two bodies attract each other1 that their mutual action is along their join and depends only upon their distance apart2 if1 in a word1 we assume the hypothesis of central forces4 ,you know that to determine the masses of the celestial bodies we use a wholly different principle4 ,the law of gravitation teaches us that the attraction of two bodies is proportional to their masses2 if r is their distance apart1 m and m' their masses1 k a constant1 their attraction will be kmm'_/r#b4 ,what we are measuring then is not mass1 the ratio of force to acceleration1 but the attracting mass2 it is not the inertia of the body1 but its attracting force4 ,this is an indirect procedure1 whose employment is not theoretically indispensable4 ,it might very well have been that attraction was inversely proportional to the square of the distance without being proportional to the product of the masses1 that it was equal to f_/r#b1 but without our having f kmm'4 ,if it were so1 we could nevertheless1 by observation of the relative motions of the heavenly bodies1 measure the masses of these bodies4 ,but have we the right to admit the hypothesis of central forces8 ,is this hypothesis rigorously exact8 ,is it certain it will never be contradicted by experiment8 ,who would dare affirm that8 ,and if we must abandon this hypothesis1 the whole edifice so laboriously erected will crumble4 ,we have no longer the right to speak of the component of the acceleration of ,a due to the action of ,b4 ,we have no means of distinguishing it from that due to the action of ,c or of another body4 ,the rule for the measurement of masses becomes inapplicable4 ,what remains then of the principle of the equality of action and reaction8 ,if the hypothesis of central forces is rejected1 this principle should evidently be enunciated thus3 the geometric resultant of all the forces applied to the various bodies of a system isolated from all external action will be null4 ,or1 in other words1 the motion of the center of gravity of this system will be rectilinear and uniform4 ,there it seems we have a means of defining mass2 the position of the center of gravity evidently depends on the values attributed to the masses2 it will be necessary to dispose of these values in such a way that the motion of the center of gravity may be rectilinear and uniform2 this will always be possible if ,newton's third law is true1 and possible in general only in a single way4 ,but there exists no system isolated from all external action2 all the parts of the universe are subject more or less to the action of all the other parts4 ,the law of the motion of the center of gravity is rigorously true only if applied to the entire universe4 ,but then1 to get from it the values of the masses1 it would be necessary to observe the motion of the center of gravity of the universe4 ,the absurdity of this consequence is manifest2 we know only relative motions2 the motion of the center of gravity of the universe will remain for us eternally unknown4 ,therefore nothing remains and our efforts have been fruitless2 we are driven to the following definition1 which is only an avowal of powerlessness3 masses are coefficients it is convenient to introduce into calculations4 ,we could reconstruct all mechanics by attributing different values to all the masses4 ,this new mechanics would not be in contradiction either with experience or with the general principles of dynamics 7principle of inertia1 proportionality of forces to masses and to accelerations1 equality of action and reaction1 rectilinear and uniform motion of the center of gravity1 principle of areas74 ,only the equations of this new mechanics would be less simple4 ,let us understand clearly3 it would only be the first terms which would be less simple1 that is those experience has already made us acquainted with2 perhaps one could alter the masses by small quantities without the complete equations gaining or losing in simplicity4 ,hertz has raised the question whether the principles of mechanics are rigorously true4 ,in the opinion of many physicists1 he says1 it is inconceivable that the remotest experience should ever change anything in the immovable principles of mechanics2 and yet1 what comes from experience may always be rectified by experience4 ,after what we have just said1 these fears will appear groundless4 ,the principles of dynamics at first appeared to us as experimental truths2 but we have been obliged to use them as definitions4 ,it is by definition that force is equal to the product of mass by acceleration2 here1 then1 is a principle which is henceforth beyond the reach of any further experiment4 ,it is in the same way by definition that action is equal to reaction4 ,but then1 it will be said1 these unverifiable principles are absolutely devoid of any significance2 experiment can not contradict them2 but they can teach us nothing useful2 then what is the use of studying dynamics8 ,this over-hasty condemnation would be unjust4 ,there is not in nature any system perfectly isolated1 perfectly removed from all external action2 but there are systems almost isolated4 ,if such a system be observed1 one may study not only the relative motion of its various parts one in reference to another1 but also the motion of its center of gravity in reference to the other parts of the universe4 ,we ascertain then that the motion of this center of gravity is almost rectilinear and uniform1 in conformity with ,newton's third law4 ,that is an experimental truth1 but it can not be invalidated by experience2 in fact1 what would a more precise experiment teach us8 ,it would teach us that the law was only almost true2 but that we knew already4 ,we can now understand how experience has been able to serve as basis for the principles of mechanics and yet will never be able to contradict them4 ,,anthropomorphic ,,mechanics4--,kirchhoff1 it will be said1 has only acted in obedience to the general tendency of mathematicians toward nominalism2 from this his ability as a physicist has not saved him4 ,he wanted a definition of force1 and he took for it the first proposition that presented itself2 but we need no definition of force3 the idea of force is primitive1 irreducible1 indefinable2 we all know what it is1 we have a direct intuition of it4 ,this direct intuition comes from the notion of effort1 which is familiar to us from infancy4 ,but first1 even though this direct intuition made known to us the real nature of force in itself1 it would be insufficient as a foundation for mechanics2 it would besides be wholly useless4 ,what is of importance is not to know what force is1 but to know how to measure it4 ,whatever does not teach us to measure it is as useless to mechanics as is1 for instance1 the subjective notion of warmth and cold to the physicist who is studying heat4 ,this subjective notion can not be translated into numbers1 therefore it is of no use2 a scientist whose skin was an absolutely bad conductor of heat and who1 consequently1 would never have felt either sensations of cold or sensations of warmth1 could read a thermometer just as well as any one else1 and that would suffice him for constructing the whole theory of heat4 ,now this immediate notion of effort is of no use to us for measuring force2 it is clear1 for instance1 that ,i should feel more fatigue in lifting a weight of fifty kilos than a man accustomed to carry burdens4 ,but more than that3 this notion of effort does not teach us the real nature of force2 it reduces itself finally to a remembrance of muscular sensations1 and it will hardly be maintained that the sun feels a muscular sensation when it draws the earth4 ,all that can there be sought is a symbol1 less precise and less convenient than the arrows the geometers use1 but just as remote from the reality4 ,anthropomorphism has played a considerable historic rle in the genesis of mechanics2 perhaps it will still at times furnish a symbol which will appear convenient to some minds2 but it can not serve as foundation for anything of a truly scientific or philosophic character4 ',,the ,,school ,,of ,,the ,,thread4'--,m4 ,andrade1 in his ,leons de mcanique physique1 has rejuvenated anthropomorphic mechanics4 ,to the school of mechanics to which ,kirchhoff belongs1 he opposes that which he bizarrely calls the school of the thread4 ,this school tries to reduce everything to the consideration of certain material systems of negligible mass1 envisaged in the state of tension and capable of transmitting considerable efforts to distant bodies1 systems of which the ideal type is the thread4 ,a thread which transmits any force is slightly elongated under the action of this force2 the direction of the thread tells us the direction of the force1 whose magnitude is measured by the elongation of the thread4 ,one may then conceive an experiment such as this4 ,a body ,a is attached to a thread2 at the other extremity of the thread any force acts which varies until the thread takes an elongation alpha2 the acceleration of the body ,a is noted2 ,a is detached and the body ,b attached to the same thread2 the same force or another force acts anew1 and is made to vary until the thread takes again the elongation alpha2 the acceleration of the body ,b is noted4 ,the experiment is then renewed with both ,a and ,b1 but so that the thread takes the elongation beta4 ,the four observed accelerations should be proportional4 ,we have thus an experimental verification of the law of acceleration above enunciated4 ,or still better1 a body is submitted to the simultaneous action of several identical threads in equal tension1 and by experiment it is sought what must be the orientations of all these threads that the body may remain in equilibrium4 ,we have then an experimental verification of the law of the composition of forces4 ,but1 after all1 what have we done8 ,we have defined the force to which the thread is subjected by the deformation undergone by this thread1 which is reasonable enough2 we have further assumed that if a body is attached to this thread1 the effort transmitted to it by the thread is equal to the action this body exercises on this thread2 after all1 we have therefore used the principle of the equality of action and reaction1 in considering it1 not as an experimental truth1 but as the very definition of force4 ,this definition is just as conventional as ,kirchhoff's1 but far less general4 ,all forces are not transmitted by threads 7besides1 to be able to compare them1 they would all have to be transmitted by identical threads74 ,even if it should be conceded that the earth is attached to the sun by some invisible thread1 at least it would be admitted that we have no means of measuring its elongation4 ,nine times out of ten1 consequently1 our definition would be at fault2 no sort of sense could be attributed to it1 and it would be necessary to fall back on ,kirchhoff's4 ,why then take this dtour8 ,you admit a certain definition of force which has a meaning only in certain particular cases4 ,in these cases you verify by experiment that it leads to the law of acceleration4 ,on the strength of this experiment1 you then take the law of acceleration as a definition of force in all the other cases4 ,would it not be simpler to consider the law of acceleration as a definition in all cases1 and to regard the experiments in question1 not as verifications of this law1 but as verifications of the principle of reaction1 or as demonstrating that the deformations of an elastic body depend only on the forces to which this body is subjected8 ,and this is without taking into account that the conditions under which your definition could be accepted are never fulfilled except imperfectly1 that a thread is never without mass1 that it is never removed from every force except the reaction of the bodies attached to its extremities4 ,andrade's ideas are nevertheless very interesting2 if they do not satisfy our logical craving1 they make us understand better the historic genesis of the fundamental ideas of mechanics4 ,the reflections they suggest show us how the human mind has raised itself from a nave anthropomorphism to the present conceptions of science4 ,we see at the start a very particular and in sum rather crude experiment2 at the finish1 a law perfectly general1 perfectly precise1 the certainty of which we regard as absolute4 ,this certainty we ourselves have bestowed upon it voluntarily1 so to speak1 by looking upon it as a convention4 ,are the law of acceleration1 the rule of the composition of forces then only arbitrary conventions8 ,conventions1 yes2 arbitrary1 no2 they would be if we lost sight of the experiments which led the creators of the science to adopt them1 and which1 imperfect as they may be1 suffice to justify them4 ,it is well that from time to time our attention is carried back to the experimental origin of these conventions4 ,,chapter ,,vii ,,relative ,,motion ,,and ,,absolute ,,motion ,,the ,,principle ,,of ,,relative ,,motion4--,the attempt has sometimes been made to attach the law of acceleration to a more general principle4 ,the motion of any system must obey the same laws1 whether it be referred to fixed axes1 or to movable axes carried along in a rectilinear and uniform motion4 ,this is the principle of relative motion1 which forces itself upon us for two reasons3 first1 the commonest experience confirms it1 and second1 the contrary hypothesis is singularly repugnant to the mind4 ,assume it then1 and consider a body subjected to a force2 the relative motion of this body1 in reference to an observer moving with a uniform velocity equal to the initial velocity of the body1 must be identical to what its absolute motion would be if it started from rest4 ,we conclude hence that its acceleration can not depend upon its absolute velocity2 the attempt has even been made to derive from this a demonstration of the law of acceleration4 ,there long were traces of this demonstration in the regulations for the degree ,b4 s ,sc4 ,it is evident that this attempt is idle4 ,the obstacle which prevented our demonstrating the law of acceleration is that we had no definition of force2 this obstacle subsists in its entirety1 since the principle invoked has not furnished us the definition we lacked4 ,the principle of relative motion is none the less highly interesting and deserves study for its own sake4 ,let us first try to enunciate it in a precise manner4 ,we have said above that the accelerations of the different bodies forming part of an isolated system depend only on their relative velocities and positions1 and not on their absolute velocities and positions1 provided the movable axes to which the relative motion is referred move uniformly in a straight line4 ,or1 if we prefer1 their accelerations depend only on the differences of their velocities and the differences of their coordinates1 and not on the absolute values of these velocities and coordinates4 ,if this principle is true for relative accelerations1 or rather for differences of acceleration1 in combining it with the law of reaction we shall thence deduce that it is still true of absolute accelerations4 ,it then remains to be seen how we may demonstrate that the differences of the accelerations depend only on the differences of the velocities and of the coordinates1 or1 to speak in mathematical language1 that these differences of coordinates satisfy differential equations of the second order4 ,can this demonstration be deduced from experiments or from a priori considerations8 ,recalling what we have said above1 the reader can answer for himself4 ,thus enunciated1 in fact1 the principle of relative motion singularly resembles what ,i called above the generalized principle of inertia2 it is not altogether the same thing1 since it is a question of the differences of coordinates and not of the coordinates themselves4 ,the new principle teaches us therefore something more than the old1 but the same discussion is applicable and would lead to the same conclusions2 it is unnecessary to return to it4 ,,newton',s ,,argument4--,here we encounter a very important and even somewhat disconcerting question4 ,i have said the principle of relative motion was for us not solely a result of experiment and that a priori every contrary hypothesis would be repugnant to the mind4 ,but then1 why is the principle true only if the motion of the movable axes is rectilinear and uniform8 ,it seems that it ought to impose itself upon us with the same force1 if this motion is varied1 or at any rate if it reduces to a uniform rotation4 ,now1 in these two cases1 the principle is not true4 ,i will not dwell long on the case where the motion of the axes is rectilinear without being uniform2 the paradox does not bear a moment's examination4 ,if ,i am on board1 and if the train1 striking any obstacle1 stops suddenly1 ,i shall be thrown against the seat in front of me1 although ,i have not been directly subjected to any force4 ,there is nothing mysterious in that2 if ,i have undergone the action of no external force1 the train itself has experienced an external impact4 ,there can be nothing paradoxical in the relative motion of two bodies being disturbed when the motion of one or the other is modified by an external cause4 ,i will pause longer on the case of relative motions referred to axes which rotate uniformly4 ,if the heavens were always covered with clouds1 if we had no means of observing the stars1 we nevertheless might conclude that the earth turns round2 we could learn this from its flattening or again by the ,foucault pendulum experiment4 ,and yet1 in this case1 would it have any meaning1 to say the earth turns round8 ,if there is no absolute space1 can one turn without turning in reference to something else8 and1 on the other hand1 how could we admit ,newton's conclusion and believe in absolute space8 ,but it does not suffice to ascertain that all possible solutions are equally repugnant to us2 we must analyze1 in each case1 the reasons for our repugnance1 so as to make our choice intelligently4 ,the long discussion which follows will therefore be excused4 ,let us resume our fiction3 thick clouds hide the stars from men1 who can not observe them and are ignorant even of their existence2 how shall these men know the earth turns round8 ,even more than our ancestors1 no doubt1 they will regard the ground which bears them as fixed and immovable2 they will await much longer the advent of a ,copernicus4 ,but in the end the ,copernicus would come--how8 ,the students of mechanics in this world would not at first be confronted with an absolute contradiction4 ,in the theory of relative motion1 besides real forces1 two fictitious forces are met which are called ordinary and compound centrifugal force4 ,our imaginary scientists could therefore explain everything by regarding these two forces as real1 and they would not see therein any contradiction of the generalized principle of inertia1 for these forces would depend1 the one on the relative positions of the various parts of the system1 as real attractions do1 the other on their relative velocities1 as real frictions do4 ,many difficulties1 however1 would soon awaken their attention2 if they succeeded in realizing an isolated system1 the center of gravity of this system would not have an almost rectilinear path4 ,they would invoke1 to explain this fact1 the centrifugal forces which they would regard as real1 and which they would attribute no doubt to the mutual actions of the bodies4 ,only they would not see these forces become null at great distances1 that is to say in proportion as the isolation was better realized2 far from it2 centrifugal force increases indefinitely with the distance4 ,this difficulty would seem to them already sufficiently great2 and yet it would not stop them long2 they would soon imagine some very subtile medium1 analogous to our ether1 in which all bodies would be immersed and which would exert a repellent action upon them4 ,but this is not all4 ,space is symmetric1 and yet the laws of motion would not show any symmetry2 they would have to distinguish between right and left4 ,it would be seen for instance that cyclones turn always in the same sense1 whereas by reason of symmetry these winds should turn indifferently in one sense and in the other4 ,if our scientists by their labor had succeeded in rendering their universe perfectly symmetric1 this symmetry would not remain1 even though there was no apparent reason why it should be disturbed in one sense rather than in the other4 ,they would get themselves out of the difficulty doubtless1 they would invent something which would be no more extraordinary than the glass spheres of ,ptolemy1 and so it would go on1 complications accumulating1 until the long-expected ,copernicus sweeps them all away at a single stroke1 saying3 ,it is much simpler to assume the earth turns round4 ,and just as our ,copernicus said to us3 ,it is more convenient to suppose the earth turns round1 since thus the laws of astronomy are expressible in a much simpler language2 this one would say3 ,it is more convenient to suppose the earth turns round1 since thus the laws of mechanics are expressible in a much simpler language4 ,this does not preclude maintaining that absolute space1 that is to say the mark to which it would be necessary to refer the earth to know whether it really moves1 has no objective existence4 ,hence1 this affirmation3 'the earth turns round' has no meaning1 since it can be verified by no experiment2 since such an experiment1 not only could not be either realized or dreamed by the boldest ,jules ,verne1 but can not be conceived of without contradiction2 or rather these two propositions3 'the earth turns round1' and1 'it is more convenient to suppose the earth turns round' have the same meaning2 there is nothing more in the one than in the other4 ,perhaps one will not be content even with that1 and will find it already shocking that among all the hypotheses1 or rather all the conventions we can make on this subject1 there is one more convenient than the others4 ,but if it has been admitted without difficulty when it was a question of the laws of astronomy1 why should it be shocking in that which concerns mechanics8 ,we have seen that the coordinates of bodies are determined by differential equations of the second order1 and that so are the differences of these coordinates4 ,this is what we have called the generalized principle of inertia and the principle of relative motion4 ,if the distances of these bodies were determined likewise by equations of the second order1 it seems that the mind ought to be entirely satisfied4 ,in what measure does the mind get this satisfaction and why is it not content with it8 ,to account for this1 we had better take a simple example4 ,i suppose a system analogous to our solar system1 but where one can not perceive fixed stars foreign to this system1 so that astronomers can observe only the mutual distances of the planets and the sun1 and not the absolute longitudes of the planets4 ,if we deduce directly from ,newton's law the differential equations which define the variation of these distances1 these equations will not be of the second order4 ,i mean that if1 besides ,newton's law1 one knew the initial values of these distances and of their derivatives with respect to the time1 that would not suffice to determine the values of these same distances at a subsequent instant4 ,there would still be lacking one datum1 and this datum might be for instance what astronomers call the area-constant4 ,but here two different points of view may be taken2 we may distinguish two sorts of constants4 ,to the eyes of the physicist the world reduces to a series of phenomena1 depending1 on the one hand1 solely upon the initial phenomena2 on the other hand1 upon the laws which bind the consequents to the antecedents4 ,if then observation teaches us that a certain quantity is a constant1 we shall have the choice between two conceptions4 ,either we shall assume that there is a law requiring this quantity not to vary1 but that by chance1 at the beginning of the ages1 it had1 rather than another1 this value it has been forced to keep ever since4 ,this quantity might then be called an accidental constant4 ,or else we shall assume1 on the contrary1 that there is a law of nature which imposes upon this quantity such a value and not such another4 ,we shall then have what we may call an essential constant4 ,for example1 in virtue of ,newton's laws1 the duration of the revolution of the earth must be constant4 ,but if it is #cff sidereal days and something over1 and not #cjj or #djj1 this is in consequence of ,i know not what initial chance4 ,this is an accidental constant4 ,if1 on the contrary1 the exponent of the distance which figures in the expression of the attractive force is equal to -#b and not to -#c1 this is not by chance1 but because ,newton's law requires it4 ,this is an essential constant4 ,i know not whether this way of giving chance its part is legitimate in itself1 and whether this distinction is not somewhat artificial2 it is certain at least that1 so long as nature shall have secrets1 this distinction will be in application extremely arbitrary and always precarious4 ,as to the area-constant1 we are accustomed to regard it as accidental4 ,is it certain our imaginary astronomers would do the same8 ,if they could have compared two different solar systems1 they would have the idea that this constant may have several different values2 but my very supposition in the beginning was that their system should appear as isolated1 and that they should observe no star foreign to it4 ,under these conditions1 they would see only one single constant which would have a single value absolutely invariable2 they would be led without any doubt to regard it as an essential constant4 ,a word in passing to forestall an objection3 the inhabitants of this imaginary world could neither observe nor define the area-constant as we do1 since the absolute longitudes escape them2 that would not preclude their being quickly led to notice a certain constant which would introduce itself naturally into their equations and which would be nothing but what we call the area-constant4 ,but then see what would happen4 ,if the area-constant is regarded as essential1 as depending upon a law of nature1 to calculate the distances of the planets at any instant it will suffice to know the initial values of these distances and those of their first derivatives4 ,from this new point of view1 the distances will be determined by differential equations of the second order4 ,yet would the mind of these astronomers be completely satisfied8 ,i do not believe so2 first1 they would soon perceive that in differentiating their equations and thus raising their order1 these equations became much simpler4 ,and above all they would be struck by the difficulty which comes from symmetry4 ,it would be necessary to assume different laws1 according as the aggregate of the planets presented the figure of a certain polyhedron or of the symmetric polyhedron1 and one would escape from this consequence only by regarding the area-constant as accidental4 ,i have taken a very special example1 since ,i have supposed astronomers who did not at all consider terrestrial mechanics1 and whose view was limited to the solar system4 ,our universe is more extended than theirs1 as we have fixed stars1 but still it too is limited1 and so we might reason on the totality of our universe as the astronomers on their solar system4 ,thus we see that finally we should be led to conclude that the equations which define distances are of an order superior to the second4 ,why should we be shocked at that1 why do we find it perfectly natural for the series of phenomena to depend upon the initial values of the first derivatives of these distances1 while we hesitate to admit that they may depend on the initial values of the second derivatives8 ,this can only be because of the habits of mind created in us by the constant study of the generalized principle of inertia and its consequences4 ,the values of the distances at any instant depend upon their initial values1 upon those of their first derivatives and also upon something else4 ,what is this something else8 ,if we will not admit that this may be simply one of the second derivatives1 we have only the choice of hypotheses4 ,either it may be supposed1 as is ordinarily done1 that this something else is the absolute orientation of the universe in space1 or the rapidity with which this orientation varies2 and this supposition may be correct2 it is certainly the most convenient solution for geometry2 it is not the most satisfactory for the philosopher1 because this orientation does not exist4 ,or it may be supposed that this something else is the position or the velocity of some invisible body2 this has been done by certain persons who have even called it the body alpha1 although we are doomed never to know anything of this body but its name4 ,this is an artifice entirely analogous to that of which ,i spoke at the end of the paragraph devoted to my reflections on the principle of inertia4 ,but1 after all1 the difficulty is artificial4 ,provided the future indications of our instruments can depend only on the indications they have given us or would have given us formerly1 this is all that is necessary4 ,now as to this we may rest easy4 ,,chapter ,,viii ,,energy ,,and ,,thermodynamics ,,energetics4--,the difficulties inherent in the classic mechanics have led certain minds to prefer a new system they call energetics4 ,energetics took its rise as an outcome of the discovery of the principle of the conservation of energy4 ,helmholtz gave it its final form4 ,it begins by defining two quantities which play the fundamental rle in this theory4 ,they are kinetic energy1 or vis viva1 and potential energy4 ,all the changes which bodies in nature can undergo are regulated by two experimental laws3 #a ,the sum of kinetic energy and potential energy is constant4 ,this is the principle of the conservation of energy4 #b ,if a system of bodies is at ,a at the time t#j and at ,b at the time t#a1 it always goes from the first situation to the second in such a way that the mean value of the difference between the two sorts of energy1 in the interval of time which separates the two epochs t#j and t#a1 may be as small as possible4 ,this is ,hamilton's principle1 which is one of the forms of the principle of least action4 ,the energetic theory has the following advantages over the classic theory3 #a ,it is less incomplete2 that is to say1 ,hamilton's principle and that of the conservation of energy teach us more than the fundamental principles of the classic theory1 and exclude certain motions not realized in nature and which would be compatible with the classic theory3 #b ,it saves us the hypothesis of atoms1 which it was almost impossible to avoid with the classic theory4 ,but it raises in its turn new difficulties3 ,the definitions of the two sorts of energy would raise difficulties almost as great as those of force and mass in the first system4 ,yet they may be gotten over more easily1 at least in the simplest cases4 ,suppose an isolated system formed of a certain number of material points2 suppose these points subjected to forces depending only on their relative position and their mutual distances1 and independent of their velocities4 ,in virtue of the principle of the conservation of energy1 a function of forces must exist4 ,in this simple case the enunciation of the principle of the conservation of energy is of extreme simplicity4 ,a certain quantity1 accessible to experiment1 must remain constant4 ,this quantity is the sum of two terms2 the first depends only on the position of the material points and is independent of their velocities2 the second is proportional to the square of these velocities4 ,this resolution can take place only in a single way4 ,the first of these terms1 which ,i shall call ,u1 will be the potential energy2 the second1 which ,i shall call ,t1 will be the kinetic energy4 ,it is true that if ,t ,u is a constant1 so is any function of ,t ,u1 ,phi7,t ,u74 ,but this function ,phi7,t ,u7 will not be the sum of two terms the one independent of the velocities1 the other proportional to the square of these velocities4 ,among the functions which remain constant there is only one which enjoys this property1 that is ,t ,u 7or a linear function of ,t ,u1 which comes to the same thing1 since this linear function may always be reduced to ,t ,u by change of unit and of origin74 ,this then is what we shall call energy2 the first term we shall call potential energy and the second kinetic energy4 ,the definition of the two sorts of energy can therefore be carried through without any ambiguity4 ,it is the same with the definition of the masses4 ,kinetic energy1 or vis viva1 is expressed very simply by the aid of the masses and the relative velocities of all the material points with reference to one of them4 ,these relative velocities are accessible to observation1 and1 when we know the expression of the kinetic energy as function of these relative velocities1 the coefficients of this expression will give us the masses4 ,thus1 in this simple case1 the fundamental ideas may be defined without difficulty4 ,but the difficulties reappear in the more complicated cases and1 for instance1 if the forces1 in lieu of depending only on the distances1 depend also on the velocities4 ,for example1 ,weber supposes the mutual action of two electric molecules to depend not only on their distance1 but on their velocity and their acceleration4 ,if material points should attract each other according to an analogous law1 ,u would depend on the velocity1 and might contain a term proportional to the square of the velocity4 ,among the terms proportional to the squares of the velocities1 how distinguish those which come from ,t or from ,u8 ,consequently1 how distinguish the two parts of energy8 ,but still more2 how define energy itself8 ,we no longer have any reason to take as definition ,t ,u rather than any other function of ,t ,u1 when the property which characterized ,t ,u has disappeared1 that1 namely1 of being the sum of two terms of a particular form4 ,but this is not all2 it is necessary to take account1 not only of mechanical energy properly so called1 but of the other forms of energy1 heat1 chemical energy1 electric energy1 etc4 ,the principle of the conservation of energy should be written3 ,t ,u ,q const4 where ,t would represent the sensible kinetic energy1 ,u the potential energy of position1 depending only on the position of the bodies1 ,q the internal molecular energy1 under the thermal1 chemic or electric form4 ,all would go well if these three terms were absolutely distinct1 if ,t were proportional to the square of the velocities1 ,u independent of these velocities and of the state of the bodies1 ,q independent of the velocities and of the positions of the bodies and dependent only on their internal state4 ,the expression for the energy could be resolved only in one single way into three terms of this form4 ,but this is not the case2 consider electrified bodies2 the electrostatic energy due to their mutual action will evidently depend upon their charge1 that is to say1 on their state2 but it will equally depend upon their position4 ,if these bodies are in motion1 they will act one upon another electrodynamically and the electrodynamic energy will depend not only upon their state and their position1 but upon their velocities4 ,we therefore no longer have any means of making the separation of the terms which should make part of ,t1 of ,u and of ,q1 and of separating the three parts of energy4 ,if 7,t ,u ,q7 is constant so is any function phi7,t ,u ,q74 ,if ,t ,u ,q were of the particular form ,i have above considered1 no ambiguity would result2 among the functions phi7,t ,u ,q7 which remain constant1 there would only be one of this particular form1 and that ,i should convene to call energy4 ,but as ,i have said1 this is not rigorously the case2 among the functions which remain constant1 there is none which can be put rigorously under this particular form2 hence1 how choose among them the one which should be called energy8 ,we no longer have anything to guide us in our choice4 ,there only remains for us one enunciation of the principle of the conservation of energy3 ,there is something which remains constant4 ,under this form it is in its turn out of the reach of experiment and reduces to a sort of tautology4 ,it is clear that if the world is governed by laws1 there will be quantities which will remain constant4 ,like ,newton's laws1 and1 for an analogous reason1 the principle of the conservation of energy1 founded on experiment1 could no longer be invalidated by it4 ,this discussion shows that in passing from the classic to the energetic system progress has been made2 but at the same time it shows this progress is insufficient4 ,another objection seems to me still more grave3 the principle of least action is applicable to reversible phenomena2 but it is not at all satisfactory in so far as irreversible phenomena are concerned2 the attempt by ,helmholtz to extend it to this kind of phenomena did not succeed and could not succeed2 in this regard everything remains to be done4 ,the very statement of the principle of least action has something about it repugnant to the mind4 ,to go from one point to another1 a material molecule1 acted upon by no force1 but required to move on a surface1 will take the geodesic line1 that is to say1 the shortest path4 ,this molecule seems to know the point whither it is to go1 to foresee the time it would take to reach it by such and such a route1 and then to choose the most suitable path4 ,the statement presents the molecule to us1 so to speak1 as a living and free being4 ,clearly it would be better to replace it by an enunciation less objectionable1 and where1 as the philosophers would say1 final causes would not seem to be substituted for efficient causes4 ,,thermodynamics4#d--,the rle of the two fundamental principles of thermodynamics in all branches of natural philosophy becomes daily more important4 ,abandoning the ambitious theories of forty years ago1 which were encumbered by molecular hypotheses1 we are trying to-day to erect upon thermodynamics alone the entire edifice of mathematical physics4 ,will the two principles of ,mayer and of ,clausius assure to it foundations solid enough for it to last some time8 ,no one doubts it2 but whence comes this confidence8 #d ,the following lines are a partial reproduction of the preface of my book ,thermodynamique4 ,an eminent physicist said to me one day propos of the law of errors3 ,all the world believes it firmly1 because the mathematicians imagine that it is a fact of observation1 and the observers that it is a theorem of mathematics4 ,it was long so for the principle of the conservation of energy4 ,it is no longer so to-day2 no one is ignorant that this is an experimental fact4 ,but then what gives us the right to attribute to the principle itself more generality and more precision than to the experiments which have served to demonstrate it8 ,this is to ask whether it is legitimate1 as is done every day1 to generalize empirical data1 and ,i shall not have the presumption to discuss this question1 after so many philosophers have vainly striven to solve it4 ,one thing is certain2 if this power were denied us1 science could not exist or1 at least1 reduced to a sort of inventory1 to the ascertaining of isolated facts1 it would have no value for us1 since it could give no satisfaction to our craving for order and harmony and since it would be at the same time incapable of foreseeing4 ,as the circumstances which have preceded any fact will probably never be simultaneously reproduced1 a first generalization is already necessary to foresee whether this fact will be reproduced again after the least of these circumstances shall be changed4 ,but every proposition may be generalized in an infinity of ways4 ,among all the generalizations possible1 we must choose1 and we can only choose the simplest4 ,we are therefore led to act as if a simple law were1 other things being equal1 more probable than a complicated law4 ,half a century ago this was frankly confessed1 and it was proclaimed that nature loves simplicity2 she has since too often given us the lie4 ,to-day we no longer confess this tendency1 and we retain only so much of it as is indispensable if science is not to become impossible4 ,in formulating a general1 simple and precise law on the basis of experiments relatively few and presenting certain divergences1 we have therefore only obeyed a necessity from which the human mind can not free itself4 ,but there is something more1 and this is why ,i dwell upon the point4 ,no one doubts that ,mayer's principle is destined to survive all the particular laws from which it was obtained1 just as ,newton's law has survived ,kepler's laws1 from which it sprang1 and which are only approximative if account be taken of perturbations4 ,why does this principle occupy thus a sort of privileged place among all the physical laws8 ,there are many little reasons for it4 ,first of all it is believed that we could not reject it or even doubt its absolute rigor without admitting the possibility of perpetual motion2 of course we are on our guard at such a prospect1 and we think ourselves less rash in affirming ,mayer's principle than in denying it4 ,that is perhaps not wholly accurate2 the impossibility of perpetual motion implies the conservation of energy only for reversible phenomena4 ,the imposing simplicity of ,mayer's principle likewise contributes to strengthen our faith4 ,in a law deduced immediately from experiment1 like ,mariotte's1 this simplicity would rather seem to us a reason for distrust2 but here this is no longer the case2 we see elements1 at first sight disparate1 arrange themselves in an unexpected order and form a harmonious whole2 and we refuse to believe that an unforeseen harmony may be a simple effect of chance4 ,it seems that our conquest is the dearer to us the more effort it has cost us1 or that we are the surer of having wrested her true secret from nature the more jealously she has hidden it from us4 ,but those are only little reasons2 to establish ,mayer's law as an absolute principle1 a more profound discussion is necessary4 ,but if this be attempted1 it is seen that this absolute principle is not even easy to state4 ,in each particular case it is clearly seen what energy is and at least a provisional definition of it can be given2 but it is impossible to find a general definition for it4 ,if we try to enunciate the principle in all its generality and apply it to the universe1 we see it vanish1 so to speak1 and nothing is left but this3 ,there is something which remains constant4 ,but has even this any meaning8 ,in the determinist hypothesis1 the state of the universe is determined by an extremely great number n of parameters which ,i shall call x#a1 x#b1444 xn4 ,as soon as the values of these n parameters at any instant are known1 their derivatives with respect to the time are likewise known and consequently the values of these same parameters at a preceding or subsequent instant can be calculated4 ,in other words1 these n parameters satisfy n differential equations of the first order4 ,these equations admit of n - #a integrals and consequently there are n - #a functions of x#a1 x#b1444 xn1 which remain constant4 ,if then we say there is something which remains constant1 we only utter a tautology4 ,we should even be puzzled to say which among all our integrals should retain the name of energy4 ,besides1 ,mayer's principle is not understood in this sense when it is applied to a limited system4 ,it is then assumed that p of our parameters vary independently1 so that we only have n - p relations1 generally linear1 between our n parameters and their derivatives4 ,to simplify the enunciation1 suppose that the sum of the work of the external forces is null1 as well as that of the quantities of heat given off to the outside4 ,then the signification of our principle will be3 ,there is a combination of these n - p relations whose first member is an exact differential2 and then this differential vanishing in virtue of our n - p relations1 its integral is a constant and this integral is called energy4 ,but how can it be possible that there are several parameters whose variations are independent8 ,that can only happen under the influence of external forces 7although we have supposed1 for simplicity1 that the algebraic sum of the effects of these forces is null74 ,in fact1 if the system were completely isolated from all external action1 the values of our n parameters at a given instant would suffice to determine the state of the system at any subsequent instant1 provided always we retain the determinist hypothesis2 we come back therefore to the same difficulty as above4 ,if the future state of the system is not entirely determined by its present state1 this is because it depends besides upon the state of bodies external to the system4 ,but then is it probable that there exist between the parameters x1 which define the state of the system1 equations independent of this state of the external bodies8 and if in certain cases we believe we can find such1 is this not solely in consequence of our ignorance and because the influence of these bodies is too slight for our experimenting to detect it8 ,if the system is not regarded as completely isolated1 it is probable that the rigorously exact expression of its internal energy will depend on the state of the external bodies4 ,again1 ,i have above supposed the sum of the external work was null1 and if we try to free ourselves from this rather artificial restriction1 the enunciation becomes still more difficult4 ,to formulate ,mayer's principle in an absolute sense1 it is therefore necessary to extend it to the whole universe1 and then we find ourselves face to face with the very difficulty we sought to avoid4 ,in conclusion1 using ordinary language1 the law of the conservation of energy can have only one signification1 which is that there is a property common to all the possibilities2 but on the determinist hypothesis there is only a single possibility1 and then the law has no longer any meaning4 ,on the indeterminist hypothesis1 on the contrary1 it would have a meaning1 even if it were taken in an absolute sense2 it would appear as a limitation imposed upon freedom4 ,but this word reminds me that ,i am digressing and am on the point of leaving the domain of mathematics and physics4 ,i check myself therefore and will stress of all this discussion only one impression1 that ,mayer's law is a form flexible enough for us to put into it almost whatever we wish4 ,by that ,i do not mean it corresponds to no objective reality1 nor that it reduces itself to a mere tautology1 since1 in each particular case1 and provided one does not try to push to the absolute1 it has a perfectly clear meaning4 ,this flexibility is a reason for believing in its permanence1 and as1 on the other hand1 it will disappear only to lose itself in a higher harmony1 we may work with confidence1 supporting ourselves upon it1 certain beforehand that our labor will not be lost4 ,almost everything ,i have just said applies to the principle of ,clausius4 ,what distinguishes it is that it is expressed by an inequality4 ,perhaps it will be said it is the same with all physical laws1 since their precision is always limited by errors of observation4 ,but they at least claim to be first approximations1 and it is hoped to replace them little by little by laws more and more precise4 ,if1 on the other hand1 the principle of ,clausius reduces to an inequality1 this is not caused by the imperfection of our means of observation1 but by the very nature of the question4 ,,general ,,conclusions ,,on ,,part ,,third ,the principles of mechanics1 then1 present themselves to us under two different aspects4 ,on the one hand1 they are truths founded on experiment and approximately verified so far as concerns almost isolated systems4 ,on the other hand1 they are postulates applicable to the totality of the universe and regarded as rigorously true4 ,if these postulates possess a generality and a certainty which are lacking to the experimental verities whence they are drawn1 this is because they reduce in the last analysis to a mere convention which we have the right to make1 because we are certain beforehand that no experiment can ever contradict it4 ,this convention1 however1 is not absolutely arbitrary2 it does not spring from our caprice2 we adopt it because certain experiments have shown us that it would be convenient4 ,thus is explained how experiment could make the principles of mechanics1 and yet why it can not overturn them4 ,compare with geometry3 ,the fundamental propositions of geometry1 as for instance ,euclid's postulate1 are nothing more than conventions1 and it is just as unreasonable to inquire whether they are true or false as to ask whether the metric system is true or false4 ,only1 these conventions are convenient1 and it is certain experiments which have taught us that4 ,at first blush1 the analogy is complete2 the rle of experiment seems the same4 ,one will therefore be tempted to say3 ,either mechanics must be regarded as an experimental science1 and then the same must hold for geometry2 or else1 on the contrary1 geometry is a deductive science1 and then one may say as much of mechanics4 ,such a conclusion would be illegitimate4 ,the experiments which have led us to adopt as more convenient the fundamental conventions of geometry bear on objects which have nothing in common with those geometry studies2 they bear on the properties of solid bodies1 on the rectilinear propagation of light4 ,they are experiments of mechanics1 experiments of optics2 they can not in any way be regarded as experiments of geometry4 ,and even the principal reason why our geometry seems convenient to us is that the different parts of our body1 our eye1 our limbs1 have the properties of solid bodies4 ,on this account1 our fundamental experiments are preeminently physiological experiments1 which bear1 not on space which is the object the geometer must study1 but on his body1 that is to say1 on the instrument he must use for this study4 ,on the contrary1 the fundamental conventions of mechanics1 and the experiments which prove to us that they are convenient1 bear on exactly the same objects or on analogous objects4 ,the conventional and general principles are the natural and direct generalization of the experimental and particular principles4 ,let it not be said that thus ,i trace artificial frontiers between the sciences2 that if ,i separate by a barrier geometry properly so called from the study of solid bodies1 ,i could just as well erect one between experimental mechanics and the conventional mechanics of the general principles4 ,in fact1 who does not see that in separating these two sciences ,i mutilate them both1 and that what will remain of conventional mechanics when it shall be isolated will be only a very small thing and can in no way be compared to that superb body of doctrine called geometry8 ,one sees now why the teaching of mechanics should remain experimental4 ,only thus can it make us comprehend the genesis of the science1 and that is indispensable for the complete understanding of the science itself4 ,besides1 if we study mechanics1 it is to apply it2 and we can apply it only if it remains objective4 ,now1 as we have seen1 what the principles gain in generality and certainty they lose in objectivity4 ,it is1 therefore1 above all with the objective side of the principles that we must be familiarized early1 and that can be done only by going from the particular to the general1 instead of the inverse4 ,the principles are conventions and disguised definitions4 ,yet they are drawn from experimental laws2 these laws have1 so to speak1 been exalted into principles to which our mind attributes an absolute value4 ,some philosophers have generalized too far2 they believed the principles were the whole science and consequently that the whole science was conventional4 ,this paradoxical doctrine1 called nominalism1 will not bear examination4 ,how can a law become a principle8 ,it expressed a relation between two real terms ,a and ,b4 ,but it was not rigorously true1 it was only approximate4 ,we introduce arbitrarily an intermediary term ,c more or less fictitious1 and ,c is by definition that which has with ,a exactly the relation expressed by the law4 ,then our law is separated into an absolute and rigorous principle which expresses the relation of ,a to ,c and an experimental law1 approximate and subject to revision1 which expresses the relation of ,c to ,b4 ,it is clear that1 however far this partition is pushed1 some laws will always be left remaining4 ,we go to enter now the domain of laws properly so called4 ,,part ,,iv ,,nature ,,chapter ,,ix ,,hypotheses ,,in ,,physics ,,the ,r,,le ,,of ,,experiment ,,and ,,generalization4--,experiment is the sole source of truth4 ,it alone can teach us anything new2 it alone can give us certainty4 ,these are two points that can not be questioned4 ,but then1 if experiment is everything1 what place will remain for mathematical physics8 ,what has experimental physics to do with such an aid1 one which seems useless and perhaps even dangerous8 ,and yet mathematical physics exists1 and has done unquestionable service4 ,we have here a fact that must be explained4 ,the explanation is that merely to observe is not enough4 ,we must use our observations1 and to do that we must generalize4 ,this is what men always have done2 only as the memory of past errors has made them more and more careful1 they have observed more and more1 and generalized less and less4 ,every age has ridiculed the one before it1 and accused it of having generalized too quickly and too navely4 ,descartes pitied the ,ionians2 ,descartes1 in his turn1 makes us smile4 ,no doubt our children will some day laugh at us4 ,but can we not then pass over immediately to the goal8 ,is not this the means of escaping the ridicule that we foresee8 ,can we not be content with just the bare experiment8 ,no1 that is impossible2 it would be to mistake utterly the true nature of science4 ,the scientist must set in order4 ,science is built up with facts1 as a house is with stones4 ,but a collection of facts is no more a science than a heap of stones is a house4 ,and above all the scientist must foresee4 ,carlyle has somewhere said something like this3 ,nothing but facts are of importance4 ,john ,lackland passed by here4 ,here is something that is admirable4 ,here is a reality for which ,i would give all the theories in the world4 ,carlyle was a fellow countryman of ,bacon2 but ,bacon would not have said that4 ,that is the language of the historian4 ,the physicist would say rather3 ,john ,lackland passed by here2 that makes no difference to me1 for he never will pass this way again4 ,we all know that there are good experiments and poor ones4 ,the latter will accumulate in vain2 though one may have made a hundred or a thousand1 a single piece of work by a true master1 by a ,pasteur1 for example1 will suffice to tumble them into oblivion4 ,bacon would have well understood this2 it is he who invented the phrase ,experimentum crucis4 ,but ,carlyle would not have understood it4 ,a fact is a fact4 ,a pupil has read a certain number on his thermometer2 he has taken no precaution2 no matter1 he has read it1 and if it is only the fact that counts1 here is a reality of the same rank as the peregrinations of ,king ,john ,lackland4 ,why is the fact that this pupil has made this reading of no interest1 while the fact that a skilled physicist had made another reading might be on the contrary very important8 ,it is because from the first reading we could not infer anything4 ,what then is a good experiment8 ,it is that which informs us of something besides an isolated fact2 it is that which enables us to foresee1 that is1 that which enables us to generalize4 ,for without generalization foreknowledge is impossible4 ,the circumstances under which one has worked will never reproduce themselves all at once4 ,the observed action then will never recur2 the only thing that can be affirmed is that under analogous circumstances an analogous action will be produced4 ,in order to foresee1 then1 it is necessary to invoke at least analogy1 that is to say1 already then to generalize4 ,no matter how timid one may be1 still it is necessary to interpolate4 ,experiment gives us only a certain number of isolated points4 ,we must unite these by a continuous line4 ,this is a veritable generalization4 ,but we do more2 the curve that we shall trace will pass between the observed points and near these points2 it will not pass through these points themselves4 ,thus one does not restrict himself to generalizing the experiments1 but corrects them2 and the physicist who should try to abstain from these corrections and really be content with the bare experiment1 would be forced to enunciate some very strange laws4 ,the bare facts1 then1 would not be enough for us2 and that is why we must have science ordered1 or rather organized4 ,it is often said experiments must be made without a preconceived idea4 ,that is impossible4 ,not only would it make all experiment barren1 but that would be attempted which could not be done4 ,every one carries in his mind his own conception of the world1 of which he can not so easily rid himself4 ,we must1 for instance1 use language2 and our language is made up only of preconceived ideas and can not be otherwise4 ,only these are unconscious preconceived ideas1 a thousand times more dangerous than the others4 ,shall we say that if we introduce others1 of which we are fully conscious1 we shall only aggravate the evil8 ,i think not4 ,i believe rather that they will serve as counterbalances to each other--,i was going to say as antidotes2 they will in general accord ill with one another--they will come into conflict with one another1 and thereby force us to regard things under different aspects4 ,this is enough to emancipate us4 ,he is no longer a slave who can choose his master4 ,thus1 thanks to generalization1 each fact observed enables us to foresee a great many others2 only we must not forget that the first alone is certain1 that all others are merely probable4 ,no matter how solidly founded a prediction may appear to us1 we are never absolutely sure that experiment will not contradict it1 if we undertake to verify it4 ,the probability1 however1 is often so great that practically we may be content with it4 ,it is far better to foresee even without certainty than not to foresee at all4 ,one must1 then1 never disdain to make a verification when opportunity offers4 ,but all experiment is long and difficult2 the workers are few2 and the number of facts that we need to foresee is immense4 ,compared with this mass the number of direct verifications that we can make will never be anything but a negligible quantity4 ,of this few that we can directly attain1 we must make the best use2 it is very necessary to get from every experiment the greatest possible number of predictions1 and with the highest possible degree of probability4 ,the problem is1 so to speak1 to increase the yield of the scientific machine4 ,let us compare science to a library that ought to grow continually4 ,the librarian has at his disposal for his purchases only insufficient funds4 ,he ought to make an effort not to waste them4 ,it is experimental physics that is entrusted with the purchases4 ,it alone1 then1 can enrich the library4 ,as for mathematical physics1 its task will be to make out the catalogue4 ,if the catalogue is well made1 the library will not be any richer1 but the reader will be helped to use its riches4 ,and even by showing the librarian the gaps in his collections1 it will enable him to make a judicious use of his funds2 which is all the more important because these funds are entirely inadequate4 ,such1 then1 is the rle of mathematical physics4 ,it must direct generalization in such a manner as to increase what ,i just now called the yield of science4 ,by what means it can arrive at this1 and how it can do it without danger1 is what remains for us to investigate4 ,,the ,,unity ,,of ,,nature4--,let us notice1 first of all1 that every generalization implies in some measure the belief in the unity and simplicity of nature4 ,as to the unity there can be no difficulty4 ,if the different parts of the universe were not like the members of one body1 they would not act on one another1 they would know nothing of one another2 and we in particular would know only one of these parts4 ,we do not ask1 then1 if nature is one1 but how it is one4 ,as for the second point1 that is not such an easy matter4 ,it is not certain that nature is simple4 ,can we without danger act as if it were8 ,there was a time when the simplicity of ,mariotte's law was an argument invoked in favor of its accuracy2 when ,fresnel himself1 after having said in a conversation with ,laplace that nature was not concerned about analytical difficulties1 felt himself obliged to make explanations1 in order not to strike too hard at prevailing opinion4 ,to-day ideas have greatly changed2 and yet1 those who do not believe that natural laws have to be simple1 are still often obliged to act as if they did4 ,they could not entirely avoid this necessity without making impossible all generalization1 and consequently all science4 ,it is clear that any fact can be generalized in an infinity of ways1 and it is a question of choice4 ,the choice can be guided only by considerations of simplicity4 ,let us take the most commonplace case1 that of interpolation4 ,we pass a continuous line1 as regular as possible1 between the points given by observation4 ,why do we avoid points making angles and too abrupt turns8 ,why do we not make our curve describe the most capricious zig-zags8 ,it is because we know beforehand1 or believe we know1 that the law to be expressed can not be so complicated as all that4 ,we may calculate the mass of ,jupiter from either the movements of its satellites1 or the perturbations of the major planets1 or those of the minor planets4 ,if we take the averages of the determinations obtained by these three methods1 we find three numbers very close together1 but different4 ,we might interpret this result by supposing that the coefficient of gravitation is not the same in the three cases4 ,the observations would certainly be much better represented4 ,why do we reject this interpretation8 ,not because it is absurd1 but because it is needlessly complicated4 ,we shall only accept it when we are forced to1 and that is not yet4 ,to sum up1 ordinarily every law is held to be simple till the contrary is proved4 ,this custom is imposed upon physicists by the causes that ,i have just explained4 ,but how shall we justify it in the presence of discoveries that show us every day new details that are richer and more complex8 ,how shall we even reconcile it with the belief in the unity of nature8 ,for if everything depends on everything1 relationships where so many diverse factors enter can no longer be simple4 ,if we study the history of science1 we see happen two inverse phenomena1 so to speak4 ,sometimes simplicity hides under complex appearances2 sometimes it is the simplicity which is apparent1 and which disguises extremely complicated realities4 ,what is more complicated than the confused movements of the planets8 ,what simpler than ,newton's law8 ,here nature1 making sport1 as ,fresnel said1 of analytical difficulties1 employs only simple means1 and by combining them produces ,i know not what inextricable tangle4 ,here it is the hidden simplicity which must be discovered4 ,examples of the opposite abound4 ,in the kinetic theory of gases1 one deals with molecules moving with great velocities1 whose paths1 altered by incessant collisions1 have the most capricious forms and traverse space in every direction4 ,the observable result is ,mariotte's simple law4 ,every individual fact was complicated4 ,the law of great numbers has reestablished simplicity in the average4 ,here the simplicity is merely apparent1 and only the coarseness of our senses prevents our perceiving the complexity4 ,many phenomena obey a law of proportionality4 ,but why8 ,because in these phenomena there is something very small4 ,the simple law observed1 then1 is only a result of the general analytical rule that the infinitely small increment of a function is proportional to the increment of the variable4 ,as in reality our increments are not infinitely small1 but very small1 the law of proportionality is only approximate1 and the simplicity is only apparent4 ,what ,i have just said applies to the rule of the superposition of small motions1 the use of which is so fruitful1 and which is the basis of optics4 ,and ,newton's law itself8 ,its simplicity1 so long undetected1 is perhaps only apparent4 ,who knows whether it is not due to some complicated mechanism1 to the impact of some subtile matter animated by irregular movements1 and whether it has not become simple only through the action of averages and of great numbers8 ,in any case1 it is difficult not to suppose that the true law contains complementary terms1 which would become sensible at small distances4 ,if in astronomy they are negligible as modifying ,newton's law1 and if the law thus regains its simplicity1 it would be only because of the immensity of celestial distances4 ,no doubt1 if our means of investigation should become more and more penetrating1 we should discover the simple under the complex1 then the complex under the simple1 then again the simple under the complex1 and so on1 without our being able to foresee what will be the last term4 ,we must stop somewhere1 and that science may be possible we must stop when we have found simplicity4 ,this is the only ground on which we can rear the edifice of our generalizations4 ,but this simplicity being only apparent1 will the ground be firm enough8 ,this is what must be investigated4 ,for that purpose1 let us see what part is played in our generalizations by the belief in simplicity4 ,we have verified a simple law in a good many particular cases2 we refuse to admit that this agreement1 so often repeated1 is simply the result of chance1 and conclude that the law must be true in the general case4 ,kepler notices that a planet's positions1 as observed by ,tycho1 are all on one ellipse4 ,never for a moment does he have the thought that by a strange play of chance ,tycho never observed the heavens except at a moment when the real orbit of the planet happened to cut this ellipse4 ,what does it matter then whether the simplicity be real1 or whether it covers a complex reality8 ,whether it is due to the influence of great numbers1 which levels down individual differences1 or to the greatness or smallness of certain quantities1 which allows us to neglect certain terms1 in no case is it due to chance4 ,this simplicity1 real or apparent1 always has a cause4 ,we can always follow1 then1 the same course of reasoning1 and if a simple law has been observed in several particular cases1 we can legitimately suppose that it will still be true in analogous cases4 ,to refuse to do this would be to attribute to chance an inadmissible rle4 ,there is1 however1 a difference4 ,if the simplicity were real and essential1 it would resist the increasing precision of our means of measure4 ,if then we believe nature to be essentially simple1 we must1 from a simplicity that is approximate1 infer a simplicity that is rigorous4 ,this is what was done formerly2 and this is what we no longer have a right to do4 ,the simplicity of ,kepler's laws1 for example1 is only apparent4 ,that does not prevent their being applicable1 very nearly1 to all systems analogous to the solar system2 but it does prevent their being rigorously exact4 ,,the ,r,,le ,,of ,,hypothesis4--,all generalization is a hypothesis4 ,hypothesis1 then1 has a necessary rle that no one has ever contested4 ,only1 it ought always1 as soon as possible and as often as possible1 to be subjected to verification4 ,and1 of course1 if it does not stand this test1 it ought to be abandoned without reserve4 ,this is what we generally do1 but sometimes with rather an ill humor4 ,well1 even this ill humor is not justified4 ,the physicist who has just renounced one of his hypotheses ought1 on the contrary1 to be full of joy2 for he has found an unexpected opportunity for discovery4 ,his hypothesis1 ,i imagine1 had not been adopted without consideration2 it took account of all the known factors that it seemed could enter into the phenomenon4 ,if the test does not support it1 it is because there is something unexpected and extraordinary2 and because there is going to be something found that is unknown and new4 ,has the discarded hypothesis1 then1 been barren8 ,far from that1 it may be said it has rendered more service than a true hypothesis4 ,not only has it been the occasion of the decisive experiment1 but1 without having made the hypothesis1 the experiment would have been made by chance1 so that nothing would have been derived from it4 ,one would have seen nothing extraordinary2 only one fact the more would have been catalogued without deducing from it the least consequence4 ,now on what condition is the use of hypothesis without danger8 ,the firm determination to submit to experiment is not enough2 there are still dangerous hypotheses2 first1 and above all1 those which are tacit and unconscious4 ,since we make them without knowing it1 we are powerless to abandon them4 ,here again1 then1 is a service that mathematical physics can render us4 ,by the precision that is characteristic of it1 it compels us to formulate all the hypotheses that we should make without it1 but unconsciously4 ,let us notice besides that it is important not to multiply hypotheses beyond measure1 and to make them only one after the other4 ,if we construct a theory based on a number of hypotheses1 and if experiment condemns it1 which of our premises is it necessary to change8 ,it will be impossible to know4 ,and inversely1 if the experiment succeeds1 shall we believe that we have demonstrated all the hypotheses at once8 ,shall we believe that with one single equation we have determined several unknowns8 ,we must equally take care to distinguish between the different kinds of hypotheses4 ,there are first those which are perfectly natural and from which one can scarcely escape4 ,it is difficult not to suppose that the influence of bodies very remote is quite negligible1 that small movements follow a linear law1 that the effect is a continuous function of its cause4 ,i will say as much of the conditions imposed by symmetry4 ,all these hypotheses form1 as it were1 the common basis of all the theories of mathematical physics4 ,they are the last that ought to be abandoned4 ,there is a second class of hypotheses1 that ,i shall term neutral4 ,in most questions the analyst assumes at the beginning of his calculations either that matter is continuous or1 on the contrary1 that it is formed of atoms4 ,he might have made the opposite assumption without changing his results4 ,he would only have had more trouble to obtain them2 that is all4 ,if1 then1 experiment confirms his conclusions1 will he think that he has demonstrated1 for instance1 the real existence of atoms8 ,in optical theories two vectors are introduced1 of which one is regarded as a velocity1 the other as a vortex4 ,here again is a neutral hypothesis1 since the same conclusions would have been reached by taking precisely the opposite4 ,the success of the experiment1 then1 can not prove that the first vector is indeed a velocity2 it can only prove one thing1 that it is a vector4 ,this is the only hypothesis that has really been introduced in the premises4 ,in order to give it that concrete appearance which the weakness of our minds requires1 it has been necessary to consider it either as a velocity or as a vortex1 in the same way that it has been necessary to represent it by a letter1 either x or y4 ,the result1 however1 whatever it may be1 will not prove that it was right or wrong to regard it as a velocity any more than it will prove that it was right or wrong to call it x and not y4 ,these neutral hypotheses are never dangerous1 if only their character is not misunderstood4 ,they may be useful1 either as devices for computation1 or to aid our understanding by concrete images1 to fix our ideas as the saying is4 ,there is1 then1 no occasion to exclude them4 ,the hypotheses of the third class are the real generalizations4 ,they are the ones that experiment must confirm or invalidate4 ,whether verified or condemned1 they will always be fruitful4 ,but for the reasons that ,i have set forth1 they will only be fruitful if they are not too numerous4 ,,origin ,,of ,,mathematical ,,physics4--,let us penetrate further1 and study more closely the conditions that have permitted the development of mathematical physics4 ,we observe at once that the efforts of scientists have always aimed to resolve the complex phenomenon directly given by experiment into a very large number of elementary phenomena4 ,this is done in three different ways3 first1 in time4 ,instead of embracing in its entirety the progressive development of a phenomenon1 the aim is simply to connect each instant with the instant immediately preceding it4 ,it is admitted that the actual state of the world depends only on the immediate past1 without being directly influenced1 so to speak1 by the memory of a distant past4 ,thanks to this postulate1 instead of studying directly the whole succession of phenomena1 it is possible to confine ourselves to writing its 'differential equation4' ,for ,kepler's laws we substitute ,newton's law4 ,next we try to analyze the phenomenon in space4 ,what experiment gives us is a confused mass of facts presented on a stage of considerable extent4 ,we must try to discover the elementary phenomenon1 which will be1 on the contrary1 localized in a very small region of space4 ,some examples will perhaps make my thought better understood4 ,if we wished to study in all its complexity the distribution of temperature in a cooling solid1 we should never succeed4 ,everything becomes simple if we reflect that one point of the solid can not give up its heat directly to a distant point2 it will give up its heat only to the points in the immediate neighborhood1 and it is by degrees that the flow of heat can reach other parts of the solid4 ,the elementary phenomenon is the exchange of heat between two contiguous points4 ,it is strictly localized1 and is relatively simple1 if we admit1 as is natural1 that it is not influenced by the temperature of molecules whose distance is sensible4 ,i bend a rod4 ,it is going to take a very complicated form1 the direct study of which would be impossible4 ,but ,i shall be able1 however1 to attack it1 if ,i observe that its flexure is a result only of the deformation of the very small elements of the rod1 and that the deformation of each of these elements depends only on the forces that are directly applied to it1 and not at all on those which may act on the other elements4 ,in all these examples1 which ,i might easily multiply1 we admit that there is no action at a distance1 or at least at a great distance4 ,this is a hypothesis4 ,it is not always true1 as the law of gravitation shows us4 ,it must1 then1 be submitted to verification4 ,if it is confirmed1 even approximately1 it is precious1 for it will enable us to make mathematical physics1 at least by successive approximations4 ,if it does not stand the test1 we must look for something else analogous2 for there are still other means of arriving at the elementary phenomenon4 ,if several bodies act simultaneously1 it may happen that their actions are independent and are simply added to one another1 either as vectors or as scalars4 ,the elementary phenomenon is then the action of an isolated body4 ,or again1 we have to deal with small movements1 or more generally with small variations1 which obey the well-known law of superposition4 ,the observed movement will then be decomposed into simple movements1 for example1 sound into its harmonics1 white light into its monochromatic components4 ,when we have discovered in what direction it is advisable to look for the elementary phenomenon1 by what means can we reach it8 ,first of all1 it will often happen that in order to detect it1 or rather to detect the part of it useful to us1 it will not be necessary to penetrate the mechanism2 the law of great numbers will suffice4 ,let us take again the instance of the propagation of heat4 ,every molecule emits rays toward every neighboring molecule4 ,according to what law1 we do not need to know4 ,if we should make any supposition in regard to this1 it would be a neutral hypothesis and consequently useless and incapable of verification4 ,and1 in fact1 by the action of averages and thanks to the symmetry of the medium1 all the differences are leveled down1 and whatever hypothesis may be made1 the result is always the same4 ,the same circumstance is presented in the theory of electricity and in that of capillarity4 ,the neighboring molecules attract and repel one another4 ,we do not need to know according to what law2 it is enough for us that this attraction is sensible only at small distances1 and that the molecules are very numerous1 that the medium is symmetrical1 and we shall only have to let the law of great numbers act4 ,here again the simplicity of the elementary phenomenon was hidden under the complexity of the resultant observable phenomenon2 but1 in its turn1 this simplicity was only apparent1 and concealed a very complex mechanism4 ,the best means of arriving at the elementary phenomenon would evidently be experiment4 ,we ought by experimental contrivance to dissociate the complex sheaf that nature offers to our researches1 and to study with care the elements as much isolated as possible4 ,for example1 natural white light would be decomposed into monochromatic lights by the aid of the prism1 and into polarized light by the aid of the polarizer4 ,unfortunately that is neither always possible nor always sufficient1 and sometimes the mind must outstrip experiment4 ,i shall cite only one example1 which has always struck me forcibly4 ,if ,i decompose white light1 ,i shall be able to isolate a small part of the spectrum1 but however small it may be1 it will retain a certain breadth4 ,likewise the natural lights1 called monochromatic1 give us a very narrow line1 but not1 however1 infinitely narrow4 ,it might be supposed that by studying experimentally the properties of these natural lights1 by working with finer and finer lines of the spectrum1 and by passing at last to the limit1 so to speak1 we should succeed in learning the properties of a light strictly monochromatic4 ,that would not be accurate4 ,suppose that two rays emanate from the same source1 that we polarize them first in two perpendicular planes1 then bring them back to the same plane of polarization1 and try to make them interfere4 ,if the light were strictly monochromatic1 they would interfere4 ,with our lights1 which are nearly monochromatic1 there will be no interference1 and that no matter how narrow the line4 ,in order to be otherwise it would have to be several million times as narrow as the finest known lines4 ,here1 then1 the passage to the limit would have deceived us4 ,the mind must outstrip the experiment1 and if it has done so with success1 it is because it has allowed itself to be guided by the instinct of simplicity4 ,the knowledge of the elementary fact enables us to put the problem in an equation4 ,nothing remains but to deduce from this by combination the complex fact that can be observed and verified4 ,this is what is called integration1 and is the business of the mathematician4 ,it may be asked why1 in physical sciences1 generalization so readily takes the mathematical form4 ,the reason is now easy to see4 ,it is not only because we have numerical laws to express2 it is because the observable phenomenon is due to the superposition of a great number of elementary phenomena all alike4 ,thus quite naturally are introduced differential equations4 ,it is not enough that each elementary phenomenon obeys simple laws2 all those to be combined must obey the same law4 ,then only can the intervention of mathematics be of use2 mathematics teaches us in fact to combine like with like4 ,its aim is to learn the result of a combination without needing to go over the combination piece by piece4 ,if we have to repeat several times the same operation1 it enables us to avoid this repetition by telling us in advance the result of it by a sort of induction4 ,i have explained this above1 in the chapter on mathematical reasoning4 ,but for this1 all the operations must be alike4 ,in the opposite case1 it would evidently be necessary to resign ourselves to doing them in reality one after another1 and mathematics would become useless4 ,it is then thanks to the approximate homogeneity of the matter studied by physicists that mathematical physics could be born4 ,in the natural sciences1 we no longer find these conditions3 homogeneity1 relative independence of remote parts1 simplicity of the elementary fact2 and this is why naturalists are obliged to resort to other methods of generalization4 ,,chapter ,x ,,the ,,theories ,,of ,,modern ,,physics ,,meaning ,,of ,,physical ,,theories4--,the laity are struck to see how ephemeral scientific theories are4 ,after some years of prosperity1 they see them successively abandoned2 they see ruins accumulate upon ruins2 they foresee that the theories fashionable to-day will shortly succumb in their turn and hence they conclude that these are absolutely idle4 ,this is what they call the bankruptcy of science4 ,their skepticism is superficial2 they give no account to themselves of the aim and the rle of scientific theories2 otherwise they would comprehend that the ruins may still be good for something4 ,no theory seemed more solid than that of ,fresnel which attributed light to motions of the ether4 ,yet now ,maxwell's is preferred4 ,does this mean the work of ,fresnel was in vain8 ,no1 because the aim of ,fresnel was not to find out whether there is really an ether1 whether it is or is not formed of atoms1 whether these atoms really move in this or that sense2 his object was to foresee optical phenomena4 ,now1 ,fresnel's theory always permits of this1 to-day as well as before ,maxwell4 ,the differential equations are always true2 they can always be integrated by the same procedures and the results of this integration always retain their value4 ,and let no one say that thus we reduce physical theories to the rle of mere practical recipes2 these equations express relations1 and if the equations remain true it is because these relations preserve their reality4 ,they teach us1 now as then1 that there is such and such a relation between some thing and some other thing2 only this something formerly we called motion2 we now call it electric current4 ,but these appellations were only images substituted for the real objects which nature will eternally hide from us4 ,the true relations between these real objects are the only reality we can attain to1 and the only condition is that the same relations exist between these objects as between the images by which we are forced to replace them4 ,if these relations are known to us1 what matter if we deem it convenient to replace one image by another4 ,that some periodic phenomenon 7an electric oscillation1 for instance7 is really due to the vibration of some atom which1 acting like a pendulum1 really moves in this or that sense1 is neither certain nor interesting4 ,but that between electric oscillation1 the motion of the pendulum and all periodic phenomena there exists a close relationship which corresponds to a profound reality2 that this relationship1 this similitude1 or rather this parallelism extends into details2 that it is a consequence of more general principles1 that of energy and that of least action2 this is what we can affirm2 this is the truth which will always remain the same under all the costumes in which we may deem it useful to deck it out4 ,numerous theories of dispersion have been proposed2 the first was imperfect and contained only a small part of truth4 ,afterwards came that of ,helmholtz2 then it was modified in various ways1 and its author himself imagined another founded on the principles of ,maxwell4 ,but1 what is remarkable1 all the scientists who came after ,helmholtz reached the same equations1 starting from points of departure in appearance very widely separated4 ,i will venture to say these theories are all true at the same time1 not only because they make us foresee the same phenomena1 but because they put in evidence a true relation1 that of absorption and anomalous dispersion4 ,what is true in the premises of these theories is what is common to all the authors2 this is the affirmation of this or that relation between certain things which some call by one name1 others by another4 ,the kinetic theory of gases has given rise to many objections1 which we could hardly answer if we pretended to see in it the absolute truth4 ,but all these objections will not preclude its having been useful1 and particularly so in revealing to us a relation true and but for it profoundly hidden1 that of the gaseous pressure and the osmotic pressure4 ,in this sense1 then1 it may be said to be true4 ,when a physicist finds a contradiction between two theories equally dear to him1 he sometimes says3 ,we will not bother about that1 but hold firmly the two ends of the chain1 though the intermediate links are hidden from us4 ,this argument of an embarrassed theologian would be ridiculous if it were necessary to attribute to physical theories the sense the laity give them4 ,in case of contradiction1 one of them at least must then be regarded as false4 ,it is no longer the same if in them be sought only what should be sought4 ,may be they both express true relations and the contradiction is only in the images wherewith we have clothed the reality4 ,to those who find we restrict too much the domain accessible to the scientist1 ,i answer3 ,these questions which we interdict to you and which you regret1 are not only insoluble1 they are illusory and devoid of meaning4 ,some philosopher pretends that all physics may be explained by the mutual impacts of atoms4 ,if he merely means there are between physical phenomena the same relations as between the mutual impacts of a great number of balls1 well and good1 that is verifiable1 that is perhaps true4 ,but he means something more2 and we think we understand it because we think we know what impact is in itself2 why8 ,simply because we have often seen games of billiards4 ,shall we think ,god1 contemplating his work1 feels the same sensations as we in watching a billiard match8 ,if we do not wish to give this bizarre sense to his assertion1 if neither do we wish the restricted sense ,i have just explained1 which is good sense1 then it has none4 ,hypotheses of this sort have therefore only a metaphorical sense4 ,the scientist should no more interdict them than the poet does metaphors2 but he ought to know what they are worth4 ,they may be useful to give a certain satisfaction to the mind1 and they will not be injurious provided they are only indifferent hypotheses4 ,these considerations explain to us why certain theories1 supposed to be abandoned and finally condemned by experiment1 suddenly arise from their ashes and recommence a new life4 ,it is because they expressed true relations2 and because they had not ceased to do so when1 for one reason or another1 we felt it necessary to enunciate the same relations in another language4 ,so they retained a sort of latent life4 ,scarcely fifteen years ago was there anything more ridiculous1 more navely antiquated1 than ,coulomb's fluids8 ,and yet here they are reappearing under the name of electrons4 ,wherein do these permanently electrified molecules differ from ,coulomb's electric molecules8 ,it is true that in the electrons the electricity is supported by a little1 a very little matter2 in other words1 they have a mass 7and yet this is now contested72 but ,coulomb did not deny mass to his fluids1 or1 if he did1 it was only with reluctance4 ,it would be rash to affirm that the belief in electrons will not again suffer eclipse2 it was none the less curious to note this unexpected resurrection4 ,but the most striking example is ,carnot's principle4 ,carnot set it up starting from false hypotheses2 when it was seen that heat is not indestructible1 but may be transformed into work1 his ideas were completely abandoned2 afterwards ,clausius returned to them and made them finally triumph4 ,carnot's theory1 under its primitive form1 expressed1 aside from true relations1 other inexact relations1 dbris of antiquated ideas2 but the presence of these latter did not change the reality of the others4 ,clausius had only to discard them as one lops off dead branches4 ,the result was the second fundamental law of thermodynamics4 ,there were always the same relations2 though these relations no longer subsisted1 at least in appearance1 between the same objects4 ,this was enough for the principle to retain its value4 ,and even the reasonings of ,carnot have not perished because of that2 they were applied to a material tainted with error2 but their form 7that is to say1 the essential7 remained correct4 ,what ,i have just said illuminates at the same time the rle of general principles such as the principle of least action1 or that of the conservation of energy4 ,these principles have a very high value2 they were obtained in seeking what there was in common in the enunciation of numerous physical laws2 they represent therefore1 as it were1 the quintessence of innumerable observations4 ,however1 from their very generality a consequence results to which ,i have called attention in ,chapter ,,viii1 namely1 that they can no longer be verified4 ,as we can not give a general definition of energy1 the principle of the conservation of energy signifies simply that there is something which remains constant4 ,well1 whatever be the new notions that future experiments shall give us about the world1 we are sure in advance that there will be something there which will remain constant and which may be called energy4 ,is this to say that the principle has no meaning and vanishes in a tautology8 ,not at all2 it signifies that the different things to which we give the name of energy are connected by a true kinship2 it affirms a real relation between them4 ,but then if this principle has a meaning1 it may be false2 it may be that we have not the right to extend indefinitely its applications1 and yet it is certain beforehand to be verified in the strict acceptation of the term2 how then shall we know when it shall have attained all the extension which can legitimately be given it8 ,just simply when it shall cease to be useful to us1 that is1 to make us correctly foresee new phenomena4 ,we shall be sure in such a case that the relation affirmed is no longer real2 for otherwise it would be fruitful2 experiment1 without directly contradicting a new extension of the principle1 will yet have condemned it4 ,,physics ,,and ,,mechanism4--,most theorists have a constant predilection for explanations borrowed from mechanics or dynamics4 ,some would be satisfied if they could explain all phenomena by motions of molecules attracting each other according to certain laws4 ,others are more exacting2 they would suppress attractions at a distance2 their molecules should follow rectilinear paths from which they could be made to deviate only by impacts4 ,others again1 like ,hertz1 suppress forces also1 but suppose their molecules subjected to geometric attachments analogous1 for instance1 to those of our linkages2 they try thus to reduce dynamics to a sort of kinematics4 ,in a word1 all would bend nature into a certain form outside of which their mind could not feel satisfied4 ,will nature be sufficiently flexible for that8 ,we shall examine this question in ,chapter ,,xii1 propos of ,maxwell's theory4 ,whenever the principles of energy and of least action are satisfied1 we shall see not only that there is always one possible mechanical explanation1 but that there is always an infinity of them4 ,thanks to a well-known theorem of ,knig's on linkages1 it could be shown that we can1 in an infinity of ways1 explain everything by attachments after the manner of ,hertz1 or also by central forces4 ,without doubt it could be demonstrated just as easily that everything can always be explained by simple impacts4 ,for that1 of course1 we need not be content with ordinary matter1 with that which falls under our senses and whose motions we observe directly4 ,either we shall suppose that this common matter is formed of atoms whose internal motions elude us1 the displacement of the totality alone remaining accessible to our senses4 ,or else we shall imagine some one of those subtile fluids which under the name of ether or under other names1 have at all times played so great a rle in physical theories4 ,often one goes further and regards the ether as the sole primitive matter or even as the only true matter4 ,the more moderate consider common matter as condensed ether1 which is nothing startling2 but others reduce still further its importance and see in it nothing more than the geometric locus of the ether's singularities4 ,for instance1 what we call matter is for ,lord ,kelvin only the locus of points where the ether is animated by vortex motions2 for ,riemann1 it was the locus of points where ether is constantly destroyed2 for other more recent authors1 ,wiechert or ,larmor1 it is the locus of points where the ether undergoes a sort of torsion of a very particular nature4 ,if the attempt is made to occupy one of these points of view1 ,i ask myself by what right shall we extend to the ether1 under pretext that this is the true matter1 mechanical properties observed in ordinary matter1 which is only false matter4 ,the ancient fluids1 caloric1 electricity1 etc41 were abandoned when it was perceived that heat is not indestructible4 ,but they were abandoned for another reason also4 ,in materializing them1 their individuality was1 so to speak1 emphasized1 a sort of abyss was opened between them4 ,this had to be filled up on the coming of a more vivid feeling of the unity of nature1 and the perception of the intimate relations which bind together all its parts4 ,not only did the old physicists1 in multiplying fluids1 create entities unnecessarily1 but they broke real ties4 ,it is not sufficient for a theory to affirm no false relations1 it must not hide true relations4 ,and does our ether really exist8 ,we know the origin of our belief in the ether4 ,if light reaches us from a distant star1 during several years it was no longer on the star and not yet on the earth2 it must then be somewhere and sustained1 so to speak1 by some material support4 ,the same idea may be expressed under a more mathematical and more abstract form4 ,what we ascertain are the changes undergone by material molecules2 we see1 for instance1 that our photographic plate feels the consequences of phenomena of which the incandescent mass of the star was the theater several years before4 ,now1 in ordinary mechanics the state of the system studied depends only on its state at an instant immediately anterior2 therefore the system satisfies differential equations4 ,on the contrary1 if we should not believe in the ether1 the state of the material universe would depend not only on the state immediately preceding1 but on states much older2 the system would satisfy equations of finite differences4 ,it is to escape this derogation of the general laws of mechanics that we have invented the ether4 ,that would still only oblige us to fill up1 with the ether1 the interplanetary void1 but not to make it penetrate the bosom of the material media themselves4 ,fizeau's experiment goes further4 ,by the interference of rays which have traversed air or water in motion1 it seems to show us two different media interpenetrating and yet changing place one with regard to the other4 ,we seem to touch the ether with the finger4 ,yet experiments may be conceived which would make us touch it still more nearly4 ,suppose ,newton's principle1 of the equality of action and reaction1 no longer true if applied to matter alone1 and that we have established it4 ,the geometric sum of all the forces applied to all the material molecules would no longer be null4 ,it would be necessary then1 if we did not wish to change all mechanics1 to introduce the ether1 in order that this action which matter appeared to experience should be counterbalanced by the reaction of matter on something4 ,or again1 suppose we discover that optical and electrical phenomena are influenced by the motion of the earth4 ,we should be led to conclude that these phenomena might reveal to us not only the relative motions of material bodies1 but what would seem to be their absolute motions4 ,again1 an ether would be necessary1 that these so-called absolute motions should not be their displacements with regard to a void space1 but their displacements with regard to something concrete4 ,shall we ever arrive at that8 ,i have not this hope1 ,i shall soon say why1 and yet it is not so absurd1 since others have had it4 ,for instance1 if the theory of ,lorentz1 of which ,i shall speak in detail further on in ,chapter ,,xiii41 were true1 ,newton's principle would not apply to matter alone1 and the difference would not be very far from being accessible to experiment4 ,on the other hand1 many researches have been made on the influence of the earth's motion4 ,the results have always been negative4 ,but these experiments were undertaken because the outcome was not sure in advance1 and1 indeed1 according to the ruling theories1 the compensation would be only approximate1 and one might expect to see precise methods give positive results4 ,i believe that such a hope is illusory2 it was none the less interesting to show that a success of this sort would open to us1 in some sort1 a new world4 ,and now ,i must be permitted a digression2 ,i must explain1 in fact1 why ,i do not believe1 despite ,lorentz1 that more precise observations can ever put in evidence anything else than the relative displacements of material bodies4 ,experiments have been made which should have disclosed the terms of the first order2 the results have been negative2 could that be by chance8 ,no one has assumed that2 a general explanation has been sought1 and ,lorentz has found it2 he has shown that the terms of the first order must destroy each other1 but not those of the second4 ,then more precise experiments were made2 they also were negative2 neither could this be the effect of chance2 an explanation was necessary2 it was found2 they always are found2 of hypotheses there is never lack4 ,but this is not enough2 who does not feel that this is still to leave to chance too great a rle8 ,would not that also be a chance1 this singular coincidence which brought it about that a certain circumstance should come just in the nick of time to destroy the terms of the first order1 and that another circumstance1 wholly different1 but just as opportune1 should take upon itself to destroy those of the second order8 ,no1 it is necessary to find an explanation the same for the one as for the other1 and then everything leads us to think that this explanation will hold good equally well for the terms of higher order1 and that the mutual destruction of these terms will be rigorous and absolute4 ,,present ,,state ,,of ,,the ,,science4--,in the history of the development of physics we distinguish two inverse tendencies4 ,on the one hand1 new bonds are continually being discovered between objects which had seemed destined to remain forever unconnected2 scattered facts cease to be strangers to one another2 they tend to arrange themselves in an imposing synthesis4 ,science advances toward unity and simplicity4 ,on the other hand1 observation reveals to us every day new phenomena2 they must long await their place and sometimes1 to make one for them1 a corner of the edifice must be demolished4 ,in the known phenomena themselves1 where our crude senses showed us uniformity1 we perceive details from day to day more varied2 what we believed simple becomes complex1 and science appears to advance toward variety and complexity4 ,of these two inverse tendencies1 which seem to triumph turn about1 which will win8 ,if it be the first1 science is possible2 but nothing proves this a priori1 and it may well be feared that after having made vain efforts to bend nature in spite of herself to our ideal of unity1 submerged by the ever-rising flood of our new riches1 we must renounce classifying them1 abandon our ideal1 and reduce science to the registration of innumerable recipes4 ,to this question we can not reply4 ,all we can do is to observe the science of to-day and compare it with that of yesterday4 ,from this examination we may doubtless draw some encouragement4 ,half a century ago1 hope ran high4 ,the discovery of the conservation of energy and of its transformations had revealed to us the unity of force4 ,thus it showed that the phenomena of heat could be explained by molecular motions4 ,what was the nature of these motions was not exactly known1 but no one doubted that it soon would be4 ,for light1 the task seemed completely accomplished4 ,in what concerns electricity1 things were less advanced4 ,electricity had just annexed magnetism4 ,this was a considerable step toward unity1 and a decisive step4 ,but how should electricity in its turn enter into the general unity1 how should it be reduced to the universal mechanism8 ,of that no one had any idea4 ,yet the possibility of this reduction was doubted by none1 there was faith4 ,finally1 in what concerns the molecular properties of material bodies1 the reduction seemed still easier1 but all the detail remained hazy4 ,in a word1 the hopes were vast and animated1 but vague4 ,to-day1 what do we see8 ,first of all1 a prime progress1 immense progress4 ,the relations of electricity and light are now known2 the three realms1 of light1 of electricity and of magnetism1 previously separated1 form now but one2 and this annexation seems final4 ,this conquest1 however1 has cost us some sacrifices4 ,the optical phenomena subordinate themselves as particular cases under the electrical phenomena2 so long as they remained isolated1 it was easy to explain them by motions that were supposed to be known in all their details1 that was a matter of course2 but now an explanation1 to be acceptable1 must be easily capable of extension to the entire electric domain4 ,now that is a matter not without difficulties4 ,the most satisfactory theory we have is that of ,lorentz1 which1 as we shall see in the last chapter1 explains electric currents by the motions of little electrified particles2 it is unquestionably the one which best explains the known facts1 the one which illuminates the greatest number of true relations1 the one of which most traces will be found in the final construction4 ,nevertheless1 it still has a serious defect1 which ,i have indicated above2 it is contrary to ,newton's law of the equality of action and reaction2 or rather1 this principle1 in the eyes of ,lorentz1 would not be applicable to matter alone2 for it to be true1 it would be necessary to take account of the action of the ether on matter and of the reaction of matter on the ether4 ,now1 from what we know at present1 it seems probable that things do not happen in this way4 ,however that may be1 thanks to ,lorentz1 ,fizeau's results on the optics of moving bodies1 the laws of normal and anomalous dispersion and of absorption find themselves linked to one another and to the other properties of the ether by bonds which beyond any doubt will never more be broken4 ,see the facility with which the new ,zeeman effect has found its place already and has even aided in classifying ,faraday's magnetic rotation which had defied ,maxwell's efforts2 this facility abundantly proves that the theory of ,lorentz is not an artificial assemblage destined to fall asunder4 ,it will probably have to be modified1 but not destroyed4 ,but ,lorentz had no aim beyond that of embracing in one totality all the optics and electrodynamics of moving bodies2 he never pretended to give a mechanical explanation of them4 ,larmor goes further2 retaining the theory of ,lorentz in essentials1 he grafts upon it1 so to speak1 ,mac,cullagh's ideas on the direction of the motions of the ether4 ,according to him1 the velocity of the ether would have the same direction and the same magnitude as the magnetic force4 ,however ingenious this attempt may be1 the defect of the theory of ,lorentz remains and is even aggravated4 ,with ,lorentz1 we do not know what are the motions of the ether2 thanks to this ignorance1 we may suppose them such that1 compensating those of matter1 they reestablish the equality of action and reaction4 ,with ,larmor1 we know the motions of the ether1 and we can ascertain that the compensation does not take place4 ,if ,larmor has failed1 as it seems to me he has1 does that mean that a mechanical explanation is impossible8 ,far from it3 ,i have said above that when a phenomenon obeys the two principles of energy and of least action1 it admits of an infinity of mechanical explanations2 so it is1 therefore1 with the optical and electrical phenomena4 ,but this is not enough3 for a mechanical explanation to be good1 it must be simple2 for choosing it among all which are possible1 there should be other reasons besides the necessity of making a choice4 ,well1 we have not as yet a theory satisfying this condition and consequently good for something4 ,must we lament this8 ,that would be to forget what is the goal sought2 this is not mechanism2 the true1 the sole aim is unity4 ,we must therefore set bounds to our ambition2 let us not try to formulate a mechanical explanation2 let us be content with showing that we could always find one if we wished to4 ,in this regard we have been successful2 the principle of the conservation of energy has received only confirmations2 a second principle has come to join it1 that of least action1 put under the form which is suitable for physics4 ,it also has always been verified1 at least in so far as concerns reversible phenomena which thus obey the equations of ,lagrange1 that is to say1 the most general laws of mechanics4 ,irreversible phenomena are much more rebellious4 ,yet these also are being coordinated1 and tend to come into unity2 the light which has illuminated them has come to us from ,carnot's principle4 ,long did thermodynamics confine itself to the study of the dilatation of bodies and their changes of state4 ,for some time past it has been growing bolder and has considerably extended its domain4 ,we owe to it the theory of the galvanic battery and that of the thermoelectric phenomena2 there is not in all physics a corner that it has not explored1 and it has attacked chemistry itself4 ,everywhere the same laws reign2 everywhere1 under the diversity of appearances1 is found again ,carnot's principle2 everywhere also is found that concept so prodigiously abstract of entropy1 which is as universal as that of energy and seems like it to cover a reality4 ,radiant heat seemed destined to escape it2 but recently we have seen that submit to the same laws4 ,in this way fresh analogies are revealed to us1 which may often be followed into detail2 ohmic resistance resembles the viscosity of liquids2 hysteresis would resemble rather the friction of solids4 ,in all cases1 friction would appear to be the type which the most various irreversible phenomena copy1 and this kinship is real and profound4 ,of these phenomena a mechanical explanation1 properly so called1 has also been sought4 ,they hardly lent themselves to it4 ,to find it1 it was necessary to suppose that the irreversibility is only apparent1 that the elementary phenomena are reversible and obey the known laws of dynamics4 ,but the elements are extremely numerous and blend more and more1 so that to our crude sight all appears to tend toward uniformity1 that is1 everything seems to go forward in the same sense without hope of return4 ,the apparent irreversibility is thus only an effect of the law of great numbers4 ,but1 only a being with infinitely subtile senses1 like ,maxwell's imaginary demon1 could disentangle this inextricable skein and turn back the course of the universe4 ,this conception1 which attaches itself to the kinetic theory of gases1 has cost great efforts and has not1 on the whole1 been fruitful2 but it may become so4 ,this is not the place to examine whether it does not lead to contradictions and whether it is in conformity with the true nature of things4 ,we signalize1 however1 ,m4 ,gouy's original ideas on the ,brownian movement4 ,according to this scientist1 this singular motion should escape ,carnot's principle4 ,the particles which it puts in swing would be smaller than the links of that so compacted skein2 they would therefore be fitted to disentangle them and hence to make the world go backward4 ,we should almost see ,maxwell's demon at work4 ,to summarize1 the previously known phenomena are better and better classified1 but new phenomena come to claim their place2 most of these1 like the ,zeeman effect1 have at once found it4 ,but we have the cathode rays1 the ,x-rays1 those of uranium and of radium4 ,herein is a whole world which no one suspected4 ,how many unexpected guests must be stowed away8 ,no one can yet foresee the place they will occupy4 ,but ,i do not believe they will destroy the general unity2 ,i think they will rather complete it4 ,on the one hand1 in fact1 the new radiations seem connected with the phenomena of luminescence2 not only do they excite fluorescence1 but they sometimes take birth in the same conditions as it4 ,nor are they without kinship with the causes which produce the electric spark under the action of the ultra-violet light4 ,finally1 and above all1 it is believed that in all these phenomena are found true ions1 animated1 it is true1 by velocities incomparably greater than in the electrolytes4 ,that is all very vague1 but it will all become more precise4 ,phosphorescence1 the action of light on the spark1 these were regions rather isolated and consequently somewhat neglected by investigators4 ,one may now hope that a new path will be constructed which will facilitate their communications with the rest of science4 ,not only do we discover new phenomena1 but in those we thought we knew1 unforeseen aspects reveal themselves4 ,in the free ether1 the laws retain their majestic simplicity2 but matter1 properly so called1 seems more and more complex2 all that is said of it is never more than approximate1 and at each instant our formulas require new terms4 ,nevertheless the frames are not broken2 the relations that we have recognized between objects we thought simple still subsist between these same objects when we know their complexity1 and it is that alone which is of importance4 ,our equations become1 it is true1 more and more complicated1 in order to embrace more closely the complexity of nature2 but nothing is changed in the relations which permit the deducing of these equations one from another4 ,in a word1 the form of these equations has persisted4 ,take1 for example1 the laws of reflection3 ,fresnel had established them by a simple and seductive theory which experiment seemed to confirm4 ,since then more precise researches have proved that this verification was only approximate2 they have shown everywhere traces of elliptic polarization4 ,but1 thanks to the help that the first approximation gave us1 we found forthwith the cause of these anomalies1 which is the presence of a transition layer2 and ,fresnel's theory has subsisted in its essentials4 ,but there is a reflection we can not help making3 ,all these relations would have remained unperceived if one had at first suspected the complexity of the objects they connect4 ,it has long been said3 ,if ,tycho had had instruments ten times more precise neither ,kepler1 nor ,newton1 nor astronomy would ever have been4 ,it is a misfortune for a science to be born too late1 when the means of observation have become too perfect4 ,this is to-day the case with physical chemistry2 its founders are embarrassed in their general grasp by third and fourth decimals2 happily they are men of a robust faith4 ,the better one knows the properties of matter the more one sees continuity reign4 ,since the labors of ,andrews and of van der ,waals1 we get an idea of how the passage is made from the liquid to the gaseous state and that this passage is not abrupt4 ,similarly1 there is no gap between the liquid and solid states1 and in the proceedings of a recent congress is to be seen1 alongside of a work on the rigidity of liquids1 a memoir on the flow of solids4 ,by this tendency no doubt simplicity loses2 some phenomenon was formerly represented by several straight lines1 now these straights must be joined by curves more or less complicated4 ,in compensation unity gains notably4 ,those cut-off categories quieted the mind1 but they did not satisfy it4 ,finally the methods of physics have invaded a new domain1 that of chemistry2 physical chemistry is born4 ,it is still very young1 but we already see that it will enable us to connect such phenomena as electrolysis1 osmosis and the motions of ions4 ,from this rapid exposition1 what shall we conclude8 ,everything considered1 we have approached unity2 we have not been as quick as was hoped fifty years ago1 we have not always taken the predicted way2 but1 finally1 we have gained ever so much ground4 ,,chapter ,,xi ,,the ,,calculus ,,of ,,probabilities ,doubtless it will be astonishing to find here thoughts about the calculus of probabilities4 ,what has it to do with the method of the physical sciences8 ,and yet the questions ,i shall raise without solving present themselves naturally to the philosopher who is thinking about physics4 ,so far is this the case that in the two preceding chapters ,i have often been led to use the words 'probability' and 'chance4' ',predicted facts1' as ,i have said above1 'can only be probable4' ,however solidly founded a prediction may seem to us to be1 we are never absolutely sure that experiment will not prove it false4 ,but the probability is often so great that practically we may be satisfied with it4 ,and a little further on ,i have added3 ,see what a rle the belief in simplicity plays in our generalizations4 ,we have verified a simple law in a great number of particular cases2 we refuse to admit that this coincidence1 so often repeated1 can be a mere effect of chance4444 ,thus in a multitude of circumstances the physicist is in the same position as the gambler who reckons up his chances4 ,as often as he reasons by induction1 he requires more or less consciously the calculus of probabilities1 and this is why ,i am obliged to introduce a parenthesis1 and interrupt our study of method in the physical sciences in order to examine a little more closely the value of this calculus1 and what confidence it merits4 ,the very name calculus of probabilities is a paradox4 ,probability opposed to certainty is what we do not know1 and how can we calculate what we do not know8 ,yet many eminent savants have occupied themselves with this calculus1 and it can not be denied that science has drawn therefrom no small advantage4 ,how can we explain this apparent contradiction8 ,has probability been defined8 ,can it even be defined8 ,and if it can not1 how dare we reason about it8 ,the definition1 it will be said1 is very simple3 the probability of an event is the ratio of the number of cases favorable to this event to the total number of possible cases4 ,a simple example will show how incomplete this definition is4 ,i throw two dice4 ,what is the probability that one of the two at least turns up a six8 ,each die can turn up in six different ways2 the number of possible cases is #f #f #cf2 the number of favorable cases is #aa2 the probability is #aa_/cf4 ,that is the correct solution4 ,but could ,i not just as well say3 ,the points which turn up on the two dice can form #f #g_/b #ba different combinations8 ,among these combinations #f are favorable2 the probability is #f_/ba4 ,now why is the first method of enumerating the possible cases more legitimate than the second8 ,in any case it is not our definition that tells us4 ,we are therefore obliged to complete this definition by saying3 '444 to the total number of possible cases provided these cases are equally probable4' ,so1 therefore1 we are reduced to defining the probable by the probable4 ,how can we know that two possible cases are equally probable8 ,will it be by a convention8 ,if we place at the beginning of each problem an explicit convention1 well and good4 ,we shall then have nothing to do but apply the rules of arithmetic and of algebra1 and we shall complete our calculation without our result leaving room for doubt4 ,but if we wish to make the slightest application of this result1 we must prove our convention was legitimate1 and we shall find ourselves in the presence of the very difficulty we thought to escape4 ,will it be said that good sense suffices to show us what convention should be adopted8 ,alas6 ,m4 ,bertrand has amused himself by discussing the following simple problem3 ,what is the probability that a chord of a circle may be greater than the side of the inscribed equilateral triangle8 ,the illustrious geometer successively adopted two conventions which good sense seemed equally to dictate and with one he found #a_/b1 with the other #a_/c4 ,the conclusion which seems to follow from all this is that the calculus of probabilities is a useless science1 and that the obscure instinct which we may call good sense1 and to which we are wont to appeal to legitimatize our conventions1 must be distrusted4 ,but neither can we subscribe to this conclusion2 we can not do without this obscure instinct4 ,without it science would be impossible1 without it we could neither discover a law nor apply it4 ,have we the right1 for instance1 to enunciate ,newton's law8 ,without doubt1 numerous observations are in accord with it2 but is not this a simple effect of chance8 ,besides how do we know whether this law1 true for so many centuries1 will still be true next year8 ,to this objection1 you will find nothing to reply1 except3 ',that is very improbable4' ,but grant the law4 ,thanks to it1 ,i believe myself able to calculate the position of ,jupiter a year from now4 ,have ,i the right to believe this8 ,who can tell if a gigantic mass of enormous velocity will not between now and that time pass near the solar system1 and produce unforeseen perturbations8 ,here again the only answer is3 ',it is very improbable4' ,from this point of view1 all the sciences would be only unconscious applications of the calculus of probabilities4 ,to condemn this calculus would be to condemn the whole of science4 ,i shall dwell lightly on the scientific problems in which the intervention of the calculus of probabilities is more evident4 ,in the forefront of these is the problem of interpolation1 in which1 knowing a certain number of values of a function1 we seek to divine the intermediate values4 ,i shall likewise mention3 the celebrated theory of errors of observation1 to which ,i shall return later2 the kinetic theory of gases1 a well-known hypothesis1 wherein each gaseous molecule is supposed to describe an extremely complicated trajectory1 but in which1 through the effect of great numbers1 the mean phenomena1 alone observable1 obey the simple laws of ,mariotte and ,gay-,lussac4 ,all these theories are based on the laws of great numbers1 and the calculus of probabilities would evidently involve them in its ruin4 ,it is true that they have only a particular interest and that1 save as far as interpolation is concerned1 these are sacrifices to which we might readily be resigned4 ,but1 as ,i have said above1 it would not be only these partial sacrifices that would be in question2 it would be the legitimacy of the whole of science that would be challenged4 ,i quite see that it might be said3 ,we are ignorant1 and yet we must act4 ,for action1 we have not time to devote ourselves to an inquiry sufficient to dispel our ignorance4 ,besides1 such an inquiry would demand an infinite time4 ,we must therefore decide without knowing2 we are obliged to do so1 hit or miss1 and we must follow rules without quite believing them4 ,what ,i know is not that such and such a thing is true1 but that the best course for me is to act as if it were true4 ,the calculus of probabilities1 and consequently science itself1 would thenceforth have merely a practical value4 ,unfortunately the difficulty does not thus disappear4 ,a gambler wants to try a coup2 he asks my advice4 ,if ,i give it to him1 ,i shall use the calculus of probabilities1 but ,i shall not guarantee success4 ,this is what ,i shall call subjective probability4 ,in this case1 we might be content with the explanation of which ,i have just given a sketch4 ,but suppose that an observer is present at the game1 that he notes all its coups1 and that the game goes on a long time4 ,when he makes a summary of his book1 he will find that events have taken place in conformity with the laws of the calculus of probabilities4 ,this is what ,i shall call objective probability1 and it is this phenomenon which has to be explained4 ,there are numerous insurance companies which apply the rules of the calculus of probabilities1 and they distribute to their shareholders dividends whose objective reality can not be contested4 ,to invoke our ignorance and the necessity to act does not suffice to explain them4 ,thus absolute skepticism is not admissible4 ,we may distrust1 but we can not condemn en bloc4 ,discussion is necessary4 ,i4 ,,classification ,,of ,,the ,,problems ,,of ,,probability4--,in order to classify the problems which present themselves propos of probabilities1 we may look at them from many different points of view1 and1 first1 from the point of view of generality4 ,i have said above that probability is the ratio of the number of favorable cases to the number of possible cases4 ,what for want of a better term ,i call the generality will increase with the number of possible cases4 ,this number may be finite1 as1 for instance1 if we take a throw of the dice in which the number of possible cases is #cf4 ,that is the first degree of generality4 ,but if we ask1 for example1 what is the probability that a point within a circle is within the inscribed square1 there are as many possible cases as there are points in the circle1 that is to say1 an infinity4 ,this is the second degree of generality4 ,generality can be pushed further still4 ,we may ask the probability that a function will satisfy a given condition4 ,there are then as many possible cases as one can imagine different functions4 ,this is the third degree of generality1 to which we rise1 for instance1 when we seek to find the most probable law in conformity with a finite number of observations4 ,we may place ourselves at a point of view wholly different4 ,if we were not ignorant1 there would be no probability1 there would be room for nothing but certainty4 ,but our ignorance can not be absolute1 for then there would no longer be any probability at all1 since a little light is necessary to attain even this uncertain science4 ,thus the problems of probability may be classed according to the greater or less depth of this ignorance4 ,in mathematics even we may set ourselves problems of probability4 ,what is the probability that the fifth decimal of a logarithm taken at random from a table is a '#i'8 ,there is no hesitation in answering that this probability is #a_/aj2 here we possess all the data of the problem4 ,we can calculate our logarithm without recourse to the table1 but we do not wish to give ourselves the trouble4 ,this is the first degree of ignorance4 ,in the physical sciences our ignorance becomes greater4 ,the state of a system at a given instant depends on two things3 ,its initial state1 and the law according to which that state varies4 ,if we know both this law and this initial state1 we shall have then only a mathematical problem to solve1 and we fall back upon the first degree of ignorance4 ,but it often happens that we know the law1 and do not know the initial state4 ,it may be asked1 for instance1 what is the present distribution of the minor planets8 ,we know that from all time they have obeyed the laws of ,kepler1 but we do not know what was their initial distribution4 ,in the kinetic theory of gases1 we assume that the gaseous molecules follow rectilinear trajectories1 and obey the laws of impact of elastic bodies4 ,but1 as we know nothing of their initial velocities1 we know nothing of their present velocities4 ,the calculus of probabilities only enables us to predict the mean phenomena which will result from the combination of these velocities4 ,this is the second degree of ignorance4 ,finally it is possible that not only the initial conditions but the laws themselves are unknown4 ,we then reach the third degree of ignorance and in general we can no longer affirm anything at all as to the probability of a phenomenon4 ,it often happens that instead of trying to guess an event1 by means of a more or less imperfect knowledge of the law1 the events may be known and we want to find the law2 or that instead of deducing effects from causes1 we wish to deduce the causes from the effects4 ,these are the problems called probability of causes1 the most interesting from the point of view of their scientific applications4 ,i play cart with a gentleman ,i know to be perfectly honest4 ,he is about to deal4 ,what is the probability of his turning up the king8 ,it is #a_/h4 ,this is a problem of the probability of effects4 ,i play with a gentleman whom ,i do not know4 ,he has dealt ten times1 and he has turned up the king six times4 ,what is the probability that he is a sharper8 ,this is a problem in the probability of causes4 ,it may be said that this is the essential problem of the experimental method4 ,i have observed n values of x and the corresponding values of y4 ,i have found that the ratio of the latter to the former is practically constant4 ,there is the event1 what is the cause8 ,is it probable that there is a general law according to which y would be proportional to x1 and that the small divergencies are due to errors of observation8 ,this is a type of question that one is ever asking1 and which we unconsciously solve whenever we are engaged in scientific work4 ,i am now going to pass in review these different categories of problems1 discussing in succession what ,i have called above subjective and objective probability4 ,,ii4 ,,probability ,,in ,,mathematics4--,the impossibility of squaring the circle has been proved since #ahhb2 but even before that date all geometers considered that impossibility as so 'probable1' that the ,academy of ,sciences rejected without examination the alas6 too numerous memoirs on this subject1 that some unhappy madmen sent in every year4 ,was the ,academy wrong8 ,evidently not1 and it knew well that in acting thus it did not run the least risk of stifling a discovery of moment4 ,the ,academy could not have proved that it was right2 but it knew quite well that its instinct was not mistaken4 ,if you had asked the ,academicians1 they would have answered3 ,we have compared the probability that an unknown savant should have found out what has been vainly sought for so long1 with the probability that there is one madman the more on the earth2 the second appears to us the greater4 ,these are very good reasons1 but there is nothing mathematical about them2 they are purely psychological4 ,and if you had pressed them further they would have added3 ,why do you suppose a particular value of a transcendental function to be an algebraic number2 and if pi were a root of an algebraic equation1 why do you suppose this root to be a period of the function sin #bx1 and not the same about the other roots of this same equation8 ,to sum up1 they would have invoked the principle of sufficient reason in its vaguest form4 ,but what could they deduce from it8 ,at most a rule of conduct for the employment of their time1 more usefully spent at their ordinary work than in reading a lucubration that inspired in them a legitimate distrust4 ,but what ,i call above objective probability has nothing in common with this first problem4 ,it is otherwise with the second problem4 ,consider the first #aj1jjj logarithms that we find in a table4 ,among these #aj1jjj logarithms ,i take one at random4 ,what is the probability that its third decimal is an even number8 ,you will not hesitate to answer #a_/b2 and in fact if you pick out in a table the third decimals of these #aj1jjj numbers1 you will find nearly as many even digits as odd4 ,or if you prefer1 let us write #aj1jjj numbers corresponding to our #aj1jjj logarithms1 each of these numbers being #a if the third decimal of the corresponding logarithm is even1 and -#a if odd4 ,then take the mean of these #aj1jjj numbers4 ,i do not hesitate to say that the mean of these #aj1jjj numbers is probably #j1 and if ,i were actually to calculate it ,i should verify that it is extremely small4 ,but even this verification is needless4 ,i might have rigorously proved that this mean is less than #j4jjc4 ,to prove this result1 ,i should have had to make a rather long calculation for which there is no room here1 and for which ,i confine myself to citing an article ,i published in the ,revue gnrale des ,sciences1 ,april #ae1 #ahii4 ,the only point to which ,i wish to call attention is the following3 in this calculation1 ,i should have needed only to rest my case on two facts1 to wit1 that the first and second derivatives of the logarithm remain1 in the interval considered1 between certain limits4 ,hence this important consequence that the property is true not only of the logarithm1 but of any continuous function whatever1 since the derivatives of every continuous function are limited4 ,if ,i was certain beforehand of the result1 it is first1 because ,i had often observed analogous facts for other continuous functions2 and next1 because ,i made in my mind1 in a more or less unconscious and imperfect manner1 the reasoning which led me to the preceding inequalities1 just as a skilled calculator before finishing his multiplication takes into account what it should come to approximately4 ,and besides1 since what ,i call my intuition was only an incomplete summary of a piece of true reasoning1 it is clear why observation has confirmed my predictions1 and why the objective probability has been in agreement with the subjective probability4 ,as a third example ,i shall choose the following problem3 ,a number u is taken at random1 and n is a given very large integer4 ,what is the probable value of sin nu8 ,this problem has no meaning by itself4 ,to give it one a convention is needed4 ,we shall agree that the probability for the number u to lie between a and a is equal to phi7a7da2 that it is therefore proportional to the infinitely small interval da1 and equal to this multiplied by a function phi7a7 depending only on a4 ,as for this function1 ,i choose it arbitrarily1 but ,i must assume it to be continuous4 ,the value of sin nu remaining the same when u increases by #bpi1 ,i may without loss of generality assume that u lies between #j and #bpi1 and ,i shall thus be led to suppose that phi7a7 is a periodic function whose period is #bpi4 ,the probable value sought is readily expressed by a simple integral1 and it is easy to show that this integral is less than #bpi,mk_/nk1 ,mk being the maximum value of the kth derivative of phi7u74 ,we see then that if the kth derivative is finite1 our probable value will tend toward #j when n increases indefinitely1 and that more rapidly than #a_/nk - #a4 ,the probable value of sin nu when n is very large is therefore naught4 ,to define this value ,i required a convention2 but the result remains the same whatever that convention may be4 ,i have imposed upon myself only slight restrictions in assuming that the function phi7a7 is continuous and periodic1 and these hypotheses are so natural that we may ask ourselves how they can be escaped4 ,examination of the three preceding examples1 so different in all respects1 has already given us a glimpse1 on the one hand1 of the rle of what philosophers call the principle of sufficient reason1 and1 on the other hand1 of the importance of the fact that certain properties are common to all continuous functions4 ,the study of probability in the physical sciences will lead us to the same result4 ,,iii4 ,,probability ,,in ,,the ,,physical ,,sciences4--,we come now to the problems connected with what ,i have called the second degree of ignorance1 those1 namely1 in which we know the law1 but do not know the initial state of the system4 ,i could multiply examples1 but will take only one4 ,what is the probable present distribution of the minor planets on the zodiac8 ,we know they obey the laws of ,kepler4 ,we may even1 without at all changing the nature of the problem1 suppose that their orbits are all circular1 and situated in the same plane1 and that we know this plane4 ,on the other hand1 we are in absolute ignorance as to what was their initial distribution4 ,however1 we do not hesitate to affirm that their distribution is now nearly uniform4 ,why8 ,let b be the longitude of a minor planet in the initial epoch1 that is to say1 the epoch zero4 ,let a be its mean motion4 ,its longitude at the present epoch1 that is to say at the epoch t1 will be at b4 ,to say that the present distribution is uniform is to say that the mean value of the sines and cosines of multiples of at b is zero4 ,why do we assert this8 ,let us represent each minor planet by a point in a plane1 to wit1 by a point whose coordinates are precisely a and b4 ,all these representative points will be contained in a certain region of the plane1 but as they are very numerous this region will appear dotted with points4 ,we know nothing else about the distribution of these points4 ,what do we do when we wish to apply the calculus of probabilities to such a question8 ,what is the probability that one or more representative points may be found in a certain portion of the plane8 ,in our ignorance1 we are reduced to making an arbitrary hypothesis4 ,to explain the nature of this hypothesis1 allow me to use1 in lieu of a mathematical formula1 a crude but concrete image4 ,let us suppose that over the surface of our plane has been spread an imaginary substance1 whose density is variable1 but varies continuously4 ,we shall then agree to say that the probable number of representative points to be found on a portion of the plane is proportional to the quantity of fictitious matter found there4 ,if we have then two regions of the plane of the same extent1 the probabilities that a representative point of one of our minor planets is found in one or the other of these regions will be to one another as the mean densities of the fictitious matter in the one and the other region4 ,here then are two distributions1 one real1 in which the representative points are very numerous1 very close together1 but discrete like the molecules of matter in the atomic hypothesis2 the other remote from reality1 in which our representative points are replaced by continuous fictitious matter4 ,we know that the latter can not be real1 but our ignorance forces us to adopt it4 ,if again we had some idea of the real distribution of the representative points1 we could arrange it so that in a region of some extent the density of this imaginary continuous matter would be nearly proportional to the number of the representative points1 or1 if you wish1 to the number of atoms which are contained in that region4 ,even that is impossible1 and our ignorance is so great that we are forced to choose arbitrarily the function which defines the density of our imaginary matter4 ,only we shall be forced to a hypothesis from which we can hardly get away1 we shall suppose that this function is continuous4 ,that is sufficient1 as we shall see1 to enable us to reach a conclusion4 ,what is at the instant t the probable distribution of the minor planets8 ,or rather what is the probable value of the sine of the longitude at the instant t1 that is to say of sin 7at b78 ,we made at the outset an arbitrary convention1 but if we adopt it1 this probable value is entirely defined4 ,divide the plane into elements of surface4 ,consider the value of sin 7at b7 at the center of each of these elements2 multiply this value by the surface of the element1 and by the corresponding density of the imaginary matter4 ,take then the sum for all the elements of the plane4 ,this sum1 by definition1 will be the probable mean value we seek1 which will thus be expressed by a double integral4 ,it may be thought at first that this mean value depends on the choice of the function which defines the density of the imaginary matter1 and that1 as this function phi is arbitrary1 we can1 according to the arbitrary choice which we make1 obtain any mean value4 ,this is not so4 ,a simple calculation shows that our double integral decreases very rapidly when t increases4 ,thus ,i could not quite tell what hypothesis to make as to the probability of this or that initial distribution2 but whatever the hypothesis made1 the result will be the same1 and this gets me out of my difficulty4 ,whatever be the function phi1 the mean value tends toward zero as t increases1 and as the minor planets have certainly accomplished a very great number of revolutions1 ,i may assert that this mean value is very small4 ,i may choose phi as ,i wish1 save always one restriction3 this function must be continuous2 and1 in fact1 from the point of view of subjective probability1 the choice of a discontinuous function would have been unreasonable4 ,for instance1 what reason could ,i have for supposing that the initial longitude might be exactly #j1 but that it could not lie between #j and #a8 ,but the difficulty reappears if we take the point of view of objective probability1 if we pass from our imaginary distribution in which the fictitious matter was supposed continuous to the real distribution in which our representative points form1 as it were1 discrete atoms4 ,the mean value of sin 7at b7 will be represented quite simply by 7#a_/n7,sigma sin 7at b71 n being the number of minor planets4 ,in lieu of a double integral referring to a continuous function1 we shall have a sum of discrete terms4 ,and yet no one will seriously doubt that this mean value is practically very small4 ,our representative points being very close together1 our discrete sum will in general differ very little from an integral4 ,an integral is the limit toward which a sum of terms tends when the number of these terms is indefinitely increased4 ,if the terms are very numerous1 the sum will differ very little from its limit1 that is to say from the integral1 and what ,i said of this latter will still be true of the sum itself4 ,nevertheless1 there are exceptions4 ,if1 for instance1 for all the minor planets1 b pi_/#b - at1 the longitude for all the planets at the time t would be pi_/#b1 and the mean value would evidently be equal to unity4 ,for this to be the case1 it would be necessary that at the epoch #j1 the minor planets must have all been lying on a spiral of peculiar form1 with its spires very close together4 ,every one will admit that such an initial distribution is extremely improbable 7and1 even supposing it realized1 the distribution would not be uniform at the present time1 for example1 on ,january #a1 #aiac1 but it would become so a few years later74 ,why then do we think this initial distribution improbable8 ,this must be explained1 because if we had no reason for rejecting as improbable this absurd hypothesis everything would break down1 and we could no longer make any affirmation about the probability of this or that present distribution4 ,once more we shall invoke the principle of sufficient reason to which we must always recur4 ,we might admit that at the beginning the planets were distributed almost in a straight line4 ,we might admit that they were irregularly distributed4 ,but it seems to us that there is no sufficient reason for the unknown cause that gave them birth to have acted along a curve so regular and yet so complicated1 which would appear to have been expressly chosen so that the present distribution would not be uniform4 ,,iv4 ,,rouge ,,et ,,noir4--,the questions raised by games of chance1 such as roulette1 are1 fundamentally1 entirely analogous to those we have just treated4 ,for example1 a wheel is partitioned into a great number of equal subdivisions1 alternately red and black4 ,a needle is whirled with force1 and after having made a great number of revolutions1 it stops before one of these subdivisions4 ,the probability that this division is red is evidently #a_/b4 ,the needle describes an angle theta1 including several complete revolutions4 ,i do not know what is the probability that the needle may be whirled with a force such that this angle should lie between theta and thetadtheta2 but ,i can make a convention4 ,i can suppose that this probability is phi7theta7dtheta4 ,as for the function phi7theta71 ,i can choose it in an entirely arbitrary manner4 ,there is nothing that can guide me in my choice1 but ,i am naturally led to suppose this function continuous4 ,let epsilon be the length 7measured on the circumference of radius #a7 of each red and black subdivision4 ,we have to calculate the integral of phi7theta7dtheta1 extending it1 on the one hand1 to all the red divisions and1 on the other hand1 to all the black divisions1 and to compare the results4 ,consider an interval #b;epsilon1 comprising a red division and a black division which follows it4 ,let ,m and m be the greatest and least values of the function phi7theta7 in this interval4 ,the integral extended to the red divisions will be smaller than ,sigma,mepsilon2 the integral extended to the black divisions will be greater than ,sigmamepsilon2 the difference will therefore be less than ,sigma7,m - m7epsilon4 ,but1 if the function theta is supposed continuous2 if1 besides1 the interval epsilon is very small with respect to the total angle described by the needle1 the difference ,m - m will be very small4 ,the difference of the two integrals will therefore be very small1 and the probability will be very nearly #a_/b4 ,we see that without knowing anything of the function theta1 ,i must act as if the probability were #a_/b4 ,we understand1 on the other hand1 why1 if1 placing myself at the objective point of view1 ,i observe a certain number of coups1 observation will give me about as many black coups as red4 ,all players know this objective law2 but it leads them into a remarkable error1 which has been often exposed1 but into which they always fall again4 ,when the red has won1 for instance1 six times running1 they bet on the black1 thinking they are playing a safe game2 because1 say they1 it is very rare that red wins seven times running4 ,in reality their probability of winning remains #a_/b4 ,observation shows1 it is true1 that series of seven consecutive reds are very rare1 but series of six reds followed by a black are just as rare4 ,they have noticed the rarity of the series of seven reds2 if they have not remarked the rarity of six reds and a black1 it is only because such series strike the attention less4 ,v4 ,,the ,,probability ,,of ,,causes4--,we now come to the problems of the probability of causes1 the most important from the point of view of scientific applications4 ,two stars1 for instance1 are very close together on the celestial sphere4 ,is this apparent contiguity a mere effect of chance8 ,are these stars1 although on almost the same visual ray1 situated at very different distances from the earth1 and consequently very far from one another8 ,or1 perhaps1 does the apparent correspond to a real contiguity8 ,this is a problem on the probability of causes4 ,i recall first that at the outset of all problems of the probability of effects that have hitherto occupied us1 we have always had to make a convention1 more or less justified4 ,and if in most cases the result was1 in a certain measure1 independent of this convention1 this was only because of certain hypotheses which permitted us to reject a priori discontinuous functions1 for example1 or certain absurd conventions4 ,we shall find something analogous when we deal with the probability of causes4 ,an effect may be produced by the cause ,a or by the cause ,b4 ,the effect has just been observed4 ,we ask the probability that it is due to the cause ,a4 ,this is an a posteriori probability of cause4 ,but ,i could not calculate it1 if a convention more or less justified did not tell me in advance what is the a priori probability for the cause ,a to come into play2 ,i mean the probability of this event for some one who had not observed the effect4 ,the better to explain myself ,i go back to the example of the game of cart mentioned above4 ,my adversary deals for the first time and he turns up a king4 ,what is the probability that he is a sharper8 ,the formulas ordinarily taught give #h_/i1 a result evidently rather surprising4 ,if we look at it closer1 we see that the calculation is made as if1 before sitting down at the table1 ,i had considered that there was one chance in two that my adversary was not honest4 ,an absurd hypothesis1 because in that case ,i should have certainly not played with him1 and this explains the absurdity of the conclusion4 ,the convention about the a priori probability was unjustified1 and that is why the calculation of the a posteriori probability led me to an inadmissible result4 ,we see the importance of this preliminary convention4 ,i shall even add that if none were made1 the problem of the a posteriori probability would have no meaning4 ,it must always be made either explicitly or tacitly4 ,pass to an example of a more scientific character4 ,i wish to determine an experimental law4 ,this law1 when ,i know it1 can be represented by a curve4 ,i make a certain number of isolated observations2 each of these will be represented by a point4 ,when ,i have obtained these different points1 ,i draw a curve between them1 striving to pass as near to them as possible and yet preserve for my curve a regular form1 without angular points1 or inflections too accentuated1 or brusque variation of the radius of curvature4 ,this curve will represent for me the probable law1 and ,i assume not only that it will tell me the values of the function intermediate between those which have been observed1 but also that it will give me the observed values themselves more exactly than direct observation4 ,this is why ,i make it pass near the points1 and not through the points themselves4 ,here is a problem in the probability of causes4 ,the effects are the measurements ,i have recorded2 they depend on a combination of two causes3 the true law of the phenomenon and the errors of observation4 ,knowing the effects1 we have to seek the probability that the phenomenon obeys this law or that1 and that the observations have been affected by this or that error4 ,the most probable law then corresponds to the curve traced1 and the most probable error of an observation is represented by the distance of the corresponding point from this curve4 ,but the problem would have no meaning if1 before any observation1 ,i had not fashioned an a priori idea of the probability of this or that law1 and of the chances of error to which ,i am exposed4 ,if my instruments are good 7and that ,i knew before making the observations71 ,i shall not permit my curve to depart much from the points which represent the rough measurements4 ,if they are bad1 ,i may go a little further away from them in order to obtain a less sinuous curve2 ,i shall sacrifice more to regularity4 ,why then is it that ,i seek to trace a curve without sinuosities8 ,it is because ,i consider a priori a law represented by a continuous function 7or by a function whose derivatives of high order are small71 as more probable than a law not satisfying these conditions4 ,without this belief1 the problem of which we speak would have no meaning2 interpolation would be impossible2 no law could be deduced from a finite number of observations2 science would not exist4 ,fifty years ago physicists considered1 other things being equal1 a simple law as more probable than a complicated law4 ,they even invoked this principle in favor of ,mariotte's law as against the experiments of ,regnault4 ,to-day they have repudiated this belief2 and yet1 how many times are they compelled to act as though they still held it6 ,however that may be1 what remains of this tendency is the belief in continuity1 and we have just seen that if this belief were to disappear in its turn1 experimental science would become impossible4 ,,vi4 ,,the ,,theory ,,of ,,errors4--,we are thus led to speak of the theory of errors1 which is directly connected with the problem of the probability of causes4 ,here again we find effects1 to wit1 a certain number of discordant observations1 and we seek to divine the causes1 which are1 on the one hand1 the real value of the quantity to be measured2 on the other hand1 the error made in each isolated observation4 ,it is necessary to calculate what is a posteriori the probable magnitude of each error1 and consequently the probable value of the quantity to be measured4 ,but as ,i have just explained1 we should not know how to undertake this calculation if we did not admit a priori1 that is to say1 before all observation1 a law of probability of errors4 ,is there a law of errors8 ,the law of errors admitted by all calculators is ,gauss's law1 which is represented by a certain transcendental curve known under the name of 'the bell4' ,but first it is proper to recall the classic distinction between systematic and accidental errors4 ,if we measure a length with too long a meter1 we shall always find too small a number1 and it will be of no use to measure several times2 this is a systematic error4 ,if we measure with an accurate meter1 we may1 however1 make a mistake2 but we go wrong1 now too much1 now too little1 and when we take the mean of a great number of measurements1 the error will tend to grow small4 ,these are accidental errors4 ,it is evident from the first that systematic errors can not satisfy ,gauss's law2 but do the accidental errors satisfy it8 ,a great number of demonstrations have been attempted2 almost all are crude paralogisms4 ,nevertheless1 we may demonstrate ,gauss's law by starting from the following hypotheses3 the error committed is the result of a great number of partial and independent errors2 each of the partial errors is very little and besides1 obeys any law of probability1 provided that the probability of a positive error is the same as that of an equal negative error4 ,it is evident that these conditions will be often but not always fulfilled1 and we may reserve the name of accidental for errors which satisfy them4 ,we see that the method of least squares is not legitimate in every case2 in general the physicists are more distrustful of it than the astronomers4 ,this is1 no doubt1 because the latter1 besides the systematic errors to which they and the physicists are subject alike1 have to control with an extremely important source of error which is wholly accidental2 ,i mean atmospheric undulations4 ,so it is very curious to hear a physicist discuss with an astronomer about a method of observation4 ,the physicist1 persuaded that one good measurement is worth more than many bad ones1 is before all concerned with eliminating by dint of precautions the least systematic errors1 and the astronomer says to him3 ',but thus you can observe only a small number of stars2 the accidental errors will not disappear4' ,what should we conclude8 ,must we continue to use the method of least squares8 ,we must distinguish4 ,we have eliminated all the systematic errors we could suspect2 we know well there are still others1 but we can not detect them2 yet it is necessary to make up our mind and adopt a definitive value which will be regarded as the probable value2 and for that it is evident the best thing to do is to apply ,gauss's method4 ,we have only applied a practical rule referring to subjective probability4 ,there is nothing more to be said4 ,but we wish to go farther and affirm that not only is the probable value so much1 but that the probable error in the result is so much4 ,this is absolutely illegitimate2 it would be true only if we were sure that all the systematic errors were eliminated1 and of that we know absolutely nothing4 ,we have two series of observations2 by applying the rule of least squares1 we find that the probable error in the first series is twice as small as in the second4 ,the second series may1 however1 be better than the first1 because the first perhaps is affected by a large systematic error4 ,all we can say is that the first series is probably better than the second1 since its accidental error is smaller1 and we have no reason to affirm that the systematic error is greater for one of the series than for the other1 our ignorance on this point being absolute4 ,,vii4 ,,conclusions4--,in the lines which precede1 ,i have set many problems without solving any of them4 ,yet ,i do not regret having written them1 because they will perhaps invite the reader to reflect on these delicate questions4 ,however that may be1 there are certain points which seem well established4 ,to undertake any calculation of probability1 and even for that calculation to have any meaning1 it is necessary to admit1 as point of departure1 a hypothesis or convention which has always something arbitrary about it4 ,in the choice of this convention1 we can be guided only by the principle of sufficient reason4 ,unfortunately this principle is very vague and very elastic1 and in the cursory examination we have just made1 we have seen it take many different forms4 ,the form under which we have met it most often is the belief in continuity1 a belief which it would be difficult to justify by apodeictic reasoning1 but without which all science would be impossible4 ,finally the problems to which the calculus of probabilities may be applied with profit are those in which the result is independent of the hypothesis made at the outset1 provided only that this hypothesis satisfies the condition of continuity4 ,,chapter ,,xii ,,optics ,,and ,,electricity ,,fresnel',s ,,theory4--,the best example#e that can be chosen of physics in the making is the theory of light and its relations to the theory of electricity4 ,thanks to ,fresnel1 optics is the best developed part of physics2 the so-called wave-theory forms a whole truly satisfying to the mind4 ,we must not1 however1 ask of it what it can not give us4 #e ,this chapter is a partial reproduction of the prefaces of two of my works3 ,thorie mathmatique de la lumire 7,paris1 ,naud1 #ahhi71 and lectricit et optique 7,paris1 ,naud1 #aija74 ,the object of mathematical theories is not to reveal to us the true nature of things2 this would be an unreasonable pretension4 ,their sole aim is to coordinate the physical laws which experiment reveals to us1 but which1 without the help of mathematics1 we should not be able even to state4 ,it matters little whether the ether really exists2 that is the affair of metaphysicians4 ,the essential thing for us is that everything happens as if it existed1 and that this hypothesis is convenient for the explanation of phenomena4 ,after all1 have we any other reason to believe in the existence of material objects8 ,that1 too1 is only a convenient hypothesis2 only this will never cease to be so1 whereas1 no doubt1 some day the ether will be thrown aside as useless4 ,but even at that day1 the laws of optics and the equations which translate them analytically will remain true1 at least as a first approximation4 ,it will always be useful1 then1 to study a doctrine that unites all these equations4 ,the undulatory theory rests on a molecular hypothesis4 ,for those who think they have thus discovered the cause under the law1 this is an advantage4 ,for the others it is a reason for distrust4 ,but this distrust seems to me as little justified as the illusion of the former4 ,these hypotheses play only a secondary part4 ,they might be sacrificed4 ,they usually are not1 because then the explanation would lose in clearness2 but that is the only reason4 ,in fact1 if we looked closer we should see that only two things are borrowed from the molecular hypotheses3 the principle of the conservation of energy and the linear form of the equations1 which is the general law of small movements1 as of all small variations4 ,this explains why most of ,fresnel's conclusions remain unchanged when we adopt the electromagnetic theory of light4 ,,maxwell',s ,,theory4--,maxwell1 we know1 connected by a close bond two parts of physics until then entirely foreign to one another1 optics and electricity4 ,by blending thus in a vaster whole1 in a higher harmony1 the optics of ,fresnel has not ceased to be alive4 ,its various parts subsist1 and their mutual relations are still the same4 ,only the language we used to express them has changed2 and1 on the other hand1 ,maxwell has revealed to us other relations1 before unsuspected1 between the different parts of optics and the domain of electricity4 ,when a ,french reader first opens ,maxwell's book1 a feeling of uneasiness and often even of mistrust mingles at first with his admiration4 ,only after a prolonged acquaintance and at the cost of many efforts does this feeling disappear4 ,there are even some eminent minds that never lose it4 ,why are the ,english scientist's ideas with such difficulty acclimatized among us8 ,it is1 no doubt1 because the education received by the majority of enlightened ,frenchmen predisposes them to appreciate precision and logic above every other quality4 ,the old theories of mathematical physics gave us in this respect complete satisfaction4 ,all our masters1 from ,laplace to ,cauchy1 have proceeded in the same way4 ,starting from clearly stated hypotheses1 they deduced all their consequences with mathematical rigor1 and then compared them with experiment4 ,it seemed their aim to give every branch of physics the same precision as celestial mechanics4 ,a mind accustomed to admire such models is hard to suit with a theory4 ,not only will it not tolerate the least appearance of contradiction1 but it will demand that the various parts be logically connected with one another1 and that the number of distinct hypotheses be reduced to minimum4 ,this is not all2 it will have still other demands1 which seem to me less reasonable4 ,behind the matter which our senses can reach1 and which experiment tells us of1 it will desire to see another1 and in its eyes the only real1 matter1 which will have only purely geometric properties1 and whose atoms will be nothing but mathematical points1 subject to the laws of dynamics alone4 ,and yet these atoms1 invisible and without color1 it will seek by an unconscious contradiction to represent to itself and consequently to identify as closely as possible with common matter4 ,then only will it be fully satisfied and imagine that it has penetrated the secret of the universe4 ,if this satisfaction is deceitful1 it is none the less difficult to renounce4 ,thus1 on opening ,maxwell1 a ,frenchman expects to find a theoretical whole as logical and precise as the physical optics based on the hypothesis of the ether2 he thus prepares for himself a disappointment which ,i should like to spare the reader by informing him immediately of what he must look for in ,maxwell1 and what he can not find there4 ,maxwell does not give a mechanical explanation of electricity and magnetism2 he confines himself to demonstrating that such an explanation is possible4 ,he shows also that optical phenomena are only a special case of electromagnetic phenomena4 ,from every theory of electricity1 one can therefore deduce immediately a theory of light4 ,the converse unfortunately is not true2 from a complete explanation of light1 it is not always easy to derive a complete explanation of electric phenomena4 ,this is not easy1 in particular1 if we wish to start from ,fresnel's theory4 ,doubtless it would not be impossible2 but nevertheless we must ask whether we are not going to be forced to renounce admirable results that we thought definitely acquired4 ,that seems a step backward2 and many good minds are not willing to submit to it4 ,when the reader shall have consented to limit his hopes1 he will still encounter other difficulties4 ,the ,english scientist does not try to construct a single edifice1 final and well ordered2 he seems rather to erect a great number of provisional and independent constructions1 between which communication is difficult and sometimes impossible4 ,take as example the chapter in which he explains electrostatic attractions by pressures and tensions in the dielectric medium4 ,this chapter might be omitted without making thereby the rest of the book less clear or complete2 and1 on the other hand1 it contains a theory complete in itself which one could understand without having read a single line that precedes or follows4 ,but it is not only independent of the rest of the work2 it is difficult to reconcile with the fundamental ideas of the book4 ,maxwell does not even attempt this reconciliation2 he merely says3 ,i have not been able to make the next step1 namely1 to account by mechanical considerations for these stresses in the dielectric4 ,this example will suffice to make my thought understood2 ,i could cite many others4 ,thus who would suspect1 in reading the pages devoted to magnetic rotary polarization1 that there is an identity between optical and magnetic phenomena8 ,one must not then flatter himself that he can avoid all contradiction2 to that it is necessary to be resigned4 ,in fact1 two contradictory theories1 provided one does not mingle them1 and if one does not seek in them the basis of things1 may both be useful instruments of research2 and perhaps the reading of ,maxwell would be less suggestive if he had not opened up to us so many new and divergent paths4 ,the fundamental idea1 however1 is thus a little obscured4 ,so far is this the case that in the majority of popularized versions it is the only point completely left aside4 ,i feel1 then1 that the better to make its importance stand out1 ,i ought to explain in what this fundamental idea consists4 ,but for that a short digression is necessary4 ,,the ,,mechanical ,,explanation ,,of ,,physical ,,phenomena4--,there is in every physical phenomenon a certain number of parameters which experiment reaches directly and allows us to measure4 ,i shall call these the parameters q4 ,observation then teaches us the laws of the variations of these parameters2 and these laws can generally be put in the form of differential equations1 which connect the parameters q with the time4 ,what is it necessary to do to give a mechanical interpretation of such a phenomenon8 ,one will try to explain it either by the motions of ordinary matter1 or by those of one or more hypothetical fluids4 ,these fluids will be considered as formed of a very great number of isolated molecules m4 ,when shall we say1 then1 that we have a complete mechanical explanation of the phenomenon8 ,it will be1 on the one hand1 when we know the differential equations satisfied by the coordinates of these hypothetical molecules m1 equations which1 moreover1 must conform to the principles of dynamics2 and1 on the other hand1 when we know the relations that define the coordinates of the molecules m as functions of the parameters q accessible to experiment4 ,these equations1 as ,i have said1 must conform to the principles of dynamics1 and1 in particular1 to the principle of the conservation of energy and the principle of least action4 ,the first of these two principles teaches us that the total energy is constant and that this energy is divided into two parts3 #a ,the kinetic energy1 or vis viva1 which depends on the masses of the hypothetical molecules m1 and their velocities1 and which ,i shall call ,t4 #b ,the potential energy1 which depends only on the coordinates of these molecules and which ,i shall call ,u4 ,it is the sum of the two energies ,t and ,u which is constant4 ,what now does the principle of least action tell us8 ,it tells us that to pass from the initial position occupied at the instant t#j to the final position occupied at the instant t#a1 the system must take such a path that1 in the interval of time that elapses between the two instants t#j and t#a1 the average value of 'the action' 7that is to say1 of the difference between the two energies ,t and ,u7 shall be as small as possible4 ,if the two functions ,t and ,u are known1 this principle suffices to determine the equations of motion4 ,among all the possible ways of passing from one position to another1 there is evidently one for which the average value of the action is less than for any other4 ,there is1 moreover1 only one2 and it results from this that the principle of least action suffices to determine the path followed and consequently the equations of motion4 ,thus we obtain what are called the equations of ,lagrange4 ,in these equations1 the independent variables are the coordinates of the hypothetical molecules m2 but ,i now suppose that one takes as variables the parameters q directly accessible to experiment4 ,the two parts of the energy must then be expressed as functions of the parameters q and of their derivatives4 ,they will evidently appear under this form to the experimenter4 ,the latter will naturally try to define the potential and the kinetic energy by the aid of quantities that he can directly observe4#f #f ,we add that ,u will depend only on the parameters q1 that ,t will depend on the parameters q and their derivatives with respect to the time and will be a homogeneous polynomial of the second degree with respect to these derivatives4 ,that granted1 the system will always go from one position to another by a path such that the average action shall be a minimum4 ,it matters little that ,t and ,u are now expressed by the aid of the parameters q and their derivatives2 it matters little that it is also by means of these parameters that we define the initial and final positions2 the principle of least action remains always true4 ,now here again1 of all the paths that lead from one position to another1 there is one for which the average action is a minimum1 and there is only one4 ,the principle of least action suffices1 then1 to determine the differential equations which define the variations of the parameters q4 ,the equations thus obtained are another form of the equations of ,lagrange4 ,to form these equations we need to know neither the relations that connect the parameters q with the coordinates of the hypothetical molecules1 nor the masses of these molecules1 nor the expression of ,u as a function of the coordinates of these molecules4 ,all we need to know is the expression of ,u as a function of the parameters1 and that of ,t as a function of the parameters q and their derivatives1 that is1 the expressions of the kinetic and of the potential energy as functions of the experimental data4 ,then we shall have one of two things3 either for a suitable choice of the functions ,t and ,u1 the equations of ,lagrange1 constructed as we have just said1 will be identical with the differential equations deduced from experiments2 or else there will exist no functions ,t and ,u1 for which this agreement takes place4 ,in the latter case it is clear that no mechanical explanation is possible4 ,the necessary condition for a mechanical explanation to be possible is therefore that we can choose the functions ,t and ,u in such a way as to satisfy the principle of least action1 which involves that of the conservation of energy4 ,this condition1 moreover1 is sufficient4 ,suppose1 in fact1 that we have found a function ,u of the parameters q1 which represents one of the parts of the energy2 that another part of the energy1 which we shall represent by ,t1 is a function of the parameters q and their derivatives1 and that it is a homogeneous polynomial of the second degree with respect to these derivatives2 and finally that the equations of ,lagrange1 formed by means of these two functions1 ,t and ,u1 conform to the data of the experiment4 ,what is necessary in order to deduce from this a mechanical explanation8 ,it is necessary that ,u can be regarded as the potential energy of a system and ,t as the vis viva of the same system4 ,there is no difficulty as to ,u1 but can ,t be regarded as the vis viva of a material system8 ,it is easy to show that this is always possible1 and even in an infinity of ways4 ,i will confine myself to referring for more details to the preface of my work1 'lectricit et optique4' ,thus if the principle of least action can not be satisfied1 no mechanical explanation is possible2 if it can be satisfied1 there is not only one1 but an infinity1 whence it follows that as soon as there is one there is an infinity of others4 ,one more observation4 ,among the quantities that experiment gives us directly1 we shall regard some as functions of the coordinates of our hypothetical molecules2 these are our parameters q4 ,we shall look upon the others as dependent not only on the coordinates1 but on the velocities1 or1 what comes to the same thing1 on the derivatives of the parameters q1 or as combinations of these parameters and their derivatives4 ,and then a question presents itself3 among all these quantities measured experimentally1 which shall we choose to represent the parameters q8 ,which shall we prefer to regard as the derivatives of these parameters8 ,this choice remains arbitrary to a very large extent2 but1 for a mechanical explanation to be possible1 it suffices if we can make the choice in such a way as to accord with the principle of least action4 ,and then ,maxwell asked himself whether he could make this choice and that of the two energies ,t and ,u1 in such a way that the electrical phenomena would satisfy this principle4 ,experiment shows us that the energy of an electromagnetic field is decomposed into two parts1 the electrostatic energy and the electrodynamic energy4 ,maxwell observed that if we regard the first as representing the potential energy ,u1 the second as representing the kinetic energy ,t2 if1 moreover1 the electrostatic charges of the conductors are considered as parameters q and the intensities of the currents as the derivatives of other parameters q2 under these conditions1 ,i say1 ,maxwell observed that the electric phenomena satisfy the principle of least action4 ,thenceforth he was certain of the possibility of a mechanical explanation4 ,if he had explained this idea at the beginning of his book instead of relegating it to an obscure part of the second volume1 it would not have escaped the majority of readers4 ,if1 then1 a phenomenon admits of a complete mechanical explanation1 it will admit of an infinity of others1 that will render an account equally well of all the particulars revealed by experiment4 ,and this is confirmed by the history of every branch of physics2 in optics1 for instance1 ,fresnel believed vibration to be perpendicular to the plane of polarization2 ,neumann regarded it as parallel to this plane4 ,an 'experimentum crucis' has long been sought which would enable us to decide between these two theories1 but it has not been found4 ,in the same way1 without leaving the domain of electricity1 we may ascertain that the theory of two fluids and that of the single fluid both account in a fashion equally satisfactory for all the observed laws of electrostatics4 ,all these facts are easily explicable1 thanks to the properties of the equations of ,lagrange which ,i have just recalled4 ,it is easy now to comprehend what is ,maxwell's fundamental idea4 ,to demonstrate the possibility of a mechanical explanation of electricity1 we need not preoccupy ourselves with finding this explanation itself2 it suffices us to know the expression of the two functions ,t and ,u1 which are the two parts of energy1 to form with these two functions the equations of ,lagrange and then to compare these equations with the experimental laws4 ,among all these possible explanations1 how make a choice for which the aid of experiment fails us8 ,a day will come perhaps when physicists will not interest themselves in these questions1 inaccessible to positive methods1 and will abandon them to the metaphysicians4 ,this day has not yet arrived2 man does not resign himself so easily to be forever ignorant of the foundation of things4 ,our choice can therefore be further guided only by considerations where the part of personal appreciation is very great2 there are1 however1 solutions that all the world will reject because of their whimsicality1 and others that all the world will prefer because of their simplicity4 ,in what concerns electricity and magnetism1 ,maxwell abstains from making any choice4 ,it is not that he systematically disdains all that is unattainable by positive methods2 the time he has devoted to the kinetic theory of gases sufficiently proves that4 ,i will add that if1 in his great work1 he develops no complete explanation1 he had previously attempted to give one in an article in the ,philosophical ,magazine4 ,the strangeness and the complexity of the hypotheses he had been obliged to make had led him afterwards to give this up4 ,the same spirit is found throughout the whole work4 ,what is essential1 that is to say what must remain common to all theories1 is made prominent2 all that would only be suitable to a particular theory is nearly always passed over in silence4 ,thus the reader finds himself in the presence of a form almost devoid of matter1 which he is at first tempted to take for a fugitive shadow not to be grasped4 ,but the efforts to which he is thus condemned force him to think and he ends by comprehending what was often rather artificial in the theoretic constructs he had previously only wondered at4 ,,chapter ,,xiii ,,electrodynamics ,the history of electrodynamics is particularly instructive from our point of view4 ,ampre entitled his immortal work1 ',thorie des phnomnes lectrodynamiques1 uniquement fonde sur l'exprience4' ,he therefore imagined that he had made no hypothesis1 but he had made them1 as we shall soon see2 only he made them without being conscious of it4 ,his successors1 on the other hand1 perceived them1 since their attention was attracted by the weak points in ,ampre's solution4 ,they made new hypotheses1 of which this time they were fully conscious2 but how many times it was necessary to change them before arriving at the classic system of to-day which is perhaps not yet final2 this we shall see4 ,i4 ,,ampere',s ,,theory4--,when ,ampre studied experimentally the mutual actions of currents1 he operated and he only could operate with closed currents4 ,it was not that he denied the possibility of open currents4 ,if two conductors are charged with positive and negative electricity and brought into communication by a wire1 a current is established going from one to the other1 which continues until the two potentials are equal4 ,according to the ideas of ,ampre's time this was an open current2 the current was known to go from the first conductor to the second1 it was not seen to return from the second to the first4 ,so ,ampre considered as open currents of this nature1 for example1 the currents of discharge of condensers2 but he could not make them the objects of his experiments because their duration is too short4 ,another sort of open current may also be imagined4 ,i suppose two conductors1 ,a and ,b1 connected by a wire ,,amb4 ,small conducting masses in motion first come in contact with the conductor ,b1 take from it an electric charge1 leave contact with ,b and move along the path ,,bna1 and1 transporting with them their charge1 come into contact with ,a and give to it their charge1 which returns then to ,b along the wire ,,amb4 ,now there we have in a sense a closed circuit1 since the electricity describes the closed circuit ,,bnamb2 but the two parts of this current are very different4 ,in the wire ,,amb1 the electricity is displaced through a fixed conductor1 like a voltaic current1 overcoming an ohmic resistance and developing heat2 we say that it is displaced by conduction4 ,in the part ,,bna1 the electricity is carried by a moving conductor2 it is said to be displaced by convection4 ,if then the current of convection is considered as altogether analogous to the current of conduction1 the circuit ,,bnamb is closed2 if1 on the contrary1 the convection current is not 'a true current' and1 for example1 does not act on the magnet1 there remains only the conduction current ,,amb1 which is open4 ,for example1 if we connect by a wire the two poles of a ,holtz machine1 the charged rotating disc transfers the electricity by convection from one pole to the other1 and it returns to the first pole by conduction through the wire4 ,but currents of this sort are very difficult to produce with appreciable intensity4 ,with the means at ,ampre's disposal1 we may say that this was impossible4 ,to sum up1 ,ampre could conceive of the existence of two kinds of open currents1 but he could operate on neither because they were not strong enough or because their duration was too short4 ,experiment therefore could only show him the action of a closed current on a closed current1 or1 more accurately1 the action of a closed current on a portion of a current1 because a current can be made to describe a closed circuit composed of a moving part and a fixed part4 ,it is possible then to study the displacements of the moving part under the action of another closed current4 ,on the other hand1 ,ampre had no means of studying the action of an open current1 either on a closed current or another open current4 #a4 ,the ,case of ,closed ,currents4--,in the case of the mutual action of two closed currents1 experiment revealed to ,ampre remarkably simple laws4 ,i recall rapidly here those which will be useful to us in the sequel3 #a ,if the intensity of the currents is kept constant1 and if the two circuits1 after having undergone any deformations and displacements whatsoever1 return finally to their initial positions1 the total work of the electrodynamic actions will be null4 ,in other words1 there is an electrodynamic potential of the two circuits1 proportional to the product of the intensities1 and depending on the form and relative position of the circuits2 the work of the electrodynamic actions is equal to the variation of this potential4 #b ,the action of a closed solenoid is null4 #c ,the action of a circuit ,c on another voltaic circuit ,c' depends only on the 'magnetic field' developed by this circuit4 ,at each point in space we can in fact define in magnitude and direction a certain force called magnetic force1 which enjoys the following properties3 7a7 ,the force exercised by ,c on a magnetic pole is applied to that pole and is equal to the magnetic force multiplied by the magnetic mass of that pole2 7b7 ,a very short magnetic needle tends to take the direction of the magnetic force1 and the couple to which it tends to reduce is proportional to the magnetic force1 the magnetic moment of the needle and the sine of the dip of the needle2 7c7 ,if the circuit ,c is displaced1 the work of the electrodynamic action exercised by ,c on ,c' will be equal to the increment of the 'flow of magnetic force' which passes through the circuit4 #b4 ,action of a ,closed ,current on a ,portion of ,current4--,ampre not having been able to produce an open current1 properly so called1 had only one way of studying the action of a closed current on a portion of current4 ,this was by operating on a circuit ,c composed of two parts1 the one fixed1 the other movable4 ,the movable part was1 for instance1 a movable wire alphabeta whose extremities alpha and beta could slide along a fixed wire4 ,in one of the positions of the movable wire1 the end alpha rested on the ,a of the fixed wire and the extremity beta on the point ,b of the fixed wire4 ,the current circulated from alpha to beta1 that is to say1 from ,a to ,b along the movable wire1 and then it returned from ,b to ,a along the fixed wire4 ,this current was therefore closed4 ,in a second position1 the movable wire having slipped1 the extremity alpha rested on another point ,a' of the fixed wire1 and the extremity beta on another point ,b' of the fixed wire4 ,the current circulated then from alpha to beta1 that is to say from ,a' to ,b' along the movable wire1 and it afterwards returned from ,b' to ,b1 then from ,b to ,a1 then finally from ,a to ,a'1 always following the fixed wire4 ,the current was therefore also closed4 ,if a like current is subjected to the action of a closed current ,c1 the movable part will be displaced just as if it were acted upon by a force4 ,ampre assumes that the apparent force to which this movable part ,,ab seems thus subjected1 representing the action of the ,c on the portion alphabeta of the current1 is the same as if alphabeta were traversed by an open current1 stopping at alpha and beta1 in place of being traversed by a closed current which after arriving at beta returns to alpha through the fixed part of the circuit4 ,this hypothesis seems natural enough1 and ,ampre made it unconsciously2 nevertheless it is not necessary1 since we shall see further on that ,helmholtz rejected it4 ,however that may be1 it permitted ,ampre1 though he had never been able to produce an open current1 to enunciate the laws of the action of a closed current on an open current1 or even on an element of current4 ,the laws are simple3 #a ,the force which acts on an element of current is applied to this element2 it is normal to the element and to the magnetic force1 and proportional to the component of this magnetic force which is normal to the element4 #b ,the action of a closed solenoid on an element of current is null4 ,but the electrodynamic potential has disappeared1 that is to say that1 when a closed current and an open current1 whose intensities have been maintained constant1 return to their initial positions1 the total work is not null4 #c4 ,continuous ,rotations4--,among electrodynamic experiments1 the most remarkable are those in which continuous rotations are produced and which are sometimes called unipolar induction experiments4 ,a magnet may turn about its axis2 a current passes first through a fixed wire1 enters the magnet by the pole ,n1 for example1 passes through half the magnet1 emerges by a sliding contact and reenters the fixed wire4 ,the magnet then begins to rotate continuously without being able ever to attain equilibrium2 this is ,faraday's experiment4 ,how is it possible8 ,if it were a question of two circuits of invariable form1 the one ,c fixed1 the other ,c' movable about an axis1 this latter could never take on continuous rotation2 in fact there is an electrodynamic potential2 there must therefore be necessarily a position of equilibrium when this potential is a maximum4 ,continuous rotations are therefore possible only when the circuit ,c' is composed of two parts3 one fixed1 the other movable about an axis1 as is the case in ,faraday's experiment4 ,here again it is convenient to draw a distinction4 ,the passage from the fixed to the movable part1 or inversely1 may take place either by simple contact 7the same point of the movable part remaining constantly in contact with the same point of the fixed part71 or by a sliding contact 7the same point of the movable part coming successively in contact with diverse points of the fixed part74 ,it is only in the second case that there can be continuous rotation4 ,this is what then happens3 ,the system tends to take a position of equilibrium2 but1 when at the point of reaching that position1 the sliding contact puts the movable part in communication with a new point of the fixed part2 it changes the connections1 it changes therefore the conditions of equilibrium1 so that the position of equilibrium fleeing1 so to say1 before the system which seeks to attain it1 rotation may take place indefinitely4 ,ampre assumes that the action of the circuit on the movable part of ,c' is the same as if the fixed part of ,c' did not exist1 and therefore as if the current passing through the movable part were open4 ,he concludes therefore that the action of a closed on an open current1 or inversely that of an open current on a closed current1 may give rise to a continuous rotation4 ,but this conclusion depends on the hypothesis ,i have enunciated and which1 as ,i said above1 is not admitted by ,helmholtz4 #d4 ,mutual ,action of ,two ,open ,currents4--,in what concerns the mutual actions of two open currents1 and in particular that of two elements of current1 all experiment breaks down4 ,ampre has recourse to hypothesis4 ,he supposes3 #a ,that the mutual action of two elements reduces to a force acting along their join2 #b ,that the action of two closed currents is the resultant of the mutual actions of their diverse elements1 which are besides the same as if these elements were isolated4 ,what is remarkable is that here again ,ampre makes these hypotheses unconsciously4 ,however that may be1 these two hypotheses1 together with the experiments on closed currents1 suffice to determine completely the law of the mutual action of two elements4 ,but then most of the simple laws we have met in the case of closed currents are no longer true4 ,in the first place1 there is no electrodynamic potential2 nor was there any1 as we have seen1 in the case of a closed current acting on an open current4 ,next there is1 properly speaking1 no magnetic force4 ,and1 in fact1 we have given above three different definitions of this force3 #a ,by the action on a magnetic pole2 #b ,by the director couple which orientates the magnetic needle2 #c ,by the action on an element of current4 ,but in the case which now occupies us1 not only these three definitions are no longer in harmony1 but each has lost its meaning1 and in fact3 #a ,a magnetic pole is no longer acted upon simply by a single force applied to this pole4 ,we have seen in fact that the force due to the action of an element of current on a pole is not applied to the pole1 but to the element2 it may moreover be replaced by a force applied to the pole and by a couple2 #b ,the couple which acts on the magnetic needle is no longer a simple director couple1 for its moment with respect to the axis of the needle is not null4 ,it breaks up into a director couple1 properly so called1 and a supplementary couple which tends to produce the continuous rotation of which we have above spoken2 #c ,finally the force acting on an element of current is not normal to this element4 ,in other words1 the unity of the magnetic force has disappeared4 ,let us see in what this unity consists4 ,two systems which exercise the same action on a magnetic pole will exert also the same action on an indefinitely small magnetic needle1 or on an element of current placed at the same point of space as this pole4 ,well1 this is true if these two systems contain only closed currents2 this would no longer be true if these two systems contained open currents4 ,it suffices to remark1 for instance1 that1 if a magnetic pole is placed at ,a and an element at ,b1 the direction of the element being along the prolongation of the sect ,,ab1 this element which will exercise no action on this pole will1 on the other hand1 exercise an action either on a magnetic needle placed at the point ,a1 or on an element of current placed at the point ,a4 #e4 ,induction4--,we know that the discovery of electrodynamic induction soon followed the immortal work of ,ampre4 ,as long as it is only a question of closed currents there is no difficulty1 and ,helmholtz has even remarked that the principle of the conservation of energy is sufficient for deducing the laws of induction from the electrodynamic laws of ,ampre4 ,but always on one condition1 as ,bertrand has well shown2 that we make besides a certain number of hypotheses4 ,the same principle again permits this deduction in the case of open currents1 although of course we can not submit the result to the test of experiment1 since we can not produce such currents4 ,if we try to apply this mode of analysis to ,ampre's theory of open currents1 we reach results calculated to surprise us4 ,in the first place1 induction can not be deduced from the variation of the magnetic field by the formula well known to savants and practicians1 and1 in fact1 as we have said1 properly speaking there is no longer a magnetic field4 ,but1 further1 if a circuit ,c is subjected to the induction of a variable voltaic system ,s1 if this system ,s be displaced and deformed in any way whatever1 so that the intensity of the currents of this system varies according to any law whatever1 but that after these variations the system finally returns to its initial situation1 it seems natural to suppose that the mean electromotive force induced in the circuit ,c is null4 ,this is true if the circuit ,c is closed and if the system ,s contains only closed currents4 ,this would no longer be true1 if one accepts the theory of ,ampre1 if there were open currents4 ,so that not only induction will no longer be the variation of the flow of magnetic force1 in any of the usual senses of the word1 but it can not be represented by the variation of anything whatever4 ,,ii4 ,,theory ,,of ,,helmholtz4--,i have dwelt upon the consequences of ,ampre's theory1 and of his method of explaining open currents4 ,it is difficult to overlook the paradoxical and artificial character of the propositions to which we are thus led4 ,one can not help thinking 'that can not be so4' ,we understand therefore why ,helmholtz was led to seek something else4 ,helmholtz rejects ,ampre's fundamental hypothesis1 to wit1 that the mutual action of two elements of current reduces to a force along their join4 ,he assumes that an element of current is not subjected to a single force1 but to a force and a couple4 ,it is just this which gave rise to the celebrated polemic between ,bertrand and ,helmholtz4 ,helmholtz replaces ,ampre's hypothesis by the following3 two elements always admit of an electrodynamic potential depending solely on their position and orientation2 and the work of the forces that they exercise1 one on the other1 is equal to the variation of this potential4 ,thus ,helmholtz can no more do without hypothesis than ,ampre2 but at least he does not make one without explicitly announcing it4 ,in the case of closed currents1 which are alone accessible to experiment1 the two theories agree4 ,in all other cases they differ4 ,in the first place1 contrary to what ,ampre supposed1 the force which seems to act on the movable portion of a closed current is not the same as would act upon this movable portion if it were isolated and constituted an open current4 ,let us return to the circuit ,c'1 of which we spoke above1 and which was formed of a movable wire alphabeta sliding on a fixed wire4 ,in the only experiment that can be made1 the movable portion alphabeta is not isolated1 but is part of a closed circuit4 ,when it passes from ,,ab to ,a',b'1 the total electrodynamic potential varies for two reasons3 #a ,it undergoes a first increase because the potential of ,a',b' with respect to the circuit ,c is not the same as that of ,,ab2 #b ,it takes a second increment because it must be increased by the potentials of the elements ,,aa'1 ,,bb' with respect to ,c4 ,it is this double increment which represents the work of the force to which the portion ,,ab seems subjected4 ,if1 on the contrary1 alphabeta were isolated1 the potential would undergo only the first increase1 and this first increment alone would measure the work of the force which acts on ,,ab4 ,in the second place1 there could be no continuous rotation without sliding contact1 and1 in fact1 that1 as we have seen propos of closed currents1 is an immediate consequence of the existence of an electrodynamic potential4 ,in ,faraday's experiment1 if the magnet is fixed and if the part of the current exterior to the magnet runs along a movable wire1 that movable part may undergo a continuous rotation4 ,but this does not mean to say that if the contacts of the wire with the magnet were suppressed1 and an open current were to run along the wire1 the wire would still take a movement of continuous rotation4 ,i have just said in fact that an isolated element is not acted upon in the same way as a movable element making part of a closed circuit4 ,another difference3 ,the action of a closed solenoid on a closed current is null according to experiment and according to the two theories4 ,its action on an open current would be null according to ,ampre2 it would not be null according to ,helmholtz4 ,from this follows an important consequence4 ,we have given above three definitions of magnetic force4 ,the third has no meaning here since an element of current is no longer acted upon by a single force4 ,no more has the first any meaning4 ,what1 in fact1 is a magnetic pole8 ,it is the extremity of an indefinite linear magnet4 ,this magnet may be replaced by an indefinite solenoid4 ,for the definition of magnetic force to have any meaning1 it would be necessary that the action exercised by an open current on an indefinite solenoid should depend only on the position of the extremity of this solenoid1 that is to say1 that the action on a closed solenoid should be null4 ,now we have just seen that such is not the case4 ,on the other hand1 nothing prevents our adopting the second definition1 which is founded on the measurement of the director couple which tends to orientate the magnetic needle4 ,but if it is adopted1 neither the effects of induction nor the electrodynamic effects will depend solely on the distribution of the lines of force in this magnetic field4 ,,iii4 ,,difficulties ,,raised ,,by ,,these ,,theories4--,the theory of ,helmholtz is in advance of that of ,ampre2 it is necessary1 however1 that all the difficulties should be smoothed away4 ,in the one as in the other1 the phrase 'magnetic field' has no meaning1 or1 if we give it one1 by a more or less artificial convention1 the ordinary laws so familiar to all electricians no longer apply2 thus the electromotive force induced in a wire is no longer measured by the number of lines of force met by this wire4 ,and our repugnance does not come alone from the difficulty of renouncing inveterate habits of language and of thought4 ,there is something more4 ,if we do not believe in action at a distance1 electrodynamic phenomena must be explained by a modification of the medium4 ,it is precisely this modification that we call 'magnetic field4' ,and then the electrodynamic effects must depend only on this field4 ,all these difficulties arise from the hypothesis of open currents4 ,,iv4 ,,maxwell',s ,,theory4--,such were the difficulties raised by the dominant theories when ,maxwell appeared1 who with a stroke of the pen made them all vanish4 ,to his mind1 in fact1 all currents are closed currents4 ,maxwell assumes that if in a dielectric the electric field happens to vary1 this dielectric becomes the seat of a particular phenomenon1 acting on the galvanometer like a current1 and which he calls current of displacement4 ,if then two conductors bearing contrary charges are put in communication by a wire1 in this wire during the discharge there is an open current of conduction2 but there are produced at the same time in the surrounding dielectric1 currents of displacement which close this current of conduction4 ,we know that ,maxwell's theory leads to the explanation of optical phenomena1 which would be due to extremely rapid electrical oscillations4 ,at that epoch such a conception was only a bold hypothesis1 which could be supported by no experiment4 ,at the end of twenty years1 ,maxwell's ideas received the confirmation of experiment4 ,hertz succeeded in producing systems of electric oscillations which reproduce all the properties of light1 and only differ from it by the length of their wave2 that is to say as violet differs from red4 ,in some measure he made the synthesis of light4 ,it might be said that ,hertz has not demonstrated directly ,maxwell's fundamental idea1 the action of the current of displacement on the galvanometer4 ,this is true in a sense4 ,what he has shown in sum is that electromagnetic induction is not propagated instantaneously as was supposed2 but with the speed of light4 ,but to suppose there is no current of displacement1 and induction is propagated with the speed of light2 or to suppose that the currents of displacement produce effects of induction1 and that the induction is propagated instantaneously1 comes to the same thing4 ,this can not be seen at the first glance1 but it is proved by an analysis of which ,i must not think of giving even a summary here4 ,v4 ,,rowland',s ,,experiment4--,but as ,i have said above1 there are two kinds of open conduction currents4 ,there are first the currents of discharge of a condenser or of any conductor whatever4 ,there are also the cases in which electric discharges describe a closed contour1 being displaced by conduction in one part of the circuit and by convection in the other part4 ,for open currents of the first sort1 the question might be considered as solved2 they were closed by the currents of displacement4 ,for open currents of the second sort1 the solution appeared still more simple4 ,it seemed that if the current were closed1 it could only be by the current of convection itself4 ,for that it sufficed to assume that a 'convection current1' that is to say a charged conductor in motion1 could act on the galvanometer4 ,but experimental confirmation was lacking4 ,it appeared difficult in fact to obtain a sufficient intensity even by augmenting as much as possible the charge and the velocity of the conductors4 ,it was ,rowland1 an extremely skillful experimenter1 who first triumphed over these difficulties4 ,a disc received a strong electrostatic charge and a very great speed of rotation4 ,an astatic magnetic system placed beside the disc underwent deviations4 ,the experiment was made twice by ,rowland1 once in ,berlin1 once in ,baltimore4 ,it was afterwards repeated by ,himstedt4 ,these physicists even announced that they had succeeded in making quantitative measurements4 ,in fact1 for twenty years ,rowland's law was admitted without objection by all physicists4 ,besides everything seemed to confirm it4 ,the spark certainly does produce a magnetic effect4 ,now does it not seem probable that the discharge by spark is due to particles taken from one of the electrodes and transferred to the other electrode with their charge8 ,is not the very spectrum of the spark1 in which we recognize the lines of the metal of the electrode1 a proof of it8 ,the spark would then be a veritable current of convection4 ,on the other hand1 it is also admitted that in an electrolyte the electricity is carried by the ions in motion4 ,the current in an electrolyte would therefore be also a current of convection2 now1 it acts on the magnetic needle4 ,the same for cathode rays4 ,crookes attributed these rays to a very subtile matter charged with electricity and moving with a very great velocity4 ,he regarded them1 in other words1 as currents of convection4 ,now these cathode rays are deviated by the magnet4 ,in virtue of the principle of action and reaction1 they should in turn deviate the magnetic needle4 ,it is true that ,hertz believed he had demonstrated that the cathode rays do not carry electricity1 and that they do not act on the magnetic needle4 ,but ,hertz was mistaken4 ,first of all1 ,perrin succeeded in collecting the electricity carried by these rays1 electricity of which ,hertz denied the existence2 the ,german scientist appears to have been deceived by effects due to the action of ,x-rays1 which were not yet discovered4 ,afterwards1 and quite recently1 the action of the cathode rays on the magnetic needle has been put in evidence4 ,thus all these phenomena regarded as currents of convection1 sparks1 electrolytic currents1 cathode rays1 act in the same manner on the galvanometer and in conformity with ,rowland's law4 ,,vi4 ,,theory ,,of ,,lorentz4--,we soon went farther4 ,according to the theory of ,lorentz1 currents of conduction themselves would be true currents of convection4 ,electricity would remain inseparably connected with certain material particles called electrons4 ,the circulation of these electrons through bodies would produce voltaic currents4 ,and what would distinguish conductors from insulators would be that the one could be traversed by these electrons while the others would arrest their movements4 ,the theory of ,lorentz is very attractive4 ,it gives a very simple explanation of certain phenomena which the earlier theories1 even ,maxwell's in its primitive form1 could not explain in a satisfactory way2 for example1 the aberration of light1 the partial carrying away of luminous waves1 magnetic polarization and the ,zeeman effect4 ,some objections still remained4 ,the phenomena of an electric system seemed to depend on the absolute velocity of translation of the center of gravity of this system1 which is contrary to the idea we have of the relativity of space4 ,supported by ,m4 ,crmieu1 ,m4 ,lippmann has presented this objection in a striking form4 ,imagine two charged conductors with the same velocity of translation2 they are relatively at rest4 ,however1 each of them being equivalent to a current of convection1 they ought to attract one another1 and by measuring this attraction we could measure their absolute velocity4 ,no6 replied the partisans of ,lorentz4 ,what we could measure in that way is not their absolute velocity1 but their relative velocity with respect to the ether1 so that the principle of relativity is safe4 ,whatever there may be in these latter objections1 the edifice of electrodynamics1 at least in its broad lines1 seemed definitively constructed4 ,everything was presented under the most satisfactory aspect4 ,the theories of ,ampre and of ,helmholtz1 made for open currents which no longer existed1 seemed to have no longer anything but a purely historic interest1 and the inextricable complications to which these theories led were almost forgotten4 ,this quiescence has been recently disturbed by the experiments of ,m4 ,crmieu1 which for a moment seemed to contradict the result previously obtained by ,rowland4 ,but fresh researches have not confirmed them1 and the theory of ,lorentz has victoriously stood the test4 ,the history of these variations will be none the less instructive2 it will teach us to what pitfalls the scientist is exposed1 and how he may hope to escape them4 99 99 99 99 99 ,,the ,,value ,,of ,,science 99 99 99 99 99 ,,translator',s ,,introduction #a4 ,does the ,scientist create ,science8--,professor ,rados of ,budapest in his report to the ,hungarian ,academy of ,science on the award to ,poincar of the ,bolyai prize of ten thousand crowns1 speaking of him as unquestionably the most powerful investigator in the domain of mathematics and mathematical physics1 characterized him as the intuitive genius drawing the inspiration for his wide-reaching researches from the exhaustless fountain of geometric and physical intuition1 yet working this inspiration out in detail with marvelous logical keenness4 ,with his brilliant creative genius was combined the capacity for sharp and successful generalization1 pushing far out the boundaries of thought in the most widely different domains1 so that his works must be ranked with the greatest mathematical achievements of all time4 ,finally1 says ,rados1 permit me to make especial mention of his intensely interesting book1 ',the ,value of ,science1' in which he in a way has laid down the scientist's creed4 ,now what is this creed8 ,sense may act as stimulus1 as suggestive1 yet not to awaken a dormant depiction1 or to educe the conception of an archetypal form1 but rather to strike the hour for creation1 to summon to work a sculptor capable of smoothing a ,venus of ,milo out of the formless clay4 ,knowledge is not a gift of bare experience1 nor even made solely out of experience4 ,the creative activity of mind is in mathematics particularly clear4 ,the axioms of geometry are conventions1 disguised definitions or unprovable hypotheses precreated by auto-active animal and human minds4 ,bertrand ,russell says of projective geometry3 ,it takes nothing from experience1 and has1 like arithmetic1 a creature of the pure intellect for its object4 ,it deals with an object whose properties are logically deduced from its definition1 not empirically discovered from data4 ,then does the scientist create science8 ,this is a question ,poincar here dissects with a master hand4 ,the physiologic-psychologic investigation of the space problem must give the meaning of the words geometric fact1 geometric reality4 ,poincar here subjects to the most successful analysis ever made the tridimensionality of our space4 #b4 ,the ,mind ,dispelling ,optical ,illusions4--,actual perception of spatial properties is accompanied by movements corresponding to its character4 ,in the case of optical illusions1 with the so-called false perceptions eye-movements are closely related4 ,but though the perceived object and its environment remain constant1 the sufficiently powerful mind can1 as we say1 dispel these illusions1 the perception itself being creatively changed4 ,photo-graphs taken at intervals during the presence of these optical illusions1 during the change1 perhaps gradual and unconscious1 in the perception1 and after these illusions have1 as the phrase is1 finally disappeared1 show quite clearly that changes in eye-movements corresponding to those internally created in perception itself successively occur4 ,what is called accuracy of movement is created by what is called correctness of perception4 ,the higher creation in the perception is the determining cause of an improvement1 a precision in the motion4 ,thus we see correct perception in the individual helping to make that cerebral organization and accurate motor adjustment on which its possibility and permanence seem in so far to depend4 ,so-called correct perception is connected with a long-continued process of perceptual education motived and initiated from within4 ,how this may take place is here illustrated at length by our author4 #c4 ,euclid not ,necessary4--,geometry is a construction of the intellect1 in application not certain but convenient4 ,as ,schiller says1 when we see these facts as clearly as the development of metageometry has compelled us to see them1 we must surely confess that the ,kantian account of space is hopelessly and demonstrably antiquated4 ,as ,royce says in ',kant's ,doctrine of the ,basis of ,mathematics1' ,that very use of intuition which ,kant regarded as geometrically ideal1 the modern geometer regards as scientifically defective1 because surreptitious4 ,no mathematical exactness without explicit proof from assumed principles--such is the motto of the modern geometer4 ,but suppose the reasoning of ,euclid purified of this comparatively surreptitious appeal to intuition4 ,suppose that the principles of geometry are made quite explicit at the outset of the treatise1 as ,pieri and ,hilbert or ,professor ,halsted or ,dr4 ,veblen makes his principles explicit in his recent treatment of geometry4 ,then1 indeed1 geometry becomes for the modern mathematician a purely rational science4 ,but very few students of the logic of mathematics at the present time can see any warrant in the analysis of geometrical truth for regarding just the ,euclidean system of principles as possessing any discoverable necessity4 ,yet the environmental and perhaps hereditary premiums on ,euclid still make even the scientist think ,euclid most convenient4 #d4 ,without ,hypotheses1 no ,science4--,nobody ever observed an equidistantial1 but also nobody ever observed a straight line4 ,emerson's ,uriel ,gave his sentiment divine ,against the being of a line4 ,line in ,nature is not found4 ,clearly not1 being an eject from man's mind4 ,what is called 'a knowledge of facts' is usually merely a subjective realization that the old hypotheses are still sufficiently elastic to serve in some domain2 that is1 with a sufficiency of conscious or unconscious omissions and doctorings and fudgings more or less wilful4 ,in the present book we see the very foundation rocks of science1 the conservation of energy and the indestructibility of matter1 beating against the bars of their cages1 seemingly anxious to take wing away into the empyrean1 to chase the once divine parallel postulate broken loose from ,euclid and ,kant4 #e4 ,what ,outcome8--,what now is the definite1 the permanent outcome8 ,what new islets raise their fronded palms in air within thought's musical domain8 ,over what age-gray barriers rise the fragrant floods of this new spring-tide1 redolent of the wolf-haunted forest of ,transylvania1 of far ,erdly's plunging river1 ,maros the bitter1 or broad mother ,volga at ,kazan8 ,what victory heralded the great rocket for which young ,lobachevski1 the widow's son1 was cast into prison8 ,what severing of age-old mental fetters symbolized young ,bolyai's cutting-off with his ,damascus blade the spikes driven into his door-post1 and strewing over the sod the thirteen ,austrian cavalry officers8 ,this book by the greatest mathematician of our time gives weightiest and most charming answer4 ,,george ,,bruce ,,halsted4 ,,introduction ,the search for truth should be the goal of our activities2 it is the sole end worthy of them4 ,doubtless we should first bend our efforts to assuage human suffering1 but why8 ,not to suffer is a negative ideal more surely attained by the annihilation of the world4 ,if we wish more and more to free man from material cares1 it is that he may be able to employ the liberty obtained in the study and contemplation of truth4 ,but sometimes truth frightens us4 ,and in fact we know that it is sometimes deceptive1 that it is a phantom never showing itself for a moment except to ceaselessly flee1 that it must be pursued further and ever further without ever being attained4 ,yet to work one must stop1 as some ,greek1 ,aristotle or another1 has said4 ,we also know how cruel the truth often is1 and we wonder whether illusion is not more consoling1 yea1 even more bracing1 for illusion it is which gives confidence4 ,when it shall have vanished1 will hope remain and shall we have the courage to achieve8 ,thus would not the horse harnessed to his treadmill refuse to go1 were his eyes not bandaged8 ,and then to seek truth it is necessary to be independent1 wholly independent4 ,if1 on the contrary1 we wish to act1 to be strong1 we should be united4 ,this is why many of us fear truth2 we consider it a cause of weakness4 ,yet truth should not be feared1 for it alone is beautiful4 ,when ,i speak here of truth1 assuredly ,i refer first to scientific truth2 but ,i also mean moral truth1 of which what we call justice is only one aspect4 ,it may seem that ,i am misusing words1 that ,i combine thus under the same name two things having nothing in common2 that scientific truth1 which is demonstrated1 can in no way be likened to moral truth1 which is felt4 ,and yet ,i can not separate them1 and whosoever loves the one can not help loving the other4 ,to find the one1 as well as to find the other1 it is necessary to free the soul completely from prejudice and from passion2 it is necessary to attain absolute sincerity4 ,these two sorts of truth when discovered give the same joy2 each when perceived beams with the same splendor1 so that we must see it or close our eyes4 ,lastly1 both attract us and flee from us2 they are never fixed3 when we think to have reached them1 we find that we have still to advance1 and he who pursues them is condemned never to know repose4 ,it must be added that those who fear the one will also fear the other2 for they are the ones who in everything are concerned above all with consequences4 ,in a word1 ,i liken the two truths1 because the same reasons make us love them and because the same reasons make us fear them4 ,if we ought not to fear moral truth1 still less should we dread scientific truth4 ,in the first place it can not conflict with ethics4 ,ethics and science have their own domains1 which touch but do not interpenetrate4 ,the one shows us to what goal we should aspire1 the other1 given the goal1 teaches us how to attain it4 ,so they can never conflict since they can never meet4 ,there can no more be immoral science than there can be scientific morals4 ,but if science is feared1 it is above all because it can not give us happiness4 ,of course it can not4 ,we may even ask whether the beast does not suffer less than man4 ,but can we regret that earthly paradise where man brute-like was really immortal in knowing not that he must die8 ,when we have tasted the apple1 no suffering can make us forget its savor4 ,we always come back to it4 ,could it be otherwise8 ,as well ask if one who has seen and is blind will not long for the light4 ,man1 then1 can not be happy through science1 but to-day he can much less be happy without it4 ,but if truth be the sole aim worth pursuing1 may we hope to attain it8 ,it may well be doubted4 ,readers of my little book ',science and ,hypothesis' already know what ,i think about the question4 ,the truth we are permitted to glimpse is not altogether what most men call by that name4 ,does this mean that our most legitimate1 most imperative aspiration is at the same time the most vain8 ,or can we1 despite all1 approach truth on some side8 ,this it is which must be investigated4 ,in the first place1 what instrument have we at our disposal for this conquest8 ,is not human intelligence1 more specifically the intelligence of the scientist1 susceptible of infinite variation8 ,volumes could be written without exhausting this subject2 ,i1 in a few brief pages1 have only touched it lightly4 ,that the geometer's mind is not like the physicist's or the naturalist's1 all the world would agree2 but mathematicians themselves do not resemble each other2 some recognize only implacable logic1 others appeal to intuition and see in it the only source of discovery4 ,and this would be a reason for distrust4 ,to minds so unlike can the mathematical theorems themselves appear in the same light8 ,truth which is not the same for all1 is it truth8 ,but looking at things more closely1 we see how these very different workers collaborate in a common task which could not be achieved without their cooperation4 ,and that already reassures us4 ,next must be examined the frames in which nature seems enclosed and which are called time and space4 ,in ',science and ,hypothesis' ,i have already shown how relative their value is2 it is not nature which imposes them upon us1 it is we who impose them upon nature because we find them convenient4 ,but ,i have spoken of scarcely more than space1 and particularly quantitative space1 so to say1 that is of the mathematical relations whose aggregate constitutes geometry4 ,i should have shown that it is the same with time as with space and still the same with 'qualitative space'2 in particular1 ,i should have investigated why we attribute three dimensions to space4 ,i may be pardoned then for taking up again these important questions4 ,is mathematical analysis1 then1 whose principal object is the study of these empty frames1 only a vain play of the mind8 ,it can give to the physicist only a convenient language2 is this not a mediocre service1 which1 strictly speaking1 could be done without2 and even is it not to be feared that this artificial language may be a veil interposed between reality and the eye of the physicist8 ,far from it2 without this language 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 is1 we shall see1 the only true objective reality4 ,the best expression of this harmony is law4 ,law is one of the most recent conquests of the human mind2 there still are people who live in the presence of a perpetual miracle and are not astonished at it4 ,on the contrary1 we it is who should be astonished at nature's regularity4 ,men demand of their gods to prove their existence by miracles2 but the eternal marvel is that there are not miracles without cease4 ,the world is divine because it is a harmony4 ,if it were ruled by caprice1 what could prove to us it was not ruled by chance8 ,this conquest of law we owe to astronomy1 and just this makes the grandeur of the science rather than the material grandeur of the objects it considers4 ,it was altogether natural1 then1 that celestial mechanics should be the first model of mathematical physics2 but since then this science has developed2 it is still developing1 even rapidly developing4 ,and it is already necessary to modify in certain points the scheme from which ,i drew two chapters of ',science and ,hypothesis4' ,in an address at the ,st4 ,louis exposition1 ,i sought to survey the road traveled2 the result of this investigation the reader shall see farther on4 ,the progress of science has seemed to imperil the best established principles1 those even which were regarded as fundamental4 ,yet nothing shows they will not be saved2 and if this comes about only imperfectly1 they will still subsist even though they are modified4 ,the advance of science is not comparable to the changes of a city1 where old edifices are pitilessly torn down to give place to new1 but to the continuous evolution of zoologic types which develop ceaselessly and end by becoming unrecognizable to the common sight1 but where an expert eye finds always traces of the prior work of the centuries past4 ,one must not think then that the old-fashioned theories have been sterile and vain4 ,were we to stop there1 we should find in these pages some reasons for confidence in the value of science1 but many more for distrusting it2 an impression of doubt would remain2 it is needful now to set things to rights4 ,some people have exaggerated the rle of convention in science2 they have even gone so far as to say that law1 that scientific fact itself1 was created by the scientist4 ,this is going much too far in the direction of nominalism4 ,no1 scientific laws are not artificial creations2 we have no reason to regard them as accidental1 though it be impossible to prove they are not4 ,does the harmony the human intelligence thinks it discovers in nature exist outside of this intelligence8 ,no1 beyond doubt a reality completely independent of the mind which conceives it1 sees or feels it1 is an impossibility4 ,a world as exterior as that1 even if it existed1 would for us be forever inaccessible4 ,but what we call objective reality is1 in the last analysis1 what is common to many thinking beings1 and could be common to all2 this common part1 we shall see1 can only be the harmony expressed by mathematical laws4 ,it is this harmony then which 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 ,,part ,i ,,the ,,mathematical ,,sciences ,,chapter ,i ,,intuition ,,and ,,logic ,,in ,,mathematics ,i ,it is impossible to study the works of the great mathematicians1 or even those of the lesser1 without noticing and distinguishing two opposite tendencies1 or rather two entirely different kinds of minds4 ,the one sort are above all preoccupied with logic2 to read their works1 one is tempted to believe they have advanced only step by step1 after the manner of a ,vauban who pushes on his trenches against the place besieged1 leaving nothing to chance4 ,the other sort are guided by intuition and at the first stroke make quick but sometimes precarious conquests1 like bold cavalrymen of the advance guard4 ,the method is not imposed by the matter treated4 ,though one often says of the first that they are analysts and calls the others geometers1 that does not prevent the one sort from remaining analysts even when they work at geometry1 while the others are still geometers even when they occupy themselves with pure analysis4 ,it is the very nature of their mind which makes them logicians or intuitionalists1 and they can not lay it aside when they approach a new subject4 ,nor is it education which has developed in them one of the two tendencies and stifled the other4 ,the mathematician is born1 not made1 and it seems he is born a geometer or an analyst4 ,i should like to cite examples and there are surely plenty2 but to accentuate the contrast ,i shall begin with an extreme example1 taking the liberty of seeking it in two living mathematicians4 ,m4 ,mray wants to prove that a binomial equation always has a root1 or1 in ordinary words1 that an angle may always be subdivided4 ,if there is any truth that we think we know by direct intuition1 it is this4 ,who could doubt that an angle may always be divided into any number of equal parts8 ,m4 ,mray does not look at it that way2 in his eyes this proposition is not at all evident and to prove it he needs several pages4 ,on the other hand1 look at ,professor ,klein3 he is studying one of the most abstract questions of the theory of functions3 to determine whether on a given ,riemann surface there always exists a function admitting of given singularities4 ,what does the celebrated ,german geometer do8 ,he replaces his ,riemann surface by a metallic surface whose electric conductivity varies according to certain laws4 ,he connects two of its points with the two poles of a battery4 ,the current1 says he1 must pass1 and the distribution of this current on the surface will define a function whose singularities will be precisely those called for by the enunciation4 ,doubtless ,professor ,klein well knows he has given here only a sketch2 nevertheless he has not hesitated to publish it2 and he would probably believe he finds in it1 if not a rigorous demonstration1 at least a kind of moral certainty4 ,a logician would have rejected with horror such a conception1 or rather he would not have had to reject it1 because in his mind it would never have originated4 ,again1 permit me to compare two men1 the honor of ,french science1 who have recently been taken from us1 but who both entered long ago into immortality4 ,i speak of ,m4 ,bertrand and ,m4 ,hermite4 ,they were scholars of the same school at the same time2 they had the same education1 were under the same influences2 and yet what a difference6 ,not only does it blaze forth in their writings2 it is in their teaching1 in their way of speaking1 in their very look4 ,in the memory of all their pupils these two faces are stamped in deathless lines2 for all who have had the pleasure of following their teaching1 this remembrance is still fresh2 it is easy for us to evoke it4 ,while speaking1 ,m4 ,bertrand is always in motion2 now he seems in combat with some outside enemy1 now he outlines with a gesture of the hand the figures he studies4 ,plainly he sees and he is eager to paint1 this is why he calls gesture to his aid4 ,with ,m4 ,hermite1 it is just the opposite2 his eyes seem to shun contact with the world2 it is not without1 it is within he seeks the vision of truth4 ,among the ,german geometers of this century1 two names above all are illustrious1 those of the two scientists who founded the general theory of functions1 ,weierstrass and ,riemann4 ,weierstrass leads everything back to the consideration of series and their analytic transformations2 to express it better1 he reduces analysis to a sort of prolongation of arithmetic2 you may turn through all his books without finding a figure4 ,riemann1 on the contrary1 at once calls geometry to his aid2 each of his conceptions is an image that no one can forget1 once he has caught its meaning4 ,more recently1 ,lie was an intuitionalist2 this might have been doubted in reading his books1 no one could doubt it after talking with him2 you saw at once that he thought in pictures4 ,madame ,kovalevski was a logician4 ,among our students we notice the same differences2 some prefer to treat their problems 'by analysis1' others 'by geometry4' ,the first are incapable of 'seeing in space1' the others are quickly tired of long calculations and become perplexed4 ,the two sorts of minds are equally necessary for the progress of science2 both the logicians and the intuitionalists have achieved great things that others could not have done4 ,who would venture to say whether he preferred that ,weierstrass had never written or that there had never been a ,riemann8 ,analysis and synthesis have then both their legitimate rles4 ,but it is interesting to study more closely in the history of science the part which belongs to each4 ,,ii ,strange6 ,if we read over the works of the ancients we are tempted to class them all among the intuitionalists4 ,and yet nature is always the same2 it is hardly probable that it has begun in this century to create minds devoted to logic4 ,if we could put ourselves into the flow of ideas which reigned in their time1 we should recognize that many of the old geometers were in tendency analysts4 ,euclid1 for example1 erected a scientific structure wherein his contemporaries could find no fault4 ,in this vast construction1 of which each piece however is due to intuition1 we may still to-day1 without much effort1 recognize the work of a logician4 ,it is not minds that have changed1 it is ideas2 the intuitional minds have remained the same2 but their readers have required of them greater concessions4 ,what is the cause of this evolution8 ,it is not hard to find4 ,intuition can not give us rigor1 nor even certainty2 this has been recognized more and more4 ,let us cite some examples4 ,we know there exist continuous functions lacking derivatives4 ,nothing is more shocking to intuition than this proposition which is imposed upon us by logic4 ,our fathers would not have failed to say3 ,it is evident that every continuous function has a derivative1 since every curve has a tangent4 ,how can intuition deceive us on this point8 ,it is because when we seek to imagine a curve we can not represent it to ourselves without width2 just so1 when we represent to ourselves a straight line1 we see it under the form of a rectilinear band of a certain breadth4 ,we well know these lines have no width2 we try to imagine them narrower and narrower and thus to approach the limit2 so we do in a certain measure1 but we shall never attain this limit4 ,and then it is clear we can always picture these two narrow bands1 one straight1 one curved1 in a position such that they encroach slightly one upon the other without crossing4 ,we shall thus be led1 unless warned by a rigorous analysis1 to conclude that a curve always has a tangent4 ,i shall take as second example ,dirichlet's principle on which rest so many theorems of mathematical physics2 to-day we establish it by reasoning very rigorous but very long2 heretofore1 on the contrary1 we were content with a very summary proof4 ,a certain integral depending on an arbitrary function can never vanish4 ,hence it is concluded that it must have a minimum4 ,the flaw in this reasoning strikes us immediately1 since we use the abstract term function and are familiar with all the singularities functions can present when the word is understood in the most general sense4 ,but it would not be the same had we used concrete images1 had we1 for example1 considered this function as an electric potential2 it would have been thought legitimate to affirm that electrostatic equilibrium can be attained4 ,yet perhaps a physical comparison would have awakened some vague distrust4 ,but if care had been taken to translate the reasoning into the language of geometry1 intermediate between that of analysis and that of physics1 doubtless this distrust would not have been produced1 and perhaps one might thus1 even to-day1 still deceive many readers not forewarned4 ,intuition1 therefore1 does not give us certainty4 ,this is why the evolution had to happen2 let us now see how it happened4 ,it was not slow in being noticed that rigor could not be introduced in the reasoning unless first made to enter into the definitions4 ,for the most part the objects treated of by mathematicians were long ill defined2 they were supposed to be known because represented by means of the senses or the imagination2 but one had only a crude image of them and not a precise idea on which reasoning could take hold4 ,it was there first that the logicians had to direct their efforts4 ,so1 in the case of incommensurable numbers4 ,the vague idea of continuity1 which we owe to intuition1 resolved itself into a complicated system of inequalities referring to whole numbers4 ,by that means the difficulties arising from passing to the limit1 or from the consideration of infinitesimals1 are finally removed4 ,to-day in analysis only whole numbers are left or systems1 finite or infinite1 of whole numbers bound together by a net of equality or inequality relations4 ,mathematics1 as they say1 is arithmetized4 ,,iii ,a first question presents itself4 ,is this evolution ended8 ,have we finally attained absolute rigor8 ,at each stage of the evolution our fathers also thought they had reached it4 ,if they deceived themselves1 do we not likewise cheat ourselves8 ,we believe that in our reasonings we no longer appeal to intuition2 the philosophers will tell us this is an illusion4 ,pure logic could never lead us to anything but tautologies2 it could create nothing new2 not from it alone can any science issue4 ,in one sense these philosophers are right2 to make arithmetic1 as to make geometry1 or to make any science1 something else than pure logic is necessary4 ,to designate this something else we have no word other than intuition4 ,but how many different ideas are hidden under this same word8 ,compare these four axioms3 7#a7 ,two quantities equal to a third are equal to one another2 7#b7 if a theorem is true of the number #a and if we prove that it is true of n #a if true for n1 then will it be true of all whole numbers2 7#c7 if on a straight the point ,c is between ,a and ,b and the point ,d between ,a and ,c1 then the point ,d will be between ,a and ,b2 7#d7 through a given point there is not more than one parallel to a given straight4 ,all four are attributed to intuition1 and yet the first is the enunciation of one of the rules of formal logic2 the second is a real synthetic a priori judgment1 it is the foundation of rigorous mathematical induction2 the third is an appeal to the imagination2 the fourth is a disguised definition4 ,intuition is not necessarily founded on the evidence of the senses2 the senses would soon become powerless2 for example1 we can not represent to ourselves a chiliagon1 and yet we reason by intuition on polygons in general1 which include the chiliagon as a particular case4 ,you know what ,poncelet understood by the principle of continuity4 ,what is true of a real quantity1 said ,poncelet1 should be true of an imaginary quantity2 what is true of the hyperbola whose asymptotes are real1 should then be true of the ellipse whose asymptotes are imaginary4 ,poncelet was one of the most intuitive minds of this century2 he was passionately1 almost ostentatiously1 so2 he regarded the principle of continuity as one of his boldest conceptions1 and yet this principle did not rest on the evidence of the senses4 ,to assimilate the hyperbola to the ellipse was rather to contradict this evidence4 ,it was only a sort of precocious and instinctive generalization which1 moreover1 ,i have no desire to defend4 ,we have then many kinds of intuition2 first1 the appeal to the senses and the imagination2 next1 generalization by induction1 copied1 so to speak1 from the procedures of the experimental sciences2 finally1 we have the intuition of pure number1 whence arose the second of the axioms just enunciated1 which is able to create the real mathematical reasoning4 ,i have shown above by examples that the first two can not give us certainty2 but who will seriously doubt the third1 who will doubt arithmetic8 ,now in the analysis of to-day1 when one cares to take the trouble to be rigorous1 there can be nothing but syllogisms or appeals to this intuition of pure number1 the only intuition which can not deceive us4 ,it may be said that to-day absolute rigor is attained4 ,,iv ,the philosophers make still another objection3 ,what you gain in rigor1 they say1 you lose in objectivity4 ,you can rise toward your logical ideal only by cutting the bonds which attach you to reality4 ,your science is infallible1 