ss^ss^ ModX CORNELL UNIVERSITY LIBRARY GIFT OF Prof. George H. Sabine MffiHEMATICS LIBRARY DATE DUE »3HS^^P^***T um '^^mii^ -mb '"'""'^^^^Q^n swf mi^' Cornell University Library Q 175.P75 The foundations of science; Science and h L^- 3 1924 012 063 537 Cornell University Library The original of tiiis book is in tine Cornell University Library. There are no known copyright restrictions in the United States on the use of the text. http://www.archive.org/details/cu31924012063537 SCIENCE AND EDUCATION A SERIES OP VOLUMES FOR THE PROMOTION OF SCIENTIFIC RESEARCH AND EDUCATIONAL PROGRESS Edited by J. McKEEN CATTELL VOLUME I-THE FOUNDATIONS OF SCIENCE UNDER THE SAME EDITORSHIP SCIENCE AND EDUCATION. A series of volumes for the promotion of scientific research and educational progress. Volume I. The Foundations of Science. By H. PoiNCABB. Containing the authorized English translation by George Bruce Halsted of "Science and Hypothesis," "The Value of Science," and ''Science and Method." Volume II. Medical Research and Education. By Richard Mills Pearce, William H. Welch, W. H. Howell, Franklin P. Mall, Lewellys F. Barker, Charles S. Minot, W. B. Cannon, W. T. Council- man, Theobald Smith, G. N. Stewart, CM. Jack- son, E. P. Lyon, James B. Herrick, John M. Dod- son, C. K. Bardeen, W. Ophiils, S. J. Meltzer, James Ewing, W. W. Keen, Henry H. Donaldson, Christ- ian A. Herter, and Henry P. Bowditch. Volume III. University Control. By J. McKben Cattell and other authors. AMERICAN MEN OF SCIENCE. A Biographical Directory. SCIENCE. A weakly journal devoted to the advancement of science. The official organ of the American Asso- ciation for the Advancement of Science. THE SCIENTIFIC MONTHLY. A monthly magazine devoted to the diffusion of science. THE AMERICAN NATURALIST. A monthly journal devoted to the biological sciences, with special refer- ence to the factors of evolution. THE SCIENCE PRESS NEW YORK GARRISON, N. Y. THE FOUNDATIONS OF SCIENCE SCIENCE AND HYPOTHESIS THE VALUE OF S.CIENCE SCIENCE AND METHOD BY H. POINCARE AUTHORIZED TRANSLATION BY GEORGE BRUCE HALSTED WITH A SPECIAL PREFACE BY POINCAE^, AND AN INTRODUCTION BY JOSIAH EOYCE, HARVARD UNIVERSITY THE SCIENCE PKESS NEW YOEK AND GARRISON, N. Y. 1921 Copyright, 1913 By The Science Press Eepbinted 1921 Pi PRESS OF THE NEW ERA PRINTING COMPANV LANCASTER, PA. CONTENTS PAGIi Henri Poincar* is Author 's Preface to the Translation 3 SCIENCE AND HYPOTHESIS Introduction by Eoyce 9 Introduction 27 Part I. Nvmber and Magmtude Chapter I. — On the Nature of Mathematical Beasoning 31 Syllogistic Deduction 31 Verification and Proof 32 Elements of Arithmetic 33 Beasoning by Recurrence 37 Induction 40 Mathematical Construction 41 Chapter II. — ^Mathematical Magnitude and Experience 43 Definition of Incommensurables 44 The Physical Continuum 46 Creation of the Mathematical Continuum 46 Measurable Magnitude 49 Various Eemarks (Curves without Tangents) 50 The Physical Continuum of Several Dimensions 52 The Mathematical Continuum of Several Dimensions 53 Part II. Siaace Chapter III. — The Non-Euclidean Geometries 55 The Bolyai-Lobachevgki Geometry 56 Eien^ann 's Geometry 57 The Surfaces of Constant Curvature 58 Interpretation of Non-Euclidean Geometries 59 The Implicit Axioms '. 60 The Fourth Geometry 62 Lie's Theorem 62 Biemann 's Geometries . . . : 63 On the Nature of Axioms 63 Chapter IV. — Space and Geometry 66 Geometric Space and Perceptual Space 66 Visual Space 67 Tactile Space and Motor Space 68 Characteristics of Perceptual Space 69 Change of State and Change of Position 70 Conditions of Compensation 72 V vi CONTENTS Solid Bodies and Geometry 72 Law of Homogeneity ' ^ The Non-Euclidean World 75 The World of Four Dimensions 78 Conclusions 79 Chaptee V. — Experience and Geometry 81 Geometry and Astronomy 81 The Law of Relativity 83 Bearing of Experiments 86 Supplement (What is a Point?) 89 Ancestral Experience • 91 Past IIL Force Ohapteb VL— The Classic Mechanics 92 The Principle of Inertia 93 The Law of Acceleration 97 Anthropomorphic Mechanics 103 The School of the Thread 104 Chapter VII. — ^Eelative Motion and Absolute Motion 107 The Principle of Eelative Motion 107 Newton's Argument 108 Chapter VIII. — ^Energy and Thermodynamics Il5 Energetics 115 Thermodynamics 119 General Conclusions on Part III > 123 Part IV. Nature Chapter IX. — Hypotheses in Physics 127 The Edle of Experiment and Generalization 127 The Unity of Nature 130 The RSle of Hypothesis 133 Origin of Mathematical Physics 136 Chapter X. — The Theories of Modern Physics 140 Meaning of Physical Theories 140 Physics and Mechanism ^ 144 Present State of the Science 148 Chapter XI. — The Calculus of Probabilities 155 Classification of the Problems of Probability 158 Probability in Mathematics 161 Probability in the Physical Sciences 163 Eouge et noir 167 The Probability of Causes 168 The Theory of Errors 170 Conclusions 172 Chapter XII. — Optics and Electricity 174 Eresnel's Theory 174 Maxwell's Theory 175 The Mechanical Explanation of Physical Phenomena 177 CONTENTS vii Chapter XIII. — ^Electrodynamics 184 Ampere's Theory 184 Closed Carrents 186 Action of a Closed Current on a Portion of Current 186 Continuous notations 188 Mutual Action of Two Open Currents 189 Induction 190 Theory of Helmholtz 191 Difficulties Raised by these Theories 193 Maxwell 's Theory 193 Rowland 's Experiment 194 The Theory of Lorentz 196 THE VALUE OF SCIENCE Translator 's Introduction 201 Boes the Scientist Create Science? 201 The Mind Dispelling Optical Illusions 202 Euclid not Necessary 202 Without Hypotheses, no Science 203 What Outcome? 203 Introduction 205 Paet I. The Mathematical Sciences Chapter I. — Intuition and Logic in Mathematics 210 Chapter II.— The Measure of Time 223 Chapter III. — The Notion of Space 235 Qualitative Geometry 238 The Physical Continuum of Several Dimensions 240 The Notion of Point 244 The Notion of Displacement 247 Visual Space 252 Chapter IV. — Space and its Three Dimensions 256 The Group of Displacements 256 Identity of Two Points 259 Tactile Space 264 Identity of the Different Spaces 268 Space and Empiricism 271 Mind and Space 273 E61e of the Semicircular Canals 276 Part II. The Physical Sciences Chapter. V. — ^Analysis and Physics 279 Chapter VI. — ^Astronomy 289 Chapter VII. — The History of Mathematical Physics 297 The Physics of Principles 299 The Physics of Central Forces 297 The Physics of the Principles 299 CHAPTiai VIII. — ^The Present Crisis of Mathematical Physics 303 The New Crisis 303 Carnot 's Principle 303 viii CONTENTS The Principle of Relativity ^"^ Newton 's Principle Lavoisier's Principle Mayer 's Principle Chapter IX.— The Future of Mathematical Physics 314 The Principles and Experiment ' ^^^ The E61e of the Analyst 314 Aberration and Astronomy 315 Electrons and Spectra 316 Conventions preceding Experiment 317 Future Mathematical Physics 319 Pakt III. The Objective Value of Science Chaptek X. — Is Science Artificial? 321 The Philosophy of LeEoy 321 Science, Eule of Action 323 The Crude Fact and the Scientific Fact 325 Nominalism and the Universal Invariant 333 Chapteb XI. — Science and Reality 340 Contingenoe and Determinism 340 Objectivity of Science 347 The Rotation of the Earth < 353 Science for Its Own Sake 354 SCIENCE AND METHOD Introduction 359 Book I. Science and the Scientist Chapter I. — The Choice of Facts 362 Chapter II. — The Future of Mathematics 369 Chapter III. — Mathematical Creation 383 Chapter IV. — Chance 395 Book II. Mathematical Beasoning Chapter I. — The Relativity of Space 413 Chapter II. — Mathematical Definitions and Teaching 480 Chapter III. — Mathematics and Logic 448 Chapter IV. — The New Logics 460 Chapter V. — The Latest Efforts of the Logisticians 472 Book III. The New Mechamics Chapter I. — ^Mechanics and Radium 486 Chapter II. — ^Mechanics and Optics 496 Chapter III. — ^The New Mechanics and Astronomy 512 Book IV. Astronomic Science Chapter I. — The Milky Way and the Theory of Gases 523 Chapter II. — ^French Geodesy 535 General Conclusions 544 Index 54.7 HENRI POINCARE Sir George Darwin, worthy son of an immortal father, said, referring to what Poineare was to him and to his work: "He must be regarded as the presiding genius — or, shall I say, my patron saint?" Henri Poineare was born April 29, 1854, at Nancy, where his father was a physician highly respected. His schooling was broken into by the war of 1870-71, to get news of which he learned to read the German newspapers. He outclassed the other boys of his age in all subjects and in 1873 passed highest into the Bcole Polytechnique, where, like John Bolyai at Maros Vasarhely, he followed the courses in mathematics without taking a note and without the syllabus. He proceeded in 1875 to the School of Mines, and was Nomme, March 26, 1879. But he won his doctorate in the University of Paris, August 1, 1879, and was appointed to teach in the Faculte des Sciences de Caen, December 1, 1879, whence he was quickly called to the Uni- versity of Paris, teaching there from October 21, 1881, until his death, July 17, 1912. So it is an error to say he started as an engineer. At the early age of thirty-two he became a member of I'Academie des Sciences, and, March 5, 1908, was chosen Membre de I'Academie Frangaise. July 1, 1909, the number of his writings was 436. His earliest publication was in 1878, and was not important. Afterward came an essay submitted in competition for the Grand Prix offered in 1880, but it did not win. Suddenly there came a change, a striking fire, a bursting forth, in February, 1881, and Poineare tells us the very minute it happened. Mount- ing an omnibus, "at the moment when I put my foot upon the step, the idea came to me, without anything in my previous thoughts seeming to foreshadow it, that the transformations I had used to define the Fuchsian functions were identical with those of non-Euclidean geometry." Thereby was opened a perspec- tive new and immense. Moreover, the magic wand of his whole X THE FOUNDATIONS OF SCIENCE life-work had been grasped, the Aladdin's lamp had been rubbed, non-Euclidean geometry, whose necromancy was to open up a new theory of our universe, whose brilliant exposition was com- menced in his book Science and Hypothesis, which has been translated into six languages and has already had a circulation of over 20,000. The non-Euclidean notion is that of the possi- bility of alternative laws of nature, which in the Introduction to the ElectriciU et Optique, 1901, is thus put: "If therefore a phenomenon admits of a complete mechanical explanation, it will admit of an infinity of others which will account equally well for all the peculiarities disclosed by experiment. ' ' 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 experience ; it is commode, con- venient ; but perhaps another may be vastly more advantageous. The old conception of true has been revised. The first expres- sion of the new idea occurs on the title page of John Bolyai's marvelous Science Absolute of Space, in the phrase "baud un- quam a priori decidenda." "With bearing on the history of the earth and moon system and the origin of double stars, in formulating the geometric criterion of stability, Poincare proved the existence of a previously un- known pear-shaped figure, with the possibility that the progres- sive deformation of this figure with increasing angular velocity might result in the breaking up of the rotating body into two detached masses. Of his treatise Les Methodes nouvelles de la Mechanique celeste. Sir George Darwin says : " It is probable that for half a century to come it will be the mine from which humbler investigators will excava,te their materials." Brilliant was his appreciation of Poincare in presenting the gold medal of the Royal Astronomical Society. The three others most akin in genius are linked with him by the Sylvester medal of the Royal Society, the Lobaehevski medal of the Physico-Mathematical Society of Kazan, and the Bolyai prize of the Hungarian Acad- emy of Sciences. His work must be reckoned with the greatest mathematical achievements of mankind. The kernel of Poincare 's power lies in an oracle Sylvester often quoted to me as from Hesiod : The whole is less than its part. HENBI POINCAEE xi He penetrates at once the divine simplicity of the perfectly general case, and thence descends, as from Olympus, to the special concrete earthly particulars. A combination of seemingly extremely simple analytic and geometric concepts gave necessary general conclusions of im- mense scope from which sprang a disconcerting wilderness of possible deductions. And so he leaves a noble, fruitful heritage. Says Love: "His right is recognized now, and it is not likely that future generations will revise the judgment, to rank among the greatest mathematicians of all time." George Bruce Halsted. SCIENCE AND HYPOTHESIS AUTHOR'S PREFACE TO THE TRANSLATION I AM exceedingly grateful to Dr. Halsted, who has been so good as to present my book to American readers in a translation, clear and faithful. 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 old. Some people love to repeat that Anglo-Saxons have not the same way of thinking as the Latins or as the Germans ; that they have quite another way of understanding mathematics or of un- derstanding physics; that this way seems to them superior to all others ; that they feel no need of changing it, nor even of know- ing the ways of other peoples. In that they would beyond question be wrong, but I do not believe that is true, or, at least, that is true no longer. 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 Europe. To be sure, each people will preserve its characteristic genius, and it would be a pity if it were otherwise, supposing such a thing possible. If the Anglo-Saxons wished to become Latins, they would never be more than bad Latins ; just as the French, in seeking to imitate them, could turn out only pretty poor Anglo-Saxons. And then the English and Americans have made scientific conquests they alone could have made ; they will make still more of which others would be incapable. It would therefore be de- plorable if there were no longer Anglo-Saxons. But continentals have on their part done things an English- man could not have done, so that there is no need either for wishing all the world Anglo-Saxon. Each has his characteristic aptitudes, and these aptitudes 3 4 SCIENCE AND HYPOTHESIS should be diverse, else would the scientific concert resemble a quartet where every one wanted to play the violin. And yet it is not bad for the violin to know what the violon- cello is playing, and vice versa. This it is that the English aiid Americans are comprehending more and more; and from this point of view the translations undertaken by Dr. Halsted are most opportune and timely. Consider first what concerns the mathematical sciences. It is frequently said the English cultivate them only in view of their applications and even that they despise those who have other aims; that speculations too abstract repel them as savor- ing of metaphysie. The English, even in mathematics, are to proceed always from the particular to the general, so that they would never have an idea of entering mathematics, as do many Germans, by the gate of the theory of aggregates. They are always to hold, so to speak, one foot in the world of the senses, and never burn the bridges keeping them in communication with reality. They thus are to be incapable of comprehending or at least of appreciat- ing certain theories more interesting than utilitarian, such as the non-Euclidean geometries. According to that, the first two parts of this book, on number and space, should seem to them void of all substance and would only baffle them. But that is not true. And first of all, are they such uncom- promising realists as has been said ? Are they absolutely refrac- tory, I do not say to metaphysie, but at least to everything metaphysical ? Eecall the name of Berkeley, born in Ireland doubtless, but immediately adopted by the English, who marked a natural and necessary stage in the development of English philosophy. Is this not enough to show they are capable of making ascen- sions otherwise than in a captive balloon? And to return to America, is not the Monist published at Chicago, that review which even to us seems bold and yet which finds readers? And in mathematics? Do you think American geometers are concerned only about applications ? Far from it. The part of the science they cultivate most devotedly is the theory of AUTHOR'S PREFACE TO TRANSLATION 5 groups of substitutions, and under its most abstract form, the farthest removed from the practical. Moreover, Dr. Halsted gives regularly each year a review of all productions relative to the non-Euclidean geometry, and he has about him a public deeply interested in his work. He has initiated this public into the ideas of Hilbert, and he has even written an elementary treatise on 'Rational Geometry,' based on the principles of the renowned German savant. To introduce this principle into teaching is surely this time to burn all bridges of reliance upon sensory intuition, and this is, I confess, a boldness which seems to me almost rashness. The American public is therefore much better prepared than has been thought for investigating the origin of the notion of space. Moreover, to analyze this concept is not to sacrifice reality to I know not what phantom. The geometric language is after aU only a language. Space is only a word that we have believed a thing. What is the origin of this word and of other words also? What things do they hide? To ask this is permissible; to forbid it would be, on the contrary, to be a dupe of words; it would be to adore a metaphysical idol, like savage peoples who prostrate themselves before a statue of wood without daring to take a look at what is within. In the study of nature, the contrast between the Anglo-Saxon spirit and the Latin spirit is still greater. The Latins seek in general to put their thought in mathe- matical form; the English prefer to express it by a material representation. Both doubtless rely only on experience for knowing the world ; when they happen to go beyond this, they consider their fore- knowledge as only provisional, and they hasten to ask its defini- tive confirmation from nature herself. But experience is not all, and the savant is not passive; he does not wait for the truth to come and find him, or for a chance meeting to bring him face to face with it. He must go to meet it, and it is for his thinking to reveal to him the way leading thither. For that there is need of an instrument; well, just there begins the difference — the instrument the Latins ordi- narily choose is not that preferred by the Anglo-Saxons. 