but it can only remain so by imprisoning itself in an ivory tower and renouncing all relation with the external world4 ,from this seclusion it must go out when it would attempt the slightest application4 ,for example1 ,i seek to show that some property pertains to some object whose concept seems to me at first indefinable1 because it is intuitive4 ,at first ,i fail or must content myself with approximate proofs2 finally ,i decide to give to my object a precise definition1 and this enables me to establish this property in an irreproachable manner4 ,and then1 say the philosophers1 it still remains to show that the object which corresponds to this definition is indeed the same made known to you by intuition2 or else that some real and concrete object whose conformity with your intuitive idea you believe you immediately recognize corresponds to your new definition4 ,only then could you affirm that it has the property in question4 ,you have only displaced the difficulty4 ,that is not exactly so2 the difficulty has not been displaced1 it has been divided4 ,the proposition to be established was in reality composed of two different truths1 at first not distinguished4 ,the first was a mathematical truth1 and it is now rigorously established4 ,the second was an experimental verity4 ,experience alone can teach us that some real and concrete object corresponds or does not correspond to some abstract definition4 ,this second verity is not mathematically demonstrated1 but neither can it be1 no more than can the empirical laws of the physical and natural sciences4 ,it would be unreasonable to ask more4 ,well1 is it not a great advance to have distinguished what long was wrongly confused8 ,does this mean that nothing is left of this objection of the philosophers8 ,that ,i do not intend to say2 in becoming rigorous1 mathematical science takes a character so artificial as to strike every one2 it forgets its historical origins2 we see how the questions can be answered1 we no longer see how and why they are put4 ,this shows us that logic is not enough2 that the science of demonstration is not all science and that intuition must retain its rle as complement1 ,i was about to say as counterpoise or as antidote of logic4 ,i have already had occasion to insist on the place intuition should hold in the teaching of the mathematical sciences4 ,without it young minds could not make a beginning in the understanding of mathematics2 they could not learn to love it and would see in it only a vain logomachy2 above all1 without intuition they would never become capable of applying mathematics4 ,but now ,i wish before all to speak of the rle of intuition in science itself4 ,if it is useful to the student it is still more so to the creative scientist4 ,v ,we seek reality1 but what is reality8 ,the physiologists tell us that organisms are formed of cells2 the chemists add that cells themselves are formed of atoms4 ,does this mean that these atoms or these cells constitute reality1 or rather the sole reality8 ,the way in which these cells are arranged and from which results the unity of the individual1 is not it also a reality much more interesting than that of the isolated elements1 and should a naturalist who had never studied the elephant except by means of the microscope think himself sufficiently acquainted with that animal8 ,well1 there is something analogous to this in mathematics4 ,the logician cuts up1 so to speak1 each demonstration into a very great number of elementary operations2 when we have examined these operations one after the other and ascertained that each is correct1 are we to think we have grasped the real meaning of the demonstration8 ,shall we have understood it even when1 by an effort of memory1 we have become able to repeat this proof by reproducing all these elementary operations in just the order in which the inventor had arranged them8 ,evidently not2 we shall not yet possess the entire reality2 that ,i know not what1 which makes the unity of the demonstration1 will completely elude us4 ,pure analysis puts at our disposal a multitude of procedures whose infallibility it guarantees2 it opens to us a thousand different ways on which we can embark in all confidence2 we are assured of meeting there no obstacles2 but of all these ways1 which will lead us most promptly to our goal8 ,who shall tell us which to choose8 ,we need a faculty which makes us see the end from afar1 and intuition is this faculty4 ,it is necessary to the explorer for choosing his route2 it is not less so to the one following his trail who wants to know why he chose it4 ,if you are present at a game of chess1 it will not suffice1 for the understanding of the game1 to know the rules for moving the pieces4 ,that will only enable you to recognize that each move has been made conformably to these rules1 and this knowledge will truly have very little value4 ,yet this is what the reader of a book on mathematics would do if he were a logician only4 ,to understand the game is wholly another matter2 it is to know why the player moves this piece rather than that other which he could have moved without breaking the rules of the game4 ,it is to perceive the inward reason which makes of this series of successive moves a sort of organized whole4 ,this faculty is still more necessary for the player himself1 that is1 for the inventor4 ,let us drop this comparison and return to mathematics4 ,for example1 see what has happened to the idea of continuous function4 ,at the outset this was only a sensible image1 for example1 that of a continuous mark traced by the chalk on a blackboard4 ,then it became little by little more refined2 ere long it was used to construct a complicated system of inequalities1 which reproduced1 so to speak1 all the lines of the original image2 this construction finished1 the centering of the arch1 so to say1 was removed1 that crude representation which had temporarily served as support and which was afterward useless was rejected2 there remained only the construction itself1 irreproachable in the eyes of the logician4 ,and yet if the primitive image had totally disappeared from our recollection1 how could we divine by what caprice all these inequalities were erected in this fashion one upon another8 ,perhaps you think ,i use too many comparisons2 yet pardon still another4 ,you have doubtless seen those delicate assemblages of silicious needles which form the skeleton of certain sponges4 ,when the organic matter has disappeared1 there remains only a frail and elegant lace-work4 ,true1 nothing is there except silica1 but what is interesting is the form this silica has taken1 and we could not understand it if we did not know the living sponge which has given it precisely this form4 ,thus it is that the old intuitive notions of our fathers1 even when we have abandoned them1 still imprint their form upon the logical constructions we have put in their place4 ,this view of the aggregate is necessary for the inventor2 it is equally necessary for whoever wishes really to comprehend the inventor4 ,can logic give it to us8 ,no2 the name mathematicians give it would suffice to prove this4 ,in mathematics logic is called analysis and analysis means division1 dissection4 ,it can have1 therefore1 no tool other than the scalpel and the microscope4 ,thus logic and intuition have each their necessary rle4 ,each is indispensable4 ,logic1 which alone can give certainty1 is the instrument of demonstration2 intuition is the instrument of invention4 ,,vi ,but at the moment of formulating this conclusion ,i am seized with scruples4 ,at the outset ,i distinguished two kinds of mathematical minds1 the one sort logicians and analysts1 the others intuitionalists and geometers4 ,well1 the analysts also have been inventors4 ,the names ,i have just cited make my insistence on this unnecessary4 ,here is a contradiction1 at least apparently1 which needs explanation4 ,and first1 do you think these logicians have always proceeded from the general to the particular1 as the rules of formal logic would seem to require of them8 ,not thus could they have extended the boundaries of science2 scientific conquest is to be made only by generalization4 ,in one of the chapters of ',science and ,hypothesis1' ,i have had occasion to study the nature of mathematical reasoning1 and ,i have shown how this reasoning1 without ceasing to be absolutely rigorous1 could lift us from the particular to the general by a procedure ,i have called mathematical induction4 ,it is by this procedure that the analysts have made science progress1 and if we examine the detail itself of their demonstrations1 we shall find it there at each instant beside the classic syllogism of ,aristotle4 ,we1 therefore1 see already that the analysts are not simply makers of syllogisms after the fashion of the scholastics4 ,besides1 do you think they have always marched step by step with no vision of the goal they wished to attain8 ,they must have divined the way leading thither1 and for that they needed a guide4 ,this guide is1 first1 analogy4 ,for example1 one of the methods of demonstration dear to analysts is that founded on the employment of dominant functions4 ,we know it has already served to solve a multitude of problems2 in what consists then the rle of the inventor who wishes to apply it to a new problem8 ,at the outset he must recognize the analogy of this question with those which have already been solved by this method2 then he must perceive in what way this new question differs from the others1 and thence deduce the modifications necessary to apply to the method4 ,but how does one perceive these analogies and these differences8 ,in the example just cited they are almost always evident1 but ,i could have found others where they would have been much more deeply hidden2 often a very uncommon penetration is necessary for their discovery4 ,the analysts1 not to let these hidden analogies escape them1 that is1 in order to be inventors1 must1 without the aid of the senses and imagination1 have a direct sense of what constitutes the unity of a piece of reasoning1 of what makes1 so to speak1 its soul and inmost life4 ,when one talked with ,m4 ,hermite1 he never evoked a sensuous image1 and yet you soon perceived that the most abstract entities were for him like living beings4 ,he did not see them1 but he perceived that they are not an artificial assemblage and that they have some principle of internal unity4 ,but1 one will say1 that still is intuition4 ,shall we conclude that the distinction made at the outset was only apparent1 that there is only one sort of mind and that all the mathematicians are intuitionalists1 at least those who are capable of inventing8 ,no1 our distinction corresponds to something real4 ,i have said above that there are many kinds of intuition4 ,i have said how much the intuition of pure number1 whence comes rigorous mathematical induction1 differs from sensible intuition to which the imagination1 properly so called1 is the principal contributor4 ,is the abyss which separates them less profound than it at first appeared8 ,could we recognize with a little attention that this pure intuition itself could not do without the aid of the senses8 ,this is the affair of the psychologist and the metaphysician and ,i shall not discuss the question4 ,but the thing's being doubtful is enough to justify me in recognizing and affirming an essential difference between the two kinds of intuition2 they have not the same object and seem to call into play two different faculties of our soul2 one would think of two search-lights directed upon two worlds strangers to one another4 ,it is the intuition of pure number1 that of pure logical forms1 which illumines and directs those we have called analysts4 ,this it is which enables them not alone to demonstrate1 but also to invent4 ,by it they perceive at a glance the general plan of a logical edifice1 and that too without the senses appearing to intervene4 ,in rejecting the aid of the imagination1 which1 as we have seen1 is not always infallible1 they can advance without fear of deceiving themselves4 ,happy1 therefore1 are those who can do without this aid6 ,we must admire them2 but how rare they are6 ,among the analysts there will then be inventors1 but they will be few4 ,the majority of us1 if we wished to see afar by pure intuition alone1 would soon feel ourselves seized with vertigo4 ,our weakness has need of a staff more solid1 and1 despite the exceptions of which we have just spoken1 it is none the less true that sensible intuition is in mathematics the most usual instrument of invention4 ,apropos of these reflections1 a question comes up that ,i have not the time either to solve or even to enunciate with the developments it would admit of4 ,is there room for a new distinction1 for distinguishing among the analysts those who above all use pure intuition and those who are first of all preoccupied with formal logic8 ,m4 ,hermite1 for example1 whom ,i have just cited1 can not be classed among the geometers who make use of the sensible intuition2 but neither is he a logician1 properly so called4 ,he does not conceal his aversion to purely deductive procedures which start from the general and end in the particular4 ,,chapter ,,ii ,,the ,,measure ,,of ,,time ,i ,so long as we do not go outside the domain of consciousness1 the notion of time is relatively clear4 ,not only do we distinguish without difficulty present sensation from the remembrance of past sensations or the anticipation of future sensations1 but we know perfectly well what we mean when we say that of two conscious phenomena which we remember1 one was anterior to the other2 or that1 of two foreseen conscious phenomena1 one will be anterior to the other4 ,when we say that two conscious facts are simultaneous1 we mean that they profoundly interpenetrate1 so that analysis can not separate them without mutilating them4 ,the order in which we arrange conscious phenomena does not admit of any arbitrariness4 ,it is imposed upon us and of it we can change nothing4 ,i have only a single observation to add4 ,for an aggregate of sensations to have become a remembrance capable of classification in time1 it must have ceased to be actual1 we must have lost the sense of its infinite complexity1 otherwise it would have remained present4 ,it must1 so to speak1 have crystallized around a center of associations of ideas which will be a sort of label4 ,it is only when they thus have lost all life that we can classify our memories in time as a botanist arranges dried flowers in his herbarium4 ,but these labels can only be finite in number4 ,on that score1 psychologic time should be discontinuous4 ,whence comes the feeling that between any two instants there are others8 ,we arrange our recollections in time1 but we know that there remain empty compartments4 ,how could that be1 if time were not a form pre-existent in our minds8 ,how could we know there were empty compartments1 if these compartments were revealed to us only by their content8 ,,ii ,but that is not all2 into this form we wish to put not only the phenomena of our own consciousness1 but those of which other consciousnesses are the theater4 ,but more1 we wish to put there physical facts1 these ,i know not what with which we people space and which no consciousness sees directly4 ,this is necessary because without it science could not exist4 ,in a word1 psychologic time is given to us and must needs create scientific and physical time4 ,there the difficulty begins1 or rather the difficulties1 for there are two4 ,think of two consciousnesses1 which are like two worlds impenetrable one to the other4 ,by what right do we strive to put them into the same mold1 to measure them by the same standard8 ,is it not as if one strove to measure length with a gram or weight with a meter8 ,and besides1 why do we speak of measuring8 ,we know perhaps that some fact is anterior to some other1 but not by how much it is anterior4 ,therefore two difficulties3 7#a7 ,can we transform psychologic time1 which is qualitative1 into a quantitative time8 7#b7 ,can we reduce to one and the same measure facts which transpire in different worlds8 ,,iii ,the first difficulty has long been noticed2 it has been the subject of long discussions and one may say the question is settled4 ,we have not a direct intuition of the equality of two intervals of time4 ,the persons who believe they possess this intuition are dupes of an illusion4 ,when ,i say1 from noon to one the same time passes as from two to three1 what meaning has this affirmation8 ,the least reflection shows that by itself it has none at all4 ,it will only have that which ,i choose to give it1 by a definition which will certainly possess a certain degree of arbitrariness4 ,psychologists could have done without this definition2 physicists and astronomers could not2 let us see how they have managed4 ,to measure time they use the pendulum and they suppose by definition that all the beats of this pendulum are of equal duration4 ,but this is only a first approximation2 the temperature1 the resistance of the air1 the barometric pressure1 make the pace of the pendulum vary4 ,if we could escape these sources of error1 we should obtain a much closer approximation1 but it would still be only an approximation4 ,new causes1 hitherto neglected1 electric1 magnetic or others1 would introduce minute perturbations4 ,in fact1 the best chronometers must be corrected from time to time1 and the corrections are made by the aid of astronomic observations2 arrangements are made so that the sidereal clock marks the same hour when the same star passes the meridian4 ,in other words1 it is the sidereal day1 that is1 the duration of the rotation of the earth1 which is the constant unit of time4 ,it is supposed1 by a new definition substituted for that based on the beats of the pendulum1 that two complete rotations of the earth about its axis have the same duration4 ,however1 the astronomers are still not content with this definition4 ,many of them think that the tides act as a check on our globe1 and that the rotation of the earth is becoming slower and slower4 ,thus would be explained the apparent acceleration of the motion of the moon1 which would seem to be going more rapidly than theory permits because our watch1 which is the earth1 is going slow4 ,,iv ,all this is unimportant1 one will say2 doubtless our instruments of measurement are imperfect1 but it suffices that we can conceive a perfect instrument4 ,this ideal can not be reached1 but it is enough to have conceived it and so to have put rigor into the definition of the unit of time4 ,the trouble is that there is no rigor in the definition4 ,when we use the pendulum to measure time1 what postulate do we implicitly admit8 ,it is that the duration of two identical phenomena is the same2 or1 if you prefer1 that the same causes take the same time to produce the same effects4 ,and at first blush1 this is a good definition of the equality of two durations4 ,but take care4 ,is it impossible that experiment may some day contradict our postulate8 ,let me explain myself4 ,i suppose that at a certain place in the world the phenomenon alpha happens1 causing as consequence at the end of a certain time the effect alpha'4 ,at another place in the world very far away from the first1 happens the phenomenon beta1 which causes as consequence the effect beta'4 ,the phenomena alpha and beta are simultaneous1 as are also the effects alpha' and beta'4 ,later1 the phenomenon alpha is reproduced under approximately the same conditions as before1 and simultaneously the phenomenon beta is also reproduced at a very distant place in the world and almost under the same circumstances4 ,the effects alpha' and beta' also take place4 ,let us suppose that the effect alpha' happens perceptibly before the effect beta'4 ,if experience made us witness such a sight1 our postulate would be contradicted4 ,for experience would tell us that the first duration alphaalpha' is equal to the first duration betabeta' and that the second duration alphaalpha' is less than the second duration betabeta'4 ,on the other hand1 our postulate would require that the two durations alphaalpha' should be equal to each other1 as likewise the two durations betabeta'4 ,the equality and the inequality deduced from experience would be incompatible with the two equalities deduced from the postulate4 ,now can we affirm that the hypotheses ,i have just made are absurd8 ,they are in no wise contrary to the principle of contradiction4 ,doubtless they could not happen without the principle of sufficient reason seeming violated4 ,but to justify a definition so fundamental ,i should prefer some other guarantee4 ,v ,but that is not all4 ,in physical reality one cause does not produce a given effect1 but a multitude of distinct causes contribute to produce it1 without our having any means of discriminating the part of each of them4 ,physicists seek to make this distinction2 but they make it only approximately1 and1 however they progress1 they never will make it except approximately4 ,it is approximately true that the motion of the pendulum is due solely to the earth's attraction2 but in all rigor every attraction1 even of ,sirius1 acts on the pendulum4 ,under these conditions1 it is clear that the causes which have produced a certain effect will never be reproduced except approximately4 ,then we should modify our postulate and our definition4 ,instead of saying3 ',the same causes take the same time to produce the same effects1' we should say3 ',causes almost identical take almost the same time to produce almost the same effects4' ,our definition therefore is no longer anything but approximate4 ,besides1 as ,m4 ,calinon very justly remarks in a recent memoir3#g ,one of the circumstances of any phenomenon is the velocity of the earth's rotation2 if this velocity of rotation varies1 it constitutes in the reproduction of this phenomenon a circumstance which no longer remains the same4 ,but to suppose this velocity of rotation constant is to suppose that we know how to measure time4 #g ,etude sur les diverses grandeurs1 ,paris1 ,gauthier-,villars1 #ahig4 ,our definition is therefore not yet satisfactory2 it is certainly not that which the astronomers of whom ,i spoke above implicitly adopt1 when they affirm that the terrestrial rotation is slowing down4 ,what meaning according to them has this affirmation8 ,we can only understand it by analyzing the proofs they give of their proposition4 ,they say first that the friction of the tides producing heat must destroy vis viva4 ,they invoke therefore the principle of vis viva1 or of the conservation of energy4 ,they say next that the secular acceleration of the moon1 calculated according to ,newton's law1 would be less than that deduced from observations unless the correction relative to the slowing down of the terrestrial rotation were made4 ,they invoke therefore ,newton's law4 ,in other words1 they define duration in the following way3 time should be so defined that ,newton's law and that of vis viva may be verified4 ,newton's law is an experimental truth2 as such it is only approximate1 which shows that we still have only a definition by approximation4 ,if now it be supposed that another way of measuring time is adopted1 the experiments on which ,newton's law is founded would none the less have the same meaning4 ,only the enunciation of the law would be different1 because it would be translated into another language2 it would evidently be much less simple4 ,so that the definition implicitly adopted by the astronomers may be summed up thus3 ,time should be so defined that the equations of mechanics may be as simple as possible4 ,in other words1 there is not one way of measuring time more true than another2 that which is generally adopted is only more convenient4 ,of two watches1 we have no right to say that the one goes true1 the other wrong2 we can only say that it is advantageous to conform to the indications of the first4 ,the difficulty which has just occupied us has been1 as ,i have said1 often pointed out2 among the most recent works in which it is considered1 ,i may mention1 besides ,m4 ,calinon's little book1 the treatise on mechanics of ,andrade4 ,,vi ,the second difficulty has up to the present attracted much less attention2 yet it is altogether analogous to the preceding2 and even1 logically1 ,i should have spoken of it first4 ,two psychological phenomena happen in two different consciousnesses2 when ,i say they are simultaneous1 what do ,i mean8 ,when ,i say that a physical phenomenon1 which happens outside of every consciousness1 is before or after a psychological phenomenon1 what do ,i mean8 ,in #aegb1 ,tycho ,brahe noticed in the heavens a new star4 ,an immense conflagration had happened in some far distant heavenly body2 but it had happened long before2 at least two hundred years were necessary for the light from that star to reach our earth4 ,this conflagration therefore happened before the discovery of ,america4 ,well1 when ,i say that2 when1 considering this gigantic phenomenon1 which perhaps had no witness1 since the satellites of that star were perhaps uninhabited1 ,i say this phenomenon is anterior to the formation of the visual image of the isle of ,espaola in the consciousness of ,christopher ,columbus1 what do ,i mean8 ,a little reflection is sufficient to understand that all these affirmations have by themselves no meaning4 ,they can have one only as the outcome of a convention4 ,,vii ,we should first ask ourselves how one could have had the idea of putting into the same frame so many worlds impenetrable to one another4 ,we should like to represent to ourselves the external universe1 and only by so doing could we feel that we understood it4 ,we know we never can attain this representation3 our weakness is too great4 ,but at least we desire the ability to conceive an infinite intelligence for which this representation could be possible1 a sort of great consciousness which should see all1 and which should classify all in its time1 as we classify1 in our time1 the little we see4 ,this hypothesis is indeed crude and incomplete1 because this supreme intelligence would be only a demigod2 infinite in one sense1 it would be limited in another1 since it would have only an imperfect recollection of the past2 and it could have no other1 since otherwise all recollections would be equally present to it and for it there would be no time4 ,and yet when we speak of time1 for all which happens outside of us1 do we not unconsciously adopt this hypothesis2 do we not put ourselves in the place of this imperfect god2 and do not even the atheists put themselves in the place where god would be if he existed8 ,what ,i have just said shows us1 perhaps1 why we have tried to put all physical phenomena into the same frame4 ,but that can not pass for a definition of simultaneity1 since this hypothetical intelligence1 even if it existed1 would be for us impenetrable4 ,it is therefore necessary to seek something else4 ,,viii ,the ordinary definitions which are proper for psychologic time would suffice us no more4 ,two simultaneous psychologic facts are so closely bound together that analysis can not separate without mutilating them4 ,is it the same with two physical facts8 ,is not my present nearer my past of yesterday than the present of ,sirius8 ,it has also been said that two facts should be regarded as simultaneous when the order of their succession may be inverted at will4 ,it is evident that this definition would not suit two physical facts which happen far from one another1 and that1 in what concerns them1 we no longer even understand what this reversibility would be2 besides1 succession itself must first be defined4 ,,ix ,let us then seek to give an account of what is understood by simultaneity or antecedence1 and for this let us analyze some examples4 ,i write a letter2 it is afterward read by the friend to whom ,i have addressed it4 ,there are two facts which have had for their theater two different consciousnesses4 ,in writing this letter ,i have had the visual image of it1 and my friend has had in his turn this same visual image in reading the letter4 ,though these two facts happen in impenetrable worlds1 ,i do not hesitate to regard the first as anterior to the second1 because ,i believe it is its cause4 ,i hear thunder1 and ,i conclude there has been an electric discharge2 ,i do not hesitate to consider the physical phenomenon as anterior to the auditory image perceived in my consciousness1 because ,i believe it is its cause4 ,behold then the rule we follow1 and the only one we can follow3 when a phenomenon appears to us as the cause of another1 we regard it as anterior4 ,it is therefore by cause that we define time2 but most often1 when two facts appear to us bound by a constant relation1 how do we recognize which is the cause and which the effect8 ,we assume that the anterior fact1 the antecedent1 is the cause of the other1 of the consequent4 ,it is then by time that we define cause4 ,how save ourselves from this petitio principii8 ,we say now post hoc1 ergo propter hoc2 now propter hoc1 ergo post hoc2 shall we escape from this vicious circle8 ,x ,let us see1 not how we succeed in escaping1 for we do not completely succeed1 but how we try to escape4 ,i execute a voluntary act ,a and ,i feel afterward a sensation ,d1 which ,i regard as a consequence of the act ,a2 on the other hand1 for whatever reason1 ,i infer that this consequence is not immediate1 but that outside my consciousness two facts ,b and ,c1 which ,i have not witnessed1 have happened1 and in such a way that ,b is the effect of ,a1 that ,c is the effect of ,b1 and ,d of ,c4 ,but why8 ,if ,i think ,i have reason to regard the four facts ,a1 ,b1 ,c1 ,d1 as bound to one another by a causal connection1 why range them in the causal order ,a ,b ,c ,d1 and at the same time in the chronologic order ,a ,b ,c ,d1 rather than in any other order8 ,i clearly see that in the act ,a ,i have the feeling of having been active1 while in undergoing the sensation ,d ,i have that of having been passive4 ,this is why ,i regard ,a as the initial cause and ,d as the ultimate effect2 this is why ,i put ,a at the beginning of the chain and ,d at the end2 but why put ,b before ,c rather than ,c before ,b8 ,if this question is put1 the reply ordinarily is3 we know that it is ,b which is the cause of ,c because we always see ,b happen before ,c4 ,these two phenomena1 when witnessed1 happen in a certain order2 when analogous phenomena happen without witness1 there is no reason to invert this order4 ,doubtless1 but take care2 we never know directly the physical phenomena ,b and ,c4 ,what we know are sensations ,b' and ,c' produced respectively by ,b and ,c4 ,our consciousness tells us immediately that ,b' precedes ,c' and we suppose that ,b and ,c succeed one another in the same order4 ,this rule appears in fact very natural1 and yet we are often led to depart from it4 ,we hear the sound of the thunder only some seconds after the electric discharge of the cloud4 ,of two flashes of lightning1 the one distant1 the other near1 can not the first be anterior to the second1 even though the sound of the second comes to us before that of the first8 ,,xi ,another difficulty2 have we really the right to speak of the cause of a phenomenon8 ,if all the parts of the universe are interchained in a certain measure1 any one phenomenon will not be the effect of a single cause1 but the resultant of causes infinitely numerous2 it is1 one often says1 the consequence of the state of the universe a moment before4 ,how enunciate rules applicable to circumstances so complex8 ,and yet it is only thus that these rules can be general and rigorous4 ,not to lose ourselves in this infinite complexity1 let us make a simpler hypothesis4 ,consider three stars1 for example1 the sun1 ,jupiter and ,saturn2 but1 for greater simplicity1 regard them as reduced to material points and isolated from the rest of the world4 ,the positions and the velocities of three bodies at a given instant suffice to determine their positions and velocities at the following instant1 and consequently at any instant4 ,their positions at the instant t determine their positions at the instant t h as well as their positions at the instant t - h4 ,even more2 the position of ,jupiter at the instant t1 together with that of ,saturn at the instant t a1 determines the position of ,jupiter at any instant and that of ,saturn at any instant4 ,the aggregate of positions occupied by ,jupiter at the instant t e and ,saturn at the instant t a e is bound to the aggregate of positions occupied by ,jupiter at the instant t and ,saturn at the instant t a1 by laws as precise as that of ,newton1 though more complicated4 ,then why not regard one of these aggregates as the cause of the other1 which would lead to considering as simultaneous the instant t of ,jupiter and the instant t a of ,saturn8 ,in answer there can only be reasons1 very strong1 it is true1 of convenience and simplicity4 ,,xii ,but let us pass to examples less artificial2 to understand the definition implicitly supposed by the savants1 let us watch them at work and look for the rules by which they investigate simultaneity4 ,i will take two simple examples1 the measurement of the velocity of light and the determination of longitude4 ,when an astronomer tells me that some stellar phenomenon1 which his telescope reveals to him at this moment1 happened1 nevertheless1 fifty years ago1 ,i seek his meaning1 and to that end ,i shall ask him first how he knows it1 that is1 how he has measured the velocity of light4 ,he has begun by supposing that light has a constant velocity1 and in particular that its velocity is the same in all directions4 ,that is a postulate without which no measurement of this velocity could be attempted4 ,this postulate could never be verified directly by experiment2 it might be contradicted by it if the results of different measurements were not concordant4 ,we should think ourselves fortunate that this contradiction has not happened and that the slight discordances which may happen can be readily explained4 ,the postulate1 at all events1 resembling the principle of sufficient reason1 has been accepted by everybody2 what ,i wish to emphasize is that it furnishes us with a new rule for the investigation of simultaneity1 entirely different from that which we have enunciated above4 ,this postulate assumed1 let us see how the velocity of light has been measured4 ,you know that ,roemer used eclipses of the satellites of ,jupiter1 and sought how much the event fell behind its prediction4 ,but how is this prediction made8 ,it is by the aid of astronomic laws2 for instance ,newton's law4 ,could not the observed facts be just as well explained if we attributed to the velocity of light a little different value from that adopted1 and supposed ,newton's law only approximate8 ,only this would lead to replacing ,newton's law by another more complicated4 ,so for the velocity of light a value is adopted1 such that the astronomic laws compatible with this value may be as simple as possible4 ,when navigators or geographers determine a longitude1 they have to solve just the problem we are discussing2 they must1 without being at ,paris1 calculate ,paris time4 ,how do they accomplish it8 ,they carry a chronometer set for ,paris4 ,the qualitative problem of simultaneity is made to depend upon the quantitative problem of the measurement of time4 ,i need not take up the difficulties relative to this latter problem1 since above ,i have emphasized them at length4 ,or else they observe an astronomic phenomenon1 such as an eclipse of the moon1 and they suppose that this phenomenon is perceived simultaneously from all points of the earth4 ,that is not altogether true1 since the propagation of light is not instantaneous2 if absolute exactitude were desired1 there would be a correction to make according to a complicated rule4 ,or else finally they use the telegraph4 ,it is clear first that the reception of the signal at ,berlin1 for instance1 is after the sending of this same signal from ,paris4 ,this is the rule of cause and effect analyzed above4 ,but how much after8 ,in general1 the duration of the transmission is neglected and the two events are regarded as simultaneous4 ,but1 to be rigorous1 a little correction would still have to be made by a complicated calculation2 in practise it is not made1 because it would be well within the errors of observation2 its theoretic necessity is none the less from our point of view1 which is that of a rigorous definition4 ,from this discussion1 ,i wish to emphasize two things3 7#a7 ,the rules applied are exceedingly various4 7#b7 ,it is difficult to separate the qualitative problem of simultaneity from the quantitative problem of the measurement of time2 no matter whether a chronometer is used1 or whether account must be taken of a velocity of transmission1 as that of light1 because such a velocity could not be measured without measuring a time4 ,,xiii ,to conclude3 ,we have not a direct intuition of simultaneity1 nor of the equality of two durations4 ,if we think we have this intuition1 this is an illusion4 ,we replace it by the aid of certain rules which we apply almost always without taking count of them4 ,but what is the nature of these rules8 ,no general rule1 no rigorous rule2 a multitude of little rules applicable to each particular case4 ,these rules are not imposed upon us and we might amuse ourselves in inventing others2 but they could not be cast aside without greatly complicating the enunciation of the laws of physics1 mechanics and astronomy4 ,we therefore choose these rules1 not because they are true1 but because they are the most convenient1 and we may recapitulate them as follows3 ,the simultaneity of two events1 or the order of their succession1 the equality of two durations1 are to be so defined that the enunciation of the natural laws may be as simple as possible4 ,in other words1 all these rules1 all these definitions are only the fruit of an unconscious opportunism4 ,,chapter ,,iii ,,the ,,notion ,,of ,,space #a4 ,introduction ,in the articles ,i have heretofore devoted to space ,i have above all emphasized the problems raised by non-,euclidean geometry1 while leaving almost completely aside other questions more difficult of approach1 such as those which pertain to the number of dimensions4 ,all the geometries ,i considered had thus a common basis1 that tridimensional continuum which was the same for all and which differentiated itself only by the figures one drew in it or when one aspired to measure it4 ,in this continuum1 primitively amorphous1 we may imagine a network of lines and surfaces1 we may then convene to regard the meshes of this net as equal to one another1 and it is only after this convention that this continuum1 become measurable1 becomes ,euclidean or non-,euclidean space4 ,from this amorphous continuum can therefore arise indifferently one or the other of the two spaces1 just as on a blank sheet of paper may be traced indifferently a straight or a circle4 ,in space we know rectilinear triangles the sum of whose angles is equal to two right angles2 but equally we know curvilinear triangles the sum of whose angles is less than two right angles4 ,the existence of the one sort is not more doubtful than that of the other4 ,to give the name of straights to the sides of the first is to adopt ,euclidean geometry2 to give the name of straights to the sides of the latter is to adopt the non-,euclidean geometry4 ,so that to ask what geometry it is proper to adopt is to ask1 to what line is it proper to give the name straight8 ,it is evident that experiment can not settle such a question2 one would not ask1 for instance1 experiment to decide whether ,i should call ,,ab or ,,cd a straight4 ,on the other hand1 neither can ,i say that ,i have not the right to give the name of straights to the sides of non-,euclidean triangles because they are not in conformity with the eternal idea of straight which ,i have by intuition4 ,i grant1 indeed1 that ,i have the intuitive idea of the side of the ,euclidean triangle1 but ,i have equally the intuitive idea of the side of the non-,euclidean triangle4 ,why should ,i have the right to apply the name of straight to the first of these ideas and not to the second8 ,wherein does this syllable form an integrant part of this intuitive idea8 ,evidently when we say that the ,euclidean straight is a true straight and that the non-,euclidean straight is not a true straight1 we simply mean that the first intuitive idea corresponds to a more noteworthy object than the second4 ,but how do we decide that this object is more noteworthy8 ,this question ,i have investigated in ',science and ,hypothesis4' ,it is here that we saw experience come in4 ,if the ,euclidean straight is more noteworthy than the non-,euclidean straight1 it is so chiefly because it differs little from certain noteworthy natural objects from which the non-,euclidean straight differs greatly4 ,but1 it will be said1 the definition of the non-,euclidean straight is artificial2 if we for a moment adopt it1 we shall see that two circles of different radius both receive the name of non-,euclidean straights1 while of two circles of the same radius one can satisfy the definition without the other being able to satisfy it1 and then if we transport one of these so-called straights without deforming it1 it will cease to be a straight4 ,but by what right do we consider as equal these two figures which the ,euclidean geometers call two circles with the same radius8 ,it is because by transporting one of them without deforming it we can make it coincide with the other4 ,and why do we say this transportation is effected without deformation8 ,it is impossible to give a good reason for it4 ,among all the motions conceivable1 there are some of which the ,euclidean geometers say that they are not accompanied by deformation2 but there are others of which the non-,euclidean geometers would say that they are not accompanied by deformation4 ,in the first1 called ,euclidean motions1 the ,euclidean straights remain ,euclidean straights and the non-,euclidean straights do not remain non-,euclidean straights2 in the motions of the second sort1 or non-,euclidean motions1 the non-,euclidean straights remain non-,euclidean straights and the ,euclidean straights do not remain ,euclidean straights4 ,it has1 therefore1 not been demonstrated that it was unreasonable to call straights the sides of non-,euclidean triangles2 it has only been shown that that would be unreasonable if one continued to call the ,euclidean motions motions without deformation2 but it has at the same time been shown that it would be just as unreasonable to call straights the sides of ,euclidean triangles if the non-,euclidean motions were called motions without deformation4 ,now when we say that the ,euclidean motions are the true motions without deformation1 what do we mean8 ,we simply mean that they are more noteworthy than the others4 ,and why are they more noteworthy8 ,it is because certain noteworthy natural bodies1 the solid bodies1 undergo motions almost similar4 ,and then when we ask3 ,can one imagine non-,euclidean space8 ,that means3 ,can we imagine a world where there would be noteworthy natural objects affecting almost the form of non-,euclidean straights1 and noteworthy natural bodies frequently undergoing motions almost similar to the non-,euclidean motions8 ,i have shown in ',science and ,hypothesis' that to this question we must answer yes4 ,it has often been observed that if all the bodies in the universe were dilated simultaneously and in the same proportion1 we should have no means of perceiving it1 since all our measuring instruments would grow at the same time as the objects themselves which they serve to measure4 ,the world1 after this dilatation1 would continue on its course without anything apprising us of so considerable an event4 ,in other words1 two worlds similar to one another 7understanding the word similitude in the sense of ,euclid1 ,book ,,vi47 would be absolutely indistinguishable4 ,but more2 worlds will be indistinguishable not only if they are equal or similar1 that is1 if we can pass from one to the other by changing the axes of coordinates1 or by changing the scale to which lengths are referred2 but they will still be indistinguishable if we can pass from one to the other by any 'point-transformation' whatever4 ,i will explain my meaning4 ,i suppose that to each point of one corresponds one point of the other and only one1 and inversely2 and besides that the coordinates of a point are continuous functions1 otherwise altogether arbitrary1 of the corresponding point4 ,i suppose besides that to each object of the first world corresponds in the second an object of the same nature placed precisely at the corresponding point4 ,i suppose finally that this correspondence fulfilled at the initial instant is maintained indefinitely4 ,we should have no means of distinguishing these two worlds one from the other4 ,the relativity of space is not ordinarily understood in so broad a sense2 it is thus1 however1 that it would be proper to understand it4 ,if one of these universes is our ,euclidean world1 what its inhabitants will call straight will be our ,euclidean straight2 but what the inhabitants of the second world will call straight will be a curve which will have the same properties in relation to the world they inhabit and in relation to the motions that they will call motions without deformation4 ,their geometry will1 therefore1 be ,euclidean geometry1 but their straight will not be our ,euclidean straight4 ,it will be its transform by the point-transformation which carries over from our world to theirs4 ,the straights of these men will not be our straights1 but they will have among themselves the same relations as our straights to one another4 ,it is in this sense ,i say their geometry will be ours4 ,if then we wish after all to proclaim that they deceive themselves1 that their straight is not the true straight1 if we still are unwilling to admit that such an affirmation has no meaning1 at least we must confess that these people have no means whatever of recognizing their error4 #b4 ,qualitative ,geometry ,all that is relatively easy to understand1 and ,i have already so often repeated it that ,i think it needless to expatiate further on the matter4 ,euclidean space is not a form imposed upon our sensibility1 since we can imagine non-,euclidean space2 but the two spaces1 ,euclidean and non-,euclidean1 have a common basis1 that amorphous continuum of which ,i spoke in the beginning4 ,from this continuum we can get either ,euclidean space or ,lobachevskian space1 just as we can1 by tracing upon it a proper graduation1 transform an ungraduated thermometer into a ,fahrenheit or a ,raumur thermometer4 ,and then comes a question3 ,is not this amorphous continuum1 that our analysis has allowed to survive1 a form imposed upon our sensibility8 ,if so1 we should have enlarged the prison in which this sensibility is confined1 but it would always be a prison4 ,this continuum has a certain number of properties1 exempt from all idea of measurement4 ,the study of these properties is the object of a science which has been cultivated by many great geometers and in particular by ,riemann and ,betti and which has received the name of analysis situs4 ,in this science abstraction is made of every quantitative idea and1 for example1 if we ascertain that on a line the point ,b is between the points ,a and ,c1 we shall be content with this ascertainment and shall not trouble to know whether the line ,,abc is straight or curved1 nor whether the length ,,ab is equal to the length ,,bc1 or whether it is twice as great4 ,the theorems of analysis situs have1 therefore1 this peculiarity1 that they would remain true if the figures were copied by an inexpert draftsman who should grossly change all the proportions and replace the straights by lines more or less sinuous4 ,in mathematical terms1 they are not altered by any 'point-transformation' whatsoever4 ,it has often been said that metric geometry was quantitative1 while projective geometry was purely qualitative4 ,that is not altogether true4 ,the straight is still distinguished from other lines by properties which remain quantitative in some respects4 ,the real qualitative geometry is1 therefore1 analysis situs4 ,the same questions which came up apropos of the truths of ,euclidean geometry1 come up anew apropos of the theorems of analysis situs4 ,are they obtainable by deductive reasoning8 ,are they disguised conventions8 ,are they experimental verities8 ,are they the characteristics of a form imposed either upon our sensibility or upon our understanding8 ,i wish simply to observe that the last two solutions exclude each other4 ,we can not admit at the same time that it is impossible to imagine space of four dimensions and that experience proves to us that space has three dimensions4 ,the experimenter puts to nature a question3 ,is it this or that8 and he can not put it without imagining the two terms of the alternative4 ,if it were impossible to imagine one of these terms1 it would be futile and besides impossible to consult experience4 ,there is no need of observation to know that the hand of a watch is not marking the hour #ae on the dial1 because we know beforehand that there are only #ab1 and we could not look at the mark #ae to see if the hand is there1 because this mark does not exist4 ,note likewise that in analysis situs the empiricists are disembarrassed of one of the gravest objections that can be leveled against them1 of that which renders absolutely vain in advance all their efforts to apply their thesis to the verities of ,euclidean geometry4 ,these verities are rigorous and all experimentation can only be approximate4 ,in analysis situs approximate experiments may suffice to give a rigorous theorem and1 for instance1 if it is seen that space can not have either two or less than two dimensions1 nor four or more than four1 we are certain that it has exactly three1 since it could not have two and a half or three and a half4 ,of all the theorems of analysis situs1 the most important is that which is expressed in saying that space has three dimensions4 ,this it is that we are about to consider1 and we shall put the question in these terms3 ,when we say that space has three dimensions1 what do we mean8 #c4 ,the ,physical ,continuum of ,several ,dimensions ,i have explained in ',science and ,hypothesis' whence we derive the notion of physical continuity and how that of mathematical continuity has arisen from it4 ,it happens that we are capable of distinguishing two impressions one from the other1 while each is indistinguishable from a third4 ,thus we can readily distinguish a weight of #ab grams from a weight of #aj grams1 while a weight of #aa grams could be distinguished from neither the one nor the other4 ,such a statement1 translated into symbols1 may be written3 ,a ,b1 ,b ,c1 ,a ,c4 ,this would be the formula of the physical continuum1 as crude experience gives it to us1 whence arises an intolerable contradiction that has been obviated by the introduction of the mathematical continuum4 ,this is a scale of which the steps 7commensurable or incommensurable numbers7 are infinite in number but are exterior to one another1 instead of encroaching on one another as do the elements of the physical continuum1 in conformity with the preceding formula4 ,the physical continuum is1 so to speak1 a nebula not resolved2 the most perfect instruments could not attain to its resolution4 ,doubtless if we measured the weights with a good balance instead of judging them by the hand1 we could distinguish the weight of #aa grams from those of #aj and #ab grams1 and our formula would become3 ,a ,b1 ,b ,c1 ,a ,c4 ,but we should always find between ,a and ,b and between ,b and ,c new elements ,d and ,e1 such that ,a ,d1 ,d ,b1 ,a ,b2 ,b ,e1 ,e ,c1 ,b ,c1 and the difficulty would only have receded and the nebula would always remain unresolved2 the mind alone can resolve it and the mathematical continuum it is which is the nebula resolved into stars4 ,yet up to this point we have not introduced the notion of the number of dimensions4 ,what is meant when we say that a mathematical continuum or that a physical continuum has two or three dimensions8 ,first we must introduce the notion of cut1 studying first physical continua4 ,we have seen what characterizes the physical continuum4 ,each of the elements of this continuum consists of a manifold of impressions2 and it may happen either that an element can not be discriminated from another element of the same continuum1 if this new element corresponds to a manifold of impressions not sufficiently different1 or1 on the contrary1 that the discrimination is possible2 finally it may happen that two elements indistinguishable from a third may1 nevertheless1 be distinguished one from the other4 ,that postulated1 if ,a and ,b are two distinguishable elements of a continuum ,c1 a series of elements may be found1 ,e#a1 ,e#b1 4441 ,en1 all belonging to this same continuum ,c and such that each of them is indistinguishable from the preceding1 that ,e#a is indistinguishable from ,a1 and ,en indistinguishable from ,b4 ,therefore we can go from ,a to ,b by a continuous route and without quitting ,c4 ,if this condition is fulfilled for any two elements ,a and ,b of the continuum ,c1 we may say that this continuum ,c is all in one piece4 ,now let us distinguish certain of the elements of ,c which may either be all distinguishable from one another1 or themselves form one or several continua4 ,the assemblage of the elements thus chosen arbitrarily among all those of ,c will form what ,i shall call the cut or the cuts4 ,take on ,c any two elements ,a and ,b4 ,either we can also find a series of elements ,e#a1 ,e#b1 4441 ,en1 such3 7#a7 that they all belong to ,c2 7#b7 that each of them is indistinguishable from the following1 ,e#a indistinguishable from ,a and ,en from ,b2 7#c7 and besides that none of the elements ,e is indistinguishable from any element of the cut4 ,or else1 on the contrary1 in each of the series ,e#a1 ,e#b1 4441 ,en satisfying the first two conditions1 there will be an element ,e indistinguishable from one of the elements of the cut4 ,in the first case we can go from ,a to ,b by a continuous route without quitting ,c and without meeting the cuts2 in the second case that is impossible4 ,if then for any two elements ,a and ,b of the continuum ,c1 it is always the first case which presents itself1 we shall say that ,c remains all in one piece despite the cuts4 ,thus1 if we choose the cuts in a certain way1 otherwise arbitrary1 it may happen either that the continuum remains all in one piece or that it does not remain all in one piece2 in this latter hypothesis we shall then say that it is divided by the cuts4 ,it will be noticed that all these definitions are constructed in setting out solely from this very simple fact1 that two manifolds of impressions sometimes can be discriminated1 sometimes can not be4 ,that postulated1 if1 to divide a continuum1 it suffices to consider as cuts a certain number of elements all distinguishable from one another1 we say that this continuum is of one dimension2 if1 on the contrary1 to divide a continuum1 it is necessary to consider as cuts a system of elements themselves forming one or several continua1 we shall say that this continuum is of several dimensions4 ,if to divide a continuum ,c1 cuts forming one or several continua of one dimension suffice1 we shall say that ,c is a continuum of two dimensions2 if cuts suffice which form one or several continua of two dimensions at most1 we shall say that ,c is a continuum of three dimensions2 and so on4 ,to justify this definition it is proper to see whether it is in this way that geometers introduce the notion of three dimensions at the beginning of their works4 ,now1 what do we see8 ,usually they begin by defining surfaces as the boundaries of solids or pieces of space1 lines as the boundaries of surfaces1 points as the boundaries of lines1 and they affirm that the same procedure can not be pushed further4 ,this is just the idea given above3 to divide space1 cuts that are called surfaces are necessary2 to divide surfaces1 cuts that are called lines are necessary2 to divide lines1 cuts that are called points are necessary2 we can go no further1 the point can not be divided1 so the point is not a continuum4 ,then lines which can be divided by cuts which are not continua will be continua of one dimension2 surfaces which can be divided by continuous cuts of one dimension will be continua of two dimensions2 finally1 space which can be divided by continuous cuts of two dimensions will be a continuum of three dimensions4 ,thus the definition ,i have just given does not differ essentially from the usual definitions2 ,i have only endeavored to give it a form applicable not to the mathematical continuum1 but to the physical continuum1 which alone is susceptible of representation1 and yet to retain all its precision4 ,moreover1 we see that this definition applies not alone to space2 that in all which falls under our senses we find the characteristics of the physical continuum1 which would allow of the same classification2 that it would be easy to find there examples of continua of four1 of five1 dimensions1 in the sense of the preceding definition2 such examples occur of themselves to the mind4 ,i should explain finally1 if ,i had the time1 that this science1 of which ,i spoke above and to which ,riemann gave the name of analysis situs1 teaches us to make distinctions among continua of the same number of dimensions and that the classification of these continua rests also on the consideration of cuts4 ,from this notion has arisen that of the mathematical continuum of several dimensions in the same way that the physical continuum of one dimension engendered the mathematical continuum of one dimension4 ,the formula ,a ,c1 ,a ,b1 ,b ,c1 which summed up the data of crude experience1 implied an intolerable contradiction4 ,to get free from it1 it was necessary to introduce a new notion while still respecting the essential characteristics of the physical continuum of several dimensions4 ,the mathematical continuum of one dimension admitted of a scale whose divisions1 infinite in number1 corresponded to the different values1 commensurable or not1 of one same magnitude4 ,to have the mathematical continuum of n dimensions1 it will suffice to take n like scales whose divisions correspond to different values of n independent magnitudes called coordinates4 ,we thus shall have an image of the physical continuum of n dimensions1 and this image will be as faithful as it can be after the determination not to allow the contradiction of which ,i spoke above4 #d4 ,the ,notion of ,point ,it seems now that the question we put to ourselves at the start is answered4 ,when we say that space has three dimensions1 it will be said1 we mean that the manifold of points of space satisfies the definition we have just given of the physical continuum of three dimensions4 ,to be content with that would be to suppose that we know what is the manifold of points of space1 or even one point of space4 ,now that is not as simple as one might think4 ,every one believes he knows what a point is1 and it is just because we know it too well that we think there is no need of defining it4 ,surely we can not be required to know how to define it1 because in going back from definition to definition a time must come when we must stop4 ,but at what moment should we stop8 ,we shall stop first when we reach an object which falls under our senses or that we can represent to ourselves2 definition then will become useless2 we do not define the sheep to a child2 we say to him3 ,see the sheep4 ,so1 then1 we should ask ourselves if it is possible to represent to ourselves a point of space4 ,those who answer yes do not reflect that they represent to themselves in reality a white spot made with the chalk on a blackboard or a black spot made with a pen on white paper1 and that they can represent to themselves only an object or rather the impressions that this object made on their senses4 ,when they try to represent to themselves a point1 they represent the impressions that very little objects made them feel4 ,it is needless to add that two different objects1 though both very little1 may produce extremely different impressions1 but ,i shall not dwell on this difficulty1 which would still require some discussion4 ,but it is not a question of that2 it does not suffice to represent one point1 it is necessary to represent a certain point and to have the means of distinguishing it from an other point4 ,and in fact1 that we may be able to apply to a continuum the rule ,i have above expounded and by which one may recognize the number of its dimensions1 we must rely upon the fact that two elements of this continuum sometimes can and sometimes can not be distinguished4 ,it is necessary therefore that we should in certain cases know how to represent to ourselves a specific element and to distinguish it from an other element4 ,the question is to know whether the point that ,i represented to myself an hour ago is the same as this that ,i now represent to myself1 or whether it is a different point4 ,in other words1 how do we know whether the point occupied by the object ,a at the instant alpha is the same as the point occupied by the object ,b at the instant beta1 or still better1 what this means8 ,i am seated in my room2 an object is placed on my table2 during a second ,i do not move1 no one touches the object4 ,i am tempted to say that the point ,a which this object occupied at the beginning of this second is identical with the point ,b which it occupies at its end4 ,not at all2 from the point ,a to the point ,b is #cj kilometers1 because the object has been carried along in the motion of the earth4 ,we can not know whether an object1 be it large or small1 has not changed its absolute position in space1 and not only can we not affirm it1 but this affirmation has no meaning and in any case can not correspond to any representation4 ,but then we may ask ourselves if the relative position of an object with regard to other objects has changed or not1 and first whether the relative position of this object with regard to our body has changed4 ,if the impressions this object makes upon us have not changed1 we shall be inclined to judge that neither has this relative position changed2 if they have changed1 we shall judge that this object has changed either in state or in relative position4 ,it remains to decide which of the two4 ,i have explained in ',science and ,hypothesis' how we have been led to distinguish the changes of position4 ,moreover1 ,i shall return to that further on4 ,we come to know1 therefore1 whether the relative position of an object with regard to our body has or has not remained the same4 ,if now we see that two objects have retained their relative position with regard to our body1 we conclude that the relative position of these two objects with regard to one another has not changed2 but we reach this conclusion only by indirect reasoning4 ,the only thing that we know directly is the relative position of the objects with regard to our body4 ,a fortiori it is only by indirect reasoning that we think we know 7and1 moreover1 this belief is delusive7 whether the absolute position of the object has changed4 ,in a word1 the system of coordinate axes to which we naturally refer all exterior objects is a system of axes invariably bound to our body1 and carried around with us4 ,it is impossible to represent to oneself absolute space2 when ,i try to represent to myself simultaneously objects and myself in motion in absolute space1 in reality ,i represent to myself my own self motionless and seeing move around me different objects and a man that is exterior to me1 but that ,i convene to call me4 ,will the difficulty be solved if we agree to refer everything to these axes bound to our body8 ,shall we know then what is a point thus defined by its relative position with regard to ourselves8 ,many persons will answer yes and will say that they 'localize' exterior objects4 ,what does this mean8 ,to localize an object simply means to represent to oneself the movements that would be necessary to reach it4 ,i will explain myself4 ,it is not a question of representing the movements themselves in space1 but solely of representing to oneself the muscular sensations which accompany these movements and which do not presuppose the preexistence of the notion of space4 ,if we suppose two different objects which successively occupy the same relative position with regard to ourselves1 the impressions that these two objects make upon us will be very different2 if we localize them at the same point1 this is simply because it is necessary to make the same movements to reach them2 apart from that1 one can not just see what they could have in common4 ,but1 given an object1 we can conceive many different series of movements which equally enable us to reach it4 ,if then we represent to ourselves a point by representing to ourselves the series of muscular sensations which accompany the movements which enable us to reach this point1 there will be many ways entirely different of representing to oneself the same point4 ,if one is not satisfied with this solution1 but wishes1 for instance1 to bring in the visual sensations along with the muscular sensations1 there will be one or two more ways of representing to oneself this same point and the difficulty will only be increased4 ,in any case the following question comes up3 ,why do we think that all these representations so different from one another still represent the same point8 ,another remark3 ,i have just said that it is to our own body that we naturally refer exterior objects2 that we carry about everywhere with us a system of axes to which we refer all the points of space and that this system of axes seems to be invariably bound to our body4 ,it should be noticed that rigorously we could not speak of axes invariably bound to the body unless the different parts of this body were themselves invariably bound to one another4 ,as this is not the case1 we ought1 before referring exterior objects to these fictitious axes1 to suppose our body brought back to the initial attitude4 #e4 ,the ,notion of ,displacement ,i have shown in ',science and ,hypothesis' the preponderant rle played by the movements of our body in the genesis of the notion of space4 ,for a being completely immovable there would be neither space nor geometry2 in vain would exterior objects be displaced about him1 the variations which these displacements would make in his impressions would not be attributed by this being to changes of position1 but to simple changes of state2 this being would have no means of distinguishing these two sorts of changes1 and this distinction1 fundamental for us1 would have no meaning for him4 ,the movements that we impress upon our members have as effect the varying of the impressions produced on our senses by external objects2 other causes may likewise make them vary2 but we are led to distinguish the changes produced by our own motions and we easily discriminate them for two reasons3 7#a7 because they are voluntary2 7#b7 because they are accompanied by muscular sensations4 ,so we naturally divide the changes that our impressions may undergo into two categories to which perhaps ,i have given an inappropriate designation3 7#a7 the internal changes1 which are voluntary and accompanied by muscular sensations2 7#b7 the external changes1 having the opposite characteristics4 ,we then observe that among the external changes are some which can be corrected1 thanks to an internal change which brings everything back to the primitive state2 others can not be corrected in this way 7it is thus that1 when an exterior object is displaced1 we may then by changing our own position replace ourselves as regards this object in the same relative position as before1 so as to reestablish the original aggregate of impressions2 if this object was not displaced1 but changed its state1 that is impossible74 ,thence comes a new distinction among external changes3 those which may be so corrected we call changes of position2 and the others1 changes of state4 ,think1 for example1 of a sphere with one hemisphere blue and the other red2 it first presents to us the blue hemisphere1 then it so revolves as to present the red hemisphere4 ,now think of a spherical vase containing a blue liquid which becomes red in consequence of a chemical reaction4 ,in both cases the sensation of red has replaced that of blue2 our senses have experienced the same impressions which have succeeded each other in the same order1 and yet these two changes are regarded by us as very different2 the first is a displacement1 the second a change of state4 ,why8 ,because in the first case it is sufficient for me to go around the sphere to place myself opposite the blue hemisphere and reestablish the original blue sensation4 ,still more2 if the two hemispheres1 in place of being red and blue1 had been yellow and green1 how should ,i have interpreted the revolution of the sphere8 ,before1 the red succeeded the blue1 now the green succeeds the yellow2 and yet ,i say that the two spheres have undergone the same revolution1 that each has turned about its axis2 yet ,i can not say that the green is to yellow as the red is to blue2 how then am ,i led to decide that the two spheres have undergone the same displacement8 ,evidently because1 in one case as in the other1 ,i am able to reestablish the original sensation by going around the sphere1 by making the same movements1 and ,i know that ,i have made the same movements because ,i have felt the same muscular sensations2 to know it1 ,i do not need1 therefore1 to know geometry in advance and to represent to myself the movements of my body in geometric space4 ,another example3 ,an object is displaced before my eye2 its image was first formed at the center of the retina2 then it is formed at the border2 the old sensation was carried to me by a nerve fiber ending at the center of the retina2 the new sensation is carried to me by another nerve fiber starting from the border of the retina2 these two sensations are qualitatively different2 otherwise1 how could ,i distinguish them8 ,why then am ,i led to decide that these two sensations1 qualitatively different1 represent the same image1 which has been displaced8 ,it is because ,i can follow the object with the eye and by a displacement of the eye1 voluntary and accompanied by muscular sensations1 bring back the image to the center of the retina and reestablish the primitive sensation4 ,i suppose that the image of a red object has gone from the center ,a to the border ,b of the retina1 then that the image of a blue object goes in its turn from the center ,a to the border ,b of the retina2 ,i shall decide that these two objects have undergone the same displacement4 ,why8 ,because in both cases ,i shall have been able to reestablish the primitive sensation1 and that to do it ,i shall have had to execute the same movement of the eye1 and ,i shall know that my eye has executed the same movement because ,i shall have felt the same muscular sensations4 ,if ,i could not move my eye1 should ,i have any reason to suppose that the sensation of red at the center of the retina is to the sensation of red at the border of the retina as that of blue at the center is to that of blue at the border8 ,i should only have four sensations qualitatively different1 and if ,i were asked if they are connected by the proportion ,i have just stated1 the question would seem to me ridiculous1 just as if ,i were asked if there is an analogous proportion between an auditory sensation1 a tactile sensation and an olfactory sensation4 ,let us now consider the internal changes1 that is1 those which are produced by the voluntary movements of our body and which are accompanied by muscular changes4 ,they give rise to the two following observations1 analogous to those we have just made on the subject of external changes4 #a4 ,i may suppose that my body has moved from one point to another1 but that the same attitude is retained2 all the parts of the body have therefore retained or resumed the same relative situation1 although their absolute situation in space may have varied4 ,i may suppose that not only has the position of my body changed1 but that its attitude is no longer the same1 that1 for instance1 my arms which before were folded are now stretched out4 ,i should therefore distinguish the simple changes of position without change of attitude1 and the changes of attitude4 ,both would appear to me under form of muscular sensations4 ,how then am ,i led to distinguish them8 ,it is that the first may serve to correct an external change1 and that the others can not1 or at least can only give an imperfect correction4 ,this fact ,i proceed to explain as ,i would explain it to some one who already knew geometry1 but it need not thence be concluded that it is necessary already to know geometry to make this distinction2 before knowing geometry ,i ascertain the fact 7experimentally1 so to speak71 without being able to explain it4 ,but merely to make the distinction between the two kinds of change1 ,i do not need to explain the fact1 it suffices me to ascertain it4 ,however that may be1 the explanation is easy4 ,suppose that an exterior object is displaced2 if we wish the different parts of our body to resume with regard to this object their initial relative position1 it is necessary that these different parts should have resumed likewise their initial relative position with regard to one another4 ,only the internal changes which satisfy this latter condition will be capable of correcting the external change produced by the displacement of that object4 ,if1 therefore1 the relative position of my eye with regard to my finger has changed1 ,i shall still be able to replace the eye in its initial relative situation with regard to the object and reestablish thus the primitive visual sensations1 but then the relative position of the finger with regard to the object will have changed and the tactile sensations will not be reestablished4 #b4 ,we ascertain likewise that the same external change may be corrected by two internal changes corresponding to different muscular sensations4 ,here again ,i can ascertain this without knowing geometry2 and ,i have no need of anything else2 but ,i proceed to give the explanation of the fact1 employing geometrical language4 ,to go from the position ,a to the position ,b ,i may take several routes4 ,to the first of these routes will correspond a series ,s of muscular sensations2 to a second route will correspond another series ,s''1 of muscular sensations which generally will be completely different1 since other muscles will be used4 ,how am ,i led to regard these two series ,s and ,s'' as corresponding to the same displacement ,,ab8 ,it is because these two series are capable of correcting the same external change4 ,apart from that1 they have nothing in common4 ,let us now consider two external changes3 alpha and beta1 which shall be1 for instance1 the rotation of a sphere half blue1 half red1 and that of a sphere half yellow1 half green2 these two changes have nothing in common1 since the one is for us the passing of blue into red and the other the passing of yellow into green4 ,consider1 on the other hand1 two series of internal changes ,s and ,s''2 like the others1 they will have nothing in common4 ,and yet ,i say that alpha and beta correspond to the same displacement1 and that ,s and ,s'' correspond also to the same displacement4 ,why8 ,simply because ,s can correct alpha as well as beta and because alpha can be corrected by ,s'' as well as by ,s4 ,and then a question suggests itself3 ,if ,i have ascertained that ,s corrects alpha and beta and that ,s'' corrects alpha1 am ,i certain that ,s'' likewise corrects beta8 ,experiment alone can teach us whether this law is verified4 ,if it were not verified1 at least approximately1 there would be no geometry1 there would be no space1 because we should have no more interest in classifying the internal and external changes as ,i have just done1 and1 for instance1 in distinguishing changes of state from changes of position4 ,it is interesting to see what has been the rle of experience in all this4 ,it has shown me that a certain law is approximately verified4 ,it has not told me how space is1 and that it satisfies the condition in question4 ,i knew1 in fact1 before all experience1 that space satisfied this condition or that it would not be2 nor have ,i any right to say that experience told me that geometry is possible2 ,i very well see that geometry is possible1 since it does not imply contradiction2 experience only tells me that geometry is useful4 #f4 ,visual ,space ,although motor impressions have had1 as ,i have just explained1 an altogether preponderant influence in the genesis of the notion of space1 which never would have taken birth without them1 it will not be without interest to examine also the rle of visual impressions and to investigate how many dimensions 'visual space' has1 and for that purpose to apply to these impressions the definition of #c4 ,a first difficulty presents itself3 consider a red color sensation affecting a certain point of the retina2 and on the other hand a blue color sensation affecting the same point of the retina4 ,it is necessary that we have some means of recognizing that these two sensations1 qualitatively different1 have something in common4 ,now1 according to the considerations expounded in the preceding paragraph1 we have been able to recognize this only by the movements of the eye and the observations to which they have given rise4 ,if the eye were immovable1 or if we were unconscious of its movements1 we should not have been able to recognize that these two sensations1 of different quality1 had something in common2 we should not have been able to disengage from them what gives them a geometric character4 ,the visual sensations1 without the muscular sensations1 would have nothing geometric1 so that it may be said there is no pure visual space4 ,to do away with this difficulty1 consider only sensations of the same nature1 red sensations1 for instance1 differing one from another only as regards the point of the retina that they affect4 ,it is clear that ,i have no reason for making such an arbitrary choice among all the possible visual sensations1 for the purpose of uniting in the same class all the sensations of the same color1 whatever may be the point of the retina affected4 ,i should never have dreamt of it1 had ,i not before learned1 by the means we have just seen1 to distinguish changes of state from changes of position1 that is1 if my eye were immovable4 ,two sensations of the same color affecting two different parts of the retina would have appeared to me as qualitatively distinct1 just as two sensations of different color4 ,in restricting myself to red sensations1 ,i therefore impose upon myself an artificial limitation and ,i neglect systematically one whole side of the question2 but it is only by this artifice that ,i am able to analyze visual space without mingling any motor sensation4 ,imagine a line traced on the retina and dividing in two its surface2 and set apart the red sensations affecting a point of this line1 or those differing from them too little to be distinguished from them4 ,the aggregate of these sensations will form a sort of cut that ,i shall call ,c1 and it is clear that this cut suffices to divide the manifold of possible red sensations1 and that if ,i take two red sensations affecting two points situated on one side and the other of the line1 ,i can not pass from one of these sensations to the other in a continuous way without passing at a certain moment through a sensation belonging to the cut4 ,if1 therefore1 the cut has n dimensions1 the total manifold of my red sensations1 or if you wish1 the whole visual space1 will have n #a4 ,now1 ,i distinguish the red sensations affecting a point of the cut ,c4 ,the assemblage of these sensations will form a new cut ,c'4 ,it is clear that this will divide the cut ,c1 always giving to the word divide the same meaning4 ,if1 therefore1 the cut ,c' has n dimensions1 the cut ,c will have n #a and the whole of visual space n #b4 ,if all the red sensations affecting the same point of the retina were regarded as identical1 the cut ,c' reducing to a single element would have #j dimensions1 and visual space would have #b4 ,and yet most often it is said that the eye gives us the sense of a third dimension1 and enables us in a certain measure to recognize the distance of objects4 ,when we seek to analyze this feeling1 we ascertain that it reduces either to the consciousness of the convergence of the eyes1 or to that of the effort of accommodation which the ciliary muscle makes to focus the image4 ,two red sensations affecting the same point of the retina will therefore be regarded as identical only if they are accompanied by the same sensation of convergence and also by the same sensation of effort of accommodation or at least by sensations of convergence and accommodation so slightly different as to be indistinguishable4 ,on this account the cut ,c' is itself a continuum and the cut ,c has more than one dimension4 ,but it happens precisely that experience teaches us that when two visual sensations are accompanied by the same sensation of convergence1 they are likewise accompanied by the same sensation of accommodation4 ,if then we form a new cut ,c'' with all those of the sensations of the cut ,c'1 which are accompanied by a certain sensation of convergence1 in accordance with the preceding law they will all be indistinguishable and may be regarded as identical4 ,therefore ,c'' will not be a continuum and will have #j dimension2 and as ,c'' divides ,c' it will thence result that ,c' has one1 ,c two and the whole visual space three dimensions4 ,but would it be the same if experience had taught us the contrary and if a certain sensation of convergence were not always accompanied by the same sensation of accommodation8 ,in this case two sensations affecting the same point of the retina and accompanied by the same sense of convergence1 two sensations which consequently would both appertain to the cut ,c''1 could nevertheless be distinguished since they would be accompanied by two different sensations of accommodation4 ,therefore ,c'' would be in its turn a continuum and would have one dimension 7at least72 then ,c' would have two1 ,c three and the whole visual space would have four dimensions4 ,will it then be said that it is experience which teaches us that space has three dimensions1 since it is in setting out from an experimental law that we have come to attribute three to it8 ,but we have therein performed1 so to speak1 only an experiment in physiology2 and as also it would suffice to fit over the eyes glasses of suitable construction to put an end to the accord between the feelings of convergence and of accommodation1 are we to say that putting on spectacles is enough to make space have four dimensions and that the optician who constructed them has given one more dimension to space8 ,evidently not2 all we can say is that experience has taught us that it is convenient to attribute three dimensions to space4 ,but visual space is only one part of space1 and in even the notion of this space there is something artificial1 as ,i have explained at the beginning4 ,the real space is motor space and this it is that we shall examine in the following chapter4 ,,chapter ,,iv ,,space ,,and ,,its ,,three ,,dimensions #a4 ,the ,group of ,displacements ,let us sum up briefly the results obtained4 ,we proposed to investigate what was meant in saying that space has three dimensions and we have asked first what is a physical continuum and when it may be said to have n dimensions4 ,if we consider different systems of impressions and compare them with one another1 we often recognize that two of these systems of impressions are indistinguishable 7which is ordinarily expressed in saying that they are too close to one another1 and that our senses are too crude1 for us to distinguish them7 and we ascertain besides that two of these systems can sometimes be discriminated from one another though indistinguishable from a third system4 ,in that case we say the manifold of these systems of impressions forms a physical continuum ,c4 ,and each of these systems is called an element of the continuum ,c4 ,how many dimensions has this continuum8 ,take first two elements ,a and ,b of ,c1 and suppose there exists a series ,sigma of elements1 all belonging to the continuum ,c1 of such a sort that ,a and ,b are the two extreme terms of this series and that each term of the series is indistinguishable from the preceding4 ,if such a series ,sigma can be found1 we say that ,a and ,b are joined to one another2 and if any two elements of ,c are joined to one another1 we say that ,c is all of one piece4 ,now take on the continuum ,c a certain number of elements in a way altogether arbitrary4 ,the aggregate of these elements will be called a cut4 ,among the various series ,sigma which join ,a to ,b1 we shall distinguish those of which an element is indistinguishable from one of the elements of the cut 7we shall say that these are they which cut the cut7 and those of which all the elements are distinguishable from all those of the cut4 ,if all the series ,sigma which join ,a to ,b cut the cut1 we shall say that ,a and ,b are separated by the cut1 and that the cut divides ,c4 ,if we can not find on ,c two elements which are separated by the cut1 we shall say that the cut does not divide ,c4 ,these definitions laid down1 if the continuum ,c can be divided by cuts which do not themselves form a continuum1 this continuum ,c has only one dimension2 in the contrary case it has several4 ,if a cut forming a continuum of #a dimension suffices to divide ,c1 ,c will have #b dimensions2 if a cut forming a continuum of #b dimensions suffices1 ,c will have #c dimensions1 etc4 ,thanks to these definitions1 we can always recognize how many dimensions any physical continuum has4 ,it only remains to find a physical continuum which is1 so to speak1 equivalent to space1 of such a sort that to every point of space corresponds an element of this continuum1 and that to points of space very near one another correspond indistinguishable elements4 ,space will have then as many dimensions as this continuum4 ,the intermediation of this physical continuum1 capable of representation1 is indispensable2 because we can not represent space to ourselves1 and that for a multitude of reasons4 ,space is a mathematical continuum1 it is infinite1 and we can represent to ourselves only physical continua and finite objects4 ,the different elements of space1 which we call points1 are all alike1 and1 to apply our definition1 it is necessary that we know how to distinguish the elements from one another1 at least if they are not too close4 ,finally absolute space is nonsense1 and it is necessary for us to begin by referring space to a system of axes invariably bound to our body 7which we must always suppose put back in the initial attitude74 ,then ,i have sought to form with our visual sensations a physical continuum equivalent to space2 that certainly is easy and this example is particularly appropriate for the discussion of the number of dimensions2 this discussion has enabled us to see in what measure it is allowable to say that 'visual space' has three dimensions4 ,only this solution is incomplete and artificial4 ,i have explained why1 and it is not on visual space but on motor space that it is necessary to bring our efforts to bear4 ,i have then recalled what is the origin of the distinction we make between changes of position and changes of state4 ,among the changes which occur in our impressions1 we distinguish1 first the internal changes1 voluntary and accompanied by muscular sensations1 and the external changes1 having opposite characteristics4 ,we ascertain that it may happen that an external change may be corrected by an internal change which reestablishes the primitive sensations4 ,the external changes1 capable of being corrected by an internal change are called changes of position1 those not capable of it are called changes of state4 ,the internal changes capable of correcting an external change are called displacements of the whole body2 the others are called changes of attitude4 ,now let alpha and beta be two external changes1 alpha' and beta' two internal changes4 ,suppose that a may be corrected either by alpha' or by beta'1 and that alpha' can correct either alpha or beta2 experience tells us then that beta' can likewise correct beta4 ,in this case we say that alpha and beta correspond to the same displacement and also that alpha' and beta' correspond to the same displacement4 ,that postulated1 we can imagine a physical continuum which we shall call the continuum or group of displacements and which we shall define in the following manner4 ,the elements of this continuum shall be the internal changes capable of correcting an external change4 ,two of these internal changes alpha' and beta' shall be regarded as indistinguishable3 7#a7 if they are so naturally1 that is1 if they are too close to one another2 7#b7 if alpha' is capable of correcting the same external change as a third internal change naturally indistinguishable from beta'4 ,in this second case1 they will be1 so to speak1 indistinguishable by convention1 ,i mean by agreeing to disregard circumstances which might distinguish them4 ,our continuum is now entirely defined1 since we know its elements and have fixed under what conditions they may be regarded as indistinguishable4 ,we thus have all that is necessary to apply our definition and determine how many dimensions this continuum has4 ,we shall recognize that it has six4 ,the continuum of displacements is1 therefore1 not equivalent to space1 since the number of dimensions is not the same2 it is only related to space4 ,now how do we know that this continuum of displacements has six dimensions8 ,we know it by experience4 ,it would be easy to describe the experiments by which we could arrive at this result4 ,it would be seen that in this continuum cuts can be made which divide it and which are continua2 that these cuts themselves can be divided by other cuts of the second order which yet are continua1 and that this would stop only after cuts of the sixth order which would no longer be continua4 ,from our definitions that would mean that the group of displacements has six dimensions4 ,that would be easy1 ,i have said1 but that would be rather long2 and would it not be a little superficial8 ,this group of displacements1 we have seen1 is related to space1 and space could be deduced from it1 but it is not equivalent to space1 since it has not the same number of dimensions2 and when we shall have shown how the notion of this continuum can be formed and how that of space may be deduced from it1 it might always be asked why space of three dimensions is much more familiar to us than this continuum of six dimensions1 and consequently doubted whether it was by this detour that the notion of space was formed in the human mind4 #b4 ,identity of ,two ,points ,what is a point8 ,how do we know whether two points of space are identical or different8 ,or1 in other words1 when ,i say3 ,the object ,a occupied at the instant alpha the point which the object ,b occupies at the instant beta1 what does that mean8 ,such is the problem we set ourselves in the preceding chapter1 #d4 ,as ,i have explained it1 it is not a question of comparing the positions of the objects ,a and ,b in absolute space2 the question then would manifestly have no meaning4 ,it is a question of comparing the positions of these two objects with regard to axes invariably bound to my body1 supposing always this body replaced in the same attitude4 ,i suppose that between the instants alpha and beta ,i have moved neither my body nor my eye1 as ,i know from my muscular sense4 ,nor have ,i moved either my head1 my arm or my hand4 ,i ascertain that at the instant alpha impressions that ,i attributed to the object ,a were transmitted to me1 some by one of the fibers of my optic nerve1 the others by one of the sensitive tactile nerves of my finger2 ,i ascertain that at the instant beta other impressions which ,i attribute to the object ,b are transmitted to me1 some by this same fiber of the optic nerve1 the others by this same tactile nerve4 ,here ,i must pause for an explanation2 how am ,i told that this impression which ,i attribute to ,a1 and that which ,i attribute to ,b1 impressions which are qualitatively different1 are transmitted to me by the same nerve8 ,must we suppose1 to take for example the visual sensations1 that ,a produces two simultaneous sensations1 a sensation purely luminous a and a colored sensation a'1 that ,b produces in the same way simultaneously a luminous sensation b and a colored sensation b'1 that if these different sensations are transmitted to me by the same retinal fiber1 a is identical with b1 but that in general the colored sensations a' and b' produced by different bodies are different8 ,in that case it would be the identity of the sensation a which accompanies a' with the sensation b which accompanies b'1 which would tell that all these sensations are transmitted to me by the same fiber4 ,however it may be with this hypothesis and although ,i am led to prefer to it others considerably more complicated1 it is certain that we are told in some way that there is something in common between these sensations a a' and b b'1 without which we should have no means of recognizing that the object ,b has taken the place of the object ,a4 ,therefore ,i do not further insist and ,i recall the hypothesis ,i have just made3 ,i suppose that ,i have ascertained that the impressions which ,i attribute to ,b are transmitted to me at the instant beta by the same fibers1 optic as well as tactile1 which1 at the instant alpha1 had transmitted to me the impressions that ,i attributed to ,a4 ,if it is so1 we shall not hesitate to declare that the point occupied by ,b at the instant beta is identical with the point occupied by ,a at the instant alpha4 ,i have just enunciated two conditions for these points being identical2 one is relative to sight1 the other to touch4 ,let us consider them separately4 ,the first is necessary1 but is not sufficient4 ,the second is at once necessary and sufficient4 ,a person knowing geometry could easily explain this in the following manner3 ,let ,o be the point of the retina where is formed at the instant alpha the image of the body ,a2 let ,m be the point of space occupied at the instant alpha by this body ,a2 let ,m' be the point of space occupied at the instant beta by the body ,b4 ,for this body ,b to form its image in ,o1 it is not necessary that the points ,m and ,m' coincide2 since vision acts at a distance1 it suffices for the three points ,o ,m ,m' to be in a straight line4 ,this condition that the two objects form their image on ,o is therefore necessary1 but not sufficient for the points ,m and ,m' to coincide4 ,let now ,p be the point occupied by my finger and where it remains1 since it does not budge4 ,as touch does not act at a distance1 if the body ,a touches my finger at the instant alpha1 it is because ,m and ,p coincide2 if ,b touches my finger at the instant beta1 it is because ,m' and ,p coincide4 ,therefore ,m and ,m' coincide4 ,thus this condition that if ,a touches my finger at the instant alpha1 ,b touches it at the instant beta1 is at once necessary and sufficient for ,m and ,m' to coincide4 ,but we who1 as yet1 do not know geometry can not reason thus2 all that we can do is to ascertain experimentally that the first condition relative to sight may be fulfilled without the second1 which is relative to touch1 but that the second can not be fulfilled without the first4 ,suppose experience had taught us the contrary1 as might well be2 this hypothesis contains nothing absurd4 ,suppose1 therefore1 that we had ascertained experimentally that the condition relative to touch may be fulfilled without that of sight being fulfilled and that1 on the contrary1 that of sight can not be fulfilled without that of touch being also4 ,it is clear that if this were so we should conclude that it is touch which may be exercised at a distance1 and that sight does not operate at a distance4 ,but this is not all2 up to this time ,i have supposed that to determine the place of an object ,i have made use only of my eye and a single finger2 but ,i could just as well have employed other means1 for example1 all my other fingers4 ,i suppose that my first finger receives at the instant alpha a tactile impression which ,i attribute to the object ,a4 ,i make a series of movements1 corresponding to a series ,s of muscular sensations4 ,after these movements1 at the instant alpha'1 my second finger receives a tactile impression that ,i attribute likewise to ,a4 ,afterward1 at the instant beta1 without my having budged1 as my muscular sense tells me1 this same second finger transmits to me anew a tactile impression which ,i attribute this time to the object ,b2 ,i then make a series of movements1 corresponding to a series ,s' of muscular sensations4 ,i know that this series ,s' is the inverse of the series ,s and corresponds to contrary movements4 ,i know this because many previous experiences have shown me that if ,i made successively the two series of movements corresponding to ,s and to ,s'1 the primitive impressions would be reestablished1 in other words1 that the two series mutually compensate4 ,that settled1 should ,i expect that at the instant beta'1 when the second series of movements is ended1 my first finger would feel a tactile impression attributable to the object ,b8 ,to answer this question1 those already knowing geometry would reason as follows3 ,there are chances that the object ,a has not budged1 between the instants alpha and alpha'1 nor the object ,b between the instants beta and beta'2 assume this4 ,at the instant alpha1 the object ,a occupied a certain point ,m of space4 ,now at this instant it touched my first finger1 and as touch does not operate at a distance1 my first finger was likewise at the point ,m4 ,i afterward made the series ,s of movements and at the end of this series1 at the instant alpha'1 ,i ascertained that the object ,a touched my second finger4 ,i thence conclude that this second finger was then at ,m1 that is1 that the movements ,s had the result of bringing the second finger to the place of the first4 ,at the instant beta the object ,b has come in contact with my second finger3 as ,i have not budged1 this second finger has remained at ,m2 therefore the object ,b has come to ,m2 by hypothesis it does not budge up to the instant beta'4 ,but between the instants beta and beta' ,i have made the movements ,s'2 as these movements are the inverse of the movements ,s1 they must have for effect bringing the first finger in the place of the second4 ,at the instant beta' this first finger will1 therefore1 be at ,m2 and as the object ,b is likewise at ,m1 this object ,b will touch my first finger4 ,to the question put1 the answer should therefore be yes4 ,we who do not yet know geometry can not reason thus2 but we ascertain that this anticipation is ordinarily realized2 and we can always explain the exceptions by saying that the object ,a has moved between the instants alpha and alpha'1 or the object ,b between the instants beta and beta'4 ,but could not experience have given a contrary result8 ,would this contrary result have been absurd in itself8 ,evidently not4 ,what should we have done then if experience had given this contrary result8 ,would all geometry thus have become impossible8 ,not the least in the world4 ,we should have contented ourselves with concluding that touch can operate at a distance4 ,when ,i say1 touch does not operate at a distance1 but sight operates at a distance1 this assertion has only one meaning1 which is as follows3 ,to recognize whether ,b occupies at the instant beta the point occupied by ,a at the instant alpha1 ,i can use a multitude of different criteria4 ,in one my eye intervenes1 in another my first finger1 in another my second finger1 etc4 ,well1 it is sufficient for the criterion relative to one of my fingers to be satisfied in order that all the others should be satisfied1 but it is not sufficient that the criterion relative to the eye should be4 ,this is the sense of my assertion4 ,i content myself with affirming an experimental fact which is ordinarily verified4 ,at the end of the preceding chapter we analyzed visual space2 we saw that to engender this space it is necessary to bring in the retinal sensations1 the sensation of convergence and the sensation of accommodation2 that if these last two were not always in accord1 visual space would have four dimensions in place of three2 we also saw that if we brought in only the retinal sensations1 we should obtain 'simple visual space1' of only two dimensions4 ,on the other hand1 consider tactile space1 limiting ourselves to the sensations of a single finger1 that is in sum to the assemblage of positions this finger can occupy4 ,this tactile space that we shall analyze in the following section and which consequently ,i ask permission not to consider further for the moment1 this tactile space1 ,i say1 has three dimensions4 ,why has space properly so called as many dimensions as tactile space and more than simple visual space8 ,it is because touch does not operate at a distance1 while vision does operate at a distance4 ,these two assertions have the same meaning and we have just seen what this is4 ,now ,i return to a point over which ,i passed rapidly in order not to interrupt the discussion4 ,how do we know that the impressions made on our retina by ,a at the instant alpha and ,b at the instant beta are transmitted by the same retinal fiber1 although these impressions are qualitatively different8 ,i have suggested a simple hypothesis1 while adding that other hypotheses1 decidedly more complex1 would seem to me more probably true4 ,here then are these hypotheses1 of which ,i have already said a word4 ,how do we know that the impressions produced by the red object ,a at the instant alpha1 and by the blue object ,b at the instant beta1 if these two objects have been imaged on the same point of the retina1 have something in common8 ,the simple hypothesis above made may be rejected and we may suppose that these two impressions1 qualitatively different1 are transmitted by two different though contiguous nervous fibers4 ,what means have ,i then of knowing that these fibers are contiguous8 ,it is probable that we should have none1 if the eye were immovable4 ,it is the movements of the eye which have told us that there is the same relation between the sensation of blue at the point ,a and the sensation of blue at the point ,b of the retina as between the sensation of red at the point ,a and the sensation of red at the point ,b4 ,they have shown us1 in fact1 that the same movements1 corresponding to the same muscular sensations1 carry us from the first to the second1 or from the third to the fourth4 ,i do not emphasize these considerations1 which belong1 as one sees1 to the question of local signs raised by ,lotze4 #c4 ,tactile ,space ,thus ,i know how to recognize the identity of two points1 the point occupied by ,a at the instant alpha and the point occupied by ,b at the instant beta1 but only on one condition1 namely1 that ,i have not budged between the instants alpha and beta4 ,that does not suffice for our object4 ,suppose1 therefore1 that ,i have moved in any manner in the interval between these two instants1 how shall ,i know whether the point occupied by ,a at the instant alpha is identical with the point occupied by ,b at the instant beta8 ,i suppose that at the instant alpha1 the object ,a was in contact with my first finger and that in the same way1 at the instant beta1 the object ,b touches this first finger2 but at the same time my muscular sense has told me that in the interval my body has moved4 ,i have considered above two series of muscular sensations ,s and ,s'1 and ,i have said it sometimes happens that we are led to consider two such series ,s and ,s' as inverse one of the other1 because we have often observed that when these two series succeed one another our primitive impressions are reestablished4 ,if then my muscular sense tells me that ,i have moved between the two instants alpha and beta1 but so as to feel successively the two series of muscular sensations ,s and ,s' that ,i consider inverses1 ,i shall still conclude1 just as if ,i had not budged1 that the points occupied by ,a at the instant alpha and by ,b at the instant beta are identical1 if ,i ascertain that my first finger touches ,a at the instant alpha1 and ,b at the instant beta4 ,this solution is not yet completely satisfactory1 as one will see4 ,let us see1 in fact1 how many dimensions it would make us attribute to space4 ,i wish to compare the two points occupied by ,a and ,b at the instants alpha and beta1 or 7what amounts to the same thing since ,i suppose that my finger touches ,a at the instant alpha and ,b at the instant beta7 ,i wish to compare the two points occupied by my finger at the two instants alpha and beta4 ,the sole means ,i use for this comparison is the series ,sigma of muscular sensations which have accompanied the movements of my body between these two instants4 ,the different imaginable series ,sigma form evidently a physical continuum of which the number of dimensions is very great4 ,let us agree1 as ,i have done1 not to consider as distinct the two series ,sigma and ,sigma ,s ,s'1 when ,s and ,s' are inverses one of the other in the sense above given to this word2 in spite of this agreement1 the aggregate of distinct series ,sigma will still form a physical continuum and the number of dimensions will be less but still very great4 ,to each of these series ,sigma corresponds a point of space2 to two series ,sigma and ,sigma' thus correspond two points ,m and ,m'4 ,the means we have hitherto used enable us to recognize that ,m and ,m' are not distinct in two cases3 7#a7 if ,sigma is identical with ,sigma'2 7#b7 if ,sigma' ,sigma ,s ,s'1 ,s and ,s' being inverses one of the other4 ,if in all the other cases we should regard ,m and ,m' as distinct1 the manifold of points would have as many dimensions as the aggregate of distinct series ,sigma1 that is1 much more than three4 ,for those who already know geometry1 the following explanation would be easily comprehensible4 ,among the imaginable series of muscular sensations1 there are those which correspond to series of movements where the finger does not budge4 ,i say that if one does not consider as distinct the series ,sigma and ,sigma sigma1 where the series sigma corresponds to movements where the finger does not budge1 the aggregate of series will constitute a continuum of three dimensions1 but that if one regards as distinct two series ,sigma and ,sigma' unless ,sigma' ,sigma ,s ,s'1 ,s and ,s' being inverses1 the aggregate of series will constitute a continuum of more than three dimensions4 ,in fact1 let there be in space a surface ,a1 on this surface a line ,b1 on this line a point ,m4 ,let ,c#j be the aggregate of all series ,sigma4 ,let ,c#a be the aggregate of all the series ,sigma1 such that at the end of corresponding movements the finger is found upon the surface ,a1 and ,c#b or ,c#c the aggregate of series ,sigma such that at the end the finger is found on ,b1 or at ,m4 ,it is clear1 first that ,c#a will constitute a cut which will divide ,c#j1 that ,c#b will be a cut which will divide ,c#a1 and ,c#c a cut which will divide ,c#b4 ,thence it results1 in accordance with our definitions1 that if ,c#c is a continuum of n dimensions1 ,c#j will be a physical continuum of n #c dimensions4 ,therefore1 let ,sigma and ,sigma' ,sigma sigma be two series forming part of ,c#c2 for both1 at the end of the movements1 the finger is found at ,m2 thence results that at the beginning and at the end of the series sigma the finger is at the same point ,m4 ,this series sigma is therefore one of those which correspond to movements where the finger does not budge4 ,if ,sigma and ,sigma sigma are not regarded as distinct1 all the series of ,c#c blend into one2 therefore ,c#c will have #j dimension1 and ,c#j will have #c1 as ,i wished to prove4 ,if1 on the contrary1 ,i do not regard ,sigma and ,sigma sigma as blending 7unless sigma ,s ,s'1 ,s and ,s' being inverses71 it is clear that ,c#c will contain a great number of series of distinct sensations2 because1 without the finger budging1 the body may take a multitude of different attitudes4 ,then ,c#c will form a continuum and ,c#j will have more than three dimensions1 and this also ,i wished to prove4 ,we who do not yet know geometry can not reason in this way2 we can only verify4 ,but then a question arises2 how1 before knowing geometry1 have we been led to distinguish from the others these series sigma where the finger does not budge8 ,it is1 in fact1 only after having made this distinction that we could be led to regard ,sigma and ,sigma sigma as identical1 and it is on this condition alone1 as we have just seen1 that we can arrive at space of three dimensions4 ,we are led to distinguish the series sigma1 because it often happens that when we have executed the movements which correspond to these series sigma of muscular sensations1 the tactile sensations which are transmitted to us by the nerve of the finger that we have called the first finger1 persist and are not altered by these movements4 ,experience alone tells us that and it alone could tell us4 ,if we have distinguished the series of muscular sensations ,s ,s' formed by the union of two inverse series1 it is because they preserve the totality of our impressions2 if now we distinguish the series sigma1 it is because they preserve certain of our impressions4 7,when ,i say that a series of muscular sensations ,s 'preserves' one of our impressions ,a1 ,i mean that we ascertain that if we feel the impression ,a1 then the muscular sensations ,s1 we still feel the impression ,a after these sensations ,s47 ,i have said above it often happens that the series sigma do not alter the tactile impressions felt by our first finger2 ,i said often1 ,i did not say always4 ,this it is that we express in our ordinary language by saying that the tactile impressions would not be altered if the finger has not moved1 on the condition that neither has the object ,a1 which was in contact with this finger1 moved4 ,before knowing geometry1 we could not give this explanation2 all we could do is to ascertain that the impression often persists1 but not always4 ,but that the impression often continues is enough to make the series sigma appear remarkable to us1 to lead us to put in the same class the series ,sigma and ,sigma sigma1 and hence not regard them as distinct4 ,under these conditions we have seen that they will engender a physical continuum of three dimensions4 ,behold then a space of three dimensions engendered by my first finger4 ,each of my fingers will create one like it4 ,it remains to consider how we are led to regard them as identical with visual space1 as identical with geometric space4 ,but one reflection before going further2 according to the foregoing1 we know the points of space1 or more generally the final situation of our body1 only by the series of muscular sensations revealing to us the movements which have carried us from a certain initial situation to this final situation4 ,but it is clear that this final situation will depend1 on the one hand1 upon these movements and1 on the other hand1 upon the initial situation from which we set out4 ,now these movements are revealed to us by our muscular sensations2 but nothing tells us the initial situation2 nothing can distinguish it for us from all the other possible situations4 ,this puts well in evidence the essential relativity of space4 #d4 ,identity of the ,different ,spaces ,we are therefore led to compare the two continua ,c and ,c' engendered1 for instance1 one by my first finger ,d1 the other by my second finger ,d'4 ,these two physical continua both have three dimensions4 ,to each element of the continuum ,c1 or1 if you prefer1 to each point of the first tactile space1 corresponds a series of muscular sensations ,sigma1 which carry me from a certain initial situation to a certain final situation4#h ,moreover1 the same point of this first space will correspond to ,sigma and ,sigma sigma1 if sigma is a series of which we know that it does not make the finger ,d move4 #h ,in place of saying that we refer space to axes rigidly bound to our body1 perhaps it would be better to say1 in conformity to what precedes1 that we refer it to axes rigidly bound to the initial situation of our body4 ,similarly to each element of the continuum ,c'1 or to each point of the second tactile space1 corresponds a series of sensations ,sigma'1 and the same point will correspond to ,sigma' and to ,sigma' sigma'1 if sigma' is a series which does not make the finger ,d' move4 ,what makes us distinguish the various series designated sigma from those called sigma' is that the first do not alter the tactile impressions felt by the finger ,d and the second preserve those the finger ,d' feels4 ,now see what we ascertain3 in the beginning my finger ,d' feels a sensation ,a'2 ,i make movements which produce muscular sensations ,s2 my finger ,d feels the impression ,a2 ,i make movements which produce a series of sensations sigma2 my finger ,d continues to feel the impression ,a1 since this is the characteristic property of the series sigma2 ,i then make movements which produce the series ,s' of muscular sensations1 inverse to ,s in the sense above given to this word4 ,i ascertain then that my finger ,d' feels anew the impression ,a'4 7,it is of course understood that ,s has been suitably chosen47 ,this means that the series ,s sigma ,s'1 preserving the tactile impressions of the finger ,d'1 is one of the series ,i have called sigma'4 ,inversely1 if one takes any series sigma'1 ,s' sigma' ,s will be one of the series that we call sigma'4 ,thus if ,s is suitably chosen1 ,s sigma ,s' will be a series sigma'1 and by making sigma vary in all possible ways1 we shall obtain all the possible series sigma'4 ,not yet knowing geometry1 we limit ourselves to verifying all that1 but here is how those who know geometry would explain the fact4 ,in the beginning my finger ,d' is at the point ,m1 in contact with the object a1 which makes it feel the impression ,a'4 ,i make the movements corresponding to the series ,s2 ,i have said that this series should be suitably chosen1 ,i should so make this choice that these movements carry the finger ,d to the point originally occupied by the finger ,d'1 that is1 to the point ,m2 this finger ,d will thus be in contact with the object a1 which will make it feel the impression ,a4 ,i then make the movements corresponding to the series sigma2 in these movements1 by hypothesis1 the position of the finger ,d does not change1 this finger therefore remains in contact with the object a and continues to feel the impression ,a4 ,finally ,i make the movements corresponding to the series ,s'4 ,as ,s' is inverse to ,s1 these movements carry the finger ,d' to the point previously occupied by the finger ,d1 that is1 to the point ,m4 ,if1 as may be supposed1 the object a has not budged1 this finger ,d' will be in contact with this object and will feel anew the impression ,a'4444 ,q4,e4,d4 ,let us see the consequences4 ,i consider a series of muscular sensations ,sigma4 ,to this series will correspond a point ,m of the first tactile space4 ,now take again the two series ,s and ,s'1 inverses of one another1 of which we have just spoken4 ,to the series ,s ,sigma ,s' will correspond a point ,n of the second tactile space1 since to any series of muscular sensations corresponds1 as we have said1 a point1 whether in the first space or in the second4 ,i am going to consider the two points ,n and ,m1 thus defined1 as corresponding4 ,what authorizes me so to do8 ,for this correspondence to be admissible1 it is necessary that if two points ,m and ,m'1 corresponding in the first space to two series ,sigma and ,sigma'1 are identical1 so also are the two corresponding points of the second space ,n and ,n'1 that is1 the two points which correspond to the two series ,s ,sigma ,s' and ,s ,sigma' ,s'4 ,now we shall see that this condition is fulfilled4 ,first a remark4 ,as ,s and ,s' are inverses of one another1 we shall have ,s ,s' #j1 and consequently ,s ,s' ,sigma ,sigma ,s ,s' ,sigma1 or again ,sigma ,s ,s' ,sigma' ,sigma ,sigma'2 but it does not follow that we have ,s ,sigma ,s' ,sigma2 because1 though we have used the addition sign to represent the succession of our sensations1 it is clear that the order of this succession is not indifferent3 we can not1 therefore1 as in ordinary addition1 invert the order of the terms2 to use abridged language1 our operations are associative1 but not commutative4 ,that fixed1 in order that ,sigma and ,sigma' should correspond to the same point ,m ,m' of the first space1 it is necessary and sufficient for us to have ,sigma' ,sigma sigma4 ,we shall then have3 ,s ,sigma' ,s' ,s ,sigma sigma ,s' ,s ,sigma ,s' ,s sigma ,s'4 ,but we have just ascertained that ,s sigma ,s' was one of the series sigma'4 ,we shall therefore have3 ,s ,sigma' ,s' ,s ,sigma ,s' sigma'1 which means that the series ,s ,sigma' ,s' and ,s ,sigma ,s' correspond to the same point ,n ,n' of the second space4 ,q4,e4,d4 ,our two spaces therefore correspond point for point2 they can be 'transformed' one into the other2 they are isomorphic4 ,how are we led to conclude thence that they are identical8 ,consider the two series sigma and ,s sigma ,s' sigma'4 ,i have said that often1 but not always1 the series sigma preserves the tactile impression ,a felt by the finger ,d2 and similarly it often happens1 but not always1 that the series sigma' preserves the tactile impression ,a' felt by the finger ,d'4 ,now ,i ascertain that it happens very often 7that is1 much more often than what ,i have just called 'often'7 that when the series sigma has preserved the impression ,a of the finger ,d1 the series sigma' preserves at the same time the impression ,a' of the finger ,d'2 and1 inversely1 that if the first impression is altered1 the second is likewise4 ,that happens very often1 but not always4 ,we interpret this experimental fact by saying that the unknown object a which gives the impression ,a to the finger ,d is identical with the unknown object a' which gives the impression ,a' to the finger ,d'4 ,and in fact when the first object moves1 which the disappearance of the impression ,a tells us1 the second likewise moves1 since the impression ,a' disappears likewise4 ,when the first object remains motionless1 the second remains motionless4 ,if these two objects are identical1 as the first is at the point ,m of the first space and the second at the point ,n of the second space1 these two points are identical4 ,this is how we are led to regard these two spaces as identical2 or better1 this is what we mean when we say that they are identical4 ,what we have just said of the identity of the two tactile spaces makes unnecessary our discussing the question of the identity of tactile space and visual space1 which could be treated in the same way4 #e4 ,space and ,empiricism ,it seems that ,i am about to be led to conclusions in conformity with empiristic ideas4 ,i have1 in fact1 sought to put in evidence the rle of experience and to analyze the experimental facts which intervene in the genesis of space of three dimensions4 ,but whatever may be the importance of these facts1 there is one thing we must not forget and to which besides ,i have more than once called attention4 ,these experimental facts are often verified but not always4 ,that evidently does not mean that space has often three dimensions1 but not always4 ,i know well that it is easy to save oneself and that1 if the facts do not verify1 it will be easily explained by saying that the exterior objects have moved4 ,if experience succeeds1 we say that it teaches us about space2 if it does not succeed1 we hie to exterior objects which we accuse of having moved2 in other words1 if it does not succeed1 it is given a fillip4 ,these fillips are legitimate2 ,i do not refuse to admit them2 but they suffice to tell us that the properties of space are not experimental truths1 properly so called4 ,if we had wished to verify other laws1 we could have succeeded also1 by giving other analogous fillips4 ,should we not always have been able to justify these fillips by the same reasons8 ,one could at most have said to us3 ',your fillips are doubtless legitimate1 but you abuse them2 why move the exterior objects so often8' ,to sum up1 experience does not prove to us that space has three dimensions2 it only proves to us that it is convenient to attribute three to it1 because thus the number of fillips is reduced to a minimum4 ,i will add that experience brings us into contact only with representative space1 which is a physical continuum1 never with geometric space1 which is a mathematical continuum4 ,at the very most it would appear to tell us that it is convenient to give to geometric space three dimensions1 so that it may have as many as representative space4 ,the empiric question may be put under another form4 ,is it impossible to conceive physical phenomena1 the mechanical phenomena1 for example1 otherwise than in space of three dimensions8 ,we should thus have an objective experimental proof1 so to speak1 independent of our physiology1 of our modes of representation4 ,but it is not so2 ,i shall not here discuss the question completely1 ,i shall confine myself to recalling the striking example given us by the mechanics of ,hertz4 ,you know that the great physicist did not believe in the existence of forces1 properly so called2 he supposed that visible material points are subjected to certain invisible bonds which join them to other invisible points and that it is the effect of these invisible bonds that we attribute to forces4 ,but that is only a part of his ideas4 ,suppose a system formed of n material points1 visible or not2 that will give in all #cn coordinates2 let us regard them as the coordinates of a single point in space of #cn dimensions4 ,this single point would be constrained to remain upon a surface 7of any number of dimensions #cn7 in virtue of the bonds of which we have just spoken2 to go on this surface from one point to another1 it would always take the shortest way2 this would be the single principle which would sum up all mechanics4 ,whatever should be thought of this hypothesis1 whether we be allured by its simplicity1 or repelled by its artificial character1 the simple fact that ,hertz was able to conceive it1 and to regard it as more convenient than our habitual hypotheses1 suffices to prove that our ordinary ideas1 and1 in particular1 the three dimensions of space1 are in no wise imposed upon mechanics with an invincible force4 #f4 ,mind and ,space ,experience1 therefore1 has played only a single rle1 it has served as occasion4 ,but this rle was none the less very important2 and ,i have thought it necessary to give it prominence4 ,this rle would have been useless if there existed an a priori form imposing itself upon our sensitivity1 and which was space of three dimensions4 ,does this form exist1 or1 if you choose1 can we represent to ourselves space of more than three dimensions8 ,and first what does this question mean8 ,in the true sense of the word1 it is clear that we can not represent to ourselves space of four1 nor space of three1 dimensions2 we can not first represent them to ourselves empty1 and no more can we represent to ourselves an object either in space of four1 or in space of three1 dimensions3 7#a7 ,because these spaces are both infinite and we can not represent to ourselves a figure in space1 that is1 the part in the whole1 without representing the whole1 and that is impossible1 because it is infinite2 7#b7 because these spaces are both mathematical continua1 and we can represent to ourselves only the physical continuum2 7#c7 because these spaces are both homogeneous1 and the frames in which we enclose our sensations1 being limited1 can not be homogeneous4 ,thus the question put can only be understood in one way2 is it possible to imagine that1 the results of the experiences related above having been different1 we might have been led to attribute to space more than three dimensions2 to imagine1 for instance1 that the sensation of accommodation might not be constantly in accord with the sensation of convergence of the eyes2 or indeed that the experiences of which we have spoken in #b1 and of which we express the result by saying 'that touch does not operate at a distance1' might have led us to an inverse conclusion4 ,and then yes evidently that is possible2 from the moment one imagines an experience1 one imagines just thereby the two contrary results it may give4 ,that is possible1 but that is difficult1 because we have to overcome a multitude of associations of ideas1 which are the fruit of a long personal experience and of the still longer experience of the race4 ,is it these associations 7or at least those of them that we have inherited from our ancestors71 which constitute this a priori form of which it is said that we have pure intuition8 ,then ,i do not see why one should declare it refractory to analysis and should deny me the right of investigating its origin4 ,when it is said that our sensations are 'extended' only one thing can be meant1 that is that they are always associated with the idea of certain muscular sensations1 corresponding to the movements which enable us to reach the object which causes them1 which enable us1 in other words1 to defend ourselves against it4 ,and it is just because this association is useful for the defense of the organism1 that it is so old in the history of the species and that it seems to us indestructible4 ,nevertheless1 it is only an association and we can conceive that it may be broken2 so that we may not say that sensation can not enter consciousness without entering in space1 but that in fact it does not enter consciousness without entering in space1 which means1 without being entangled in this association4 ,no more can ,i understand one's saying that the idea of time is logically subsequent to space1 since we can represent it to ourselves only under the form of a straight line2 as well say that time is logically subsequent to the cultivation of the prairies1 since it is usually represented armed with a scythe4 ,that one can not represent to himself simultaneously the different parts of time1 goes without saying1 since the essential character of these parts is precisely not to be simultaneous4 ,that does not mean that we have not the intuition of time4 ,so far as that goes1 no more should we have that of space1 because neither can we represent it1 in the proper sense of the word1 for the reasons ,i have mentioned4 ,what we represent to ourselves under the name of straight is a crude image which as ill resembles the geometric straight as it does time itself4 ,why has it been said that every attempt to give a fourth dimension to space always carries this one back to one of the other three8 ,it is easy to understand4 ,consider our muscular sensations and the 'series' they may form4 ,in consequence of numerous experiences1 the ideas of these series are associated together in a very complex woof1 our series are classed4 ,allow me1 for convenience of language1 to express my thought in a way altogether crude and even inexact by saying that our series of muscular sensations are classed in three classes corresponding to the three dimensions of space4 ,of course this classification is much more complicated than that1 but that will suffice to make my reasoning understood4 ,if ,i wish to imagine a fourth dimension1 ,i shall suppose another series of muscular sensations1 making part of a fourth class4 ,but as all my muscular sensations have already been classed in one of the three pre-existent classes1 ,i can only represent to myself a series belonging to one of these three classes1 so that my fourth dimension is carried back to one of the other three4 ,what does that prove8 ,this3 that it would have been necessary first to destroy the old classification and replace it by a new one in which the series of muscular sensations should have been distributed into four classes4 ,the difficulty would have disappeared4 ,it is presented sometimes under a more striking form4 ,suppose ,i am enclosed in a chamber between the six impassable boundaries formed by the four walls1 the floor and the ceiling2 it will be impossible for me to get out and to imagine my getting out4 ,pardon1 can you not imagine that the door opens1 or that two of these walls separate8 ,but of course1 you answer1 one must suppose that these walls remain immovable4 ,yes1 but it is evident that ,i have the right to move2 and then the walls that we suppose absolutely at rest will be in motion with regard to me4 ,yes1 but such a relative motion can not be arbitrary2 when objects are at rest1 their relative motion with regard to any axes is that of a rigid solid2 now1 the apparent motions that you imagine are not in conformity with the laws of motion of a rigid solid4 ,yes1 but it is experience which has taught us the laws of motion of a rigid solid2 nothing would prevent our imagining them different4 ,to sum up1 for me to imagine that ,i get out of my prison1 ,i have only to imagine that the walls seem to open1 when ,i move4 ,i believe1 therefore1 that if by space is understood a mathematical continuum of three dimensions1 were it otherwise amorphous1 it is the mind which constructs it1 but it does not construct it out of nothing2 it needs materials and models4 ,these materials1 like these models1 preexist within it4 ,but there is not a single model which is imposed upon it2 it has choice2 it may choose1 for instance1 between space of four and space of three dimensions4 ,what then is the rle of experience8 ,it gives the indications following which the choice is made4 ,another thing3 whence does space get its quantitative character8 ,it comes from the rle which the series of muscular sensations play in its genesis4 ,these are series which may repeat themselves1 and it is from their repetition that number comes2 it is because they can repeat themselves indefinitely that space is infinite4 ,and finally we have seen1 at the end of section #c1 that it is also because of this that space is relative4 ,so it is repetition which has given to space its essential characteristics2 now1 repetition supposes time2 this is enough to tell that time is logically anterior to space4 #g4 ,rle of the ,semicircular ,canals ,i have not hitherto spoken of the rle of certain organs to which the physiologists attribute with reason a capital importance1 ,i mean the semicircular canals4 ,numerous experiments have sufficiently shown that these canals are necessary to our sense of orientation2 but the physiologists are not entirely in accord2 two opposing theories have been proposed1 that of ,mach-,delage and that of ,m4 de ,cyon4 ,m4 de ,cyon is a physiologist who has made his name illustrious by important discoveries on the innervation of the heart2 ,i can not1 however1 agree with his ideas on the question before us4 ,not being a physiologist1 ,i hesitate to criticize the experiments he has directed against the adverse theory of ,mach-,delage2 it seems to me1 however1 that they are not convincing1 because in many of them the total pressure was made to vary in one of the canals1 while1 physiologically1 what varies is the difference between the pressures on the two extremities of the canal2 in others the organs were subjected to profound lesions1 which must alter their functions4 ,besides1 this is not important2 the experiments1 if they were irreproachable1 might be convincing against the old theory4 ,they would not be convincing for the new theory4 ,in fact1 if ,i have rightly understood the theory1 my explaining it will be enough for one to understand that it is impossible to conceive of an experiment confirming it4 ,the three pairs of canals would have as sole function to tell us that space has three dimensions4 ,japanese mice have only two pairs of canals2 they believe1 it would seem1 that space has only two dimensions1 and they manifest this opinion in the strangest way2 they put themselves in a circle1 and1 so ordered1 they spin rapidly around4 ,the lampreys1 having only one pair of canals1 believe that space has only one dimension1 but their manifestations are less turbulent4 ,it is evident that such a theory is inadmissible4 ,the sense-organs are designed to tell us of changes which happen in the exterior world4 ,we could not understand why the ,creator should have given us organs destined to cry without cease3 ,remember that space has three dimensions1 since the number of these three dimensions is not subject to change4 ,we must1 therefore1 come back to the theory of ,mach-,delage4 ,what the nerves of the canals can tell us is the difference of pressure on the two extremities of the same canal1 and thereby3 7#a7 the direction of the vertical with regard to three axes rigidly bound to the head2 7#b7 the three components of the acceleration of translation of the center of gravity of the head2 7#c7 the centrifugal forces developed by the rotation of the head2 7#d7 the acceleration of the motion of rotation of the head4 ,it follows from the experiments of ,m4 ,delage that it is this last indication which is much the most important2 doubtless because the nerves are less sensible to the difference of pressure itself than to the brusque variations of this difference4 ,the first three indications may thus be neglected4 ,knowing the acceleration of the motion of rotation of the head at each instant1 we deduce from it1 by an unconscious integration1 the final orientation of the head1 referred to a certain initial orientation taken as origin4 ,the circular canals contribute1 therefore1 to inform us of the movements that we have executed1 and that on the same ground as the muscular sensations4 ,when1 therefore1 above we speak of the series ,s or of the series ,sigma1 we should say1 not that these were series of muscular sensations alone1 but that they were series at the same time of muscular sensations and of sensations due to the semicircular canals4 ,apart from this addition1 we should have nothing to change in what precedes4 ,in the series ,s and ,sigma1 these sensations of the semicircular canals evidently hold a very important place4 ,yet alone they would not suffice1 because they can tell us only of the movements of the head2 they tell us nothing of the relative movements of the body or of the members in regard to the head4 ,and more1 it seems that they tell us only of the rotations of the head and not of the translations it may undergo4 ,,part ,,ii ,,the ,,physical ,,sciences ,,chapter ,v ,,analysis ,,and ,,physics ,i ,you have doubtless often been asked of what good is mathematics and whether these delicate constructions entirely mind-made are not artificial and born of our caprice4 ,among those who put this question ,i should make a distinction2 practical people ask of us only the means of money-making4 ,these merit no reply2 rather would it be proper to ask of them what is the good of accumulating so much wealth and whether1 to get time to acquire it1 we are to neglect art and science1 which alone give us souls capable of enjoying it1 'and for life's sake to sacrifice all reasons for living4' ,besides1 a science made solely in view of applications is impossible2 truths are fecund only if bound together4 ,if we devote ourselves solely to those truths whence we expect an immediate result1 the intermediary links are wanting and there will no longer be a chain4 ,the men most disdainful of theory get from it1 without suspecting it1 their daily bread2 deprived of this food1 progress would quickly cease1 and we should soon congeal into the immobility of old ,china4 ,but enough of uncompromising practicians6 ,besides these1 there are those who are only interested in nature and who ask us if we can enable them to know it better4 ,to answer these1 we have only to show them the two monuments already rough-hewn1 ,celestial ,mechanics and ,mathematical ,physics4 ,they would doubtless concede that these structures are well worth the trouble they have cost us4 ,but this is not enough4 ,mathematics has a triple aim4 ,it must furnish an instrument for the study of nature4 ,but that is not all3 it has a philosophic aim and1 ,i dare maintain1 an esthetic aim4 ,it must aid the philosopher to fathom the notions of number1 of space1 of time4 ,and above all1 its adepts find therein delights analogous to those given by painting and music4 ,they admire the delicate harmony of numbers and forms2 they marvel when a new discovery opens to them an unexpected perspective2 and has not the joy they thus feel the esthetic character1 even though the senses take no part therein8 ,only a privileged few are called to enjoy it fully1 it is true1 but is not this the case for all the noblest arts8 ,this is why ,i do not hesitate to say that mathematics deserves to be cultivated for its own sake1 and the theories inapplicable to physics as well as the others4 ,even if the physical aim and the esthetic aim were not united1 we ought not to sacrifice either4 ,but more3 these two aims are inseparable and the best means of attaining one is to aim at the other1 or at least never to lose sight of it4 ,this is what ,i am about to try to demonstrate in setting forth the nature of the relations between the pure science and its applications4 ,the mathematician should not be for the physicist a mere purveyor of formulas2 there should be between them a more intimate collaboration4 ,mathematical physics and pure analysis are not merely adjacent powers1 maintaining good neighborly relations2 they mutually interpenetrate and their spirit is the same4 ,this will be better understood when ,i have shown what physics gets from mathematics and what mathematics1 in return1 borrows from physics4 ,,ii ,the physicist can not ask of the analyst to reveal to him a new truth2 the latter could at most only aid him to foresee it4 ,it is a long time since one still dreamt of forestalling experiment1 or of constructing the entire world on certain premature hypotheses4 ,since all those constructions in which one yet took a nave delight it is an age1 to-day only their ruins remain4 ,all laws are therefore deduced from experiment2 but to enunciate them1 a special language is needful2 ordinary language is too poor1 it is besides too vague1 to express relations so delicate1 so rich1 and so precise4 ,this therefore is one reason why the physicist can not do without mathematics2 it furnishes him the only language he can speak4 ,and a well-made language is no indifferent thing2 not to go beyond physics1 the unknown man who invented the word heat devoted many generations to error4 ,heat has been treated as a substance1 simply because it was designated by a substantive1 and it has been thought indestructible4 ,on the other hand1 he who invented the word electricity had the unmerited good fortune to implicitly endow physics with a new law1 that of the conservation of electricity1 which1 by a pure chance1 has been found exact1 at least until now4 ,well1 to continue the simile1 the writers who embellish a language1 who treat it as an object of art1 make of it at the same time a more supple instrument1 more apt for rendering shades of thought4 ,we understand1 then1 how the analyst1 who pursues a purely esthetic aim1 helps create1 just by that1 a language more fit to satisfy the physicist4 ,but this is not all3 law springs from experiment1 but not immediately4 ,experiment is individual1 the law deduced from it is general2 experiment is only approximate1 the law is precise1 or at least pretends to be4 ,experiment is made under conditions always complex1 the enunciation of the law eliminates these complications4 ,this is what is called 'correcting the systematic errors4' ,in a word1 to get the law from experiment1 it is necessary to generalize2 this is a necessity imposed upon the most circumspect observer4 ,but how generalize8 ,every particular truth may evidently be extended in an infinity of ways4 ,among these thousand routes opening before us1 it is necessary to make a choice1 at least provisional2 in this choice1 what shall guide us8 ,it can only be analogy4 ,but how vague is this word6 ,primitive man knew only crude analogies1 those which strike the senses1 those of colors or of sounds4 ,he never would have dreamt of likening light to radiant heat4 ,what has taught us to know the true1 profound analogies1 those the eyes do not see but reason divines8 ,it is the mathematical spirit1 which disdains matter to cling only to pure form4 ,this it is which has taught us to give the same name to things differing only in material1 to call by the same name1 for instance1 the multiplication of quaternions and that of whole numbers4 ,if quaternions1 of which ,i have just spoken1 had not been so promptly utilized by the ,english physicists1 many persons would doubtless see in them only a useless fancy1 and yet1 in teaching us to liken what appearances separate1 they would have already rendered us more apt to penetrate the secrets of nature4 ,such are the services the physicist should expect of analysis2 but for this science to be able to render them1 it must be cultivated in the broadest fashion without immediate expectation of utility--the mathematician must have worked as artist4 ,what we ask of him is to help us to see1 to discern our way in the labyrinth which opens before us4 ,now1 he sees best who stands highest4 ,examples abound1 and ,i limit myself to the most striking4 ,the first will show us how to change the language suffices to reveal generalizations not before suspected4 ,when ,newton's law has been substituted for ,kepler's we still know only elliptic motion4 ,now1 in so far as concerns this motion1 the two laws differ only in form2 we pass from one to the other by a simple differentiation4 ,and yet from ,newton's law may be deduced by an immediate generalization all the effects of perturbations and the whole of celestial mechanics4 ,if1 on the other hand1 ,kepler's enunciation had been retained1 no one would ever have regarded the orbits of the perturbed planets1 those complicated curves of which no one has ever written the equation1 as the natural generalizations of the ellipse4 ,the progress of observations would only have served to create belief in chaos4 ,the second example is equally deserving of consideration4 ,when ,maxwell began his work1 the laws of electro-dynamics admitted up to his time accounted for all the known facts4 ,it was not a new experiment which came to invalidate them4 ,but in looking at them under a new bias1 ,maxwell saw that the equations became more symmetrical when a term was added1 and besides1 this term was too small to produce effects appreciable with the old methods4 ,you know that ,maxwell's a priori views awaited for twenty years an experimental confirmation2 or1 if you prefer1 ,maxwell was twenty years ahead of experiment4 ,how was this triumph obtained8 ,it was because ,maxwell was profoundly steeped in the sense of mathematical symmetry2 would he have been so1 if others before him had not studied this symmetry for its own beauty8 ,it was because ,maxwell was accustomed to 'think in vectors1' and yet it was through the theory of imaginaries 7neomonics7 that vectors were introduced into analysis4 ,and those who invented imaginaries hardly suspected the advantage which would be obtained from them for the study of the real world1 of this the name given them is proof sufficient4 ,in a word1 ,maxwell was perhaps not an able analyst1 but this ability would have been for him only a useless and bothersome baggage4 ,on the other hand1 he had in the highest degree the intimate sense of mathematical analogies4 ,therefore it is that he made good mathematical physics4 ,maxwell's example teaches us still another thing4 ,how should the equations of mathematical physics be treated8 ,should we simply deduce all the consequences and regard them as intangible realities8 ,far from it2 what they should teach us above all is what can and what should be changed4 ,it is thus that we get from them something useful4 ,the third example goes to show us how we may perceive mathematical analogies between phenomena which have physically no relation either apparent or real1 so that the laws of one of these phenomena aid us to divine those of the other4 ,the very same equation1 that of ,laplace1 is met in the theory of ,newtonian attraction1 in that of the motion of liquids1 in that of the electric potential1 in that of magnetism1 in that of the propagation of heat and in still many others4 ,what is the result8 ,these theories seem images copied one from the other2 they are mutually illuminating1 borrowing their language from each other2 ask electricians if they do not felicitate themselves on having invented the phrase flow of force1 suggested by hydrodynamics and the theory of heat4 ,thus mathematical analogies not only may make us foresee physical analogies1 but besides do not cease to be useful when these latter fail4 ,to sum up1 the aim of mathematical physics is not only to facilitate for the physicist the numerical calculation of certain constants or the integration of certain differential equations4 ,it is besides1 it is above all1 to reveal to him the hidden harmony of things in making him see them in a new way4 ,of all the parts of analysis1 the most elevated1 the purest1 so to speak1 will be the most fruitful in the hands of those who know how to use them4 ,,iii ,let us now see what analysis owes to physics4 ,it would be necessary to have completely forgotten the history of science not to remember that the desire to understand nature has had on the development of mathematics the most constant and happiest influence4 ,in the first place the physicist sets us problems whose solution he expects of us4 ,but in proposing them to us1 he has largely paid us in advance for the service we shall render him1 if we solve them4 ,if ,i may be allowed to continue my comparison with the fine arts1 the pure mathematician who should forget the existence of the exterior world would be like a painter who knew how to harmoniously combine colors and forms1 but who lacked models4 ,his creative power would soon be exhausted4 ,the combinations which numbers and symbols may form are an infinite multitude4 ,in this multitude how shall we choose those which are worthy to fix our attention8 ,shall we let ourselves be guided solely by our caprice8 ,this caprice1 which itself would besides soon tire1 would doubtless carry us very far apart and we should quickly cease to understand each other4 ,but this is only the smaller side of the question4 ,physics will doubtless prevent our straying1 but it will also preserve us from a danger much more formidable2 it will prevent our ceaselessly going around in the same circle4 ,history proves that physics has not only forced us to choose among problems which came in a crowd2 it has imposed upon us such as we should without it never have dreamed of4 ,however varied may be the imagination of man1 nature is still a thousand times richer4 ,to follow her we must take ways we have neglected1 and these paths lead us often to summits whence we discover new countries4 ,what could be more useful6 ,it is with mathematical symbols as with physical realities2 it is in comparing the different aspects of things that we are able to comprehend their inner harmony1 which alone is beautiful and consequently worthy of our efforts4 ,the first example ,i shall cite is so old we are tempted to forget it2 it is nevertheless the most important of all4 ,the sole natural object of mathematical thought is the whole number4 ,it is the external world which has imposed the continuum upon us1 which we doubtless have invented1 but which it has forced us to invent4 ,without it there would be no infinitesimal analysis2 all mathematical science would reduce itself to arithmetic or to the theory of substitutions4 ,on the contrary1 we have devoted to the study of the continuum almost all our time and all our strength4 ,who will regret it2 who will think that this time and this strength have been wasted8 ,analysis unfolds before us infinite perspectives that arithmetic never suspects2 it shows us at a glance a majestic assemblage whose array is simple and symmetric2 on the contrary1 in the theory of numbers1 where reigns the unforeseen1 the view is1 so to speak1 arrested at every step4 ,doubtless it will be said that outside of the whole number there is no rigor1 and consequently no mathematical truth2 that the whole number hides everywhere1 and that we must strive to render transparent the screens which cloak it1 even if to do so we must resign ourselves to interminable repetitions4 ,let us not be such purists and let us be grateful to the continuum1 which1 if all springs from the whole number1 was alone capable of making so much proceed therefrom4 ,need ,i also recall that ,m4 ,hermite obtained a surprising advantage from the introduction of continuous variables into the theory of numbers8 ,thus the whole number's own domain is itself invaded1 and this invasion has established order where disorder reigned4 ,see what we owe to the continuum and consequently to physical nature4 ,fourier's series is a precious instrument of which analysis makes continual use1 it is by this means that it has been able to represent discontinuous functions2 ,fourier invented it to solve a problem of physics relative to the propagation of heat4 ,if this problem had not come up naturally1 we should never have dared to give discontinuity its rights2 we should still long have regarded continuous functions as the only true functions4 ,the notion of function has been thereby considerably extended and has received from some logician-analysts an unforeseen development4 ,these analysts have thus adventured into regions where reigns the purest abstraction and have gone as far away as possible from the real world4 ,yet it is a problem of physics which has furnished them the occasion4 ,after ,fourier's series1 other analogous series have entered the domain of analysis2 they have entered by the same door2 they have been imagined in view of applications4 ,the theory of partial differential equations of the second order has an analogous history4 ,it has been developed chiefly by and for physics4 ,but it may take many forms1 because such an equation does not suffice to determine the unknown function1 it is necessary to adjoin to it complementary conditions which are called conditions at the limits2 whence many different problems4 ,if the analysts had abandoned themselves to their natural tendencies1 they would never have known but one1 that which ,madame ,kovalevski has treated in her celebrated memoir4 ,but there are a multitude of others which they would have ignored4 ,each of the theories of physics1 that of electricity1 that of heat1 presents us these equations under a new aspect4 ,it may1 therefore1 be said that without these theories we should not know partial differential equations4 ,it is needless to multiply examples4 ,i have given enough to be able to conclude3 when physicists ask of us the solution of a problem1 it is not a duty-service they impose upon us1 it is on the contrary we who owe them thanks4 ,,iv ,but this is not all2 physics not only gives us the occasion to solve problems2 it aids us to find the means thereto1 and that in two ways4 ,it makes us foresee the solution2 it suggests arguments to us4 ,i have spoken above of ,laplace's equation which is met in a multitude of diverse physical theories4 ,it is found again in geometry1 in the theory of conformal representation and in pure analysis1 in that of imaginaries4 ,in this way1 in the study of functions of complex variables1 the analyst1 alongside of the geometric image1 which is his usual instrument1 finds many physical images which he may make use of with the same success4 ,thanks to these images1 he can see at a glance what pure deduction would show him only successively4 ,he masses thus the separate elements of the solution1 and by a sort of intuition divines before being able to demonstrate4 ,to divine before demonstrating6 ,need ,i recall that thus have been made all the important discoveries8 ,how many are the truths that physical analogies permit us to present and that we are not in condition to establish by rigorous reasoning6 ,for example1 mathematical physics introduces a great number of developments in series4 ,no one doubts that these developments converge2 but the mathematical certitude is lacking4 ,these are so many conquests assured for the investigators who shall come after us4 ,on the other hand1 physics furnishes us not alone solutions2 it furnishes us besides1 in a certain measure1 arguments4 ,it will suffice to recall how ,felix ,klein1 in a question relative to ,riemann surfaces1 has had recourse to the properties of electric currents4 ,it is true1 the arguments of this species are not rigorous1 in the sense the analyst attaches to this word4 ,and here a question arises3 ,how can a demonstration not sufficiently rigorous for the analyst suffice for the physicist8 ,it seems there can not be two rigors1 that rigor is or is not1 and that1 where it is not there can not be deduction4 ,this apparent paradox will be better understood by recalling under what conditions number is applied to natural phenomena4 ,whence come in general the difficulties encountered in seeking rigor8 ,we strike them almost always in seeking to establish that some quantity tends to some limit1 or that some function is continuous1 or that it has a derivative4 ,now the numbers the physicist measures by experiment are never known except approximately2 and besides1 any function always differs as little as you choose from a discontinuous function1 and at the same time it differs as little as you choose from a continuous function4 ,the physicist may1 therefore1 at will suppose that the function studied is continuous1 or that it is discontinuous2 that it has or has not a derivative2 and may do so without fear of ever being contradicted1 either by present experience or by any future experiment4 ,we see that with such liberty he makes sport of difficulties which stop the analyst4 ,he may always reason as if all the functions which occur in his calculations were entire polynomials4 ,thus the sketch which suffices for physics is not the deduction which analysis requires4 ,it does not follow thence that one can not aid in finding the other4 ,so many physical sketches have already been transformed into rigorous demonstrations that to-day this transformation is easy4 ,there would be plenty of examples did ,i not fear in citing them to tire the reader4 ,i hope ,i have said enough to show that pure analysis and mathematical physics may serve one another without making any sacrifice one to the other1 and that each of these two sciences should rejoice in all which elevates its associate4 ,,chapter ,,vi ,,astronomy ,governments and parliaments must find that astronomy is one of the sciences which cost most dear3 the least instrument costs hundreds of thousands of dollars1 the least observatory costs millions2 each eclipse carries with it supplementary appropriations4 ,and all that for stars which are so far away1 which are complete strangers to our electoral contests1 and in all probability will never take any part in them4 ,it must be that our politicians have retained a remnant of idealism1 a vague instinct for what is grand2 truly1 ,i think they have been calumniated2 they should be encouraged and shown that this instinct does not deceive them1 that they are not dupes of that idealism4 ,we might indeed speak to them of navigation1 of which no one can underestimate the importance1 and which has need of astronomy4 ,but this would be to take the question by its smaller side4 ,astronomy is useful because it raises us above ourselves2 it is useful because it is grand2 that is what we should say4 ,it shows us how small is man's body1 how great his mind1 since his intelligence can embrace the whole of this dazzling immensity1 where his body is only an obscure point1 and enjoy its silent harmony4 ,thus we attain the consciousness of our power1 and this is something which can not cost too dear1 since this consciousness makes us mightier4 ,but what ,i should wish before all to show is1 to what point astronomy has facilitated the work of the other sciences1 more directly useful1 since it has given us a soul capable of comprehending nature4 ,think how diminished humanity would be if1 under heavens constantly overclouded1 as ,jupiter's must be1 it had forever remained ignorant of the stars4 ,do you think that in such a world we should be what we are8 ,i know well that under this somber vault we should have been deprived of the light of the sun1 necessary to organisms like those which inhabit the earth4 ,but if you please1 we shall assume that these clouds are phosphorescent and emit a soft and constant light4 ,since we are making hypotheses1 another will cost no more4 ,well6 ,i repeat my question3 ,do you think that in such a world we should be what we are8 ,the stars send us not only that visible and gross light which strikes our bodily eyes1 but from them also comes to us a light far more subtle1 which illuminates our minds and whose effects ,i shall try to show you4 ,you know what man was on the earth some thousands of years ago1 and what he is to-day4 ,isolated amidst a nature where everything was a mystery to him1 terrified at each unexpected manifestation of incomprehensible forces1 he was incapable of seeing in the conduct of the universe anything but caprice2 he attributed all phenomena to the action of a multitude of little genii1 fantastic and exacting1 and to act on the world he sought to conciliate them by means analogous to those employed to gain the good graces of a minister or a deputy4 ,even his failures did not enlighten him1 any more than to-day a beggar refused is discouraged to the point of ceasing to beg4 ,to-day we no longer beg of nature2 we command her1 because we have discovered certain of her secrets and shall discover others each day4 ,we command her in the name of laws she can not challenge1 because they are hers2 these laws we do not madly ask her to change1 we are the first to submit to them4 ,nature can only be governed by obeying her4 ,what a change must our souls have undergone to pass from the one state to the other6 ,does any one believe that1 without the lessons of the stars1 under the heavens perpetually overclouded that ,i have just supposed1 they would have changed so quickly8 ,would the metamorphosis have been possible1 or at least would it not have been much slower8 ,and first of all1 astronomy it is which taught that there are laws4 ,the ,chaldeans1 who were the first to observe the heavens with some attention1 saw that this multitude of luminous points is not a confused crowd wandering at random1 but rather a disciplined army4 ,doubtless the rules of this discipline escaped them1 but the harmonious spectacle of the starry night sufficed to give them the impression of regularity1 and that was in itself already a great thing4 ,besides1 these rules were discerned by ,hipparchus1 ,ptolemy1 ,copernicus1 ,kepler1 one after another1 and finally1 it is needless to recall that ,newton it was who enunciated the oldest1 the most precise1 the most simple1 the most general of all natural laws4 ,and then1 taught by this example1 we have seen our little terrestrial world better and1 under the apparent disorder1 there also we have found again the harmony that the study of the heavens had revealed to us4 ,it also is regular1 it also obeys immutable laws1 but they are more complicated1 in apparent conflict one with another1 and an eye untrained by other sights would have seen there only chaos and the reign of chance or caprice4 ,if we had not known the stars1 some bold spirits might perhaps have sought to foresee physical phenomena2 but their failures would have been frequent1 and they would have excited only the derision of the vulgar2 do we not see1 that even in our day the meteorologists sometimes deceive themselves1 and that certain persons are inclined to laugh at them4 ,how often would the physicists1 disheartened by so many checks1 have fallen into discouragement1 if they had not had1 to sustain their confidence1 the brilliant example of the success of the astronomers6 ,this success showed them that nature obeys laws2 it only remained to know what laws2 for that they only needed patience1 and they had the right to demand that the sceptics should give them credit4 ,this is not all3 astronomy has not only taught us that there are laws1 but that from these laws there is no escape1 that with them there is no possible compromise4 ,how much time should we have needed to comprehend that fact1 if we had known only the terrestrial world1 where each elemental force would always seem to us in conflict with other forces8 ,astronomy has taught us that the laws are infinitely precise1 and that if those we enunciate are approximative1 it is because we do not know them well4 ,aristotle1 the most scientific mind of antiquity1 still accorded a part to accident1 to chance1 and seemed to think that the laws of nature1 at least here below1 determine only the large features of phenomena4 ,how much has the ever-increasing precision of astronomical predictions contributed to correct such an error1 which would have rendered nature unintelligible6 ,but are these laws not local1 varying in different places1 like those which men make2 does not that which is truth in one corner of the universe1 on our globe1 for instance1 or in our little solar system1 become error a little farther away8 ,and then could it not be asked whether laws depending on space do not also depend upon time1 whether they are not simple habitudes1 transitory1 therefore1 and ephemeral8 ,again it is astronomy that answers this question4 ,consider the double stars2 all describe conics2 thus1 as far as the telescope carries1 it does not reach the limits of the domain which obeys ,newton's law4 ,even the simplicity of this law is a lesson for us2 how many complicated phenomena are contained in the two lines of its enunciation2 persons who do not understand celestial mechanics may form some idea of it at least from the size of the treatises devoted to this science2 and then it may be hoped that the complication of physical phenomena likewise hides from us some simple cause still unknown4 ,it is therefore astronomy which has shown us what are the general characteristics of natural laws2 but among these characteristics there is one1 the most subtle and the most important of all1 which ,i shall ask leave to stress4 ,how was the order of the universe understood by the ancients2 for instance1 by ,pythagoras1 ,plato or ,aristotle8 ,it was either an immutable type fixed once for all1 or an ideal to which the world sought to approach4 ,kepler himself still thought thus when1 for instance1 he sought whether the distances of the planets from the sun had not some relation to the five regular polyhedrons4 ,this idea contained nothing absurd1 but it was sterile1 since nature is not so made4 ,newton has shown us that a law is only a necessary relation between the present state of the world and its immediately subsequent state4 ,all the other laws since discovered are nothing else2 they are in sum1 differential equations2 but it is astronomy which furnished the first model for them1 without which we should doubtless long have erred4 ,astronomy has also taught us to set at naught appearances4 ,the day ,copernicus proved that what was thought the most stable was in motion1 that what was thought moving was fixed1 he showed us how deceptive could be the infantile reasonings which spring directly from the immediate data of our senses4 ,true1 his ideas did not easily triumph1 but since this triumph there is no longer a prejudice so inveterate that we can not shake it off4 ,how can we estimate the value of the new weapon thus won8 ,the ancients thought everything was made for man1 and this illusion must be very tenacious1 since it must ever be combated4 ,yet it is necessary to divest oneself of it2 or else one will be only an eternal myope1 incapable of seeing the truth4 ,to comprehend nature one must be able to get out of self1 so to speak1 and to contemplate her from many different points of view2 otherwise we never shall know more than one side4 ,now1 to get out of self is what he who refers everything to himself can not do4 ,who delivered us from this illusion8 ,it was those who showed us that the earth is only one of the smallest planets of the solar system1 and that the solar system itself is only an imperceptible point in the infinite spaces of the stellar universe4 ,at the same time astronomy taught us not to be afraid of big numbers4 ,this was needful1 not only for knowing the heavens1 but to know the earth itself2 and was not so easy as it seems to us to-day4 ,let us try to go back and picture to ourselves what a ,greek would have thought if told that red light vibrates four hundred millions of millions of times per second4 ,without any doubt1 such an assertion would have appeared to him pure madness1 and he never would have lowered himself to test it4 ,to-day a hypothesis will no longer appear absurd to us because it obliges us to imagine objects much larger or smaller than those our senses are capable of showing us1 and we no longer comprehend those scruples which arrested our predecessors and prevented them from discovering certain truths simply because they were afraid of them4 ,but why8 ,it is because we have seen the heavens enlarging and enlarging without cease2 because we know that the sun is #aej millions of kilometers from the earth and that the distances of the nearest stars are hundreds of thousands of times greater yet4 ,habituated to the contemplation of the infinitely great1 we have become apt to comprehend the infinitely small4 ,thanks to the education it has received1 our imagination1 like the eagle's eye that the sun does not dazzle1 can look truth in the face4 ,was ,i wrong in saying that it is astronomy which has made us a soul capable of comprehending nature2 that under heavens always overcast and starless1 the earth itself would have been for us eternally unintelligible2 that we should there have seen only caprice and disorder2 and that1 not knowing the world1 we should never have been able to subdue it8 ,what science could have been more useful8 ,and in thus speaking ,i put myself at the point of view of those who only value practical applications4 ,certainly1 this point of view is not mine2 as for me1 on the contrary1 if ,i admire the conquests of industry1 it is above all because if they free us from material cares1 they will one day give to all the leisure to contemplate nature4 ,i do not say3 ,science is useful1 because it teaches us to construct machines4 ,i say3 ,machines are useful1 because in working for us1 they will some day leave us more time to make science4 ,but finally it is worth remarking that between the two points of view there is no antagonism1 and that man having pursued a disinterested aim1 all else has been added unto him4 ,auguste ,comte has said somewhere1 that it would be idle to seek to know the composition of the sun1 since this knowledge would be of no use to sociology4 ,how could he be so short-sighted8 ,have we not just seen that it is by astronomy that1 to speak his language1 humanity has passed from the theological to the positive state8 ,he found an explanation for that because it had happened4 ,but how has he not understood that what remained to do was not less considerable and would be not less profitable8 ,physical astronomy1 which he seems to condemn1 has already begun to bear fruit1 and it will give us much more1 for it only dates from yesterday4 ,first was discovered the nature of the sun1 what the founder of positivism wished to deny us1 and there bodies were found which exist on the earth1 but had here remained undiscovered2 for example1 helium1 that gas almost as light as hydrogen4 ,that already contradicted ,comte4 ,but to the spectroscope we owe a lesson precious in a quite different way2 in the most distant stars1 it shows us the same substances4 ,it might have been asked whether the terrestrial elements were not due to some chance which had brought together more tenuous atoms to construct of them the more complex edifice that the chemists call atom2 whether1 in other regions of the universe1 other fortuitous meetings had not engendered edifices entirely different4 ,now we know that this is not so1 that the laws of our chemistry are the general laws of nature1 and that they owe nothing to the chance which caused us to be born on the earth4 ,but1 it will be said1 astronomy has given to the other sciences all it can give them1 and now that the heavens have procured for us the instruments which enable us to study terrestrial nature1 they could without danger veil themselves forever4 ,after what we have just said1 is there still need to answer this objection8 ,one could have reasoned the same in ,ptolemy's time2 then also men thought they knew everything1 and they still had almost everything to learn4 ,the stars are majestic laboratories1 gigantic crucibles1 such as no chemist could dream4 ,there reign temperatures impossible for us to realize4 ,their only defect is being a little far away2 but the telescope will soon bring them near to us1 and then we shall see how matter acts there4 ,what good fortune for the physicist and the chemist6 ,matter will there exhibit itself to us under a thousand different states1 from those rarefied gases which seem to form the nebul and which are luminous with ,i know not what glimmering of mysterious origin1 even to the incandescent stars and to the planets so near and yet so different4 ,perchance even1 the stars will some day teach us something about life2 that seems an insensate dream and ,i do not at all see how it can be realized2 but1 a hundred years ago1 would not the chemistry of the stars have also appeared a mad dream8 ,but limiting our views to horizons less distant1 there still will remain to us promises less contingent and yet sufficiently seductive4 ,if the past has given us much1 we may rest assured that the future will give us still more4 ,in sum1 it is incredible how useful belief in astrology has been to humanity4 ,if ,kepler and ,tycho ,brahe made a living1 it was because they sold to nave kings predictions founded on the conjunctions of the stars4 ,if these princes had not been so credulous1 we should perhaps still believe that nature obeys caprice1 and we should still wallow in ignorance4 ,,chapter ,,vii ,,the ,,history ,,of ,,mathematical ,,physics ,the ,past and the ,future of ,physics4--,what is the present state of mathematical physics8 ,what are the problems it is led to set itself8 ,what is its future8 ,is its orientation about to be modified8 ,ten years hence will the aim and the methods of this science appear to our immediate successors in the same light as to ourselves2 or1 on the contrary1 are we about to witness a profound transformation8 ,such are the questions we are forced to raise in entering to-day upon our investigation4 ,if it is easy to propound them3 to answer is difficult4 ,if we felt tempted to risk a prediction1 we should easily resist this temptation1 by thinking of all the stupidities the most eminent savants of a hundred years ago would have uttered1 if some one had asked them what the science of the nineteenth century would be4 ,they would have thought themselves bold in their predictions1 and after the event1 how very timid we should have found them4 ,do not1 therefore1 expect of me any prophecy4 ,but if1 like all prudent physicians1 ,i shun giving a prognosis1 yet ,i can not dispense with a little diagnostic2 well1 yes1 there are indications of a serious crisis1 as if we might expect an approaching transformation4 ,still1 be not too anxious3 we are sure the patient will not die of it1 and we may even hope that this crisis will be salutary1 for the history of the past seems to guarantee us this4 ,this crisis1 in fact1 is not the first1 and to understand it1 it is important to recall those which have preceded4 ,pardon then a brief historical sketch4 ,the ,physics of ,central ,forces4--,mathematical physics1 as we know1 was born of celestial mechanics1 which gave birth to it at the end of the eighteenth century1 at the moment when it itself attained its complete development4 ,during its first years especially1 the infant strikingly resembled its mother4 ,the astronomic universe is formed of masses1 very great1 no doubt1 but separated by intervals so immense that they appear to us only as material points4 ,these points attract each other inversely as the square of the distance1 and this attraction is the sole force which influences their movements4 ,but if our senses were sufficiently keen to show us all the details of the bodies which the physicist studies1 the spectacle thus disclosed would scarcely differ from the one the astronomer contemplates4 ,there also we should see material points1 separated from one another by intervals1 enormous in comparison with their dimensions1 and describing orbits according to regular laws4 ,these infinitesimal stars are the atoms4 ,like the stars proper1 they attract or repel each other1 and this attraction or this repulsion1 following the straight line which joins them1 depends only on the distance4 ,the law according to which this force varies as function of the distance is perhaps not the law of ,newton1 but it is an analogous law2 in place of the exponent -#b1 we have probably a different exponent1 and it is from this change of exponent that arises all the diversity of physical phenomena1 the variety of qualities and of sensations1 all the world1 colored and sonorous1 which surrounds us2 in a word1 all nature4 ,such is the primitive conception in all its purity4 ,it only remains to seek in the different cases what value should be given to this exponent in order to explain all the facts4 ,it is on this model that ,laplace1 for example1 constructed his beautiful theory of capillarity2 he regards it only as a particular case of attraction1 or1 as he says1 of universal gravitation1 and no one is astonished to find it in the middle of one of the five volumes of the ',mcanique cleste4' ,more recently ,briot believes he penetrated the final secret of optics in demonstrating that the atoms of ether attract each other in the inverse ratio of the sixth power of the distance2 and ,maxwell himself1 does he not say somewhere that the atoms of gases repel each other in the inverse ratio of the fifth power of the distance8 ,we have the exponent -#f1 or -#e1 in place of the exponent -#b1 but it is always an exponent4 ,among the theories of this epoch1 one alone is an exception1 that of ,fourier2 in it are indeed atoms acting at a distance one upon the other2 they mutually transmit heat1 but they do not attract1 they never budge4 ,from this point of view1 ,fourier's theory must have appeared to the eyes of his contemporaries1 to those of ,fourier himself1 as imperfect and provisional4 ,this conception was not without grandeur2 it was seductive1 and many among us have not finally renounced it2 they know that one will attain the ultimate elements of things only by patiently disentangling the complicated skein that our senses give us2 that it is necessary to advance step by step1 neglecting no intermediary2 that our fathers were wrong in wishing to skip stations2 but they believe that when one shall have arrived at these ultimate elements1 there again will be found the majestic simplicity of celestial mechanics4 ,neither has this conception been useless2 it has rendered us an inestimable service1 since it has contributed to make precise the fundamental notion of the physical law4 ,i will explain myself2 how did the ancients understand law8 ,it was for them an internal harmony1 static1 so to say1 and immutable2 or else it was like a model that nature tried to imitate4 ,for us a law is something quite different2 it is a constant relation between the phenomenon of to-day and that of to-morrow2 in a word1 it is a differential equation4 ,behold the ideal form of physical law2 well1 it is ,newton's law which first clothed it forth4 ,if then one has acclimated this form in physics1 it is precisely by copying as far as possible this law of ,newton1 that is by imitating celestial mechanics4 ,this is1 moreover1 the idea ,i have tried to bring out in ,chapter ,,vi4 ,the ,physics of the ,principles4--,nevertheless1 a day arrived when the conception of central forces no longer appeared sufficient1 and this is the first of those crises of which ,i just now spoke4 ,what was done then8 ,the attempt to penetrate into the detail of the structure of the universe1 to isolate the pieces of this vast mechanism1 to analyze one by one the forces which put them in motion1 was abandoned1 and we were content to take as guides certain general principles the express object of which is to spare us this minute study4 ,how so8 ,suppose we have before us any machine2 the initial wheel work and the final wheel work alone are visible1 but the transmission1 the intermediary machinery by which the movement is communicated from one to the other1 is hidden in the interior and escapes our view2 we do not know whether the communication is made by gearing or by belts1 by connecting-rods or by other contrivances4 ,do we say that it is impossible for us to understand anything about this machine so long as we are not permitted to take it to pieces8 ,you know well we do not1 and that the principle of the conservation of energy suffices to determine for us the most interesting point4 ,we easily ascertain that the final wheel turns ten times less quickly than the initial wheel1 since these two wheels are visible2 we are able thence to conclude that a couple applied to the one will be balanced by a couple ten times greater applied to the other4 ,for that there is no need to penetrate the mechanism of this equilibrium and to know how the forces compensate each other in the interior of the machine2 it suffices to be assured that this compensation can not fail to occur4 ,well1 in regard to the universe1 the principle of the conservation of energy is able to render us the same service4 ,the universe is also a machine1 much more complicated than all those of industry1 of which almost all the parts are profoundly hidden from us2 but in observing the motion of those that we can see1 we are able1 by the aid of this principle1 to draw conclusions which remain true whatever may be the details of the invisible mechanism which animates them4 ,the principle of the conservation of energy1 or ,mayer's principle1 is certainly the most important1 but it is not the only one2 there are others from which we can derive the same advantage4 ,these are3 ,carnot's principle1 or the principle of the degradation of energy4 ,newton's principle1 or the principle of the equality of action and reaction4 ,the principle of relativity1 according to which the laws of physical phenomena must be the same for a stationary observer as for an observer carried along in a uniform motion of translation2 so that we have not and can not have any means of discerning whether or not we are carried along in such a motion4 ,the principle of the conservation of mass1 or ,lavoisier's principle4 ,i will add the principle of least action4 ,the application of these five or six general principles to the different physical phenomena is sufficient for our learning of them all that we could reasonably hope to know of them4 ,the most remarkable example of this new mathematical physics is1 beyond question1 ,maxwell's electromagnetic theory of light4 ,we know nothing as to what the ether is1 how its molecules are disposed1 whether they attract or repel each other2 but we know that this medium transmits at the same time the optical perturbations and the electrical perturbations2 we know that this transmission must take place in conformity with the general principles of mechanics1 and that suffices us for the establishment of the equations of the electromagnetic field4 ,these principles are results of experiments boldly generalized2 but they seem to derive from their very generality a high degree of certainty4 ,in fact1 the more general they are1 the more frequent are the opportunities to check them1 and the verifications multiplying1 taking the most varied1 the most unexpected forms1 end by no longer leaving place for doubt4 ,utility of the ,old ,physics4--,such is the second phase of the history of mathematical physics and we have not yet emerged from it4 ,shall we say that the first has been useless8 that during fifty years science went the wrong way1 and that there is nothing left but to forget so many accumulated efforts that a vicious conception condemned in advance to failure8 ,not the least in the world4 ,do you think the second phase could have come into existence without the first8 ,the hypothesis of central forces contained all the principles2 it involved them as necessary consequences2 it involved both the conservation of energy and that of masses1 and the equality of action and reaction1 and the law of least action1 which appeared1 it is true1 not as experimental truths1 but as theorems2 the enunciation of which had at the same time something more precise and less general than under their present form4 ,it is the mathematical physics of our fathers which has familiarized us little by little with these various principles2 which has habituated us to recognize them under the different vestments in which they disguise themselves4 ,they have been compared with the data of experience1 it has been seen how it was necessary to modify their enunciation to adapt them to these data2 thereby they have been extended and consolidated4 ,thus they came to be regarded as experimental truths2 the conception of central forces became then a useless support1 or rather an embarrassment1 since it made the principles partake of its hypothetical character4 ,the frames then have not broken1 because they are elastic2 but they have enlarged2 our fathers1 who established them1 did not labor in vain1 and we recognize in the science of to-day the general traits of the sketch which they traced4 ,,chapter ,,viii ,,the ,,present ,,crisis ,,of ,,mathematical ,,physics ,the ,new ,crisis4--,are we now about to enter upon a third period8 ,are we on the eve of a second crisis8 ,these principles on which we have built all1 are they about to crumble away in their turn8 ,this has been for some time a pertinent question4 ,when ,i speak thus1 you no doubt think of radium1 that grand revolutionist of the present time1 and in fact ,i shall come back to it presently2 but there is something else4 ,it is not alone the conservation of energy which is in question2 all the other principles are equally in danger1 as we shall see in passing them successively in review4 ,carnot's ,principle4--,let us commence with the principle of ,carnot4 ,this is the only one which does not present itself as an immediate consequence of the hypothesis of central forces2 more than that1 it seems1 if not to directly contradict that hypothesis1 at least not to be reconciled with it without a certain effort4 ,if physical phenomena were due exclusively to the movements of atoms whose mutual attraction depended only on the distance1 it seems that all these phenomena should be reversible2 if all the initial velocities were reversed1 these atoms1 always subjected to the same forces1 ought to go over their trajectories in the contrary sense1 just as the earth would describe in the retrograde sense this same elliptic orbit which it describes in the direct sense1 if the initial conditions of its motion had been reversed4 ,on this account1 if a physical phenomenon is possible1 the inverse phenomenon should be equally so1 and one should be able to reascend the course of time4 ,now1 it is not so in nature1 and this is precisely what the principle of ,carnot teaches us2 heat can pass from the warm body to the cold body2 it is impossible afterward to make it take the inverse route and to reestablish differences of temperature which have been effaced4 ,motion can be wholly dissipated and transformed into heat by friction2 the contrary transformation can never be made except partially4 ,we have striven to reconcile this apparent contradiction4 ,if the world tends toward uniformity1 this is not because its ultimate parts1 at first unlike1 tend to become less and less different2 it is because1 shifting at random1 they end by blending4 ,for an eye which should distinguish all the elements1 the variety would remain always as great2 each grain of this dust preserves its originality and does not model itself on its neighbors2 but as the blend becomes more and more intimate1 our gross senses perceive only the uniformity4 ,this is why1 for example1 temperatures tend to a level1 without the possibility of going backwards4 ,a drop of wine falls into a glass of water2 whatever may be the law of the internal motion of the liquid1 we shall soon see it colored of a uniform rosy tint1 and however much from this moment one may shake it afterwards1 the wine and the water do not seem capable of again separating4 ,here we have the type of the irreversible physical phenomenon3 to hide a grain of barley in a heap of wheat1 this is easy2 afterwards to find it again and get it out1 this is practically impossible4 ,all this ,maxwell and ,boltzmann have explained2 but the one who has seen it most clearly1 in a book too little read because it is a little difficult to read1 is ,gibbs1 in his ,elementary ,principles of ,statistical ,mechanics4' ,for those who take this point of view1 ,carnot's principle is only an imperfect principle1 a sort of concession to the infirmity of our senses2 it is because our eyes are too gross that we do not distinguish the elements of the blend2 it is because our hands are too gross that we can not force them to separate2 the imaginary demon of ,maxwell1 who is able to sort the molecules one by one1 could well constrain the world to return backward4 ,can it return of itself8 ,that is not impossible2 that is only infinitely improbable4 ,the chances are that we should wait a long time for the concourse of circumstances which would permit a retrogradation2 but sooner or later they will occur1 after years whose number it would take millions of figures to write4 ,these reservations1 however1 all remained theoretic2 they were not very disquieting1 and ,carnot's principle retained all its practical value4 ,but here the scene changes4 ,the biologist1 armed with his microscope1 long ago noticed in his preparations irregular movements of little particles in suspension2 this is the ,brownian movement4 ,he first thought this was a vital phenomenon1 but soon he saw that the inanimate bodies danced with no less ardor than the others2 then he turned the matter over to the physicists4 ,unhappily1 the physicists remained long uninterested in this question2 one concentrates the light to illuminate the microscopic preparation1 thought they2 with light goes heat2 thence inequalities of temperature and in the liquid interior currents which produce the movements referred to4 ,it occurred to ,m4 ,gouy to look more closely1 and he saw1 or thought he saw1 that this explanation is untenable1 that the movements become brisker as the particles are smaller1 but that they are not influenced by the mode of illumination4 ,if then these movements never cease1 or rather are reborn without cease1 without borrowing anything from an external source of energy1 what ought we to believe8 ,to be sure1 we should not on this account renounce our belief in the conservation of energy1 but we see under our eyes now motion transformed into heat by friction1 now inversely heat changed into motion1 and that without loss since the movement lasts forever4 ,this is the contrary of ,carnot's principle4 ,if this be so1 to see the world return backward1 we no longer have need of the infinitely keen eye of ,maxwell's demon2 our microscope suffices4 ,bodies too large1 those1 for example1 which are a tenth of a millimeter1 are hit from all sides by moving atoms1 but they do not budge1 because these shocks are very numerous and the law of chance makes them compensate each other2 but the smaller particles receive too few shocks for this compensation to take place with certainty and are incessantly knocked about4 ,and behold already one of our principles in peril4 ,the ,principle of ,relativity4--,let us pass to the principle of relativity2 this not only is confirmed by daily experience1 not only is it a necessary consequence of the hypothesis of central forces1 but it is irresistibly imposed upon our good sense1 and yet it also is assailed4 ,consider two electrified bodies2 though they seem to us at rest1 they are both carried along by the motion of the earth2 an electric charge in motion1 ,rowland has taught us1 is equivalent to a current2 these two charged bodies are1 therefore1 equivalent to two parallel currents of the same sense and these two currents should attract each other4 ,in measuring this attraction1 we shall measure the velocity of the earth2 not its velocity in relation to the sun or the fixed stars1 but its absolute velocity4 ,i well know what will be said3 ,it is not its absolute velocity that is measured1 it is its velocity in relation to the ether4 ,how unsatisfactory that is6 ,is it not evident that from the principle so understood we could no longer infer anything8 ,it could no longer tell us anything just because it would no longer fear any contradiction4 ,if we succeed in measuring anything1 we shall always be free to say that this is not the absolute velocity1 and if it is not the velocity in relation to the ether1 it might always be the velocity in relation to some new unknown fluid with which we might fill space4 ,indeed1 experiment has taken upon itself to ruin this interpretation of the principle of relativity2 all attempts to measure the velocity of the earth in relation to the ether have led to negative results4 ,this time experimental physics has been more faithful to the principle than mathematical physics2 the theorists1 to put in accord their other general views1 would not have spared it2 but experiment has been stubborn in confirming it4 ,the means have been varied2 finally ,michelson pushed precision to its last limits2 nothing came of it4 ,it is precisely to explain this obstinacy that the mathematicians are forced to-day to employ all their ingenuity4 ,their task was not easy1 and if ,lorentz has got through it1 it is only by accumulating hypotheses4 ,the most ingenious idea was that of local time4 ,imagine two observers who wish to adjust their timepieces by optical signals2 they exchange signals1 but as they know that the transmission of light is not instantaneous1 they are careful to cross them4 ,when station ,b perceives the signal from station ,a1 its clock should not mark the same hour as that of station ,a at the moment of sending the signal1 but this hour augmented by a constant representing the duration of the transmission4 ,suppose1 for example1 that station ,a sends its signal when its clock marks the hour ,o1 and that station ,b perceives it when its clock marks the hour t4 ,the clocks are adjusted if the slowness equal to t represents the duration of the transmission1 and to verify it1 station ,b sends in its turn a signal when its clock marks ,o2 then station ,a should perceive it when its clock marks t4 ,the timepieces are then adjusted4 ,and in fact they mark the same hour at the same physical instant1 but on the one condition1 that the two stations are fixed4 ,otherwise the duration of the transmission will not be the same in the two senses1 since the station ,a1 for example1 moves forward to meet the optical perturbation emanating from ,b1 whereas the station ,b flees before the perturbation emanating from ,a4 ,the watches adjusted in that way will not mark1 therefore1 the true time2 they will mark what may be called the local time1 so that one of them will be slow of the other4 ,it matters little1 since we have no means of perceiving it4 ,all the phenomena which happen at ,a1 for example1 will be late1 but all will be equally so1 and the observer will not perceive it1 since his watch is slow2 so1 as the principle of relativity requires1 he will have no means of knowing whether he is at rest or in absolute motion4 ,unhappily1 that does not suffice1 and complementary hypotheses are necessary2 it is necessary to admit that bodies in motion undergo a uniform contraction in the sense of the motion4 ,one of the diameters of the earth1 for example1 is shrunk by one two-hundred-millionth in consequence of our planet's motion1 while the other diameter retains its normal length4 ,thus the last little differences are compensated4 ,and then1 there is still the hypothesis about forces4 ,forces1 whatever be their origin1 gravity as well as elasticity1 would be reduced in a certain proportion in a world animated by a uniform translation2 or1 rather1 this would happen for the components perpendicular to the translation2 the components parallel would not change4 ,resume1 then1 our example of two electrified bodies2 these bodies repel each other1 but at the same time if all is carried along in a uniform translation1 they are equivalent to two parallel currents of the same sense which attract each other4 ,this electrodynamic attraction diminishes1 therefore1 the electrostatic repulsion1 and the total repulsion is feebler than if the two bodies were at rest4 ,but since to measure this repulsion we must balance it by another force1 and all these other forces are reduced in the same proportion1 we perceive nothing4 ,thus all seems arranged1 but are all the doubts dissipated8 ,what would happen if one could communicate by non-luminous signals whose velocity of propagation differed from that of light8 ,if1 after having adjusted the watches by the optical procedure1 we wished to verify the adjustment by the aid of these new signals1 we should observe discrepancies which would render evident the common translation of the two stations4 ,and are such signals inconceivable1 if we admit with ,laplace that universal gravitation is transmitted a million times more rapidly than light8 ,thus1 the principle of relativity has been valiantly defended in these latter times1 but the very energy of the defense proves how serious was the attack4 ,newton's ,principle4--,let us speak now of the principle of ,newton1 on the equality of action and reaction4 ,this is intimately bound up with the preceding1 and it seems indeed that the fall of the one would involve that of the other4 ,thus we must not be astonished to find here the same difficulties4 ,electrical phenomena1 according to the theory of ,lorentz1 are due to the displacements of little charged particles1 called electrons1 immersed in the medium we call ether4 ,the movements of these electrons produce perturbations in the neighboring ether2 these perturbations propagate themselves in every direction with the velocity of light1 and in turn other electrons1 originally at rest1 are made to vibrate when the perturbation reaches the parts of the ether which touch them4 ,the electrons1 therefore1 act on one another1 but this action is not direct1 it is accomplished through the ether as intermediary4 ,under these conditions can there be compensation between action and reaction1 at least for an observer who should take account only of the movements of matter1 that is1 of the electrons1 and who should be ignorant of those of the ether that he could not see8 ,evidently not4 ,even if the compensation should be exact1 it could not be simultaneous4 ,the perturbation is propagated with a finite velocity2 it1 therefore1 reaches the second electron only when the first has long ago entered upon its rest4 ,this second electron1 therefore1 will undergo1 after a delay1 the action of the first1 but will certainly not at that moment react upon it1 since around this first electron nothing any longer budges4 ,the analysis of the facts permits us to be still more precise4 ,imagine1 for example1 a ,hertzian oscillator1 like those used in wireless telegraphy2 it sends out energy in every direction2 but we can provide it with a parabolic mirror1 as ,hertz did with his smallest oscillators1 so as to send all the energy produced in a single direction4 ,what happens then according to the theory8 ,the apparatus recoils1 as if it were a cannon and the projected energy a ball2 and that is contrary to the principle of ,newton1 since our projectile here has no mass1 it is not matter1 it is energy4 ,the case is still the same1 moreover1 with a beacon light provided with a reflector1 since light is nothing but a perturbation of the electromagnetic field4 ,this beacon light should recoil as if the light it sends out were a projectile4 ,what is the force that should produce this recoil8 ,it is what is called the ,maxwell-,bartholi pressure4 ,it is very minute1 and it has been difficult to put it in evidence even with the most sensitive radiometers2 but it suffices that it exists4 ,if all the energy issuing from our oscillator falls on a receiver1 this will act as if it had received a mechanical shock1 which will represent in a sense the compensation of the oscillator's recoil2 the reaction will be equal to the action1 but it will not be simultaneous2 the receiver will move on1 but not at the moment when the oscillator recoils4 ,if the energy propagates itself indefinitely without encountering a receiver1 the compensation will never occur4 ,shall we say that the space which separates the oscillator from the receiver and which the perturbation must pass over in going from the one to the other is not void1 that it is full not only of ether1 but of air1 or even in the interplanetary spaces of some fluid subtile but still ponderable2 that this matter undergoes the shock like the receiver at the moment when the energy reaches it1 and recoils in its turn when the perturbation quits it8 ,that would save ,newton's principle1 but that is not true4 ,if energy in its diffusion remained always attached to some material substratum1 then matter in motion would carry along light with it1 and ,fizeau has demonstrated that it does nothing of the sort1 at least for air4 ,michelson and ,morley have since confirmed this4 ,it might be supposed also that the movements of matter proper are exactly compensated by those of the ether2 but that would lead us to the same reflections as before now4 ,the principle so understood will explain everything1 since1 whatever might be the visible movements1 we always could imagine hypothetical movements which compensate them4 ,but if it is able to explain everything1 this is because it does not enable us to foresee anything2 it does not enable us to decide between the different possible hypotheses1 since it explains everything beforehand4 ,it therefore becomes useless4 ,and then the suppositions that it would be necessary to make on the movements of the ether are not very satisfactory4 ,if the electric charges double1 it would be natural to imagine that the velocities of the diverse atoms of ether double also2 but1 for the compensation1 it would be necessary that the mean velocity of the ether quadruple4 ,this is why ,i have long thought that these consequences of theory1 contrary to ,newton's principle1 would end some day by being abandoned1 and yet the recent experiments on the movements of the electrons issuing from radium seem rather to confirm them4 ,lavoisier's ,principle4--,i arrive at the principle of ,lavoisier on the conservation of mass4 ,certainly1 this is one not to be touched without unsettling all mechanics4 ,and now certain persons think that it seems true to us only because in mechanics merely moderate velocities are considered1 but that it would cease to be true for bodies animated by velocities comparable to that of light4 ,now these velocities are believed at present to have been realized2 the cathode rays and those of radium may be formed of very minute particles or of electrons which are displaced with velocities smaller no doubt than that of light1 but which might be its one tenth or one third4 ,these rays can be deflected1 whether by an electric field1 or by a magnetic field1 and we are able1 by comparing these deflections1 to measure at the same time the velocity of the electrons and their mass 7or rather the relation of their mass to their charge74 ,but when it was seen that these velocities approached that of light1 it was decided that a correction was necessary4 ,these molecules1 being electrified1 can not be displaced without agitating the ether2 to put them in motion it is necessary to overcome a double inertia1 that of the molecule itself and that of the ether4 ,the total or apparent mass that one measures is composed1 therefore1 of two parts3 the real or mechanical mass of the molecule and the electrodynamic mass representing the inertia of the ether4 ,the calculations of ,abraham and the experiments of ,kaufmann have then shown that the mechanical mass1 properly so called1 is null1 and that the mass of the electrons1 or1 at least1 of the negative electrons1 is of exclusively electrodynamic origin4 ,this is what forces us to change the definition of mass2 we can not any longer distinguish mechanical mass and electrodynamic mass1 since then the first would vanish2 there is no mass other than electrodynamic inertia4 ,but in this case the mass can no longer be constant2 it augments with the velocity1 and it even depends on the direction1 and a body animated by a notable velocity will not oppose the same inertia to the forces which tend to deflect it from its route1 as to those which tend to accelerate or to retard its progress4 ,there is still a resource2 the ultimate elements of bodies are electrons1 some charged negatively1 the others charged positively4 ,the negative electrons have no mass1 this is understood2 but the positive electrons1 from the little we know of them1 seem much greater4 ,perhaps they have1 besides their electrodynamic mass1 a true mechanical mass4 ,the real mass of a body would1 then1 be the sum of the mechanical masses of its positive electrons1 the negative electrons not counting2 mass so defined might still be constant4 ,alas6 this resource also evades us4 ,recall what we have said of the principle of relativity and of the efforts made to save it4 ,and it is not merely a principle which it is a question of saving1 it is the indubitable results of the experiments of ,michelson4 ,well1 as was above seen1 ,lorentz1 to account for these results1 was obliged to suppose that all forces1 whatever their origin1 were reduced in the same proportion in a medium animated by a uniform translation2 this is not sufficient2 it is not enough that this take place for the real forces1 it must also be the same for the forces of inertia2 it is therefore necessary1 he says1 that the masses of all the particles be influenced by a translation to the same degree as the electromagnetic masses of the electrons4 ,so the mechanical masses must vary in accordance with the same laws as the electrodynamic masses2 they can not1 therefore1 be constant4 ,need ,i point out that the fall of ,lavoisier's principle involves that of ,newton's8 ,this latter signifies that the center of gravity of an isolated system moves in a straight line2 but if there is no longer a constant mass1 there is no longer a center of gravity1 we no longer know even what this is4 ,this is why ,i said above that the experiments on the cathode rays appeared to justify the doubts of ,lorentz concerning ,newton's principle4 ,from all these results1 if they were confirmed1 would arise an entirely new mechanics1 which would be1 above all1 characterized by this fact1 that no velocity could surpass that of light1#i any more than any temperature can fall below absolute zero4 #i ,because bodies would oppose an increasing inertia to the causes which would tend to accelerate their motion2 and this inertia would become infinite when one approached the velocity of light4 ,no more for an observer1 carried along himself in a translation he does not suspect1 could any apparent velocity surpass that of light2 and this would be then a contradiction1 if we did not recall that this observer would not use the same clocks as a fixed observer1 but1 indeed1 clocks marking 'local time4' ,here we are then facing a question ,i content myself with stating4 ,if there is no longer any mass1 what becomes of ,newton's law8 ,mass has two aspects3 it is at the same time a coefficient of inertia and an attracting mass entering as factor into ,newtonian attraction4 ,if the coefficient of inertia is not constant1 can the attracting mass be8 ,that is the question4 ,mayer's ,principle4--,at least1 the principle of the conservation of energy yet remained to us1 and this seemed more solid4 ,shall ,i recall to you how it was in its turn thrown into discredit8 ,this event has made more noise than the preceding1 and it is in all the memoirs4 ,from the first words of ,becquerel1 and1 above all1 when the ,curies had discovered radium1 it was seen that every radioactive body was an inexhaustible source of radiation4 ,its activity seemed to subsist without alteration throughout the months and the years4 ,this was in itself a strain on the principles2 these radiations were in fact energy1 and from the same morsel of radium this issued and forever issued4 ,but these quantities of energy were too slight to be measured2 at least that was the belief and we were not much disquieted4 ,the scene changed when ,curie bethought himself to put radium in a calorimeter2 it was then seen that the quantity of heat incessantly created was very notable4 ,the explanations proposed were numerous2 but in such case we can not say1 the more the better4 ,in so far as no one of them has prevailed over the others1 we can not be sure there is a good one among them4 ,since some time1 however1 one of these explanations seems to be getting the upper hand and we may reasonably hope that we hold the key to the mystery4 ,sir ,w4 ,ramsay has striven to show that radium is in process of transformation1 that it contains a store of energy enormous but not inexhaustible4 ,the transformation of radium then would produce a million times more heat than all known transformations2 radium would wear itself out in #a1bej years2 this is quite short1 and you see that we are at least certain to have this point settled some hundreds of years from now4 ,while waiting1 our doubts remain4 ,,chapter ,,ix ,,the ,,future ,,of ,,mathematical ,,physics ,the ,principles and ,experiment4--,in the midst of so much ruin1 what remains standing8 ,the principle of least action is hitherto intact1 and ,larmor appears to believe that it will long survive the others2 in reality1 it is still more vague and more general4 ,in presence of this general collapse of the principles1 what attitude will mathematical physics take8 ,and first1 before too much excitement1 it is proper to ask if all that is really true4 ,all these derogations to the principles are encountered only among infinitesimals2 the microscope is necessary to see the ,brownian movement2 electrons are very light2 radium is very rare1 and one never has more than some milligrams of it at a time4 ,and1 then1 it may be asked whether1 besides the infinitesimal seen1 there was not another infinitesimal unseen counterpoise to the first4 ,so there is an interlocutory question1 and1 as it seems1 only experiment can solve it4 ,we shall1 therefore1 only have to hand over the matter to the experimenters1 and1 while waiting for them to finally decide the debate1 not to preoccupy ourselves with these disquieting problems1 and to tranquilly continue our work as if the principles were still uncontested4 ,certes1 we have much to do without leaving the domain where they may be applied in all security2 we have enough to employ our activity during this period of doubts4 ,the ,rle of the ,analyst4--,and as to these doubts1 is it indeed true that we can do nothing to disembarrass science of them8 ,it must indeed be said1 it is not alone experimental physics that has given birth to them2 mathematical physics has well contributed4 ,it is the experimenters who have seen radium throw out energy1 but it is the theorists who have put in evidence all the difficulties raised by the propagation of light across a medium in motion2 but for these it is probable we should not have become conscious of them4 ,well1 then1 if they have done their best to put us into this embarrassment1 it is proper also that they help us to get out of it4 ,they must subject to critical examination all these new views ,i have just outlined before you1 and abandon the principles only after having made a loyal effort to save them4 ,what can they do in this sense8 ,that is what ,i will try to explain4 ,it is a question before all of endeavoring to obtain a more satisfactory theory of the electrodynamics of bodies in motion4 ,it is there especially1 as ,i have sufficiently shown above1 that difficulties accumulate4 ,it is useless to heap up hypotheses1 we can not satisfy all the principles at once2 so far1 one has succeeded in safeguarding some only on condition of sacrificing the others2 but all hope of obtaining better results is not yet lost4 ,let us take1 then1 the theory of ,lorentz1 turn it in all senses1 modify it little by little1 and perhaps everything will arrange itself4 ,thus in place of supposing that bodies in motion undergo a contraction in the sense of the motion1 and that this contraction is the same whatever be the nature of these bodies and the forces to which they are otherwise subjected1 could we not make a more simple and natural hypothesis8 ,we might imagine1 for example1 that it is the ether which is modified when it is in relative motion in reference to the material medium which penetrates it1 that1 when it is thus modified1 it no longer transmits perturbations with the same velocity in every direction4 ,it might transmit more rapidly those which are propagated parallel to the motion of the medium1 whether in the same sense or in the opposite sense1 and less rapidly those which are propagated perpendicularly4 ,the wave surfaces would no longer be spheres1 but ellipsoids1 and we could dispense with that extraordinary contraction of all bodies4 ,i cite this only as an example1 since the modifications that might be essayed would be evidently susceptible of infinite variation4 ,aberration and ,astronomy4--,it is possible also that astronomy may some day furnish us data on this point2 she it was in the main who raised the question in making us acquainted with the phenomenon of the aberration of light4 ,if we make crudely the theory of aberration1 we reach a very curious result4 ,the apparent positions of the stars differ from their real positions because of the earth's motion1 and as this motion is variable1 these apparent positions vary4 ,the real position we can not ascertain1 but we can observe the variations of the apparent position4 ,the observations of the aberration show us1 therefore1 not the earth's motion1 but the variations of this motion2 they can not1 therefore1 give us information about the absolute motion of the earth4 ,at least this is true in first approximation1 but the case would be no longer the same if we could appreciate the thousandths of a second4 ,then it would be seen that the amplitude of the oscillation depends not alone on the variation of the motion1 a variation which is well known1 since it is the motion of our globe on its elliptic orbit1 but on the mean value of this motion1 so that the constant of aberration would not be quite the same for all the stars1 and the differences would tell us the absolute motion of the earth in space4 ,this1 then1 would be1 under another form1 the ruin of the principle of relativity4 ,we are far1 it is true1 from appreciating the thousandth of a second1 but1 after all1 say some1 the earth's total absolute velocity is perhaps much greater than its relative velocity with respect to the sun4 ,if1 for example1 it were #cjj kilometers per second in place of #cj1 this would suffice to make the phenomenon observable4 ,i believe that in reasoning thus one admits a too simple theory of aberration4 ,michelson has shown us1 ,i have told you1 that the physical procedures are powerless to put in evidence absolute motion2 ,i am persuaded that the same will be true of the astronomic procedures1 however far precision be carried4 ,however that may be1 the data astronomy will furnish us in this regard will some day be precious to the physicist4 ,meanwhile1 ,i believe that the theorists1 recalling the experience of ,michelson1 may anticipate a negative result1 and that they would accomplish a useful work in constructing a theory of aberration which would explain this in advance4 ,electrons and ,spectra4--,this dynamics of electrons can be approached from many sides1 but among the ways leading thither is one which has been somewhat neglected1 and yet this is one of those which promise us the most surprises4 ,it is movements of electrons which produce the lines of the emission spectra2 this is proved by the ,zeeman effect2 in an incandescent body what vibrates is sensitive to the magnet1 therefore electrified4 ,this is a very important first point1 but no one has gone farther4 ,why are the lines of the spectrum distributed in accordance with a regular law8 ,these laws have been studied by the experimenters in their least details2 they are very precise and comparatively simple4 ,a first study of these distributions recalls the harmonics encountered in acoustics2 but the difference is great4 ,not only are the numbers of vibrations not the successive multiples of a single number1 but we do not even find anything analogous to the roots of those transcendental equations to which we are led by so many problems of mathematical physics3 that of the vibrations of an elastic body of any form1 that of the ,hertzian oscillations in a generator of any form1 the problem of ,fourier for the cooling of a solid body4 ,the laws are simpler1 but they are of wholly other nature1 and to cite only one of these differences1 for the harmonics of high order1 the number of vibrations tends toward a finite limit1 instead of increasing indefinitely4 ,that has not yet been accounted for1 and ,i believe that there we have one of the most important secrets of nature4 ,a ,japanese physicist1 ,m4 ,nagaoka1 has recently proposed an explanation2 according to him1 atoms are composed of a large positive electron surrounded by a ring formed of a great number of very small negative electrons4 ,such is the planet ,saturn with its rings4 ,this is a very interesting attempt1 but not yet wholly satisfactory2 this attempt should be renewed4 ,we will penetrate1 so to speak1 into the inmost recess of matter4 ,and from the particular point of view which we to-day occupy1 when we know why the vibrations of incandescent bodies differ thus from ordinary elastic vibrations1 why the electrons do not behave like the matter which is familiar to us1 we shall better comprehend the dynamics of electrons and it will be perhaps more easy for us to reconcile it with the principles4 ,conventions ,preceding ,experiment4--,suppose1 now1 that all these efforts fail1 and1 after all1 ,i do not believe they will1 what must be done8 ,will it be necessary to seek to mend the broken principles by giving what we ,french call a coup de pouce8 ,that evidently is always possible1 and ,i retract nothing of what ,i have said above4 ,have you not written1 you might say if you wished to seek a quarrel with me--have you not written that the principles1 though of experimental origin1 are now unassailable by experiment because they have become conventions8 ,and now you have just told us that the most recent conquests of experiment put these principles in danger4 ,well1 formerly ,i was right and to-day ,i am not wrong4 ,formerly ,i was right1 and what is now happening is a new proof of it4 ,take1 for example1 the calorimetric experiment of ,curie on radium4 ,is it possible to reconcile it with the principle of the conservation of energy8 ,this has been attempted in many ways4 ,but there is among them one ,i should like you to notice2 this is not the explanation which tends to-day to prevail1 but it is one of those which have been proposed4 ,it has been conjectured that radium was only an intermediary1 that it only stored radiations of unknown nature which flashed through space in every direction1 traversing all bodies1 save radium1 without being altered by this passage and without exercising any action upon them4 ,radium alone took from them a little of their energy and afterward gave it out to us in various forms4 ,what an advantageous explanation1 and how convenient6 ,first1 it is unverifiable and thus irrefutable4 ,then again it will serve to account for any derogation whatever to ,mayer's principle2 it answers in advance not only the objection of ,curie1 but all the objections that future experimenters might accumulate4 ,this new and unknown energy would serve for everything4 ,this is just what ,i said1 and therewith we are shown that our principle is unassailable by experiment4 ,but then1 what have we gained by this stroke8 ,the principle is intact1 but thenceforth of what use is it8 ,it enabled us to foresee that in such or such circumstance we could count on such a total quantity of energy2 it limited us2 but now that this indefinite provision of new energy is placed at our disposal1 we are no longer limited by anything2 and1 as ,i have written in ',science and ,hypothesis1' if a principle ceases to be fecund1 experiment without contradicting it directly will nevertheless have condemned it4 ,future ,mathematical ,physics4--,this1 therefore1 is not what would have to be done2 it would be necessary to rebuild anew4 ,if we were reduced to this necessity2 we could moreover console ourselves4 ,it would not be necessary thence to conclude that science can weave only a ,penelope's web1 that it can raise only ephemeral structures1 which it is soon forced to demolish from top to bottom with its own hands4 ,as ,i have said1 we have already passed through a like crisis4 ,i have shown you that in the second mathematical physics1 that of the principles1 we find traces of the first1 that of central forces2 it will be just the same if we must know a third4 ,just so with the animal that exuviates1 that breaks its too narrow carapace and makes itself a fresh one2 under the new envelope one will recognize the essential traits of the organism which have persisted4 ,we can not foresee in what way we are about to expand2 perhaps it is the kinetic theory of gases which is about to undergo development and serve as model to the others4 ,then the facts which first appeared to us as simple thereafter would be merely resultants of a very great number of elementary facts which only the laws of chance would make cooperate for a common end4 ,physical law would then assume an entirely new aspect2 it would no longer be solely a differential equation1 it would take the character of a statistical law4 ,perhaps1 too1 we shall have to construct an entirely new mechanics that we only succeed in catching a glimpse of1 where1 inertia increasing with the velocity1 the velocity of light would become an impassable limit4 ,the ordinary mechanics1 more simple1 would remain a first approximation1 since it would be true for velocities not too great1 so that the old dynamics would still be found under the new4 ,we should not have to regret having believed in the principles1 and even1 since velocities too great for the old formulas would always be only exceptional1 the surest way in practise would be still to act as if we continued to believe in them4 ,they are so useful1 it would be necessary to keep a place for them4 ,to determine to exclude them altogether would be to deprive oneself of a precious weapon4 ,i hasten to say in conclusion that we are not yet there1 and as yet nothing proves that the principles will not come forth from out the fray victorious and intact4#aj #aj ,these considerations on mathematical physics are borrowed from my ,st4 ,louis address4 ,,part ,,iii ,,the ,,objective ,,value ,,of ,,science ,,chapter ,x ,,is ,,science ,,artificial8 #a4 ,the ,philosophy of ,m4 ,le,roy ,there are many reasons for being sceptics2 should we push this scepticism to the very end or stop on the way8 ,to go to the end is the most tempting solution1 the easiest and that which many have adopted1 despairing of saving anything from the shipwreck4 ,among the writings inspired by this tendency it is proper to place in the first rank those of ,m4 ,le,roy4 ,this thinker is not only a philosopher and a writer of the greatest merit1 but he has acquired a deep knowledge of the exact and physical sciences1 and even has shown rare powers of mathematical invention4 ,let us recapitulate in a few words his doctrine1 which has given rise to numerous discussions4 ,science consists only of conventions1 and to this circumstance solely does it owe its apparent certitude2 the facts of science and1 a fortiori1 its laws are the artificial work of the scientist2 science therefore can teach us nothing of the truth2 it can only serve us as rule of action4 ,here we recognize the philosophic theory known under the name of nominalism2 all is not false in this theory2 its legitimate domain must be left it1 but out of this it should not be allowed to go4 ,this is not all2 ,m4 ,le,roy's doctrine is not only nominalistic2 it has besides another characteristic which it doubtless owes to ,m4 ,bergson1 it is anti-intellectualistic4 ,according to ,m4 ,le,roy1 the intellect deforms all it touches1 and that is still more true of its necessary instrument 'discourse4' ,there is reality only in our fugitive and changing impressions1 and even this reality1 when touched1 vanishes4 ,and yet ,m4 ,le,roy is not a sceptic2 if he regards the intellect as incurably powerless1 it is only to give more scope to other sources of knowledge1 to the heart1 for instance1 to sentiment1 to instinct or to faith4 ,however great my esteem for ,m4 ,le,roy's talent1 whatever the ingenuity of this thesis1 ,i can not wholly accept it4 ,certes1 ,i am in accord on many points with ,m4 ,le,roy1 and he has even cited1 in support of his view1 various passages of my writings which ,i am by no means disposed to reject4 ,i think myself only the more bound to explain why ,i can not go with him all the way4 ,m4 ,le,roy often complains of being accused of scepticism4 ,he could not help being1 though this accusation is probably unjust4 ,are not appearances against him8 ,nominalist in doctrine1 but realist at heart1 he seems to escape absolute nominalism only by a desperate act of faith4 ,the fact is that anti-intellectualistic philosophy in rejecting analysis and 'discourse1' just by that condemns itself to being intransmissible2 it is a philosophy essentially internal1 or1 at the very least1 only its negations can be transmitted2 what wonder then that for an external observer it takes the shape of scepticism8 ,therein lies the weak point of this philosophy2 if it strives to remain faithful to itself1 its energy is spent in a negation and a cry of enthusiasm4 ,each author may repeat this negation and this cry1 may vary their form1 but without adding anything4 ,and1 yet1 would it not be more logical in remaining silent8 ,see1 you have written long articles2 for that1 it was necessary to use words4 ,and therein have you not been much more 'discursive' and consequently much farther from life and truth than the animal who simply lives without philosophizing8 ,would not this animal be the true philosopher8 ,however1 because no painter has made a perfect portrait1 should we conclude that the best painting is not to paint8 ,when a zoologist dissects an animal1 certainly he 'alters it4' ,yes1 in dissecting it1 he condemns himself to never know all of it2 but in not dissecting it1 he would condemn himself to never know anything of it and consequently to never see anything of it4 ,certes1 in man are other forces besides his intellect2 no one has ever been mad enough to deny that4 ,the first comer makes these blind forces act or lets them act2 the philosopher must speak of them2 to speak of them1 he must know of them the little that can be known1 he should therefore see them act4 ,how8 ,with what eyes1 if not with his intellect8 ,heart1 instinct1 may guide it1 but not render it useless2 they may direct the look1 but not replace the eye4 ,it may be granted that the heart is the workman1 and the intellect only the instrument4 ,yet is it an instrument not to be done without1 if not for action1 at least for philosophizing8 ,therefore a philosopher really anti-intellectualistic is impossible4 ,perhaps we shall have to declare for the supremacy of action2 always it is our intellect which will thus conclude2 in allowing precedence to action it will thus retain the superiority of the thinking reed4 ,this also is a supremacy not to be disdained4 ,pardon these brief reflections and pardon also their brevity1 scarcely skimming the question4 ,the process of intellectualism is not the subject ,i wish to treat3 ,i wish to speak of science1 and about it there is no doubt2 by definition1 so to speak1 it will be intellectualistic or it will not be at all4 ,precisely the question is1 whether it will be4 #b4 ,science1 ,rule of ,action ,for ,m4 ,le,roy1 science is only a rule of action4 ,we are powerless to know anything and yet we are launched1 we must act1 and at all hazards we have established rules4 ,it is the aggregate of these rules that is called science4 ,it is thus that men1 desirous of diversion1 have instituted rules of play1 like those of tric-trac for instance1 which1 better than science itself1 could rely upon the proof by universal consent4 ,it is thus likewise that1 unable to choose1 but forced to choose1 we toss up a coin1 head or tail to win4 ,the rule of tric-trac is indeed a rule of action like science1 but does any one think the comparison just and not see the difference8 ,the rules of the game are arbitrary conventions and the contrary convention might have been adopted1 which would have been none the less good4 ,on the contrary1 science is a rule of action which is successful1 generally at least1 and ,i add1 while the contrary rule would not have succeeded4 ,if ,i say1 to make hydrogen cause an acid to act on zinc1 ,i formulate a rule which succeeds2 ,i could have said1 make distilled water act on gold2 that also would have been a rule1 only it would not have succeeded4 ,if1 therefore1 scientific 'recipes' have a value1 as rule of action1 it is because we know they succeed1 generally at least4 ,but to know this is to know something and then why tell us we can know nothing8 ,science foresees1 and it is because it foresees that it can be useful and serve as rule of action4 ,i well know that its previsions are often contradicted by the event2 that shows that science is imperfect1 and if ,i add that it will always remain so1 ,i am certain that this is a prevision which1 at least1 will never be contradicted4 ,always the scientist is less often mistaken than a prophet who should predict at random4 ,besides the progress though slow is continuous1 so that scientists1 though more and more bold1 are less and less misled4 ,this is little1 but it is enough4 ,i well know that ,m4 ,le,roy has somewhere said that science was mistaken oftener than one thought1 that comets sometimes played tricks on astronomers1 that scientists1 who apparently are men1 did not willingly speak of their failures1 and that1 if they should speak of them1 they would have to count more defeats than victories4 ,that day1 ,m4 ,le,roy evidently overreached himself4 ,if science did not succeed1 it could not serve as rule of action2 whence would it get its value8 ,because it is 'lived1' that is1 because we love it and believe in it8 ,the alchemists had recipes for making gold1 they loved them and had faith in them1 and yet our recipes are the good ones1 although our faith be less lively1 because they succeed4 ,there is no escape from this dilemma2 either science does not enable us to foresee1 and then it is valueless as rule of action2 or else it enables us to foresee1 in a fashion more or less imperfect1 and then it is not without value as means of knowledge4 ,it should not even be said that action is the goal of science2 should we condemn studies of the star ,sirius1 under pretext that we shall probably never exercise any influence on that star8 ,to my eyes1 on the contrary1 it is the knowledge which is the end1 and the action which is the means4 ,if ,i felicitate myself on the industrial development1 it is not alone because it furnishes a facile argument to the advocates of science2 it is above all because it gives to the scientist faith in himself and also because it offers him an immense field of experience where he clashes against forces too colossal to be tampered with4 ,without this ballast1 who knows whether he would not quit solid ground1 seduced by the mirage of some scholastic novelty1 or whether he would not despair1 believing he had fashioned only a dream8 #c4 ,the ,crude ,fact and the ,scientific ,fact ,what was most paradoxical in ,m4 ,le,roy's thesis was that affirmation that the scientist creates the fact2 this was at the same time its essential point and it is one of those which have been most discussed4 ,perhaps1 says he 7,i well believe that this was a concession71 it is not the scientist that creates the fact in the rough2 it is at least he who creates the scientific fact4 ,this distinction between the fact in the rough and the scientific fact does not by itself appear to me illegitimate4 ,but ,i complain first that the boundary has not been traced either exactly or precisely2 and then that the author has seemed to suppose that the crude fact1 not being scientific1 is outside of science4 ,finally1 ,i can not admit that the scientist creates without restraint the scientific fact1 since it is the crude fact which imposes it upon him4 ,the examples given by ,m4 ,le,roy have greatly astonished me4 ,the first is taken from the notion of atom4 ,the atom chosen as example of fact6 ,i avow that this choice has so disconcerted me that ,i prefer to say nothing about it4 ,i have evidently misunderstood the author's thought and ,i could not fruitfully discuss it4 ,the second case taken as example is that of an eclipse where the crude phenomenon is a play of light and shadow1 but where the astronomer can not intervene without introducing two foreign elements1 to wit1 a clock and ,newton's law4 ,finally1 ,m4 ,le,roy cites the rotation of the earth2 it has been answered3 but this is not a fact1 and he has replied3 it was one for ,galileo1 who affirmed it1 as for the inquisitor1 who denied it4 ,it always remains that this is not a fact in the same sense as those just spoken of and that to give them the same name is to expose one's self to many confusions4 ,here then are four degrees3 #a4 ,it grows dark1 says the clown4 #b4 ,the eclipse happened at nine o'clock1 says the astronomer4 #c4 ,the eclipse happened at the time deducible from the tables constructed according to ,newton's law1 says he again4 #d4 ,that results from the earth's turning around the sun1 says ,galileo finally4 ,where then is the boundary between the fact in the rough and the scientific fact8 ,to read ,m4 ,le,roy one would believe that it is between the first and the second stage1 but who does not see that there is a greater distance from the second to the third1 and still more from the third to the fourth4 ,allow me to cite two examples which perhaps will enlighten us a little4 ,i observe the deviation of a galvanometer by the aid of a movable mirror which projects a luminous image or spot on a divided scale4 ,the crude fact is this3 ,i see the spot displace itself on the scale1 and the scientific fact is this3 a current passes in the circuit4 ,or again3 when ,i make an experiment ,i should subject the result to certain corrections1 because ,i know ,i must have made errors4 ,these errors are of two kinds1 some are accidental and these ,i shall correct by taking the mean2 the others are systematic and ,i shall be able to correct those only by a thorough study of their causes4 ,the first result obtained is then the fact in the rough1 while the scientific fact is the final result after the finished corrections4 ,reflecting on this latter example1 we are led to subdivide our second stage1 and in place of saying3 #b4 ,the eclipse happened at nine o'clock1 we shall say3 #b;a4 ,the eclipse happened when my clock pointed to nine1 and #b;b4 ,my clock being ten minutes slow1 the eclipse happened at ten minutes past nine4 ,and this is not all3 the first stage also should be subdivided1 and not between these two subdivisions will be the least distance2 it is necessary to distinguish between the impression of obscurity felt by one witnessing an eclipse1 and the affirmation3 ,it grows dark1 which this impression extorts from him4 ,in a sense it is the first which is the only true fact in the rough1 and the second is already a sort of scientific fact4 ,now then our scale has six stages1 and even though there is no reason for halting at this figure1 there we shall stop4 ,what strikes me at the start is this4 ,at the first of our six stages1 the fact1 still completely in the rough1 is1 so to speak1 individual1 it is completely distinct from all other possible facts4 ,from the second stage1 already it is no longer the same4 ,the enunciation of the fact would suit an infinity of other facts4 ,so soon as language intervenes1 ,i have at my command only a finite number of terms to express the shades1 in number infinite1 that my impressions might cover4 ,when ,i say3 ,it grows dark1 that well expresses the impressions ,i feel in being present at an eclipse2 but even in obscurity a multitude of shades could be imagined1 and if1 instead of that actually realized1 had happened a slightly different shade1 yet ,i should still have enunciated this other fact by saying3 ,it grows dark4 ,second remark3 even at the second stage1 the enunciation of a fact can only be true or false4 ,this is not so of any proposition2 if this proposition is the enunciation of a convention1 it can not be said that this enunciation is true1 in the proper sense of the word1 since it could not be true apart from me and is true only because ,i wish it to be4 ,when1 for instance1 ,i say the unit for length is the meter1 this is a decree that ,i promulgate1 it is not something ascertained which forces itself upon me4 ,it is the same1 as ,i think ,i have elsewhere shown1 when it is a question1 for example1 of ,euclid's postulate4 ,when ,i am asked3 ,is it growing dark8 ,i always know whether ,i ought to reply yes or no4 ,although an infinity of possible facts may be susceptible of this same enunciation1 it grows dark1 ,i shall always know whether the fact realized belongs or does not belong among those which answer to this enunciation4 ,facts are classed in categories1 and if ,i am asked whether the fact that ,i ascertain belongs or does not belong in such a category1 ,i shall not hesitate4 ,doubtless this classification is sufficiently arbitrary to leave a large part to man's freedom or caprice4 ,in a word1 this classification is a convention4 ,this convention being given1 if ,i am asked3 ,is such a fact true8 ,i shall always know what to answer1 and my reply will be imposed upon me by the witness of my senses4 ,if therefore1 during an eclipse1 it is asked3 ,is it growing dark8 all the world will answer yes4 ,doubtless those speaking a language where bright was called dark1 and dark bright1 would answer no4 ,but of what importance is that8 ,in the same way1 in mathematics1 when ,i have laid down the definitions1 and the postulates which are conventions1 a theorem henceforth can only be true or false4 ,but to answer the question3 ,is this theorem true8 it is no longer to the witness of my senses that ,i shall have recourse1 but to reasoning4 ,a statement of fact is always verifiable1 and for the verification we have recourse either to the witness of our senses1 or to the memory of this witness4 ,this is properly what characterizes a fact4 ,if you put the question to me3 ,is such a fact true8 ,i shall begin by asking you1 if there is occasion1 to state precisely the conventions1 by asking you1 in other words1 what language you have spoken2 then once settled on this point1 ,i shall interrogate my senses and shall answer yes or no4 ,but it will be my senses that will have made answer1 it will not be you when you say to me3 ,i have spoken to you in ,english or in ,french4 ,is there something to change in all that when we pass to the following stages8 ,when ,i observe a galvanometer1 as ,i have just said1 if ,i ask an ignorant visitor3 ,is the current passing8 he looks at the wire to try to see something pass2 but if ,i put the same question to my assistant who understands my language1 he will know ,i mean3 ,does the spot move8 and he will look at the scale4 ,what difference is there then between the statement of a fact in the rough and the statement of a scientific fact8 ,the same difference as between the statement of the same crude fact in ,french and in ,german4 ,the scientific statement is the translation of the crude statement into a language which is distinguished above all from the common ,german or ,french1 because it is spoken by a very much smaller number of people4 ,yet let us not go too fast4 ,to measure a current ,i may use a very great number of types of galvanometers or besides an electrodynamometer4 ,and then when ,i shall say there is running in this circuit a current of so many amperes1 that will mean3 if ,i adapt to this circuit such a galvanometer ,i shall see the spot come to the division a2 but that will mean equally3 if ,i adapt to this circuit such an electrodynamometer1 ,i shall see the spot go to the division b4 ,and that will mean still many other things1 because the current can manifest itself not only by mechanical effects1 but by effects chemical1 thermal1 luminous1 etc4 ,here then is one same statement which suits a very great number of facts absolutely different4 ,why8 ,it is because ,i assume a law according to which1 whenever such a mechanical effect shall happen1 such a chemical effect will happen also4 ,previous experiments1 very numerous1 have never shown this law to fail1 and then ,i have understood that ,i could express by the same statement two facts so invariably bound one to the other4 ,when ,i am asked3 ,is the current passing8 ,i can understand that that means3 ,will such a mechanical effect happen8 ,but ,i can understand also3 ,will such a chemical effect happen8 ,i shall then verify either the existence of the mechanical effect1 or that of the chemical effect2 that will be indifferent1 since in both cases the answer must be the same4 ,and if the law should one day be found false8 ,if it was perceived that the concordance of the two effects1 mechanical and chemical1 is not constant8 ,that day it would be necessary to change the scientific language to free it from a grave ambiguity4 ,and after that8 ,is it thought that ordinary language by aid of which are expressed the facts of daily life is exempt from ambiguity8 ,shall we thence conclude that the facts of daily life are the work of the grammarians8 ,you ask me3 ,is there a current8 ,i try whether the mechanical effect exists1 ,i ascertain it and ,i answer3 ,yes1 there is a current4 ,you understand at once that that means that the mechanical effect exists1 and that the chemical effect1 that ,i have not investigated1 exists likewise4 ,imagine now1 supposing an impossibility1 the law we believe true1 not to be1 and the chemical effect not to exist4 ,under this hypothesis there will be two distinct facts1 the one directly observed and which is true1 the other inferred and which is false4 ,it may strictly be said that we have created the second4 ,so that error is the part of man's personal collaboration in the creation of the scientific fact4 ,but if we can say that the fact in question is false1 is this not just because it is not a free and arbitrary creation of our mind1 a disguised convention1 in which case it would be neither true nor false4 ,and in fact it was verifiable2 ,i had not made the verification1 but ,i could have made it4 ,if ,i answered amiss1 it was because ,i chose to reply too quickly1 without having asked nature1 who alone knew the secret4 ,when1 after an experiment1 ,i correct the accidental and systematic errors to bring out the scientific fact1 the case is the same2 the scientific fact will never be anything but the crude fact translated into another language4 ,when ,i shall say3 ,it is such an hour1 that will be a short way of saying3 ,there is such a relation between the hour indicated by my clock1 and the hour it marked at the moment of the passing of such a star and such another star across the meridian4 ,and this convention of language once adopted1 when ,i shall be asked3 ,is it such an hour8 it will not depend upon me to answer yes or no4 ,let us pass to the stage before the last3 the eclipse happened at the hour given by the tables deduced from ,newton's laws4 ,this is still a convention of language which is perfectly clear for those who know celestial mechanics or simply for those who have the tables calculated by the astronomers4 ,i am asked3 ,did the eclipse happen at the hour predicted8 ,i look in the nautical almanac1 ,i see that the eclipse was announced for nine o'clock and ,i understand that the question means3 ,did the eclipse happen at nine o'clock8 ,there still we have nothing to change in our conclusions4 ,the scientific fact is only the crude fact translated into a convenient language4 ,it is true that at the last stage things change4 ,does the earth rotate8 ,is this a verifiable fact8 ,could ,galileo and the ,grand ,inquisitor1 to settle the matter1 appeal to the witness of their senses8 ,on the contrary1 they were in accord about the appearances1 and whatever had been the accumulated experiences1 they would have remained in accord with regard to the appearances without ever agreeing on their interpretation4 ,it is just on that account that they were obliged to have recourse to procedures of discussion so unscientific4 ,this is why ,i think they did not disagree about a fact3 we have not the right to give the same name to the rotation of the earth1 which was the object of their discussion1 and to the facts crude or scientific we have hitherto passed in review4 ,after what precedes1 it seems superfluous to investigate whether the fact in the rough is outside of science1 because there can neither be science without scientific fact1 nor scientific fact without fact in the rough1 since the first is only the translation of the second4 ,and then1 has one the right to say that the scientist creates the scientific fact8 ,first of all1 he does not create it from nothing1 since he makes it with the fact in the rough4 ,consequently he does not make it freely and as he chooses4 ,however able the worker may be1 his freedom is always limited by the properties of the raw material on which he works4 ,after all1 what do you mean when you speak of this free creation of the scientific fact and when you take as example the astronomer who intervenes actively in the phenomenon of the eclipse by bringing his clock8 ,do you mean3 ,the eclipse happened at nine o'clock2 but if the astronomer had wished it to happen at ten1 that depended only on him1 he had only to advance his clock an hour8 ,but the astronomer1 in perpetrating that bad joke1 would evidently have been guilty of an equivocation4 ,when he tells me3 ,the eclipse happened at nine1 ,i understand that nine is the hour deduced from the crude indication of the pendulum by the usual series of corrections4 ,if he has given me solely that crude indication1 or if he has made corrections contrary to the habitual rules1 he has changed the language agreed upon without forewarning me4 ,if1 on the contrary1 he took care to forewarn me1 ,i have nothing to complain of1 but then it is always the same fact expressed in another language4 ,in sum1 all the scientist creates in a fact is the language in which he enunciates it4 ,if he predicts a fact1 he will employ this language1 and for all those who can speak and understand it1 his prediction is free from ambiguity4 ,moreover1 this prediction once made1 it evidently does not depend upon him whether it is fulfilled or not4 ,what then remains of ,m4 ,le,roy's thesis8 ,this remains3 the scientist intervenes actively in choosing the facts worth observing4 ,an isolated fact has by itself no interest2 it becomes interesting if one has reason to think that it may aid in the prediction of other facts2 or better1 if1 having been predicted1 its verification is the confirmation of a law4 ,who shall choose the facts which1 corresponding to these conditions1 are worthy the freedom of the city in science8 ,this is the free activity of the scientist4 ,and that is not all4 ,i have said that the scientific fact is the translation of a crude fact into a certain language2 ,i should add that every scientific fact is formed of many crude facts4 ,this is sufficiently shown by the examples cited above4 ,for instance1 for the hour of the eclipse my clock marked the hour alpha at the instant of the eclipse2 it marked the hour beta at the moment of the last transit of the meridian of a certain star that we take as origin of right ascensions2 it marked the hour gamma at the moment of the preceding transit of this same star4 ,there are three distinct facts 7still it will be noticed that each of them results itself from two simultaneous facts in the rough2 but let us pass this over74 ,in place of that ,i say3 ,the eclipse happened at the hour #bd 7alpha-beta7_/7beta-gamma71 and the three facts are combined in a single scientific fact4 ,i have concluded that the three readings1 alpha1 beta1 gamma made on my clock at three different moments lacked interest and that the only thing interesting was the combination 7alpha-beta7_/7beta-gamma7 of the three4 ,in this conclusion is found the free activity of my mind4 ,but ,i have thus used up my power2 ,i can not make this combination 7alpha-beta7_/7beta-gamma7 have such a value and not such another1 since ,i can not influence either the value of alpha1 or that of beta1 or that of gamma1 which are imposed upon me as crude facts4 ,in sum1 facts are facts1 and if it happens that they satisfy a prediction1 this is not an effect of our free activity4 ,there is no precise frontier between the fact in the rough and the scientific fact2 it can only be said that such an enunciation of fact is more crude or1 on the contrary1 more scientific than such another4 #d4 ',nominalism' and 'the ,universal ,invariant' ,if from facts we pass to laws1 it is clear that the part of the free activity of the scientist will become much greater4 ,but did not ,m4 ,le,roy make it still too great8 ,this is what we are about to examine4 ,recall first the examples he has given4 ,when ,i say3 ,phosphorus melts at #dd1 ,i think ,i am enunciating a law2 in reality it is just the definition of phosphorus2 if one should discover a body which1 possessing otherwise all the properties of phosphorus1 did not melt at #dd1 we should give it another name1 that is all1 and the law would remain true4 ,just so when ,i say3 ,heavy bodies falling freely pass over spaces proportional to the squares of the times1 ,i only give the definition of free fall4 ,whenever the condition shall not be fulfilled1 ,i shall say that the fall is not free1 so that the law will never be wrong4 ,it is clear that if laws were reduced to that1 they could not serve in prediction2 then they would be good for nothing1 either as means of knowledge or as principle of action4 ,when ,i say3 ,phosphorus melts at #dd1 ,i mean by that3 ,all bodies possessing such or such a property 7to wit1 all the properties of phosphorus1 save fusing-point7 fuse at #dd4 ,so understood1 my proposition is indeed a law1 and this law may be useful to me1 because if ,i meet a body possessing these properties ,i shall be able to predict that it will fuse at #dd4 ,doubtless the law may be found to be false4 ,then we shall read in the treatises on chemistry3 ,there are two bodies which chemists long confounded under the name of phosphorus2 these two bodies differ only by their points of fusion4 ,that would evidently not be the first time for chemists to attain to the separation of two bodies they were at first not able to distinguish2 such1 for example1 are neodymium and praseodymium1 long confounded under the name of didymium4 ,i do not think the chemists much fear that a like mischance will ever happen to phosphorus4 ,and if1 to suppose the impossible1 it should happen1 the two bodies would probably not have identically the same density1 identically the same specific heat1 etc41 so that after having determined with care the density1 for instance1 one could still foresee the fusion point4 ,it is1 moreover1 unimportant2 it suffices to remark that there is a law1 and that this law1 true or false1 does not reduce to a tautology4 ,will it be said that if we do not know on the earth a body which does not fuse at #dd while having all the other properties of phosphorus1 we can not know whether it does not exist on other planets8 ,doubtless that may be maintained1 and it would then be inferred that the law in question1 which may serve as a rule of action to us who inhabit the earth1 has yet no general value from the point of view of knowledge1 and owes its interest only to the chance which has placed us on this globe4 ,this is possible1 but1 if it were so1 the law would be valueless1 not because it reduced to a convention1 but because it would be false4 ,the same is true in what concerns the fall of bodies4 ,it would do me no good to have given the name of free fall to falls which happen in conformity with ,galileo's law1 if ,i did not know that elsewhere1 in such circumstances1 the fall will be probably free or approximately free4 ,that then is a law which may be true or false1 but which does not reduce to a convention4 ,suppose the astronomers discover that the stars do not exactly obey ,newton's law4 ,they will have the choice between two attitudes2 they may say that gravitation does not vary exactly as the inverse of the square of the distance1 or else they may say that gravitation is not the only force which acts on the stars and that there is in addition a different sort of force4 ,in the second case1 ,newton's law will be considered as the definition of gravitation4 ,this will be the nominalist attitude4 ,the choice between the two attitudes is free1 and is made from considerations of convenience1 though these considerations are most often so strong that there remains practically little of this freedom4 ,we can break up this proposition3 7#a7 ,the stars obey ,newton's law1 into two others2 7#b7 gravitation obeys ,newton's law2 7#c7 gravitation is the only force acting on the stars4 ,in this case proposition 7#b7 is no longer anything but a definition and is beyond the test of experiment2 but then it will be on proposition 7#c7 that this check can be exercised4 ,this is indeed necessary1 since the resulting proposition 7#a7 predicts verifiable facts in the rough4 ,it is thanks to these artifices that by an unconscious nominalism the scientists have elevated above the laws what they call principles4 ,when a law has received a sufficient confirmation from experiment1 we may adopt two attitudes3 either we may leave this law in the fray2 it will then remain subjected to an incessant revision1 which without any doubt will end by demonstrating that it is only approximative4 ,or else we may elevate it into a principle by adopting conventions such that the proposition may be certainly true4 ,for that the procedure is always the same4 ,the primitive law enunciated a relation between two facts in the rough1 ,a and ,b2 between these two crude facts is introduced an abstract intermediary ,c1 more or less fictitious 7such was in the preceding example the impalpable entity1 gravitation74 ,and then we have a relation between ,a and ,c that we may suppose rigorous and which is the principle2 and another between ,c and ,b which remains a law subject to revision4 ,the principle1 henceforth crystallized1 so to speak1 is no longer subject to the test of experiment4 ,it is not true or false1 it is convenient4 ,great advantages have often been found in proceeding in that way1 but it is clear that if all the laws had been transformed into principles nothing would be left of science4 ,every law may be broken up into a principle and a law1 but thereby it is very clear that1 however far this partition be pushed1 there will always remain laws4 ,nominalism has therefore limits1 and this is what one might fail to recognize if one took to the very letter ,m4 ,le,roy's assertions4 ,a rapid review of the sciences will make us comprehend better what are these limits4 ,the nominalist attitude is justified only when it is convenient2 when is it so8 ,experiment teaches us relations between bodies2 this is the fact in the rough2 these relations are extremely complicated4 ,instead of envisaging directly the relation of the body ,a and the body ,b1 we introduce between them an intermediary1 which is space1 and we envisage three distinct relations3 that of the body ,a with the figure ,a' of space1 that of the body ,b with the figure ,b' of space1 that of the two figures ,a' and ,b' to each other4 ,why is this detour advantageous8 ,because the relation of ,a and ,b was complicated1 but differed little from that of ,a' and ,b'1 which is simple2 so that this complicated relation may be replaced by the simple relation between ,a' and ,b' and by two other relations which tell us that the differences between ,a and ,a'1 on the one hand1 between ,b and ,b'1 on the other hand1 are very small4 ,for example1 if ,a and ,b are two natural solid bodies which are displaced with slight deformation1 we envisage two movable rigid figures ,a' and ,b'4 ,the laws of the relative displacement of these figures ,a' and ,b' will be very simple2 they will be those of geometry4 ,and we shall afterward add that the body ,a1 which always differs very little from ,a'1 dilates from the effect of heat and bends from the effect of elasticity4 ,these dilatations and flexions1 just because they are very small1 will be for our mind relatively easy to study4 ,just imagine to what complexities of language it would have been necessary to be resigned if we had wished to comprehend in the same enunciation the displacement of the solid1 its dilatation and its flexure8 ,the relation between ,a and ,b was a rough law1 and was broken up2 we now have two laws which express the relations of ,a and ,a'1 of ,b and ,b'1 and a principle which expresses that of ,a' with ,b'4 ,it is the aggregate of these principles that is called geometry4 ,two other remarks4 ,we have a relation between two bodies ,a and ,b1 which we have replaced by a relation between two figures ,a' and ,b'2 but this same relation between the same two figures ,a' and ,b' could just as well have replaced advantageously a relation between two other bodies ,a'' and ,b''1 entirely different from ,a and ,b4 ,and that in many ways4 ,if the principles of geometry had not been invented1 after having studied the relation of ,a and ,b1 it would be necessary to begin again ab ovo the study of the relation of ,a'' and ,b''4 ,that is why geometry is so precious4 ,a geometrical relation can advantageously replace a relation which1 considered in the rough state1 should be regarded as mechanical1 it can replace another which should be regarded as optical1 etc4 ,yet let no one say3 ,but that proves geometry an experimental science2 in separating its principles from laws whence they have been drawn1 you artificially separate it itself from the sciences which have given birth to it4 ,the other sciences have likewise principles1 but that does not preclude our having to call them experimental4 ,it must be recognized that it would have been difficult not to make this separation that is pretended to be artificial4 ,we know the rle that the kinematics of solid bodies has played in the genesis of geometry2 should it then be said that geometry is only a branch of experimental kinematics8 ,but the laws of the rectilinear propagation of light have also contributed to the formation of its principles4 ,must geometry be regarded both as a branch of kinematics and as a branch of optics8 ,i recall besides that our ,euclidean space which is the proper object of geometry has been chosen1 for reasons of convenience1 from among a certain number of types which preexist in our mind and which are called groups4 ,if we pass to mechanics1 we still see great principles whose origin is analogous1 and1 as their 'radius of action1' so to speak1 is smaller1 there is no longer reason to separate them from mechanics proper and to regard this science as deductive4 ,in physics1 finally1 the rle of the principles is still more diminished4 ,and in fact they are only introduced when it is of advantage4 ,now they are advantageous precisely because they are few1 since each of them very nearly replaces a great number of laws4 ,therefore it is not of interest to multiply them4 ,besides an outcome is necessary1 and for that it is needful to end by leaving abstraction to take hold of reality4 ,such are the limits of nominalism1 and they are narrow4 ,m4 ,le,roy has insisted1 however1 and he has put the question under another form4 ,since the enunciation of our laws may vary with the conventions that we adopt1 since these conventions may modify even the natural relations of these laws1 is there in the manifold of these laws something independent of these conventions and which may1 so to speak1 play the rle of universal invariant8 ,for instance1 the fiction has been introduced of beings who1 having been educated in a world different from ours1 would have been led to create a non-,euclidean geometry4 ,if these beings were afterward suddenly transported into our world1 they would observe the same laws as we1 but they would enunciate them in an entirely different way4 ,in truth there would still be something in common between the two enunciations1 but this is because these beings do not yet differ enough from us4 ,beings still more strange may be imagined1 and the part common to the two systems of enunciations will shrink more and more4 ,will it thus shrink in convergence toward zero1 or will there remain an irreducible residue which will then be the universal invariant sought8 ,the question calls for precise statement4 ,is it desired that this common part of the enunciations be expressible in words8 ,it is clear1 then1 that there are not words common to all languages1 and we can not pretend to construct ,i know not what universal invariant which should be understood both by us and by the fictitious non-,euclidean geometers of whom ,i have just spoken2 no more than we can construct a phrase which can be understood both by ,germans who do not understand ,french and by ,french who do not understand ,german4 ,but we have fixed rules which permit us to translate the ,french enunciations into ,german1 and inversely4 ,it is for that that grammars and dictionaries have been made4 ,there are also fixed rules for translating the ,euclidean language into the non-,euclidean language1 or1 if there are not1 they could be made4 ,and even if there were neither interpreter nor dictionary1 if the ,germans and the ,french1 after having lived centuries in separate worlds1 found themselves all at once in contact1 do you think there would be nothing in common between the science of the ,german books and that of the ,french books8 ,the ,french and the ,germans would certainly end by understanding each other1 as the ,american ,indians ended by understanding the language of their conquerors after the arrival of the ,spanish4 ,but1 it will be said1 doubtless the ,french would be capable of understanding the ,germans even without having learned ,german1 but this is because there remains between the ,french and the ,germans something in common1 since both are men4 ,we should still attain to an understanding with our hypothetical non-,euclideans1 though they be not men1 because they would still retain something human4 ,but in any case a minimum of humanity is necessary4 ,this is possible1 but ,i shall observe first that this little humanness which would remain in the non-,euclideans would suffice not only to make possible the translation of a little of their language1 but to make possible the translation of all their language4 ,now1 that there must be a minimum is what ,i concede2 suppose there exists ,i know not what fluid which penetrates between the molecules of our matter1 without having any action on it and without being subject to any action coming from it4 ,suppose beings sensible to the influence of this fluid and insensible to that of our matter4 ,it is clear that the science of these beings would differ absolutely from ours and that it would be idle to seek an 'invariant' common to these two sciences4 ,or again1 if these beings rejected our logic and did not admit1 for instance1 the principle of contradiction4 ,but truly ,i think it without interest to examine such hypotheses4 ,and then1 if we do not push whimsicality so far1 if we introduce only fictitious beings having senses analogous to ours and sensible to the same impressions1 and moreover admitting the principles of our logic1 we shall then be able to conclude that their language1 however different from ours it may be1 would always be capable of translation4 ,now the possibility of translation implies the existence of an invariant4 ,to translate is precisely to disengage this invariant4 ,thus1 to decipher a cryptogram is to seek what in this document remains invariant1 when the letters are permuted4 ,what now is the nature of this invariant it is easy to understand1 and a word will suffice us4 ,the invariant laws are the relations between the crude facts1 while the relations between the 'scientific facts' remain always dependent on certain conventions4 ,,chapter ,,xi ,,science ,,and ,,reality #e4 ,contingence and ,determinism ,i do not intend to treat here the question of the contingence of the laws of nature1 which is evidently insoluble1 and on which so much has already been written4 ,i only wish to call attention to what different meanings have been given to this word1 contingence1 and how advantageous it would be to distinguish them4 ,if we look at any particular law1 we may be certain in advance that it can only be approximate4 ,it is1 in fact1 deduced from experimental verifications1 and these verifications were and could be only approximate4 ,we should always expect that more precise measurements will oblige us to add new terms to our formulas2 this is what has happened1 for instance1 in the case of ,mariotte's law4 ,moreover the statement of any law is necessarily incomplete4 ,this enunciation should comprise the enumeration of all the antecedents in virtue of which a given consequent can happen4 ,i should first describe all the conditions of the experiment to be made and the law would then be stated3 ,if all the conditions are fulfilled1 the phenomenon will happen4 ,but we shall be sure of not having forgotten any of these conditions only when we shall have described the state of the entire universe at the instant t2 all the parts of this universe may1 in fact1 exercise an influence more or less great on the phenomenon which must happen at the instant t dt4 ,now it is clear that such a description could not be found in the enunciation of the law2 besides1 if it were made1 the law would become incapable of application2 if one required so many conditions1 there would be very little chance of their ever being all realized at any moment4 ,then as one can never be certain of not having forgotten some essential condition1 it can not be said3 ,if such and such conditions are realized1 such a phenomenon will occur2 it can only be said3 ,if such and such conditions are realized1 it is probable that such a phenomenon will occur1 very nearly4 ,take the law of gravitation1 which is the least imperfect of all known laws4 ,it enables us to foresee the motions of the planets4 ,when ,i use it1 for instance1 to calculate the orbit of ,saturn1 ,i neglect the action of the stars1 and in doing so ,i am certain of not deceiving myself1 because ,i know that these stars are too far away for their action to be sensible4 ,i announce1 then1 with a quasi-certitude that the coordinates of ,saturn at such an hour will be comprised between such and such limits4 ,yet is that certitude absolute8 ,could there not exist in the universe some gigantic mass1 much greater than that of all the known stars and whose action could make itself felt at great distances8 ,that mass might be animated by a colossal velocity1 and after having circulated from all time at such distances that its influence had remained hitherto insensible to us1 it might come all at once to pass near us4 ,surely it would produce in our solar system enormous perturbations that we could not have foreseen4 ,all that can be said is that such an event is wholly improbable1 and then1 instead of saying3 ,saturn will be near such a point of the heavens1 we must limit ourselves to saying3 ,saturn will probably be near such a point of the heavens4 ,although this probability may be practically equivalent to certainty1 it is only a probability4 ,for all these reasons1 no particular law will ever be more than approximate and probable4 ,scientists have never failed to recognize this truth2 only they believe1 right or wrong1 that every law may be replaced by another closer and more probable1 that this new law will itself be only provisional1 but that the same movement can continue indefinitely1 so that science in progressing will possess laws more and more probable1 that the approximation will end by differing as little as you choose from exactitude and the probability from certitude4 ,if the scientists who think thus are right1 still could it be said that the laws of nature are contingent1 even though each law1 taken in particular1 may be qualified as contingent8 ,or must one require1 before concluding the contingence of the natural laws1 that this progress have an end1 that the scientist finish some day by being arrested in his search for a closer and closer approximation1 and that1 beyond a certain limit1 he thereafter meet in nature only caprice8 ,in the conception of which ,i have just spoken 7and which ,i shall call the scientific conception71 every law is only a statement imperfect and provisional1 but it must one day be replaced by another1 a superior law1 of which it is only a crude image4 ,no place therefore remains for the intervention of a free will4 ,it seems to me that the kinetic theory of gases will furnish us a striking example4 ,you know that in this theory all the properties of gases are explained by a simple hypothesis2 it is supposed that all the gaseous molecules move in every direction with great velocities and that they follow rectilineal paths which are disturbed only when one molecule passes very near the sides of the vessel or another molecule4 ,the effects our crude senses enable us to observe are the mean effects1 and in these means1 the great deviations compensate1 or at least it is very improbable that they do not compensate2 so that the observable phenomena follow simple laws such as that of ,mariotte or of ,gay-,lussac4 ,but this compensation of deviations is only probable4 ,the molecules incessantly change place and in these continual displacements the figures they form pass successively through all possible combinations4 ,singly these combinations are very numerous2 almost all are in conformity with ,mariotte's law1 only a few deviate from it4 ,these also will happen1 only it would be necessary to wait a long time for them4 ,if a gas were observed during a sufficiently long time it would certainly be finally seen to deviate1 for a very short time1 from ,mariotte's law4 ,how long would it be necessary to wait8 ,if it were desired to calculate the probable number of years1 it would be found that this number is so great that to write only the number of places of figures employed would still require half a score places of figures4 ,no matter2 enough that it may be done4 ,i do not care to discuss here the value of this theory4 ,it is evident that if it be adopted1 ,mariotte's law will thereafter appear only as contingent1 since a day will come when it will not be true4 ,and yet1 think you the partisans of the kinetic theory are adversaries of determinism8 ,far from it2 they are the most ultra of mechanists4 ,their molecules follow rigid paths1 from which they depart only under the influence of forces which vary with the distance1 following a perfectly determinate law4 ,there remains in their system not the smallest place either for freedom1 or for an evolutionary factor1 properly so-called1 or for anything whatever that could be called contingence4 ,i add1 to avoid mistake1 that neither is there any evolution of ,mariotte's law itself2 it ceases to be true after ,i know not how many centuries2 but at the end of a fraction of a second it again becomes true and that for an incalculable number of centuries4 ,and since ,i have pronounced the word evolution1 let us clear away another mistake4 ,it is often said3 ,who knows whether the laws do not evolve and whether we shall not one day discover that they were not at the ,carboniferous epoch what they are to-day8 ,what are we to understand by that8 ,what we think we know about the past state of our globe1 we deduce from its present state4 ,and how is this deduction made8 ,it is by means of laws supposed known4 ,the law1 being a relation between the antecedent and the consequent1 enables us equally well to deduce the consequent from the antecedent1 that is1 to foresee the future1 and to deduce the antecedent from the consequent1 that is1 to conclude from the present to the past4 ,the astronomer who knows the present situation of the stars can from it deduce their future situation by ,newton's law1 and this is what he does when he constructs ephemerides2 and he can equally deduce from it their past situation4 ,the calculations he thus can make can not teach him that ,newton's law will cease to be true in the future1 since this law is precisely his point of departure2 not more can they tell him it was not true in the past4 ,still1 in what concerns the future1 his ephemerides can one day be tested and our descendants will perhaps recognize that they were false4 ,but in what concerns the past1 the geologic past which had no witnesses1 the results of his calculation1 like those of all speculations where we seek to deduce the past from the present1 escape by their very nature every species of test4 ,so that if the laws of nature were not the same in the ,carboniferous age as at the present epoch1 we shall never be able to know it1 since we can know nothing of this age1 only what we deduce from the hypothesis of the permanence of these laws4 ,perhaps it will be said that this hypothesis might lead to contradictory results and that we shall be obliged to abandon it4 ,thus1 in what concerns the origin of life1 we may conclude that there have always been living beings1 since the present world shows us always life springing from life2 and we may also conclude that there have not always been1 since the application of the existent laws of physics to the present state of our globe teaches us that there was a time when this globe was so warm that life on it was impossible4 ,but contradictions of this sort can always be removed in two ways2 it may be supposed that the actual laws of nature are not exactly what we have assumed2 or else it may be supposed that the laws of nature actually are what we have assumed1 but that it has not always been so4 ,it is evident that the actual laws will never be sufficiently well known for us not to be able to adopt the first of these two solutions and for us to be constrained to infer the evolution of natural laws4 ,on the other hand1 suppose such an evolution2 assume1 if you wish1 that humanity lasts sufficiently long for this evolution to have witnesses4 ,the same antecedent shall produce1 for instance1 different consequents at the ,carboniferous epoch and at the ,quaternary4 ,that evidently means that the antecedents are closely alike2 if all the circumstances were identical1 the ,carboniferous epoch would be indistinguishable from the ,quaternary4 ,evidently this is not what is supposed4 ,what remains is that such antecedent1 accompanied by such accessory circumstance1 produces such consequent2 and that the same antecedent1 accompanied by such other accessory circumstance1 produces such other consequent4 ,time does not enter into the affair4 ,the law1 such as ill-informed science would have stated it1 and which would have affirmed that this antecedent always produces this consequent1 without taking account of the accessory circumstances1 this law1 which was only approximate and probable1 must be replaced by another law more approximate and more probable1 which brings in these accessory circumstances4 ,we always come back1 therefore1 to that same process which we have analyzed above1 and if humanity should discover something of this sort1 it would not say that it is the laws which have evoluted1 but the circumstances which have changed4 ,here1 therefore1 are several different senses of the word contingence4 ,m4 ,le,roy retains them all and he does not sufficiently distinguish them1 but he introduces a new one4 ,experimental laws are only approximate1 and if some appear to us as exact1 it is because we have artificially transformed them into what ,i have above called a principle4 ,we have made this transformation freely1 and as the caprice which has determined us to make it is something eminently contingent1 we have communicated this contingence to the law itself4 ,it is in this sense that we have the right to say that determinism supposes freedom1 since it is freely that we become determinists4 ,perhaps it will be found that this is to give large scope to nominalism and that the introduction of this new sense of the word contingence will not help much to solve all those questions which naturally arise and of which we have just been speaking4 ,i do not at all wish to investigate here the foundations of the principle of induction2 ,i know very well that ,i should not succeed2 it is as difficult to justify this principle as to get on without it4 ,i only wish to show how scientists apply it and are forced to apply it4 ,when the same antecedent recurs1 the same consequent must likewise recur2 such is the ordinary statement4 ,but reduced to these terms this principle could be of no use4 ,for one to be able to say that the same antecedent recurred1 it would be necessary for the circumstances all to be reproduced1 since no one is absolutely indifferent1 and for them to be exactly reproduced4 ,and1 as that will never happen1 the principle can have no application4 ,we should therefore modify the enunciation and say3 ,if an antecedent ,a has once produced a consequent ,b1 an antecedent ,a'1 slightly different from ,a1 will produce a consequent ,b'1 slightly different from ,b4 ,but how shall we recognize that the antecedents ,a and ,a' are 'slightly different'8 ,if some one of the circumstances can be expressed by a number1 and this number has in the two cases values very near together1 the sense of the phrase 'slightly different' is relatively clear2 the principle then signifies that the consequent is a continuous function of the antecedent4 ,and as a practical rule1 we reach this conclusion that we have the right to interpolate4 ,this is in fact what scientists do every day1 and without interpolation all science would be impossible4 ,yet observe one thing4 ,the law sought may be represented by a curve4 ,experiment has taught us certain points of this curve4 ,in virtue of the principle we have just stated1 we believe these points may be connected by a continuous graph4 ,we trace this graph with the eye4 ,new experiments will furnish us new points of the curve4 ,if these points are outside of the graph traced in advance1 we shall have to modify our curve1 but not to abandon our principle4 ,through any points1 however numerous they may be1 a continuous curve may always be passed4 ,doubtless1 if this curve is too capricious1 we shall be shocked 7and we shall even suspect errors of experiment71 but the principle will not be directly put at fault4 ,furthermore1 among the circumstances of a phenomenon1 there are some that we regard as negligible1 and we shall consider ,a and ,a' as slightly different if they differ only by these accessory circumstances4 ,for instance1 ,i have ascertained that hydrogen unites with oxygen under the influence of the electric spark1 and ,i am certain that these two gases will unite anew1 although the longitude of ,jupiter may have changed considerably in the interval4 ,we assume1 for instance1 that the state of distant bodies can have no sensible influence on terrestrial phenomena1 and that seems in fact requisite1 but there are cases where the choice of these practically indifferent circumstances admits of more arbitrariness or1 if you choose1 requires more tact4 ,one more remark3 ,the principle of induction would be inapplicable if there did not exist in nature a great quantity of bodies like one another1 or almost alike1 and if we could not infer1 for instance1 from one bit of phosphorus to another bit of phosphorus4 ,if we reflect on these considerations1 the problem of determinism and of contingence will appear to us in a new light4 ,suppose we were able to embrace the series of all phenomena of the universe in the whole sequence of time4 ,we could envisage what might be called the sequences2 ,i mean relations between antecedent and consequent4 ,i do not wish to speak of constant relations or laws1 ,i envisage separately 7individually1 so to speak7 the different sequences realized4 ,we should then recognize that among these sequences there are no two altogether alike4 ,but1 if the principle of induction1 as we have just stated it1 is true1 there will be those almost alike and that can be classed alongside one another4 ,in other words1 it is possible to make a classification of sequences4 ,it is to the possibility and the legitimacy of such a classification that determinism1 in the end1 reduces4 ,this is all that the preceding analysis leaves of it4 ,perhaps under this modest form it will seem less appalling to the moralist4 ,it will doubtless be said that this is to come back by a detour to ,m4 ,le,roy's conclusion which a moment ago we seemed to reject3 we are determinists voluntarily4 ,and in fact all classification supposes the active intervention of the classifier4 ,i agree that this may be maintained1 but it seems to me that this detour will not have been useless and will have contributed to enlighten us a little4 #f4 ,objectivity of ,science ,i arrive at the question set by the title of this article3 ,what is the objective value of science8 ,and first what should we understand by objectivity8 ,what guarantees the objectivity of the world in which we live is that this world is common to us with other thinking beings4 ,through the communications that we have with other men1 we receive from them ready-made reasonings2 we know that these reasonings do not come from us and at the same time we recognize in them the work of reasonable beings like ourselves4 ,and as these reasonings appear to fit the world of our sensations1 we think we may infer that these reasonable beings have seen the same thing as we2 thus it is we know we have not been dreaming4 ,such1 therefore1 is the first condition of objectivity2 what is objective must be common to many minds and consequently transmissible from one to the other1 and as this transmission can only come about by that 'discourse' which inspires so much distrust in ,m4 ,le,roy1 we are even forced to conclude3 no discourse1 no objectivity4 ,the sensations of others will be for us a world eternally closed4 ,we have no means of verifying that the sensation ,i call red is the same as that which my neighbor calls red4 ,suppose that a cherry and a red poppy produce on me the sensation ,a and on him the sensation ,b and that1 on the contrary1 a leaf produces on me the sensation ,b and on him the sensation ,a4 ,it is clear we shall never know anything about it2 since ,i shall call red the sensation ,a and green the sensation ,b1 while he will call the first green and the second red4 ,in compensation1 what we shall be able to ascertain is that1 for him as for me1 the cherry and the red poppy produce the same sensation1 since he gives the same name to the sensations he feels and ,i do the same4 ,sensations are therefore intransmissible1 or rather all that is pure quality in them is intransmissible and forever impenetrable4 ,but it is not the same with relations between these sensations4 ,from this point of view1 all that is objective is devoid of all quality and is only pure relation4 ,certes1 ,i shall not go so far as to say that objectivity is only pure quantity 7this would be to particularize too far the nature of the relations in question71 but we understand how some one could have been carried away into saying that the world is only a differential equation4 ,with due reserve regarding this paradoxical proposition1 we must nevertheless admit that nothing is objective which is not transmissible1 and consequently that the relations between the sensations can alone have an objective value4 ,perhaps it will be said that the esthetic emotion1 which is common to all mankind1 is proof that the qualities of our sensations are also the same for all men and hence are objective4 ,but if we think about this1 we shall see that the proof is not complete2 what is proved is that this emotion is aroused in ,john as in ,james by the sensations to which ,james and ,john give the same name or by the corresponding combinations of these sensations2 either because this emotion is associated in ,john with the sensation ,a1 which ,john calls red1 while parallelly it is associated in ,james with the sensation ,b1 which ,james calls red2 or better because this emotion is aroused1 not by the qualities themselves of the sensations1 but by the harmonious combination of their relations of which we undergo the unconscious impression4 ,such a sensation is beautiful1 not because it possesses such a quality1 but because it occupies such a place in the woof of our associations of ideas1 so that it can not be excited without putting in motion the 'receiver' which is at the other end of the thread and which corresponds to the artistic emotion4 ,whether we take the moral1 the esthetic or the scientific point of view1 it is always the same thing4 ,nothing is objective except what is identical for all2 now we can only speak of such an identity if a comparison is possible1 and can be translated into a 'money of exchange' capable of transmission from one mind to another4 ,nothing1 therefore1 will have objective value except what is transmissible by 'discourse1' that is1 intelligible4 ,but this is only one side of the question4 ,an absolutely disordered aggregate could not have objective value since it would be unintelligible1 but no more can a well-ordered assemblage have it1 if it does not correspond to sensations really experienced4 ,it seems to me superfluous to recall this condition1 and ,i should not have dreamed of it1 if it had not lately been maintained that physics is not an experimental science4 ,although this opinion has no chance of being adopted either by physicists or by philosophers1 it is well to be warned so as not to let oneself slip over the declivity which would lead thither4 ,two conditions are therefore to be fulfilled1 and if the first separates reality#aa from the dream1 the second distinguishes it from the romance4 #aa ,i here use the word real as a synonym of objective2 ,i thus conform to common usage2 perhaps ,i am wrong1 our dreams are real1 but they are not objective4 ,now what is science8 ,i have explained in the preceding article1 it is before all a classification1 a manner of bringing together facts which appearances separate1 though they were bound together by some natural and hidden kinship4 ,science1 in other words1 is a system of relations4 ,now we have just said1 it is in the relations alone that objectivity must be sought2 it would be vain to seek it in beings considered as isolated from one another4 ,to say that science can not have objective value since it teaches us only relations1 this is to reason backward1 since1 precisely1 it is relations alone which can be regarded as objective4 ,external objects1 for instance1 for which the word object was invented1 are really objects and not fleeting and fugitive appearances1 because they are not only groups of sensations1 but groups cemented by a constant bond4 ,it is this bond1 and this bond alone1 which is the object in itself1 and this bond is a relation4 ,therefore1 when we ask what is the objective value of science1 that does not mean3 ,does science teach us the true nature of things8 but it means3 ,does it teach us the true relations of things8 ,to the first question1 no one would hesitate to reply1 no2 but ,i think we may go farther2 not only science can not teach us the nature of things2 but nothing is capable of teaching it to us1 and if any god knew it1 he could not find words to express it4 ,not only can we not divine the response1 but if it were given to us we could understand nothing of it2 ,i ask myself even whether we really understand the question4 ,when1 therefore1 a scientific theory pretends to teach us what heat is1 or what is electricity1 or life1 it is condemned beforehand2 all it can give us is only a crude image4 ,it is1 therefore1 provisional and crumbling4 ,the first question being out of reason1 the second remains4 ,can science teach us the true relations of things8 ,what it joins together should that be put asunder1 what it puts asunder should that be joined together8 ,to understand the meaning of this new question1 it is needful to refer to what was said above on the conditions of objectivity4 ,have these relations an objective value8 ,that means3 ,are these relations the same for all8 ,will they still be the same for those who shall come after us8 ,it is clear that they are not the same for the scientist and the ignorant person4 ,but that is unimportant1 because if the ignorant person does not see them all at once1 the scientist may succeed in making him see them by a series of experiments and reasonings4 ,the thing essential is that there are points on which all those acquainted with the experiments made can reach accord4 ,the question is to know whether this accord will be durable and whether it will persist for our successors4 ,it may be asked whether the unions that the science of to-day makes will be confirmed by the science of to-morrow4 ,to affirm that it will be so we can not invoke any a priori reason2 but this is a question of fact1 and science has already lived long enough for us to be able to find out by asking its history whether the edifices it builds stand the test of time1 or whether they are only ephemeral constructions4 ,now what do we see8 ,at the first blush1 it seems to us that the theories last only a day and that ruins upon ruins accumulate4 ,to-day the theories are born1 to-morrow they are the fashion1 the day after to-morrow they are classic1 the fourth day they are superannuated1 and the fifth they are forgotten4 ,but if we look more closely1 we see that what thus succumb are the theories properly so called1 those which pretend to teach us what things are4 ,but there is in them something which usually survives4 ,if one of them taught us a true relation1 this relation is definitively acquired1 and it will be found again under a new disguise in the other theories which will successively come to reign in place of the old4 ,take only a single example3 ,the theory of the undulations of the ether taught us that light is a motion2 to-day fashion favors the electromagnetic theory which teaches us that light is a current4 ,we do not consider whether we could reconcile them and say that light is a current1 and that this current is a motion4 ,as it is probable in any case that this motion would not be identical with that which the partisans of the old theory presume1 we might think ourselves justified in saying that this old theory is dethroned4 ,and yet something of it remains1 since between the hypothetical currents which ,maxwell supposes there are the same relations as between the hypothetical motions that ,fresnel supposed4 ,there is1 therefore1 something which remains over and this something is the essential4 ,this it is which explains how we see the present physicists pass without any embarrassment from the language of ,fresnel to that of ,maxwell4 ,doubtless many connections that were believed well established have been abandoned1 but the greatest number remain and it would seem must remain4 ,and for these1 then1 what is the measure of their objectivity8 ,well1 it is precisely the same as for our belief in external objects4 ,these latter are real in this1 that the sensations they make us feel appear to us as united to each other by ,i know not what indestructible cement and not by the hazard of a day4 ,in the same way science reveals to us between phenomena other bonds finer but not less solid2 these are threads so slender that they long remained unperceived1 but once noticed there remains no way of not seeing them2 they are therefore not less real than those which give their reality to external objects2 small matter that they are more recently known1 since neither can perish before the other4 ,it may be said1 for instance1 that the ether is no less real than any external body2 to say this body exists is to say there is between the color of this body1 its taste1 its smell1 an intimate bond1 solid and persistent2 to say the ether exists is to say there is a natural kinship between all the optical phenomena1 and neither of the two propositions has less value than the other4 ,and the scientific syntheses have in a sense even more reality than those of the ordinary senses1 since they embrace more terms and tend to absorb in them the partial syntheses4 ,it will be said that science is only a classification and that a classification can not be true1 but convenient4 ,but it is true that it is convenient1 it is true that it is so not only for me1 but for all men2 it is true that it will remain convenient for our descendants2 it is true finally that this can not be by chance4 ,in sum1 the sole objective reality consists in the relations of things whence results the universal harmony4 ,doubtless these relations1 this harmony1 could not be conceived outside of a mind which conceives them4 ,but they are nevertheless objective because they are1 will become1 or will remain1 common to all thinking beings4 ,this will permit us to revert to the question of the rotation of the earth which will give us at the same time a chance to make clear what precedes by an example4 #g4 ,the ,rotation of the ,earth 444 ,therefore1 have ,i said in ,science and ,hypothesis1 this affirmation1 the earth turns round1 has no meaning 444 or rather these two propositions1 the earth turns round1 and1 it is more convenient to suppose that the earth turns round1 have one and the same meaning4 ,these words have given rise to the strangest interpretations4 ,some have thought they saw in them the rehabilitation of ,ptolemy's system1 and perhaps the justification of ,galileo's condemnation4 ,those who had read attentively the whole volume could not1 however1 delude themselves4 ,this truth1 the earth turns round1 was put on the same footing as ,euclid's postulate1 for example4 ,was that to reject it8 ,but better2 in the same language it may very well be said3 ,these two propositions1 the external world exists1 or1 it is more convenient to suppose that it exists1 have one and the same meaning4 ,so the hypothesis of the rotation of the earth would have the same degree of certitude as the very existence of external objects4 ,but after what we have just explained in the fourth part1 we may go farther4 ,a physical theory1 we have said1 is by so much the more true as it puts in evidence more true relations4 ,in the light of this new principle1 let us examine the question which occupies us4 ,no1 there is no absolute space2 these two contradictory propositions3 ',the earth turns round' and ',the earth does not turn round' are1 therefore1 neither of them more true than the other4 ,to affirm one while denying the other1 in the kinematic sense1 would be to admit the existence of absolute space4 ,but if the one reveals true relations that the other hides from us1 we can nevertheless regard it as physically more true than the other1 since it has a richer content4 ,now in this regard no doubt is possible4 ,behold the apparent diurnal motion of the stars1 and the diurnal motion of the other heavenly bodies1 and besides1 the flattening of the earth1 the rotation of ,foucault's pendulum1 the gyration of cyclones1 the trade-winds1 what not else8 ,for the ,ptolemaist all these phenomena have no bond between them2 for the ,copernican they are produced by the one same cause4 ,in saying1 the earth turns round1 ,i affirm that all these phenomena have an intimate relation1 and that is true1 and that remains true1 although there is not and can not be absolute space4 ,so much for the rotation of the earth upon itself2 what shall we say of its revolution around the sun8 ,here again1 we have three phenomena which for the ,ptolemaist are absolutely independent and which for the ,copernican are referred back to the same origin2 they are the apparent displacements of the planets on the celestial sphere1 the aberration of the fixed stars1 the parallax of these same stars4 ,is it by chance that all the planets admit an inequality whose period is a year1 and that this period is precisely equal to that of aberration1 precisely equal besides to that of parallax8 ,to adopt ,ptolemy's system is to answer1 yes2 to adopt that of ,copernicus is to answer1 no2 this is to affirm that there is a bond between the three phenomena1 and that also is true1 although there is no absolute space4 ,in ,ptolemy's system1 the motions of the heavenly bodies can not be explained by the action of central forces1 celestial mechanics is impossible4 ,the intimate relations that celestial mechanics reveals to us between all the celestial phenomena are true relations2 to affirm the immobility of the earth would be to deny these relations1 that would be to fool ourselves4 ,the truth for which ,galileo suffered remains1 therefore1 the truth1 although it has not altogether the same meaning as for the vulgar1 and its true meaning is much more subtle1 more profound and more rich4 #h4 ,science for ,its ,own ,sake ,not against ,m4 ,le,roy do ,i wish to defend science for its own sake2 maybe this is what he condemns1 but this is what he cultivates1 since he loves and seeks truth and could not live without it4 ,but ,i have some thoughts to express4 ,we can not know all facts and it is necessary to choose those which are worthy of being known4 ,according to ,tolstoi1 scientists make this choice at random1 instead of making it1 which would be reasonable1 with a view to practical applications4 ,on the contrary1 scientists think that certain facts are more interesting than others1 because they complete an unfinished harmony1 or because they make one foresee a great number of other facts4 ,if they are wrong1 if this hierarchy of facts that they implicitly postulate is only an idle illusion1 there could be no science for its own sake1 and consequently there could be no science4 ,as for me1 ,i believe they are right1 and1 for example1 ,i have shown above what is the high value of astronomical facts1 not because they are capable of practical applications1 but because they are the most instructive of all4 ,it is only through science and art that civilization is of value4 ,some have wondered at the formula3 science for its own sake2 and yet it is as good as life for its own sake1 if life is only misery2 and even as happiness for its own sake1 if we do not believe that all pleasures are of the same quality1 if we do not wish to admit that the goal of civilization is to furnish alcohol to people who love to drink4 ,every act should have an aim4 ,we must suffer1 we must work1 we must pay for our place at the game1 but this is for seeing's sake2 or at the very least that others may one day see4 ,all that is not thought is pure nothingness2 since we can think only thoughts and all the words we use to speak of things can express only thoughts1 to say there is something other than thought1 is therefore an affirmation which can have no meaning4 ,and yet--strange contradiction for those who believe in time--geologic history shows us that life is only a short episode between two eternities of death1 and that1 even in this episode1 conscious thought has lasted and will last only a moment4 ,thought is only a gleam in the midst of a long night4 ,but it is this gleam which is everything4 99 99 99 99 99 ,,science ,,and ,,method 99 99 99 99 99 ,,introduction ,i bring together here different studies relating more or less directly to questions of scientific methodology4 ,the scientific method consists in observing and experimenting2 if the scientist had at his disposal infinite time1 it would only be necessary to say to him3 ',look and notice well'2 but1 as there is not time to see everything1 and as it is better not to see than to see wrongly1 it is necessary for him to make choice4 ,the first question1 therefore1 is how he should make this choice4 ,this question presents itself as well to the physicist as to the historian2 it presents itself equally to the mathematician1 and the principles which should guide each are not without analogy4 ,the scientist conforms to them instinctively1 and one can1 reflecting on these principles1 foretell the future of mathematics4 ,we shall understand them better yet if we observe the scientist at work1 and first of all it is necessary to know the psychologic mechanism of invention and1 in particular1 that of mathematical creation4 ,observation of the processes of the work of the mathematician is particularly instructive for the psychologist4 ,in all the sciences of observation account must be taken of the errors due to the imperfections of our senses and our instruments4 ,luckily1 we may assume that1 under certain conditions1 these errors are in part self-compensating1 so as to disappear in the average2 this compensation is due to chance4 ,but what is chance8 ,this idea is difficult to justify or even to define2 and yet what ,i have just said about the errors of observation1 shows that the scientist can not neglect it4 ,it therefore is necessary to give a definition as precise as possible of this concept1 so indispensable yet so illusive4 ,these are generalities applicable in sum to all the sciences2 and for example the mechanism of mathematical invention does not differ sensibly from the mechanism of invention in general4 ,later ,i attack questions relating more particularly to certain special sciences and first to pure mathematics4 ,in the chapters devoted to these1 ,i have to treat subjects a little more abstract4 ,i have first to speak of the notion of space2 every one knows space is relative1 or rather every one says so1 but many think still as if they believed it absolute2 it suffices to reflect a little however to perceive to what contradictions they are exposed4 ,the questions of teaching have their importance1 first in themselves1 then because reflecting on the best way to make new ideas penetrate virgin minds is at the same time reflecting on how these notions were acquired by our ancestors1 and consequently on their true origin1 that is to say1 in reality on their true nature4 ,why do children usually understand nothing of the definitions which satisfy scientists8 ,why is it necessary to give them others8 ,this is the question ,i set myself in the succeeding chapter and whose solution should1 ,i think1 suggest useful reflections to the philosophers occupied with the logic of the sciences4 ,on the other hand1 many geometers believe we can reduce mathematics to the rules of formal logic4 ,unheard-of efforts have been made to do this2 to accomplish it1 some have not hesitated1 for example1 to reverse the historic order of the genesis of our conceptions and to try to explain the finite by the infinite4 ,i believe ,i have succeeded in showing1 for all those who attack the problem unprejudiced1 that here there is a fallacious illusion4 ,i hope the reader will understand the importance of the question and pardon me the aridity of the pages devoted to it4 ,the concluding chapters relative to mechanics and astronomy will be easier to read4 ,mechanics seems on the point of undergoing a complete revolution4 ,ideas which appeared best established are assailed by bold innovators4 ,certainly it would be premature to decide in their favor at once simply because they are innovators4 ,but it is of interest to make known their doctrines1 and this is what ,i have tried to do4 ,as far as possible ,i have followed the historic order2 for the new ideas would seem too astonishing unless we saw how they arose4 ,astronomy offers us majestic spectacles and raises gigantic problems4 ,we can not dream of applying to them directly the experimental method2 our laboratories are too small4 ,but analogy with phenomena these laboratories permit us to attain may nevertheless guide the astronomer4 ,the ,milky ,way1 for example1 is an assemblage of suns whose movements seem at first capricious4 ,but may not this assemblage be compared to that of the molecules of a gas1 whose properties the kinetic theory of gases has made known to us8 ,it is thus by a roundabout way that the method of the physicist may come to the aid of the astronomer4 ,finally ,i have endeavored to give in a few lines the history of the development of ,french geodesy2 ,i have shown through what persevering efforts1 and often what dangers1 the geodesists have procured for us the knowledge we have of the figure of the earth4 ,is this then a question of method8 ,yes1 without doubt1 this history teaches us in fact by what precautions it is necessary to surround a serious scientific operation and how much time and pains it costs to conquer one new decimal4 ,,book ,i ,,science ,,and ,,the ,,scientist ,,chapter ,i ,,the ,,choice ,,of ,,facts ,tolstoi somewhere explains why 'science for its own sake' is in his eyes an absurd conception4 ,we can not know all facts1 since their number is practically infinite4 ,it is necessary to choose2 then we may let this choice depend on the pure caprice of our curiosity2 would it not be better to let ourselves be guided by utility1 by our practical and above all by our moral needs2 have we nothing better to do than to count the number of lady-bugs on our planet8 ,it is clear the word utility has not for him the sense men of affairs give it1 and following them most of our contemporaries4 ,little cares he for industrial applications1 for the marvels of electricity or of automobilism1 which he regards rather as obstacles to moral progress2 utility for him is solely what can make man better4 ,for my part1 it need scarce be said1 ,i could never be content with either the one or the other ideal2 ,i want neither that plutocracy grasping and mean1 nor that democracy goody and mediocre1 occupied solely in turning the other cheek1 where would dwell sages without curiosity1 who1 shunning excess1 would not die of disease1 but would surely die of ennui4 ,but that is a matter of taste and is not what ,i wish to discuss4 ,the question nevertheless remains and should fix our attention2 if our choice can only be determined by caprice or by immediate utility1 there can be no science for its own sake1 and consequently no science4 ,but is that true8 ,that a choice must be made is incontestable2 whatever be our activity1 facts go quicker than we1 and we can not catch them2 while the scientist discovers one fact1 there happen milliards of milliards in a cubic millimeter of his body4 ,to wish to comprise nature in science would be to want to put the whole into the part4 ,but scientists believe there is a hierarchy of facts and that among them may be made a judicious choice4 ,they are right1 since otherwise there would be no science1 yet science exists4 ,one need only open the eyes to see that the conquests of industry which have enriched so many practical men would never have seen the light1 if these practical men alone had existed and if they had not been preceded by unselfish devotees who died poor1 who never thought of utility1 and yet had a guide far other than caprice4 ,as ,mach says1 these devotees have spared their successors the trouble of thinking4 ,those who might have worked solely in view of an immediate application would have left nothing behind them1 and1 in face of a new need1 all must have been begun over again4 ,now most men do not love to think1 and this is perhaps fortunate when instinct guides them1 for most often1 when they pursue an aim which is immediate and ever the same1 instinct guides them better than reason would guide a pure intelligence4 ,but instinct is routine1 and if thought did not fecundate it1 it would no more progress in man than in the bee or ant4 ,it is needful then to think for those who love not thinking1 and1 as they are numerous1 it is needful that each of our thoughts be as often useful as possible1 and this is why a law will be the more precious the more general it is4 ,this shows us how we should choose3 the most interesting facts are those which may serve many times2 these are the facts which have a chance of coming up again4 ,we have been so fortunate as to be born in a world where there are such4 ,suppose that instead of #fj chemical elements there were #fj milliards of them1 that they were not some common1 the others rare1 but that they were uniformly distributed4 ,then1 every time we picked up a new pebble there would be great probability of its being formed of some unknown substance2 all that we knew of other pebbles would be worthless for it2 before each new object we should be as the new-born babe2 like it we could only obey our caprices or our needs4 ,biologists would be just as much at a loss if there were only individuals and no species and if heredity did not make sons like their fathers4 ,in such a world there would be no science2 perhaps thought and even life would be impossible1 since evolution could not there develop the preservational instincts4 ,happily it is not so2 like all good fortune to which we are accustomed1 this is not appreciated at its true worth4 ,which then are the facts likely to reappear8 ,they are first the simple facts4 ,it is clear that in a complex fact a thousand circumstances are united by chance1 and that only a chance still much less probable could reunite them anew4 ,but are there any simple facts8 ,and if there are1 how recognize them8 ,what assurance is there that a thing we think simple does not hide a dreadful complexity8 ,all we can say is that we ought to prefer the facts which seem simple to those where our crude eye discerns unlike elements4 ,and then one of two things3 either this simplicity is real1 or else the elements are so intimately mingled as not to be distinguishable4 ,in the first case there is chance of our meeting anew this same simple fact1 either in all its purity or entering itself as element in a complex manifold4 ,in the second case this intimate mixture has likewise more chances of recurring than a heterogeneous assemblage2 chance knows how to mix1 it knows not how to disentangle1 and to make with multiple elements a well-ordered edifice in which something is distinguishable1 it must be made expressly4 ,the facts which appear simple1 even if they are not so1 will therefore be more easily revived by chance4 ,this it is which justifies the method instinctively adopted by the scientist1 and what justifies it still better1 perhaps1 is that oft-recurring facts appear to us simple1 precisely because we are used to them4 ,but where is the simple fact8 ,scientists have been seeking it in the two extremes1 in the infinitely great and in the infinitely small4 ,the astronomer has found it because the distances of the stars are immense1 so great that each of them appears but as a point1 so great that the qualitative differences are effaced1 and because a point is simpler than a body which has form and qualities4 ,the physicist on the other hand has sought the elementary phenomenon in fictively cutting up bodies into infinitesimal cubes1 because the conditions of the problem1 which undergo slow and continuous variation in passing from one point of the body to another1 may be regarded as constant in the interior of each of these little cubes4 ,in the same way the biologist has been instinctively led to regard the cell as more interesting than the whole animal1 and the outcome has shown his wisdom1 since cells belonging to organisms the most different are more alike1 for the one who can recognize their resemblances1 than are these organisms themselves4 ,the sociologist is more embarrassed2 the elements1 which for him are men1 are too unlike1 too variable1 too capricious1 in a word1 too complex2 besides1 history never begins over again4 ,how then choose the interesting fact1 which is that which begins again8 ,method is precisely the choice of facts2 it is needful then to be occupied first with creating a method1 and many have been imagined1 since none imposes itself1 so that sociology is the science which has the most methods and the fewest results4 ,therefore it is by the regular facts that it is proper to begin2 but after the rule is well established1 after it is beyond all doubt1 the facts in full conformity with it are erelong without interest since they no longer teach us anything new4 ,it is then the exception which becomes important4 ,we cease to seek resemblances2 we devote ourselves above all to the differences1 and among the differences are chosen first the most accentuated1 not only because they are the most striking1 but because they will be the most instructive4 ,a simple example will make my thought plainer3 ,suppose one wishes to determine a curve by observing some of its points4 ,the practician who concerns himself only with immediate utility would observe only the points he might need for some special object4 ,these points would be badly distributed on the curve2 they would be crowded in certain regions1 rare in others1 so that it would be impossible to join them by a continuous line1 and they would be unavailable for other applications4 ,the scientist will proceed differently2 as he wishes to study the curve for itself1 he will distribute regularly the points to be observed1 and when enough are known he will join them by a regular line and then he will have the entire curve4 ,but for that how does he proceed8 ,if he has determined an extreme point of the curve1 he does not stay near this extremity1 but goes first to the other end2 after the two extremities the most instructive point will be the mid-point1 and so on4 ,so when a rule is established we should first seek the cases where this rule has the greatest chance of failing4 ,thence1 among other reasons1 come the interest of astronomic facts1 and the interest of the geologic past2 by going very far away in space or very far away in time1 we may find our usual rules entirely overturned1 and these grand overturnings aid us the better to see or the better to understand the little changes which may happen nearer to us1 in the little corner of the world where we are called to live and act4 ,we shall better know this corner for having traveled in distant countries with which we have nothing to do4 ,but what we ought to aim at is less the ascertainment of resemblances and differences than the recognition of likenesses hidden under apparent divergences4 ,particular rules seem at first discordant1 but looking more closely we see in general that they resemble each other2 different as to matter1 they are alike as to form1 as to the order of their parts4 ,when we look at them with this bias1 we shall see them enlarge and tend to embrace everything4 ,and this it is which makes the value of certain facts which come to complete an assemblage and to show that it is the faithful image of other known assemblages4 ,i will not further insist1 but these few words suffice to show that the scientist does not choose at random the facts he observes4 ,he does not1 as ,tolstoi says1 count the lady-bugs1 because1 however interesting lady-bugs may be1 their number is subject to capricious variations4 ,he seeks to condense much experience and much thought into a slender volume2 and that is why a little book on physics contains so many past experiences and a thousand times as many possible experiences whose result is known beforehand4 ,but we have as yet looked at only one side of the question4 ,the scientist does not study nature because it is useful2 he studies it because he delights in it1 and he delights in it because it is beautiful4 ,if nature were not beautiful1 it would not be worth knowing1 and if nature were not worth knowing1 life would not be worth living4 ,of course ,i do not here speak of that beauty which strikes the senses1 the beauty of qualities and of appearances2 not that ,i undervalue such beauty1 far from it1 but it has nothing to do with science2 ,i mean that profounder beauty which comes from the harmonious order of the parts and which a pure intelligence can grasp4 ,this it is which gives body1 a structure so to speak1 to the iridescent appearances which flatter our senses1 and without this support the beauty of these fugitive dreams would be only imperfect1 because it would be vague and always fleeting4 ,on the contrary1 intellectual beauty is sufficient unto itself1 and it is for its sake1 more perhaps than for the future good of humanity1 that the scientist devotes himself to long and difficult labors4 ,it is1 therefore1 the quest of this especial beauty1 the sense of the harmony of the cosmos1 which makes us choose the facts most fitting to contribute to this harmony1 just as the artist chooses from among the features of his model those which perfect the picture and give it character and life4 ,and we need not fear that this instinctive and unavowed prepossession will turn the scientist aside from the search for the true4 ,one may dream a harmonious world1 but how far the real world will leave it behind6 ,the greatest artists that ever lived1 the ,greeks1 made their heavens2 how shabby it is beside the true heavens1 ours6 ,and it is because simplicity1 because grandeur1 is beautiful1 that we preferably seek simple facts1 sublime facts1 that we delight now to follow the majestic course of the stars1 now to examine with the microscope that prodigious littleness which is also a grandeur1 now to seek in geologic time the traces of a past which attracts because it is far away4 ,we see too that the longing for the beautiful leads us to the same choice as the longing for the useful4 ,and so it is that this economy of thought1 this economy of effort1 which is1 according to ,mach1 the constant tendency of science1 is at the same time a source of beauty and a practical advantage4 ,the edifices that we admire are those where the architect has known how to proportion the means to the end1 where the columns seem to carry gaily1 without effort1 the weight placed upon them1 like the gracious caryatids of the ,erechtheum4 ,whence comes this concordance8 ,is it simply that the things which seem to us beautiful are those which best adapt themselves to our intelligence1 and that consequently they are at the same time the implement this intelligence knows best how to use8 ,or is there here a play of evolution and natural selection8 ,have the peoples whose ideal most conformed to their highest interest exterminated the others and taken their place8 ,all pursued their ideals without reference to consequences1 but while this quest led some to destruction1 to others it gave empire4 ,one is tempted to believe it4 ,if the ,greeks triumphed over the barbarians and if ,europe1 heir of ,greek thought1 dominates the world1 it is because the savages loved loud colors and the clamorous tones of the drum which occupied only their senses1 while the ,greeks loved the intellectual beauty which hides beneath sensuous beauty1 and this intellectual beauty it is which makes intelligence sure and strong4 ,doubtless such a triumph would horrify ,tolstoi1 and he would not like to acknowledge that it might be truly useful4 ,but this disinterested quest of the true for its own beauty is sane also and able to make man better4 ,i well know that there are mistakes1 that the thinker does not always draw thence the serenity he should find therein1 and even that there are scientists of bad character4 ,must we1 therefore1 abandon science and study only morals8 ,what6 ,do you think the moralists themselves are irreproachable when they come down from their pedestal8 ,,chapter ,,ii ,,the ,,future ,,of ,,mathematics ,to foresee the future of mathematics1 the true method is to study its history and its present state4 ,is this not for us mathematicians in a way a professional procedure8 ,we are accustomed to extrapolate1 which is a means of deducing the future from the past and present1 and as we well know what this amounts to1 we run no risk of deceiving ourselves about the range of the results it gives us4 ,we have had hitherto prophets of evil4 ,they blithely reiterate that all problems capable of solution have already been solved1 and that nothing is left but gleaning4 ,happily the case of the past reassures us4 ,often it was thought all problems were solved or at least an inventory was made of all admitting solution4 ,and then the sense of the word solution enlarged1 the insoluble problems became the most interesting of all1 and others unforeseen presented themselves4 ,for the ,greeks a good solution was one employing only ruler and compasses2 then it became one obtained by the extraction of roots1 then one using only algebraic or logarithmic functions4 ,the pessimists thus found themselves always outflanked1 always forced to retreat1 so that at present ,i think there are no more4 ,my intention1 therefore1 is not to combat them1 as they are dead2 we well know that mathematics will continue to develop1 but the question is how1 in what direction8 ,you will answer1 'in every direction1' and that is partly true2 but if it were wholly true it would be a little appalling4 ,our riches would soon become encumbering and their accumulation would produce a medley as impenetrable as the unknown true was for the ignorant4 ,the historian1 the physicist1 even1 must make a choice among facts2 the head of the scientist1 which is only a corner of the universe1 could never contain the universe entire2 so that among the innumerable facts nature offers1 some will be passed by1 others retained4 ,just so1 a fortiori1 in mathematics2 no more can the geometer hold fast pell-mell all the facts presenting themselves to him2 all the more because he it is1 almost ,i had said his caprice1 that creates these facts4 ,he constructs a wholly new combination by putting together its elements2 nature does not in general give it to him ready made4 ,doubtless it sometimes happens that the mathematician undertakes a problem to satisfy a need in physics2 that the physicist or engineer asks him to calculate a number for a certain application4 ,shall it be said that we geometers should limit ourselves to awaiting orders1 and1 in place of cultivating our science for our own delectation1 try only to accommodate ourselves to the wants of our patrons8 ,if mathematics has no other object besides aiding those who study nature1 it is from these we should await orders4 ,is this way of looking at it legitimate8 ,certainly not2 if we had not cultivated the exact sciences for themselves1 we should not have created mathematics the instrument1 and the day the call came from the physicist we should have been helpless4 ,nor do the physicists wait to study a phenomenon until some urgent need of material life has made it a necessity for them2 and they are right4 ,if the scientists of the eighteenth century had neglected electricity as being in their eyes only a curiosity without practical interest1 we should have had in the twentieth century neither telegraphy1 nor electro-chemistry1 nor electro-technics4 ,the physicists1 compelled to choose1 are therefore not guided in their choice solely by utility4 ,how then do they choose between the facts of nature8 ,we have explained it in the preceding chapter3 the facts which interest them are those capable of leading to the discovery of a law1 and so they are analogous to many other facts which do not seem to us isolated1 but closely grouped with others4 ,the isolated fact attracts all eyes1 those of the layman as well as of the scientist4 ,but what the genuine physicist alone knows how to see1 is the bond which unites many facts whose analogy is profound but hidden4 ,the story of ,newton's apple is probably not true1 but it is symbolic2 let us speak of it then as if it were true4 ,well then1 we must believe that before ,newton plenty of men had seen apples fall2 not one knew how to conclude anything therefrom4 ,facts would be sterile were there not minds capable of choosing among them1 discerning those behind which something was hidden1 and of recognizing what is hiding1 minds which under the crude fact perceive the soul of the fact4 ,we find just the same thing in mathematics4 ,from the varied elements at our disposal we can get millions of different combinations2 but one of these combinations1 in so far as it is isolated1 is absolutely void of value4 ,often we have taken great pains to construct it1 but it serves no purpose1 if not perhaps to furnish a task in secondary education4 ,quite otherwise will it be when this combination shall find place in a class of analogous combinations and we shall have noticed this analogy4 ,we are no longer in the presence of a fact1 but of a law4 ,and upon that day the real discoverer will not be the workman who shall have patiently built up certain of these combinations2 it will be he who brings to light their kinship4 ,the first will have seen merely the crude fact1 only the other will have perceived the soul of the fact4 ,often to fix this kinship it suffices him to make a new word1 and this word is creative4 ,the history of science furnishes us a crowd of examples familiar to all4 ,the celebrated ,vienna philosopher ,mach has said that the rle of science is to produce economy of thought1 just as machines produce economy of effort4 ,and that is very true4 ,the savage reckons on his fingers or by heaping pebbles4 ,in teaching children the multiplication table we spare them later innumerable pebble bunchings4 ,some one has already found out1 with pebbles or otherwise1 that #f times #g is #db and has had the idea of noting the result1 and so we need not do it over again4 ,he did not waste his time even if he reckoned for pleasure3 his operation took him only two minutes2 it would have taken in all two milliards if a milliard men had had to do it over after him4 ,the importance of a fact then is measured by its yield1 that is to say1 by the amount of thought it permits us to spare4 ,in physics the facts of great yield are those entering into a very general law1 since from it they enable us to foresee a great number of others1 and just so it is in mathematics4 ,suppose ,i have undertaken a complicated calculation and laboriously reached a result3 ,i shall not be compensated for my trouble if thereby ,i have not become capable of foreseeing the results of other analogous calculations and guiding them with a certainty that avoids the gropings to which one must be resigned in a first attempt4 ,on the other hand1 ,i shall not have wasted my time if these gropings themselves have ended by revealing to me the profound analogy of the problem just treated with a much more extended class of other problems2 if they have shown me at once the resemblances and differences of these1 if in a word they have made me perceive the possibility of a generalization4 ,then it is not a new result ,i have won1 it is a new power4 ,the simple example that comes first to mind is that of an algebraic formula which gives us the solution of a type of numeric problems when finally we replace the letters by numbers4 ,thanks to it1 a single algebraic calculation saves us the pains of ceaselessly beginning over again new numeric calculations4 ,but this is only a crude example2 we all know there are analogies inexpressible by a formula and all the more precious4 ,a new result is of value1 if at all1 when in unifying elements long known but hitherto separate and seeming strangers one to another it suddenly introduces order where apparently disorder reigned4 ,it then permits us to see at a glance each of these elements and its place in the assemblage4 ,this new fact is not merely precious by itself1 but it alone gives value to all the old facts it combines4 ,our mind is weak as are the senses2 it would lose itself in the world's complexity were this complexity not harmonious2 like a near-sighted person1 it would see only the details and would be forced to forget each of these details before examining the following1 since it would be incapable of embracing all4 ,the only facts worthy our attention are those which introduce order into this complexity and so make it accessible4 ,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 results2 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 ,again the comparison of the ,erechtheum comes to my mind1 but ,i must not use it too often4 ,it is for the same reason that1 when a rather long calculation has led to some simple and striking result1 we are not satisfied until we have shown that we should have been able to foresee1 if not this entire result1 at least its most characteristic traits4 ,why8 ,what prevents our being content with a calculation which has told us1 it seems1 all we wished to know8 ,it is because1 in analogous cases1 the long calculation might not again avail1 and that this is not so about the reasoning often half intuitive which would have enabled us to foresee4 ,this reasoning being short1 we see at a single glance all its parts1 so that we immediately perceive what must be changed to adapt it to all the problems of the same nature which can occur4 ,and then it enables us to foresee if the solution of these problems will be simple1 it shows us at least if the calculation is worth undertaking4 ,what we have just said suffices to show how vain it would be to seek to replace by any mechanical procedure the free initiative of the mathematician4 ,to obtain a result of real value1 it is not enough to grind out calculations1 or to have a machine to put things in order2 it is not order alone1 it is unexpected order1 which is worth while4 ,the machine may gnaw on the crude fact1 the soul of the fact will always escape it4 ,since the middle of the last century1 mathematicians are more and more desirous of attaining absolute rigor2 they are right1 and this tendency will be more and more accentuated4 ,in mathematics rigor is not everything1 but without it there is nothing4 ,a demonstration which is not rigorous is nothingness4 ,i think no one will contest this truth4 ,but if it were taken too literally1 we should be led to conclude that before #ahbj1 for example1 there was no mathematics2 this would be manifestly excessive2 the geometers of that time understood voluntarily what we explain by prolix discourse4 ,this does not mean that they did not see it at all2 but they passed over it too rapidly1 and to see it well would have necessitated taking the pains to say it4 ,but is it always needful to say it so many times8 ,those who were the first to emphasize exactness before all else have given us arguments that we may try to imitate2 but if the demonstrations of the future are to be built on this model1 mathematical treatises will be very long2 and if ,i fear the lengthenings1 it is not solely because ,i deprecate encumbering libraries1 but because ,i fear that in being lengthened out1 our demonstrations may lose that appearance of harmony whose usefulness ,i have just explained4 ,the economy of thought is what we should aim at1 so it is not enough to supply models for imitation4 ,it is needful for those after us to be able to dispense with these models and1 in place of repeating an argument already made1 summarize it in a few words4 ,and this has already been attained at times4 ,for instance1 there was a type of reasoning found everywhere1 and everywhere alike4 ,they were perfectly exact but long4 ,then all at once the phrase 'uniformity of convergence' was hit upon and this phrase made those arguments needless2 we were no longer called upon to repeat them1 since they could be understood4 ,those who conquer difficulties then do us a double service3 first they teach us to do as they at need1 but above all they enable us as often as possible to avoid doing as they1 yet without sacrifice of exactness4 ,we have just seen by one example the importance of words in mathematics1 but many others could be cited4 ,it is hard to believe how much a well-chosen word can economize thought1 as ,mach says4 ,perhaps ,i have already said somewhere that mathematics is the art of giving the same name to different things4 ,it is proper that these things1 differing in matter1 be alike in form1 that they may1 so to speak1 run in the same mold4 ,when the language has been well chosen1 we are astonished to see that all the proofs made for a certain object apply immediately to many new objects2 there is nothing to change1 not even the words1 since the names have become the same4 ,a well-chosen word usually suffices to do away with the exceptions from which the rules stated in the old way suffer2 this is why we have created negative quantities1 imaginaries1 points at infinity1 and what not4 ,and exceptions1 we must not forget1 are pernicious because they hide the laws4 ,well1 this is one of the characteristics by which we recognize the facts which yield great results4 ,they are those which allow of these happy innovations of language4 ,the crude fact then is often of no great interest2 we may point it out many times without having rendered great service to science4 ,it takes value only when a wiser thinker perceives the relation for which it stands1 and symbolizes it by a word4 ,moreover the physicists do just the same4 ,they have invented the word 'energy1' and this word has been prodigiously fruitful1 because it also made the law by eliminating the exceptions1 since it gave the same name to things differing in matter and like in form4 ,among words that have had the most fortunate influence ,i would select 'group' and 'invariant4' ,they have made us see the essence of many mathematical reasonings2 they have shown us in how many cases the old mathematicians considered groups without knowing it1 and how1 believing themselves far from one another1 they suddenly found themselves near without knowing why4 ,to-day we should say that they had dealt with isomorphic groups4 ,we now know that in a group the matter is of little interest1 the form alone counts1 and that when we know a group we thus know all the isomorphic groups2 and thanks to these words 'group' and 'isomorphism1' which condense in a few syllables this subtile rule and quickly make it familiar to all minds1 the transition is immediate and can be done with every economy of thought effort4 ,the idea of group besides attaches to that of transformation4 ,why do we put such a value on the invention of a new transformation8 ,because from a single theorem it enables us to get ten or twenty2 it has the same value as a zero adjoined to the right of a whole number4 ,this then it is which has hitherto determined the direction of mathematical advance1 and just as certainly will determine it in the future4 ,but to this end the nature of the problems which come up contributes equally4 ,we can not forget what must be our aim4 ,in my opinion this aim is double4 ,our science borders upon both philosophy and physics1 and we work for our two neighbors2 so we have always seen and shall still see mathematicians advancing in two opposite directions4 ,on the one hand1 mathematical science must reflect upon itself1 and that is useful since reflecting on itself is reflecting on the human mind which has created it1 all the more because it is the very one of its creations for which it has borrowed least from without4 ,this is why certain mathematical speculations are useful1 such as those devoted to the study of the postulates1 of unusual geometries1 of peculiar functions4 ,the more these speculations diverge from ordinary conceptions1 and consequently from nature and applications1 the better they show us what the human mind can create when it frees itself more and more from the tyranny of the external world1 the better therefore they let us know it in itself4 ,but it is toward the other side1 the side of nature1 that we must direct the bulk of our army4 ,there we meet the physicist or the engineer1 who says to us3 ,please integrate this differential equation for me2 ,i might need it in a week in view of a construction which should be finished by that time4 ,this equation1 we answer1 does not come under one of the integrable types2 you know there are not many4 ,yes1 ,i know2 but then what good are you8 ,usually to understand each other is enough2 the engineer in reality does not need the integral in finite terms2 he needs to know the general look of the integral function1 or he simply wants a certain number which could readily be deduced from this integral if it were known4 ,usually it is not known1 but the number can be calculated without it if we know exactly what number the engineer needs and with what approximation4 ,formerly an equation was considered solved only when its solution had been expressed by aid of a finite number of known functions2 but that is possible scarcely once in a hundred times4 ,what we always can do1 or rather what we should always seek to do1 is to solve the problem qualitatively so to speak2 that is to say1 seek to know the general form of the curve which represents the unknown function4 ,it remains to find the quantitative solution of the problem2 but if the unknown can not be determined by a finite calculation1 it may always be represented by a convergent infinite series which enables us to calculate it4 ,can that be regarded as a true solution8 ,we are told that ,newton sent ,leibnitz an anagram almost like this3 aaaaabbbeeeeij1 etc4 ,leibnitz naturally understood nothing at all of it2 but we1 who have the key1 know that this anagram meant1 translated into modern terms3 ,i can integrate all differential equations2 and we are tempted to say that ,newton had either great luck or strange delusions4 ,he merely wished to say he could form 7by the method of indeterminate coefficients7 a series of powers formally satisfying the proposed equation4 ,such a solution would not satisfy us to-day1 and for two reasons3 because the convergence is too slow and because the terms follow each other without obeying any law4 ,on the contrary1 the series ,theta seems to us to leave nothing to be desired1 first because it converges very quickly 7this is for the practical man who wishes to get at a number as quickly as possible7 and next because we see at a glance the law of the terms 7this is to satisfy the esthetic need of the theorist74 ,but then there are no longer solved problems and others which are not2 there are only problems more or less solved1 according as they are solved by a series converging more or less rapidly1 or ruled by a law more or less harmonious4 ,it often happens however that an imperfect solution guides us toward a better one4 ,sometimes the series converges so slowly that the computation is impracticable and we have only succeeded in proving the possibility of the problem4 ,and then the engineer finds this a mockery1 and justly1 since it will not aid him to complete his construction by the date fixed4 ,he little cares to know if it will benefit engineers of the twenty-second century4 ,but as for us1 we think differently and we are sometimes happier to have spared our grandchildren a day's work than to have saved our contemporaries an hour4 ,sometimes by groping1 empirically1 so to speak1 we reach a formula sufficiently convergent4 ,what more do you want8 says the engineer4 ,and yet1 in spite of all1 we are not satisfied2 we should have liked to foresee that convergence4 ,why8 ,because if we had known how to foresee it once1 we would know how to foresee it another time4 ,we have succeeded2 that is a small matter in our eyes if we can not validly expect to do so again4 ,in proportion as science develops1 its total comprehension becomes more difficult2 then we seek to cut it in pieces and to be satisfied with one of these pieces3 in a word1 to specialize4 ,if we went on in this way1 it would be a grievous obstacle to the progress of science4 ,as we have said1 it is by unexpected union between its diverse parts that it progresses4 ,to specialize too much would be to forbid these drawings together4 ,it is to be hoped that congresses like those of ,heidelberg and ,rome1 by putting us in touch with one another1 will open for us vistas over neighboring domains and oblige us to compare them with our own1 to range somewhat abroad from our own little village2 thus they will be the best remedy for the danger just mentioned4 ,but ,i have lingered too long over generalities2 it is time to enter into detail4 ,let us pass in review the various special sciences which combined make mathematics2 let us see what each has accomplished1 whither it tends and what we may hope from it4 ,if the preceding views are correct1 we should see that the greatest advances in the past have happened when two of these sciences have united1 when we have become conscious of the similarity of their form1 despite the difference of their matter1 when they have so modeled themselves upon each other that each could profit by the other's conquests4 ,we should at the same time foresee in combinations of the same sort the progress of the future4 ,,arithmetic ,progress in arithmetic has been much slower than in algebra and analysis1 and it is easy to see why4 ,the feeling of continuity is a precious guide which the arithmetician lacks2 each whole number is separated from the others--it has1 so to speak1 its own individuality4 ,each of them is a sort of exception and this is why general theorems are rarer in the theory of numbers2 this is also why those which exist are more hidden and longer elude the searchers4 ,if arithmetic is behind algebra and analysis1 the best thing for it to do is to seek to model itself upon these sciences so as to profit by their advance4 ,the arithmetician ought therefore to take as guide the analogies with algebra4 ,these analogies are numerous and if1 in many cases1 they have not yet been studied sufficiently closely to become utilizable1 they at least have long been foreseen1 and even the language of the two sciences shows they have been recognized4 ,thus we speak of transcendent numbers and thus we account for the future classification of these numbers already having as model the classification of transcendent functions1 and still we do not as yet very well see how to pass from one classification to the other2 but had it been seen1 it would already have been accomplished and would no longer be the work of the future4 ,the first example that comes to my mind is the theory of congruences1 where is found a perfect parallelism to the theory of algebraic equations4 ,surely we shall succeed in completing this parallelism1 which must hold for instance between the theory of algebraic curves and that of congruences with two variables4 ,and when the problems relative to congruences with several variables shall be solved1 this will be a first step toward the solution of many questions of indeterminate analysis4 ,,algebra ,the theory of algebraic equations will still long hold the attention of geometers2 numerous and very different are the sides whence it may be attacked4 ,we need not think algebra is ended because it gives us rules to form all possible combinations2 it remains to find the interesting combinations1 those which satisfy such and such a condition4 ,thus will be formed a sort of indeterminate analysis where the unknowns will no longer be whole numbers1 but polynomials4 ,this time it is algebra which will model itself upon arithmetic1 following the analogy of the whole number to the integral polynomial with any coefficients or to the integral polynomial with integral coefficients4 ,,geometry ,it looks as if geometry could contain nothing which is not already included in algebra or analysis2 that geometric facts are only algebraic or analytic facts expressed in another language4 ,it might then be thought that after our review there would remain nothing more for us to say relating specially to geometry4 ,this would be to fail to recognize the importance of well-constructed language1 not to comprehend what is added to the things themselves by the method of expressing these things and consequently of grouping them4 ,first the geometric considerations lead us to set ourselves new problems2 these may be1 if you choose1 analytic problems1 but such as we never would have set ourselves in connection with analysis4 ,analysis profits by them however1 as it profits by those it has to solve to satisfy the needs of physics4 ,a great advantage of geometry lies in the fact that in it the senses can come to the aid of thought1 and help find the path to follow1 and many minds prefer to put the problems of analysis into geometric form4 ,unhappily our senses can not carry us very far1 and they desert us when we wish to soar beyond the classical three dimensions4 ,does this mean that1 beyond the restricted domain wherein they seem to wish to imprison us1 we should rely only on pure analysis and that all geometry of more than three dimensions is vain and objectless8 ,the greatest masters of a preceding generation would have answered 'yes'2 to-day we are so familiarized with this notion that we can speak of it1 even in a university course1 without arousing too much astonishment4 ,but what good is it8 ,that is easy to see3 ,first it gives us a very convenient terminology1 which expresses concisely what the ordinary analytic language would say in prolix phrases4 ,moreover1 this language makes us call like things by the same name and emphasize analogies it will never again let us forget4 ,it enables us therefore still to find our way in this space which is too big for us and which we can not see1 always recalling visible space1 which is only an imperfect image of it doubtless1 but which is nevertheless an image4 ,here again1 as in all the preceding examples1 it is analogy with the simple which enables us to comprehend the complex4 ,this geometry of more than three dimensions is not a simple analytic geometry2 it is not purely quantitative1 but qualitative also1 and it is in this respect above all that it becomes interesting4 ,there is a science called analysis situs and which has for its object the study of the positional relations of the different elements of a figure1 apart from their sizes4 ,this geometry is purely qualitative2 its theorems would remain true if the figures1 instead of being exact1 were roughly imitated by a child4 ,we may also make an analysis situs of more than three dimensions4 ,the importance of analysis situs is enormous and can not be too much emphasized2 the advantage obtained from it by ,riemann1 one of its chief creators1 would suffice to prove this4 ,we must achieve its complete construction in the higher spaces2 then we shall have an instrument which will enable us really to see in hyperspace and supplement our senses4 ,the problems of analysis situs would perhaps not have suggested themselves if the analytic language alone had been spoken2 or rather1 ,i am mistaken1 they would have occurred surely1 since their solution is essential to a crowd of questions in analysis1 but they would have come singly1 one after another1 and without our being able to perceive their common bond4 ,,cantorism ,i have spoken above of our need to go back continually to the first principles of our science1 and of the advantage of this for the study of the human mind4 ,this need has inspired two endeavors which have taken a very prominent place in the most recent annals of mathematics4 ,the first is ,cantorism1 which has rendered our science such conspicuous service4 ,cantor introduced into science a new way of considering mathematical infinity4 ,one of the characteristic traits of ,cantorism is that in place of going up to the general by building up constructions more and more complicated and defining by construction1 it starts from the genus supremum and defines only1 as the scholastics would have said1 per genus proximum et differentiam specificam4 ,thence comes the horror it has sometimes inspired in certain minds1 for instance in ,hermite1 whose favorite idea was to compare the mathematical to the natural sciences4 ,with most of us these prejudices have been dissipated1 but it has come to pass that we have encountered certain paradoxes1 certain apparent contradictions that would have delighted ,zeno1 the ,eleatic and the school of ,megara4 ,and then each must seek the remedy4 ,for my part1 ,i think1 and ,i am not the only one1 that the important thing is never to introduce entities not completely definable in a finite number of words4 ,whatever be the cure adopted1 we may promise ourselves the joy of the doctor called in to follow a beautiful pathologic case4 ,,the ,,investigation ,,of ,,the ,,postulates ,on the other hand1 efforts have been made to enumerate the axioms and postulates1 more or less hidden1 which serve as foundation to the different theories of mathematics4 ,professor ,hilbert has obtained the most brilliant results4 ,it seems at first that this domain would be very restricted and there would be nothing more to do when the inventory should be ended1 which could not take long4 ,but when we shall have enumerated all1 there will be many ways of classifying all2 a good librarian always finds something to do1 and each new classification will be instructive for the philosopher4 ,here ,i end this review which ,i could not dream of making complete4 ,i think these examples will suffice to show by what mechanism the mathematical sciences have made their progress in the past and in what direction they must advance in the future4 ,,chapter ,,iii ,,mathematical ,,creation ,the genesis of mathematical creation is a problem which should intensely interest the psychologist4 ,it is the activity in which the human mind seems to take least from the outside world1 in which it acts or seems to act only of itself and on itself1 so that in studying the procedure of geometric thought we may hope to reach what is most essential in man's mind4 ,this has long been appreciated1 and some time back the journal called ,l'enseignement mathmatique1 edited by ,laisant and ,fehr1 began an investigation of the mental habits and methods of work of different mathematicians4 ,i had finished the main outlines of this article when the results of that inquiry were published1 so ,i have hardly been able to utilize them and shall confine myself to saying that the majority of witnesses confirm my conclusions2 ,i do not say all1 for when the appeal is to universal suffrage unanimity is not to be hoped4 ,a first fact should surprise us1 or rather would surprise us if we were not so used to it4 ,how does it happen there are people who do not understand mathematics8 ,if mathematics invokes only the rules of logic1 such as are accepted by all normal minds2 if its evidence is