6 SCIENCE AND HYPOTHESIS For a Latin, truth can be expressed only by equations; it must obey laws simple, logical, symmetric and fitted to satisfy minds in love with mathematical elegance. The Anglo-Saxon to depict a phenomenon will first be en- grossed in making a model, and he will make it with common materials, such as our crude, unaided senses show us them. He also makes a hypothesis, he assumes implicitly that nature, in her finest elements, is the same as in the complicated aggregates which alone are within the reach of our senses. He concludes from the body to the atom. Both therefore make hypotheses, and this indeed is necessary, since no scientist has ever been able to get on without them. The essential thing is never to make them unconsciously. From this point of view again, it would be well for these two sorts of physicists to know something of each other; in study- ing the work of minds so unlike their own, they will immedi- ately recognize that in this work there has been an accumulation of hypotheses. Doubtless this will not suffice to make them comprehend that they on their part have made just as many ; each sees the mote without seeing the beam ; but by their criticisms they vrill warn their rivals, and it may be supposed these will not fail to render them the same service. The English procedure often seems to us crude, the analogies they think they discover to us seem at times superficial ; they are not sufficiently interlocked, not precise enough; they sometimes permit incoherences, contradictions in terms, which shock a geo- metric spirit and which the employment of the mathematical method would immediately have put in evidence. But most often it is, on the other hand, very fortunate that they have not per- ceived these contradictions; 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 it. And then these very contradictions, when they end by per- ceiving them, have the advantage of showing them the hypothet- ical character of their conceptions, whereas the mathematical method, by its apparent rigor and inflexible course, often inspires in us a confidence nothing warrants, and prevents our looking about us. AUTHOR'S PBEFACE TO TRANSLATION 7 From another point of view, however, the two conceptions are very unlike, and if all must be said, they are very unlike because of a common fault. The English wish to make the world out "of what we see. I mean what we see with the unaided eye, not the microscope, nor that still more subtile microscope, the human head guided by scientific induction. The Latin wants to make it out of formulas, but these for- mulas are still the quintessenced expression of what we see. In a word, both would make the unknown out of the known, and their excuse is that there is no way of doing otherwise. And yet is this legitimate, if the unknown be the simple and the known the complex? Shall we not get of the simple a false idea, if we think it like the complex, or worse yet if we strive to make it out of elements which are themselves compounds? Is not each great advance accomplished precisely the day some one has discovered under the complex aggregate shown by our senses something far more simple, not even resembling it — as when Newton replaced Kepler's three laws by the single law of gravitation, which was something simpler, equivalent, yet unlike ? One is justified in asking if we are not on the eve of just such a revolution or one even more important. Matter seems on the point of losing its mass, its solidest attribute, and resolving itself into electrons. Mechanics must then give place to a broader conception which will explain it, but which it will not explain. So it was in vain the attempt was made in England to con- struct the ether by material models, or in France to apply to it the laws of dynamics. The ether it is, the unknown, which explains matter, the known; matter is incapable of explaining the ether. POINCAB^. INTRODUCTION BY PEOFESSOE JOSIAH EOTCE Habvakd University The treatise of a master needs no commendation through the words of a mere learner. But, since my friend and former fellow student, the translator of this volume, has joined with another of my colleagues, Professor Cattell, in asking me to undertake the task of calling the attention of my fellow students to the importance and to the scope of M. Poincare's volume, I accept the office, not as one competent to pass judgment upon the book, but simply as a learner, desirous to increase the number of those amongst us who are already interested in the type of researches to which M. Poincare has so notably contributed. The branches of inquiry collectively known as the Philosophy of Science have undergone great changes since the appearance of Herbert Spencer's First Principles, 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 knowledge. The summary which M. Poincare gives, at the outset of his own introduction to the present work, where he states the view which the 'superficial observer' takes of scientific truth, suggests, not indeed Spencer's own most characteristic theories, 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 experience. It was well known to them, indeed, that experience is a constant guide, and an inexhaustible source both of novel scientific results and of unsolved problems; but the fundamental Spencerian principles of science, such as 'the persistence of force,' the 'rhythm of motion' and the rest, were treated by Spencer himself as demonstrably objective, although 9 10 SCIENCE AND HYPOTHESIS indeed 'relative' truths, capable of being tested once for all by the 'inconceivability of the opposite,' and certain to hold true for the whole 'knowable' universe. Thus, whether one dwelt upon the results of such a mathematical procedure as that to which M. Poincare refers in his opening paragraphs, or whether, like Spen- cer himself, one applied the 'first principles' to regions of less exact science, this confidence that a certain orthodoxy regarding the principles of science was established forever was characteristic of the followers of the movement in question. Experience, lighted up by reason, seemed to them to have predetermined for all future time certain great theoretical results regarding the real constitution of the 'knowable' cosmos. Whoever doubted this doubted 'the verdict of science.' Some of us well remember how, when Stallo's 'Principles and Theories of Modern Physics' first appeared, this sense of scien- tific orthodoxy was shocked amongst many of our American read- ers and teachers of science. 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 presump- tion in speaking with the freedom that he there used regarding such sacred possessions of humanity as the fundamental concepts of physics. That very book, however, has quite lately been translated into German as a valuable contribution to some of the most recent efforts to reconstitute a modern 'philosophy of nature.' And whatever may be otherwise thought of Stallo's critical methods, or of his results, there can be no doubt that, at the present moment, if his book were to appear for the first time, 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 gases, or on account of its call for a logical rearrangement of the fundamental concepts of the theory of energy. We are no longer able so easily to know heretics at first sight. For we now appear to stand in this position: The control of natural phenomena, which through the sciences men have attained, grows daily vaster and more detailed, and in its de- tails more assured. Phenomena men know and predict better than ever. But regarding the most general theories, and the INTRODUCTION 11 most fundamental, of science, there is no longer any notable scientific orthodoxy. Thus, as knowledge grows firmer and wider, conceptual construction becomes less rigid. The field of the theoretical philosophy of nature — ^yes, the field of the logic of science — this whole region is to-day an open one. Whoever will work there must indeed accept the verdict of experience regard- ing what happens in the natural world. So far he is indeed bound. But he may undertake without hindrance from mere tradition the task of trying afresh to reduce what happens to conceptual unity. 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 defined ; but that is not because the foundations of geometry are now viewed as completely settled, beyond controversy, nor yet because the 'persistence of force' has been finally so defined as to make the 'opposite inconceiv- able ' and the doctrine of energy beyond the reach of novel formu- lations. No, the circle-squarers and the inventors of devices for perpetual motion are to-day discredited, not because of any unorthodoxy of their general philosophy of nature, 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 interpre- tations. Certain properties of the irrational number ir are known, in sufficient multitude to justify the mathematician in declining to listen to the arguments of the circle-squarer ; but, despite great advances, and despite the assured results of Dede- Mnd, of Cantor, of Weierstrass and of various others, the gen- eral theory of the logic of the numbers, rational and irrational, still presents several important features of great obscurity ; and the philosophy of the concepts of geometry yet remains, in sev- eral very notable respects, unconquered territory, despite the work of Hilbert and of Fieri, and of our author himself. The ordinary inventors of the perpetual motion machines still stand in conflict with accepted generalizations; but nobody knows as yet what the final form of the theory of energy will be, nor can any one say precisely what place the phenomena of the radioac- tive bodies will occupy in that theory. The alchemists would not 12 SCIENCE AND HYPOTHESIS be welcome workers in modern laboratories; yet some sorts of transformation and of evolution of the elements are to-day matters which theory can find it convenient, upon occasion, to treat as more or less exactly definable possibilities; while some newly observed phenomena tend to indicate, not indeed that the ancient hopes of the alchemists were well founded, but that the ultimate constitution of matter is something more fluent, less in- variant, than the theoretical orthodoxy of a recent period sup- posed. Again, regarding the foundations of biology, a theoret- ical orthodoxy grows less possible, less definable, less conceiv- able (even as a hope) the more knowledge advances. Once 'mechanism' and 'vitalism' were mutually contradictory theories regarding the ultimate constitution of living bodies. Now they are obviously becoming more and more 'points of view,' diverse but not necessarily conflicting. 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 obects, you may assume, in the modern 'pragmatic' sense, the attitude of a 'neo- vitalist. ' So far, however, as you are able to lay stress, with good results, upon the many ways in which the life processes can be assimilated to those studied in physics and in chemistry, you work as if you were a partisan of 'mechanics.' In any case, your special science prospers by reason of the empirical discov- eries that you make. And your theories, whatever they are, must not run counter to any positive empirical results. But otherwise, scientific orthodoxy no longer predetermines what alone it is respectable for you to think about the nature of living substance. This gain in the freedom of theory, coming, as it does, side by side with a constant increase of a positive knowledge of nature, lends itself to various interpretations, and raises various obvious questions. II One of the most natural of these interpretations, one of the most obvious of these questions, may be readily stated. Is not the lesson of all these recent discussions simply this, that general theories are simply vain, that a philosophy of nature is an idle INTRODUCTION 13 dream, and that the results of science are coextensive with the range of actual empirical observation and of successful predic- tion ? If this is indeed the lesson, then the decline of theoretical orthodoxy in science is — ^like the eclipse of dogma in religion — merely a further lesson in pure positivism, another proof that man does best when he limits himself to thinking about what can be found in human experience, and in trying to plan what can be done to make human life more controllable and more reason- able. What we are free to do as we please — is it any longer a serious business ? What we are free to think as we please — is it of any further interest to one who is in search of truth? If certain general theories are mere conceptual constructions, which to-day are, and to-morrow are cast into the oven, why dignify them by the name of philosophy? Has science any place for such theories? Why be a 'neo-vitalist,' or an 'evolutionist,' or an ' atomist, ' or an ' Energetiker ' ? Why not say, plainly : ' ' Such and such phenomena, thus and thus described, have been ob- served; such and such experiences are to be expected, 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 chance; so much then with reasonable assurance we know; all else is silence — or else is some matter to be tested by another experiment?" Why not limit our philosophy of science strictly to such a counsel of resig- nation? Why not substitute, for the old scientific orthodoxy, simply a confession of ignorance, and a resolution to devote our- selves to the business of enlarging the bounds of actual em- pirical knowledge? Such comments upon the situation just characterized are fre- quently made. Unfortunately, they seem not to content the very age whose revolt from the orthodoxy of traditional theory, whose uncertainty about all theoretical formulations, and whose vast wealth of empirical discoveries and of rapidly advancing special researches, would seem most to justify tliese very com- ments. Never has there been better reason than there is to-day to be content, if rational man could be content, with a pure pos- itivism. The splendid triumphs of special research in the most various fields, the constant increase in our practical control over 14 SCIENCE AND HYPOTHESIS nature— these, our positive and growing possessions, 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 universe. "Why not 'take the cash and let the credit go'? Why pursue the elusive theoretical 'unifica- tion' any further, when what we daily get from our sciences is an increasing wealth of detailed information and of practical guidance ? As a fact, however, the known answer of our own age to these very obvious comments is a constant multiplication of new efforts towards large and unifying theories. If theoretical ortho- doxy is no longer clearly definable, theoretical construction was never more rife. The history of the doctrine of evolution, even in its most recent phases, when the theoretical uncertainties re- garding the 'factors of evolution' are most insisted upon, is full of illustrations of this remarkable union of scepticism in critical work with courage regarding the use of the scientific imagination. The history of those controversies regarding theoretical physics, some of whose principal phases M. Poincare, in his book, sketches with the hand of the master, is another illustration of the con- sciousness of the time. Men have their freedom of thought in these regions; and they feel the need of making constant and constructive use of this freedom. And the men who most feel this need are by no means in the majority of cases professional metaphysicians — or students who, like myself, have to view all these controversies amongst the scientific theoreticians from without as learners. These large theoretical constructions are due, on the contrary, in a great many cases to special workers, who have been driven to the freedom of philosophy by the oppres- sion of experience, 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 science. Why, then, does science actually need general theories, despite the fact that these theories inevitably alter and pass away? What is the service of a philosophy of science, 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 generation? Why must that which endlessly grows, INTBODUCTION 15 namely, man's knowledge of the phenomenal order of nature, be constantly united in men's minds with that which is certain to decay, namely, the theoretical formulation of special koowl- edge in more or less completely unified systems of doctrine? I understand our author's volume to be in the main an answer to this question. To be sure, the compact and manifold teachings which this text contains relate to a great many dif- ferent special issues. A student interested in the problems of the philosophy of mathematics, or in the theory of probabilities, or in the nature and ofSce of mathematical physics, or in still other problems belonging to the wide field here discussed, may find what he wants here and there in the text, even in case the general issues which give the volume its unity mean little to him, or even if he differs from the author's views regarding the principal issues of the book. But in the main, 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 fact. The result of the book is a substantial justification of the scientific utility of theoretical con- struction — an abandonment of dogma, but a vindication of the rights of the constructive reason. Ill The most notable of the results of our author's investigation of the logic of scientific theories relates, as I understand his work, to a topic which the present state of logical investigation, just summarized, makes especially important, but which has thus far been very inadequately treated in the text-books of inductive logic. The useful hypotheses of science are of two kinds : 1. The hypotheses which are valuable precisely because they are either verifiable or else refutable through a definite appeal to the tests furnished by experience ; and 2. The hypotheses which, despite the fact that experience sug- gests them, are valuable despite, or even because, of the fact that experience can neither confirm nor refute them. The contrast between these two kinds of hypotheses is a prominent topic of our author's discussion. Hypotheses of the general type which I have here placed first 16 SCIENCE AND HYPOTHESIS 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 characterize. The value of such hypotheses is indeed undoubted. 