based on principles common to all men1 and that none could deny without being mad1 how does it come about that so many persons are here refractory8 ,that not every one can invent is nowise mysterious4 ,that not every one can retain a demonstration once learned may also pass4 ,but that not every one can understand mathematical reasoning when explained appears very surprising when we think of it4 ,and yet those who can follow this reasoning only with difficulty are in the majority3 that is undeniable1 and will surely not be gainsaid by the experience of secondary-school teachers4 ,and further3 how is error possible in mathematics8 ,a sane mind should not be guilty of a logical fallacy1 and yet there are very fine minds who do not trip in brief reasoning such as occurs in the ordinary doings of life1 and who are incapable of following or repeating without error the mathematical demonstrations which are longer1 but which after all are only an accumulation of brief reasonings wholly analogous to those they make so easily4 ,need we add that mathematicians themselves are not infallible8 ,the answer seems to me evident4 ,imagine a long series of syllogisms1 and that the conclusions of the first serve as premises of the following3 we shall be able to catch each of these syllogisms1 and it is not in passing from premises to conclusion that we are in danger of deceiving ourselves4 ,but between the moment in which we first meet a proposition as conclusion of one syllogism1 and that in which we reencounter it as premise of another syllogism occasionally some time will elapse1 several links of the chain will have unrolled2 so it may happen that we have forgotten it1 or worse1 that we have forgotten its meaning4 ,so it may happen that we replace it by a slightly different proposition1 or that1 while retaining the same enunciation1 we attribute to it a slightly different meaning1 and thus it is that we are exposed to error4 ,often the mathematician uses a rule4 ,naturally he begins by demonstrating this rule2 and at the time when this proof is fresh in his memory he understands perfectly its meaning and its bearing1 and he is in no danger of changing it4 ,but subsequently he trusts his memory and afterward only applies it in a mechanical way2 and then if his memory fails him1 he may apply it all wrong4 ,thus it is1 to take a simple example1 that we sometimes make slips in calculation because we have forgotten our multiplication table4 ,according to this1 the special aptitude for mathematics would be due only to a very sure memory or to a prodigious force of attention4 ,it would be a power like that of the whist-player who remembers the cards played2 or1 to go up a step1 like that of the chess-player who can visualize a great number of combinations and hold them in his memory4 ,every good mathematician ought to be a good chess-player1 and inversely2 likewise he should be a good computer4 ,of course that sometimes happens2 thus ,gauss was at the same time a geometer of genius and a very precocious and accurate computer4 ,but there are exceptions2 or rather ,i err2 ,i can not call them exceptions without the exceptions being more than the rule4 ,gauss it is1 on the contrary1 who was an exception4 ,as for myself1 ,i must confess1 ,i am absolutely incapable even of adding without mistakes4 ,in the same way ,i should be but a poor chess-player2 ,i would perceive that by a certain play ,i should expose myself to a certain danger2 ,i would pass in review several other plays1 rejecting them for other reasons1 and then finally ,i should make the move first examined1 having meantime forgotten the danger ,i had foreseen4 ,in a word1 my memory is not bad1 but it would be insufficient to make me a good chess-player4 ,why then does it not fail me in a difficult piece of mathematical reasoning where most chess-players would lose themselves8 ,evidently because it is guided by the general march of the reasoning4 ,a mathematical demonstration is not a simple juxtaposition of syllogisms1 it is syllogisms placed in a certain order1 and the order in which these elements are placed is much more important than the elements themselves4 ,if ,i have the feeling1 the intuition1 so to speak1 of this order1 so as to perceive at a glance the reasoning as a whole1 ,i need no longer fear lest ,i forget one of the elements1 for each of them will take its allotted place in the array1 and that without any effort of memory on my part4 ,it seems to me then1 in repeating a reasoning learned1 that ,i could have invented it4 ,this is often only an illusion2 but even then1 even if ,i am not so gifted as to create it by myself1 ,i myself re-invent it in so far as ,i repeat it4 ,we know that this feeling1 this intuition of mathematical order1 that makes us divine hidden harmonies and relations1 can not be possessed by every one4 ,some will not have either this delicate feeling so difficult to define1 or a strength of memory and attention beyond the ordinary1 and then they will be absolutely incapable of understanding higher mathematics4 ,such are the majority4 ,others will have this feeling only in a slight degree1 but they will be gifted with an uncommon memory and a great power of attention4 ,they will learn by heart the details one after another2 they can understand mathematics and sometimes make applications1 but they cannot create4 ,others1 finally1 will possess in a less or greater degree the special intuition referred to1 and then not only can they understand mathematics even if their memory is nothing extraordinary1 but they may become creators and try to invent with more or less success according as this intuition is more or less developed in them4 ,in fact1 what is mathematical creation8 ,it does not consist in making new combinations with mathematical entities already known4 ,any one could do that1 but the combinations so made would be infinite in number and most of them absolutely without interest4 ,to create consists precisely in not making useless combinations and in making those which are useful and which are only a small minority4 ,invention is discernment1 choice4 ,how to make this choice ,i have before explained2 the mathematical facts worthy of being studied are those which1 by their analogy with other facts1 are capable of leading us to the knowledge of a mathematical law just as experimental facts lead us to the knowledge of a physical law4 ,they are those which reveal to us unsuspected kinship between other facts1 long known1 but wrongly believed to be strangers to one another4 ,among chosen combinations the most fertile will often be those formed of elements drawn from domains which are far apart4 ,not that ,i mean as sufficing for invention the bringing together of objects as disparate as possible2 most combinations so formed would be entirely sterile4 ,but certain among them1 very rare1 are the most fruitful of all4 ,to invent1 ,i have said1 is to choose2 but the word is perhaps not wholly exact4 ,it makes one think of a purchaser before whom are displayed a large number of samples1 and who examines them1 one after the other1 to make a choice4 ,here the samples would be so numerous that a whole lifetime would not suffice to examine them4 ,this is not the actual state of things4 ,the sterile combinations do not even present themselves to the mind of the inventor4 ,never in the field of his consciousness do combinations appear that are not really useful1 except some that he rejects but which have to some extent the characteristics of useful combinations4 ,all goes on as if the inventor were an examiner for the second degree who would only have to question the candidates who had passed a previous examination4 ,but what ,i have hitherto said is what may be observed or inferred in reading the writings of the geometers1 reading reflectively4 ,it is time to penetrate deeper and to see what goes on in the very soul of the mathematician4 ,for this1 ,i believe1 ,i can do best by recalling memories of my own4 ,but ,i shall limit myself to telling how ,i wrote my first memoir on ,fuchsian functions4 ,i beg the reader's pardon2 ,i am about to use some technical expressions1 but they need not frighten him1 for he is not obliged to understand them4 ,i shall say1 for example1 that ,i have found the demonstration of such a theorem under such circumstances4 ,this theorem will have a barbarous name1 unfamiliar to many1 but that is unimportant2 what is of interest for the psychologist is not the theorem but the circumstances4 ,for fifteen days ,i strove to prove that there could not be any functions like those ,i have since called ,fuchsian functions4 ,i was then very ignorant2 every day ,i seated myself at my work table1 stayed an hour or two1 tried a great number of combinations and reached no results4 ,one evening1 contrary to my custom1 ,i drank black coffee and could not sleep4 ,ideas rose in crowds2 ,i felt them collide until pairs interlocked1 so to speak1 making a stable combination4 ,by the next morning ,i had established the existence of a class of ,fuchsian functions1 those which come from the hypergeometric series2 ,i had only to write out the results1 which took but a few hours4 ,then ,i wanted to represent these functions by the quotient of two series2 this idea was perfectly conscious and deliberate1 the analogy with elliptic functions guided me4 ,i asked myself what properties these series must have if they existed1 and ,i succeeded without difficulty in forming the series ,i have called theta-,fuchsian4 ,just at this time ,i left ,caen1 where ,i was then living1 to go on a geologic excursion under the auspices of the school of mines4 ,the changes of travel made me forget my mathematical work4 ,having reached ,coutances1 we entered an omnibus to go some place or other4 ,at the moment when ,i put my foot on the step the idea came to me1 without anything in my former thoughts seeming to have paved the way for it1 that the transformations ,i had used to define the ,fuchsian functions were identical with those of non-,euclidean geometry4 ,i did not verify the idea2 ,i should not have had time1 as1 upon taking my seat in the omnibus1 ,i went on with a conversation already commenced1 but ,i felt a perfect certainty4 ,on my return to ,caen1 for conscience' sake ,i verified the result at my leisure4 ,then ,i turned my attention to the study of some arithmetical questions apparently without much success and without a suspicion of any connection with my preceding researches4 ,disgusted with my failure1 ,i went to spend a few days at the seaside1 and thought of something else4 ,one morning1 walking on the bluff1 the idea came to me1 with just the same characteristics of brevity1 suddenness and immediate certainty1 that the arithmetic transformations of indeterminate ternary quadratic forms were identical with those of non-,euclidean geometry4 ,returned to ,caen1 ,i meditated on this result and deduced the consequences4 ,the example of quadratic forms showed me that there were ,fuchsian groups other than those corresponding to the hypergeometric series2 ,i saw that ,i could apply to them the theory of theta-,fuchsian series and that consequently there existed ,fuchsian functions other than those from the hypergeometric series1 the ones ,i then knew4 ,naturally ,i set myself to form all these functions4 ,i made a systematic attack upon them and carried all the outworks1 one after another4 ,there was one however that still held out1 whose fall would involve that of the whole place4 ,but all my efforts only served at first the better to show me the difficulty1 which indeed was something4 ,all this work was perfectly conscious4 ,thereupon ,i left for ,mont-,valrien1 where ,i was to go through my military service2 so ,i was very differently occupied4 ,one day1 going along the street1 the solution of the difficulty which had stopped me suddenly appeared to me4 ,i did not try to go deep into it immediately1 and only after my service did ,i again take up the question4 ,i had all the elements and had only to arrange them and put them together4 ,so ,i wrote out my final memoir at a single stroke and without difficulty4 ,i shall limit myself to this single example2 it is useless to multiply them4 ,in regard to my other researches ,i would have to say analogous things1 and the observations of other mathematicians given in ,l'enseignement mathmatique would only confirm them4 ,most striking at first is this appearance of sudden illumination1 a manifest sign of long1 unconscious prior work4 ,the rle of this unconscious work in mathematical invention appears to me incontestable1 and traces of it would be found in other cases where it is less evident4 ,often when one works at a hard question1 nothing good is accomplished at the first attack4 ,then one takes a rest1 longer or shorter1 and sits down anew to the work4 ,during the first half-hour1 as before1 nothing is found1 and then all of a sudden the decisive idea presents itself to the mind4 ,it might be said that the conscious work has been more fruitful because it has been interrupted and the rest has given back to the mind its force and freshness4 ,but it is more probable that this rest has been filled out with unconscious work and that the result of this work has afterward revealed itself to the geometer just as in the cases ,i have cited2 only the revelation1 instead of coming during a walk or a journey1 has happened during a period of conscious work1 but independently of this work which plays at most a rle of excitant1 as if it were the goad stimulating the results already reached during rest1 but remaining unconscious1 to assume the conscious form4 ,there is another remark to be made about the conditions of this unconscious work3 it is possible1 and of a certainty it is only fruitful1 if it is on the one hand preceded and on the other hand followed by a period of conscious work4 ,these sudden inspirations 7and the examples already cited sufficiently prove this7 never happen except after some days of voluntary effort which has appeared absolutely fruitless and whence nothing good seems to have come1 where the way taken seems totally astray4 ,these efforts then have not been as sterile as one thinks2 they have set agoing the unconscious machine and without them it would not have moved and would have produced nothing4 ,the need for the second period of conscious work1 after the inspiration1 is still easier to understand4 ,it is necessary to put in shape the results of this inspiration1 to deduce from them the immediate consequences1 to arrange them1 to word the demonstrations1 but above all is verification necessary4 ,i have spoken of the feeling of absolute certitude accompanying the inspiration2 in the cases cited this feeling was no deceiver1 nor is it usually4 ,but do not think this a rule without exception2 often this feeling deceives us without being any the less vivid1 and we only find it out when we seek to put on foot the demonstration4 ,i have especially noticed this fact in regard to ideas coming to me in the morning or evening in bed while in a semi-hypnagogic state4 ,such are the realities2 now for the thoughts they force upon us4 ,the unconscious1 or1 as we say1 the subliminal self plays an important rle in mathematical creation2 this follows from what we have said4 ,but usually the subliminal self is considered as purely automatic4 ,now we have seen that mathematical work is not simply mechanical1 that it could not be done by a machine1 however perfect4 ,it is not merely a question of applying rules1 of making the most combinations possible according to certain fixed laws4 ,the combinations so obtained would be exceedingly numerous1 useless and cumbersome4 ,the true work of the inventor consists in choosing among these combinations so as to eliminate the useless ones or rather to avoid the trouble of making them1 and the rules which must guide this choice are extremely fine and delicate4 ,it is almost impossible to state them precisely2 they are felt rather than formulated4 ,under these conditions1 how imagine a sieve capable of applying them mechanically8 ,a first hypothesis now presents itself3 the subliminal self is in no way inferior to the conscious self2 it is not purely automatic2 it is capable of discernment2 it has tact1 delicacy2 it knows how to choose1 to divine4 ,what do ,i say8 ,it knows better how to divine than the conscious self1 since it succeeds where that has failed4 ,in a word1 is not the subliminal self superior to the conscious self8 ,you recognize the full importance of this question4 ,boutroux in a recent lecture has shown how it came up on a very different occasion1 and what consequences would follow an affirmative answer4 7,see also1 by the same author1 ,science et ,religion1 pp4 #cac ff47 ,is this affirmative answer forced upon us by the facts ,i have just given8 ,i confess that1 for my part1 ,i should hate to accept it4 ,reexamine the facts then and see if they are not compatible with another explanation4 ,it is certain that the combinations which present themselves to the mind in a sort of sudden illumination1 after an unconscious working somewhat prolonged1 are generally useful and fertile combinations1 which seem the result of a first impression4 ,does it follow that the subliminal self1 having divined by a delicate intuition that these combinations would be useful1 has formed only these1 or has it rather formed many others which were lacking in interest and have remained unconscious8 ,in this second way of looking at it1 all the combinations would be formed in consequence of the automatism of the subliminal self1 but only the interesting ones would break into the domain of consciousness4 ,and this is still very mysterious4 ,what is the cause that1 among the thousand products of our unconscious activity1 some are called to pass the threshold1 while others remain below8 ,is it a simple chance which confers this privilege8 ,evidently not2 among all the stimuli of our senses1 for example1 only the most intense fix our attention1 unless it has been drawn to them by other causes4 ,more generally the privileged unconscious phenomena1 those susceptible of becoming conscious1 are those which1 directly or indirectly1 affect most profoundly our emotional sensibility4 ,it may be surprising to see emotional sensibility invoked propos of mathematical demonstrations which1 it would seem1 can interest only the intellect4 ,this would be to forget the feeling of mathematical beauty1 of the harmony of numbers and forms1 of geometric elegance4 ,this is a true esthetic feeling that all real mathematicians know1 and surely it belongs to emotional sensibility4 ,now1 what are the mathematic entities to which we attribute this character of beauty and elegance1 and which are capable of developing in us a sort of esthetic emotion8 ,they are those whose elements are harmoniously disposed so that the mind without effort can embrace their totality while realizing the details4 ,this harmony is at once a satisfaction of our esthetic needs and an aid to the mind1 sustaining and guiding2 ,and at the same time1 in putting under our eyes a well-ordered whole1 it makes us foresee a mathematical law4 ,now1 as we have said above1 the only mathematical facts worthy of fixing our attention and capable of being useful are those which can teach us a mathematical law4 ,so that we reach the following conclusion3 ,the useful combinations are precisely the most beautiful1 ,i mean those best able to charm this special sensibility that all mathematicians know1 but of which the profane are so ignorant as often to be tempted to smile at it4 ,what happens then8 ,among the great numbers of combinations blindly formed by the subliminal self1 almost all are without interest and without utility2 but just for that reason they are also without effect upon the esthetic sensibility4 ,consciousness will never know them2 only certain ones are harmonious1 and1 consequently1 at once useful and beautiful4 ,they will be capable of touching this special sensibility of the geometer of which ,i have just spoken1 and which1 once aroused1 will call our attention to them1 and thus give them occasion to become conscious4 ,this is only a hypothesis1 and yet here is an observation which may confirm it3 when a sudden illumination seizes upon the mind of the mathematician1 it usually happens that it does not deceive him1 but it also sometimes happens1 as ,i have said1 that it does not stand the test of verification2 well1 we almost always notice that this false idea1 had it been true1 would have gratified our natural feeling for mathematical elegance4 ,thus it is this special esthetic sensibility which plays the rle of the delicate sieve of which ,i spoke1 and that sufficiently explains why the one lacking it will never be a real creator4 ,yet all the difficulties have not disappeared4 ,the conscious self is narrowly limited1 and as for the subliminal self we know not its limitations1 and this is why we are not too reluctant in supposing that it has been able in a short time to make more different combinations than the whole life of a conscious being could encompass4 ,yet these limitations exist4 ,is it likely that it is able to form all the possible combinations1 whose number would frighten the imagination8 ,nevertheless that would seem necessary1 because if it produces only a small part of these combinations1 and if it makes them at random1 there would be small chance that the good1 the one we should choose1 would be found among them4 ,perhaps we ought to seek the explanation in that preliminary period of conscious work which always precedes all fruitful unconscious labor4 ,permit me a rough comparison4 ,figure the future elements of our combinations as something like the hooked atoms of ,epicurus4 ,during the complete repose of the mind1 these atoms are motionless1 they are1 so to speak1 hooked to the wall2 so this complete rest may be indefinitely prolonged without the atoms meeting1 and consequently without any combination between them4 ,on the other hand1 during a period of apparent rest and unconscious work1 certain of them are detached from the wall and put in motion4 ,they flash in every direction through the space 7,i was about to say the room7 where they are enclosed1 as would1 for example1 a swarm of gnats or1 if you prefer a more learned comparison1 like the molecules of gas in the kinematic theory of gases4 ,then their mutual impacts may produce new combinations4 ,what is the rle of the preliminary conscious work8 ,it is evidently to mobilize certain of these atoms1 to unhook them from the wall and put them in swing4 ,we think we have done no good1 because we have moved these elements a thousand different ways in seeking to assemble them1 and have found no satisfactory aggregate4 ,but1 after this shaking up imposed upon them by our will1 these atoms do not return to their primitive rest4 ,they freely continue their dance4 ,now1 our will did not choose them at random2 it pursued a perfectly determined aim4 ,the mobilized atoms are therefore not any atoms whatsoever2 they are those from which we might reasonably expect the desired solution4 ,then the mobilized atoms undergo impacts which make them enter into combinations among themselves or with other atoms at rest which they struck against in their course4 ,again ,i beg pardon1 my comparison is very rough1 but ,i scarcely know how otherwise to make my thought understood4 ,however it may be1 the only combinations that have a chance of forming are those where at least one of the elements is one of those atoms freely chosen by our will4 ,now1 it is evidently among these that is found what ,i called the good combination4 ,perhaps this is a way of lessening the paradoxical in the original hypothesis4 ,another observation4 ,it never happens that the unconscious work gives us the result of a somewhat long calculation all made1 where we have only to apply fixed rules4 ,we might think the wholly automatic subliminal self particularly apt for this sort of work1 which is in a way exclusively mechanical4 ,it seems that thinking in the evening upon the factors of a multiplication we might hope to find the product ready made upon our awakening1 or again that an algebraic calculation1 for example a verification1 would be made unconsciously4 ,nothing of the sort1 as observation proves4 ,all one may hope from these inspirations1 fruits of unconscious work1 is a point of departure for such calculations4 ,as for the calculations themselves1 they must be made in the second period of conscious work1 that which follows the inspiration1 that in which one verifies the results of this inspiration and deduces their consequences4 ,the rules of these calculations are strict and complicated4 ,they require discipline1 attention1 will1 and therefore consciousness4 ,in the subliminal self1 on the contrary1 reigns what ,i should call liberty1 if we might give this name to the simple absence of discipline and to the disorder born of chance4 ,only1 this disorder itself permits unexpected combinations4 ,i shall make a last remark3 when above ,i made certain personal observations1 ,i spoke of a night of excitement when ,i worked in spite of myself4 ,such cases are frequent1 and it is not necessary that the abnormal cerebral activity be caused by a physical excitant as in that ,i mentioned4 ,it seems1 in such cases1 that one is present at his own unconscious work1 made partially perceptible to the over-excited consciousness1 yet without having changed its nature4 ,then we vaguely comprehend what distinguishes the two mechanisms or1 if you wish1 the working methods of the two egos4 ,and the psychologic observations ,i have been able thus to make seem to me to confirm in their general outlines the views ,i have given4 ,surely they have need of it1 for they are and remain in spite of all very hypothetical3 the interest of the questions is so great that ,i do not repent of having submitted them to the reader4 ,,chapter ,,iv ,,chance ,i ,how dare we speak of the laws of chance8 ,is not chance the antithesis of all law8 ,so says ,bertrand at the beginning of his ,calcul des probabiltits4 ,probability is opposed to certitude2 so it is what we do not know and consequently it seems what we could not calculate4 ,here is at least apparently a contradiction1 and about it much has already been written4 ,and first1 what is chance8 ,the ancients distinguished between phenomena seemingly obeying harmonious laws1 established once for all1 and those which they attributed to chance2 these were the ones unpredictable because rebellious to all law4 ,in each domain the precise laws did not decide everything1 they only drew limits between which chance might act4 ,in this conception the word chance had a precise and objective meaning2 what was chance for one was also chance for another and even for the gods4 ,but this conception is not ours to-day4 ,we have become absolute determinists1 and even those who want to reserve the rights of human free will let determinism reign undividedly in the inorganic world at least4 ,every phenomenon1 however minute1 has a cause2 and a mind infinitely powerful1 infinitely well-informed about the laws of nature1 could have foreseen it from the beginning of the centuries4 ,if such a mind existed1 we could not play with it at any game of chance2 we should always lose4 ,in fact for it the word chance would not have any meaning1 or rather there would be no chance4 ,it is because of our weakness and our ignorance that the word has a meaning for us4 ,and1 even without going beyond our feeble humanity1 what is chance for the ignorant is not chance for the scientist4 ,chance is only the measure of our ignorance4 ,fortuitous phenomena are1 by definition1 those whose laws we do not know4 ,but is this definition altogether satisfactory8 ,when the first ,chaldean shepherds followed with their eyes the movements of the stars1 they knew not as yet the laws of astronomy2 would they have dreamed of saying that the stars move at random8 ,if a modern physicist studies a new phenomenon1 and if he discovers its law ,tuesday1 would he have said ,monday that this phenomenon was fortuitous8 ,moreover1 do we not often invoke what ,bertrand calls the laws of chance1 to predict a phenomenon8 ,for example1 in the kinetic theory of gases we obtain the known laws of ,mariotte and of ,gay-,lussac by means of the hypothesis that the velocities of the molecules of gas vary irregularly1 that is to say at random4 ,all physicists will agree that the observable laws would be much less simple if the velocities were ruled by any simple elementary law whatsoever1 if the molecules were1 as we say1 organized1 if they were subject to some discipline4 ,it is due to chance1 that is to say1 to our ignorance1 that we can draw our conclusions2 and then if the word chance is simply synonymous with ignorance what does that mean8 ,must we therefore translate as follows8 ,you ask me to predict for you the phenomena about to happen4 ,if1 unluckily1 ,i knew the laws of these phenomena ,i could make the prediction only by inextricable calculations and would have to renounce attempting to answer you2 but as ,i have the good fortune not to know them1 ,i will answer you at once4 ,and what is most surprising1 my answer will be right4 ,so it must well be that chance is something other than the name we give our ignorance1 that among phenomena whose causes are unknown to us we must distinguish fortuitous phenomena about which the calculus of probabilities will provisionally give information1 from those which are not fortuitous and of which we can say nothing so long as we shall not have determined the laws governing them4 ,for the fortuitous phenomena themselves1 it is clear that the information given us by the calculus of probabilities will not cease to be true upon the day when these phenomena shall be better known4 ,the director of a life insurance company does not know when each of the insured will die1 but he relies upon the calculus of probabilities and on the law of great numbers1 and he is not deceived1 since he distributes dividends to his stockholders4 ,these dividends would not vanish if a very penetrating and very indiscreet physician should1 after the policies were signed1 reveal to the director the life chances of the insured4 ,this doctor would dissipate the ignorance of the director1 but he would have no influence on the dividends1 which evidently are not an outcome of this ignorance4 ,,ii ,to find a better definition of chance we must examine some of the facts which we agree to regard as fortuitous1 and to which the calculus of probabilities seems to apply2 we then shall investigate what are their common characteristics4 ,the first example we select is that of unstable equilibrium2 if a cone rests upon its apex1 we know well that it will fall1 but we do not know toward what side2 it seems to us chance alone will decide4 ,if the cone were perfectly symmetric1 if its axis were perfectly vertical1 if it were acted upon by no force other than gravity1 it would not fall at all4 ,but the least defect in symmetry will make it lean slightly toward one side or the other1 and if it leans1 however little1 it will fall altogether toward that side4 ,even if the symmetry were perfect1 a very slight tremor1 a breath of air could make it incline some seconds of arc2 this will be enough to determine its fall and even the sense of its fall which will be that of the initial inclination4 ,a very slight cause1 which escapes us1 determines a considerable effect which we can not help seeing1 and then we say this effect is due to chance4 ,if we could know exactly the laws of nature and the situation of the universe at the initial instant1 we should be able to predict exactly the situation of this same universe at a subsequent instant4 ,but even when the natural laws should have no further secret for us1 we could know the initial situation only approximately4 ,if that permits us to foresee the subsequent situation with the same degree of approximation1 this is all we require1 we say the phenomenon has been predicted1 that it is ruled by laws4 ,but this is not always the case2 it may happen that slight differences in the initial conditions produce very great differences in the final phenomena2 a slight error in the former would make an enormous error in the latter4 ,prediction becomes impossible and we have the fortuitous phenomenon4 ,our second example will be very analogous to the first and we shall take it from meteorology4 ,why have the meteorologists such difficulty in predicting the weather with any certainty8 ,why do the rains1 the tempests themselves seem to us to come by chance1 so that many persons find it quite natural to pray for rain or shine1 when they would think it ridiculous to pray for an eclipse8 ,we see that great perturbations generally happen in regions where the atmosphere is in unstable equilibrium4 ,the meteorologists are aware that this equilibrium is unstable1 that a cyclone is arising somewhere2 but where they can not tell2 one-tenth of a degree more or less at any point1 and the cyclone bursts here and not there1 and spreads its ravages over countries it would have spared4 ,this we could have foreseen if we had known that tenth of a degree1 but the observations were neither sufficiently close nor sufficiently precise1 and for this reason all seems due to the agency of chance4 ,here again we find the same contrast between a very slight cause1 unappreciable to the observer1 and important effects1 which are sometimes tremendous disasters4 ,let us pass to another example1 the distribution of the minor planets on the zodiac4 ,their initial longitudes may have been any longitudes whatever2 but their mean motions were different and they have revolved for so long a time that we may say they are now distributed at random along the zodiac4 ,very slight initial differences between their distances from the sun1 or1 what comes to the same thing1 between their mean motions1 have ended by giving enormous differences between their present longitudes4 ,an excess of the thousandth of a second in the daily mean motion will give in fact a second in three years1 a degree in ten thousand years1 an entire circumference in three or four million years1 and what is that to the time which has passed since the minor planets detached themselves from the nebula of ,laplace8 ,again therefore we see a slight cause and a great effect2 or better1 slight differences in the cause and great differences in the effect4 ,the game of roulette does not take us as far as might seem from the preceding example4 ,assume a needle to be turned on a pivot over a dial divided into a hundred sectors alternately red and black4 ,if it stops on a red sector ,i win2 if not1 ,i lose4 ,evidently all depends upon the initial impulse ,i give the needle4 ,the needle will make1 suppose1 ten or twenty turns1 but it will stop sooner or not so soon1 according as ,i shall have pushed it more or less strongly4 ,it suffices that the impulse vary only by a thousandth or a two thousandth to make the needle stop over a black sector or over the following red one4 ,these are differences the muscular sense can not distinguish and which elude even the most delicate instruments4 ,so it is impossible for me to foresee what the needle ,i have started will do1 and this is why my heart throbs and ,i hope everything from luck4 ,the difference in the cause is imperceptible1 and the difference in the effect is for me of the highest importance1 since it means my whole stake4 ,,iii ,permit me1 in this connection1 a thought somewhat foreign to my subject4 ,some years ago a philosopher said that the future is determined by the past1 but not the past by the future2 or1 in other words1 from knowledge of the present we could deduce the future1 but not the past2 because1 said he1 a cause can have only one effect1 while the same effect might be produced by several different causes4 ,it is clear no scientist can subscribe to this conclusion4 ,the laws of nature bind the antecedent to the consequent in such a way that the antecedent is as well determined by the consequent as the consequent by the antecedent4 ,but whence came the error of this philosopher8 ,we know that in virtue of ,carnot's principle physical phenomena are irreversible and the world tends toward uniformity4 ,when two bodies of different temperature come in contact1 the warmer gives up heat to the colder2 so we may foresee that the temperature will equalize4 ,but once equal1 if asked about the anterior state1 what can we answer8 ,we might say that one was warm and the other cold1 but not be able to divine which formerly was the warmer4 ,and yet in reality the temperatures will never reach perfect equality4 ,the difference of the temperatures only tends asymptotically toward zero4 ,there comes a moment when our thermometers are powerless to make it known4 ,but if we had thermometers a thousand times1 a hundred thousand times as sensitive1 we should recognize that there still is a slight difference1 and that one of the bodies remains a little warmer than the other1 and so we could say this it is which formerly was much the warmer4 ,so then there are1 contrary to what we found in the former examples1 great differences in cause and slight differences in effect4 ,flammarion once imagined an observer going away from the earth with a velocity greater than that of light2 for him time would have changed sign4 ,history would be turned about1 and ,waterloo would precede ,austerlitz4 ,well1 for this observer1 effects and causes would be inverted2 unstable equilibrium would no longer be the exception4 ,because of the universal irreversibility1 all would seem to him to come out of a sort of chaos in unstable equilibrium4 ,all nature would appear to him delivered over to chance4 ,,iv ,now for other examples where we shall see somewhat different characteristics4 ,take first the kinetic theory of gases4 ,how should we picture a receptacle filled with gas8 ,innumerable molecules1 moving at high speeds1 flash through this receptacle in every direction4 ,at every instant they strike against its walls or each other1 and these collisions happen under the most diverse conditions4 ,what above all impresses us here is not the littleness of the causes1 but their complexity1 and yet the former element is still found here and plays an important rle4 ,if a molecule deviated right or left from its trajectory1 by a very small quantity1 comparable to the radius of action of the gaseous molecules1 it would avoid a collision or sustain it under different conditions1 and that would vary the direction of its velocity after the impact1 perhaps by ninety degrees or by a hundred and eighty degrees4 ,and this is not all2 we have just seen that it is necessary to deflect the molecule before the clash by only an infinitesimal1 to produce its deviation after the collision by a finite quantity4 ,if then the molecule undergoes two successive shocks1 it will suffice to deflect it before the first by an infinitesimal of the second order1 for it to deviate after the first encounter by an infinitesimal of the first order1 and after the second hit1 by a finite quantity4 ,and the molecule will not undergo merely two shocks2 it will undergo a very great number per second4 ,so that if the first shock has multiplied the deviation by a very large number ,a1 after n shocks it will be multiplied by ,an4 ,it will therefore become very great not merely because ,a is large1 that is to say because little causes produce big effects1 but because the exponent n is large1 that is to say because the shocks are very numerous and the causes very complex4 ,take a second example4 ,why do the drops of rain in a shower seem to be distributed at random8 ,this is again because of the complexity of the causes which determine their formation4 ,ions are distributed in the atmosphere4 ,for a long while they have been subjected to air-currents constantly changing1 they have been caught in very small whirlwinds1 so that their final distribution has no longer any relation to their initial distribution4 ,suddenly the temperature falls1 vapor condenses1 and each of these ions becomes the center of a drop of rain4 ,to know what will be the distribution of these drops and how many will fall on each paving-stone1 it would not be sufficient to know the initial situation of the ions1 it would be necessary to compute the effect of a thousand little capricious air-currents4 ,and again it is the same if we put grains of powder in suspension in water4 ,the vase is ploughed by currents whose law we know not1 we only know it is very complicated4 ,at the end of a certain time the grains will be distributed at random1 that is to say uniformly1 in the vase2 and this is due precisely to the complexity of these currents4 ,if they obeyed some simple law1 if for example the vase revolved and the currents circulated around the axis of the vase1 describing circles1 it would no longer be the same1 since each grain would retain its initial altitude and its initial distance from the axis4 ,we should reach the same result in considering the mixing of two liquids or of two fine-grained powders4 ,and to take a grosser example1 this is also what happens when we shuffle playing-cards4 ,at each stroke the cards undergo a permutation 7analogous to that studied in the theory of substitutions74 ,what will happen8 ,the probability of a particular permutation 7for example1 that bringing to the nth place the card occupying the phi7n7th place before the permutation7 depends upon the player's habits4 ,but if this player shuffles the cards long enough1 there will be a great number of successive permutations1 and the resulting final order will no longer be governed by aught but chance2 ,i mean to say that all possible orders will be equally probable4 ,it is to the great number of successive permutations1 that is to say to the complexity of the phenomenon1 that this result is due4 ,a final word about the theory of errors4 ,here it is that the causes are complex and multiple4 ,to how many snares is not the observer exposed1 even with the best instrument6 ,he should apply himself to finding out the largest and avoiding them4 ,these are the ones giving birth to systematic errors4 ,but when he has eliminated those1 admitting that he succeeds1 there remain many small ones which1 their effects accumulating1 may become dangerous4 ,thence come the accidental errors2 and we attribute them to chance because their causes are too complicated and too numerous4 ,here again we have only little causes1 but each of them would produce only a slight effect2 it is by their union and their number that their effects become formidable4 ,v ,we may take still a third point of view1 less important than the first two and upon which ,i shall lay less stress4 ,when we seek to foresee an event and examine its antecedents1 we strive to search into the anterior situation4 ,this could not be done for all parts of the universe and we are content to know what is passing in the neighborhood of the point where the event should occur1 or what would appear to have some relation to it4 ,an examination can not be complete and we must know how to choose4 ,but it may happen that we have passed by circumstances which at first sight seemed completely foreign to the foreseen happening1 to which one would never have dreamed of attributing any influence and which nevertheless1 contrary to all anticipation1 come to play an important rle4 ,a man passes in the street going to his business2 some one knowing the business could have told why he started at such a time and went by such a street4 ,on the roof works a tiler4 ,the contractor employing him could in a certain measure foresee what he would do4 ,but the passer-by scarcely thinks of the tiler1 nor the tiler of him2 they seem to belong to two worlds completely foreign to one another4 ,and yet the tiler drops a tile which kills the man1 and we do not hesitate to say this is chance4 ,our weakness forbids our considering the entire universe and makes us cut it up into slices4 ,we try to do this as little artificially as possible4 ,and yet it happens from time to time that two of these slices react upon each other4 ,the effects of this mutual action then seem to us to be due to chance4 ,is this a third way of conceiving chance8 ,not always2 in fact most often we are carried back to the first or the second4 ,whenever two worlds usually foreign to one another come thus to react upon each other1 the laws of this reaction must be very complex4 ,on the other hand1 a very slight change in the initial conditions of these two worlds would have been sufficient for the reaction not to have happened4 ,how little was needed for the man to pass a second later or the tiler to drop his tile a second sooner4 ,,vi ,all we have said still does not explain why chance obeys laws4 ,does the fact that the causes are slight or complex suffice for our foreseeing1 if not their effects in each case1 at least what their effects will be1 on the average8 ,to answer this question we had better take up again some of the examples already cited4 ,i shall begin with that of the roulette4 ,i have said that the point where the needle will stop depends upon the initial push given it4 ,what is the probability of this push having this or that value8 ,i know nothing about it1 but it is difficult for me not to suppose that this probability is represented by a continuous analytic function4 ,the probability that the push is comprised between alpha and alpha epsilon will then be sensibly equal to the probability of its being comprised between alpha epsilon and alpha #b;epsilon1 provided epsilon be very small4 ,this is a property common to all analytic functions4 ,minute variations of the function are proportional to minute variations of the variable4 ,but we have assumed that an exceedingly slight variation of the push suffices to change the color of the sector over which the needle finally stops4 ,from alpha to alphaepsilon it is red1 from alphaepsilon to alpha#b;epsilon it is black2 the probability of each red sector is therefore the same as of the following black1 and consequently the total probability of red equals the total probability of black4 ,the datum of the question is the analytic function representing the probability of a particular initial push4 ,but the theorem remains true whatever be this datum1 since it depends upon a property common to all analytic functions4 ,from this it follows finally that we no longer need the datum4 ,what we have just said for the case of the roulette applies also to the example of the minor planets4 ,the zodiac may be regarded as an immense roulette on which have been tossed many little balls with different initial impulses varying according to some law4 ,their present distribution is uniform and independent of this law1 for the same reason as in the preceding case4 ,thus we see why phenomena obey the laws of chance when slight differences in the causes suffice to bring on great differences in the effects4 ,the probabilities of these slight differences may then be regarded as proportional to these differences themselves1 just because these differences are minute1 and the infinitesimal increments of a continuous function are proportional to those of the variable4 ,take an entirely different example1 where intervenes especially the complexity of the causes4 ,suppose a player shuffles a pack of cards4 ,at each shuffle he changes the order of the cards1 and he may change them in many ways4 ,to simplify the exposition1 consider only three cards4 ,the cards which before the shuffle occupied respectively the places #abc1 may after the shuffle occupy the places #abc1 #bca1 #cab1 #cba1 #acb1 #bac4 ,each of these six hypotheses is possible and they have respectively for probabilities3 p#a1 p#b1 p#c1 p#d1 p#e1 p#f4 ,the sum of these six numbers equals #a2 but this is all we know of them2 these six probabilities depend naturally upon the habits of the player which we do not know4 ,at the second shuffle and the following1 this will recommence1 and under the same conditions2 ,i mean that p#d for example represents always the probability that the three cards which occupied after the nth shuffle and before the n #ath the places #abc1 occupy the places #cba after the n #ath shuffle4 ,and this remains true whatever be the number n1 since the habits of the player and his way of shuffling remain the same4 ,but if the number of shuffles is very great1 the cards which before the first shuffle occupied the places #abc may1 after the last shuffle1 occupy the places #abc1 #bca1 #cab1 #cba1 #acb1 #bac and the probability of these six hypotheses will be sensibly the same and equal to #a_/f2 and this will be true whatever be the numbers p#a 444 p#f which we do not know4 ,the great number of shuffles1 that is to say the complexity of the causes1 has produced uniformity4 ,this would apply without change if there were more than three cards1 but even with three cards the demonstration would be complicated2 let it suffice to give it for only two cards4 ,then we have only two possibilities #ab1 #ba with the probabilities p#a and p#b #a - p#a4 ,suppose n shuffles and suppose ,i win one franc if the cards are finally in the initial order and lose one if they are finally inverted4 ,then1 my mathematical expectation will be 7p#a - p#b7n4 ,the difference p#a - p#b is certainly less than #a2 so that if n is very great my expectation will be zero2 we need not learn p#a and p#b to be aware that the game is equitable4 ,there would always be an exception if one of the numbers p#a and p#b was equal to #a and the other naught4 ,then it would not apply because our initial hypotheses would be too simple4 ,what we have just seen applies not only to the mixing of cards1 but to all mixings1 to those of powders and of liquids2 and even to those of the molecules of gases in the kinetic theory of gases4 ,to return to this theory1 suppose for a moment a gas whose molecules can not mutually clash1 but may be deviated by hitting the insides of the vase wherein the gas is confined4 ,if the form of the vase is sufficiently complex the distribution of the molecules and that of the velocities will not be long in becoming uniform4 ,but this will not be so if the vase is spherical or if it has the shape of a cuboid4 ,why8 ,because in the first case the distance from the center to any trajectory will remain constant2 in the second case this will be the absolute value of the angle of each trajectory with the faces of the cuboid4 ,so we see what should be understood by conditions too simple2 they are those which conserve something1 which leave an invariant remaining4 ,are the differential equations of the problem too simple for us to apply the laws of chance8 ,this question would seem at first view to lack precise meaning2 now we know what it means4 ,they are too simple if they conserve something1 if they admit a uniform integral4 ,if something in the initial conditions remains unchanged1 it is clear the final situation can no longer be independent of the initial situation4 ,we come finally to the theory of errors4 ,we know not to what are due the accidental errors1 and precisely because we do not know1 we are aware they obey the law of ,gauss4 ,such is the paradox4 ,the explanation is nearly the same as in the preceding cases4 ,we need know only one thing3 that the errors are very numerous1 that they are very slight1 that each may be as well negative as positive4 ,what is the curve of probability of each of them8 ,we do not know2 we only suppose it is symmetric4 ,we prove then that the resultant error will follow ,gauss's law1 and this resulting law is independent of the particular laws which we do not know4 ,here again the simplicity of the result is born of the very complexity of the data4 ,,vii ,but we are not through with paradoxes4 ,i have just recalled the figment of ,flammarion1 that of the man going quicker than light1 for whom time changes sign4 ,i said that for him all phenomena would seem due to chance4 ,that is true from a certain point of view1 and yet all these phenomena at a given moment would not be distributed in conformity with the laws of chance1 since the distribution would be the same as for us1 who1 seeing them unfold harmoniously and without coming out of a primal chaos1 do not regard them as ruled by chance4 ,what does that mean8 ,for ,lumen1 ,flammarion's man1 slight causes seem to produce great effects2 why do not things go on as for us when we think we see grand effects due to little causes8 ,would not the same reasoning be applicable in his case8 ,let us return to the argument4 ,when slight differences in the causes produce vast differences in the effects1 why are these effects distributed according to the laws of chance8 ,suppose a difference of a millimeter in the cause produces a difference of a kilometer in the effect4 ,if ,i win in case the effect corresponds to a kilometer bearing an even number1 my probability of winning will be #a_/b4 ,why8 ,because to make that1 the cause must correspond to a millimeter with an even number4 ,now1 according to all appearance1 the probability of the cause varying between certain limits will be proportional to the distance apart of these limits1 provided this distance be very small4 ,if this hypothesis were not admitted there would no longer be any way of representing the probability by a continuous function4 ,what now will happen when great causes produce small effects8 ,this is the case where we should not attribute the phenomenon to chance and where on the contrary ,lumen would attribute it to chance4 ,to a difference of a kilometer in the cause would correspond a difference of a millimeter in the effect4 ,would the probability of the cause being comprised between two limits n kilometers apart still be proportional to n8 ,we have no reason to suppose so1 since this distance1 n kilometers1 is great4 ,but the probability that the effect lies between two limits n millimeters apart will be precisely the same1 so it will not be proportional to n1 even though this distance1 n millimeters1 be small4 ,there is no way therefore of representing the law of probability of effects by a continuous curve4 ,this curve1 understand1 may remain continuous in the analytic sense of the word2 to infinitesimal variations of the abscissa will correspond infinitesimal variations of the ordinate4 ,but practically it will not be continuous1 since very small variations of the ordinate would not correspond to very small variations of the abscissa4 ,it would become impossible to trace the curve with an ordinary pencil2 that is what ,i mean4 ,so what must we conclude8 ,lumen has no right to say that the probability of the cause 7his cause1 our effect7 should be represented necessarily by a continuous function4 ,but then why have we this right8 ,it is because this state of unstable equilibrium which we have been calling initial is itself only the final outcome of a long previous history4 ,in the course of this history complex causes have worked a great while3 they have contributed to produce the mixture of elements and they have tended to make everything uniform at least within a small region2 they have rounded off the corners1 smoothed down the hills and filled up the valleys4 ,however capricious and irregular may have been the primitive curve given over to them1 they have worked so much toward making it regular that finally they deliver over to us a continuous curve4 ,and this is why we may in all confidence assume its continuity4 ,lumen would not have the same reasons for such a conclusion4 ,for him complex causes would not seem agents of equalization and regularity1 but on the contrary would create only inequality and differentiation4 ,he would see a world more and more varied come forth from a sort of primitive chaos4 ,the changes he could observe would be for him unforeseen and impossible to foresee4 ,they would seem to him due to some caprice or another2 but this caprice would be quite different from our chance1 since it would be opposed to all law1 while our chance still has its laws4 ,all these points call for lengthy explications1 which perhaps would aid in the better comprehension of the irreversibility of the universe4 ,,viii ,we have sought to define chance1 and now it is proper to put a question4 ,has chance thus defined1 in so far as this is possible1 objectivity8 ,it may be questioned4 ,i have spoken of very slight or very complex causes4 ,but what is very little for one may be very big for another1 and what seems very complex to one may seem simple to another4 ,in part ,i have already answered by saying precisely in what cases differential equations become too simple for the laws of chance to remain applicable4 ,but it is fitting to examine the matter a little more closely1 because we may take still other points of view4 ,what means the phrase 'very slight'8 ,to understand it we need only go back to what has already been said4 ,a difference is very slight1 an interval is very small1 when within the limits of this interval the probability remains sensibly constant4 ,and why may this probability be regarded as constant within a small interval8 ,it is because we assume that the law of probability is represented by a continuous curve1 continuous not only in the analytic sense1 but practically continuous1 as already explained4 ,this means that it not only presents no absolute hiatus1 but that it has neither salients nor reentrants too acute or too accentuated4 ,and what gives us the right to make this hypothesis8 ,we have already said it is because1 since the beginning of the ages1 there have always been complex causes ceaselessly acting in the same way and making the world tend toward uniformity without ever being able to turn back4 ,these are the causes which little by little have flattened the salients and filled up the reentrants1 and this is why our probability curves now show only gentle undulations4 ,in milliards of milliards of ages another step will have been made toward uniformity1 and these undulations will be ten times as gentle2 the radius of mean curvature of our curve will have become ten times as great4 ,and then such a length as seems to us to-day not very small1 since on our curve an arc of this length can not be regarded as rectilineal1 should on the contrary at that epoch be called very little1 since the curvature will have become ten times less and an arc of this length may be sensibly identified with a sect4 ,thus the phrase 'very slight' remains relative2 but it is not relative to such or such a man1 it is relative to the actual state of the world4 ,it will change its meaning when the world shall have become more uniform1 when all things shall have blended still more4 ,but then doubtless men can no longer live and must give place to other beings--should ,i say far smaller or far larger8 ,so that our criterion1 remaining true for all men1 retains an objective sense4 ,and on the other hand what means the phrase 'very complex'8 ,i have already given one solution1 but there are others4 ,complex causes we have said produce a blend more and more intimate1 but after how long a time will this blend satisfy us8 ,when will it have accumulated sufficient complexity8 ,when shall we have sufficiently shuffled the cards8 ,if we mix two powders1 one blue1 the other white1 there comes a moment when the tint of the mixture seems to us uniform because of the feebleness of our senses2 it will be uniform for the presbyte1 forced to gaze from afar1 before it will be so for the myope4 ,and when it has become uniform for all eyes1 we still could push back the limit by the use of instruments4 ,there is no chance for any man ever to discern the infinite variety which1 if the kinetic theory is true1 hides under the uniform appearance of a gas4 ,and yet if we accept ,gouy's ideas on the ,brownian movement1 does not the microscope seem on the point of showing us something analogous8 ,this new criterion is therefore relative like the first2 and if it retains an objective character1 it is because all men have approximately the same senses1 the power of their instruments is limited1 and besides they use them only exceptionally4 ,,ix ,it is just the same in the moral sciences and particularly in history4 ,the historian is obliged to make a choice among the events of the epoch he studies2 he recounts only those which seem to him the most important4 ,he therefore contents himself with relating the most momentous events of the sixteenth century1 for example1 as likewise the most remarkable facts of the seventeenth century4 ,if the first suffice to explain the second1 we say these conform to the laws of history4 ,but if a great event of the seventeenth century should have for cause a small fact of the sixteenth century which no history reports1 which all the world has neglected1 then we say this event is due to chance4 ,this word has therefore the same sense as in the physical sciences2 it means that slight causes have produced great effects4 ,the greatest bit of chance is the birth of a great man4 ,it is only by chance that meeting of two germinal cells1 of different sex1 containing precisely1 each on its side1 the mysterious elements whose mutual reaction must produce the genius4 ,one will agree that these elements must be rare and that their meeting is still more rare4 ,how slight a thing it would have required to deflect from its route the carrying spermatozoon4 ,it would have sufficed to deflect it a tenth of a millimeter and ,napoleon would not have been born and the destinies of a continent would have been changed4 ,no example can better make us understand the veritable characteristics of chance4 ,one more word about the paradoxes brought out by the application of the calculus of probabilities to the moral sciences4 ,it has been proven that no ,chamber of ,deputies will ever fail to contain a member of the opposition1 or at least such an event would be so improbable that we might without fear wager the contrary1 and bet a million against a sou4 ,condorcet has striven to calculate how many jurors it would require to make a judicial error practically impossible4 ,if we had used the results of this calculation1 we should certainly have been exposed to the same disappointments as in betting1 on the faith of the calculus1 that the opposition would never be without a representative4 ,the laws of chance do not apply to these questions4 ,if justice be not always meted out to accord with the best reasons1 it uses less than we think the method of ,bridoye4 ,this is perhaps to be regretted1 for then the system of ,condorcet would shield us from judicial errors4 ,what is the meaning of this8 ,we are tempted to attribute facts of this nature to chance because their causes are obscure2 but this is not true chance4 ,the causes are unknown to us1 it is true1 and they are even complex2 but they are not sufficiently so1 since they conserve something4 ,we have seen that this it is which distinguishes causes 'too simple4' ,when men are brought together they no longer decide at random and independently one of another2 they influence one another4 ,multiplex causes come into action4 ,they worry men1 dragging them to right or left1 but one thing there is they can not destroy1 this is their ,panurge flock-of-sheep habits4 ,and this is an invariant4 ,x ,difficulties are indeed involved in the application of the calculus of probabilities to the exact sciences4 ,why are the decimals of a table of logarithms1 why are those of the number pi distributed in accordance with the laws of chance8 ,elsewhere ,i have already studied the question in so far as it concerns logarithms1 and there it is easy4 ,it is clear that a slight difference of argument will give a slight difference of logarithm1 but a great difference in the sixth decimal of the logarithm4 ,always we find again the same criterion4 ,but as for the number pi1 that presents more difficulties1 and ,i have at the moment nothing worth while to say4 ,there would be many other questions to resolve1 had ,i wished to attack them before solving that which ,i more specially set myself4 ,when we reach a simple result1 when we find for example a round number1 we say that such a result can not be due to chance1 and we seek1 for its explanation1 a non-fortuitous cause4 ,and in fact there is only a very slight probability that among #aj1jjj numbers chance will give a round number2 for example1 the number #aj1jjj4 ,this has only one chance in #aj1jjj4 ,but there is only one chance in #aj1jjj for the occurrence of any other one number2 and yet this result will not astonish us1 nor will it be hard for us to attribute it to chance2 and that simply because it will be less striking4 ,is this a simple illusion of ours1 or are there cases where this way of thinking is legitimate8 ,we must hope so1 else were all science impossible4 ,when we wish to check a hypothesis1 what do we do8 ,we can not verify all its consequences1 since they would be infinite in number2 we content ourselves with verifying certain ones and if we succeed we declare the hypothesis confirmed1 because so much success could not be due to chance4 ,and this is always at bottom the same reasoning4 ,i can not completely justify it here1 since it would take too much time2 but ,i may at least say that we find ourselves confronted by two hypotheses1 either a simple cause or that aggregate of complex causes we call chance4 ,we find it natural to suppose that the first should produce a simple result1 and then1 if we find that simple result1 the round number for example1 it seems more likely to us to be attributable to the simple cause which must give it almost certainly1 than to chance which could only give it once in #aj1jjj times4 ,it will not be the same if we find a result which is not simple2 chance1 it is true1 will not give this more than once in #aj1jjj times2 but neither has the simple cause any more chance of producing it4 ,,book ,,ii ,,mathematical ,,reasoning ,,chapter ,i ,,the ,,relativity ,,of ,,space ,i ,it is impossible to represent to oneself empty space2 all our efforts to imagine a pure space1 whence should be excluded the changing images of material objects1 can result only in a representation where vividly colored surfaces1 for example1 are replaced by lines of faint coloration1 and we can not go to the very end in this way without all vanishing and terminating in nothingness4 ,thence comes the irreducible relativity of space4 ,whoever speaks of absolute space uses a meaningless phrase4 ,this is a truth long proclaimed by all who have reflected upon the matter1 but which we are too often led to forget4 ,i am at a determinate point in ,paris1 place du ,panthon for instance1 and ,i say3 ,i shall come back here to-morrow4 ,if ,i be asked3 ,do you mean you will return to the same point of space1 ,i shall be tempted to answer3 yes2 and yet ,i shall be wrong1 since by to-morrow the earth will have journeyed hence1 carrying with it the place du ,panthon1 which will have traveled over more than two million kilometers4 ,and if ,i tried to speak more precisely1 ,i should gain nothing1 since our globe has run over these two million kilometers in its motion with relation to the sun1 while the sun in its turn is displaced with reference to the ,milky ,way1 while the ,milky ,way itself is doubtless in motion without our being able to perceive its velocity4 ,so that we are completely ignorant1 and always shall be1 of how much the place du ,panthon is displaced in a day4 ,in sum1 ,i meant to say3 ,to-morrow ,i shall see again the dome and the pediment of the ,panthon1 and if there were no ,panthon my phrase would be meaningless and space would vanish4 ,this is one of the most commonplace forms of the principle of the relativity of space2 but there is another1 upon which ,delbeuf has particularly insisted4 ,suppose that in the night all the dimensions of the universe become a thousand times greater3 the world will have remained similar to itself1 giving to the word similitude the same meaning as in ,euclid1 ,book ,,vi4 ,only what was a meter long will measure thenceforth a kilometer1 what was a millimeter long will become a meter4 ,the bed whereon ,i lie and my body itself will be enlarged in the same proportion4 ,when ,i awake to-morrow morning1 what sensation shall ,i feel in presence of such an astounding transformation8 ,well1 ,i shall perceive nothing at all4 ,the most precise measurements will be incapable of revealing to me anything of this immense convulsion1 since the measures ,i use will have varied precisely in the same proportion as the objects ,i seek to measure4 ,in reality1 this convulsion exists only for those who reason as if space were absolute4 ,if ,i for a moment have reasoned as they do1 it is the better to bring out that their way of seeing implies contradiction4 ,in fact it would be better to say that1 space being relative1 nothing at all has happened1 which is why we have perceived nothing4 ,has one the right1 therefore1 to say he knows the distance between two points8 ,no1 since this distance could undergo enormous variations without our being able to perceive them1 provided the other distances have varied in the same proportion4 ,we have just seen that when ,i say3 ,i shall be here to-morrow1 this does not mean3 ,to-morrow ,i shall be at the same point of space where ,i am to-day1 but rather3 ,to-morrow ,i shall be at the same distance from the ,panthon as to-day4 ,and we see that this statement is no longer sufficient and that ,i should say3 ,to-morrow and to-day my distance from the ,panthon will be equal to the same number of times the height of my body4 ,but this is not all2 ,i have supposed the dimensions of the world to vary1 but that at least the world remained always similar to itself4 ,we might go much further1 and one of the most astonishing theories of modern physics furnishes us the occasion4 ,according to ,lorentz and ,fitzgerald1 all the bodies borne along in the motion of the earth undergo a deformation4 ,this deformation is1 in reality1 very slight1 since all dimensions parallel to the movement of the earth diminish by a hundred millionth1 while the dimensions perpendicular to this movement are unchanged4 ,but it matters little that it is slight1 that it exists suffices for the conclusion ,i am about to draw4 ,and besides1 ,i have said it was slight1 but in reality ,i know nothing about it2 ,i have myself been victim of the tenacious illusion which makes us believe we conceive an absolute space2 ,i have thought of the motion of the earth in its elliptic orbit around the sun1 and ,i have allowed thirty kilometers as its velocity4 ,but its real velocity 7,i mean1 this time1 not its absolute velocity1 which is meaningless1 but its velocity with relation to the ether71 ,i do not know that1 and have no means of knowing it3 it is perhaps1 #aj1 #ajj times greater1 and then the deformation will be #ajj1 #aj1jjj times more4 ,can we show this deformation8 ,evidently not2 here is a cube with edge one meter2 in consequence of the earth's displacement it is deformed1 one of its edges1 that parallel to the motion1 becomes smaller1 the others do not change4 ,if ,i wish to assure myself of it by aid of a meter measure1 ,i shall measure first one of the edges perpendicular to the motion and shall find that my standard meter fits this edge exactly2 and in fact neither of these two lengths is changed1 since both are perpendicular to the motion4 ,then ,i wish to measure the other edge1 that parallel to the motion2 to do this ,i displace my meter and turn it so as to apply it to the edge4 ,but the meter1 having changed orientation and become parallel to the motion1 has undergone1 in its turn1 the deformation1 so that though the edge be not a meter long1 it will fit exactly1 ,i shall find out nothing4 ,you ask then of what use is the hypothesis of ,lorentz and of ,fitzgerald if no experiment can permit of its verification8 ,it is my exposition that has been incomplete2 ,i have spoken only of measurements that can be made with a meter2 but we can also measure a length by the time it takes light to traverse it1 on condition we suppose the velocity of light constant and independent of direction4 ,lorentz could have accounted for the facts by supposing the velocity of light greater in the direction of the earth's motion than in the perpendicular direction4 ,he preferred to suppose that the velocity is the same in these different directions but that the bodies are smaller in the one than in the other4 ,if the wave surfaces of light had undergone the same deformations as the material bodies we should never have perceived the ,lorentz-,fitzgerald deformation4 ,in either case1 it is not a question of absolute magnitude1 but of the measure of this magnitude by means of some instrument2 this instrument may be a meter1 or the path traversed by light2 it is only the relation of the magnitude to the instrument that we measure2 and if this relation is altered1 we have no way of knowing whether it is the magnitude or the instrument which has changed4 ,but what ,i wish to bring out is1 that in this deformation the world has not remained similar to itself2 squares have become rectangles1 circles ellipses1 spheres ellipsoids4 ,and yet we have no way of knowing whether this deformation be real4 ,evidently one could go much further3 in place of the ,lorentz-,fitzgerald deformation1 whose laws are particularly simple1 we could imagine any deformation whatsoever4 ,bodies could be deformed according to any laws1 as complicated as we might wish1 we never should notice it provided all bodies without exception were deformed according to the same laws4 ,in saying1 all bodies without exception1 ,i include of course our own body and the light rays emanating from different objects4 ,if we look at the world in one of those mirrors of complicated shape which deform objects in a bizarre way1 the mutual relations of the different parts of this world would not be altered2 if1 in fact two real objects touch1 their images likewise seem to touch4 ,of course when we look in such a mirror we see indeed the deformation1 but this is because the real world subsists alongside of its deformed image2 and then even were this real world hidden from us1 something there is could not be hidden1 ourself2 we could not cease to see1 or at least to feel1 our body and our limbs which have not been deformed and which continue to serve us as instruments of measure4 ,but if we imagine our body itself deformed in the same way as if seen in the mirror1 these instruments of measure in their turn will fail us and the deformation will no longer be ascertainable4 ,consider in the same way two worlds images of one another2 to each object ,p of the world ,a corresponds in the world ,b an object ,p'1 its image2 the coordinates of this image ,p' are determinate functions of those of the object ,p2 moreover these functions may be any whatsoever2 ,i only suppose them chosen once for all4 ,between the position of ,p and that of ,p' there is a constant relation2 what this relation is1 matters not2 enough that it be constant4 ,well1 these two worlds will be indistinguishable one from the other4 ,i mean the first will be for its inhabitants what the second is for its4 ,and so it will be as long as the two worlds remain strangers to each other4 ,suppose we lived in world ,a1 we shall have constructed our science and in particular our geometry2 during this time the inhabitants of world ,b will have constructed a science1 and as their world is the image of ours1 their geometry will also be the image of ours or1 better1 it will be the same4 ,but if for us some day a window is opened upon world ,b1 how we shall pity them3 ,poor things1 we shall say1 they think they have made a geometry1 but what they call so is only a grotesque image of ours2 their straights are all twisted1 their circles are humped1 their spheres have capricious inequalities4 ,and we shall never suspect they say the same of us1 and one never will know who is right4 ,we see in how broad a sense should be understood the relativity of space2 space is in reality amorphous and the things which are therein alone give it a form4 ,what then should be thought of that direct intuition we should have of the straight or of distance8 ,so little have we intuition of distance in itself that in the night1 as we have said1 a distance might become a thousand times greater without our being able to perceive it1 if all other distances had undergone the same alteration4 ,and even in a night the world ,b might be substituted for the world ,a without our having any way of knowing it1 and then the straight lines of yesterday would have ceased to be straight and we should never notice4 ,one part of space is not by itself and in the absolute sense of the word equal to another part of space2 because if so it is for us1 it would not be for the dwellers in world ,b2 and these have just as much right to reject our opinion as we to condemn theirs4 ,i have elsewhere shown what are the consequences of these facts from the viewpoint of the idea we should form of non-,euclidean geometry and other analogous geometries2 to that ,i do not care to return2 and to-day ,i shall take a somewhat different point of view4 ,,ii ,if this intuition of distance1 of direction1 of the straight line1 if this direct intuition of space in a word does not exist1 whence comes our belief that we have it8 ,if this is only an illusion1 why is this illusion so tenacious8 ,it is proper to examine into this4 ,we have said there is no direct intuition of size and we can only arrive at the relation of this magnitude to our instruments of measure4 ,we should therefore not have been able to construct space if we had not had an instrument to measure it2 well1 this instrument to which we relate everything1 which we use instinctively1 it is our own body4 ,it is in relation to our body that we place exterior objects1 and the only spatial relations of these objects that we can represent are their relations to our body4 ,it is our body which serves us1 so to speak1 as system of axes of coordinates4 ,for example1 at an instant alpha1 the presence of the object ,a is revealed to me by the sense of sight2 at another instant1 beta1 the presence of another object1 ,b1 is revealed to me by another sense1 that of hearing or of touch1 for instance4 ,i judge that this object ,b occupies the same place as the object ,a4 ,what does that mean8 ,first that does not signify that these two objects occupy1 at two different moments1 the same point of an absolute space1 which even if it existed would escape our cognition1 since1 between the instants alpha and beta1 the solar system has moved and we can not know its displacement4 ,that means these two objects occupy the same relative position with reference to our body4 ,but even this1 what does it mean8 ,the impressions that have come to us from these objects have followed paths absolutely different1 the optic nerve for the object ,a1 the acoustic nerve for the object ,b4 ,they have nothing in common from the qualitative point of view4 ,the representations we are able to make of these two objects are absolutely heterogeneous1 irreducible one to the other4 ,only ,i know that to reach the object ,a ,i have just to extend the right arm in a certain way2 even when ,i abstain from doing it1 ,i represent to myself the muscular sensations and other analogous sensations which would accompany this extension1 and this representation is associated with that of the object ,a4 ,now1 ,i likewise know ,i can reach the object ,b by extending my right arm in the same manner1 an extension accompanied by the same train of muscular sensations4 ,and when ,i say these two objects occupy the same place1 ,i mean nothing more4 ,i also know ,i could have reached the object ,a by another appropriate motion of the left arm and ,i represent to myself the muscular sensations which would have accompanied this movement2 and by this same motion of the left arm1 accompanied by the same sensations1 ,i likewise could have reached the object ,b4 ,and that is very important1 since thus ,i can defend myself against dangers menacing me from the object ,a or the object ,b4 ,with each of the blows we can be hit1 nature has associated one or more parries which permit of our guarding ourselves4 ,the same parry may respond to several strokes2 and so it is1 for instance1 that the same motion of the right arm would have allowed us to guard at the instant alpha against the object ,a and at the instant beta against the object ,b4 ,just so1 the same stroke can be parried in several ways1 and we have said1 for instance1 the object ,a could be reached indifferently either by a certain movement of the right arm or by a certain movement of the left arm4 ,all these parries have nothing in common except warding off the same blow1 and this it is1 and nothing else1 which is meant when we say they are movements terminating at the same point of space4 ,just so1 these objects1 of which we say they occupy the same point of space1 have nothing in common1 except that the same parry guards against them4 ,or1 if you choose1 imagine innumerable telegraph wires1 some centripetal1 others centrifugal4 ,the centripetal wires warn us of accidents happening without2 the centrifugal wires carry the reparation4 ,connections are so established that when a centripetal wire is traversed by a current this acts on a relay and so starts a current in one of the centrifugal wires1 and things are so arranged that several centripetal wires may act on the same centrifugal wire if the same remedy suits several ills1 and that a centripetal wire may agitate different centrifugal wires1 either simultaneously or in lieu one of the other when the same ill may be cured by several remedies4 ,it is this complex system of associations1 it is this table of distribution1 so to speak1 which is all our geometry or1 if you wish1 all in our geometry that is instinctive4 ,what we call our intuition of the straight line or of distance is the consciousness we have of these associations and of their imperious character4 ,and it is easy to understand whence comes this imperious character itself4 ,an association will seem to us by so much the more indestructible as it is more ancient4 ,but these associations are not1 for the most part1 conquests of the individual1 since their trace is seen in the new-born babe3 they are conquests of the race4 ,natural selection had to bring about these conquests by so much the more quickly as they were the more necessary4 ,on this account1 those of which we speak must have been of the earliest in date1 since without them the defense of the organism would have been impossible4 ,from the time when the cellules were no longer merely juxtaposed1 but were called upon to give mutual aid1 it was needful that a mechanism organize analogous to what we have described1 so that this aid miss not its way1 but forestall the peril4 ,when a frog is decapitated1 and a drop of acid is placed on a point of its skin1 it seeks to wipe off the acid with the nearest foot1 and1 if this foot be amputated1 it sweeps it off with the foot of the opposite side4 ,there we have the double parry of which ,i have just spoken1 allowing the combating of an ill by a second remedy1 if the first fails4 ,and it is this multiplicity of parries1 and the resulting coordination1 which is space4 ,we see to what depths of the unconscious we must descend to find the first traces of these spatial associations1 since only the inferior parts of the nervous system are involved4 ,why be astonished then at the resistance we oppose to every attempt made to dissociate what so long has been associated8 ,now1 it is just this resistance that we call the evidence for the geometric truths2 this evidence is nothing but the repugnance we feel toward breaking with very old habits which have always proved good4 ,,iii ,the space so created is only a little space extending no farther than my arm can reach2 the intervention of the memory is necessary to push back its limits4 ,there are points which will remain out of my reach1 whatever effort ,i make to stretch forth my hand2 if ,i were fastened to the ground like a hydra polyp1 for instance1 which can only extend its tentacles1 all these points would be outside of space1 since the sensations we could experience from the action of bodies there situated1 would be associated with the idea of no movement allowing us to reach them1 of no appropriate parry4 ,these sensations would not seem to us to have any spatial character and we should not seek to localize them4 ,but we are not fixed to the ground like the lower animals2 we can1 if the enemy be too far away1 advance toward him first and extend the hand when we are sufficiently near4 ,this is still a parry1 but a parry at long range4 ,on the other hand1 it is a complex parry1 and into the representation we make of it enter the representation of the muscular sensations caused by the movements of the legs1 that of the muscular sensations caused by the final movement of the arm1 that of the sensations of the semicircular canals1 etc4 ,we must1 besides1 represent to ourselves1 not a complex of simultaneous sensations1 but a complex of successive sensations1 following each other in a determinate order1 and this is why ,i have just said the intervention of memory was necessary4 ,notice moreover that1 to reach the same point1 ,i may approach nearer the mark to be attained1 so as to have to stretch my arm less4 ,what more8 ,it is not one1 it is a thousand parries ,i can oppose to the same danger4 ,all these parries are made of sensations which may have nothing in common and yet we regard them as defining the same point of space1 since they may respond to the same danger and are all associated with the notion of this danger4 ,it is the potentiality of warding off the same stroke which makes the unity of these different parries1 as it is the possibility of being parried in the same way which makes the unity of the strokes so different in kind1 which may menace us from the same point of space4 ,it is this double unity which makes the individuality of each point of space1 and1 in the notion of point1 there is nothing else4 ,the space before considered1 which might be called restricted space1 was referred to coordinate axes bound to my body2 these axes were fixed1 since my body did not move and only my members were displaced4 ,what are the axes to which we naturally refer the extended space8 that is to say the new space just defined4 ,we define a point by the sequence of movements to be made to reach it1 starting from a certain initial position of the body4 ,the axes are therefore fixed to this initial position of the body4 ,but the position ,i call initial may be arbitrarily chosen among all the positions my body has successively occupied2 if the memory more or less unconscious of these successive positions is necessary for the genesis of the notion of space1 this memory may go back more or less far into the past4 ,thence results in the definition itself of space a certain indetermination1 and it is precisely this indetermination which constitutes its relativity4 ,there is no absolute space1 there is only space relative to a certain initial position of the body4 ,for a conscious being fixed to the ground like the lower animals1 and consequently knowing only restricted space1 space would still be relative 7since it would have reference to his body71 but this being would not be conscious of this relativity1 because the axes of reference for this restricted space would be unchanging6 ,doubtless the rock to which this being would be fettered would not be motionless1 since it would be carried along in the movement of our planet2 for us consequently these axes would change at each instant2 but for him they would be changeless4 ,we have the faculty of referring our extended space now to the position ,a of our body1 considered as initial1 again to the position ,b1 which it had some moments afterward1 and which we are free to regard in its turn as initial2 we make therefore at each instant unconscious transformations of coordinates4 ,this faculty would be lacking in our imaginary being1 and from not having traveled1 he would think space absolute4 ,at every instant1 his system of axes would be imposed upon him2 this system would have to change greatly in reality1 but for him it would be always the same1 since it would be always the only system4 ,quite otherwise is it with us1 who at each instant have many systems between which we may choose at will1 on condition of going back by memory more or less far into the past4 ,this is not all2 restricted space would not be homogeneous2 the different points of this space could not be regarded as equivalent1 since some could be reached only at the cost of the greatest efforts1 while others could be easily attained4 ,on the contrary1 our extended space seems to us homogeneous1 and we say all its points are equivalent4 ,what does that mean8 ,if we start from a certain place ,a1 we can1 from this position1 make certain movements1 ,m1 characterized by a certain complex of muscular sensations4 ,but1 starting from another position1 ,b1 we make movements ,m' characterized by the same muscular sensations4 ,let a1 then1 be the situation of a certain point of the body1 the end of the index finger of the right hand for example1 in the initial position ,a1 and b the situation of this same index when1 starting from this position ,a1 we have made the motions ,m4 ,afterwards1 let a' be the situation of this index in the position ,b1 and b' its situation when1 starting from the position ,b1 we have made the motions ,m'4 ,well1 ,i am accustomed to say that the points of space a and b are related to each other just as the points a' and b'1 and this simply means that the two series of movements ,m and ,m' are accompanied by the same muscular sensations4 ,and as ,i am conscious that1 in passing from the position ,a to the position ,b1 my body has remained capable of the same movements1 ,i know there is a point of space related to the point a' just as any point b is to the point a1 so that the two points a and a' are equivalent4 ,this is what is called the homogeneity of space4 ,and1 at the same time1 this is why space is relative1 since its properties remain the same whether it be referred to the axes ,a or to the axes ,b4 ,so that the relativity of space and its homogeneity are one sole and same thing4 ,now1 if ,i wish to pass to the great space1 which no longer serves only for me1 but where ,i may lodge the universe1 ,i get there by an act of imagination4 ,i imagine how a giant would feel who could reach the planets in a few steps2 or1 if you choose1 what ,i myself should feel in presence of a miniature world where these planets were replaced by little balls1 while on one of these little balls moved a liliputian ,i should call myself4 ,but this act of imagination would be impossible for me had ,i not previously constructed my restricted space and my extended space for my own use4 ,,iv ,why now have all these spaces three dimensions8 ,go back to the table of distribution of which we have spoken4 ,we have on the one side the list of the different possible dangers2 designate them by ,a#a1 ,a#b1 etc42 and1 on the other side1 the list of the different remedies which ,i shall call in the same way ,b#a1 ,b#b1 etc4 ,we have then connections between the contact studs or push buttons of the first list and those of the second1 so that when1 for instance1 the announcer of danger ,a#c functions1 it will put or may put in action the relay corresponding to the parry ,b#d4 ,as ,i have spoken above of centripetal or centrifugal wires1 ,i fear lest one see in all this1 not a simple comparison1 but a description of the nervous system4 ,such is not my thought1 and that for several reasons3 first ,i should not permit myself to put forth an opinion on the structure of the nervous system which ,i do not know1 while those who have studied it speak only circumspectly2 again because1 despite my incompetence1 ,i well know this scheme would be too simplistic2 and finally because on my list of parries1 some would figure very complex1 which might even1 in the case of extended space1 as we have seen above1 consist of many steps followed by a movement of the arm4 ,it is not a question then of physical connection between two real conductors but of psychologic association between two series of sensations4 ,if ,a#a and ,a#b for instance are both associated with the parry ,b#a1 and if ,a#a is likewise associated with the parry ,b#b1 it will generally happen that ,a#b and ,b#b will also themselves be associated4 ,if this fundamental law were not generally true1 there would exist only an immense confusion and there would be nothing resembling a conception of space or a geometry4 ,how in fact have we defined a point of space4 ,we have done it in two ways3 it is on the one hand the aggregate of announcers ,a in connection with the same parry ,b2 it is on the other hand the aggregate of parries ,b in connection with the same announcer ,a4 ,if our law was not true1 we should say ,a#a and ,a#b correspond to the same point since they are both in connection with ,b#a2 but we should likewise say they do not correspond to the same point1 since ,a#a would be in connection with ,b#b and the same would not be true of ,a#b4 ,this would be a contradiction4 ,but1 from another side1 if the law were rigorously and always true1 space would be very different from what it is4 ,we should have categories strongly contrasted between which would be portioned out on the one hand the announcers ,a1 on the other hand the parries ,b2 these categories would be excessively numerous1 but they would be entirely separated one from another4 ,space would be composed of points very numerous1 but discrete2 it would be discontinuous4 ,there would be no reason for ranging these points in one order rather than another1 nor consequently for attributing to space three dimensions4 ,but it is not so2 permit me to resume for a moment the language of those who already know geometry2 this is quite proper since this is the language best understood by those ,i wish to make understand me4 ,when ,i desire to parry the stroke1 ,i seek to attain the point whence comes this blow1 but it suffices that ,i approach quite near4 ,then the parry ,b#a may answer for ,a#a and for ,a#b1 if the point which corresponds to ,b#a is sufficiently near both to that corresponding to ,a#a and to that corresponding to ,a#b4 ,but it may happen that the point corresponding to another parry ,b#b may be sufficiently near to the point corresponding to ,a#a and not sufficiently near the point corresponding to ,a#b2 so that the parry ,b#b may answer for ,a#a without answering for ,a#b4 ,for one who does not yet know geometry1 this translates itself simply by a derogation of the law stated above4 ,and then things will happen thus3 ,two parries ,b#a and ,b#b will be associated with the same warning ,a#a and with a large number of warnings which we shall range in the same category as ,a#a and which we shall make correspond to the same point of space4 ,but we may find warnings ,a#b which will be associated with ,b#b without being associated with ,b#a1 and which in compensation will be associated with ,b#c1 which ,b#c was not associated with ,a#a1 and so forth1 so that we may write the series ,b#a1 ,a#a1 ,b#b1 ,a#b1 ,b#c1 ,a#c1 ,b#d1 ,a#d1 where each term is associated with the following and the preceding1 but not with the terms several places away4 ,needless to add that each of the terms of these series is not isolated1 but forms part of a very numerous category of other warnings or of other parries which have the same connections as it1 and which may be regarded as belonging to the same point of space4 ,the fundamental law1 though admitting of exceptions1 remains therefore almost always true4 ,only1 in consequence of these exceptions1 these categories1 in place of being entirely separated1 encroach partially one upon another and mutually penetrate in a certain measure1 so that space becomes continuous4 ,on the other hand1 the order in which these categories are to be ranged is no longer arbitrary1 and if we refer to the preceding series1 we see it is necessary to put ,b#b between ,a#a and ,a#b and consequently between ,b#a and ,b#c and that we could not for instance put it between ,b#c and ,b#d4 ,there is therefore an order in which are naturally arranged our categories which correspond to the points of space1 and experience teaches us that this order presents itself under the form of a table of triple entry1 and this is why space has three dimensions4 ,v ,so the characteristic property of space1 that of having three dimensions1 is only a property of our table of distribution1 an internal property of the human intelligence1 so to speak4 ,it would suffice to destroy certain of these connections1 that is to say of the associations of ideas to give a different table of distribution1 and that might be enough for space to acquire a fourth dimension4 ,some persons will be astonished at such a result4 ,the external world1 they will think1 should count for something4 ,if the number of dimensions comes from the way we are made1 there might be thinking beings living in our world1 but who might be made differently from us and who would believe space has more or less than three dimensions4 ,has not ,m4 de ,cyon said that the ,japanese mice1 having only two pair of semicircular canals1 believe that space is two-dimensional8 ,and then this thinking being1 if he is capable of constructing a physics1 would he not make a physics of two or of four dimensions1 and which in a sense would still be the same as ours1 since it would be the description of the same world in another language8 ,it seems in fact that it would be possible to translate our physics into the language of geometry of four dimensions2 to attempt this translation would be to take great pains for little profit1 and ,i shall confine myself to citing the mechanics of ,hertz where we have something analogous4 ,however1 it seems that the translation would always be less simple than the text1 and that it would always have the air of a translation1 that the language of three dimensions seems the better fitted to the description of our world1 although this description can be rigorously made in another idiom4 ,besides1 our table of distribution was not made at random4 ,there is connection between the warning ,a#a and the parry ,b#a1 this is an internal property of our intelligence2 but why this connection8 ,it is because the parry ,b#a affords means effectively to guard against the danger ,a#a2 and this is a fact exterior to us1 this is a property of the exterior world4 ,our table of distribution is therefore only the translation of an aggregate of exterior facts2 if it has three dimensions1 this is because it has adapted itself to a world having certain properties2 and the chief of these properties is that there exist natural solids whose displacements follow sensibly the laws we call laws of motion of rigid solids4 ,if therefore the language of three dimensions is that which permits us most easily to describe our world1 we should not be astonished2 this language is copied from our table of distribution2 and it is in order to be able to live in this world that this table has been established4 ,i have said we could conceive1 living in our world1 thinking beings whose table of distribution would be four-dimensional and who consequently would think in hyperspace4 ,it is not certain however that such beings1 admitting they were born there1 could live there and defend themselves against the thousand dangers by which they would there be assailed4 ,,vi ,a few remarks to end with4 ,there is a striking contrast between the roughness of this primitive geometry1 reducible to what ,i call a table of distribution1 and the infinite precision of the geometers' geometry4 ,and yet this is born of that2 but not of that alone2 it must be made fecund by the faculty we have of constructing mathematical concepts1 such as that of group1 for instance2 it was needful to seek among the pure concepts that which best adapts itself to this rough space whose genesis ,i have sought to explain and which is common to us and the higher animals4 ,the evidence for certain geometric postulates1 we have said1 is only our repugnance to renouncing very old habits4 ,but these postulates are infinitely precise1 while these habits have something about them essentially pliant4 ,when we wish to think1 we need postulates infinitely precise1 since this is the only way to avoid contradiction2 but among all the possible systems of postulates1 there are some we dislike to choose because they are not sufficiently in accord with our habits2 however pliant1 however elastic they may be1 these have a limit of elasticity4 ,we see that if geometry is not an experimental science1 it is a science born apropos of experience2 that we have created the space it studies1 but adapting it to the world wherein we live4 ,we have selected the most convenient space1 but experience has guided our choice2 as this choice has been unconscious1 we think it has been imposed upon us2 some say experience imposes it1 others that we are born with our space ready made2 we see from the preceding considerations1 what in these two opinions is the part of truth1 what of error4 ,in this progressive education whose outcome has been the construction of space1 it is very difficult to determine what is the part of the individual1 what the part of the race4 ,how far could one of us1 transported from birth to an entirely different world1 where were dominant1 for instance1 bodies moving in conformity to the laws of motion of non-,euclidean solids1 renounce the ancestral space to build a space completely new8 ,the part of the race seems indeed preponderant2 yet if to it we owe rough space1 the soft space ,i have spoken of1 the space of the higher animals1 is it not to the unconscious experience of the individual we owe the infinitely precise space of the geometer8 ,this is a question not easy to solve4 ,yet we cite a fact showing that the space our ancestors have bequeathed us still retains a certain plasticity4 ,some hunters learn to shoot fish under water1 though the image of these fish be turned up by refraction4 ,besides they do it instinctively3 they therefore have learned to modify their old instinct of direction2 or1 if you choose1 to substitute for the association ,a#a1 ,b#a1 another association ,a#a1 ,b#b1 because experience showed them the first would not work4 ,,chapter ,,ii ,,mathematical ,,definitions ,,and ,,teaching #a4 ,i should speak here of general definitions in mathematics2 at least that is the title1 but it will be impossible to confine myself to the subject as strictly as the rule of unity of action would require2 ,i shall not be able to treat it without touching upon a few other related questions1 and if thus ,i am forced from time to time to walk on the bordering flower-beds on the right or left1 ,i pray you bear with me4 ,what is a good definition8 ,for the philosopher or the scientist it is a definition which applies to all the objects defined1 and only those2 it is the one satisfying the rules of logic4 ,but in teaching it is not that2 a good definition is one understood by the scholars4 ,how does it happen that so many refuse to understand mathematics8 ,is that not something of a paradox8 ,lo and behold6 a science appealing only to the fundamental principles of logic1 to the principle of contradiction1 for instance1 to that which is the skeleton1 so to speak1 of our intelligence1 to that of which we can not divest ourselves without ceasing to think1 and there are people who find it obscure6 and they are even in the majority6 ,that they are incapable of inventing may pass1 but that they do not understand the demonstrations shown them1 that they remain blind when we show them a light which seems to us flashing pure flame1 this it is which is altogether prodigious4 ,and yet there is no need of a wide experience with examinations to know that these blind men are in no wise exceptional beings4 ,this is a problem not easy to solve1 but which should engage the attention of all those wishing to devote themselves to teaching4 ,what is it1 to understand8 ,has this word the same meaning for all the world8 ,to understand the demonstration of a theorem1 is that to examine successively each of the syllogisms composing it and to ascertain its correctness1 its conformity to the rules of the game8 ,likewise1 to understand a definition1 is this merely to recognize that one already knows the meaning of all the terms employed and to ascertain that it implies no contradiction8 ,for some1 yes2 when they have done this1 they will say3 ,i understand4 ,for the majority1 no4 ,almost all are much more exacting2 they wish to know not merely whether all the syllogisms of a demonstration are correct1 but why they link together in this order rather than another4 ,in so far as to them they seem engendered by caprice and not by an intelligence always conscious of the end to be attained1 they do not believe they understand4 ,doubtless they are not themselves just conscious of what they crave and they could not formulate their desire1 but if they do not get satisfaction1 they vaguely feel that something is lacking4 ,then what happens8 ,in the beginning they still perceive the proofs one puts under their eyes2 but as these are connected only by too slender a thread to those which precede and those which follow1 they pass without leaving any trace in their head2 they are soon forgotten2 a moment bright1 they quickly vanish in night eternal4 ,when they are farther on1 they will no longer see even this ephemeral light1 since the theorems lean one upon another and those they would need are forgotten2 thus it is they become incapable of understanding mathematics4 ,this is not always the fault of their teacher2 often their mind1 which needs to perceive the guiding thread1 is too lazy to seek and find it4 ,but to come to their aid1 we first must know just what hinders them4 ,others will always ask of what use is it2 they will not have understood if they do not find about them1 in practise or in nature1 the justification of such and such a mathematical concept4 ,under each word they wish to put a sensible image2 the definition must evoke this image1 so that at each stage of the demonstration they may see it transform and evolve4 ,only upon this condition do they comprehend and retain4 ,often these deceive themselves2 they do not listen to the reasoning1 they look at the figures2 they think they have understood and they have only seen4 #b4 ,how many different tendencies6 ,must we combat them8 ,must we use them8 ,and if we wish to combat them1 which should be favored8 ,must we show those content with the pure logic that they have seen only one side of the matter8 ,or need we say to those not so cheaply satisfied that what they demand is not necessary8 ,in other words1 should we constrain the young people to change the nature of their minds8 ,such an attempt would be vain2 we do not possess the philosopher's stone which would enable us to transmute one into another the metals confided to us2 all we can do is to work with them1 adapting ourselves to their properties4 ,many children are incapable of becoming mathematicians1 to whom however it is necessary to teach mathematics2 and the mathematicians themselves are not all cast in the same mold4 ,to read their works suffices to distinguish among them two sorts of minds1 the logicians like ,weierstrass for example1 the intuitives like ,riemann4 ,there is the same difference among our students4 ,the one sort prefer to treat their problems 'by analysis' as they say1 the others 'by geometry4' ,it is useless to seek to change anything of that1 and besides would it be desirable8 ,it is well that there are logicians and that there are intuitives2 who would dare say whether he preferred that ,weierstrass had never written or that there never had been a ,riemann8 ,we must therefore resign ourselves to the diversity of minds1 or better we must rejoice in it4 #c4 ,since the word understand has many meanings1 the definitions which will be best understood by some will not be best suited to others4 ,we have those which seek to produce an image1 and those where we confine ourselves to combining empty forms1 perfectly intelligible1 but purely intelligible1 which abstraction has deprived of all matter4 ,i know not whether it be necessary to cite examples4 ,let us cite them1 anyhow1 and first the definition of fractions will furnish us an extreme case4 ,in the primary schools1 to define a fraction1 one cuts up an apple or a pie2 it is cut up mentally of course and not in reality1 because ,i do not suppose the budget of the primary instruction allows of such prodigality4 ,at the ,normal ,school1 on the other hand1 or at the college1 it is said3 a fraction is the combination of two whole numbers separated by a horizontal bar2 we define by conventions the operations to which these symbols may be submitted2 it is proved that the rules of these operations are the same as in calculating with whole numbers1 and we ascertain finally that multiplying the fraction1 according to these rules1 by the denominator gives the numerator4 ,this is all very well because we are addressing young people long familiarized with the notion of fractions through having cut up apples or other objects1 and whose mind1 matured by a hard mathematical education1 has come little by little to desire a purely logical definition4 ,but the dbutant to whom one should try to give it1 how dumfounded6 ,such also are the definitions found in a book justly admired and greatly honored1 the ,foundations of ,geometry by ,hilbert4 ,see in fact how he begins3 ,we think three systems of ,,things which we shall call points1 straights and planes4 ,what are these 'things'8 ,we know not1 nor need we know2 it would even be a pity to seek to know2 all we have the right to know of them is what the assumptions tell us2 this for example3 ,two distinct points always determine a straight1 which is followed by this remark3 in place of determine1 we may say the two points are on the straight1 or the straight goes through these two points or joins the two points4 ,thus 'to be on a straight' is simply defined as synonymous with 'determine a straight4' ,behold a book of which ,i think much good1 but which ,i should not recommend to a school boy4 ,yet ,i could do so without fear1 he would not read much of it4 ,i have taken extreme examples and no teacher would dream of going that far4 ,but even stopping short of such models1 does he not already expose himself to the same danger8 ,suppose we are in a class2 the professor dictates3 the circle is the locus of points of the plane equidistant from an interior point called the center4 ,the good scholar writes this phrase in his note-book2 the bad scholar draws faces2 but neither understands2 then the professor takes the chalk and draws a circle on the board4 ,ah6 think the scholars1 why did he not say at once3 a circle is a ring1 we should have understood4 ,doubtless the professor is right4 ,the scholars' definition would have been of no avail1 since it could serve for no demonstration1 since besides it would not give them the salutary habit of analyzing their conceptions4 ,but one should show them that they do not comprehend what they think they know1 lead them to be conscious of the roughness of their primitive conception1 and of themselves to wish it purified and made precise4 #d4 ,i shall return to these examples2 ,i only wished to show you the two opposed conceptions2 they are in violent contrast4 ,this contrast the history of science explains4 ,if we read a book written fifty years ago1 most of the reasoning we find there seems lacking in rigor4 ,then it was assumed a continuous function can change sign only by vanishing2 to-day we prove it4 ,it was assumed the ordinary rules of calculation are applicable to incommensurable numbers2 to-day we prove it4 ,many other things were assumed which sometimes were false4 ,we trusted to intuition2 but intuition can not give rigor1 nor even certainty2 we see this more and more4 ,it tells us for instance that every curve has a tangent1 that is to say that every continuous function has a derivative1 and that is false4 ,and as we sought certainty1 we had to make less and less the part of intuition4 ,what has made necessary this evolution8 ,we have not been slow to perceive that rigor could not be established in the reasonings1 if it were not first put into the definitions4 ,the objects occupying mathematicians were long ill defined2 we thought we knew them because we represented them with the senses or the imagination2 but we had of them only a rough image and not a precise concept upon which reasoning could take hold4 ,it is there that the logicians would have done well to direct their efforts4 ,so for the incommensurable number1 the vague idea of continuity1 which we owe to intuition1 has resolved itself into a complicated system of inequalities bearing on whole numbers4 ,thus have finally vanished all those difficulties which frightened our fathers when they reflected upon the foundations of the infinitesimal calculus4 ,to-day only whole numbers are left in analysis1 or systems finite or infinite of whole numbers1 bound by a plexus of equalities and inequalities4 ,mathematics we say is arithmetized4 #e4 ,but do you think mathematics has attained absolute rigor without making any sacrifice8 ,not at all2 what it has gained in rigor it has lost in objectivity4 ,it is by separating itself from reality that it has acquired this perfect purity4 ,we may freely run over its whole domain1 formerly bristling with obstacles1 but these obstacles have not disappeared4 ,they have only been moved to the frontier1 and it would be necessary to vanquish them anew if we wished to break over this frontier to enter the realm of the practical4 ,we had a vague notion1 formed of incongruous elements1 some a priori1 others coming from experiences more or less digested2 we thought we knew1 by intuition1 its principal properties4 ,to-day we reject the empiric elements1 retaining only the a priori2 one of the properties serves as definition and all the others are deduced from it by rigorous reasoning4 ,this is all very well1 but it remains to be proved that this property1 which has become a definition1 pertains to the real objects which experience had made known to us and whence we drew our vague intuitive notion4 ,to prove that1 it would be necessary to appeal to experience1 or to make an effort of intuition1 and if we could not prove it1 our theorems would be perfectly rigorous1 but perfectly useless4 ,logic sometimes makes monsters4 ,since half a century we have seen arise a crowd of bizarre functions which seem to try to resemble as little as possible the honest functions which serve some purpose4 ,no longer continuity1 or perhaps continuity1 but no derivatives1 etc4 ,nay more1 from the logical point of view1 it is these strange functions which are the most general1 those one meets without seeking no longer appear except as particular case4 ,there remains for them only a very small corner4 ,heretofore when a new function was invented1 it was for some practical end2 to-day they are invented expressly to put at fault the reasonings of our fathers1 and one never will get from them anything more than that4 ,if logic were the sole guide of the teacher1 it would be necessary to begin with the most general functions1 that is to say with the most bizarre4 ,it is the beginner that would have to be set grappling with this teratologic museum4 ,if you do not do it1 the logicians might say1 you will achieve rigor only by stages4 #f4 ,yes1 perhaps1 but we can not make so cheap of reality1 and ,i mean not only the reality of the sensible world1 which however has its worth1 since it is to combat against it that nine tenths of your students ask of you weapons4 ,there is a reality more subtile1 which makes the very life of the mathematical beings1 and which is quite other than logic4 ,our body is formed of cells1 and the cells of atoms2 are these cells and these atoms then all the reality of the human body8 ,the way these cells are arranged1 whence results the unity of the individual1 is it not also a reality and much more interesting8 ,a naturalist who never had studied the