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 science. One usually either fails to admit their presence in scientific work, or else remains silent as to the reasons of their usefulness. Our author's treatment of the work of science is therefore especially marked by the fact that he ex- plicitly makes prominent both the existence and the scientific importance of hypotheses of this second type. They occupy in his discussion a place somewhat analogous to each of the two dis- tinct positions occupied by the 'categories' and the 'forms of sensibility,' on the one hand, and by the 'regulative principles of the reason,' on the other hand, in the Kantian theory of our knowledge of nature. That is, these hypotheses which can neither be confirmed nor refuted by experience appear, in M. Poincare's account, partly (like the conception of ' continuous quantity') 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 variety; and partly (like the larger organizing con- cepts of science) as principles regarding the structure of the world in its wholeness ; i. e., as principles in the light of which we try to interpret our experience, so as to give to it a totality and an inclusive unity such as Euclidean space, or such as the world of the theory of energy is conceived to possess. Thus viewed, M. Poincare's logical theory of this second class of hypotheses under- takes to accomplish, with modern means and in the light of to-day's issues, a part of what Kant endeavored to accomplish in his theory of scientific knowledge with the limited means which were at his disposal. 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 nature, an interpretation rather than a portrayal or a prediction of the objective facts of nature, an INTRODUCTION 17 adjustment of our conceptions of things to the internal needs of our intelligence, rather than a grasping of things as they are in themselves. To be sure, M. Poincare's view, in this portion of his work, obviously differs, meanwhile, from that of Kant, as well as this agrees, in a measure, with the spirit of the Kantian epistemology. I do not mean therefore to class our author as a Kantian. For Kant, the interpretations imposed by the 'forms of sensibility,' and by the 'categories of the understanding,' upon our doctrine of nature are rigidly predetermined by the unalterable 'form' of our intellectual powers. We 'must' thus view facts, whatever the data of sense must be. This, of course, is not M. Poincare's view. 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 'regulative,' but is 'a priori,' and so unchangeable. For M. Poineare, on the contrary, all this adjustment of our interpre- tations of experience to the needs of our intellect is something far less rigid and unalterable, and is constantly subject to the suggestions of experience. We must indeed interpret in our own way*; but our way is itself only relatively determinate; it is essentially more or less plastic ; other interpretations of experience are conceivable. Those that we use are merely the ones found to be most convenient. But this convenience is not absolute neces- sity. Unverifiable and irrefutable hypotheses in science are in- deed, in general, indispensable aids to the organization and to the guidance of our interpretation of experience. But it is expe- rience itself which points out to us what lines of interpretation will prove most convenient. Instead of Kant's rigid list of a priori 'forms,' we consequently have in M. Poincare's account a set of conventions, neither wholly subjective and arbitrary, nor yet imposed upon us unambiguously by the external compulsion of experience. The organization of science, so far as this organ- ization is due to hypotheses of the kind here in question, thus resembles that of a constitutional government — neither abso- lutely necessary, nor yet determined apart from the will of the subjects, nor yet accidental — a free, yet not a capricious estab- lishment of good order, in conformity with empirical needs. 18 SCIENCE AND HYPOTHESIS Characteristic remains, however, for our author, as, in his decidedly contrasting way, for Kant, the thought that without principles which at every stage transcend precise confirmation through such experience as is then accessible the organization of experience is impossible. Whether one views these principles as conventions or as a priori 'forms,' they may therefore be de- scribed as hypotheses, but as hypotheses that, while lying at the basis of our actual physical sciences, at once refer to experience and help us in dealing with experience, and are yet neither con- firmed nor refuted by the experiences which we possess or which we can hope to attain. Three special instances or classes of instances, according to our author 's account, may be used as illustrations of this general type of hypotheses. They are: (1) The hypothesis of the exist- ence of continuous extensive quanta in nature; (2) The prin- ciples of geometry; (3) The principles of mechanics and of the general theory of energy. In case of each of these special types of hypotheses we are at first disposed, apart from reflection, to say that we find the world to be thus or thus, so that, for instance, we can confirm the thesis according to which nature contains continuous magnitudes; or can prove or disprove the physical truth of the postulates of Euclidean geometry ; or can confirm by definite experience the objective validity of the principles of mechanics. A closer examination reveals, according to our author, the incorrectness of all such opinions. Hypotheses of these various special types are needed; and their usefulness can be empirically shown. They are in touch with experience; and that they are not merely arbitrary conventions is also verifiable. They are not a priori necessities ; and we can easily conceive in- telligent beings whose experience could be best interpreted with- out using these hypotheses. Tet these hypotheses are not sub- ject to direct confirmation or refutation by experience. They stand then in sharp contrast to the scientific hypotheses of the other, and more frequently recognized, type, i. e., to the hy- potheses which can be tested by a definite appeal to experience. To these other hypotheses our author attaches, of course, great importance. His treatment of them is full of a living apprecia- tion of the significance of empirical investigation. But the cen- INTBODVCTION 19 tral problem of the logic of science tlras becomes the problem of the relation between the two fundamentally distinct types of hypotheses, i. e., between those which can not be verified or re- futed through experience, and those which can be empirically tested. IV The detailed treatment which M. Poincare gives to the problem thus defined must be learned from his text. It is no part of my purpose to expound, to defend or to traverse any of his special conclusions regarding this matter. Yet I can not avoid observ- ing that, while M. Poincare strictly confines his illustrations and his expressions of opinion to those regions of science wherein, as special investigator, he is himself most at home, 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 M. Poincare 's methods to the study of the concepts and presuppositions of the organic and of the historical and social sciences, than when one confines one's attention, as our author here does, to the physical sciences. It belongs to the province of an introduction like the present to point out, however briefly and inadequately, that the significance of our author's ideas extends far beyond the scope to which he chooses to confine their discussion. The historical sciences, and in fact all those sciences such as geology, and such as the evolutionary sciences in general, un- dertake theoretical constructions which relate to past time. Hy- potheses relating to the more or less remote past stand, however, in a position which is very interesting from the point of view of the logic of science. Directly speaking, no such hypothesis is capable of confirmation or of refutation, because we can not return into the past to verify by our own experience what then happened. Yet indirectly, such hypotheses may lead to predic- tions of coming experience. These latter will be subject to con- trol. Thus, Schliemann's confidence that the legend of Troy had a definite historical foundation led to predictions regarding what certain excavations would reveal. In a sense somewhat different from that which filled Schliemann's enthusiastic mind, these pre- dictions proved verifiable. The result has been a considerable 20 SCIENCE AND HYPOTHESIS change in the attitude of historians toward the legend of Troy. Geological investigation leads to predictions regarding the order of the strata or the course of mineral veins in a district, regard- ing the fossils which may be discovered in given formations, and so on. These hypotheses are subject to the control of experience. The various theories of evolutionary doctrine include many hy- potheses capable of confirmation and of refutation by empirical tests. Yet, despite all such empirical control, it still remains true that whenever a science is mainly concerned with the remote past, whether this science be archeology, or geology, or anthro- pology, or Old Testament history, the principal theoretical con- structions always include features which no appeal to present or to accessible future experience can ever definitely test. Hence the suspicion with which students of experimental science often regard the theoretical constructions of their confreres of the sci- ences that deal with the past. The origin of the races of men, of man himself, of life, of species, of the planet ; the hypotheses of anthropologists, of archeologists, of students of 'higher criti- cism' — aU 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 science. Yet no one can doubt the im- portance and the inevitableness of endeavoring to apply scientific method to these regions also. Science needs theories regarding the past history of the world. And no one who looks closer into the methods of these sciences of past time can doubt that verifi- able and unverifiable hypotheses are in all these regions inevitably interwoven; so that, while experience is always the guide, the attitude of the investigator towards experience is determined by interests which have to be partially due to what I should call that 'internal meaning,' that human interest in rational theoret- ical construction which inspires the scientific inquiry; and the theoretical constructions which prevail in such sciences are neither unbiased reports of the actual constitution of an external reality, nor yet arbitrary constructions of fancy. These con- structions in fact resemble in a measure those which M. Poincare in this book has analyzed in the case of geometry. They are constructions molded, but not predetermined in their details, by experience. We report facts ; we let the facts speak ; but we, as INTRODUCTION 21 •we investigate, in the popular phrase, 'talk back' to the facts. We interpret as well as report. Man is not merely made for science, but science is made for man. It expresses his deepest intellectual needs, as well as his careful observations. It is an effort to bring internal meanings into harmony with external verifications. It attempts therefore to control, as well as to submit, to conceive with rational unity, as weU as to accept data. Its arts are those directed towards self-possession as well as towards an imitation of the outer reality which we find. It seeks therefore a disciplined freedom of thought. The discipline is as essential as the freedom; but the latter has also its place. The theories of science are human, as well as objective, inter- nally rational, as weU as (when that is possible) subject to ex- ternal tests. In a field very different from that of the historical sciences, namely, in a science of observation and of experiment, which is at the same time an organic science, 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 research, but which apparently resembles in a measure the type of hypotheses of which M. Poincare speaks when he characterizes the principles of mechanics and of the theory of energy. I venture to call attention here to this con- ception, which seems to me to illustrate M. Poincare 's view of the functions of hypothesis in scientific work. The modern science of pathology is usually regarded as dating from the earlier researches of Virchow, whose 'Cellular Path- ology' was the outcome of a very careful and elaborate induc- tion. Virchow, himself, felt a strong aversion to mere specula- tion. He endeavored to keep close to observation, and to relieve medical science from the control of fantastic theories, such as those of the Naturphilosophen had been. Yet Virchow 's re- searches were, as early as 1847, or still earlier, already under the guidance of a theoretical presupposition which he himself states as follows: "We have learned to recognize," he says, "that dis- eases are not autonomous organisms, that they are no entities • that have entered into the body, that they are no parasites which take root in the body, but that they merely show us the course of 22 SCIENCE AND HYPOTHESIS the vital processes under altered conditions" ('dass sie nur Ablauf der Lebenserseheinungen unter veranderten Bedingun- gen darstellen'). The enormous importance of this theoretical presupposition for all the early successes of modern pathological investigation is generally recognized by the experts. I do not doubt this opinion. It appears to be a commonplace of the history of this science. 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 bacteriology. The question arose whether the theoretical foundations of Virchow's pathology had not been set aside. 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 recognized. Yet to the end of his own career Virchow stoutly maintained that in all its essential significance his own fundamental principle remained quite un- touched by the newer discoveries. And, as a fact, this view could indeed be maintained. For if diseases proved to be the consequences of the presence of parasites, the diseases them- selves, so far as they belonged to the diseased organism, were still not the parasites, but were, as before, the reaction of the organism to the veranderte Bedingungen which the presence of the parasites entailed. So Virchow could well insist. And if the famous principle in question is only stated with sufficient generality, it amounts simply to saying that if a disease in- volves a change in an organism, and if this change is subject to law at all, then the nature of the organism and the reaction of the organism to whatever it is which causes the disease must be understood in ease the disease is to be understood. For this very reason, however, Virchow's theoretical principle in its most general form could he neither confirmed nor refuted hy experience. It would remain empirically irrefutable, so far as I can see, even if we should learn that the devil was the true cause of all diseases. For the devil himself would then simply predetermine the veranderte Bedingungen to which the diseased organism would be reacting. Let buUets or bacteria, poisons or compressed air, or the devil be the Bedingungen to which a diseased organism reacts, the postulate that Virchow INTBODUCTION 23 states in the passage just quoted will remain irrefutable, if only this postulate be interpreted to meet the case. For the principle in question merely says that whatever entity it may be, bullet, or poison, or devil, that affects the organism, the disease is not that entity, but is the resulting alteration in the process of the organism. I insist, then, that this principle of Virehow's is no trial sup- position, no scientific hypothesis in the narrower sense — capable of being submitted to precise empirical tests. It is, on the contrary, a very precious leading idea, a theoretical interpre- tation of phenomena, in the light of which observations are to be made — 'a regulative principle' of research. It is equivalent to a resolution to search for those detailed connections which link the processes of disease to the normal process of the organism. Such a search undertakes to find the true unity, whatever that may prove to be, wherein the pathological and the normal proc- esses are linked. Now without some such leading idea, the cellu- lar pathology itself could never have been reached ; because the empirical facts in question would never have been observed. Hence this principle of Virehow's was indispensable to the growth of his science. Yet it was not a verifiable and not a re- futable hypothesis. One value of unverifiable and irrefutable hypotheses of this type lies, then, in the sort of empirical inquiries which they initiate, inspire, organize and guide. In these inquiries hypotheses in the narrower sense, that is, trial propositions which are to be submitted to definite empirical con- trol, are indeed everywhere present. And the use of the other sort of principles lies wholly in their application to experience. Yet without what I have just proposed to call the 'leading ideas' of a science, that is, its principles of an unverifiable and irre- futable character, suggested, but not to be finally tested, by experience, the hypotheses in the narrower sense would lack that guidance which, as M. Poincare has shown, the larger ideas of science give to empirical investigation. V I have dwelt, no doubt, at too great length upon one aspect only of our author's varied and well-balanced discussion of the 24 SCIENCE AND HYPOTHESIS problems and concepts of scientific theory. Of the hypotheses in the narrower sense and of the value of direct empirical control, he has also spoken with the authority and the originality which belong to his position. And in dealing with the foundations of mathematics he has raised one or two questions of great philo- sophical import into which I have no time, even if I had the right, to enter here. In particular, in speaking of the essence of mathematical reasoning, and of the difficult problem of