elephant except in the microscope1 would he think he knew the animal adequately8 ,it is the same in mathematics4 ,when the logician shall have broken up each demonstration into a multitude of elementary operations1 all correct1 he still will not possess the whole reality2 this ,i know not what which makes the unity of the demonstration will completely escape him4 ,in the edifices built up by our masters1 of what use to admire the work of the mason if we can not comprehend the plan of the architect8 ,now pure logic can not give us this appreciation of the total effect2 this we must ask of intuition4 ,take for instance the idea of continuous function4 ,this is at first only a sensible image1 a mark traced by the chalk on the blackboard4 ,little by little it is refined2 we use it to construct a complicated system of inequalities1 which reproduces all the features of the primitive image2 when all is done1 we have removed the centering1 as after the construction of an arch2 this rough representation1 support thenceforth useless1 has disappeared and there remains only the edifice itself1 irreproachable in the eyes of the logician4 ,and yet1 if the professor did not recall the primitive image1 if he did not restore momentarily the centering1 how could the student divine by what caprice all these inequalities have been scaffolded in this fashion one upon another8 ,the definition would be logically correct1 but it would not show him the veritable reality4 #g4 ,so back we must return2 doubtless it is hard for a master to teach what does not entirely satisfy him2 but the satisfaction of the master is not the unique object of teaching2 we should first give attention to what the mind of the pupil is and to what we wish it to become4 ,zoologists maintain that the embryonic development of an animal recapitulates in brief the whole history of its ancestors throughout geologic time4 ,it seems it is the same in the development of minds4 ,the teacher should make the child go over the path his fathers trod2 more rapidly1 but without skipping stations4 ,for this reason1 the history of science should be our first guide4 ,our fathers thought they knew what a fraction was1 or continuity1 or the area of a curved surface2 we have found they did not know it4 ,just so our scholars think they know it when they begin the serious study of mathematics4 ,if without warning ,i tell them3 ,no1 you do not know it2 what you think you understand1 you do not understand2 ,i must prove to you what seems to you evident1 and if in the demonstration ,i support myself upon premises which to them seem less evident than the conclusion1 what shall the unfortunates think8 ,they will think that the science of mathematics is only an arbitrary mass of useless subtilities2 either they will be disgusted with it1 or they will play it as a game and will reach a state of mind like that of the ,greek sophists4 ,later1 on the contrary1 when the mind of the scholar1 familiarized with mathematical reasoning1 has been matured by this long frequentation1 the doubts will arise of themselves and then your demonstration will be welcome4 ,it will awaken new doubts1 and the questions will arise successively to the child1 as they arose successively to our fathers1 until perfect rigor alone can satisfy him4 ,to doubt everything does not suffice1 one must know why he doubts4 #h4 ,the principal aim of mathematical teaching is to develop certain faculties of the mind1 and among them intuition is not the least precious4 ,it is through it that the mathematical world remains in contact with the real world1 and if pure mathematics could do without it1 it would always be necessary to have recourse to it to fill up the chasm which separates the symbol from reality4 ,the practician will always have need of it1 and for one pure geometer there should be a hundred practicians4 ,the engineer should receive a complete mathematical education1 but for what should it serve him8 ,to see the different aspects of things and see them quickly2 he has no time to hunt mice4 ,it is necessary that1 in the complex physical objects presented to him1 he should promptly recognize the point where the mathematical tools we have put in his hands can take hold4 ,how could he do it if we should leave between instruments and objects the deep chasm hollowed out by the logicians8 #i4 ,besides the engineers1 other scholars1 less numerous1 are in their turn to become teachers2 they therefore must go to the very bottom2 a knowledge deep and rigorous of the first principles is for them before all indispensable4 ,but this is no reason not to cultivate in them intuition2 for they would get a false idea of the science if they never looked at it except from a single side1 and besides they could not develop in their students a quality they did not themselves possess4 ,for the pure geometer himself1 this faculty is necessary2 it is by logic one demonstrates1 by intuition one invents4 ,to know how to criticize is good1 to know how to create is better4 ,you know how to recognize if a combination is correct2 what a predicament if you have not the art of choosing among all the possible combinations4 ,logic tells us that on such and such a way we are sure not to meet any obstacle2 it does not say which way leads to the end4 ,for that it is necessary to see the end from afar1 and the faculty which teaches us to see is intuition4 ,without it the geometer would be like a writer who should be versed in grammar but had no ideas4 ,now how could this faculty develop if1 as soon as it showed itself1 we chase it away and proscribe it1 if we learn to set it at naught before knowing the good of it4 ,and here permit a parenthesis to insist upon the importance of written exercises4 ,written compositions are perhaps not sufficiently emphasized in certain examinations1 at the polytechnic school1 for instance4 ,i am told they would close the door against very good scholars who have mastered the course1 thoroughly understanding it1 and who nevertheless are incapable of making the slightest application4 ,i have just said the word understand has several meanings3 such students only understand in the first way1 and we have seen that suffices neither to make an engineer nor a geometer4 ,well1 since choice must be made1 ,i prefer those who understand completely4 #aj4 ,but is the art of sound reasoning not also a precious thing1 which the professor of mathematics ought before all to cultivate8 ,i take good care not to forget that4 ,it should occupy our attention and from the very beginning4 ,i should be distressed to see geometry degenerate into ,i know not what tachymetry of low grade and ,i by no means subscribe to the extreme doctrines of certain ,german ,oberlehrer4 ,but there are occasions enough to exercise the scholars in correct reasoning in the parts of mathematics where the inconveniences ,i have pointed out do not present themselves4 ,there are long chains of theorems where absolute logic has reigned from the very first and1 so to speak1 quite naturally1 where the first geometers have given us models we should constantly imitate and admire4 ,it is in the exposition of first principles that it is necessary to avoid too much subtility2 there it would be most discouraging and moreover useless4 ,we can not prove everything and we can not define everything2 and it will always be necessary to borrow from intuition2 what does it matter whether it be done a little sooner or a little later1 provided that in using correctly premises it has furnished us1 we learn to reason soundly4 #aa4 ,is it possible to fulfill so many opposing conditions8 ,is this possible in particular when it is a question of giving a definition8 ,how find a concise statement satisfying at once the uncompromising rules of logic1 our desire to grasp the place of the new notion in the totality of the science1 our need of thinking with images8 ,usually it will not be found1 and this is why it is not enough to state a definition2 it must be prepared for and justified4 ,what does that mean8 ,you know it has often been said3 every definition implies an assumption1 since it affirms the existence of the object defined4 ,the definition then will not be justified1 from the purely logical point of view1 until one shall have proved that it involves no contradiction1 neither in the terms1 nor with the verities previously admitted4 ,but this is not enough2 the definition is stated to us as a convention2 but most minds will revolt if we wish to impose it upon them as an arbitrary convention4 ,they will be satisfied only when you have answered numerous questions4 ,usually mathematical definitions1 as ,m4 ,liard has shown1 are veritable constructions built up wholly of more simple notions4 ,but why assemble these elements in this way when a thousand other combinations were possible8 ,is it by caprice8 ,if not1 why had this combination more right to exist than all the others8 ,to what need does it respond8 ,how was it foreseen that it would play an important rle in the development of the science1 that it would abridge our reasonings and our calculations8 ,is there in nature some familiar object which is so to speak the rough and vague image of it8 ,this is not all2 if you answer all these questions in a satisfactory manner1 we shall see indeed that the new-born had the right to be baptized2 but neither is the choice of a name arbitrary2 it is needful to explain by what analogies one has been guided and that if analogous names have been given to different things1 these things at least differ only in material and are allied in form2 that their properties are analogous and so to say parallel4 ,at this cost we may satisfy all inclinations4 ,if the statement is correct enough to please the logician1 the justification will satisfy the intuitive4 ,but there is still a better procedure2 wherever possible1 the justification should precede the statement and prepare for it2 one should be led on to the general statement by the study of some particular examples4 ,still another thing3 each of the parts of the statement of a definition has as aim to distinguish the thing to be defined from a class of other neighboring objects4 ,the definition will be understood only when you have shown1 not merely the object defined1 but the neighboring objects from which it is proper to distinguish it1 when you have given a grasp of the difference and when you have added explicitly3 this is why in stating the definition ,i have said this or that4 ,but it is time to leave generalities and examine how the somewhat abstract principles ,i have expounded may be applied in arithmetic1 geometry1 analysis and mechanics4 ,,arithmetic #ab4 ,the whole number is not to be defined2 in return1 one ordinarily defines the operations upon whole numbers2 ,i believe the scholars learn these definitions by heart and attach no meaning to them4 ,for that there are two reasons3 first they are made to learn them too soon1 when their mind as yet feels no need of them2 then these definitions are not satisfactory from the logical point of view4 ,a good definition for addition is not to be found just simply because we must stop and can not define everything4 ,it is not defining addition to say it consists in adding4 ,all that can be done is to start from a certain number of concrete examples and say3 the operation we have performed is called addition4 ,for subtraction it is quite otherwise2 it may be logically defined as the operation inverse to addition2 but should we begin in that way8 ,here also start with examples1 show on these examples the reciprocity of the two operations2 thus the definition will be prepared for and justified4 ,just so again for multiplication2 take a particular problem2 show that it may be solved by adding several equal numbers2 then show that we reach the result more quickly by a multiplication1 an operation the scholars already know how to do by routine and out of that the logical definition will issue naturally4 ,division is defined as the operation inverse to multiplication2 but begin by an example taken from the familiar notion of partition and show on this example that multiplication reproduces the dividend4 ,there still remain the operations on fractions4 ,the only difficulty is for multiplication4 ,it is best to expound first the theory of proportion2 from it alone can come a logical definition2 but to make acceptable the definitions met at the beginning of this theory1 it is necessary to prepare for them by numerous examples taken from classic problems of the rule of three1 taking pains to introduce fractional data4 ,neither should we fear to familiarize the scholars with the notion of proportion by geometric images1 either by appealing to what they remember if they have already studied geometry1 or in having recourse to direct intuition1 if they have not studied it1 which besides will prepare them to study it4 ,finally ,i shall add that after defining multiplication of fractions1 it is needful to justify this definition by showing that it is commutative1 associative and distributive1 and calling to the attention of the auditors that this is established to justify the definition4 ,one sees what a rle geometric images play in all this2 and this rle is justified by the philosophy and the history of the science4 ,if arithmetic had remained free from all admixture of geometry1 it would have known only the whole number2 it is to adapt itself to the needs of geometry that it invented anything else4 ,,geometry ,in geometry we meet forthwith the notion of the straight line4 ,can the straight line be defined8 ,the well-known definition1 the shortest path from one point to another1 scarcely satisfies me4 ,i should start simply with the ruler and show at first to the scholar how one may verify a ruler by turning2 this verification is the true definition of the straight line2 the straight line is an axis of rotation4 ,next he should be shown how to verify the ruler by sliding and he would have one of the most important properties of the straight line4 ,as to this other property of being the shortest path from one point to another1 it is a theorem which can be demonstrated apodictically1 but the demonstration is too delicate to find a place in secondary teaching4 ,it will be worth more to show that a ruler previously verified fits on a stretched thread4 ,in presence of difficulties like these one need not dread to multiply assumptions1 justifying them by rough experiments4 ,it is needful to grant these assumptions1 and if one admits a few more of them than is strictly necessary1 the evil is not very great2 the essential thing is to learn to reason soundly on the assumptions admitted4 ,uncle ,sarcey1 who loved to repeat1 often said that at the theater the spectator accepts willingly all the postulates imposed upon him at the beginning1 but the curtain once raised1 he becomes uncompromising on the logic4 ,well1 it is just the same in mathematics4 ,for the circle1 we may start with the compasses2 the scholars will recognize at the first glance the curve traced2 then make them observe that the distance of the two points of the instrument remains constant1 that one of these points is fixed and the other movable1 and so we shall be led naturally to the logical definition4 ,the definition of the plane implies an axiom and this need not be hidden4 ,take a drawing board and show that a moving ruler may be kept constantly in complete contact with this plane and yet retain three degrees of freedom4 ,compare with the cylinder and the cone1 surfaces on which an applied straight retains only two degrees of freedom2 next take three drawing boards2 show first that they will glide while remaining applied to one another and this with three degrees of freedom2 and finally to distinguish the plane from the sphere1 show that two of these boards which fit a third will fit each other4 ,perhaps you are surprised at this incessant employment of moving things2 this is not a rough artifice2 it is much more philosophic than one would at first think4 ,what is geometry for the philosopher8 ,it is the study of a group4 ,and what group8 ,that of the motions of solid bodies4 ,how define this group then without moving some solids8 ,should we retain the classic definition of parallels and say parallels are two coplanar straights which do not meet1 however far they be prolonged8 ,no1 since this definition is negative1 since it is unverifiable by experiment1 and consequently can not be regarded as an immediate datum of intuition4 ,no1 above all because it is wholly strange to the notion of group1 to the consideration of the motion of solid bodies which is1 as ,i have said1 the true source of geometry4 ,would it not be better to define first the rectilinear translation of an invariable figure1 as a motion wherein all the points of this figure have rectilinear trajectories2 to show that such a translation is possible by making a square glide on a ruler8 ,from this experimental ascertainment1 set up as an assumption1 it would be easy to derive the notion of parallel and ,euclid's postulate itself4 ,,mechanics ,i need not return to the definition of velocity1 or acceleration1 or other kinematic notions2 they may be advantageously connected with that of the derivative4 ,i shall insist1 on the other hand1 upon the dynamic notions of force and mass4 ,i am struck by one thing3 how very far the young people who have received a high-school education are from applying to the real world the mechanical laws they have been taught4 ,it is not only that they are incapable of it2 they do not even think of it4 ,for them the world of science and the world of reality are separated by an impervious partition wall4 ,if we try to analyze the state of mind of our scholars1 this will astonish us less4 ,what is for them the real definition of force8 ,not that which they recite1 but that which1 crouching in a nook of their mind1 from there directs it wholly4 ,here is the definition3 forces are arrows with which one makes parallelograms4 ,these arrows are imaginary things which have nothing to do with anything existing in nature4 ,this would not happen if they had been shown forces in reality before representing them by arrows4 ,how shall we define force8 ,i think ,i have elsewhere sufficiently shown there is no good logical definition4 ,there is the anthropomorphic definition1 the sensation of muscular effort2 this is really too rough and nothing useful can be drawn from it4 ,here is how we should go3 first1 to make known the genus force1 we must show one after the other all the species of this genus2 they are very numerous and very different2 there is the pressure of fluids on the insides of the vases wherein they are contained2 the tension of threads2 the elasticity of a spring2 the gravity working on all the molecules of a body2 friction2 the normal mutual action and reaction of two solids in contact4 ,this is only a qualitative definition2 it is necessary to learn to measure force4 ,for that begin by showing that one force may be replaced by another without destroying equilibrium2 we may find the first example of this substitution in the balance and ,borda's double weighing4 ,then show that a weight may be replaced1 not only by another weight1 but by force of a different nature2 for instance1 ,prony's brake permits replacing weight by friction4 ,from all this arises the notion of the equivalence of two forces4 ,the direction of a force must be defined4 ,if a force ,f is equivalent to another force ,f' applied to the body considered by means of a stretched string1 so that ,f may be replaced by ,f' without affecting the equilibrium1 then the point of attachment of the string will be by definition the point of application of the force ,f'1 and that of the equivalent force ,f2 the direction of the string will be the direction of the force ,f' and that of the equivalent force ,f4 ,from that1 pass to the comparison of the magnitude of forces4 ,if a force can replace two others with the same direction1 it equals their sum2 show for example that a weight of #bj grams may replace two #aj-;gram weights4 ,is this enough8 ,not yet4 ,we now know how to compare the intensity of two forces which have the same direction and same point of application2 we must learn to do it when the directions are different4 ,for that1 imagine a string stretched by a weight and passing over a pulley2 we shall say that the tensor of the two legs of the string is the same and equal to the tension weight4 ,this definition of ours enables us to compare the tensions of the two pieces of our string1 and1 using the preceding definitions1 to compare any two forces having the same direction as these two pieces4 ,it should be justified by showing that the tension of the last piece of the string remains the same for the same tensor weight1 whatever be the number and the disposition of the reflecting pulleys4 ,it has still to be completed by showing this is only true if the pulleys are frictionless4 ,once master of these definitions1 it is to be shown that the point of application1 the direction and the intensity suffice to determine a force2 that two forces for which these three elements are the same are always equivalent and may always be replaced by one another1 whether in equilibrium or in movement1 and this whatever be the other forces acting4 ,it must be shown that two concurrent forces may always be replaced by a unique resultant2 and that this resultant remains the same1 whether the body be at rest or in motion and whatever be the other forces applied to it4 ,finally it must be shown that forces thus defined satisfy the principle of the equality of action and reaction4 ,experiment it is1 and experiment alone1 which can teach us all that4 ,it will suffice to cite certain common experiments1 which the scholars make daily without suspecting it1 and to perform before them a few experiments1 simple and well chosen4 ,it is after having passed through all these meanders that one may represent forces by arrows1 and ,i should even wish that in the development of the reasonings return were made from time to time from the symbol to the reality4 ,for instance it would not be difficult to illustrate the parallelogram of forces by aid of an apparatus formed of three strings1 passing over pulleys1 stretched by weights and in equilibrium while pulling on the same point4 ,knowing force1 it is easy to define mass2 this time the definition should be borrowed from dynamics2 there is no way of doing otherwise1 since the end to be attained is to give understanding of the distinction between mass and weight4 ,here again1 the definition should be led up to by experiments2 there is in fact a machine which seems made expressly to show what mass is1 ,atwood's machine2 recall also the laws of the fall of bodies1 that the acceleration of gravity is the same for heavy as for light bodies1 and that it varies with the latitude1 etc4 ,now1 if you tell me that all the methods ,i extol have long been applied in the schools1 ,i shall rejoice over it more than be surprised at it4 ,i know that on the whole our mathematical teaching is good4 ,i do not wish it overturned2 that would even distress me4 ,i only desire betterments slowly progressive4 ,this teaching should not be subjected to brusque oscillations under the capricious blast of ephemeral fads4 ,in such tempests its high educative value would soon founder4 ,a good and sound logic should continue to be its basis4 ,the definition by example is always necessary1 but it should prepare the way for the logical definition1 it should not replace it2 it should at least make this wished for1 in the cases where the true logical definition can be advantageously given only in advanced teaching4 ,understand that what ,i have here said does not imply giving up what ,i have written elsewhere4 ,i have often had occasion to criticize certain definitions ,i extol to-day4 ,these criticisms hold good completely4 ,these definitions can only be provisory4 ,but it is by way of them that we must pass4 ,,chapter ,,iii ,,mathematics ,,and ,,logic ,,introduction ,can mathematics be reduced to logic without having to appeal to principles peculiar to mathematics8 ,there is a whole school1 abounding in ardor and full of faith1 striving to prove it4 ,they have their own special language1 which is without words1 using only signs4 ,this language is understood only by the initiates1 so that commoners are disposed to bow to the trenchant affirmations of the adepts4 ,it is perhaps not unprofitable to examine these affirmations somewhat closely1 to see if they justify the peremptory tone with which they are presented4 ,but to make clear the nature of the question it is necessary to enter upon certain historical details and in particular to recall the character of the works of ,cantor4 ,since long ago the notion of infinity had been introduced into mathematics2 but this infinite was what philosophers call a becoming4 ,the mathematical infinite was only a quantity capable of increasing beyond all limit3 it was a variable quantity of which it could not be said that it had passed all limits1 but only that it could pass them4 ,cantor has undertaken to introduce into mathematics an actual infinite1 that is to say a quantity which not only is capable of passing all limits1 but which is regarded as having already passed them4 ,he has set himself questions like these3 ,are there more points in space than whole numbers8 ,are there more points in space than points in a plane8 etc4 ,and then the number of whole numbers1 that of the points of space1 etc41 constitutes what he calls a transfinite cardinal number1 that is to say a cardinal number greater than all the ordinary cardinal numbers4 ,and he has occupied himself in comparing these transfinite cardinal numbers4 ,in arranging in a proper order the elements of an aggregate containing an infinity of them1 he has also imagined what he calls transfinite ordinal numbers upon which ,i shall not dwell4 ,many mathematicians followed his lead and set a series of questions of the sort4 ,they so familiarized themselves with transfinite numbers that they have come to make the theory of finite numbers depend upon that of ,cantor's cardinal numbers4 ,in their eyes1 to teach arithmetic in a way truly logical1 one should begin by establishing the general properties of transfinite cardinal numbers1 then distinguish among them a very small class1 that of the ordinary whole numbers4 ,thanks to this dtour1 one might succeed in proving all the propositions relative to this little class 7that is to say all our arithmetic and our algebra7 without using any principle foreign to logic4 ,this method is evidently contrary to all sane psychology2 it is certainly not in this way that the human mind proceeded in constructing mathematics2 so its authors do not dream1 ,i think1 of introducing it into secondary teaching4 ,but is it at least logic1 or1 better1 is it correct8 ,it may be doubted4 ,the geometers who have employed it are however very numerous4 ,they have accumulated formulas and they have thought to free themselves from what was not pure logic by writing memoirs where the formulas no longer alternate with explanatory discourse as in the books of ordinary mathematics1 but where this discourse has completely disappeared4 ,unfortunately they have reached contradictory results1 what are called the cantorian antinomies1 to which we shall have occasion to return4 ,these contradictions have not discouraged them and they have tried to modify their rules so as to make those disappear which had already shown themselves1 without being sure1 for all that1 that new ones would not manifest themselves4 ,it is time to administer justice on these exaggerations4 ,i do not hope to convince them2 for they have lived too long in this atmosphere4 ,besides1 when one of their demonstrations has been refuted1 we are sure to see it resurrected with insignificant alterations1 and some of them have already risen several times from their ashes4 ,such long ago was the ,lernan hydra with its famous heads which always grew again4 ,hercules got through1 since his hydra had only nine heads1 or eleven2 but here there are too many1 some in ,england1 some in ,germany1 in ,italy1 in ,france1 and he would have to give up the struggle4 ,so ,i appeal only to men of good judgment unprejudiced4 ,i ,in these latter years numerous works have been published on pure mathematics and the philosophy of mathematics1 trying to separate and isolate the logical elements of mathematical reasoning4 ,these works have been analyzed and expounded very clearly by ,m4 ,couturat in a book entitled3 ,the ,principles of ,mathematics4 ,for ,m4 ,couturat1 the new works1 and in particular those of ,russell and ,peano1 have finally settled the controversy1 so long pending between ,leibnitz and ,kant4 ,they have shown that there are no synthetic judgments a priori 7,kant's phrase to designate judgments which can neither be demonstrated analytically1 nor reduced to identities1 nor established experimentally71 they have shown that mathematics is entirely reducible to logic and that intuition here plays no rle4 ,this is what ,m4 ,couturat has set forth in the work just cited2 this he says still more explicitly in his ,kant jubilee discourse1 so that ,i heard my neighbor whisper3 ,i well see this is the centenary of ,kant's death4 ,can we subscribe to this conclusive condemnation8 ,i think not1 and ,i shall try to show why4 ,,ii ,what strikes us first in the new mathematics is its purely formal character3 ,we think1 says ,hilbert1 three sorts of things1 which we shall call points1 straights and planes4 ,we convene that a straight shall be determined by two points1 and that in place of saying this straight is determined by these two points1 we may say it passes through these two points1 or that these two points are situated on this straight4 ,what these things are1 not only we do not know1 but we should not seek to know4 ,we have no need to1 and one who never had seen either point or straight or plane could geometrize as well as we4 ,that the phrase to pass through1 or the phrase to be situated upon may arouse in us no image1 the first is simply a synonym of to be determined and the second of to determine4 ,thus1 be it understood1 to demonstrate a theorem1 it is neither necessary nor even advantageous to know what it means4 ,the geometer might be replaced by the logic piano imagined by ,stanley ,jevons2 or1 if you choose1 a machine might be imagined where the assumptions were put in at one end1 while the theorems came out at the other1 like the legendary ,chicago machine where the pigs go in alive and come out transformed into hams and sausages4 ,no more than these machines need the mathematician know what he does4 ,i do not make this formal character of his geometry a reproach to ,hilbert4 ,this is the way he should go1 given the problem he set himself4 ,he wished to reduce to a minimum the number of the fundamental assumptions of geometry and completely enumerate them2 now1 in reasonings where our mind remains active1 in those where intuition still plays a part1 in living reasonings1 so to speak1 it is difficult not to introduce an assumption or a postulate which passes unperceived4 ,it is therefore only after having carried back all the geometric reasonings to a form purely mechanical that he could be sure of having accomplished his design and finished his work4 ,what ,hilbert did for geometry1 others have tried to do for arithmetic and analysis4 ,even if they had entirely succeeded1 would the ,kantians be finally condemned to silence8 ,perhaps not1 for in reducing mathematical thought to an empty form1 it is certainly mutilated4 ,even admitting it were established that all the theorems could be deduced by procedures purely analytic1 by simple logical combinations of a finite number of assumptions1 and that these assumptions are only conventions2 the philosopher would still have the right to investigate the origins of these conventions1 to see why they have been judged preferable to the contrary conventions4 ,and then the logical correctness of the reasonings leading from the assumptions to the theorems is not the only thing which should occupy us4 ,the rules of perfect logic1 are they the whole of mathematics8 ,as well say the whole art of playing chess reduces to the rules of the moves of the pieces4 ,among all the constructs which can be built up of the materials furnished by logic1 choice must be made2 the true geometer makes this choice judiciously because he is guided by a sure instinct1 or by some vague consciousness of ,i know not what more profound and more hidden geometry1 which alone gives value to the edifice constructed4 ,to seek the origin of this instinct1 to study the laws of this deep geometry1 felt1 not stated1 would also be a fine employment for the philosophers who do not want logic to be all4 ,but it is not at this point of view ,i wish to put myself1 it is not thus ,i wish to consider the question4 ,the instinct mentioned is necessary for the inventor1 but it would seem at first we might do without it in studying the science once created4 ,well1 what ,i wish to investigate is if it be true that1 the principles of logic once admitted1 one can1 ,i do not say discover1 but demonstrate1 all the mathematical verities without making a new appeal to intuition4 ,,iii ,i once said no to this question3#ab should our reply be modified by the recent works8 ,my saying no was because the principle of complete induction seemed to me at once necessary to the mathematician and irreducible to logic4 ,the statement of this principle is3 ,if a property be true of the number #a1 and if we establish that it is true of n #a provided it be of n1 it will be true of all the whole numbers4 ,therein ,i see the mathematical reasoning par excellence4 ,i did not mean to say1 as has been supposed1 that all mathematical reasonings can be reduced to an application of this principle4 ,examining these reasonings closely1 we there should see applied many other analogous principles1 presenting the same essential characteristics4 ,in this category of principles1 that of complete induction is only the simplest of all and this is why ,i have chosen it as type4 #ab ,see ,science and ,hypothesis1 chapter ,i4 ,the current name1 principle of complete induction1 is not justified4 ,this mode of reasoning is none the less a true mathematical induction which differs from ordinary induction only by its certitude4 ,,iv ,,definitions ,,and ,,assumptions ,the existence of such principles is a difficulty for the uncompromising logicians2 how do they pretend to get out of it8 ,the principle of complete induction1 they say1 is not an assumption properly so called or a synthetic judgment a priori2 it is just simply the definition of whole number4 ,it is therefore a simple convention4 ,to discuss this way of looking at it1 we must examine a little closely the relations between definitions and assumptions4 ,let us go back first to an article by ,m4 ,couturat on mathematical definitions which appeared in l',enseignement mathmatique1 a magazine published by ,gauthier-,villars and by ,georg at ,geneva4 ,we shall see there a distinction between the direct definition and the definition by postulates4 ,the definition by postulates1 says ,m4 ,couturat1 applies not to a single notion1 but to a system of notions2 it consists in enumerating the fundamental relations which unite them and which enable us to demonstrate all their other properties2 these relations are postulates4 ,if previously have been defined all these notions but one1 then this last will be by definition the thing which verifies these postulates4 ,thus certain indemonstrable assumptions of mathematics would be only disguised definitions4 ,this point of view is often legitimate2 and ,i have myself admitted it in regard for instance to ,euclid's postulate4 ,the other assumptions of geometry do not suffice to completely define distance2 the distance then will be1 by definition1 among all the magnitudes which satisfy these other assumptions1 that which is such as to make ,euclid's postulate true4 ,well the logicians suppose true for the principle of complete induction what ,i admit for ,euclid's postulate2 they want to see in it only a disguised definition4 ,but to give them this right1 two conditions must be fulfilled4 ,stuart ,mill says every definition implies an assumption1 that by which the existence of the defined object is affirmed4 ,according to that1 it would no longer be the assumption which might be a disguised definition1 it would on the contrary be the definition which would be a disguised assumption4 ,stuart ,mill meant the word existence in a material and empirical sense2 he meant to say that in defining the circle we affirm there are round things in nature4 ,under this form1 his opinion is inadmissible4 ,mathematics is independent of the existence of material objects2 in mathematics the word exist can have only one meaning1 it means free from contradiction4 ,thus rectified1 ,stuart ,mill's thought becomes exact2 in defining a thing1 we affirm that the definition implies no contradiction4 ,if therefore we have a system of postulates1 and if we can demonstrate that these postulates imply no contradiction1 we shall have the right to consider them as representing the definition of one of the notions entering therein4 ,if we can not demonstrate that1 it must be admitted without proof1 and that then will be an assumption2 so that1 seeking the definition under the postulate1 we should find the assumption under the definition4 ,usually1 to show that a definition implies no contradiction1 we proceed by example1 we try to make an example of a thing satisfying the definition4 ,take the case of a definition by postulates2 we wish to define a notion ,a1 and we say that1 by definition1 an ,a is anything for which certain postulates are true4 ,if we can prove directly that all these postulates are true of a certain object ,b1 the definition will be justified2 the object ,b will be an example of an ,a4 ,we shall be certain that the postulates are not contradictory1 since there are cases where they are all true at the same time4 ,but such a direct demonstration by example is not always possible4 ,to establish that the postulates imply no contradiction1 it is then necessary to consider all the propositions deducible from these postulates considered as premises1 and to show that1 among these propositions1 no two are contradictory4 ,if these propositions are finite in number1 a direct verification is possible4 ,this case is infrequent and uninteresting4 ,if these propositions are infinite in number1 this direct verification can no longer be made2 recourse must be had to procedures where in general it is necessary to invoke just this principle of complete induction which is precisely the thing to be proved4 ,this is an explanation of one of the conditions the logicians should satisfy1 and further on we shall see they have not done it4 ,v ,there is a second4 ,when we give a definition1 it is to use it4 ,we therefore shall find in the sequel of the exposition the word defined2 have we the right to affirm1 of the thing represented by this word1 the postulate which has served for definition8 ,yes1 evidently1 if the word has retained its meaning1 if we do not attribute to it implicitly a different meaning4 ,now this is what sometimes happens and it is usually difficult to perceive it2 it is needful to see how this word comes into our discourse1 and if the gate by which it has entered does not imply in reality a definition other than that stated4 ,this difficulty presents itself in all the applications of mathematics4 ,the mathematical notion has been given a definition very refined and very rigorous2 and for the pure mathematician all doubt has disappeared2 but if one wishes to apply it to the physical sciences for instance1 it is no longer a question of this pure notion1 but of a concrete object which is often only a rough image of it4 ,to say that this object satisfies1 at least approximately1 the definition1 is to state a new truth1 which experience alone can put beyond doubt1 and which no longer has the character of a conventional postulate4 ,but without going beyond pure mathematics1 we also meet the same difficulty4 ,you give a subtile definition of numbers2 then1 once this definition given1 you think no more of it2 because1 in reality1 it is not it which has taught you what number is2 you long ago knew that1 and when the word number further on is found under your pen1 you give it the same sense as the first comer4 ,to know what is this meaning and whether it is the same in this phrase or that1 it is needful to see how you have been led to speak of number and to introduce this word into these two phrases4 ,i shall not for the moment dilate upon this point1 because we shall have occasion to return to it4 ,thus consider a word of which we have given explicitly a definition ,a2 afterwards in the discourse we make a use of it which implicitly supposes another definition ,b4 ,it is possible that these two definitions designate the same thing4 ,but that this is so is a new truth which must either be demonstrated or admitted as an independent assumption4 ,we shall see farther on that the logicians have not fulfilled the second condition any better than the first4 ,,vi ,the definitions of number are very numerous and very different2 ,i forego the enumeration even of the names of their authors4 ,we should not be astonished that there are so many4 ,if one among them was satisfactory1 no new one would be given4 ,if each new philosopher occupying himself with this question has thought he must invent another one1 this was because he was not satisfied with those of his predecessors1 and he was not satisfied with them because he thought he saw a petitio principii4 ,i have always felt1 in reading the writings devoted to this problem1 a profound feeling of discomfort2 ,i was always expecting to run against a petitio principii1 and when ,i did not immediately perceive it1 ,i feared ,i had overlooked it4 ,this is because it is impossible to give a definition without using a sentence1 and difficult to make a sentence without using a number word1 or at least the word several1 or at least a word in the plural4 ,and then the declivity is slippery and at each instant there is risk of a fall into petitio principii4 ,i shall devote my attention in what follows only to those of these definitions where the petitio principii is most ably concealed4 ,,vii ,,pasigraphy ,the symbolic language created by ,peano plays a very grand rle in these new researches4 ,it is capable of rendering some service1 but ,i think ,m4 ,couturat attaches to it an exaggerated importance which must astonish ,peano himself4 ,the essential element of this language is certain algebraic signs which represent the different conjunctions3 if1 and1 or1 therefore4 ,that these signs may be convenient is possible2 but that they are destined to revolutionize all philosophy is a different matter4 ,it is difficult to admit that the word if acquires1 when written ,c1 a virtue it had not when written if4 ,this invention of ,peano was first called pasigraphy1 that is to say the art of writing a treatise on mathematics without using a single word of ordinary language4 ,this name defined its range very exactly4 ,later1 it was raised to a more eminent dignity by conferring on it the title of logistic4 ,this word is1 it appears1 employed at the ,military ,academy1 to designate the art of the quartermaster of cavalry1 the art of marching and cantoning troops2 but here no confusion need be feared1 and it is at once seen that this new name implies the design of revolutionizing logic4 ,we may see the new method at work in a mathematical memoir by ,burali-,forti1 entitled3 ,una ,questione sui numeri transfiniti1 inserted in ,volume ,,xi of the ,rendiconti del circolo matematico di ,palermo4 ,i begin by saying this memoir is very interesting1 and my taking it here as example is precisely because it is the most important of all those written in the new language4 ,besides1 the uninitiated may read it1 thanks to an ,italian interlinear translation4 ,what constitutes the importance of this memoir is that it has given the first example of those antinomies met in the study of transfinite numbers and making since some years the despair of mathematicians4 ,the aim1 says ,burali-,forti1 of this note is to show there may be two transfinite numbers 7ordinals71 a and b1 such that a is neither equal to1 greater than1 nor less than b4 ,to reassure the reader1 to comprehend the considerations which follow1 he has no need of knowing what a transfinite ordinal number is4 ,now1 ,cantor had precisely proved that between two transfinite numbers as between two finite1 there can be no other relation than equality or inequality in one sense or the other4 ,but it is not of the substance of this memoir that ,i wish to speak here2 that would carry me much too far from my subject2 ,i only wish to consider the form1 and just to ask if this form makes it gain much in rigor and whether it thus compensates for the efforts it imposes upon the writer and the reader4 ,first we see ,burali-,forti define the number #a as follows3 #a iota,t',ko7n7u1 h7epsilon7uepsilon,un71 a definition eminently fitted to give an idea of the number #a to persons who had never heard speak of it4 ,i understand ,peanian too ill to dare risk a critique1 but still ,i fear this definition contains a petitio principii1 considering that ,i see the figure #a in the first member and ,un in letters in the second4 ,however that may be1 ,burali-,forti starts from this definition and1 after a short calculation1 reaches the equation3 7#bg7 #a;epsilon,no1 which tells us that ,one is a number4 ,and since we are on these definitions of the first numbers1 we recall that ,m4 ,couturat has also defined #j and #a4 ,what is zero8 ,it is the number of elements of the null class4 ,and what is the null class8 ,it is that containing no element4 ,to define zero by null1 and null by no1 is really to abuse the wealth of language2 so ,m4 ,couturat has introduced an improvement in his definition1 by writing3 #j iota,lambda3phix ,lambdainverted c,lambda 7xepsilonphix71 which means3 zero is the number of things satisfying a condition never satisfied4 ,but as never means in no case ,i do not see that the progress is great4 ,i hasten to add that the definition ,m4 ,couturat gives of the number #a is more satisfactory4 ,one1 says he in substance1 is the number of elements in a class in which any two elements are identical4 ,it is more satisfactory1 ,i have said1 in this sense that to define #a1 he does not use the word one2 in compensation1 he uses the word two4 ,but ,i fear1 if asked what is two1 ,m4 ,couturat would have to use the word one4 ,,viii ,but to return to the memoir of ,burali-,forti2 ,i have said his conclusions are in direct opposition to those of ,cantor4 ,now1 one day ,m4 ,hadamard came to see me and the talk fell upon this antinomy4 ,burali-,forti's reasoning1 ,i said1 does it not seem to you irreproachable8 ,no1 and on the contrary ,i find nothing to object to in that of ,cantor4 ,besides1 ,burali-,forti had no right to speak of the aggregate of all the ordinal numbers4 ,pardon1 he had the right1 since he could always put ,omega ,t'7,no1epsilon74 ,i should like to know who was to prevent him1 and can it be said a thing does not exist1 when we have called it ,omega8 ,it was in vain1 ,i could not convince him 7which besides would have been sad1 since he was right74 ,was it merely because ,i do not speak the ,peanian with enough eloquence8 ,perhaps2 but between ourselves ,i do not think so4 ,thus1 despite all this pasigraphic apparatus1 the question was not solved4 ,what does that prove8 ,in so far as it is a question only of proving one a number1 pasigraphy suffices1 but if a difficulty presents itself1 if there is an antinomy to solve1 pasigraphy becomes impotent4 ,,chapter ,,iv ,,the ,,new ,,logics ,i ,the ,russell ,logic ,to justify its pretensions1 logic had to change4 ,we have seen new logics arise of which the most interesting is that of ,russell4 ,it seems he has nothing new to write about formal logic1 as if ,aristotle there had touched bottom4 ,but the domain ,russell attributes to logic is infinitely more extended than that of the classic logic1 and he has put forth on the subject views which are original and at times well warranted4 ,first1 ,russell subordinates the logic of classes to that of propositions1 while the logic of ,aristotle was above all the logic of classes and took as its point of departure the relation of subject to predicate4 ,the classic syllogism1 ,socrates is a man1 etc41 gives place to the hypothetical syllogism3 ,if ,a is true1 ,b is true2 now if ,b is true1 ,c is true1 etc4 ,and this is1 ,i think1 a most happy idea1 because the classic syllogism is easy to carry back to the hypothetical syllogism1 while the inverse transformation is not without difficulty4 ,and then this is not all4 ,russell's logic of propositions is the study of the laws of combination of the conjunctions if1 and1 or1 and the negation not4 ,in adding here two other conjunctions1 and and or1 ,russell opens to logic a new field4 ,the symbols and1 or follow the same laws as the two signs and 1 that is to say the commutative associative and distributive laws4 ,thus and represents logical multiplication1 while or represents logical addition4 ,this also is very interesting4 ,russell reaches the conclusion that any false proposition implies all other propositions true or false4 ,m4 ,couturat says this conclusion will at first seem paradoxical4 ,it is sufficient however to have corrected a bad thesis in mathematics to recognize how right ,russell is4 ,the candidate often is at great pains to get the first false equation2 but that once obtained1 it is only sport then for him to accumulate the most surprising results1 some of which even may be true4 ,,ii ,we see how much richer the new logic is than the classic logic2 the symbols are multiplied and allow of varied combinations which are no longer limited in number4 ,has one the right to give this extension to the meaning of the word logic8 ,it would be useless to examine this question and to seek with ,russell a mere quarrel about words4 ,grant him what he demands2 but be not astonished if certain verities declared irreducible to logic in the old sense of the word find themselves now reducible to logic in the new sense--something very different4 ,a great number of new notions have been introduced1 and these are not simply combinations of the old4 ,russell knows this1 and not only at the beginning of the first chapter1 ',the ,logic of ,propositions1' but at the beginning of the second and third1 ',the ,logic of ,classes' and ',the ,logic of ,relations1' he introduces new words that he declares indefinable4 ,and this is not all2 he likewise introduces principles he declares indemonstrable4 ,but these indemonstrable principles are appeals to intuition1 synthetic judgments a priori4 ,we regard them as intuitive when we meet them more or less explicitly enunciated in mathematical treatises2 have they changed character because the meaning of the word logic has been enlarged and we now find them in a book entitled ,treatise on ,logic8 ,they have not changed nature2 they have only changed place4 ,,iii ,could these principles be considered as disguised definitions8 ,it would then be necessary to have some way of proving that they imply no contradiction4 ,it would be necessary to establish that1 however far one followed the series of deductions1 he would never be exposed to contradicting himself4 ,we might attempt to reason as follows3 ,we can verify that the operations of the new logic applied to premises exempt from contradiction can only give consequences equally exempt from contradiction4 ,if therefore after n operations we have not met contradiction1 we shall not encounter it after n #a4 ,thus it is impossible that there should be a moment when contradiction begins1 which shows we shall never meet it4 ,have we the right to reason in this way8 ,no1 for this would be to make use of complete induction2 and remember1 we do not yet know the principle of complete induction4 ,we therefore have not the right to regard these assumptions as disguised definitions and only one resource remains for us1 to admit a new act of intuition for each of them4 ,moreover ,i believe this is indeed the thought of ,russell and ,m4 ,couturat4 ,thus each of the nine indefinable notions and of the twenty indemonstrable propositions 7,i believe if it were ,i that did the counting1 ,i should have found some more7 which are the foundation of the new logic1 logic in the broad sense1 presupposes a new and independent act of our intuition and 7why not say it87 a veritable synthetic judgment a priori4 ,on this point all seem agreed1 but what ,russell claims1 and what seems to me doubtful1 is that after these appeals to intuition1 that will be the end of it2 we need make no others and can build all mathematics without the intervention of any new element4 ,,iv ,m4 ,couturat often repeats that this new logic is altogether independent of the idea of number4 ,i shall not amuse myself by counting how many numeral adjectives his exposition contains1 both cardinal and ordinal1 or indefinite adjectives such as several4 ,we may cite1 however1 some examples3 ,the logical product of two or more propositions is44442 ,all propositions are capable only of two values1 true and false2 ,the relative product of two relations is a relation2 ,a relation exists between two terms1 etc41 etc4 ,sometimes this inconvenience would not be unavoidable1 but sometimes also it is essential4 ,a relation is incomprehensible without two terms2 it is impossible to have the intuition of the relation1 without having at the same time that of its two terms1 and without noticing they are two1 because1 if the relation is to be conceivable1 it is necessary that there be two and only two4 ,v ,arithmetic ,i reach what ,m4 ,couturat calls the ordinal theory which is the foundation of arithmetic properly so called4 ,m4 ,couturat begins by stating ,peano's five assumptions1 which are independent1 as has been proved by ,peano and ,padoa4 #a4 ,zero is an integer4 #b4 ,zero is not the successor of any integer4 #c4 ,the successor of an integer is an integer4 ,to this it would be proper to add1 ,every integer has a successor4 #d4 ,two integers are equal if their successors are4 ,the fifth assumption is the principle of complete induction4 ,m4 ,couturat considers these assumptions as disguised definitions2 they constitute the definition by postulates of zero1 of successor1 and of integer4 ,but we have seen that for a definition by postulates to be acceptable we must be able to prove that it implies no contradiction4 ,is this the case here8 ,not at all4 ,the demonstration can not be made by example4 ,we can not take a part of the integers1 for instance the first three1 and prove they satisfy the definition4 ,if ,i take the series #j1 #a1 #b1 ,i see it fulfils the assumptions #a1 #b1 #d and #e2 but to satisfy assumption #c it still is necessary that #c be an integer1 and consequently that the series #j1 #a1 #b1 #c1 fulfil the assumptions2 we might prove that it satisfies assumptions #a1 #b1 #d1 #e1 but assumption #c requires besides that #d be an integer and that the series #j1 #a1 #b1 #c1 #d fulfil the assumptions1 and so on4 ,it is therefore impossible to demonstrate the assumptions for certain integers without proving them for all2 we must give up proof by example4 ,it is necessary then to take all the consequences of our assumptions and see if they contain no contradiction4 ,if these consequences were finite in number1 this would be easy2 but they are infinite in number2 they are the whole of mathematics1 or at least all arithmetic4 ,what then is to be done8 ,perhaps strictly we could repeat the reasoning of number ,,iii4 ,but as we have said1 this reasoning is complete induction1 and it is precisely the principle of complete induction whose justification would be the point in question4 ,,vi ,the ,logic of ,hilbert ,i come now to the capital work of ,hilbert which he communicated to the ,congress of ,mathematicians at ,heidelberg1 and of which a ,french translation by ,m4 ,pierre ,boutroux appeared in l',enseignement mathmatique1 while an ,english translation due to ,halsted appeared in ,the ,monist4#ac ,in this work1 which contains profound thoughts1 the author's aim is analogous to that of ,russell1 but on many points he diverges from his predecessor4 #ac ',the ,foundations of ,logic and ,arithmetic1' ,monist1 ,,xv41 #cch-ceb4 ,but1 he says 7,monist1 p4 #cdj71 on attentive consideration we become aware that in the usual exposition of the laws of logic certain fundamental concepts of arithmetic are already employed2 for example1 the concept of the aggregate1 in part also the concept of number4 ,we fall thus into a vicious circle and therefore to avoid paradoxes a partly simultaneous development of the laws of logic and arithmetic is requisite4 ,we have seen above that what ,hilbert says of the principles of logic in the usual exposition applies likewise to the logic of ,russell4 ,so for ,russell logic is prior to arithmetic2 for ,hilbert they are 'simultaneous4' ,we shall find further on other differences still greater1 but we shall point them out as we come to them4 ,i prefer to follow step by step the development of ,hilbert's thought1 quoting textually the most important passages4 ,let us take as the basis of our consideration first of all a thought-thing #a 7one7 7p4 #cda74 ,notice that in so doing we in no wise imply the notion of number1 because it is understood that #a is here only a symbol and that we do not at all seek to know its meaning4 ,the taking of this thing together with itself respectively two1 three or more times4444 ,ah6 this time it is no longer the same2 if we introduce the words 'two1' 'three1' and above all 'more1' 'several1' we introduce the notion of number2 and then the definition of finite whole number which we shall presently find1 will come too late4 ,our author was too circumspect not to perceive this begging of the question4 ,so at the end of his work he tries to proceed to a truly patching-up process4 ,hilbert then introduces two simple objects #a and 1 and considers all the combinations of these two objects1 all the combinations of their combinations1 etc4 ,it goes without saying that we must forget the ordinary meaning of these two signs and not attribute any to them4 ,afterwards he separates these combinations into two classes1 the class of the existent and the class of the non-existent1 and till further orders this separation is entirely arbitrary4 ,every affirmative statement tells us that a certain combination belongs to the class of the existent2 every negative statement tells us that a certain combination belongs to the class of the non-existent4 ,,vii ,note now a difference of the highest importance4 ,for ,russell any object whatsoever1 which he designates by x1 is an object absolutely undetermined and about which he supposes nothing2 for ,hilbert it is one of the combinations formed with the symbols #a and 2 he could not conceive of the introduction of anything other than combinations of objects already defined4 ,moreover ,hilbert formulates his thought in the neatest way1 and ,i think ,i must reproduce in extenso his statement 7p4 #cdh73 ,in the assumptions the arbitraries 7as equivalent for the concept 'every' and 'all' in the customary logic7 represent only those thought-things and their combinations with one another1 which at this stage are laid down as fundamental or are to be newly defined4 ,therefore in the deduction of inferences from the assumptions1 the arbitraries1 which occur in the assumptions1 can be replaced only by such thought-things and their combinations4 ,also we must duly remember1 that through the super-addition and making fundamental of a new thought-thing the preceding assumptions undergo an enlargement of their validity1 and where necessary1 are to be subjected to a change in conformity with the sense4 ,the contrast with ,russell's view-point is complete4 ,for this philosopher we may substitute for x not only objects already known1 but anything4 ,russell is faithful to his point of view1 which is that of comprehension4 ,he starts from the general idea of being1 and enriches it more and more while restricting it1 by adding new qualities4 ,hilbert on the contrary recognizes as possible beings only combinations of objects already known2 so that 7looking at only one side of his thought7 we might say he takes the view-point of extension4 ,,viii ,let us continue with the exposition of ,hilbert's ideas4 ,he introduces two assumptions which he states in his symbolic language but which signify1 in the language of the uninitiated1 that every quality is equal to itself and that every operation performed upon two identical quantities gives identical results4 ,so stated1 they are evident1 but thus to present them would be to misrepresent ,hilbert's thought4 ,for him mathematics has to combine only pure symbols1 and a true mathematician should reason upon them without preconceptions as to their meaning4 ,so his assumptions are not for him what they are for the common people4 ,he considers them as representing the definition by postulates of the symbol 77 heretofore void of all signification4 ,but to justify this definition we must show that these two assumptions lead to no contradiction4 ,for this ,hilbert used the reasoning of our number ,,iii1 without appearing to perceive that he is using complete induction4 ,,ix ,the end of ,hilbert's memoir is altogether enigmatic and ,i shall not lay stress upon it4 ,contradictions accumulate2 we feel that the author is dimly conscious of the petitio principii he has committed1 and that he seeks vainly to patch up the holes in his argument4 ,what does this mean8 ,at the point of proving that the definition of the whole number by the assumption of complete induction implies no contradiction1 ,hilbert withdraws as ,russell and ,couturat withdrew1 because the difficulty is too great4 ,x ,geometry ,geometry1 says ,m4 ,couturat1 is a vast body of doctrine wherein the principle of complete induction does not enter4 ,that is true in a certain measure2 we can not say it is entirely absent1 but it enters very slightly4 ,if we refer to the ,rational ,geometry of ,dr4 ,halsted 7,new ,york1 ,john ,wiley and ,sons1 #aijd7 built up in accordance with the principles of ,hilbert1 we see the principle of induction enter for the first time on page #aad 7unless ,i have made an oversight1 which is quite possible74#ad #ad ,second ed41 #aijg1 p4 #hf2 ,french ed41 #aiaa1 p4 #ig4 ,g4 ,b4 ,h4 ,so geometry1 which only a few years ago seemed the domain where the reign of intuition was uncontested1 is to-day the realm where the logicians seem to triumph4 ,nothing could better measure the importance of the geometric works of ,hilbert and the profound impress they have left on our conceptions4 ,but be not deceived4 ,what is after all the fundamental theorem of geometry8 ,it is that the assumptions of geometry imply no contradiction1 and this we can not prove without the principle of induction4 ,how does ,hilbert demonstrate this essential point8 ,by leaning upon analysis and through it upon arithmetic and through it upon the principle of induction4 ,and if ever one invents another demonstration1 it will still be necessary to lean upon this principle1 since the possible consequences of the assumptions1 of which it is necessary to show that they are not contradictory1 are infinite in number4 ,,xi ,conclusion ,our conclusion straightway is that the principle of induction can not be regarded as the disguised definition of the entire world4 ,here are three truths3 7#a7 ,the principle of complete induction2 7#b7 ,euclid's postulate2 7#c7 the physical law according to which phosphorus melts at #dd 7cited by ,m4 ,le ,roy74 ,these are said to be three disguised definitions3 the first1 that of the whole number2 the second1 that of the straight line2 the third1 that of phosphorus4 ,i grant it for the second2 ,i do not admit it for the other two4 ,i must explain the reason for this apparent inconsistency4 ,first1 we have seen that a definition is acceptable only on condition that it implies no contradiction4 ,we have shown likewise that for the first definition this demonstration is impossible2 on the other hand1 we have just recalled that for the second ,hilbert has given a complete proof4 ,as to the third1 evidently it implies no contradiction4 ,does this mean that the definition guarantees1 as it should1 the existence of the object defined8 ,we are here no longer in the mathematical sciences1 but in the physical1 and the word existence has no longer the same meaning4 ,it no longer signifies absence of contradiction2 it means objective existence4 ,you already see a first reason for the distinction ,i made between the three cases2 there is a second4 ,in the applications we have to make of these three concepts1 do they present themselves to us as defined by these three postulates8 ,the possible applications of the principle of induction are innumerable2 take1 for example1 one of those we have expounded above1 and where it is sought to prove that an aggregate of assumptions can lead to no contradiction4 ,for this we consider one of the series of syllogisms we may go on with in starting from these assumptions as premises4 ,when we have finished the nth syllogism1 we see we can make still another and this is the n #ath4 ,thus the number n serves to count a series of successive operations2 it is a number obtainable by successive additions4 ,this therefore is a number from which we may go back to unity by successive subtractions4 ,evidently we could not do this if we had n n - #a1 since then by subtraction we should always obtain again the same number4 ,so the way we have been led to consider this number n implies a definition of the finite whole number and this definition is the following3 ,a finite whole number is that which can be obtained by successive additions2 it is such that n is not equal to n - #a4 ,that granted1 what do we do8 ,we show that if there has been no contradiction up to the nth syllogism1 no more will there be up to the n #ath1 and we conclude there never will be4 ,you say3 ,i have the right to draw this conclusion1 since the whole numbers are by definition those for which a like reasoning is legitimate4 ,but that implies another definition of the whole number1 which is as follows3 ,a whole number is that on which we may reason by recurrence4 ,in the particular case it is that of which we may say that1 if the absence of contradiction up to the time of a syllogism of which the number is an integer carries with it the absence of contradiction up to the time of the syllogism whose number is the following integer1 we need fear no contradiction for any of the syllogisms whose number is an integer4 ,the two definitions are not identical2 they are doubtless equivalent1 but only in virtue of a synthetic judgment a priori2 we can not pass from one to the other by a purely logical procedure4 ,consequently we have no right to adopt the second1 after having introduced the whole number by a way that presupposes the first4 ,on the other hand1 what happens with regard to the straight line8 ,i have already explained this so often that ,i hesitate to repeat it again1 and shall confine myself to a brief recapitulation of my thought4 ,we have not1 as in the preceding case1 two equivalent definitions logically irreducible one to the other4 ,we have only one expressible in words4 ,will it be said there is another which we feel without being able to word it1 since we have the intuition of the straight line or since we represent to ourselves the straight line8 ,first of all1 we can not represent it to ourselves in geometric space1 but only in representative space1 and then we can represent to ourselves just as well the objects which possess the other properties of the straight line1 save that of satisfying ,euclid's postulate4 ,these objects are 'the non-,euclidean straights1' which from a certain point of view are not entities void of sense1 but circles 7true circles of true space7 orthogonal to a certain sphere4 ,if1 among these objects equally capable of representation1 it is the first 7the ,euclidean straights7 which we call straights1 and not the latter 7the non-,euclidean straights71 this is properly by definition4 ,and arriving finally at the third example1 the definition of phosphorus1 we see the true definition would be3 ,phosphorus is the bit of matter ,i see in yonder flask4 ,,xii ,and since ,i am on this subject1 still another word4 ,of the phosphorus example ,i said3 ,this proposition is a real verifiable physical law1 because it means that all bodies having all the other properties of phosphorus1 save its point of fusion1 melt like it at #dd4 ,and it was answered3 ,no1 this law is not verifiable1 because if it were shown that two bodies resembling phosphorus melt one at #dd and the other at #ej1 it might always be said that doubtless1 besides the point of fusion1 there is some other unknown property by which they differ4 ,that was not quite what ,i meant to say4 ,i should have written1 ,all bodies possessing such and such properties finite in number 7to wit1 the properties of phosphorus stated in the books on chemistry1 the fusion-point excepted7 melt at #dd4 ,and the better to make evident the difference between the case of the straight and that of phosphorus1 one more remark4 ,the straight has in nature many images more or less imperfect1 of which the chief are the light rays and the rotation axis of the solid4 ,suppose we find the ray of light does not satisfy ,euclid's postulate 7for example by showing that a star has a negative parallax71 what shall we do8 ,shall we conclude that the straight being by definition the trajectory of light does not satisfy the postulate1 or1 on the other hand1 that the straight by definition satisfying the postulate1 the ray of light is not straight8 ,assuredly we are free to adopt the one or the other definition and consequently the one or the other conclusion2 but to adopt the first would be stupid1 because the ray of light probably satisfies only imperfectly not merely ,euclid's postulate1 but the other properties of the straight line1 so that if it deviates from the ,euclidean straight1 it deviates no less from the rotation axis of solids which is another imperfect image of the straight line2 while finally it is doubtless subject to change1 so that such a line which yesterday was straight will cease to be straight to-morrow if some physical circumstance has changed4 ,suppose now we find that phosphorus does not melt at #dd1 but at #dc4i4 ,shall we conclude that phosphorus being by definition that which melts at #dd1 this body that we did call phosphorus is not true phosphorus1 or1 on the other hand1 that phosphorous melts at #dc4i8 ,here again we are free to adopt the one or the other definition and consequently the one or the other conclusion2 but to adopt the first would be stupid because we can not be changing the name of a substance every time we determine a new decimal of its fusion-point4 ,,xiii ,to sum up1 ,russell and ,hilbert have each made a vigorous effort2 they have each written a work full of original views1 profound and often well warranted4 ,these two works give us much to think about and we have much to learn from them4 ,among their results1 some1 many even1 are solid and destined to live4 ,but to say that they have finally settled the debate between ,kant and ,leibnitz and ruined the ,kantian theory of mathematics is evidently incorrect4 ,i do not know whether they really believed they had done it1 but if they believed so1 they deceived themselves4 ,,chapter ,v ,,the ,,latest ,,efforts ,,of ,,the ,,logisticians ,i ,the logicians have attempted to answer the preceding considerations4 ,for that1 a transformation of logistic was necessary1 and ,russell in particular has modified on certain points his original views4 ,without entering into the details of the debate1 ,i should like to return to the two questions to my mind most important3 ,have the rules of logistic demonstrated their fruitfulness and infallibility8 ,is it true they afford means of proving the principle of complete induction without any appeal to intuition8 ,,ii ,the ,infallibility of ,logistic ,on the question of fertility1 it seems ,m4 ,couturat has nave illusions4 ,logistic1 according to him1 lends invention 'stilts and wings1' and on the next page3 ,ten years ago1 ,peano published the first edition of his ,formulaire4 ,how is that1 ten years of wings and not to have flown6 ,i have the highest esteem for ,peano1 who has done very pretty things 7for instance his 'space-filling curve1' a phrase now discarded72 but after all he has not gone further nor higher nor quicker than the majority of wingless mathematicians1 and would have done just as well with his legs4 ,on the contrary ,i see in logistic only shackles for the inventor4 ,it is no aid to conciseness--far from it1 and if twenty-seven equations were necessary to establish that #a is a number1 how many would be needed to prove a real theorem8 ,if we distinguish1 with ,whitehead1 the individual x1 the class of which the only member is x and which shall be called iotax1 then the class of which the only member is the class of which the only member is x and which shall be called mux1 do you think these distinctions1 useful as they may be1 go far to quicken our pace8 ,logistic forces us to say all that is ordinarily left to be understood2 it makes us advance step by step2 this is perhaps surer but not quicker4 ,it is not wings you logisticians give us1 but leading-strings4 ,and then we have the right to require that these leading-strings prevent our falling4 ,this will be their only excuse4 ,when a bond does not bear much interest1 it should at least be an investment for a father of a family4 ,should your rules be followed blindly8 ,yes1 else only intuition could enable us to distinguish among them2 but then they must be infallible2 for only in an infallible authority can one have a blind confidence4 ,this1 therefore1 is for you a necessity4 ,infallible you shall be1 or not at all4 ,you have no right to say to us3 ,it is true we make mistakes1 but so do you4 ,for us to blunder is a misfortune1 a very great misfortune2 for you it is death4 ,nor may you ask3 ,does the infallibility of arithmetic prevent errors in addition8 ,the rules of calculation are infallible1 and yet we see those blunder who do not apply these rules2 but in checking their calculation it is at once seen where they went wrong4 ,here it is not at all the case2 the logicians have applied their rules1 and they have fallen into contradiction2 and so true is this1 that they are preparing to change these rules and to sacrifice the notion of class4 ,why change them if they were infallible8 ,we are not obliged1 you say1 to solve hic et nunc all possible problems4 ,oh1 we do not ask so much of you4 ,if1 in face of a problem1 you would give no solution1 we should have nothing to say2 but on the contrary you give us two of them and those contradictory1 and consequently at least one false2 this it is which is failure4 ,russell seeks to reconcile these contradictions1 which can only be done1 according to him1 by restricting or even sacrificing the notion of class4 ,and ,m4 ,couturat1 discovering the success of his attempt1 adds3 ,if the logicians succeed where others have failed1 ,m4 ,poincar will remember this phrase1 and give the honor of the solution to logistic4 ,but no6 ,logistic exists1 it has its code which has already had four editions2 or rather this code is logistic itself4 ,is ,mr4 ,russell preparing to show that one at least of the two contradictory reasonings has transgressed the code8 ,not at all2 he is preparing to change these laws and to abrogate a certain number of them4 ,if he succeeds1 ,i shall give the honor of it to ,russell's intuition and not to the ,peanian logistic which he will have destroyed4 ,,iii ,the ,liberty of ,contradiction ,i made two principal objections to the definition of whole number adopted in logistic4 ,what says ,m4 ,couturat to the first of these objections8 ,what does the word exist mean in mathematics8 ,it means1 ,i said1 to be free from contradiction4 ,this ,m4 ,couturat contests4 ,logical existence1 says he1 is quite another thing from the absence of contradiction4 ,it consists in the fact that a class is not empty4 ,to say3 a's exist1 is1 by definition1 to affirm that the class a is not null4 ,and doubtless to affirm that the class a is not null1 is1 by definition1 to affirm that a's exist4 ,but one of the two affirmations is as denuded of meaning as the other1 if they do not both signify1 either that one may see or touch a's which is the meaning physicists or naturalists give them1 or that one may conceive an a without being drawn into contradictions1 which is the meaning given them by logicians and mathematicians4 ,for ,m4 ,couturat1 it is not non-contradiction that proves existence1 but it is existence that proves non-contradiction4 ,to establish the existence of a class1 it is necessary therefore to establish1 by an example1 that there is an individual belonging to this class3 ,but1 it will be said1 how is the existence of this individual proved8 ,must not this existence be established1 in order that the existence of the class of which it is a part may be deduced8 ,well1 no2 however paradoxical may appear the assertion1 we never demonstrate the existence of an individual4 ,individuals1 just because they are individuals1 are always considered as existent4444 ,we never have to express that an individual exists1 absolutely speaking1 but only that it exists in a class4 ,m4 ,couturat finds his own assertion paradoxical1 and he will certainly not be the only one4 ,yet it must have a meaning4 ,it doubtless means that the existence of an individual1 alone in the world1 and of which nothing is affirmed1 can not involve contradiction2 in so far as it is all alone it evidently will not embarrass any one4 ,well1 so let it be2 we shall admit the existence of the individual1 'absolutely speaking1' but nothing more4 ,it remains to prove the existence of the individual 'in a class1' and for that it will always be necessary to prove that the affirmation1 ,such an individual belongs to such a class1 is neither contradictory in itself1 nor to the other postulates adopted4 ,it is then1 continues ,m4 ,couturat1 arbitrary and misleading to maintain that a definition is valid only if we first prove it is not contradictory4 ,one could not claim in prouder and more energetic terms the liberty of contradiction4 ,in any case1 the onus probandi rests upon those who believe that these principles are contradictory4 ,postulates are presumed to be compatible until the contrary is proved1 just as the accused person is presumed innocent4 ,needless to add that ,i do not assent to this claim4 ,but1 you say1 the demonstration you require of us is impossible1 and you can not ask us to jump over the moon4 ,pardon me2 that is impossible for you1 but not for us1 who admit the principle of induction as a synthetic judgment a priori4 ,and that would be necessary for you1 as for us4 ,to demonstrate that a system of postulates implies no contradiction1 it is necessary to apply the principle of complete induction2 this mode of reasoning not only has nothing 'bizarre' about it1 but it is the only correct one4 ,it is not 'unlikely' that it has ever been employed2 and it is not hard to find 'examples and precedents' of it4 ,i have cited two such instances borrowed from ,hilbert's article4 ,he is not the only one to have used it1 and those who have not done so have been wrong4 ,what ,i have blamed ,hilbert for is not his having recourse to it 7a born mathematician such as he could not fail to see a demonstration was necessary and this the only one possible71 but his having recourse without recognizing the reasoning by recurrence4 ,,iv ,the ,second ,objection ,i pointed out a second error of logistic in ,hilbert's article4 ,to-day ,hilbert is excommunicated and ,m4 ,couturat no longer regards him as of the logistic cult2 so he asks if ,i have found the same fault among the orthodox4 ,no1 ,i have not seen it in the pages ,i have read2 ,i know not whether ,i should find it in the three hundred pages they have written which ,i have no desire to read4 ,only1 they must commit it the day they wish to make any application of mathematics4 ,this science has not as sole object the eternal contemplation of its own navel2 it has to do with nature and some day it will touch it4 ,then it will be necessary to shake off purely verbal definitions and to stop paying oneself with words4 ,to go back to the example of ,hilbert3 always the point at issue is reasoning by recurrence and the question of knowing whether a system of postulates is not contradictory4 ,m4 ,couturat will doubtless say that then this does not touch him1 but it perhaps will interest those who do not claim1 as he does1 the liberty of contradiction4 ,we wish to establish1 as above1 that we shall never encounter contradiction after any number of deductions whatever1 provided this number be finite4 ,for that1 it is necessary to apply the principle of induction4 ,should we here understand by finite number every number to which by definition the principle of induction applies8 ,evidently not1 else we should be led to most embarrassing consequences4 ,to have the right to lay down a system of postulates1 we must be sure they are not contradictory4 ,this is a truth admitted by most scientists2 ,i should have written by all before reading ,m4 ,couturat's last article4 ,but what does this signify8 ,does it mean that we must be sure of not meeting contradiction after a finite number of propositions1 the finite number being by definition that which has all properties of recurrent nature1 so that if one of these properties fails--if1 for instance1 we come upon a contradiction--we shall agree to say that the number in question is not finite8 ,in other words1 do we mean that we must be sure not to meet contradictions1 on condition of agreeing to stop just when we are about to encounter one8 ,to state such a proposition is enough to condemn it4 ,so1 ,hilbert's reasoning not only assumes the principle of induction1 but it supposes that this principle is given us not as a simple definition1 but as a synthetic judgment a priori4 ,to sum up3 ,a demonstration is necessary4 ,the only demonstration possible is the proof by recurrence4 ,this is legitimate only if we admit the principle of induction and if we regard it not as a definition but as a synthetic judgment4 ,v ,the ,cantor ,antinomies ,now to examine ,russell's new memoir4 ,this memoir was written with the view to conquer the difficulties raised by those ,cantor antinomies to which frequent allusion has already been made4 ,cantor thought he could construct a science of the infinite2 others went on in the way he opened1 but they soon ran foul of strange contradictions4 ,these antinomies are already numerous1 but the most celebrated are3 #a4 ,the ,burali-,forti antinomy2 #b4 ,the ,zermelo-,knig antinomy2 #c4 ,the ,richard antinomy4 ,cantor proved that the ordinal numbers 7the question is of transfinite ordinal numbers1 a new notion introduced by him7 can be ranged in a linear series2 that is to say that of two unequal ordinals one is always less than the other4 ,burali-,forti proves the contrary2 and in fact he says in substance that if one could range all the ordinals in a linear series1 this series would define an ordinal greater than all the others2 we could afterwards adjoin #a and would obtain again an ordinal which would be still greater1 and this is contradictory4 ,we shall return later to the ,zermelo-,knig antinomy which is of a slightly different nature4 ,the ,richard antinomy#ae is as follows3 ,consider all the decimal numbers definable by a finite number of words2 these decimal numbers form an aggregate ,e1 and it is easy to see that this aggregate is countable1 that is to say we can number the different decimal numbers of this assemblage from #a to infinity4 ,suppose the numbering effected1 and define a number ,n as follows3 ,if the nth decimal of the nth number of the assemblage ,e is #j1 #a1 #b1 #c1 #d1 #e1 #f1 #g1 #h1 #i the nth decimal of ,n shall be3 #a1 #b1 #c1 #d1 #e1 #f1 #g1 #h1 #a1 #a #ae ,revue gnrale des sciences1 ,june #cj1 #aije4 ,as we see1 ,n is not equal to the nth number of ,e1 and as n is arbitrary1 ,n does not appertain to ,e and yet ,n should belong to this assemblage since we have defined it with a finite number of words4 ,we shall later see that ,m4 ,richard has himself given with much sagacity the explanation of his paradox and that this extends1 mutatis mutandis1 to the other like paradoxes4 ,again1 ,russell cites another quite amusing paradox3 ,what is the least whole number which can not be defined by a phrase composed of less than a hundred ,english words8 ,this number exists2 and in fact the numbers capable of being defined by a like phrase are evidently finite in number since the words of the ,english language are not infinite in number4 ,therefore among them will be one less than all the others4 ,and1 on the other hand1 this number does not exist1 because its definition implies contradiction4 ,this number1 in fact1 is defined by the phrase in italics which is composed of less than a hundred ,english words2 and by definition this number should not be capable of definition by a like phrase4 ,,vi ,zigzag ,theory and ,no-class ,theory ,what is ,mr4 ,russell's attitude in presence of these contradictions8 ,after having analyzed those of which we have just spoken1 and cited still others1 after having given them a form recalling ,epimenides1 he does not hesitate to conclude3 ,a propositional function of one variable does not always determine a class4 ,a propositional function 7that is to say a definition7 does not always determine a class4 ,a 'propositional function' or 'norm' may be 'non-predicative4' ,and this does not mean that these non-predicative propositions determine an empty class1 a null class2 this does not mean that there is no value of x satisfying the definition and capable of being one of the elements of the class4 ,the elements exist1 but they have no right to unite in a syndicate to form a class4 ,but this is only the beginning and it is needful to know how to recognize whether a definition is or is not predicative4 ,to solve this problem ,russell hesitates between three theories which he calls ,a4 ,the zigzag theory2 ,b4 ,the theory of limitation of size2 ,c4 ,the no-class theory4 ,according to the zigzag theory definitions 7propositional functions7 determine a class when they are very simple and cease to do so only when they are complicated and obscure4 ,who1 now1 is to decide whether a definition may be regarded as simple enough to be acceptable8 ,to this question there is no answer1 if it be not the loyal avowal of a complete inability3 ,the rules which enable us to recognize whether these definitions are predicative would be extremely complicated and can not commend themselves by any plausible reason4 ,this is a fault which might be remedied by greater ingenuity or by using distinctions not yet pointed out4 ,but hitherto in seeking these rules1 ,i have not been able to find any other directing principle than the absence of contradiction4 ,this theory therefore remains very obscure2 in this night a single light--the word zigzag4 ,what ,russell calls the 'zigzaginess' is doubtless the particular characteristic which distinguishes the argument of ,epimenides4 ,according to the theory of limitation of size1 a class would cease to have the right to exist if it were too extended4 ,perhaps it might be infinite1 but it should not be too much so4 ,but we always meet again the same difficulty2 at what precise moment does it begin to be too much so8 ,of course this difficulty is not solved and ,russell passes on to the third theory4 ,in the no-classes theory it is forbidden to speak the word 'class' and this word must be replaced by various periphrases4 ,what a change for logistic which talks only of classes and classes of classes6 ,it becomes necessary to remake the whole of logistic4 ,imagine how a page of logistic would look upon suppressing all the propositions where it is a question of class4 ,there would only be some scattered survivors in the midst of a blank page4 ,apparent rari nantes in gurgite vasto4 ,be that as it may1 we see how ,russell hesitates and the modifications to which he submits the fundamental principles he has hitherto adopted4 ,criteria are needed to decide whether a definition is too complex or too extended1 and these criteria can only be justified by an appeal to intuition4 ,it is toward the no-classes theory that ,russell finally inclines4 ,be that as it may1 logistic is to be remade and it is not clear how much of it can be saved4 ,needless to add that ,cantorism and logistic are alone under consideration2 real mathematics1 that which is good for something1 may continue to develop in accordance with its own principles without bothering about the storms which rage outside it1 and go on step by step with its usual conquests which are final and which it never has to abandon4 ,,vii ,the ,true ,solution ,what choice ought we to make among these different theories8 ,it seems to me that the solution is contained in a letter of ,m4 ,richard of which ,i have spoken above1 to be found in the ,revue gnrale des sciences of ,june #cj1 #aije4 ,after having set forth the antinomy we have called ,richard's antinomy1 he gives its explanation4 ,recall what has already been said of this antinomy4 ,e is the aggregate of all the numbers definable by a finite number of words1 without introducing the notion of the aggregate ,e itself4 ,else the definition of ,e would contain a vicious circle2 we must not define ,e by the aggregate ,e itself4 ,now we have defined ,n with a finite number of words1 it is true1 but with the aid of the notion of the aggregate ,e4 ,and this is why ,n is not part of ,e4 ,in the example selected by ,m4 ,richard1 the conclusion presents itself with complete evidence and the evidence will appear still stronger on consulting the text of the letter itself4 ,but the same explanation holds good for the other antinomies1 as is easily verified4 ,thus the definitions which should be regarded as not predicative are those which contain a vicious circle4 ,and the preceding examples sufficiently show what ,i mean by that4 ,is it this which ,russell calls the 'zigzaginess'8 ,i put the question without answering it4 ,,viii ,the ,demonstrations of the ,principle of ,induction ,let us now examine the pretended demonstrations of the principle of induction and in particular those of ,whitehead and of ,burali-,forti4 ,we shall speak of ,whitehead's first1 and take advantage of certain new terms happily introduced by ,russell in his recent memoir4 ,call recurrent class every class containing zero1 and containing n #a if it contains n4 ,call inductive number every number which is a part of all the recurrent classes4 ,upon what condition will this latter definition1 which plays an essential rle in ,whitehead's proof1 be 'predicative' and consequently acceptable8 ,in accordance with what has been said1 it is necessary to understand by all the recurrent classes1 all those in whose definition the notion of inductive number does not enter4 ,else we fall again upon the vicious circle which has engendered the antinomies4 ,now ,whitehead has not taken this precaution4 ,whitehead's reasoning is therefore fallacious2 it is the same which led to the antinomies4 ,it was illegitimate when it gave false results2 it remains illegitimate when by chance it leads to a true result4 ,a definition containing a vicious circle defines nothing4 ,it is of no use to say1 we are sure1 whatever meaning we may give to our definition1 zero at least belongs to the class of inductive numbers2 it is not a question of knowing whether this class is void1 but whether it can be rigorously deliminated4 ,a 'non-predicative' class is not an empty class1 it is a class whose boundary is undetermined4 ,needless to add that this particular objection leaves in force the general objections applicable to all the demonstrations4 ,,ix ,burali-,forti has given another demonstration4#af ,but he is obliged to assume two postulates3 ,first1 there always exists at least one infinite class4 ,the second is thus expressed3 uepsilon,k7,k - iota,lambda7 inverted c u v'u4 ,the first postulate is not more evident than the principle to be proved4 ,the second not only is not evident1 but it is false1 as ,whitehead has shown2 as moreover any recruit would see at the first glance1 if the axiom had been stated in intelligible language1 since it means that the number of combinations which can be formed with several objects is less than the number of these objects4 #af ,in his article ',le classi finite1' ,atti di ,torino1 ,vol4 ,,xxxii4 ,x ,zermelo's ,assumption ,a famous demonstration by ,zermelo rests upon the following assumption3 ,in any aggregate 7or the same in each aggregate of an assemblage of aggregates7 we can always choose at random an element 7even if this assemblage of aggregates should contain an infinity of aggregates74 ,this assumption had been applied a thousand times without being stated1 but1 once stated1 it aroused doubts4 ,some mathematicians1 for instance ,m4 ,borel1 resolutely reject it2 others admire it4 ,let us see what1 according to his last article1 ,russell thinks of it4 ,he does not speak out1 but his reflections are very suggestive4 ,and first a picturesque example3 ,suppose we have as many pairs of shoes as there are whole numbers1 and so that we can number the pairs from one to infinity1 how many shoes shall we have8 ,will the number of shoes be equal to the number of pairs8 ,yes1 if in each pair the right shoe is distinguishable from the left2 it will in fact suffice to give the number #bn - #a to the right shoe of the nth pair1 and the number #bn to the left shoe of the nth pair4 ,no1 if the right shoe is just like the left1 because a similar operation would become impossible--unless we admit ,zermelo's assumption1 since then we could choose at random in each pair the shoe to be regarded as the right4 ,,xi ,conclusions ,a demonstration truly founded upon the principles of analytic logic will be composed of a series of propositions4 ,some1 serving as premises1 will be identities or definitions2 the others will be deduced from the premises step by step4 ,but though the bond between each proposition and the following is immediately evident1 it will not at first sight appear how we get from the first to the last1 which we may be tempted to regard as a new truth4 ,but if we replace successively the different expressions therein by their definition and if this operation be carried as far as possible1 there will finally remain only identities1 so that all will reduce to an immense tautology4 ,logic therefore remains sterile unless made fruitful by intuition4 ,this ,i wrote long ago2 logistic professes the contrary and thinks it has proved it by actually proving new truths4 ,by what mechanism8 ,why in applying to their reasonings the procedure just described--namely1 replacing the terms defined by their definitions--do we not see them dissolve into identities like ordinary reasonings8 ,it is because this procedure is not applicable to them4 ,and why8 ,because their definitions are not predicative and present this sort of hidden vicious circle which ,i have pointed out above2 non-predicative definitions can not be substituted for the terms defined4 ,under these conditions logistic is not sterile1 it engenders antinomies4 ,it is the belief in the existence of the actual infinite which has given birth to those non-predicative definitions4 ,let me explain4 ,in these definitions the word 'all' figures1 as is seen in the examples cited above4 ,the word 'all' has a very precise meaning when it is a question of a finite number of objects2 to have another one1 when the objects are infinite in number1 would require there being an actual 7given complete7 infinity4 ,otherwise all these objects could not be conceived as postulated anteriorly to their definition1 and then if the definition of a notion ,n depends upon all the objects ,a1 it may be infected with a vicious circle1 if among the objects ,a are some indefinable without the intervention of the notion ,n itself4 ,the rules of formal logic express simply the properties of all possible classifications4 ,but for them to be applicable it is necessary that these classifications be immutable and that we have no need to modify them in the course of the reasoning4 ,if we have to classify only a finite number of objects1 it is easy to keep our classifications without change4 ,if the objects are indefinite in number1 that is to say if one is constantly exposed to seeing new and unforeseen objects arise1 it may happen that the appearance of a new object may require the classification to be modified1 and thus it is we are exposed to antinomies4 ,there is no actual 7given complete7 infinity4 ,the ,cantorians have forgotten this1 and they have fallen into contradiction4 ,it is true that ,cantorism has been of service1 but this was when applied to a real problem whose terms were precisely defined1 and then we could advance without fear4 ,logistic also forgot it1 like the ,cantorians1 and encountered the same difficulties4 ,but the question is to know whether they went this way by accident or whether it was a necessity for them4 ,for me1 the question is not doubtful2 belief in an actual infinity is essential in the ,russell logic4 ,it is just this which distinguishes it from the ,hilbert logic4 ,hilbert takes the view-point of extension1 precisely in order to avoid the ,cantorian antinomies4 ,russell takes the view-point of comprehension4 ,consequently for him the genus is anterior to the species1 and the summum genus is anterior to all4 ,that would not be inconvenient if the summum genus was finite2 but if it is infinite1 it is necessary to postulate the infinite1 that is to say to regard the infinite as actual 7given complete74 ,and we have not only infinite classes2 when we pass from the genus to the species in restricting the concept by new conditions1 these conditions are still infinite in number4 ,because they express generally that the envisaged object presents such or such a relation with all the objects of an infinite class4 ,but that is ancient history4 ,russell has perceived the peril and takes counsel4 ,he is about to change everything1 and1 what is easily understood1 he is preparing not only to introduce new principles which shall allow of operations formerly forbidden1 but he is preparing to forbid operations he formerly thought legitimate4 ,not content to adore what he burned1 he is about to burn what he adored1 which is more serious4 ,he does not add a new wing to the building1 he saps its foundation4 ,the old logistic is dead1 so much so that already the zigzag theory and the no-classes theory are disputing over the succession4 ,to judge of the new1 we shall await its coming4 ,,book ,,iii ,,the ,,new ,,mechanics ,,chapter ,i ,,mechanics ,,and ,,radium ,i ,introduction ,the general principles of ,dynamics1 which have1 since ,newton1 served as foundation for physical science1 and which appeared immovable1 are they on the point of being abandoned or at least profoundly modified8 ,this is what many people have been asking themselves for some years4 ,according to them1 the discovery of radium has overturned the scientific dogmas we believed the most solid3 on the one hand1 the impossibility of the transmutation of metals2 on the other hand1 the fundamental postulates of mechanics4 ,perhaps one is too hasty in considering these novelties as finally established1 and breaking our idols of yesterday2 perhaps it would be proper1 before taking sides1 to await experiments more numerous and more convincing4 ,none the less is it necessary1 from to-day1 to know the new doctrines and the arguments1 already very weighty1 upon which they rest4 ,in few words let us first recall in what those principles consist3 ,a4 ,the motion of a material point isolated and apart from all exterior force is straight and uniform2 this is the principle of inertia3 without force no acceleration2 ,b4 ,the acceleration of a moving point has the same direction as the resultant of all the forces to which it is subjected2 it is equal to the quotient of this resultant by a coefficient called mass of the moving point4 ,the mass of a moving point1 so defined1 is a constant2 it does not depend upon the velocity acquired by this point2 it is the same whether the force1 being parallel to this velocity1 tends only to accelerate or to retard the motion of the point1 or whether1 on the contrary1 being perpendicular to this velocity1 it tends to make this motion deviate toward the right1 or the left1 that is to say to curve the trajectory2 ,c4 ,all the forces affecting a material point come from the action of other material points2 they depend only upon the relative positions and velocities of these different material points4 ,combining the two principles ,b and ,c1 we reach the principle of relative motion1 in virtue of which the laws of the motion of a system are the same whether we refer this system to fixed axes1 or to moving axes animated by a straight and uniform motion of translation1 so that it is impossible to distinguish absolute motion from a relative motion with reference to such moving axes2 ,d4 ,if a material point ,a acts upon another material point ,b1 the body ,b reacts upon ,a1 and these two actions are two equal and directly opposite forces4 ,this is the principle of the equality of action and reaction1 or1 more briefly1 the principle of reaction4 ,astronomic observations and the most ordinary physical phenomena seem to have given of these principles a confirmation complete1 constant and very precise4 ,this is true1 it is now said1 but it is because we have never operated with any but very small velocities2 ,mercury1 for example1 the fastest of the planets1 goes scarcely #ajj kilometers a second4 ,would this planet act the same if it went a thousand times faster8 ,we see there is yet no need to worry2 whatever may be the progress of automobilism1 it will be long before we must give up applying to our machines the classic principles of dynamics4 ,how then have we come to make actual speeds a thousand times greater than that of ,mercury1 equal1 for instance1 to a tenth or a third of the velocity of light1 or approaching still more closely to that velocity8 ,it is by aid of the cathode rays and the rays from radium4 ,we know that radium emits three kinds of rays1 designated by the three ,greek letters alpha1 beta1 gamma2 in what follows1 unless the contrary be expressly stated1 it will always be a question of the beta rays1 which are analogous to the cathode rays4 ,after the discovery of the cathode rays two theories appeared4 ,crookes attributed the phenomena to a veritable molecular bombardment2 ,hertz1 to special undulations of the ether4 ,this was a renewal of the debate which divided physicists a century ago about light2 ,crookes took up the emission theory1 abandoned for light2 ,hertz held to the undulatory theory4 ,the facts seem to decide in favor of ,crookes4 ,it has been recognized1 in the first place1 that the cathode rays carry with them a negative electric charge2 they are deviated by a magnetic field and by an electric field2 and these deviations are precisely such as these same fields would produce upon projectiles animated by a very high velocity and strongly charged with electricity4 ,these two deviations depend upon two quantities3 one the velocity1 the other the relation of the electric charge of the projectile to its mass2 we cannot know the absolute value of this mass1 nor that of the charge1 but only their relation2 in fact1 it is clear that if we double at the same time the charge and the mass1 without changing the velocity1 we shall double the force which tends to deviate the projectile1 but1 as its mass is also doubled1 the acceleration and deviation observable will not be changed4 ,the observation of the two deviations will give us therefore two equations to determine these two unknowns4 ,we find a velocity of from #aj1jjj to #cj1jjj kilometers a second2 as to the ratio of the charge to the mass1 it is very great4 ,we may compare it to the corresponding ratio in regard to the hydrogen ion in electrolysis2 we then find that a cathodic projectile carries about a thousand times more electricity than an equal mass of hydrogen would carry in an electrolyte4 ,to confirm these views1 we need a direct measurement of this velocity to compare with the velocity so calculated4 ,old experiments of ,j4 ,j4 ,thomson had given results more than a hundred times too small2 but they were exposed to certain causes of error4 ,the question was taken up again by ,wiechert in an arrangement where the ,hertzian oscillations were utilized2 results were found agreeing with the theory1 at least as to order of magnitude2 it would be of great interest to repeat these experiments4 ,however that may be1 the theory of undulations appears powerless to account for this complex of facts4 ,the same calculations made with reference to the beta rays of radium have given velocities still greater3 #ajj1jjj or #bjj1jjj kilometers or more yet4 ,these velocities greatly surpass all those we know4 ,it is true that light has long been known to go #cjj1jjj kilometers a second2 but it is not a carrying of matter1 while1 if we adopt the emission theory for the cathode rays1 there would be material molecules really impelled at the velocities in question1 and it is proper to investigate whether the ordinary laws of mechanics are still applicable to them4 ,,ii ,mass ,longitudinal and ,mass ,transversal ,we know that electric currents produce the phenomena of induction1 in particular self-induction4 ,when a current increases1 there develops an electromotive force of self-induction which tends to oppose the current2 on the contrary1 when the current decreases1 the electromotive force of self-induction tends to maintain the current4 ,the self-induction therefore opposes every variation of the intensity of the current1 just as in mechanics the inertia of a body opposes every variation of its velocity4 ,self-induction is a veritable inertia4 ,everything happens as if the current could not establish itself without putting in motion the surrounding ether and as if the inertia of this ether tended1 in consequence1 to keep constant the intensity of this current4 ,it would be requisite to overcome this inertia to establish the current1 it would be necessary to overcome it again to make the current cease4 ,a cathode ray1 which is a rain of projectiles charged with negative electricity1 may be likened to a current2 doubtless this current differs1 at first sight at least1 from the currents of ordinary conduction1 where the matter does not move and where the electricity circulates through the matter4 ,this is a current of convection1 where the electricity1 attached to a material vehicle1 is carried along by the motion of this vehicle4 ,but ,rowland has proved that currents of convection produce the same magnetic effects as currents of conduction2 they should produce also the same effects of induction4 ,first1 if this were not so1 the principle of the conservation of energy would be violated2 besides1 ,crmieu and ,pender have employed a method putting in evidence directly these effects of induction4 ,if the velocity of a cathode corpuscle varies1 the intensity of the corresponding current will likewise vary2 and there will develop effects of self-induction which will tend to oppose this variation4 ,these corpuscles should therefore possess a double inertia3 first their own proper inertia1 and then the apparent inertia1 due to self-induction1 which produces the same effects4 ,they will therefore have a total apparent mass1 composed of their real mass and of a fictitious mass of electromagnetic origin4 ,calculation shows that this fictitious mass varies with the velocity1 and that the force of inertia of self-induction is not the same when the velocity of the projectile accelerates or slackens1 or when it is deviated2 therefore so it is with the force of the total apparent inertia4 ,the total apparent mass is therefore not the same when the real force applied to the corpuscle is parallel to its velocity and tends to accelerate the motion as when it is perpendicular to this velocity and tends to make the direction vary4 ,it is necessary therefore to distinguish the total longitudinal mass from the total transversal mass4 ,these two total masses depend1 moreover1 upon the velocity4 ,this follows from the theoretical work of ,abraham4 ,in the measurements of which we speak in the preceding section1 what is it we determine in measuring the two deviations8 ,it is the velocity on the one hand1 and on the other hand the ratio of the charge to the total transversal mass4 ,how1 under these conditions1 can we make out in this total mass the part of the real mass and that of the fictitious electromagnetic mass8 ,if we had only the cathode rays properly so called1 it could not be dreamed of2 but happily we have the rays of radium which1 as we have seen1 are notably swifter4 ,these rays are not all identical and do not behave in the same way under the action of an electric field and a magnetic field4 ,it is found that the electric deviation is a function of the magnetic deviation1 and we are able1 by receiving on a sensitive plate radium rays which have been subjected to the action of the two fields1 to photograph the curve which represents the relation between these two deviations4 ,this is what ,kaufmann has done1 deducing from it the relation between the velocity and the ratio of the charge to the total apparent mass1 a ratio we shall call epsilon4 ,one might suppose there are several species of rays1 each characterized by a fixed velocity1 by a fixed charge and by a fixed mass4 ,but this hypothesis is improbable2 why1 in fact1 would all the corpuscles of the same mass take always the same velocity8 ,it is more natural to suppose that the charge as well as the real mass are the same for all the projectiles1 and that these differ only by their velocity4 ,if the ratio epsilon is a function of the velocity1 this is not because the real mass varies with this velocity2 but1 since the fictitious electromagnetic mass depends upon this velocity1 the total apparent mass1 alone observable1 must depend upon it1 though the real mass does not depend upon it and may be constant4 ,the calculations of ,abraham let us know the law according to which the fictitious mass varies as a function of the velocity2 ,kaufmann's experiment lets us know the law of variation of the total mass4 ,the comparison of these two laws will enable us therefore to determine the ratio of the real mass to the total mass4 ,such is the method ,kaufmann used to determine this ratio4 ,the result is highly surprising3 the real mass is naught4 ,this has led to conceptions wholly unexpected4 ,what had only been proved for cathode corpuscles was extended to all bodies4 ,what we call mass would be only semblance2 all inertia would be of electromagnetic origin4 ,but then mass would no longer be constant1 it would augment with the velocity2 sensibly constant for velocities up to #a1jjj kilometers a second1 it then would increase and would become infinite for the velocity of light4 ,the transversal mass would no longer be equal to the longitudinal3 they would only be nearly equal if the velocity is not too great4 ,the principle ,b of mechanics would no longer be true4 ,,iii ,the ,canal ,rays ,at the point where we now are1 this conclusion might seem premature4 ,can one apply to all matter what has been proved only for such light corpuscles1 which are a mere emanation of matter and perhaps not true matter8 ,but before entering upon this question1 a word must be said of another sort of rays4 ,i refer to the canal rays1 the ,kanalstrahlen of ,goldstein4 ,the cathode1 together with the cathode rays charged with negative electricity1 emits canal rays charged with positive electricity4 ,in general1 these canal rays not being repelled by the cathode1 are confined to the immediate neighborhood of this cathode1 where they constitute the chamois cushion1' not very easy to perceive2 but1 if the cathode is pierced with holes and if it almost completely blocks up the tube1 the canal rays spread back of the cathode1 in the direction opposite to that of the cathode rays1 and it becomes possible to study them4 ,it is thus that it has been possible to show their positive charge and to show that the magnetic and electric deviations still exist1 as for the cathode rays1 but are much feebler4 ,radium likewise emits rays analogous to the canal rays1 and relatively very absorbable1 called alpha rays4 ,we can1 as for the cathode rays1 measure the two deviations and thence deduce the velocity and the ratio epsilon4 ,the results are less constant than for the cathode rays1 but the velocity is less1 as well as the ratio epsilon2 the positive corpuscles are less charged than the negative2 or if1 which is more natural1 we suppose the charges equal and of opposite sign1 the positive corpuscles are much the larger4 ,these corpuscles1 charged the ones positively1 the others negatively1 have been called electrons4 ,,iv ,the ,theory of ,lorentz ,but the electrons do not merely show us their existence in these rays where they are endowed with enormous velocities4 ,we shall see them in very different rles1 and it is they that account for the principal phenomena of optics and electricity4 ,the brilliant synthesis about to be noticed is due to ,lorentz4 ,matter is formed solely of electrons carrying enormous charges1 and1 if it seems to us neutral1 this is because the charges of opposite sign of these electrons compensate each other4 ,we may imagine1 for example1 a sort of solar system formed of a great positive electron1 around which gravitate numerous little planets1 the negative electrons1 attracted by the electricity of opposite name which charges the central electron4 ,the negative charges of these planets would balance the positive charge of this sun1 so that the algebraic sum of all these charges would be naught4 ,all these electrons swim in the ether4 ,the ether is everywhere identically the same1 and perturbations in it are propagated according to the same laws as light or the ,hertzian oscillations in vacuo4 ,there is nothing but electrons and ether4 ,when a luminous wave enters a part of the ether where electrons are numerous1 these electrons are put in motion under the influence of the perturbation of the ether1 and they then react upon the ether4 ,so would be explained refraction1 dispersion1 double refraction and absorption4 ,just so1 if for any cause an electron be put in motion1 it would trouble the ether around it and would give rise to luminous waves1 and this would explain the emission of light by incandescent bodies4 ,in certain bodies1 the metals for example1 we should have fixed electrons1 between which would circulate moving electrons enjoying perfect liberty1 save that of going out from the metallic body and breaking the surface which separates it from the exterior void or from the air1 or from any other non-metallic body4 ,these movable electrons behave then1 within the metallic body1 as do1 according to the kinetic theory of gases1 the molecules of a gas within the vase where this gas is confined4 ,but1 under the influence of a difference of potential1 the negative movable electrons would tend to go all to one side1 and the positive movable electrons to the other4 ,this is what would produce electric currents1 and this is why these bodies would be conductors4 ,on the other hand1 the velocities of our electrons would be the greater the higher the temperature1 if we accept the assimilation with the kinetic theory of gases4 ,when one of these movable electrons encounters the surface of the metallic body1 whose boundary it can not pass1 it is reflected like a billiard ball which has hit