what makes possible novel results in the field of pure mathematics, M. Poincare defends a thesis regarding the office of 'demonstration by recurrence' — a thesis which is indeed disputable, which has been disputed and which I myself should be disposed, so far as I at present understand the matter, to modify in some respects, even in accepting the spirit of our author's assertion. Yet there can be no doubt of the importance of this thesis, and of the fact that it defines a characteristic that is indeed fundamental in a wide range of mathematical research. The philosophical prob- lems that lie at the basis of recurrent proofs and processes are, as I have elsewhere argued, of the most fundamental importance. These, then, are a few hints relating to the significance of our author's discussion, 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 done. Of the person and of the life-work of our author a few words are here, in conclusion, still in place, addressed, not to the stu- dents of his own science, to whom his position is well known, but to the general reader who may seek guidance in these pages. Jules Henri Poincare was born at Nancy, in 1854, the son of a professor in the Faculty of Medicine at Nancy. He studied at the lElcole Polytechnique and at the ficole des Mines, and later received his doctorate in mathematics in 1879. In 1883 he began courses of instruction in mathematics at the ficole Polytechnique ; in 1886 received a professorship of mathe- matical physics in the Faculty of Sciences at Paris; then became member of the Academy of Sciences at Paris, in 1887, and devoted his life to instruction and investigation in the regions of pure mathematics, of mathematical physics and of celestial mechanics. His list of published treatises relating to INTBODUCTION 25 various branches of his chosen sciences is long; and his ori- ginal memoirs have included several momentous investigations, which have gone far to transform more than one branch of research. His presence at the International Congress of Arts and Science in St. Louis was one of the most noticeable features of that remarkable gathering of distinguished foreign guests. In Poincare the reader meets, then, not one who is primarily a speculative student of general problems for their own sake, but an original investigator of the highest rank in several distinct, although interrelated, branches of modern research. The theory of functions — a highly recondite region of pure mathematics — owes to him advances of the first importance, for instance, the definition of a new type of functions. The 'problem of the three bodies, ' a famous and fundamental problem of celestial mechanics, has received from his studies a treatment whose significance has been recognized by the highest authorities. His international reputation has been confirmed by the conferring of more than one important prize for his researches. His membership in the most eminent learned societies of various nations is widely extended ; his volumes bearing upon various branches of mathematics and of mathematical physics are used by special students in all parts of the learned world ; in brief, he is, as geometer, as analyst and as a theoretical physicist, a leader of his age. Meanwhile, as contributor to the philosophical discussion of the bases and methods of science, M. Poincare has long been active. When, in 1893, the admirable Bevue de Metaphysique et de Morale began to appear, M. Poincare was soon found amongst the most satisfactory of the contributors to the work of that journal, whose office it has especially beeii to bring philosophy and the various special sciences (both natural and moral) into a closer mutual understanding. The discussions brought to- gether in the present volume are in large part the outcome of M. Poincare 's contributions to the Bevue de Metaphysique et de Morale. The reader of M. Poincare 's book is in presence, then, of a great special investigator who is also a philosopher. SCIENCE AND HYPOTHESIS INTRODUCTION For a superficial observer, scientific truth is beyond the possi- bility of doubt ; the logic of science is infallible, and if the scien- tists are sometimes mistaken, this is only from their mistaking its rules. ' ' The mathematical verities flow from a small number of self- evident propositions by a chain of impeccable reasonings ; they impose themselves not only on us, but on nature itself. They fetter, so to speak, the Creator and only permit him to choose between some relatively few solutions. A few experiments then will suffice to let us know what choice he has made. Prom each experiment a crowd of consequences will follow by a series of mathematical deductions, and thus each experiment will make known to us a corner of the universe. ' ' Behold what is for many people in the world, for scholars get- ting their first notions of physics, the origin of scientific certi- tude. This is what they suppose to be the role of experimenta- tion and mathematics. This same conception, a hundred years ago, was held by many savants who dreamed of constructing the world with as little as possible taken from experiment. On a little more reflection it was perceived how great a place hypothesis occupies ; that the mathematician can not do without it, still less the experimenter. And then it was doubted if all these constructions were really solid, and believed that a breath would overthrow them. To be skeptical in this fashion is still to be superficial. To doubt everything and to believe everything 1 are two equally convenient solutions ; each saves us from , thinking. Instead of pronouncing a summary condemnation, we ought therefore to examine with care the role of hypothesis ; we shall then recognize, not only that it is necessary, but that usually it is 27 28 SCIENCE AND HYPOTHESIS legitimate. We shall also see that there are several sorts of hy- potheses ; that some are verifiable, and once confirmed by experi- ment become fruitful truths; that others, powerless to lead us astray, may be useful to us In fixing our ideas; that others, finally, are hypotheses only in appearance and are reducible to disguised definitions or conventions. These last are met with above all in mathematics and the related sciences. Thence precisely it is that these sciences get their rigor; these conventions are the work of the free activity of our mind, which, in this domain, recognizes no obstacle. Here our mind can affirm, since it decrees ; but let us understand that while these decrees are imposed upon our science, which, without them, would be impossible, they are not imposed upon nature. Are they then arbitrary? No, else were they sterile. Experi- ment leaves us our freedom of choice, but it guides us by aiding us to discern the easiest way. Our decrees are therefore like those of a prince, absolute but wise, who consults his council of state. Some people have been struck by this character of free conven- tion recognizable in certain fundamental principles of the sciences. They have wished to generalize beyond measure, and, at the same time, they have forgotten that liberty is not license. Thus they have reached what is called nominalism, and have asked themselves if the savant is not the dupe of his own defi- nitions and if the world he thinks he discovers is not simply created by his own caprice.^ Under these conditions science would be certain, but deprived of significance. If this were so, science would be powerless. Now every day we see it work under our very eyes. That could not be if it taught us nothing of reality. Still, the things themselves are not what it can reach, as the naive dogmatists think, but only the relations between things. Outside of these relations there is no knowable reality. Such is the conclusion to which we shall come, but for that we must review the series of sciences from arithmetic and geometry to mechanics and experimental physics. iSee Le Eoy, 'Science et Philosophie, ' Revue de M4taphysique et de Morale, 1901. INTBODUCTION 29 What is the nature of mathematical reasoning? Is is really- deductive, as is commonly supposed? A deeper analysis shows us that it is not, that it partakes in a certain measure of the nature of inductive reasoning, and just because of this is it so fruitful. None the less does it retain its character of rigor absolute; this is the first thing that had to be shown. Knowing better now one of the instruments which mathemat- ics puts into the hands of the investigator, we had to analyze an- other fundamental notion, that of mathematical magnitude. Do we find it in nature, or do we ourselves introduce it there 1 And, in this latter case, do we not risk marring everything? Com- paring the rough data of our senses with that extremely complex and subtile concept which mathematicians call magnitude, we are forced to recognize a difference ; this frame into which we wish to force everything is of our own construction; but we have not made it at random. We have made it, so to speak, by measure and therefore we can make the facts fit into it without changing what is essential in them. Another frame which we impose on the world is space. Whence come the first principles of geometry? Are they im- posed on us by logic ? Lobachevski has proved not, by creating non-Euclidean geometry. Is space revealed to us by our senses ? Still no, for the space our senses could show us differs absolutely from that of the geometer. Is experience the source of geom- etry ? A deeper discussion will show us it is not. We therefore conclude that the first principles of geometry are only conven- tions ; but these conventions are not arbitrary and if transported into another world (that I call the non-Euclidean world and seek to imagine), then we should have been led to adopt others. In mechanics we should be led to analogous conclusions, and should see that the principles of this science, though more di- rectly based on experiment, still partake of the conventional character of the geometric postulates. Thus far nominalism triumphs ; but now we arrive at the physical sciences, properly so called. Here the scene changes; we meet another sort of hy- potheses and we see their fertility. Without doubt, at first blush, the theories seem to us fragile, and the history of science proves to us how ephemeral they are ; yet they do not entirely perish, 4 30 SCIENCE AND HYPOTHESIS and of each of them something remains. It is this something we must seek to disentangle, since there and there alone is the veritable reality. 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 reproduced. If all these circumstances could be reproduced at once, this prin- ciple could be applied without fear ; but that will never happen ; some of these circumstances will always be lacking. Are we absolutely sure they are unimportant? Evidently not. That may be probable, it can not be rigorously certain. Hence the important role the notion of probability plays in the physical sciences. The calculus of probabilities is therefore not merely a recreation or a guide to players of baccarat, and we must seek to go deeper with its foundations. Under this head I have been able to give only very incomplete results, so strongly does this vague instinct which lets us discern probability defy analysis. After a study of the conditions under which the physicist works, I have thought proper to show him at work. For that I have taken instances from the history of optics and of electricity. We shall see whence have sprung the ideas of Fresnel, of Max- well, and what unconscious hypotheses were made by Ampere and the other founders of electrodynamics. PART I NUMBER AND MAGNITUDE CHAPTER I On the Natuee of Mathematical Reasoning The very possibility of the science of mathematics seems an insoluble contradiction. If this science is deductive only in appearance, whence does it derive that perfect rigor no one dreams of doubting? If, on the contrary, all the propositions it enunciates can be deduced one from another by the rules of formal logic, why is not mathematics reduced to an immense tautology 1 The syllogism can teach us nothing essentially new, and, if everything is to spring from the principle of identity, everything should be capable of being reduced to it. Shall we then admit that the enunciations of all those theorems which fill I so many volumes are nothing but devious ways of saying J. is A ? ! "Without doubt, we can go back to the axioms, which are at the source of all these reasonings. If we decide that these can not be reduced to the principle of contradiction, if still less we see in them experimental facts which could not partake of mathe- matical necessity, we have yet the resource of classing them among synthetic a priori judgments. This is not to solve the diffi- culty, but only to baptize it ; and even if the nature of synthetic judgments were for us no mystery, the contradiction would not have disappeared, it would only have moved back ; syllogistic rea- soning remains incapable of adding anything to the data given it ; these data reduce themselves to a few axioms, and we should find nothing else in the conclusions. No theorem could be new if no new axiom intervened in its demonstration; reasoning could give us only the immediately 31 32 SCIENCE AND HYPOTHESIS evident verities borrowed from direct intuition ; it would be only an intermediary parasite, and therefore should we not have good reason to ask whether the whole syllogistic apparatus did not serve solely to disguise our borrowing? The contradiction will strike us the more if we open any book on mathematics; on every page the author will announce his in- tention of generalizing some proposition already known. Does the mathematical method proceed from the particular to the gen- eral, and, if so, how then can it be called deductive? If finally the science of number were purely analytic, or could be analytically derived from a small number of synthetic judgments, it seems that a mind sufSciently powerful could at a glance perceive aU its truths; nay more, we might even hope that some day one would invent to express them a language suffi- ciently simple to have them appear self-evident to an ordinary intelligence. If we refuse to admit these consequences, it must be conceded that mathematical reasoning has of itself a sort of creative virtue and consequently differs from the syllogism. The difference must even be profound. We shall not, for example, 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 results. All these modes of reasoning, whether or not they be reducible to the syllogism properly so caUed, retain the analytic character, and just because of that are powerless. II The discussion is old ; Leibnitz tried to prove 2 and 2 make 4 ; let us look a moment at his demonstration. I wiU suppose the number 1 defined and also the operation X -f 1 which consists in adding unity to a given number x. These definitions, whatever they be, do not enter into th^ course of the reasoning. I define then the numbers 2, 3 and 4 by the equalities (1) l-hl = 2; (2) 2-fl = 3; (3) 3-fl = 4. In the same way, I define the operation x + 2 by the relation: MATHEMATICAL BEASONING 33 (4) a; + 2=(a; + l)+l. That presupposed, we have 2 + 1 + 1 = 3 + 1 (Definition 2), 3 + 1 = 4 (Definition 3), 2 + 2=(2 + l)+l (Definition 4), ■whence 2 + 2 = 4 Q.E.D. It can not be denied that this reasoning is purely analytic. But ask any mathematician : ' That is not a demonstration prop- erly so called,' he will say to you: 'that is a verification.' "Wei have confined ourselves to comparing two purely conventional i definitions and have ascertained their identity ; we have learned j nothing new. Verification differs from true demonstration pre- cisely because it is purely analytic and because it is sterile. It is sterile because the conclusion is nothing but the premises trans- lated into another language. On the contrary, true demonstration is fruitful because the conclusion here is in a sense more general than the premises. The equality 2 + 2 = 4 is thus susceptible of a verification only because it is particular. Every particular enunciation in mathematics can always be verified in this same way. But if mathematics could be reduced to a series of such verifications, it would not be a science. So a chess-player, for example, does not create a science in winning a game. There is no science apart from the general. It may even be said the very object of the exact sciences is to spare us these direct verifications. Ill Let us, therefore, see the geometer at work and seek to catch his process. The task is not without difficulty ; it does not suffice to open a work at random and analyze any demonstration in it. "We must first exclude geometry, where the question is com- plicated by arduous problems relative to the role of the postu- lates, to the nature and the origin of the notion of space. For analogous reasons we can not turn to the infinitesimal analysis. 34 SCIEyCE AND HYPOTHESIS We must seek mathematical thought where it has remained pure, that is, in arithmetic. A choice still is necessary; in the higher parts of the theory of numbers, the primitive mathematical notions have already un- dergone an elaboration so profound that it becomes difficult to analyze them. It is, therefore, at the beginning of arithmetic that we must expect to find the explanation we seek, but it happens that pre- cisely in the demonstration of the most elementary theorems the authors of the classic treatises have shown the least precision and rigor. "We must not impute this to them as a crime; they have yielded to a necessity ; beginners are not prepared for real mathe- matical rigor ; they would see in it only useless and irksome sub- tleties ; it would be a waste of time to try prematurely to make them more exacting; they must pass over rapidly, but without skipping stations, the road traversed slowly by the founders of the science. Why is so long a preparation necessary to become habituated to this perfect rigor, which, it would seem, should naturally im- press itself upon aU good minds? This is a logical and psy- chological problem well worthy of study. But we shall not take it up ; it is foreign to our purpose ; all I wish to insist on is that, not to faU of our purpose, we must recast the demonstrations of the most elementary theorems and give them, not the crude form in which they are left, so as not to harass beginners, but the form that will satisfy a skilled geometer. DEFmrriON op Addition. — I suppose already defined the operation x-\-l, which consists in adding the number 1 to a given number x. This definition, whatever it be, does not enter into our sub- sequent reasoning. We now have to define the operation x-\-a, which consists in adding the number a to a given number x. Supposing we have defined the operation s-f (a — 1), the operation x-\-a will be defined by the equality (1) x + a=lx+ {a — l)]+i. MATHEMATICAL REASONING 35 We shall know then what x-{- a is when we know what x-{- (a — 1) is, and as I have supposed that to start with we knew what x-\-l is, we can define successively and 'by recur- rence ' the operations x-\-2, x-\-3, etc. This definition deserves a moment's attention; it is of a par- ticular nature which already distinguishes it from the purely logical definition; the equality (1) contains an infinity of dis- tinct definitions, each having a meaning only when one knows the preceding. Peopeeties of Addition. — Associativity. — I say that » + (6 + c) = (o + 6) + c. In fact the theorem is true for c = 1 ; it is then written a+(,b + l) = (,a + l)+l, which, apart from the difference of notation, is nothing but the equality (1), by which I have just defined addition. Supposing the theorem true for c = y,l say it will be true for C = y + 1. In fact, supposing (o-i-6) +y — a+^b + y), it follows that Ua + b)+y-]+l = la+ (6-1- 7)1+1 or by definition (1) (a + l) + (y + 1) =a + {b + y + 1) =a + [b + (7 + 1)1, which shows, by a series of purely analytic deductions, that the theorem is true for y -f- 1. Being true for c ^ 1, we thus see successively that so it is for c = 2, for = 3, etc. Commutativity. — 1° I say that The theorem is evidently true for a = l; we can verify by purely analytic reasoning that if it is true for o=y it will be true f or a = y -f 1 ; f or then (7-t-l)-f-l = (l+7) -l-i = i-F(7 + i); now it is true for a = l, therefore it will be true for a = 2, for = 3, etc., which is expressed by saying that the enunciated proposition is demonstrated by recurrence. 36 SCIENCE AND HYPOTHESIS 2° I say that a -\- b ■=::'b -\- a. The theorem has just been demonstrated for & = 1 ; it can be verified analytically that if it is true for 6 = ^, it will be true for 6 = ^ + 1. The proposition is therefore established by recurrence. Definition op Multipucation. — ^We shall define multiplica- tion by the equalities. (1) aXl = o. (2) oX6 = [oX (6 — l)] + a. Like equality (1), equality (2) contains an infinity of defini- tions ; having defined a X 1, it enables us to define successively : a X 2, a X 3, etc. Properties of Multiplication. — Distnbutivity. — I say that (o -f- 6) X c= (o X c) -1- (6 X c). We verify analytically that the equality is true f or c == 1 ; then that if the theorem is true for c = y, it will be true for c == y -f- 1. The proposition is, therefore, demonstrated by recurrence. Commutativity. — 1° I say that a X 1 = 1 X o. The theorem is evident for = 1. We verify analytically that if it is true for a = a, it will be true for a = a -|- 1. 2° I say that a X 6 = 6 X o. The theorem has just been proven for & = 1. We could verify analytically that if it is true for 6=/3, it will be true for IV Here I stop this monotonous series of reasonings. But this very monotony has the better brought out the procedure which is uniform and is met again at each step. This procedure is the demonstration by redurrence. We first establish a theorem for 7i = l; then we show that if it is true of M — 1, it is true of n, and thence conclude that it is true for all the whole numbers. MATHEMATICAL REASONING 37 We have just seen how it may be used to demonstrate the rules of addition and multiplication, that is to say, the rules of the algebraic calculus; this calculus is an instrument of transforma- tion, which lends itself to many more differing combinations than does the simple syllogism; but it is still an instrument purely analytic, and incapable of teaching us anything new. If mathe- matics had no other instrument, it would therefore be forth- with arrested in its development; but it has recourse anew to the same procedure, that is, to reasoning by recurrence, and it is able to continue its forward march. If we look closely, at every step we meet again this mode of reasoning, either in the simple form we have just given it, or under a form more or less modified. Here then we have the mathematical reasoning par excellence, and we must examine it more closely. V The essential characteristic of reasoning by recurrence is that it contains, condensed, so to speak, in a single formula, an infinity of syllogisms. That this may the better be seen, I will state one after another these syllogisms which are, if you will allow me the expression, arranged in 'cascade.' These are of course hypothetical syllogisms. The theorem is true of the number 1. Now, if it is true of 1, it is true of 2. ' Therefore it is true of 2. Now, if it is true of 2, it is true of 3. Therefore it is true of 3, and so on. We see that the conclusion of each syllogism serves as minor to the following. Furthermore the majors of all our syllogisms can be reduced to a single formula. If the theorem is true of w — 1, so it is of ii. We see, then, that in reasoning by recurrence we confine our- selves to stating the minor of the first syllogism, and the general formula which contains as particular cases all the majors. This never-ending series of syllogisms is thus reduced to a phrase of a few lines. 38 SCIENCE AND HYPOTHESIS It is now easy to comprehend why every particular conse- quence of a theorem can, as I have explained above, be verified by purely analytic procedures. If instead of showing that our theorem is true- of all num- bers, we only wish to show it true of the number 6, for example, it will suffice for us to establish the first 5 syllogisms of our cas- cade ; 9 would be necessary if we wished to prove the theorem for the number 10 ; more would be needed for a larger number ; but, however great this number might be, we should always end by reaching it, and the analytic verification would be possible. And yet, however far we thus might go, we could never rise to the general theorem, applicable to all numbers, which alone can be the object of science. To reach this, an infinity of syl- logisms would be necessary ; it would be necessary to overleap an abyss that the patience of the analyst, restricted to the resources of formal logic alone, never could fill up. 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 truths. The answer is now easy; a chess-player is able to combine four moves, five moves, in advance, but, however extraordinary he may be, he will never prepare more than a finite number of them; if he applies his faculties to arithmetic, he wiU not be able to perceive its general truths by a single direct intuition ; to arrive at the smallest theorem he can not dispense with the aid of reasoning by recurrence, for this is an instrument which enables us to pass from the finite to the infinite. This instrument is always useful, for, allowing us to overleap at a bound as many stages as we wish, it spares us verifications, long, irksome and monotonous, which would quickly become im- practicable. But it becomes indispensable as soon as we aim at the general theorem, to which analytic verification would bring us continually nearer without ever enabling us to reach it. In this domain of arithmetic, we may think ourselves very far from the infinitesimal analysis, and yet, as we have just seen, the idea of the mathematical infinite already plays a preponder- ant role, and without it there would be no science, because there would be nothing general. MATHEMATICAL SEASONING 39 VI The judgment on which reasoning by recurrence rests can be put under other forms; we may say, for example, that in an infinite collection of different whole numbers there is always one which is less than all the others. We can easily pass from one enunciation to the other and thus get the illusion of having demonstrated the legitimacy of reason- ing by recurrence. But we shall always be arrested, we shall always arrive at an undemonstrable axiom which will be in reality only the proposition to be proved translated into another language. "We can not therefore escape the conclusion that the rule of reasoning by recurrence is irreducible to the principle of con- tradiction. Neither can this rule come to us from experience; experience could teach us that the rule is true for the first ten or hundred numbers; for example, it can not attain to the indefinite series of numbers, but only to a portion of this serifes, more or less long but always limited. Now if it were only a question of that, the principle of con- tradiction would suffice ; it would always allow of our developing as many syllogisms as we wished ; it is only when it is a question of including an infinity of them in a single formula, it is only before the infinite that this principle fails, and there too, experi- ence becomes powerless. This rule, inaccessible to analytic demonstration and to experience, is the veritable type of the synthetic a priori judgment. On the other hand, we can not think of seeing in it a convention, as in some of the postulates of geometry. Why then does this judgment force itself upon us with an irresistible evidence? 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 possible. The mind has a direct intuition of this power, and experience can only give occasion for using it and thereby becoming conscious of it. But, one will say, if raw experience can not legitimatize reasoning by recurrence, is it so of experiment aided by indue- 40 SCIENCE AND HYPOTHESIS tion ? We see successively that a theorem is true of the number 1, of the number 2, of the number 3 and so on ; the law is evident, we say, and it has the same warranty as every physical law based on observations, whose number is very great but limited. Here is, it must be admitted, a striking analogy with the usual procedures of induction. But there is an essential difference. Induction applied to the physical sciences is always uncertain, because it rests on the belief in a general order of the universe, an order outside of us. Mathematical induction, that is, demon- stration by recurrence, on the contrary, imposes itself necessarily because it is only the affirmation of a property of the mind itself. VII Mathematicians, as I have said before, always endeavor to generalize the propositions they have obtained, and, to seek no other example, we have just proved the equality : a+l=l+a and afterwards used it to establish the equality a + h:=h + a which is manifestly more general. Mathematics can, therefore, like the other sciences, proceed from the particular to the general. This is a fact which would have appeared incomprehensible to us at the outset of this study, but which is no longer mys- terious to us, since we have ascertained the analogies between demonstration by recurrence and ordinary induction. Without doubt recurrent reasoning in mathematics and in- ductive reasoning in physics rest on different foundations, but their march is parallel, they advance in the same sense, that is to say, from the particular to the general. Let us examine the case a little more closely. To demonstrate the equality a + 2 = 2 + a it suffices to twice apply the rule (1) a+l = i + a and write (2) o + 2 = a-t-l + l = l-f-a-f-l = l + i+a = 2-|-o. MATHEMATICAL BEASONING 41 The equality (2) thus deduced in purely analytic way from the equality (1) is, however, not simply a particular case of it; it is something quite different. 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 word. The two members of the equality (2) are simply combinations more complicated than the two members of the equality (1), and analysis only serves to separate the elements which enter into these combinations and to study their relations. Mathematicians proceed therefore ' by construction, ' they ' con- struct' combinations more and more complicated. Coming back then by the analysis of these combinations, of these aggregates, so to speak, to their primitive elements, they perceive the rela- tions of these elements and from them deduce the relations of the aggregates themselves. This is a purely analytical proceeding, but it is not, however, a proceeding from the general to the particular, because evi- dently the aggregates can not be regarded as more particular than their elements. Great importance, and justly, has been attached to this pro- cedure of 'construction,' and some have tried to see in it the necessary and sufficient condition for the progress of the exact sciences. Necessary, without doubt ; but sufficient, no. For a construction to be useful and not a vain toil for the mind, that it may serve as stepping-stone to one wishing to mount, it must first of all possess a sort of unity enabling us to see in it something besides the juxtaposition of its elements. Or, more exactly, there must be some advantage in considering the construction rather than its elements themselves. What can this advantage be ? Why reason on a polygon, for instance, which is always de- composable into triangles, and not on the elementary triangles? It is because there are properties appertaining to polygons of any number of sides and that may be immediately applied to any particular polygon. Usually, on the contrary, it is only at the cost of the most 42 SCIENCE AND HTPOTHESIS prolonged exertions that they could be found by studying directly the relations of the elementary triangles. The knowl- edge of the general theorem spares us these efforts. A construction, therefore, becomes interesting only when it can be ranged beside other analogous constructions, forming spe- cies of the same genus. If the quadrilateral is something besides the juxtaposition of two triangles, this is because it belongs to the genus polygon. Moreover, one must be able to demonstrate the properties of the genus without being forced to establish them successively for each of the species. To attain that, we must necessarily mount from the particular to the general, ascending one or more steps. The analytic procedure 'by construction' does not oblige us to descend, but it leaves us at the same level. "We can ascend only by mathematical induction, which alone can teach us something new. Without the aid of this induction, different in certain respects from physical induction, but quite as fertile, construction would be powerless to create science. Observe finally that this induction is possible only if the same operation can be repeated indefinitely. That is why the theory of chess can never become a science, for the different moves of the same game do not resemble one another. CHAPTER II Mathematical Magnitude and Experience To learn what mathematicians understand by a continuum, one should not inquire of geometry. The geometer always seeks to represent to himself more or less the figures he studies, but his representations are for him only instruments; in making geometry he uses space just as he does chalk- so too much weight should not be attached to non-essentials, often of no more im- portance than the whiteness of the chalk. The pure analyst has not this rock to fear. He has disen- gaged the science of mathematics from aU foreign elements, and can answer our question: 'What exactly is this continuum about which mathematicians reason?' Many analysts who reflect on their art have answered already ; Monsieur Tannery, for example, in his Introduction a, la theorie des fonctions d'une variable. Let us start from the scale of whole numbers; between two consecutive steps, intercalate one or more intermediary steps, then between these new steps still others, and so on indefinitely. Thus we shall have an unlimited number of terms; these will be the numbers called fractional, rational or commensurable. But this is not yet enough ; between these terms, which, however, are already infinite in number, it is still necessary to intercalate others called irrational or incommensurable. A remark before going further. The continuum so conceived is only a collection of individuals ranged in a certain order, infinite in number, it is true, but exterior to one another. This is not the ordinary con- ception, wherein is supposed between the elements of the con- tinuum a sort of intimate bond which makes of them a whole, | where the point does not exist before the line, but the line before ' the point. Of the celebrated formula, 'the continuum is unity i in multiplicity,' only the multiplicity remains, the unity has disappeared. The analysts are none the less right in defining their continuum as they do, for they always reason on just this as soon as they pique themselves on their rigor. But this is 43 44 SCIENCE AND HTPOTHESIS enough to apprise us that the veritable mathematical continuum is a very different thing from that of the physicists and that of the metaphysicians. It may also be said perhaps that the mathematicians who are content with this definition are dupes of words, that it is neces- sary to say precisely what each of these intermediary steps is, to explain how they are to be intercalated and to demonstrate that it is possible to do it. But that would be wrong ; the only prop- erty of these steps which is used in their reasonings^ is that of being before or after such and such steps; therefore also this alone should occur in the definition. So how the intermediary terms should be intercalated need not concern us ; on the other hand, no one will doubt the possi- (bility of this operation, unless from forgetting that possible, in 'the language of geometers, simply means free from contradiction. Our definition, however, is not yet complete, and I return to it after this over-long digression. Definition op Incommensueables. — The mathematicians of the Berlin school, Kronecker in particular, have devoted them- selves to constructing this continuous scale of fractional and irra- tional numbers without using any material other than the whole number. The mathematical continuum would be, in this view, a pure creation of the mind, where experience would have no part. The notion of the rational number seeming to them to present no difficulty, they have chiefly striven to define the incommen- surable number. But before producing here their definition, I must make a remark to forestall the astonishment