the cushion1 and its velocity undergoes a sudden change of direction4 ,but when an electron changes direction1 as we shall see further on1 it becomes the source of a luminous wave1 and this is why hot metals are incandescent4 ,in other bodies1 the dielectrics and the transparent bodies1 the movable electrons enjoy much less freedom4 ,they remain as if attached to fixed electrons which attract them4 ,the farther they go away from them the greater becomes this attraction and tends to pull them back4 ,they therefore can make only small excursions2 they can no longer circulate1 but only oscillate about their mean position4 ,this is why these bodies would not be conductors2 moreover they would most often be transparent1 and they would be refractive1 since the luminous vibrations would be communicated to the movable electrons1 susceptible of oscillation1 and thence a perturbation would result4 ,i can not here give the details of the calculations2 ,i confine myself to saying that this theory accounts for all the known facts1 and has predicted new ones1 such as the ,zeeman effect4 ,v ,mechanical ,consequences ,we now may face two hypotheses3 #a ,the positive electrons have a real mass1 much greater than their fictitious electromagnetic mass2 the negative electrons alone lack real mass4 ,we might even suppose that apart from electrons of the two signs1 there are neutral atoms which have only their real mass4 ,in this case1 mechanics is not affected2 there is no need of touching its laws2 the real mass is constant2 simply1 motions are deranged by the effects of self-induction1 as has always been known2 moreover1 these perturbations are almost negligible1 except for the negative electrons which1 not having real mass1 are not true matter4 #b ,but there is another point of view2 we may suppose there are no neutral atoms1 and the positive electrons lack real mass just as the negative electrons4 ,but then1 real mass vanishing1 either the word mass will no longer have any meaning1 or else it must designate the fictitious electromagnetic mass2 in this case1 mass will no longer be constant1 the transversal mass will no longer be equal to the longitudinal1 the principles of mechanics will be overthrown4 ,first a word of explanation4 ,we have said that1 for the same charge1 the total mass of a positive electron is much greater than that of a negative4 ,and then it is natural to think that this difference is explained by the positive electron having1 besides its fictitious mass1 a considerable real mass2 which takes us back to the first hypothesis4 ,but we may just as well suppose that the real mass is null for these as for the others1 but that the fictitious mass of the positive electron is much the greater since this electron is much the smaller4 ,i say advisedly3 much the smaller4 ,and1 in fact1 in this hypothesis inertia is exclusively electromagnetic in origin2 it reduces itself to the inertia of the ether2 the electrons are no longer anything by themselves2 they are solely holes in the ether and around which the ether moves2 the smaller these holes are1 the more will there be of ether1 the greater1 consequently1 will be the inertia of the ether4 ,how shall we decide between these two hypotheses8 ,by operating upon the canal rays as ,kaufmann did upon the beta rays8 ,this is impossible2 the velocity of these rays is much too slight4 ,should each therefore decide according to his temperament1 the conservatives going to one side and the lovers of the new to the other8 ,perhaps1 but1 to fully understand the arguments of the innovators1 other considerations must come in4 ,,chapter ,,ii ,,mechanics ,,and ,,optics ,i ,aberration ,you know in what the phenomenon of aberration1 discovered by ,bradley1 consists4 ,the light issuing from a star takes a certain time to go through a telescope2 during this time1 the telescope1 carried along by the motion of the earth1 is displaced4 ,if therefore the telescope were pointed in the true direction of the star1 the image would be formed at the point occupied by the crossing of the threads of the network when the light has reached the objective2 and this crossing would no longer be at this same point when the light reached the plane of the network4 ,we would therefore be led to mis-point the telescope to bring the image upon the crossing of the threads4 ,thence results that the astronomer will not point the telescope in the direction of the absolute velocity of the light1 that is to say toward the true position of the star1 but just in the direction of the relative velocity of the light with reference to the earth1 that is to say toward what is called the apparent position of the star4 ,the velocity of light is known2 we might therefore suppose that we have the means of calculating the absolute velocity of the earth4 7,i shall soon explain my use here of the word absolute47 ,nothing of the sort2 we indeed know the apparent position of the star we observe2 but we do not know its true position2 we know the velocity of the light only in magnitude and not in direction4 ,if therefore the absolute velocity of the earth were straight and uniform1 we should never have suspected the phenomenon of aberration2 but it is variable2 it is composed of two parts3 the velocity of the solar system1 which is straight and uniform2 the velocity of the earth with reference to the sun1 which is variable4 ,if the velocity of the solar system1 that is to say if the constant part existed alone1 the observed direction would be invariable4 ,this position that one would thus observe is called the mean apparent position of the star4 ,taking account now at the same time of the two parts of the velocity of the earth1 we shall have the actual apparent position1 which describes a little ellipse around the mean apparent position1 and it is this ellipse that we observe4 ,neglecting very small quantities1 we shall see that the dimensions of this ellipse depend only upon the ratio of the velocity of the earth with reference to the sun to the velocity of light1 so that the relative velocity of the earth with regard to the sun has alone come in4 ,but wait6 ,this result is not exact1 it is only approximate2 let us push the approximation a little farther4 ,the dimensions of the ellipse will depend then upon the absolute velocity of the earth4 ,let us compare the major axes of the ellipse for the different stars3 we shall have1 theoretically at least1 the means of determining this absolute velocity4 ,that would be perhaps less shocking than it at first seems2 it is a question1 in fact1 not of the velocity with reference to an absolute void1 but of the velocity with regard to the ether1 which is taken by definition as being absolutely at rest4 ,besides1 this method is purely theoretical4 ,in fact1 the aberration is very small2 the possible variations of the ellipse of aberration are much smaller yet1 and1 if we consider the aberration as of the first order1 they should therefore be regarded as of the second order3 about a millionth of a second2 they are absolutely inappreciable for our instruments4 ,we shall finally see1 further on1 why the preceding theory should be rejected1 and why we could not determine this absolute velocity even if our instruments were ten thousand times more precise6 ,one might imagine some other means1 and in fact1 so one has4 ,the velocity of light is not the same in water as in air2 could we not compare the two apparent positions of a star seen through a telescope first full of air1 then full of water8 ,the results have been negative2 the apparent laws of reflection and refraction are not altered by the motion of the earth4 ,this phenomenon is capable of two explanations3 #a ,it might be supposed that the ether is not at rest1 but that it is carried along by the body in motion4 ,it would then not be astonishing that the phenomena of refraction are not altered by the motion of the earth1 since all1 prisms1 telescopes and ether1 are carried along together in the same translation4 ,as to the aberration itself1 it would be explained by a sort of refraction happening at the surface of separation of the ether at rest in the interstellar spaces and the ether carried along by the motion of the earth4 ,it is upon this hypothesis 7bodily carrying along of the ether7 that is founded the theory of ,hertz on the electrodynamics of moving bodies4 #b ,fresnel1 on the contrary1 supposes that the ether is at absolute rest in the void1 at rest almost absolute in the air1 whatever be the velocity of this air1 and that it is partially carried along by refractive media4 ,lorentz has given to this theory a more satisfactory form4 ,for him1 the ether is at rest1 only the electrons are in motion2 in the void1 where it is only a question of the ether1 in the air1 where this is almost the case1 the carrying along is null or almost null2 in refractive media1 where perturbation is produced at the same time by vibrations of the ether and those of electrons put in swing by the agitation of the ether1 the undulations are partially carried along4 ,to decide between the two hypotheses1 we have ,fizeau's experiment1 comparing by measurements of the fringes of interference1 the velocity of light in air at rest or in motion4 ,these experiments have confirmed ,fresnel's hypothesis of partial carrying along4 ,they have been repeated with the same result by ,michelson4 ,the theory of ,hertz must therefore be rejected4 ,,ii ,the ,principle of ,relativity ,but if the ether is not carried along by the motion of the earth1 is it possible to show1 by means of optical phenomena1 the absolute velocity of the earth1 or rather its velocity with respect to the unmoving ether8 ,experiment has answered negatively1 and yet the experimental procedures have been varied in all possible ways4 ,whatever be the means employed there will never be disclosed anything but relative velocities2 ,i mean the velocities of certain material bodies with reference to other material bodies4 ,in fact1 if the source of light and the apparatus of observation are on the earth and participate in its motion1 the experimental results have always been the same1 whatever be the orientation of the apparatus with reference to the orbital motion of the earth4 ,if astronomic aberration happens1 it is because the source1 a star1 is in motion with reference to the observer4 ,the hypotheses so far made perfectly account for this general result1 if we neglect very small quantities of the order of the square of the aberration4 ,the explanation rests upon the notion of local time1 introduced by ,lorentz1 which ,i shall try to make clear4 ,suppose two observers1 placed one at ,a1 the other at ,b1 and wishing to set their watches by means of optical signals4 ,they agree that ,b shall send a signal to ,a when his watch marks an hour determined upon1 and ,a is to put his watch to that hour the moment he sees the signal4 ,if this alone were done1 there would be a systematic error1 because as the light takes a certain time t to go from ,b to ,a1 ,a's watch would be behind ,b's the time t4 ,this error is easily corrected4 ,it suffices to cross the signals4 ,a in turn must signal ,b1 and1 after this new adjustment1 ,b's watch will be behind ,a's the time t4 ,then it will be sufficient to take the arithmetic mean of the two adjustments4 ,but this way of doing supposes that light takes the same time to go from ,a to ,b as to return from ,b to ,a4 ,that is true if the observers are motionless2 it is no longer so if they are carried along in a common translation1 since then ,a1 for example1 will go to meet the light coming from ,b1 while ,b will flee before the light coming from ,a4 ,if therefore the observers are borne along in a common translation and if they do not suspect it1 their adjustment will be defective2 their watches will not indicate the same time2 each will show the local time belonging to the point where it is4 ,the two observers will have no way of perceiving this1 if the unmoving ether can transmit to them only luminous signals all of the same velocity1 and if the other signals they might send are transmitted by media carried along with them in their translation4 ,the phenomenon each observes will be too soon or too late2 it would be seen at the same instant only if the translation did not exist2 but as it will be observed with a watch that is wrong1 this will not be perceived and the appearances will not be altered4 ,it results from this that the compensation is easy to explain so long as we neglect the square of the aberration1 and for a long time the experiments were not sufficiently precise to warrant taking account of it4 ,but the day came when ,michelson imagined a much more delicate procedure3 he made rays interfere which had traversed different courses1 after being reflected by mirrors2 each of the paths approximating a meter and the fringes of interference permitting the recognition of a fraction of a thousandth of a millimeter1 the square of the aberration could no longer be neglected1 and yet the results were still negative4 ,therefore the theory required to be completed1 and it has been by the ,lorentz-,fitzgerald hypothesis4 ,these two physicists suppose that all bodies carried along in a translation undergo a contraction in the sense of this translation1 while their dimensions perpendicular to this translation remain unchanged4 ,this contraction is the same for all bodies2 moreover1 it is very slight1 about one two-hundred-millionth for a velocity such as that of the earth4 ,furthermore our measuring instruments could not disclose it1 even if they were much more precise2 our measuring rods in fact undergo the same contraction as the objects to be measured4 ,if the meter exactly fits when applied to a body1 if we point the body and consequently the meter in the sense of the motion of the earth1 it will not cease to exactly fit in another orientation1 and that although the body and the meter have changed in length as well as orientation1 and precisely because the change is the same for one as for the other4 ,but it is quite different if we measure a length1 not now with a meter1 but by the time taken by light to pass along it1 and this is just what ,michelson has done4 ,a body1 spherical when at rest1 will take thus the form of a flattened ellipsoid of revolution when in motion2 but the observer will always think it spherical1 since he himself has undergone an analogous deformation1 as also all the objects serving as points of reference4 ,on the contrary1 the surfaces of the waves of light1 remaining rigorously spherical1 will seem to him elongated ellipsoids4 ,what happens then8 ,suppose an observer and a source of light carried along together in the translation3 the wave surfaces emanating from the source will be spheres having as centers the successive positions of the source2 the distance from this center to the actual position of the source will be proportional to the time elapsed after the emission1 that is to say to the radius of the sphere4 ,all these spheres are therefore homothetic one to the other1 with relation to the actual position ,s of the source4 ,but1 for our observer1 because of the contraction1 all these spheres will seem elongated ellipsoids1 and all these ellipsoids will moreover be homothetic1 with reference to the point ,s2 the excentricity of all these ellipsoids is the same and depends solely upon the velocity of the earth4 ,we shall so select the law of contraction that the point ,s may be at the focus of the meridian section of the ellipsoid4 ,this time the compensation is rigorous1 and this it is which explains ,michelson's experiment4 ,i have said above that1 according to the ordinary theories1 observations of the astronomic aberration would give us the absolute velocity of the earth1 if our instruments were a thousand times more precise4 ,i must modify this statement4 ,yes1 the observed angles would be modified by the effect of this absolute velocity1 but the graduated circles we use to measure the angles would be deformed by the translation3 they would become ellipses2 thence would result an error in regard to the angle measured1 and this second error would exactly compensate the first4 ,this ,lorentz-,fitzgerald hypothesis seems at first very extraordinary2 all we can say for the moment1 in its favor1 is that it is only the immediate translation of ,michelson's experimental result1 if we define lengths by the time taken by light to run along them4 ,however that may be1 it is impossible to escape the impression that the principle of relativity is a general law of nature1 that one will never be able by any imaginable means to show any but relative velocities1 and ,i mean by that not only the velocities of bodies with reference to the ether1 but the velocities of bodies with regard to one another4 ,too many different experiments have given concordant results for us not to feel tempted to attribute to this principle of relativity a value comparable to that1 for example1 of the principle of equivalence4 ,in any case1 it is proper to see to what consequences this way of looking at things would lead us and then to submit these consequences to the control of experiment4 ,,iii ,the ,principle of ,reaction ,let us see what the principle of the equality of action and reaction becomes in the theory of ,lorentz4 ,consider an electron ,a which for any cause begins to move2 it produces a perturbation in the ether2 at the end of a certain time1 this perturbation reaches another electron ,b1 which will be disturbed from its position of equilibrium4 ,in these conditions there can not be equality between action and reaction1 at least if we do not consider the ether1 but only the electrons1 which alone are observable1 since our matter is made of electrons4 ,in fact it is the electron ,a which has disturbed the electron ,b2 even in case the electron ,b should react upon ,a1 this reaction could be equal to the action1 but in no case simultaneous1 since the electron ,b can begin to move only after a certain time1 necessary for the propagation4 ,submitting the problem to a more exact calculation1 we reach the following result3 ,suppose a ,hertz discharger placed at the focus of a parabolic mirror to which it is mechanically attached2 this discharger emits electromagnetic waves1 and the mirror reflects all these waves in the same direction2 the discharger therefore will radiate energy in a determinate direction4 ,well1 the calculation shows that the discharger recoils like a cannon which has shot out a projectile4 ,in the case of the cannon1 the recoil is the natural result of the equality of action and reaction4 ,the cannon recoils because the projectile upon which it has acted reacts upon it4 ,but here it is no longer the same4 ,what has been sent out is no longer a material projectile3 it is energy1 and energy has no mass3 it has no counterpart4 ,and1 in place of a discharger1 we could have considered just simply a lamp with a reflector concentrating its rays in a single direction4 ,it is true that1 if the energy sent out from the discharger or from the lamp meets a material object1 this object receives a mechanical push as if it had been hit by a real projectile1 and this push will be equal to the recoil of the discharger and of the lamp1 if no energy has been lost on the way and if the object absorbs the whole of the energy4 ,therefore one is tempted to say that there still is compensation between the action and the reaction4 ,but this compensation1 even should it be complete1 is always belated4 ,it never happens if the light1 after leaving its source1 wanders through interstellar spaces without ever meeting a material body2 it is incomplete1 if the body it strikes is not perfectly absorbent4 ,are these mechanical actions too small to be measured1 or are they accessible to experiment8 ,these actions are nothing other than those due to the ,maxwell-,bartholi pressures2 ,maxwell had predicted these pressures from calculations relative to electrostatics and magnetism2 ,bartholi reached the same result by thermodynamic considerations4 ,this is how the tails of comets are explained4 ,little particles detach themselves from the nucleus of the comet2 they are struck by the light of the sun1 which pushes them back as would a rain of projectiles coming from the sun4 ,the mass of these particles is so little that this repulsion sweeps it away against the ,newtonian attraction2 so in moving away from the sun they form the tails4 ,the direct experimental verification was not easy to obtain4 ,the first endeavor led to the construction of the radiometer4 ,but this instrument turns backward1 in the sense opposite to the theoretic sense1 and the explanation of its rotation1 since discovered1 is wholly different4 ,at last success came1 by making the vacuum more complete1 on the one hand1 and on the other by not blackening one of the faces of the paddles and directing a pencil of luminous rays upon one of the faces4 ,the radiometric effects and the other disturbing causes are eliminated by a series of pains-taking precautions1 and one obtains a deviation which is very minute1 but which is1 it would seem1 in conformity with the theory4 ,the same effects of the ,maxwell-,bartholi pressure are forecast likewise by the theory of ,hertz of which we have before spoken1 and by that of ,lorentz4 ,but there is a difference4 ,suppose that the energy1 under the form of light1 for example1 proceeds from a luminous source to any body through a transparent medium4 ,the ,maxwell-,bartholi pressure will act1 not alone upon the source at the departure1 and on the body lit up at the arrival1 but upon the matter of the transparent medium which it traverses4 ,at the moment when the luminous wave reaches a new region of this medium1 this pressure will push forward the matter there distributed and will put it back when the wave leaves this region4 ,so that the recoil of the source has for counterpart the forward movement of the transparent matter which is in contact with this source2 a little later1 the recoil of this same matter has for counterpart the forward movement of the transparent matter which lies a little further on1 and so on4 ,only1 is the compensation perfect8 ,is the action of the ,maxwell-,bartholi pressure upon the matter of the transparent medium equal to its reaction upon the source1 and that whatever be this matter8 ,or is this action by so much the less as the medium is less refractive and more rarefied1 becoming null in the void8 ,if we admit the theory of ,hertz1 who regards matter as mechanically bound to the ether1 so that the ether may be entirely carried along by matter1 it would be necessary to answer yes to the first question and no to the second4 ,there would then be perfect compensation1 as required by the principle of the equality of action and reaction1 even in the least refractive media1 even in the air1 even in the interplanetary void1 where it would suffice to suppose a residue of matter1 however subtile4 ,if on the contrary we admit the theory of ,lorentz1 the compensation1 always imperfect1 is insensible in the air and becomes null in the void4 ,but we have seen above that ,fizeau's experiment does not permit of our retaining the theory of ,hertz2 it is necessary therefore to adopt the theory of ,lorentz1 and consequently to renounce the principle of reaction4 ,,iv ,consequences of the ,principle of ,relativity ,we have seen above the reasons which impel us to regard the principle of relativity as a general law of nature4 ,let us see to what consequences this principle would lead1 should it be regarded as finally demonstrated4 ,first1 it obliges us to generalize the hypothesis of ,lorentz and ,fitzgerald on the contraction of all bodies in the sense of the translation4 ,in particular1 we must extend this hypothesis to the electrons themselves4 ,abraham considered these electrons as spherical and indeformable2 it will be necessary for us to admit that these electrons1 spherical when in repose1 undergo the ,lorentz contraction when in motion and take then the form of flattened ellipsoids4 ,this deformation of the electrons will influence their mechanical properties4 ,in fact ,i have said that the displacement of these charged electrons is a veritable current of convection and that their apparent inertia is due to the self-induction of this current3 exclusively as concerns the negative electrons2 exclusively or not1 we do not yet know1 for the positive electrons4 ,well1 the deformation of the electrons1 a deformation which depends upon their velocity1 will modify the distribution of the electricity upon their surface1 consequently the intensity of the convection current they produce1 consequently the laws according to which the self-induction of this current will vary as a function of the velocity4 ,at this price1 the compensation will be perfect and will conform to the requirements of the principle of relativity1 but only upon two conditions3 #a ,that the positive electrons have no real mass1 but only a fictitious electromagnetic mass2 or at least that their real mass1 if it exists1 is not constant and varies with the velocity according to the same laws as their fictitious mass2 #b ,that all forces are of electromagnetic origin1 or at least that they vary with the velocity according to the same laws as the forces of electromagnetic origin4 ,it still is ,lorentz who has made this remarkable synthesis2 stop a moment and see what follows therefrom4 ,first1 there is no more matter1 since the positive electrons no longer have real mass1 or at least no constant real mass4 ,the present principles of our mechanics1 founded upon the constancy of mass1 must therefore be modified4 ,again1 an electromagnetic explanation must be sought of all the known forces1 in particular of gravitation1 or at least the law of gravitation must be so modified that this force is altered by velocity in the same way as the electromagnetic forces4 ,we shall return to this point4 ,all that appears1 at first sight1 a little artificial4 ,in particular1 this deformation of electrons seems quite hypothetical4 ,but the thing may be presented otherwise1 so as to avoid putting this hypothesis of deformation at the foundation of the reasoning4 ,consider the electrons as material points and ask how their mass should vary as function of the velocity not to contravene the principle of relativity4 ,or1 still better1 ask what should be their acceleration under the influence of an electric or magnetic field1 that this principle be not violated and that we come back to the ordinary laws when we suppose the velocity very slight4 ,we shall find that the variations of this mass1 or of these accelerations1 must be as if the electron underwent the ,lorentz deformation4 ,v ,kaufmann's ,experiment ,we have before us1 then1 two theories3 one where the electrons are indeformable1 this is that of ,abraham2 the other where they undergo the ,lorentz deformation4 ,in both cases1 their mass increases with the velocity1 becoming infinite when this velocity becomes equal to that of light2 but the law of the variation is not the same4 ,the method employed by ,kaufmann to bring to light the law of variation of the mass seems therefore to give us an experimental means of deciding between the two theories4 ,unhappily1 his first experiments were not sufficiently precise for that2 so he decided to repeat them with more precautions1 and measuring with great care the intensity of the fields4 ,under their new form they are in favor of the theory of ,abraham4 ,then the principle of relativity would not have the rigorous value we were tempted to attribute to it2 there would no longer be reason for believing the positive electrons denuded of real mass like the negative electrons4 ,however1 before definitely adopting this conclusion1 a little reflection is necessary4 ,the question is of such importance that it is to be wished ,kaufmann's experiment were repeated by another experimenter4#ag ,unhappily1 this experiment is very delicate and could be carried out successfully only by a physicist of the same ability as ,kaufmann4 ,all precautions have been properly taken and we hardly see what objection could be made4 #ag ,at the moment of going to press we learn that ,m4 ,bucherer has repeated the experiment1 taking new precautions1 and that he has obtained1 contrary to ,kaufmann1 results confirming the views of ,lorentz4 ,there is one point however to which ,i wish to draw attention3 that is to the measurement of the electrostatic field1 a measurement upon which all depends4 ,this field was produced between the two armatures of a condenser2 and1 between these armatures1 there was to be made an extremely perfect vacuum1 in order to obtain a complete isolation4 ,then the difference of potential of the two armatures was measured1 and the field obtained by dividing this difference by the distance apart of the armatures4 ,that supposes the field uniform2 is this certain8 ,might there not be an abrupt fall of potential in the neighborhood of one of the armatures1 of the negative armature1 for example8 ,there may be a difference of potential at the meeting of the metal and the vacuum1 and it may be that this difference is not the same on the positive side and on the negative side2 what would lead me to think so is the electric valve effects between mercury and vacuum4 ,however slight the probability that it is so1 it seems that it should be considered4 ,,vi ,the ,principle of ,inertia ,in the new dynamics1 the principle of inertia is still true1 that is to say that an isolated electron will have a straight and uniform motion4 ,at least this is generally assumed2 however1 ,lindemann has made objections to this view2 ,i do not wish to take part in this discussion1 which ,i can not here expound because of its too difficult character4 ,in any case1 slight modifications to the theory would suffice to shelter it from ,lindemann's objections4 ,we know that a body submerged in a fluid experiences1 when in motion1 considerable resistance1 but this is because our fluids are viscous2 in an ideal fluid1 perfectly free from viscosity1 the body would stir up behind it a liquid hill1 a sort of wake2 upon departure1 a great effort would be necessary to put it in motion1 since it would be necessary to move not only the body itself1 but the liquid of its wake4 ,but1 the motion once acquired1 it would perpetuate itself without resistance1 since the body1 in advancing1 would simply carry with it the perturbation of the liquid1 without the total vis viva of the liquid augmenting4 ,everything would happen therefore as if its inertia was augmented4 ,an electron advancing in the ether would behave in the same way3 around it1 the ether would be stirred up1 but this perturbation would accompany the body in its motion2 so that1 for an observer carried along with the electron1 the electric and magnetic fields accompanying this electron would appear invariable1 and would change only if the velocity of the electron varied4 ,an effort would therefore be necessary to put the electron in motion1 since it would be necessary to create the energy of these fields2 on the contrary1 once the movement acquired1 no effort would be necessary to maintain it1 since the created energy would only have to go along behind the electron as a wake4 ,this energy1 therefore1 could only augment the inertia of the electron1 as the agitation of the liquid augments that of the body submerged in a perfect fluid4 ,and anyhow1 the negative electrons at least have no other inertia except that4 ,in the hypothesis of ,lorentz1 the vis viva1 which is only the energy of the ether1 is not proportional to v#b4 ,doubtless if v is very slight1 the vis viva is sensibly proportional to v#b1 the quantity of motion sensibly proportional to v1 the two masses sensibly constant and equal to each other4 ,but when the velocity tends toward the velocity of light1 the vis viva1 the quantity of motion and the two masses increase beyond all limit4 ,in the hypothesis of ,abraham1 the expressions are a little more complicated2 but what we have just said remains true in essentials4 ,so the mass1 the quantity of motion1 the vis viva become infinite when the velocity is equal to that of light4 ,thence results that no body can attain in any way a velocity beyond that of light4 ,and in fact1 in proportion as its velocity increases1 its mass increases1 so that its inertia opposes to any new increase of velocity a greater and greater obstacle4 ,a question then suggests itself3 let us admit the principle of relativity2 an observer in motion would not have any means of perceiving his own motion4 ,if therefore no body in its absolute motion can exceed the velocity of light1 but may approach it as nearly as you choose1 it should be the same concerning its relative motion with reference to our observer4 ,and then we might be tempted to reason as follows3 ,the observer may attain a velocity of #bjj1jjj kilometers2 the body in its relative motion with reference to the observer may attain the same velocity2 its absolute velocity will then be #djj1jjj kilometers1 which is impossible1 since this is beyond the velocity of light4 ,this is only a seeming1 which vanishes when account is taken of how ,lorentz evaluates local time4 ,,vii ,the ,wave of ,acceleration ,when an electron is in motion1 it produces a perturbation in the ether surrounding it2 if its motion is straight and uniform1 this perturbation reduces to the wake of which we have spoken in the preceding section4 ,but it is no longer the same1 if the motion be curvilinear or varied4 ,the perturbation may then be regarded as the superposition of two others1 to which ,langevin has given the names wave of velocity and wave of acceleration4 ,the wave of velocity is only the wave which happens in uniform motion4 ,as to the wave of acceleration1 this is a perturbation altogether analogous to light waves1 which starts from the electron at the instant when it undergoes an acceleration1 and which is then propagated by successive spherical waves with the velocity of light4 ,whence follows3 in a straight and uniform motion1 the energy is wholly conserved2 but1 when there is an acceleration1 there is loss of energy1 which is dissipated under the form of luminous waves and goes out to infinity across the ether4 ,however1 the effects of this wave of acceleration1 in particular the corresponding loss of energy1 are in most cases negligible1 that is to say not only in ordinary mechanics and in the motions of the heavenly bodies1 but even in the radium rays1 where the velocity is very great without the acceleration being so4 ,we may then confine ourselves to applying the laws of mechanics1 putting the force equal to the product of acceleration by mass1 this mass1 however1 varying with the velocity according to the laws explained above4 ,we then say the motion is quasi-stationary4 ,it would not be the same in all cases where the acceleration is great1 of which the chief are the following3 #a ,in incandescent gases certain electrons take an oscillatory motion of very high frequency2 the displacements are very small1 the velocities are finite1 and the accelerations very great2 energy is then communicated to the ether1 and this is why these gases radiate light of the same period as the oscillations of the electron2 #b ,inversely1 when a gas receives light1 these same electrons are put in swing with strong accelerations and they absorb light2 #c ,in the ,hertz discharger1 the electrons which circulate in the metallic mass undergo1 at the instant of the discharge1 an abrupt acceleration and take then an oscillatory motion of high frequency4 ,thence results that a part of the energy radiates under the form of ,hertzian waves2 #d ,in an incandescent metal1 the electrons enclosed in this metal are impelled with great velocity2 upon reaching the surface of the metal1 which they can not get through1 they are reflected and thus undergo a considerable acceleration4 ,this is why the metal emits light4 ,the details of the laws of the emission of light by dark bodies are perfectly explained by this hypothesis2 #e ,finally when the cathode rays strike the anticathode1 the negative electrons1 constituting these rays1 which are impelled with very great velocity1 are abruptly arrested4 ,because of the acceleration they thus undergo1 they produce undulations in the ether4 ,this1 according to certain physicists1 is the origin of the ,rntgen rays1 which would only be light rays of very short wave-length4 ,,chapter ,,iii ,,the ,,new ,,mechanics ,,and ,,astronomy ,i ,gravitation ,mass may be defined in two ways3 #a ,by the quotient of the force by the acceleration2 this is the true definition of the mass1 which measures the inertia of the body4 #b ,by the attraction the body exercises upon an exterior body1 in virtue of ,newton's law4 ,we should therefore distinguish the mass coefficient of inertia and the mass coefficient of attraction4 ,according to ,newton's law1 there is rigorous proportionality between these two coefficients4 ,but that is demonstrated only for velocities to which the general principles of dynamics are applicable4 ,now1 we have seen that the mass coefficient of inertia increases with the velocity2 should we conclude that the mass coefficient of attraction increases likewise with the velocity and remains proportional to the coefficient of inertia1 or1 on the contrary1 that this coefficient of attraction remains constant8 ,this is a question we have no means of deciding4 ,on the other hand1 if the coefficient of attraction depends upon the velocity1 since the velocities of two bodies which mutually attract are not in general the same1 how will this coefficient depend upon these two velocities8 ,upon this subject we can only make hypotheses1 but we are naturally led to investigate which of these hypotheses would be compatible with the principle of relativity4 ,there are a great number of them2 the only one of which ,i shall here speak is that of ,lorentz1 which ,i shall briefly expound4 ,consider first electrons at rest4 ,two electrons of the same sign repel each other and two electrons of contrary sign attract each other2 in the ordinary theory1 their mutual actions are proportional to their electric charges2 if therefore we have four electrons1 two positive ,a and ,a'1 and two negative ,b and ,b'1 the charges of these four being the same in absolute value1 the repulsion of ,a for ,a' will be1 at the same distance1 equal to the repulsion of ,b for ,b' and equal also to the attraction of ,a for ,b'1 or of ,a' for ,b4 ,if therefore ,a and ,b are very near each other1 as also ,a' and ,b'1 and we examine the action of the system ,a ,b upon the system ,a' ,b'1 we shall have two repulsions and two attractions which will exactly compensate each other and the resulting action will be null4 ,now1 material molecules should just be regarded as species of solar systems where circulate the electrons1 some positive1 some negative1 and in such a way that the algebraic sum of all the charges is null4 ,a material molecule is therefore wholly analogous to the system ,a ,b of which we have spoken1 so that the total electric action of two molecules one upon the other should be null4 ,but experiment shows us that these molecules attract each other in consequence of ,newtonian gravitation2 and then we may make two hypotheses3 we may suppose gravitation has no relation to the electrostatic attractions1 that it is due to a cause entirely different1 and is simply something additional2 or else we may suppose the attractions are not proportional to the charges and that the attraction exercised by a charge #a upon a charge -#a is greater than the mutual repulsion of two #a charges1 or two -#a charges4 ,in other words1 the electric field produced by the positive electrons and that which the negative electrons produce might be superposed and yet remain distinct4 ,the positive electrons would be more sensitive to the field produced by the negative electrons than to the field produced by the positive electrons2 the contrary would be the case for the negative electrons4 ,it is clear that this hypothesis somewhat complicates electrostatics1 but that it brings back into it gravitation4 ,this was1 in sum1 ,franklin's hypothesis4 ,what happens now if the electrons are in motion8 ,the positive electrons will cause a perturbation in the ether and produce there an electric and magnetic field4 ,the same will be the case for the negative electrons4 ,the electrons1 positive as well as negative1 undergo then a mechanical impulsion by the action of these different fields4 ,in the ordinary theory1 the electromagnetic field1 due to the motion of the positive electrons1 exercises1 upon two electrons of contrary sign and of the same absolute charge1 equal actions with contrary sign4 ,we may then without inconvenience not distinguish the field due to the motion of the positive electrons and the field due to the motion of the negative electrons and consider only the algebraic sum of these two fields1 that is to say the resulting field4 ,in the new theory1 on the contrary1 the action upon the positive electrons of the electromagnetic field due to the positive electrons follows the ordinary laws2 it is the same with the action upon the negative electrons of the field due to the negative electrons4 ,let us now consider the action of the field due to the positive electrons upon the negative electrons 7or inversely72 it will still follow the same laws1 but with a different coefficient4 ,each electron is more sensitive to the field created by the electrons of contrary name than to the field created by the electrons of the same name4 ,such is the hypothesis of ,lorentz1 which reduces to ,franklin's hypothesis for slight velocities2 it will therefore explain1 for these small velocities1 ,newton's law4 ,moreover1 as gravitation goes back to forces of electrodynamic origin1 the general theory of ,lorentz will apply1 and consequently the principle of relativity will not be violated4 ,we see that ,newton's law is no longer applicable to great velocities and that it must be modified1 for bodies in motion1 precisely in the same way as the laws of electrostatics for electricity in motion4 ,we know that electromagnetic perturbations spread with the velocity of light4 ,we may therefore be tempted to reject the preceding theory upon remembering that gravitation spreads1 according to the calculations of ,laplace1 at least ten million times more quickly than light1 and that consequently it can not be of electromagnetic origin4 ,the result of ,laplace is well known1 but one is generally ignorant of its signification4 ,laplace supposed that1 if the propagation of gravitation is not instantaneous1 its velocity of spread combines with that of the body attracted1 as happens for light in the phenomenon of astronomic aberration1 so that the effective force is not directed along the straight joining the two bodies1 but makes with this straight a small angle4 ,this is a very special hypothesis1 not well justified1 and1 in any case1 entirely different from that of ,lorentz4 ,laplace's result proves nothing against the theory of ,lorentz4 ,,ii ,comparison with ,astronomic ,observations ,can the preceding theories be reconciled with astronomic observations8 ,first of all1 if we adopt them1 the energy of the planetary motions will be constantly dissipated by the effect of the wave of acceleration4 ,from this would result that the mean motions of the stars would constantly accelerate1 as if these stars were moving in a resistant medium4 ,but this effect is exceedingly slight1 far too much so to be discerned by the most precise observations4 ,the acceleration of the heavenly bodies is relatively slight1 so that the effects of the wave of acceleration are negligible and the motion may be regarded as quasi stationary4 ,it is true that the effects of the wave of acceleration constantly accumulate1 but this accumulation itself is so slow that thousands of years of observation would be necessary for it to become sensible4 ,let us therefore make the calculation considering the motion as quasi-stationary1 and that under the three following hypotheses3 ,a4 ,admit the hypothesis of ,abraham 7electrons indeformable7 and retain ,newton's law in its usual form2 ,b4 ,admit the hypothesis of ,lorentz about the deformation of electrons and retain the usual ,newton's law2 ,c4 ,admit the hypothesis of ,lorentz about electrons and modify ,newton's law as we have done in the preceding paragraph1 so as to render it compatible with the principle of relativity4 ,it is in the motion of ,mercury that the effect will be most sensible1 since this planet has the greatest velocity4 ,tisserand formerly made an analogous calculation1 admitting ,weber's law2 ,i recall that ,weber had sought to explain at the same time the electrostatic and electrodynamic phenomena in supposing that electrons 7whose name was not yet invented7 exercise1 one upon another1 attractions and repulsions directed along the straight joining them1 and depending not only upon their distances1 but upon the first and second derivatives of these distances1 consequently upon their velocities and their accelerations4 ,this law of ,weber1 different enough from those which to-day tend to prevail1 none the less presents a certain analogy with them4 ,tisserand found that1 if the ,newtonian attraction conformed to ,weber's law there resulted1 for ,mercury's perihelion1 secular variation of #ad1 of the same sense as that which has been observed and could not be explained1 but smaller1 since this is #ch4 ,let us recur to the hypotheses ,a1 ,b and ,c1 and study first the motion of a planet attracted by a fixed center4 ,the hypotheses ,b and ,c are no longer distinguished1 since1 if the attracting point is fixed1 the field it produces is a purely electrostatic field1 where the attraction varies inversely as the square of the distance1 in conformity with ,coulomb's electrostatic law1 identical with that of ,newton4 ,the vis viva equation holds good1 taking for vis viva the new definition2 in the same way1 the equation of areas is replaced by another equivalent to it2 the moment of the quantity of motion is a constant1 but the quantity of motion must be defined as in the new dynamics4 ,the only sensible effect will be a secular motion of the perihelion4 ,with the theory of ,lorentz1 we shall find1 for this motion1 half of what ,weber's law would give2 with the theory of ,abraham1 two fifths4 ,if now we suppose two moving bodies gravitating around their common center of gravity1 the effects are very little different1 though the calculations may be a little more complicated4 ,the motion of ,mercury's perihelion would therefore be #g in the theory of ,lorentz and #e4f in that of ,abraham4 ,the effect moreover is proportional to 7n#c77;a#b71 where n is the star's mean motion and a the radius of its orbit4 ,for the planets1 in virtue of ,kepler's law1 the effect varies then inversely as sqrt7a#e72 it is therefore insensible1 save for ,mercury4 ,it is likewise insensible for the moon though n is great1 because a is extremely small2 in sum1 it is five times less for ,venus1 and six hundred times less for the moon than for ,mercury4 ,we may add that as to ,venus and the earth1 the motion of the perihelion 7for the same angular velocity of this motion7 would be much more difficult to discern by astronomic observations1 because the excentricity of their orbits is much less than for ,mercury4 ,to sum up1 the only sensible effect upon astronomic observations would be a motion of ,mercury's perihelion1 in the same sense as that which has been observed without being explained1 but notably slighter4 ,that can not be regarded as an argument in favor of the new dynamics1 since it will always be necessary to seek another explanation for the greater part of ,mercury's anomaly2 but still less can it be regarded as an argument against it4 ,,iii ,the ,theory of ,lesage ,it is interesting to compare these considerations with a theory long since proposed to explain universal gravitation4 ,suppose that1 in the interplanetary spaces1 circulate in every direction1 with high velocities1 very tenuous corpuscles4 ,a body isolated in space will not be affected1 apparently1 by the impacts of these corpuscles1 since these impacts are equally distributed in all directions4 ,but if two bodies ,a and ,b are present1 the body ,b will play the rle of screen and will intercept part of the corpuscles which1 without it1 would have struck ,a4 ,then1 the impacts received by ,a in the direction opposite that from ,b will no longer have a counterpart1 or will now be only partially compensated1 and this will push ,a toward ,b4 ,such is the theory of ,lesage2 and we shall discuss it1 taking first the view-point of ordinary mechanics4 ,first1 how should the impacts postulated by this theory take place2 is it according to the laws of perfectly elastic bodies1 or according to those of bodies devoid of elasticity1 or according to an intermediate law8 ,the corpuscles of ,lesage can not act as perfectly elastic bodies2 otherwise the effect would be null1 since the corpuscles intercepted by the body ,b would be replaced by others which would have rebounded from ,b1 and calculation proves that the compensation would be perfect4 ,it is necessary then that the impact make the corpuscles lose energy1 and this energy should appear under the form of heat4 ,but how much heat would thus be produced8 ,note that attraction passes through bodies2 it is necessary therefore to represent to ourselves the earth1 for example1 not as a solid screen1 but as formed of a very great number of very small spherical molecules1 which play individually the rle of little screens1 but between which the corpuscles of ,lesage may freely circulate4 ,so1 not only the earth is not a solid screen1 but it is not even a cullender1 since the voids occupy much more space than the plenums4 ,to realize this1 recall that ,laplace has demonstrated that attraction1 in traversing the earth1 is weakened at most by one ten-millionth part1 and his proof is perfectly satisfactory3 in fact1 if attraction were absorbed by the body it traverses1 it would no longer be proportional to the masses2 it would be relatively weaker for great bodies than for small1 since it would have a greater thickness to traverse4 ,the attraction of the sun for the earth would therefore be relatively weaker than that of the sun for the moon1 and thence would result1 in the motion of the moon1 a very sensible inequality4 ,we should therefore conclude1 if we adopt the theory of ,lesage1 that the total surface of the spherical molecules which compose the earth is at most the ten-millionth part of the total surface of the earth4 ,darwin has proved that the theory of ,lesage only leads exactly to ,newton's law when we postulate particles entirely devoid of elasticity4 ,the attraction exerted by the earth on a mass #a at a distance #a will then be proportional1 at the same time1 to the total surface ,s of the spherical molecules composing it1 to the velocity v of the corpuscles1 to the square root of the density rho of the medium formed by the corpuscles4 ,the heat produced will be proportional to ,s1 to the density rho1 and to the cube of the velocity v4 ,but it is necessary to take account of the resistance experienced by a body moving in such a medium2 it can not move1 in fact1 without going against certain impacts1 in fleeing1 on the contrary1 before those coming in the opposite direction1 so that the compensation realized in the state of rest can no longer subsist4 ,the calculated resistance is proportional to ,s1 to rho and to v2 now1 we know that the heavenly bodies move as if they experienced no resistance1 and the precision of observations permits us to fix a limit to the resistance of the medium4 ,this resistance varying as ,srhov1 while the attraction varies as ,ssqrt7rhov71 we see that the ratio of the resistance to the square of the attraction is inversely as the product ,sv4 ,we have therefore a lower limit of the product ,sv4 ,we have already an upper limit of ,s 7by the absorption of attraction by the body it traverses72 we have therefore a lower limit of the velocity v1 which must be at least #bdajag times that of light4 ,from this we are able to deduce rho and the quantity of heat produced2 this quantity would suffice to raise the temperature #ajbf degrees a second2 the earth would receive in a given time #ajbj times more heat than the sun emits in the same time2 ,i am not speaking of the heat the sun sends to the earth1 but of that it radiates in all directions4 ,it is evident the earth could not long stand such a rgime4 ,we should not be led to results less fantastic if1 contrary to ,darwin's views1 we endowed the corpuscles of ,lesage with an elasticity imperfect without being null4 ,in truth1 the vis viva of these corpuscles would not be entirely converted into heat1 but the attraction produced would likewise be less1 so that it would be only the part of this vis viva converted into heat1 which would contribute to produce the attraction and that would come to the same thing2 a judicious employment of the theorem of the viriel would enable us to account for this4 ,the theory of ,lesage may be transformed2 suppress the corpuscles and imagine the ether overrun in all senses by luminous waves coming from all points of space4 ,when a material object receives a luminous wave1 this wave exercises upon it a mechanical action due to the ,maxwell-,bartholi pressure1 just as if it had received the impact of a material projectile4 ,the waves in question could therefore play the rle of the corpuscles of ,lesage4 ,this is what is supposed1 for example1 by ,m4 ,tommasina4 ,the difficulties are not removed for all that2 the velocity of propagation can be only that of light1 and we are thus led1 for the resistance of the medium1 to an inadmissible figure4 ,besides1 if the light is all reflected1 the effect is null1 just as in the hypothesis of the perfectly elastic corpuscles4 ,that there should be attraction1 it is necessary that the light be partially absorbed2 but then there is production of heat4 ,the calculations do not differ essentially from those made in the ordinary theory of ,lesage1 and the result retains the same fantastic character4 ,on the other hand1 attraction is not absorbed by the body it traverses1 or hardly at all2 it is not so with the light we know4 ,light which would produce the ,newtonian attraction would have to be considerably different from ordinary light and be1 for example1 of very short wave length4 ,this does not count that1 if our eyes were sensible of this light1 the whole heavens should appear to us much more brilliant than the sun1 so that the sun would seem to us to stand out in black1 otherwise the sun would repel us instead of attracting us4 ,for all these reasons1 light which would permit of the explanation of attraction would be much more like ,rntgen rays than like ordinary light4 ,and besides1 the ,x-rays would not suffice2 however penetrating they may seem to us1 they could not pass through the whole earth2 it would be necessary therefore to imagine ,x'-rays much more penetrating than the ordinary ,x-rays4 ,moreover a part of the energy of these ,x'-rays would have to be destroyed1 otherwise there would be no attraction4 ,if you do not wish it transformed into heat1 which would lead to an enormous heat production1 you must suppose it radiated in every direction under the form of secondary rays1 which might be called ,x'' and which would have to be much more penetrating still than the ,x'-rays1 otherwise they would in their turn derange the phenomena of attraction4 ,such are the complicated hypotheses to which we are led when we try to give life to the theory of ,lesage4 ,but all we have said presupposes the ordinary laws of mechanics4 ,will things go better if we admit the new dynamics8 ,and first1 can we conserve the principles of relativity8 ,let us give at first to the theory of ,lesage its primitive form1 and suppose space ploughed by material corpuscles2 if these corpuscles were perfectly elastic1 the laws of their impact would conform to this principle of relativity1 but we know that then their effect would be null4 ,we must therefore suppose these corpuscles are not elastic1 and then it is difficult to imagine a law of impact compatible with the principle of relativity4 ,besides1 we should still find a production of considerable heat1 and yet a very sensible resistance of the medium4 ,if we suppress these corpuscles and revert to the hypothesis of the ,maxwell-,bartholi pressure1 the difficulties will not be less4 ,this is what ,lorentz himself has attempted in his ,memoir to the ,amsterdam ,academy of ,sciences of ,april #be1 #aijj4 ,consider a system of electrons immersed in an ether permeated in every sense by luminous waves2 one of these electrons1 struck by one of these waves1 begins to vibrate2 its vibration will be synchronous with that of light2 but it may have a difference of phase1 if the electron absorbs a part of the incident energy4 ,in fact1 if it absorbs energy1 this is because the vibration of the ether impels the electron2 the electron must therefore be slower than the ether4 ,an electron in motion is analogous to a convection current2 therefore every magnetic field1 in particular that due to the luminous perturbation itself1 must exert a mechanical action upon this electron4 ,this action is very slight2 moreover1 it changes sign in the current of the period2 nevertheless1 the mean action is not null if there is a difference of phase between the vibrations of the electron and those of the ether4 ,the mean action is proportional to this difference1 consequently to the energy absorbed by the electron4 ,i can not here enter into the detail of the calculations2 suffice it to say only that the final result is an attraction of any two electrons1 varying inversely as the square of the distance and proportional to the energy absorbed by the two electrons4 ,therefore there can not be attraction without absorption of light and1 consequently1 without production of heat1 and this it is which determined ,lorentz to abandon this theory1 which1 at bottom1 does not differ from that of ,lesage-,maxwell-,bartholi4 ,he would have been much more dismayed still if he had pushed the calculation to the end4 ,he would have found that the temperature of the earth would have to increase #ajab degrees a second4 ,,iv ,conclusions ,i have striven to give in few words an idea as complete as possible of these new doctrines2 ,i have sought to explain how they took birth2 otherwise the reader would have had ground to be frightened by their boldness4 ,the new theories are not yet demonstrated2 far from it2 only they rest upon an aggregate of probabilities sufficiently weighty for us not to have the right to treat them with disregard4 ,new experiments will doubtless teach us what we should finally think of them4 ,the knotty point of the question lies in ,kaufmann's experiment and those that may be undertaken to verify it4 ,in conclusion1 permit me a word of warning4 ,suppose that1 after some years1 these theories undergo new tests and triumph2 then our secondary education will incur a great danger2 certain professors will doubtless wish to make a place for the new theories4 ,novelties are so attractive1 and it is so hard not to seem highly advanced6 ,at least there will be the wish to open vistas to the pupils and1 before teaching them the ordinary mechanics1 to let them know it has had its day and was at best good enough for that old dolt ,laplace4 ,and then they will not form the habit of the ordinary mechanics4 ,is it well to let them know this is only approximative8 ,yes2 but later1 when it has penetrated to their very marrow1 when they shall have taken the bent of thinking only through it1 when there shall no longer be risk of their unlearning it1 then one may1 without inconvenience1 show them its limits4 ,it is with the ordinary mechanics that they must live2 this alone will they ever have to apply4 ,whatever be the progress of automobilism1 our vehicles will never attain speeds where it is not true4 ,the other is only a luxury1 and we should think of the luxury only when there is no longer any risk of harming the necessary4 ,,book ,,iv ,,astronomic ,,science ,,chapter ,i ,,the ,,milky ,,way ,,and ,,the ,,theory ,,of ,,gases ,the considerations to be here developed have scarcely as yet drawn the attention of astronomers2 there is hardly anything to cite except an ingenious idea of ,lord ,kelvin's1 which has opened a new field of research1 but still waits to be followed out4 ,nor have ,i original results to impart1 and all ,i can do is to give an idea of the problems presented1 but which no one hitherto has undertaken to solve4 ,every one knows how a large number of modern physicists represent the constitution of gases2 gases are formed of an innumerable multitude of molecules which1 at high speeds1 cross and crisscross in every direction4 ,these molecules probably act at a distance one upon another1 but this action decreases very rapidly with distance1 so that their trajectories remain sensibly straight2 they cease to be so only when two molecules happen to pass very near to each other2 in this case1 their mutual attraction or repulsion makes them deviate to right or left4 ,this is what is sometimes called an impact2 but the word impact is not to be understood in its usual sense2 it is not necessary that the two molecules come into contact1 it suffices that they approach sufficiently near each other for their mutual attractions to become sensible4 ,the laws of the deviation they undergo are the same as for a veritable impact4 ,it seems at first that the disorderly impacts of this innumerable dust can engender only an inextricable chaos before which analysis must recoil4 ,but the law of great numbers1 that supreme law of chance1 comes to our aid2 in presence of a semi-disorder1 we must despair1 but in extreme disorder1 this statistical law reestablishes a sort of mean order where the mind can recover4 ,it is the study of this mean order which constitutes the kinetic theory of gases2 it shows us that the velocities of the molecules are equally distributed among all the directions1 that the rapidity of these velocities varies from one molecule to another1 but that even this variation is subject to a law called ,maxwell's law4 ,this law tells us how many of the molecules move with such and such a velocity4 ,as soon as the gas departs from this law1 the mutual impacts of the molecules1 in modifying the rapidity and direction of their velocities1 tend to bring it promptly back4 ,physicists have striven1 not without success1 to explain in this way the experimental properties of gases2 for example ,mariotte's law4 ,consider now the milky way2 there also we see an innumerable dust2 only the grains of this dust are not atoms1 they are stars2 these grains move also with high velocities2 they act at a distance one upon another1 but this action is so slight at great distance that their trajectories are straight2 and yet1 from time to time1 two of them may approach near enough to be deviated from their path1 like a comet which has passed too near ,jupiter4 ,in a word1 to the eyes of a giant for whom our suns would be as for us our atoms1 the milky way would seem only a bubble of gas4 ,such was ,lord ,kelvin's leading idea4 ,what may be drawn from this comparison8 ,in how far is it exact8 ,this is what we are to investigate together2 but before reaching a definite conclusion1 and without wishing to prejudge it1 we foresee that the kinetic theory of gases will be for the astronomer a model he should not follow blindly1 but from which he may advantageously draw inspiration4 ,up to the present1 celestial mechanics has attacked only the solar system or certain systems of double stars4 ,before the assemblage presented by the milky way1 or the agglomeration of stars1 or the resolvable nebulae it recoils1 because it sees therein only chaos4 ,but the milky way is not more complicated than a gas2 the statistical methods founded upon the calculus of probabilities applicable to a gas are also applicable to it4 ,before all1 it is important to grasp the resemblance of the two cases1 and their difference4 ,lord ,kelvin has striven to determine in this manner the dimensions of the milky way2 for that we are reduced to counting the stars visible in our telescopes2 but we are not sure that behind the stars we see1 there are not others we do not see2 so that what we should measure in this way would not be the size of the milky way1 it would be the range of our instruments4 ,the new theory comes to offer us other resources4 ,in fact1 we know the motions of the stars nearest us1 and we can form an idea of the rapidity and direction of their velocities4 ,if the ideas above set forth are exact1 these velocities should follow ,maxwell's law1 and their mean value will tell us1 so to speak1 that which corresponds to the temperature of our fictitious gas4 ,but this temperature depends itself upon the dimensions of our gas bubble4 ,in fact1 how will a gaseous mass let loose in the void act1 if its elements attract one another according to ,newton's law8 ,it will take a spherical form2 moreover1 because of gravitation1 the density will be greater at the center1 the pressure also will increase from the surface to the center because of the weight of the outer parts drawn toward the center2 finally1 the temperature will increase toward the center3 the temperature and the pressure being connected by the law called adiabatic1 as happens in the successive layers of our atmosphere4 ,at the surface itself1 the pressure will be null1 and it will be the same with the absolute temperature1 that is to say with the velocity of the molecules4 ,a question comes here3 ,i have spoken of the adiabatic law1 but this law is not the same for all gases1 since it depends upon the ratio of their two specific heats2 for the air and like gases1 this ratio is #a4db2 but is it to air that it is proper to liken the milky way8 ,evidently not1 it should be regarded as a mono-atomic gas1 like mercury vapor1 like argon1 like helium1 that is to say that the ratio of the specific heats should be taken equal to #a4ff4 ,and1 in fact1 one of our molecules would be for example the solar system2 but the planets are very small personages1 the sun alone counts1 so that our molecule is indeed mono-atomic4 ,and even if we take a double star1 it is probable that the action of a strange star which might approach it would become sufficiently sensible to deviate the motion of general translation of the system much before being able to trouble the relative orbits of the two components2 the double star1 in a word1 would act like an indivisible atom4 ,however that may be1 the pressure1 and consequently the temperature1 at the center of the gaseous sphere would be by so much the greater as the sphere was larger since the pressure increases by the weight of all the superposed layers4 ,we may suppose that we are nearly at the center of the milky way1 and by observing the mean proper velocity of the stars1 we shall know that which corresponds to the central temperature of our gaseous sphere and we shall determine its radius4 ,we may get an idea of the result by the following considerations3 make a simpler hypothesis3 the milky way is spherical1 and in it the masses are distributed in a homogeneous manner2 thence results that the stars in it describe ellipses having the same center4 ,if we suppose the velocity becomes nothing at the surface1 we may calculate this velocity at the center by the equation of vis viva4 ,thus we find that this velocity is proportional to the radius of the sphere and to the square root of its density4 ,if the mass of this sphere was that of the sun and its radius that of the terrestrial orbit1 this velocity would be 7it is easy to see7 that of the earth in its orbit4 ,but in the case we have supposed1 the mass of the sun should be distributed in a sphere of radius #a1jjj1jjj times greater1 this radius being the distance of the nearest stars2 the density is therefore #ajah times less2 now1 the velocities are of the same order1 therefore it is necessary that the radius be #aji times greater1 be #a1jjj times the distance of the nearest stars1 which would give about a thousand millions of stars in the milky way4 ,but you will say these hypothesis differ greatly from the reality2 first1 the milky way is not spherical and we shall soon return to this point1 and again the kinetic theory of gases is not compatible with the hypothesis of a homogeneous sphere4 ,but in making the exact calculation according to this theory1 we should find a different result1 doubtless1 but of the same order of magnitude2 now in such a problem the data are so uncertain that the order of magnitude is the sole end to be aimed at4 ,and here a first remark presents itself2 ,lord ,kelvin's result1 which ,i have obtained again by an approximative calculation1 agrees sensibly with the evaluations the observers have made with their telescopes2 so that we must conclude we are very near to piercing through the milky way4 ,but that enables us to answer another question4 ,there are the stars we see because they shine2 but may there not be dark stars circulating in the interstellar spaces whose existence might long remain unknown8 ,but then1 what ,lord ,kelvin's method would give us would be the total number of stars1 including the dark stars2 as his figure is comparable to that the telescope gives1 this means there is no dark matter1 or at least not so much as of shining matter4 ,before going further1 we must look at the problem from another angle4 ,is the milky way thus constituted truly the image of a gas properly so called8 ,you know ,crookes has introduced the notion of a fourth state of matter1 where gases having become too rarefied are no longer true gases and become what he calls radiant matter4 ,considering the slight density of the milky way1 is it the image of gaseous matter or of radiant matter8 ,the consideration of what is called the free path will furnish us the answer4 ,the trajectory of a gaseous molecule may be regarded as formed of straight segments united by very small arcs corresponding to the successive impacts4 ,the length of each of these segments is what is called the free path2 of course this length is not the same for all the segments and for all the molecules2 but we may take a mean2 this is what is called the mean path4 ,this is the greater the less the density of the gas4 ,the matter will be radiant if the mean path is greater than the dimensions of the receptacle wherein the gas is enclosed1 so that a molecule has a chance to go across the whole receptacle without undergoing an impact2 if the contrary be the case1 it is gaseous4 ,from this it follows that the same fluid may be radiant in a little receptacle and gaseous in a big one2 this perhaps is why1 in a ,crookes tube1 it is necessary to make the vacuum by so much the more complete as the tube is larger4 ,how is it then for the milky way8 ,this is a mass of gas of which the density is very slight1 but whose dimensions are very great2 has a star chances of traversing it without undergoing an impact1 that is to say without passing sufficiently near another star to be sensibly deviated from its route6 ,what do we mean by sufficiently near8 ,that is perforce a little arbitrary2 take it as the distance from the sun to ,neptune1 which would represent a deviation of a dozen degrees2 suppose therefore each of our stars surrounded by a protective sphere of this radius2 could a straight pass between these spheres8 ,at the mean distance of the stars of the milky way1 the radius of these spheres will be seen under an angle of about a tenth of a second2 and we have a thousand millions of stars4 ,put upon the celestial sphere a thousand million little circles of a tenth of a second radius4 ,are the chances that these circles will cover a great number of times the celestial sphere8 ,far from it2 they will cover only its sixteen thousandth part4 ,so the milky way is not the image of gaseous matter1 but of ,crookes' radiant matter4 ,nevertheless1 as our foregoing conclusions are happily not at all precise1 we need not sensibly modify them4 ,but there is another difficulty3 the milky way is not spherical1 and we have reasoned hitherto as if it were1 since this is the form of equilibrium a gas isolated in space would take4 ,to make amends1 agglomerations of stars exist whose form is globular and to which would better apply what we have hitherto said4 ,herschel has already endeavored to explain their remarkable appearances4 ,he supposed the stars of the aggregates uniformly distributed1 so that an assemblage is a homogeneous sphere2 each star would then describe an ellipse and all these orbits would be passed over in the same time1 so that at the end of a period the aggregate would take again its primitive configuration and this configuration would be stable4 ,unluckily1 the aggregates do not appear to be homogeneous2 we see a condensation at the center1 we should observe it even were the sphere homogeneous1 since it is thicker at the center2 but it would not be so accentuated4 ,we may therefore rather compare an aggregate to a gas in adiabatic equilibrium1 which takes the spherical form because this is the figure of equilibrium of a gaseous mass4 ,but1 you will say1 these aggregates are much smaller than the milky way1 of which they even in probability make part1 and even though they be more dense1 they will rather present something analogous to radiant matter2 now1 gases attain their adiabatic equilibrium only through innumerable impacts of the molecules4 ,that might perhaps be adjusted4 ,suppose the stars of the aggregate have just enough energy for their velocity to become null when they reach the surface2 then they may traverse the aggregate without impact1 but arrived at the surface they will go back and will traverse it anew2 after a great number of crossings1 they will at last be deviated by an impact2 under these conditions1 we should still have a matter which might be regarded as gaseous2 if perchance there had been in the aggregate stars whose velocity was greater1 they have long gone away out of it1 they have left it never to return4 ,for all these reasons1 it would be interesting to examine the known aggregates1 to seek to account for the law of the densities1 and to see if it is the adiabatic law of gases4 ,but to return to the milky way2 it is not spherical and would rather be represented as a flattened disc4 ,it is clear then that a mass starting without velocity from the surface will reach the center with different velocities1 according as it starts from the surface in the neighborhood of the middle of the disc or just on the border of the disc2 the velocity would be notably greater in the latter case4 ,now1 up to the present1 we have supposed that the proper velocities of the stars1 those we observe1 must be comparable to those which like masses would attain2 this involves a certain difficulty4 ,we have given above a value for the dimensions of the milky way1 and we have deduced it from the observed proper velocities which are of the same order of magnitude as that of the earth in its orbit2 but which is the dimension we have thus measured8 ,is it the thickness8 ,is it the radius of the disc8 ,it is doubtless something intermediate2 but what can we say then of the thickness itself1 or of the radius of the disc8 ,data are lacking to make the calculation2 ,i shall confine myself to giving a glimpse of the possibility of basing an evaluation at least approximate upon a deeper discussion of the proper motions4 ,and then we find ourselves facing two hypotheses3 either the stars of the milky way are impelled by velocities for the most part parallel to the galactic plane1 but otherwise distributed uniformly in all directions parallel to this plane4 ,if this be so1 observation of the proper motions should show a preponderance of components parallel to the milky way2 this is to be determined1 because ,i do not know whether a systematic discussion has ever been made from this view-point4 ,on the other hand1 such an equilibrium could only be provisory1 since because of impacts the molecules1 ,i mean the stars1 would in the long run acquire notable velocities in the sense perpendicular to the milky way and would end by swerving from its plane1 so that the system would tend toward the spherical form1 the only figure of equilibrium of an isolated gaseous mass4 ,or else the whole system is impelled by a common rotation1 and for that reason is flattened like the earth1 like ,jupiter1 like all bodies that twirl4 ,only1 as the flattening is considerable1 the rotation must be rapid2 rapid doubtless1 but it must be understood in what sense this word is used4 ,the density of the milky way is #ajbc times less than that of the sun2 a velocity of rotation sqrt7#ajbe7 times less than that of the sun1 for it would1 therefore1 be the equivalent so far as concerns flattening2 a velocity #ajab times slower than that of the earth1 say a thirtieth of a second of arc in a century1 would be a very rapid rotation1 almost too rapid for stable equilibrium to be possible4 ,in this hypothesis1 the observable proper motions would appear to us uniformly distributed1 and there would no longer be a preponderance of components parallel to the galactic plane4 ,they will tell us nothing about the rotation itself1 since we belong to the turning system4 ,if the spiral nebul are other milky ways1 foreign to ours1 they are not borne along in this rotation1 and we might study their proper motions4 ,it is true they are very far away2 if a nebula has the dimensions of the milky way and if its apparent radius is for example #bj1 its distance is #aj1jjj times the radius of the milky way4 ,but that makes no difference1 since it is not about the translation of our system that we ask information from them1 but about its rotation4 ,the fixed stars1 by their apparent motion1 reveal to us the diurnal rotation of the earth1 though their distance is immense4 ,unluckily1 the possible rotation of the milky way1 however rapid it may be relatively1 is very slow viewed absolutely1 and besides the pointings on nebul can not be very precise2 therefore thousands of years of observations would be necessary to learn anything4 ,however that may be1 in this second hypothesis1 the figure of the milky way would be a figure of final equilibrium4 ,i shall not further discuss the relative value of these two hypotheses since there is a third which is perhaps more probable4 ,we know that among the irresolvable nebul1 several kinds may be distinguished3 the irregular nebul like that of ,orion1 the planetary and annular nebul1 the spiral nebul4 ,the spectra of the first two families have been determined1 they are discontinuous2 these nebul are therefore not formed of stars2 besides1 their distribution on the heavens seems to depend upon the milky way2 whether they have a tendency to go away from it1 or on the contrary to approach it1 they make therefore a part of the system4 ,on the other hand1 the spiral nebul are generally considered as independent of the milky way2 it is supposed that they1 like it1 are formed of a multitude of stars1 that they are1 in a word1 other milky ways very far away from ours4 ,the recent investigations of ,stratonoff tend to make us regard the milky way itself as a spiral nebula1 and this is the third hypothesis of which ,i wish to speak4 ,how can we explain the very singular appearances presented by the spiral nebul1 which are too regular and too constant to be due to chance8 ,first of all1 to take a look at one of these representations is enough to see that the mass is in rotation2 we may even see what the sense of the rotation is2 all the spiral radii are curved in the same sense2 it is evident that the moving wing lags behind the pivot and that fixes the sense of the rotation4 ,but this is not all2 it is evident that these nebul can not be likened to a gas at rest1 nor even to a gas in relative equilibrium under the sway of a uniform rotation2 they are to be compared to a gas in permanent motion in which internal currents prevail4 ,suppose1 for example1 that the rotation of the central nucleus is rapid 7you know what ,i mean by this word71 too rapid for stable equilibrium2 then at the equator the centrifugal force will drive it away over the attraction1 and the stars will tend to break away at the equator and will form divergent currents2 but in going away1 as their moment of rotation remains constant1 while the radius vector augments1 their angular velocity will diminish1 and this is why the moving wing seems to lag back4 ,from this point of view1 there would not be a real permanent motion1 the central nucleus would constantly lose matter which would go out of it never to return1 and would drain away progressively4 ,but we may modify the hypothesis4 ,in proportion as it goes away1 the star loses its velocity and ends by stopping2 at this moment attraction regains possession of it and leads it back toward the nucleus2 so there will be centripetal currents4 ,we must suppose the centripetal currents are the first rank and the centrifugal currents the second rank1 if we adopt the comparison with a troop in battle executing a change of front2 and1 in fact1 it is necessary that the composite centrifugal force be compensated by the attraction exercised by the central layers of the swarm upon the extreme layers4 ,besides1 at the end of a certain time a permanent rgime establishes itself2 the swarm being curved1 the attraction exercised upon the pivot by the moving wing tends to slow up the pivot and that of the pivot upon the moving wing tends to accelerate the advance of this wing which no longer augments its lag1 so that finally all the radii end by turning with a uniform velocity4 ,we may still suppose that the rotation of the nucleus is quicker than that of the radii4 ,a question remains2 why do these centripetal and centrifugal swarms tend to concentrate themselves in radii instead of disseminating themselves a little everywhere8 ,why do these rays distribute themselves regularly8 ,if the swarms concentrate themselves1 it is because of the attraction exercised by the already existing swarms upon the stars which go out from the nucleus in their neighborhood4 ,after an inequality is produced1 it tends to accentuate itself in this way4 ,why do the rays distribute themselves regularly8 ,that is less obvious4 ,suppose there is no rotation1 that all the stars are in two planes at right angles1 in such a way that their distribution is symmetric with regard to these two planes4 ,by symmetry1 there would be no reason for their going out of these planes1 nor for the symmetry changing4 ,this configuration would give us therefore equilibrium1 but this would be an unstable equilibrium4 ,if on the contrary1 there is rotation1 we shall find an analogous configuration of equilibrium with four curved rays1 equal to each other and intersecting at #ij1 and if the rotation is sufficiently rapid1 this equilibrium is stable4 ,i am not in position to make this more precise3 enough if you see that these spiral forms may perhaps some day be explained by only the law of gravitation and statistical consideration recalling those of the theory of gases4 ,what has been said of internal currents shows it is of interest to discuss systematically the aggregate of proper motions2 this may be done in a hundred years1 when the second edition is issued of the chart of the heavens and compared with the first1 that we now are making4 ,but1 in conclusion1 ,i wish to call your attention to a question1 that of the age of the milky way or the nebul4 ,if what we think we see is confirmed1 we can get an idea of it4 ,that sort of statistical equilibrium of which gases give us the model is established only in consequence of a great number of impacts4 ,if these impacts are rare1 it can come about only after a very long time2 if really the milky way 7or at least the agglomerations which are contained in it71 if the nebul have attained this equilibrium1 this means they are very old1 and we shall have an inferior limit of their age4 ,likewise we should have of it a superior limit2 this equilibrium is not final and can not last always4 ,our spiral nebul would be comparable to gases impelled by permanent motions2 but gases in motion are viscous and their velocities end by wearing out4 ,what here corresponds to the viscosity 7and which depends upon the chances of impact of the molecules7 is excessively slight1 so that the present rgime may persist during an extremely long time1 yet not forever1 so that our milky ways can not live eternally nor become infinitely old4 ,and this is not all4 ,consider our atmosphere3 at the surface must reign a temperature infinitely small and the velocity of the molecules there is near zero4 ,but this is a question only of the mean velocity2 as a consequence of impacts1 one of these molecules may acquire 7rarely1 it is true7 an enormous velocity1 and then it will rush out of the atmosphere1 and once out1 it will never return2 therefore our atmosphere drains off thus with extreme slowness4 ,the milky way also from time to time loses a star by the same mechanism1 and that likewise limits its duration4 ,well1 it is certain that if we compute in this manner the age of the milky way1 we shall get enormous figures4 ,but here a difficulty presents itself4 ,certain physicists1 relying upon other considerations1 reckon that suns can have only an ephemeral existence1 about fifty million years2 our minimum would be much greater than that4 ,must we believe that the evolution of the milky way began when the matter was still dark8 ,but how have the stars composing it reached all at the same time adult age1 an age so briefly to endure8 ,or must they reach there all successively1 and are those we see only a feeble minority compared with those extinguished or which shall one day light up8 ,but how reconcile that with what we have said above on the absence of a noteworthy proportion of dark matter8 ,should we abandon one of the two hypotheses1 and which8 ,i confine myself to pointing out the difficulty without pretending to solve it2 ,i shall end therefore with a big interrogation point4 ,however1 it is interesting to set problems1 even when their solution seems very far away4 ,,chapter ,,ii ,,french ,,geodesy ,every one understands our interest in knowing the form and dimensions of our earth2 but some persons will perhaps be surprised at the exactitude sought after4 ,is this a useless luxury8 ,what good are the efforts so expended by the geodesist8 ,should this question be put to a congressman1 ,i suppose he would say3 ,i am led to believe that geodesy is one of the most useful of the sciences2 because it is one of those costing us most dear4 ,i shall try to give you an answer a little more precise4 ,the great works of art1 those of peace as well as those of war1 are not to be undertaken without long studies which save much groping1 miscalculation and useless expense4 ,these studies can only be based upon a good map4 ,but a map will be only a valueless phantasy if constructed without basing it upon a solid framework4 ,as well make stand a human body minus the skeleton4 ,now1 this framework is given us by geodesic measurements2 so1 without geodesy1 no good map2 without a good map1 no great public works4 ,these reasons will doubtless suffice to justify much expense2 but these are arguments for practical men4 ,it is not upon these that it is proper to insist here2 there are others higher and1 everything considered1 more important4 ,so we shall put the question otherwise2 can geodesy aid us the better to know nature8 ,does it make us understand its unity and harmony8 ,in reality an isolated fact is of slight value1 and the conquests of science are precious only if they prepare for new conquests4 ,if therefore a little hump were discovered on the terrestrial ellipsoid1 this discovery would be by itself of no great interest4 ,on the other hand1 it would become precious if1 in seeking the cause of this hump1 we hoped to penetrate new secrets4 ,well1 when1 in the eighteenth century1 ,maupertuis and ,la ,condamine braved such opposite climates1 it was not solely to learn the shape of our planet1 it was a question of the whole world-system4 ,if the earth was flattened1 ,newton triumphed and with him the doctrine of gravitation and the whole modern celestial mechanics4 ,and to-day1 a century and a half after the victory of the ,newtonians1 think you geodesy has nothing more to teach us8 ,we know not what is within our globe4 ,the shafts of mines and borings have let us know a layer of #a or #b kilometers thickness1 that is to say1 the millionth part of the total mass2 but what is beneath8 ,of all the extraordinary journeys dreamed by ,jules ,verne1 perhaps that to the center of the earth took us to regions least explored4 ,but these deep-lying rocks we can not reach1 exercise from afar their attraction which operates upon the pendulum and deforms the terrestrial spheroid4 ,geodesy can therefore weigh them from afar1 so to speak1 and tell us of their distribution4 ,thus will it make us really see those mysterious regions which ,jules ,verne only showed us in imagination4 ,this is not an empty illusion4 ,m4 ,faye1 comparing all the measurements1 has reached a result well calculated to surprise us4 ,under the oceans1 in the depths1 are rocks of very great density2 under the continents1 on the contrary1 are empty spaces4 ,new observations will modify perhaps the details of these conclusions4 ,in any case1 our venerated dean has shown us where to search and what the geodesist may teach the geologist1 desirous of knowing the interior constitution of the earth1 and even the thinker wishing to speculate upon the past and the origin of this planet4 ,and now1 why have ,i entitled this chapter ,french ,geodesy8 ,it is because1 in each country1 this science has taken1 more than all others1 perhaps1 a national character4 ,it is easy to see why4 ,there must be rivalry4 ,the scientific rivalries are always courteous1 or at least almost always2 in any case1 they are necessary1 because they are always fruitful4 ,well1 in those enterprises which require such long efforts and so many collaborators1 the individual is effaced1 in spite of himself1 of course2 no one has the right to say3 this is my work4 ,therefore it is not between men1 but between nations that rivalries go on4 ,so we are led to seek what has been the part of ,france4 ,her part ,i believe we are right to be proud of4 ,at the beginning of the eighteenth century1 long discussions arose between the ,newtonians who believed the earth flattened1 as the theory of gravitation requires1 and ,cassini1 who1 deceived by inexact measurements1 believed our globe elongated4 ,only direct observation could settle the question4 ,it was our ,academy of ,sciences that undertook this task1 gigantic for the epoch4 ,while ,maupertuis and ,clairaut measured a degree of meridian under the polar circle1 ,bouguer and ,la ,condamine went toward the ,andes ,mountains1 in regions then under ,spain which to-day are the ,republic of ,ecuador4 ,our envoys were exposed to great hardships4 ,traveling was not as easy as at present4 ,truly1 the country where ,maupertuis operated was not a desert and he even enjoyed1 it is said1 among the ,laplanders those sweet satisfactions of the heart that real arctic voyagers never know4 ,it was almost the region where1 in our days1 comfortable steamers carry1 each summer1 hosts of tourists and young ,english people4 ,but in those days ,cook's agency did not exist and ,maupertuis really believed he had made a polar expedition4 ,perhaps he was not altogether wrong4 ,the ,russians and the ,swedes carry out to-day analogous measurements at ,spitzbergen1 in a country where there is real ice-cap4 ,but they have quite other resources1 and the difference of time makes up for that of latitude4 ,the name of ,maupertuis has reached us much scratched by the claws of ,doctor ,akakia2 the scientist had the misfortune to displease ,voltaire1 who was then the king of mind4 ,he was first praised beyond measure2 but the flatteries of kings are as much to be dreaded as their displeasure1 because the days after are terrible4 ,voltaire himself knew something of this4 ,voltaire called ,maupertuis1 my amiable master in thinking1 marquis of the polar circle1 dear flattener out of the world and ,cassini1 and even1 flattery supreme1 ,sir ,isaac ,maupertuis2 he wrote him3 ,only the king of ,prussia do ,i put on a level with you2 he only lacks being a geometer4 ,but soon the scene changes1 he no longer speaks of deifying him1 as in days of yore the ,argonauts1 or of calling down from ,olympus the council of the gods to contemplate his works1 but of chaining him up in a madhouse4 ,he speaks no longer of his sublime mind1 but of his despotic pride1 plated with very little science and much absurdity4 ,i care not to relate these comico-heroic combats2 but permit me some reflections on two of ,voltaire's verses4 ,in his ',discourse on ,moderation' 7no question of moderation in praise and criticism71 the poet has written3 ,you have confirmed in regions drear ,what ,newton discerned without going abroad4 ,these two verses 7which replace the hyperbolic praises of the first period7 are very unjust1 and doubtless ,voltaire was too enlightened not to know it4 ,then1 only those discoveries were esteemed which could be made without leaving one's house4 ,to-day1 it would rather be theory that one would make light of4 ,this is to misunderstand the aim of science4 ,is nature governed by caprice1 or does harmony rule there8 ,that is the question4 ,it is when it discloses to us this harmony that science is beautiful and so worthy to be cultivated4 ,but whence can come to us this revelation1 if not from the accord of a theory with experiment8 ,to seek whether this accord exists or if it fails1 this therefore is our aim4 ,consequently these two terms1 which we must compare1 are as indispensable the one as the other4 ,to neglect one for the other would be nonsense4 ,isolated1 theory would be empty1 experiment would be blind2 each would be useless and without interest4 ,maupertuis therefore deserves his share of glory4 ,truly1 it will not equal that of ,newton1 who had received the spark divine2 nor even that of his collaborator ,clairaut4 ,yet it is not to be despised1 because his work was necessary1 and if ,france1 outstripped by ,england in the seventeenth century1 has so well taken her revenge in the century following1 it is not alone to the genius of ,clairauts1 d',alemberts1 ,laplaces that she owes it2 it is also to the long patience of the ,maupertuis and the ,la ,condamines4 ,we reach what may be called the second heroic period of geodesy4 ,france is torn within4 ,all ,europe is armed against her2 it would seem that these gigantic combats might absorb all her forces4 ,far from it2 she still has them for the service of science4 ,the men of that time recoiled before no enterprise1 they were men of faith4 ,delambre and ,mchain were commissioned to measure an arc going from ,dunkerque to ,barcelona4 ,this time there was no going to ,lapland or to ,peru2 the hostile squadrons had closed to us the ways thither4 ,but1 though the expeditions are less distant1 the epoch is so troubled that the obstacles1 the perils even1 are just as great4 ,in ,france1 ,delambre had to fight against the ill-will of suspicious municipalities4 ,one knows that the steeples1 which are visible from so far1 and can be aimed at with precision1 often serve as signal points to geodesists4 ,but in the region ,delambre traversed there were no longer any steeples4 ,a certain proconsul had passed there1 and boasted of knocking down all the steeples rising proudly above the humble abode of the sans-culottes4 ,pyramids then were built of planks and covered with white cloth to make them more visible4 ,that was quite another thing3 with white cloth6 ,what was this rash person who1 upon our heights so recently set free1 dared to raise the hateful standard of the counter-revolution8 ,it was necessary to border the white cloth with blue and red bands4 ,mchain operated in ,spain2 the difficulties were other2 but they were not less4 ,the ,spanish peasants were hostile4 ,there steeples were not lacking3 but to install oneself in them with mysterious and perhaps diabolic instruments1 was it not sacrilege8 ,the revolutionists were allies of ,spain1 but allies smelling a little of the stake4 ,without cease1 writes ,mchain1 they threaten to butcher us4 ,fortunately1 thanks to the exhortations of the priests1 to the pastoral letters of the bishops1 these ferocious ,spaniards contented themselves with threatening4 ,some years after ,mchain made a second expedition into ,spain3 he proposed to prolong the meridian from ,barcelona to the ,balearics4 ,this was the first time it had been attempted to make the triangulations overpass a large arm of the sea by observing signals installed upon some high mountain of a far-away isle4 ,the enterprise was well conceived and well prepared2 it failed however4 ,the ,french scientist encountered all sorts of difficulties of which he complains bitterly in his correspondence4 ,hell1 he writes1 perhaps with some exaggeration--hell and all the scourges it vomits upon the earth1 tempests1 war1 the plague and black intrigues are therefore unchained against me6 ,the fact is that he encountered among his collaborators more of proud obstinacy than of good will and that a thousand accidents retarded his work4 ,the plague was nothing1 the fear of the plague was much more redoubtable2 all these isles were on their guard against the neighboring isles and feared lest they should receive the scourge from them4 ,mchain obtained permission to disembark only after long weeks upon the condition of covering all his papers with vinegar2 this was the antisepsis of that time4 ,disgusted and sick1 he had just asked to be recalled1 when he died4 ,arago and ,biot it was who had the honor of taking up the unfinished work and carrying it on to completion4 ,thanks to the support of the ,spanish government1 to the protection of several bishops and1 above all1 to that of a famous brigand chief1 the operations went rapidly forward4 ,they were successfully completed1 and ,biot had returned to ,france when the storm burst4 ,it was the moment when all ,spain took up arms to defend her independence against ,france4 ,why did this stranger climb the mountains to make signals8 ,it was evidently to call the ,french army4 ,arago was able to escape the populace only by becoming a prisoner4 ,in his prison1 his only distraction was reading in the ,spanish papers the account of his own execution4 ,the papers of that time sometimes gave out news prematurely4 ,he had at least the consolation of learning that he died with courage and like a ,christian4 ,even the prison was no longer safe2 he had to escape and reach ,algiers4 ,there1 he embarked for ,marseilles on an ,algerian vessel4 ,this ship was captured by a ,spanish corsair1 and behold ,arago carried back to ,spain and dragged from dungeon to dungeon1 in the midst of vermin and in the most shocking wretchedness4 ,if it had only been a question of his subjects and his guests1 the dey would have said nothing4 ,but there were on board two lions1 a present from the ,african sovereign to ,napoleon4 ,the dey threatened war4 ,the vessel and the prisoners were released4 ,the port should have been properly reached1 since they had on board an astronomer2 but the astronomer was seasick1 and the ,algerian seamen1 who wished to make ,marseilles1 came out at ,bougie4 ,thence ,arago went to ,algiers1 traversing ,kabylia on foot in the midst of a thousand perils4 ,he was long detained in ,africa and threatened with the convict prison4 ,finally he was able to get back to ,france2 his observations1 which he had preserved and safeguarded under his shirt1 and1 what is still more remarkable1 his instruments had traversed unhurt these terrible adventures4 ,up to this point1 not only did ,france hold the foremost place1 but she occupied the stage almost alone4 ,in the years which follow1 she has not been inactive and our staff-office map is a model4 ,however1 the new methods of observation and calculation have come to us above all from ,germany and ,england4 ,it is only in the last forty years that ,france has regained her rank4 ,she owes it to a scientific officer1 ,general ,perrier1 who has successfully executed an enterprise truly audacious1 the junction of ,spain and ,africa4 ,stations were installed on four peaks upon the two sides of the ,mediterranean4 ,for long months they awaited a calm and limpid atmosphere4 ,at last was seen the little thread of light which had traversed #cjj kilometers over the sea4 ,the undertaking had succeeded4 ,to-day have been conceived projects still more bold4 ,from a mountain near ,nice will be sent signals to ,corsica1 not now for geodesic determinations1 but to measure the velocity of light4 ,the distance is only #bjj kilometers2 but the ray of light is to make the journey there and return1 after reflection by a mirror installed in ,corsica4 ,and it should not wander on the way1 for it must return exactly to the point of departure4 ,ever since1 the activity of ,french geodesy has never slackened4 ,we have no more such astonishing adventures to tell2 but the scientific work accomplished is immense4 ,the territory of ,france beyond the sea1 like that of the mother country1 is covered by triangles measured with precision4 ,we have become more and more exacting and what our fathers admired does not satisfy us to-day4 ,but in proportion as we seek more exactitude1 the difficulties greatly increase2 we are surrounded by snares and must be on our guard against a thousand unsuspected causes of error4 ,it is needful1 therefore1 to create instruments more and more faultless4 ,here again ,france has not let herself be distanced4 ,our appliances for the measurement of bases and angles leave nothing to desire1 and1 ,i may also mention the pendulum of ,colonel ,defforges1 which enables us to determine gravity with a precision hitherto unknown4 ,the future of ,french geodesy is at present in the hands of the ,geographic ,service of the army1 successively directed by ,general ,bassot and ,general ,berthaut4 ,we can not sufficiently congratulate ourselves upon it4 ,for success in geodesy1 scientific aptitudes are not enough2 it is necessary to be capable of standing long fatigues in all sorts of climates2 the chief must be able to win obedience from his collaborators and to make obedient his native auxiliaries4 ,these are military qualities4 ,besides1 one knows that1 in our army1 science has always marched shoulder to shoulder with courage4 ,i add that a military organization assures the indispensable unity of action4 ,it would be more difficult to reconcile the rival pretensions of scientists jealous of their independence1 solicitous of what they call their fame1 and who yet must work in concert1 though separated by great distances4 ,among the geodesists of former times there were often discussions1 of which some aroused long echoes4 ,the ,academy long resounded with the quarrel of ,bouguer and ,la ,condamine4 ,i do not mean to say that soldiers are exempt from passion1 but discipline imposes silence upon a too sensitive self-esteem4 ,several foreign governments have called upon our officers to organize their geodesic service3 this is proof that the scientific influence of ,france abroad has not declined4 ,our hydrographic engineers contribute also to the common achievement a glorious contingent4 ,the survey of our coasts1 of our colonies1 the study of the tides1 offer them a vast domain of research4 ,finally ,i may mention the general leveling of ,france which is carried out by the ingenious and precise methods of ,m4 ,lallemand4 ,with such men we are sure of the future4 ,moreover1 work for them will not be lacking2 our colonial empire opens for them immense expanses illy explored4 ,that is not all3 the ,international ,geodetic ,association has recognized the necessity of a new measurement of the arc of ,quito1 determined in days of yore by ,la ,condamine4 ,it is ,france that has been charged with this operation2 she had every right to it1 since our ancestors had made1 so to speak1 the scientific conquest of the ,cordilleras4 ,besides1 these rights have not been contested and our government has undertaken to exercise them4 ,captains ,maurain and ,lacombe completed a first reconnaissance1 and the rapidity with which they accomplished their mission1 crossing the roughest regions and climbing the most precipitous summits1 is worthy of all praise4 ,it won the admiration of ,general ,alfaro1 ,president of the ,republic of ,ecuador1 who called them 'los hombres de hierro1' the men of iron4 ,the final commission then set out under the command of ,lieutenant-,colonel 7then ,major7 ,bourgeois4 ,the results obtained have justified the hopes entertained4 ,but our officers have encountered unforeseen difficulties due to the climate4 ,more than once1 one of them has been forced to remain several months at an altitude of #d1jjj meters1 in the clouds and the snow1 without seeing anything of the signals he had to aim at and which refused to unmask themselves4 ,but thanks to their perseverance and courage1 there resulted from this only a delay and an increase of expense1 without the exactitude of the measurements suffering therefrom4 ,,general ,,conclusions ,what ,i have sought to explain in the preceding pages is how the scientist should guide himself in choosing among the innumerable facts offered to his curiosity1 since indeed the natural limitations of his mind compel him to make a choice1 even though a choice be always a sacrifice4 ,i have expounded it first by general considerations1 recalling on the one hand the nature of the problem to be solved and on the other hand seeking to better comprehend that of the human mind1 which is the principal instrument of the solution4 ,i then have explained it by examples2 ,i have not multiplied them indefinitely2 ,i also have had to make a choice1 and ,i have chosen naturally the questions ,i had studied most4 ,others would doubtless have made a different choice2 but what difference1 because ,i believe they would have reached the same conclusions4 ,there is a hierarchy of facts2 some have no reach2 they teach us nothing but themselves4 ,the scientist who has ascertained them has learned nothing but a fact1 and has not become more capable of foreseeing new facts4 ,such facts1 it seems1 come once1 but are not destined to reappear4 ,there are1 on the other hand1 facts of great yield2 each of them teaches us a new law4 ,and since a choice must be made1 it is to these that the scientist should devote himself4 ,doubtless this classification is relative and depends upon the weakness of our mind4 ,the facts of slight outcome are the complex facts1 upon which various circumstances may exercise a sensible influence1 circumstances too numerous and too diverse for us to discern them all4 ,but ,i should rather say that these are the facts we think complex1 since the intricacy of these circumstances surpasses the range of our mind4 ,doubtless a mind vaster and finer than ours would think differently of them4 ,but what matter2 we can not use that superior mind1 but only our own4 ,the facts of great outcome are those we think simple2 may be they really are so1 because they are influenced only by a small number of well-defined circumstances1 may be they take on an appearance of simplicity because the various circumstances upon which they depend obey the laws of chance and so come to mutually compensate4 ,and this is what happens most often4 ,and so we have been obliged to examine somewhat more closely what chance is4 ,facts where the laws of chance apply become easy of access to the scientist who would be discouraged before the extraordinary complication of the problems where these laws are not applicable4 ,we have seen that these considerations apply not only to the physical sciences1 but to the mathematical sciences4 ,the method of demonstration is not the same for the physicist and the mathematician4 ,but the methods of invention are very much alike4 ,in both cases they consist in passing up from the fact to the law1 and in finding the facts capable of leading to a law4 ,to bring out this point1 ,i have shown the mind of the mathematician at work1 and under three forms3 the mind of the mathematical inventor and creator2 that of the unconscious geometer who among our far distant ancestors1 or in the misty years of our infancy1 has constructed for us our instinctive notion of space2 that of the adolescent to whom the teachers of secondary education unveil the first principles of the science1 seeking to give understanding of the fundamental definitions4 ,everywhere we have seen the rle of intuition and of the spirit of generalization without which these three stages of mathematicians1 if ,i may so express myself1 would be reduced to an equal impotence4 ,and in the demonstration itself1 the logic is not all2 the true mathematical reasoning is a veritable induction1 different in many regards from the induction of physics1 but proceeding like it from the particular to the general4 ,all the efforts that have been made to reverse this order and to carry back mathematical induction to the rules of logic have eventuated only in failures1 illy concealed by the employment of a language inaccessible to the uninitiated4 ,the examples ,i have taken from the physical sciences have shown us very different cases of facts of great outcome4 ,an experiment of ,kaufmann on radium rays revolutionizes at the same time mechanics1 optics and astronomy4 ,why8 ,because in proportion as these sciences have developed1 we have the better recognized the bonds uniting them1 and then we have perceived a species of general design of the chart of universal science4 ,there are facts common to several sciences1 which seem the common source of streams diverging in all directions and which are comparable to that knoll of ,saint ,gothard whence spring waters which fertilize four different valleys4 ,and then we can make choice of facts with more discernment than our predecessors who regarded these valleys as distinct and separated by impassable barriers4 ,it is always simple facts which must be chosen1 but among these simple facts we must prefer those which are situated upon these sorts of knolls of ,saint ,gothard of which ,i have just spoken4 ,and when sciences have no direct bond1 they still mutually throw light upon one another by analogy4 ,when we studied the laws obeyed by gases we knew we had attacked a fact of great outcome2 and yet this outcome was still estimated beneath its value1 since gases are1 from a certain point of view1 the image of the milky way1 and those facts which seemed of interest only for the physicist1 ere long opened new vistas to astronomy quite unexpected4 ,and finally when the geodesist sees it is necessary to move his telescope some seconds to see a signal he has set up with great pains1 this is a very small fact2 but this is a fact of great outcome1 not only because this reveals to him the existence of a small protuberance upon the terrestrial globe1 that little hump would be by itself of no great interest1 but because this protuberance gives him information about the distribution of matter in the interior of the globe1 and through that about the past of our planet1 about its future1 about the laws of its development4 99 99 99 99 99 ,,index aberration of light1 #cae1 #dif ,abraham1 #caa1 #dij-a1 #eje-g1 #eji1 #eae-f absolute motion1 #ajg orientation1 #hc space1 #he1 #ic1 #bdf1 #beg1 #cec acceleration1 #id1 #ih1 #dhf1 #eji accidental constant1 #aab errors1 #aga1 #djb accommodation of the eyes1 #fg-h action at a distance1 #acg addition1 #cd aim of mathematics1 #bhj alchemists1 #aa ,alfaro1 #edc algebra1 #cgi analogy1 #bbj analysis1 #bah-i1 #bgi analysis situs1 #ec1 #bci1 #cha analyst1 #baj1 #bba ancestral experience1 #ia ,andrade1 #ic1 #ajd1 #bbh ,andrews1 #aec angle sum of triangle1 #eh ,anglo-,saxons1 #c antinomies1 #ddi1 #deg1 #dgg ,arago1 #edj-a ,aristotle1 #bje1 #bib1 #dfj arithmetic1 #cd1 #cgi1 #dda1 #dfc associativity1 #ce assumptions1 #dea1 #dec astronomy1 #ha1 #bhi1 #cae1 #eab ,atwood1 #ddf axiom1 #fj1 #fc1 #fe1 #bae ,bacon1 #abh ,bartholi1 #ejc ,bassot1 #edb beauty1 #cdi1 #cfh ,becquerel1 #cab ,beltrami1 #ef1 #eh ,bergson1 #cba ,berkeley1 #d ,berthaut1 #edb ,bertrand1 #aef1 #aij1 #baa1 #cie ,betti1 #bci ,biot1 #edj bodies1 solid1 #gb ,boltzmann1 #cjd ,bolyai1 #ef1 #bja1 #bjc ,borel1 #dhb ,bouguer1 #ecg1 #edb ,bourgeois1 #edc ,boutroux1 #cij1 #dfd ,bradley1 #dif ,briot1 #bih ,brownian movement1 #aeb1 #daj ,bucherer1 #ejg ,burali-,forti1 #deg-i1 #dgg1 #dha-b ,caen1 #chg-h ,calinon1 #bbh canal rays1 #dia-b canals1 semicircular1 #bgf ,cantor1 #aa1 #ddh-i1 #deg1 #dei1 #dgg ,cantorism1 #cha1 #chb1 #dhj1 #dhd capillarity1 #bih ,carlyle1 #abh ,carnot's principle1 #adc1 #aea1 #cjj1 #cjc-e1 #cii ,cassini1 #ecg cathode rays1 #dhg-ib cells1 #bag center of gravity1 #ajc central forces1 #big ,chaldeans1 #bij chance1 #cie1 #djh change of position1 #gj state1 #gj chemistry of the stars1 #bie circle-squarers1 #aa ,clairaut1 #ecg-h ,clausius1 #aai1 #abc1 #adc color sensation1 #beb ,columbus1 #bbh commutativity1 #ce-f compensation1 #gb complete induction1 #dj ,comte1 #bid ,condorcet1 #daa contingence1 #cdj continuity1 #agc continuum1 #dc amorphous1 #bch mathematical1 #df physical1 #df1 #bdj tridimensional1 #bdj convention1 #ej1 #ic1 #ajf1 #abe1 #agc1 #bjh1 #cag1 #ddj1 #dea convergence1 #fg-h coordinates1 #bdd ,copernicus1 #aji1 #bia1 #ced ,coulomb1 #adc1 #eaf ,couturat1 #dej1 #dec1 #def1 #dfj1 #dfb-c1 #dfg1 #dgb-f creation1 mathematical1 #chc creed1 #a ,crmieu1 #afh-i1 #dij crisis1 #cjc ,crookes1 #aie1 #dhh1 #ebg-h crude fact1 #cbf1 #ccj ,curie1 #cab-c1 #cah current1 #ahf curvature1 #eh-i curve1 #bac1 #cdf curves without tangents1 #ea cut1 #eb1 #bef cyclones1 #cec d',alembert1 #ech ,darwin1 #eah-i ,de ,cyon1 #bgf1 #dbg ,dedekind1 #dd-e ,defforges1 #edb definitions1 #dcj1 #dec deformation1 #gc1 #dae ,delage1 #bgg ,delambre1 #eci ,delbeuf1 #dad ,descartes1 #abg determinism1 #abc1 #cdj dictionary1 #ei didymium1 #ccc dilatation1 #gf dimensions1 #ec1 #fh1 #gh1 #bda1 #bef1 #dbf direction1 #fi ,dirichlet1 #bac dispersion1 #ada displacement1 #gc1 #gg1 #bdg1 #bef distance1 #ei1 #bib distributivity1 #cf ,du ,bois-,reymond1 #ej earth1 rotation of1 #cbf1 #cec eclipse1 #cbf electricity1 #agd electrified bodies1 #aag electrodynamic attraction1 #cjh induction1 #ahh mass1 #caa electrodynamics1 #ahd1 #bhb electromagnetic theory of light1 #cja electrons1 #caf1 #dib-d1 #eje-h1 #eaj1 #eab-d elephant1 #bag1 #dcf ellipse1 #bae ,emerson1 #bjc empiricism1 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#bai ,stallo1 #aj stars1 #bib statistical mechanics1 #cjd straight1 #fb1 #hb1 #bcf1 #dcc1 #dej1 #dgj ,stratonoff1 #eca surfaces1 #eh systematic errors1 #aga tactile space1 #fh1 #bfd ,tait1 #ih tangent1 #ea ,tannery1 #dc teaching1 #dcj1 #dcg thermodynamics1 #aae1 #aai ,thomson1 #ih1 #dhh thread1 #ajd time1 #bbc equality1 #bbe local1 #cjf1 #cjg measure of1 #bbc-d ,tisserand1 #eae-f ,tolstoi1 #ced1 #cfb1 #cfh ,tommasina1 #eai ,transylvania1 #bjc triangle1 #eh angle sum of1 #eh truth1 #bje ,tycho ,brahe1 #acc1 #aec1 #bbh unity of nature1 #acj universal invariant1 #ccc ,uriel1 #bjc van der ,waals1 #aec ,vauban1 #baj ,veblen1 #bjc velocity of light1 #bcb1 #cab ,venus of ,milo1 #bja verification1 #cc ,virchow1 #ba visual impressions1 #beb space1 #fg1 #beb ,volga1 #bjc ,voltaire1 #ecg-h ,weber1 #aag1 #eae-f ,weierstrass1 #aa1 #bab1 #dcb ,whitehead1 #dgb1 #dha-b whole numbers1 #dd ,wiechert1 #ade1 #dhh x-rays1 #aeb1 #eaa1 #ebj ,zeeman effect1 #aeb1 #aif1 #cag1 #did ,zeno1 #chb ,zermelo1 #dgg1 #dhb-c zigzag theory1 #dgh zodiac1 #cih1 #djd 99 99 99 99 99 ,transcriber's ,note3 ,the ,greek alphabets are represented within square brackets4 ,for example1 alpha stands for first ,greek alphabet alpha4 ,square root of a number is represented using the symbol 7sqrt74 ,that is to say1 sqrt7#be7 stands for square root of #be4 ,the superscript is shown with carat 77 symbol1 example1 #aje stands for #eth power of #aj4 ,similarly1 subscript is represented by underscore 77 symbol4 ,for instance1 n#c stands for letter n with subscript #c4