it is sure to arouse in readers unfamiliar with the customs of geometers. Mathematicians study not objects, but relations between ob- jects; the replacement of these objects by others is therefore j indifferent to them, provided the relations do not change. The J matter is for them unimportant, the form alone interests them. Without recalling this, it would scarcely be comprehensible that Dedekind should designate by the name incommensurable number a mere symbol, that is to say, something very different iWith those contained in the special conventions which serve to define addition and of which we shall speak later. MATHEMATICAL MAGNITUDE AND EXPERIENCE 45 from the ordinary idea of a quantity, which should be measurable and almost tangible. Let us see now what Dedekind's definition is: The commensurable numbers can in an infinity of ways be partitioned into two classes, such that any number of the first class is greater than any number of the second class. It may happen that among the numbers of the first class there is one smaller than all the others; if, for example, we range in the first class all numbers greater than 2, and 2 itself, and in the second class all numbers less than 2, it is clear that 2 will be the least of all numbers of the fiif^t class. The number 2 may be chosen as symbol of this partition. It may happen, on the contrary, that among the numbers of the second class is one greater than all the others; this is the case, for example, if the first class comprehends all numbers greater than 2, and the second all numbers less than 2, and 2 itself. Here again the number 2 may be chosen as symbol of this partition. But it may equally well happen that neither is there in the first class a number less than all the others, nor in the second class a number greater than all the others. Suppose, for ex- ample, we put in the first class all commensurable numbers whose squares are greater than 2 and in the second all whose squares are less than 2. There is none whose square is precisely 2. Evi- dently there is not in the first class a number less than all the others, for, however near the square of a number may be to 2, we can always find a commensurable number whose square is still closer to 2. In DedeMnd's view, the incommensurable number V2 or (2)i is nothing but the symbol of this particular mode of partition of commensurable numbers; and to each mode of partition cor- responds thus a number, commensurable or not, which serves as its symbol. But to be content with this would be to forget too far the origin of these symbols; it remains to explain how we have been led to attribute to them a sort of concrete existence, and, besides, 5 46 SCIENCE AND HYPOTHESIS does not the difficulty begin even for the fractional numbers themselves? Should we have the notion of these numbers if we had not previously known a matter that we conceive as infinitely divisible, that is to say, a continuum? The Phtsical Continuum.— We ask ourselves then if the notion of the mathematical continuum is not simply drawn from experience. If it were, the raw data of experience, which are our sensations, would be susceptible of measurement. We might be tempted to believe they really are so, since in these latter days the attempt has been made to measure them and a law has even been formulated, known as Fechner's law, according to which sensation is proportional to the logarithm of the stimulus. But if we examine more closely the experiments by which it has been sought to establish this law, we shall be led to a diametrically opposite conclusion. It has been observed, for ex- ample, that a weight A of 10 grams and a weight B of 11 grams produce identical sensations, that the weight B is just as indis- tinguishable from a weight C of 12 grams, but that the weight A is easily distinguished from the weight C. Thus the raw results of experience may be expressed by the following relations : A—B, Bz=C, AiT + U + Q). If T -\-U -\-Q were of the particular form I have above considered, no ambiguity would result; among the functions ^{T -\-U -\-Q) which remain constant, there would only be one of this particular form, and that I should convene to call energy. But as I have said, this is not rigorously the case; among the functions which remain constant, there is none which can be put rigorously under this particular form ; hence, how choose among them the one which should be called energy? We no longer have anything to guide us in our choice. There only remains for us one enunciation of the principle of the conservation of energy: There is something which remains constant. Under this form it is in its turn out of the reach of experiment and reduces to a sort of tautology. It is clear that if the world is governed by laws, there will be quantities which will remain constant. Like Newton's laws, and, for an analogous reason, the principle of the conservation of energy, founded on experiment, could no longer be invalidated by it. This discussion shows that in passing from the classic to the energetic system progress has been made; but at the same time it shows this progress is insufficient. Another objection seems to me still more grave: the prin- ciple of least action is applicable to reversible phenomena ; but it is not at all satisfactory in so far as irreversible phenomena are concerned ; the attempt by Helmholtz to extend it to this kind of phenomena did not succeed and could not succeed; in this regard everything remains to be done. The very statement of the prin- ciple of least action has something about it repugnant to the mind. To go from one point to another, a material molecule, acted upon by no force, but required to move on a surface, will take the geodesic line, that is to say, the shortest path. ENERGY AND THERMODYNAMICS 119 This molecule seems to know the point whither it is to go, to foresee the time it would take to reach it by such and such a route, and then to choose the most suitable path. The state- ment presents the molecule to us, so to speak, as a living and free being. Clearly it would be better to replace it by an enun- ciation less objectionable, and where, as the philosophers would say, final causes would not seem to be substituted for efficient causes. Thermodynamics.^ — The role of the two fundamental prin- ciples of thermodynamics in all branches of natural philosophy becomes daily more important. Abandoning the ambitious the- ories of forty years ago, which were encumbered by molecular hypotheses, we are trying to-day to erect upon thermodynamics alone the entire edifice of mathematical physics. Will the two principles of Mayer and of Clausius assure to it foundations solid enough for it to last some time? No one doubts it; but whence comes this confidence? An eminent physicist said to me one day d propos of the law of errors: "All the world believes it firmly, because the mathe- maticians imagine that it is a fact of observation, and the ob- servers that it is a theorem of mathematics. ' ' It was long so for the principle of the conservation of energy. It is no longer so to-day ; no one is ignorant that this is an experimental fact. 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 it? This is to ask whether it is legitimate, as is done every day, to generalize empirical data, and I shall not have the presumption to discuss this ques- tion, after so many philosophers have vainly striven to solve it. One thing is certain ; if this power were denied us, science could not exist or, at least, reduced to a sort of inventory, to the ascertaining of isolated facts, it would have no value for us, since it could give no satisfaction to our craving for order and harmony and since it would be at the same time incapable of foreseeing. As the circumstances which have preceded any fact will probably never be simultaneously reproduced, a first general- iThe following lines are a partial reproduction of the preface of my book Thermodynamique. 120 SCIENCE AND ETPOTEESIS ization is already necessary to foresee whether this fact will be reproduced again after the least of these circumstances shall be changed. But every proposition may be generalized in an infinity of ways. Among all the generalizations possible, we must choose, and we can only choose the simplest. "We are therefore led to act as if a simple law were, other things being equal, more probable than a complicated law. Half a century ago this was frankly confessed, and it was proclaimed that nature loves simplicity; she has since too often given us the lie. To-day we no longer confess this tendency, and we retain only so much of it as is indispensable if science is not to become impossible. In formulating a general, simple and precise law on the basis of experiments relatively few and presenting certain divergences, we have therefore only obeyed a necessity from which the human mind can not free itself. But there is something more, and this is why I dwell upon the point. No one doubts that Mayer's principle is destined to survive all the particular laws from which it was obtained, just as New- ton's law has survived Kepler's laws, from which it sprang, and which are only approximative if account be taken of perturbations. Why does this principle occupy thus a sort of privileged place among all the physical laws? There are many little reasons for it. First of all it is believed that we could not reject it or even doubt its absolute rigor without admitting the possibility of per- petual motion ; of course we are on our guard at such a prospect, and we think ourselves less rash in affirming Mayer's principle than in denying it. That is perhaps not wholly accurate ; the impossibility of per- petual motion implies the conservation of energy only for re- versible phenomena. The imposing simplicity of Mayer's principle likewise con- tributes to strengthen our faith. In a law deduced immediately from experiment, like Mariotte's, this simplicity would rather ENERGY AND THEBM0DYNAMIC8 121 seem to us 'a reason for distrust ; but here this is no longer the case; we see elements, at first sight disparate, arrange them- selves in an unexpected order and form a harmonious whole ; and we refuse to believe that an unforeseen harmony may be a simple effect of chance. It seems that our conquest is the dearer to us the more effort it has cost us, or that we are the surer of having wrested her true secret from nature the more jealously she has hidden it from us. But those are only little reasons; to establish Mayer's law as an absolute principle, a more profound discussion is necessary. But if this be attempted, it is seen that this absolute principle is not even easy to state. In each particular case it is clearly seen what energy is and at least a provisional definition of it can be given; but it is im- possible to find a general definition for it. If we try to enunciate the principle in all its generality and apply it to the universe, we see it vanish, so to speak, and nothing is left but this: There is something which remains constant. But has even this any meaning? In the determinist hypoth- esis, the state of the universe is determined by an extremely great number n of parameters which I shall call x^, x^, . . . x,,. As soon as the values of these n parameters at any instant are known, 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 calculated. In other words, these n parameters satisfy n differential equations of the first order. These equations admit of ?i — 1 integrals and consequently there are n — 1 functions of x.^, x^, ... Xn, which remain constant. // then we say there is something which remains constant, we only utter a tautology. We should even be puzzled to say which among all our integrals should retain the name of energy. Besides, Mayer's principle is not understood in this sense when it is applied to a limited system. It is then assumed that p of our parameters vary independently, so that we only have n^~p relations, generally linear, between our n parameters and their derivatives. 122 SCIENCE AND HYPOTHESIS To simplify the enunciation, suppose that the sum of the work of the external forces is null, as well as that of the quan- tities of heat given off to the outside. Then the signification of our principle will be : There is a combination of these n — p relations whose first member is an exact differential; and then this differential vanish- ing in virtue of our n — p relations, its integral is a constant and this integral is called energy. But how can it be possible that there are several parameters whose variations are independent ? That can only happen under the influence of external forces (although we have supposed, for simplicity, that the algebraic sum of the effects of these forces is null). In fact, if the system were completely isolated from all external action, the values of our n parameters at a given instant would suffice to determine the state of the system at any subsequent instant, provided always we retain the determinist hypothesis; we come back therefore to the same difficulty as above. If the future state of the system is not entirely determined by its present state, this is because it depends besides upon the state of bodies external to the system. But then is it probable that there exist between the parameters x, which define the state of the system, equations independent of this state of the external bodies ? and if in certain cases we believe we can find such, is this not solely in consequence of our ignorance and because the influ- ence of these bodies is too slight for our experimenting to detect it? If the system is not regarded as completely isolated, it is probable that the rigorously exact expression of its internal energy will depend on the state of the external bodies. Again, I have above supposed the sum of the external work was null, and if we try to free ourselves from this rather artificial restric- tion, the enunciation becomes still more difficult. To formulate Mayer's principle in an absolute sense, it is therefore necessary to extend it to the whole universe, and then we find ourselves face to face with the very difficulty we sought to avoid. In conclusion, using ordinary language, the law of the con- ENERGY AND THEBMODYNAMICS 123 servation of energy can have only one signification, which is that there is a property conunon to all the possibilities ; but on the determinist hypothesis there is only a single possibility, and then the law has no longer any meaning. On the indeterminist hypothesis, on the contrary, it would have a meaning, even if it were taken in an absolute sense; it would appear as a limitation imposed upon freedom. But this word reminds me that I am digressing and am on the point of leaving the domain of mathematics and physics. I check myself therefore and will stress of all this discussion only one impression, that Mayer's law is a form flexible enough for us to put into it almost whatever we wish. By that 1 do not mean it corresponds to no objective reality, nor that it reduces itself to a mere tautology, since, in each particular case, and provided one does not try to push to the absolute, it has a perfectly clear meaning. This flexibility is a reason for believing in its permanence, and as, on the other hand, it will disappear only to lose itself in a higher harmony, we may work with confidence, supporting ourselves upon it, certain beforehand that our labor will not be lost. Almost everything I have just said applies to the principle of Clausius. What distinguishes it is that it is expressed by an inequality. Perhaps it will be said it is the same with all physical laws, since their precision is always limited by errors of observation. But they at least claim to be first approxima- tions, and it is hoped to replace them little by little by laws more and more precise. If, on the other hand, the principle of Clau- sius reduces to an inequality, this is not caused by the imper- fection of our means of observation, but by the very nature of the question. General Conclusions on Pabt Thied The principles of mechanics, then, present themselves to us under two different aspects. On the one hand, they are truths founded on experiment and approximately verified so far as concerns almost isolated systems. On the other hand, they are 124 SCIENCE AND HYPOTHESIS postulates applicable to the totality of the universe and regarded as rigorously true. If these postulates possess a generality and a certainty which are lacking to the experimental verities whence they are drawn, this is because they reduce in the last analysis to a mere con- vention which we have the right to make, because we are certain beforehand that no experiment can ever contradict it. This convention, however, is not absolutely arbitrary; it does not spring from our caprice ; we adopt it because certain experi- ments have shown us that it would be convenient. Thus is explained how experiment could make the principles of mechanics, and yet why it can not overturn them. Compare with geometry: The fundamental propositions of geometry, as for instance Euclid's postulate, are nothing more than conventions, and it is just as unreasonable to inquire whether they are true or false as to ask whether the metric sys- tem is true or false. Only, these conventions are convenient, and it is certain experi- ments which have taught us that. At first blush, the analogy is complete; the role of experi- ment seems the same. One will therefore be teiapted to say: Either mechanics must be regarded as an experimental science, and then the same must hold for geometry; or else, on the con- trary, geometry is a deductive science, and then one may say as much of mechanics. Such a conclusion would be illegitimate. 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 studies ; they bear on the properties of solid bodies, on the rectilinear propagation of light. They are experiments of mechanics, experiments of optics; they can not in any way be regarded as experiments of geometry. And even the principal reason why our geometry seems convenient to us is that the different parts of our body, our eye, our limbs, have the properties of solid bodies. On this account, our funda- mental experiments are preeminently physiological experiments, which bear, not on space which is the object the geometer must ENERGY AND THERMODYNAMICS 125 study, but on his body, that is to say, on the instrument he must use for this study. On the contrary, the fundamental conventions of mechanics, and the experiments which prove to us that they are convenient, bear on exactly the same objects or on analogous objects. The conventional and general principles are the natural and direct generalization of the experimental and particular principles. Let it not be said that thus I trace artificial frontiers between the sciences; that if I separate by a barrier geometry properly so called from the study of solid bodies, I could just as well erect one between experimental mechanics and the conventional me- chanics of the general principles. In fact, who does not see that in separating these two sciences I mutilate them both, 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 com- pared to that superb body of doctrine called geometry? One sees now why the teaching of mechanics should remain experimental. Only thus can it make us comprehend the genesis of the science, and that is indispensable for the complete understanding of the science itself. Besides, if we study mechanics, it is to apply it; and we can apply it only if it remains objective. Now, as we have seen, what the principles gain in generality and certainty they lose in objec- tivity. It is, therefore, above all with the objective side of the principles that we must be familiarized early, and that can be done only by going from the particular to the general, instead of the inverse. The principles are conventions and disguised definitions. Yet they are drawn from experimental laws; these laws have, so to speak, been exalted into principles to which our mind attri- butes an absolujie value. Some philosophers have generalized too far; they believed the principles were the whole science and consequently that the whole science was conventional. This paradoxical doctrine, called nominalism, will not bear examination. 10 126 SCIENCE AND HYPOTHESIS How can a law become a principle? It expressed a relation between two real terms A and B. But it was not rigorously true, it was only approximate. We introduce arbitrarily an inter- mediary term C more or less fictitious, and C is by defimtion that which has with A exactly the relation expressed by the law. Then our law is separated into an absolute and rigorous prin- ciple which expresses the relation of A to C and an experimental law, approximate and subject to revision, which expresses the relation of C to B. It is clear that, however far this partition is pushed, some laws will always be left remaining. We go to enter now the domain of laws properly so called. PART IV NATURE CHAPTER IX Hypotheses in Physics The R6le op Experiment and Generalization. — Experiment is the sole source of truth. It alone can teach us anything new ; it alone can give us certainty. These are two points that can not be questioned. But then, if experiment is everything, what place will remain for mathematical physics ? What has experimental physics to do with such an aid, one which seems useless and perhaps even dangerous ? And yet mathematical physics exists, and has done unquestion- able service. "We have here a fact that must be explained. The explanation is that merely to observe is not enough. We must use our observations, and to do that we must generalize. This is what men always have done ; only as the memory of past errors has made them more and more careful, they have observed more and more, and generalized less and less. Every age has ridiculed the one before it, and accused it of having generalized too quickly and too naively. Descartes pitied the lonians; Descartes, in his turn, makes us smile. No doubt our children will some day laugh at us. But can we not then pass over immediately to the goal? Is not this the means of escaping the ridicule that we foresee ? Can we not be content with just the bare experiment? No, that is impossible; it would be to mistake utterly the true nature of science. The scientist must set in order. Science is built up with facts, as a house is with stones. But a collection of facts is no more a science than a heap of stones is a house. 127 128 SCIENCE AND HYPOTHESIS And above all the scientist must foresee. Carlyle has some- where said something like this: "Nothing but facts are of im- portance. John Lackland passed by here. Here is something that is admirable. Here is a reality for which I would give all the theories in the world. ' ' Carlyle was a fellow countryman of Bacon ; but Bacon would not have said that. That is the language of the historian. The physicist would say rather : ' ' John Lack- land passed by here; that makes no difference to me, for he never will pass this way again. " We all know that there are good experiments and poor ones. The latter will accumulate in vain ; though one may have made a hundred or a thousand, a single piece of work by a true master, by a Pasteur, for example, will suffice to tumble them into oblivion. Bacon would have well understood this ; it is he who invented the phrase Experimentum crucis. But Carlyle would not have under- stood it. A fact is a fact. A pupil has read a certain number on his thermometer ; he has taken no precaution ; no matter, he has read it, and if it is only the fact that counts, here is a reality of the same rank as the peregrinations of King John Lackland. Why is the fact that this pupil has made this reading of no interest, while the fact that a skilled physicist had made another reading might be on the contrary very important 1 It is because from the first reading we could not infer anything. What then is a good experiment? It is that which informs us of something besides an isolated fact ; it is that which enables us to foresee, that is, that which enables us to generalize. For without generalization foreknowledge is impossible. The circumstances under which one has worked will never reproduce themselves all at once. The observed action then will never recur ; the only thing that can be affirmed is that under analogous cir- cumstances an analogous action will be produced. In order to foresee, then, it is necessary to invoke at least analogy, that is to say, already then to generalize. No matter how timid one may be, still it is necessary to inter- polate. Experiment gives us only a certain number of isolated points. We must unite these by a continuous line. This is a veritable generalization. But we do more ; the curve that we shall trace will pass between the observed points and near these points ; HYPOTHESES IN PHYSICS 129 it will not pass through these points themselves. Thus one does not restrict himself to generalizing the experiments, but corrects them ; and the physicist who should try to abstain from these cor- rections and really be content with the bare experiment, would be forced to enunciate some very strange laws. The bare facts, then, would not be enough for us ; and that is why we must have science ordered, or rather organized. It is often said experiments must be made without a pre- conceived idea. That is impossible. Not only would it make all experiment barren, but that would be attempted which could not be done. Every one carries in his mind his own conception of the world, of which he can not so easily rid himself. We must, - for instance, use language ; and our language is made up only of preconceived ideas and can not be otherwise. Only these are unconscious preconceived ideas, a thousand times. more dangerous - than the others. Shall we say that if we introduce others, of which we are fully conscious, we shall only aggravate the evil? I think not. I believe rather that they will serve as counterbalances to each other — I was going to say as antidotes ; they will in general accord, ill with one another — they will come into conflict with one another, and thereby force us to regard things under different aspects. This is enough to emancipate us. He is no longer a slave who can choose his master. Thus, thanks to generalization, each fact observed enables us ^ to foresee a great many others ; only we must not forget that the first alone is certain, that all others are merely probable. No matter how solidly founded a prediction may appear to us, we are never absolutely sure that experiment will not contradict it, if we undertake to verify it. The probability, however, is often so great that practically we may be content with it. It is far better to foresee even without certainty than not to foresee at all. One must, then, never disdain to make a verification when opportunity offers. But all experiment is long and difficult ; the workers are few ; and the number of facts that we need to foresee is immense. Compared with this mass the number of direct verifi- cations that we can make will never be anything but a negligible quantity. 130 SCIENCE AND HYPOTHESIS ■ Of this few that we can directly attain, we must make the best use ; it is very necessary to get from every experiment the greatest possible number of predictions, and with the highest possible degree of probability. The problem is, so to speak, to increase the yield of the scientific machine. Let us compare science to a library that ought to grow continu- ally. The librarian has at his disposal for his purchases only insufSeient funds. He ought to make an effort not to waste them. It is experimental physics that is entrusted with the purchases. It alone, then, can enrich the library. As for mathematical physics, its task will be to make out the catalogue. If the catalogue is well made, the library will not be any richer, but the reader will be helped to use its riches. And even by showing the librarian the gaps in his collections, it will enable him to make a judicious use of his funds ; which is all the more important because these funds are entirely inadequate. Such, then, is the role of mathematical physics. It must direct generalization in such a manner as to increase what I just now called the yield of science. By what means it can arrive at this, and how it can do it without danger, is what remains for us to investigate. The Unity of Nature. — Let us notice, first of all, that every generalization implies in some measure the belief in the unity and simplicity of nature. As to the unity there can be no diffi- culty. If the different parts of the universe were not like the members of one body, they would not act on one another, they would know nothing of one another ; and we in particular would know only one of these parts. "We do not ask, then, if nature is one, but how it is one. As for the second point, that is not such an easy matter. It is not certain that nature is simple. Can we without danger act as if it were 1 There was a time when the simplicity of Mariotte's law was an argument invoked in favor of its accuracy ; when Fresnel him- self, after having said in a conversation with Laplace that nature was not concerned about analytical difficulties, felt himself obliged to make explanations, in order not to strike too hard at prevailing opinion. HYPOTHESES IN PHYSICS 131 To-day idea/S have greatly changed ; and yet, those who do not believe that natural laws have to be simple, are stiU often obliged to act as if they did. They coidd not entirely avoid this necessity without making impossible all generalization, and consequently all science. It is clear that any fact can be generalized in an infinity of ways, and it is a question of choice. The choice can be guided only by considerations of simplicity. Let us take the most com- monplace case, that of interpolation. "We pass a continuous line, as regular as possible, between the points given by observation. Why do we avoid points making angles and too abrupt turns? Why do we not make our curve describe the most capricious zig- zags ? It is because we know beforehand, or believe we know, that the law to be expressed can not be so complicated as all that. We may calculate the mass of Jupiter from either the move- ments of its satellites, or the perturbations of the major planets, or those of the minor planets. If we take the averages of the determinations obtained by these three methods, we find three numbers very close together, but different. We might interpret this result by supposing that the coefficient of gravitation is not the same in the three cases. The observations would certainly be much better represented. Why do we reject this interpretation? Not because it is absurd, but because it is needlessly complicated. We shall only accept it when we are forced to, and that is not yet. To sum up, ordinarily every law is held to be simple till the ' contrary is proved. This custom is imposed upon physicists by the causes that I have just explained. But how shall we justify it in the presence of discoveries that show us every day new details that are richer and more complex? How shall we even reconcile it with the belief in the unity of nature? For if everything depends on everything, relationships where so many diverse factors enter can no longer be simple. If we study the history of science, we see happen two inverse phenomena, so to speak. Sometimes simplicity hides under com- plex appearances ; sometimes it is the simplicity which is appar- ent, and which disguises extremely complicated realities. What is more complicated than the confused movements of 132 SCIENCE AND HYPOTHESIS I the planets? What simpler than Newton's law? Here nature, I making sport, as Fresnel said, of analytical difficulties, employs only simple means, and by comhining them produces I know not ! what inextricable tangle. Here it is the hidden simplicity which must be discovered. Examples of the opposite abound. In the kinetic theory of gases, one deals with molecules moving with great velocities, whose paths, altered by incessant collisions, have the most capri- cious forms and traverse space in every direction. The observable result is Mariotte's simple law. Every individual fact was com- plicated. The law of great numbers has reestablished simplicity I in the average. Here the simplicity is merely apparent, and only the coarseness of our senses prevents our perceiving the complexity. Many phenomena obey a law of proportionality. But why? Because in these phenomena there is something very small. The ' simple law observed, then, is only a result of the general ana- lytical rule that the infinitely small increment of a function is proportional to the increment of the variable. As in reality our increments are not infinitely small, but very small, the law of proportionality is only approximate, and the simplicity is only apparent. What I have just said applies to the rule of the super- position of smaU motions, the use of which is so fruitful, and which is the basis of optics. And Newton's law itself? Its simplicity, so long undetected, is perhaps only apparent. Who knows whether it is not due to some complicated mechanism, to the impact of some subtile matter animated by irregular movements, and whether it has not become simple only through the action of averages and of great num- bers? In any ease, it is difficult not to suppose that the true law contains complementary terms, which would become sensible at small distances. If in astronomy they are negligible as modify- ing Newton's law, and if the law thus regains its simplicity, it would be only because of the immensity of celestial distances. No doubt, if our means of investigation should become more and more penetrating, we should discover the simple under the complex, then the complex under the simple, then again the simple under the complex, and so on, without our being able to foresee what will be the last term. HYPOTHESES IN PHYSICS 133 We must stop somewhere, and that science may be possible, we must stop when we have found simplicity. This is the only ground on which we can rear the edifice of our generalizations. But this simplicity being only apparent, wiU the ground be firm enough? This is what must be investigated. For that purpose, let us see what part is played in our gener- alizations by the belief in simplicity. "We have verified a simple law in a good many particular cases ; we refuse to admit that this agreement, so often repeated, is simply the result of chance, and conclude that the law must be true in the general case. Kepler notices that a planet's positions, as observed by Tycho, are all on one ellipse. 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 ellipse. What does it matter then whether the simplicity be real, or ' whether it covers a complex reality? Whether it is due to the influence of great numbers, which levels down individual differ- ences, or to the greatness or smallness of certain quantities, which allows us to neglect certain terms, in no case is it due to chance. This simplicity, real or apparent, always has a cause. We can always follow, then, the same course of reasoning, and if a simple law has been observed in several particular cases, we can legiti- mately suppose that it will still be true in analogous eases. To refuse to do this would be to attribute to chance an inadmis- sible role. There is, however, a difference. If the simplicity were real and essential, it would resist the increasing precision of our means of measure. If then we believe nature to be essentially simple, we must, from a simplicity that is approximate, infer a simplicity;, that is rigorous. This is what was done formerly; and this is what we no longer have a right to do. The simplicity of Kepler's laws, for example, is only apparent. That does not prevent their being applicable, very nearly, to all systems analogous to the solar system ; but it does prevent their being rigorously exact. The E6le of Hypothesis. — ^AU generalization is a hypothesis. Hypothesis, then, has a necessary role that no one has ever con- 134 SCIENCE AND HYPOTHESIS tested. Only, it ought always, as soon as possible and as often as possible, to be subjected to verification. And, of course, if it does not stand this test, it ought to be abandoned without reserve. This is what we generally do, but sometimes with rather an ill humor. "Well, even this ill humor is not justified. The physicist who has just renounced one of his hypotheses ought, on the contrary, to be full of joy; for he has found an unexpected opportunity for discovery. His hypothesis, I imagine, had not been adopted without consideration; it took account of all the known factors that it seemed could enter into the phenomenon. If the test does not support it, it is because there is something unexpected and extraordinary ; and because there is going to be something found that is unknown and new. Has the discarded hypothesis, then, been barren? Far from that, it may be said it has rendered more service than a true hypothesis. Not only has it been the occasion of the decisive experiment, but, without having made the hypothesis, the experi- ment would have been made by chance, so that nothing would have been derived from it. One would have seen nothing ex- traordinary ; only one fact the more would have been catalogued without deducing from it the least consequence. Now on what condition is the use of hypothesis without danger? The firm determination to submit to experiment is not enough ; there are stiU dangerous hypotheses; first, and above all, those which are tacit and unconscious. Since we make them without iknowing it, we are powerless to abandon them. Here again, then, is a service that mathematical physics can render us. By the precision that is characteristic of it, it compels us to formulate aU the hypotheses that we should make without it, but uncon- sciously. Let us notice besides that it is important not to multiply hypotheses beyond measure, and to make them only one after the other. If we construct a theory based on a number of hypotheses, and if experiment condemns it, which of our premises is it neces- sary to change? It will be impossible to know. And inversely, if the experiment succeeds, shall we believe that we have demon- HYPOTHESES IN PHTSICS 135 strated all the hypotheses at once? Shall we believe that with one single equation we have determined several unknowns? "We must equally take care to distinguish between the different kinds of hypotheses. There are first those which are perfectly natural and from which one can scarcely escape. It is difficult not to suppose that the influence of bodies very remote is quite negligible, that small movements follow a linear law, that the effect is a continuous function of its cause. I will say as much of the conditions imposed by symmetry. All these hypotheses form, as it were, the common basis of all the theories of mathe- matical physics. They are the last that ought to be abandoned. There is a second class of hypotheses, that I shall term neutral. In most questions the analyst assumes at the beginning of his calculations either that matter is continuous or, on the contrary, that it is formed of atoms. He might have made the opposite assumption without changing his results. He would only have had more trouble to obtain them ; that is all. If, then, experiment confirms his conclusions, will he think that he has demonstrated, for instance, the real existence of atoms ? In optical theories two vectors are introduced, of which one is regarded as a velocity, the other as a vortex. Here again is a neutral hypothesis, since the same conclusions would have been reached by taking precisely the opposite. The success of the experiment, then, can not prove that the first vector is indeed a velocity; it can only prove one thing, that it is a vector. This is the only hypothesis that has really been introduced in the premises. In order to give it that concrete appearance which the ) weakness of our minds requires, it has been necessary to consider it either as a velocity or as a vortex, in the same way that it has been necessary to represent it by a letter, either x or y. The result, however, whatever it may be, 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 y. These neutral hypotheses are never dangerous, if only their character is not misunderstood. They may be useful, either as devices for computation, or to aid our understanding by concrete images, to fix our ideas as the saying is. There is, then, no occa- sion to exclude them. 136 SCIENCE AND HYPOTHESIS The hypotheses of the third class are the real generalizations. They are the ones that experiment must confirm or invalidate. Whether verified or condemned, they will always be fruitful. But for the reasons that I have set forth, they wiU only be fruit- ful if they are not too numerous. Origin of Mathematical Physics. — Let us penetrate further, and study more closely the conditions that have permitted the development of mathematical physics. 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 num- ber of elementary phenomena. This is done in three different ways : first, in time. Instead of embracing in its entirety the progressive development of a phenomenon, the aim is simply to connect each instant with the instant immediately preceding it. It is admitted that the actual state of the world depends only on the immediate past, without being directly influenced, so to speak, by the memory of a distant past. Thanks to this postulate, instead of studying directly the whole succession of phenomena, it is possible to confine ourselves to writing its 'differential equation.' For Kepler's laws we sub- stitute Newton's law. Next we try to analyze the phenomenon in space. What ex- periment gives us is a confused mass of facts presented on a. stage of considerable extent. We must try to discover the ele- mentary phenomenon, which will be, on the contrary, localized in a very small region of space. Some examples will perhaps make my thought better under- stood. If we wished to study in all its complexity the distribu- tion of temperature in a cooling solid, we should never succeed. Everything becomes simple if we reflect that one point of the solid can not give up its heat directly to a distant point ; it will give up its heat only to the points in the immediate neighbor- hood, and it is by degrees that the flow of heat can reach other parts of the solid. The elementary phenomenon is the exchange of heat between two contiguous points. It is strictly localized, and is relatively simple, if we admit, as is natural, that it is not influenced by the temperature of molecules whose distance is sensible. HYPOTHESES IN PHYSICS 137 I bend a rod. It is going to take a very complicated form, the direct study of which would be impossible. But I shall be able, however, to attack it, if I observe that its flexure is a result only of the deformation of the very small elements of the rod, and that the deformation of each of these elements depends only on the forces that are directly applied to it, and not at all on those which may act on the other elements. In all these examples, which I might easily multiply, we admit that there is no action at a distance, or at least at a great distance. This is a hypothesis. It is not always true, as the law of gravitation shows us. It must, then, be submitted to veri- fication. If it is confirmed, even approximately, it is precious, for it will enable us to make mathematical physics, at least by successive approximations. If it does not stand the test, we must look for something else analogous; for there are still other means of arriving at the elementary phenomenon. If several bodies act simultaneously, it may happen that their actions are independent and are simply added to one another, either as vectors or as scalars. The ele- mentary phenomenon is then the action of an isolated body. Or again, we have to deal with small movements, or more generally with small variations, which obey the well-known law of super- position. The observed movement will then be decomposed into simple movements, for example, sound into its harmonics, white light into its monochromatic components. When we have discovered in what direction it is advisable to look for the elementary phenomenon, by what means can we reach it ? First of all, it will often happen that in order to detect it, or rather to detect the part of it useful to us, it will not be neces- sary to penetrate the mechanism ; the law of great numbers will suffice. Let us take again the instance of the propagation of heat. Every molecule emits rays toward every neighboring molecule. According to what law, we do not need to know. If we should make any supposition in regard to this, it would be a neutral hypothesis and consequently useless and incapable of verification. And, in fact, by the action of averages and thanks to the sym- 138 SCIENCE AND HYPOTHESIS metry of the medium, all the differences are leveled down, and whatever hjrpothesis may be made, the result is always the same. The same circumstance is presented in the theory of electricity and in that of capillarity. The neighboring molecules attract and repel one another. We do not need to know according to what law ; it is enough for us that this attraction is sensible only at small distances, that the molecules are very numerous, that the medium is symmetrical, and we shall only have to let the law of great numbers act. Here again the simplicity of the elementary phenomenon was hidden under the complexity of the resultant observable phe- nomenon ; but, in its turn, this simplicity was only apparent, and concealed a very complex mechanism. The best means of arriving at the elementary phenomenon would evidently be experiment. We ought by experimental con- trivance to dissociate the complex sheaf that nature offers to our researches, and to study with care the elements as much isolated as possible. For example, natural white light would be decom- posed into monochromatic lights by the aid of the prism, and into polarized light by the aid of the polarizer. Unfortunately that is neither always possible nor always suffi- cient, and sometimes the mind must outstrip experiment. I shall cite only one example, which has always struck me forcibly. If I decompose white light, I shall be able to isolate a small part of the speptrum, but however small it may be, it will retain a certain breadth. Likewise the natural lights, called monochro- matic, give us a very narrow line, but not, however, infinitely narrow. It might be supposed that by studying experimentally the properties of these natural lights, by working with finer and finer lines of the spectrum, and by passing at last to the limit, so to speak, we should succeed in learning the properties. of a light strictly monochromatic. That would not be accurate. Suppose that two rays emanate from the same source, that we polarize them first in two perpen- dicular planes, then bring them back to the same plane of polari- zation, and try to make them interfere. If the light were strictly monochromatic, they would interfere. With our lights, which are nearly monochromatic, there will be no interference, and HYPOTHESES IN PHYSICS 139 that no matter how narrow the line. In order to be otherwise it would have to be several million times as narrow as the finest known lines. Here, then, the passage to the limit would have deceived us. The mind must outstrip the experiment, and if it has done so with success, it is because it has allowed itself to be guided by the instinct of simplicity. i The knowledge of the elementary fact enables us to put the problem in an equation. Nothing remains but to deduce from this by combination the complex fact that can be observed and verified. This is what is called integration, and is the business of the mathematician. It may be asked why, in physical sciences, generalization so readily takes the mathematical form. The reason is now easy to see. It is not only because we have numerical laws to express ; it is because the observable phenomenon is due to the superposition of a great number of elementary phenomena all alike. Thus quite naturally are introduced differential equations. It is not enough that each elementary phenomenon obeys sim- f pie laws ; all those to be combined must obey the same law. Then / only can the intervention of mathematics be of use ; mathematics teaches us in fact to combine like with like. Its aim is to learn the result of a combination without needing to go over the com- bination piece by piece. If we have to repeat several times the same operation, it enables us to avoid this repetition by telling us in advance the result of it by a sort of induction. I have ex- plained this above, ia the chapter on mathematical reasoning. But, for this, all the operations must be aUke. In the opposite case, it would evidently be necessary to resign ourselves to doing them in reality one after another, and mathematics would become useless. It is then thanks to the approximate homogeneity of the ' matter studied by physicists, that mathematical physics could be born. In the natural sciences, we no longer find these conditions: homogeneity, relative independence of remote parts, simplicity of the elementary fact; and this is why naturalists are obliged to resort to other methods of generalization. CHAPTER X The Theories of Modern Physics Meaning op Physical Theories. — The laity are struck to see how ephemeral scientific theories are. After some years of prosperity, they see them successively abandoned ; they see ruins accumulate upon ruins ; they foresee that the theories fashionable to-day will shortly succumb in their turn and hence they con- clude that these are absolutely idle. This is what they call the iankruptcy of science. Their scepticism is superficial ; they give no account to them- selves of the aim and the role of scientific theories; otherwise they would comprehend that the ruins may still be good for something. No theory seemed more solid than that of Fresnel which attributed light to motions of the ether. Yet now Maxwell's is preferred. Does this mean the work of Fresnel was in vain? No, because the aim of Fresnel was not to find out whether there is really an ether, whether it is or is not formed of atoms, whether these atoms really move in this or that sense ; his object was to foresee optical phenomena. Now, Fresnel 's theory always permits of this, to-day as well as before Maxwell. The differential equations are always true; they can always be integrated by the same procedures and the results of this integration always retain their value. And let no one say that thus we reduce physical theories to the role of mere practical recipes; these equations express rela- tions, and if the equations remain true it is because these rela- tions preserve their reality. They teach us, now as then, that there is such and such a relation between some thing and some other thing; only this something formerly we called motion; we now call it electric current. But these appellations were only images substituted for the real objects which nature will eternally hide from us. The true relations between these real objects are the only reality we can attain to, and the only condition is that 140 THE THEOBIES OF MODERN PHYSICS 141 the same relations exist between these objects as between the images by which we are forced to replace them. If these rela- tions are known to us, what matter if we deem it convenient to replace one image by another. That some periodic phenomenon (an electric oscillation, for instance) is really due to the vibration of some atom which, act- ing like a pendulum, really moves in this or that sense, is neither certain nor interesting. But that between electric oscillation, the motion of the pendulum and all periodic phenomena there exists a close relationship which corresponds to a profound real- ity ; that this relationship, this similitude, or rather this parallel- ism extends into details ; that it is a consequence of more general principles, that of energy and that of least action; this is what we can affirm; this is the truth which wiU. always remain the same under all the costumes in which we may deem it useful to deck it out. Numerous theories of dispersion have been proposed; the first was imperfect and contained only a small part of truth. Afterwards came that of Helmholtz ; then it was modified in vari- ous ways, and its author himself imagined another founded on the principles of Maxwell. But, what is remarkable, all the sci- entists who came after Helmholtz reached the same equations, starting from points of departure in appearance very widely separated. I will venture to say these theories are all true at the same time, not only because they make us foresee the same phenomena, but because they put in evidence a true relation, that of absorption and anomalous dispersion. What is true in thef premises of these theories is what is common to all the authors ; \ this is the affirmation of this or that relation between certain things which some call by one name, others by another. ' The kinetic theory of gases has given rise to many objections, which we could hardly answer if we pretended to see in it the absolute truth. But all these objections will not preclude its having been useful, and particularly so in revealing to us a relation true and but for it profoundly hidden, that of the gaseous pressure and the osmotic pressure. In this sense, then, it may be said to be true. When a physicist finds a contradiction between two theories 11 142 SCIENCE AND HYPOTHESIS equally dear to him, he sometimes says: "We will not bother about that, but hold firmly the two ends of the chain, though the intermediate links are hidden from us." This argument of an embarrassed theologian would be ridiculous if it were necessary to attribute to physical theories the sense the laity give them. In case of contradiction, one of them at least must then be re- garded as false. It is no longer the same if in them be sought only what should be sought. May be they both express true relations and the contradiction is only in the images wherewith we have clothed the reality. To those who find we restrict too much the domain accessible to the scientist, I answer: These questions which we interdict to you and which you regret, are not only insoluble, they are illusory and devoid of meaning. Some philosopher pretends that aU physics may be explained by the mutual impacts of atoms. If he merely means there are between physical phenomena the same relations as between the mutual impacts of a great number of balls, well and good, that is verifiable, that is perhaps true. But he means something more ; and we think we understand it because we think we know what impact is in itself; why? Simply because we have often seen games of biUiards. Shall we think God, contemplating his work, feels the same sensations as we in watching a billiard match ? If we do not wish to give this bizarre sense to his asser- tion, if neither do we wish the restricted sense I have just ex- plained, which is good sense, then it has none. Hypotheses of this sort have therefore only a metaphorical sense. The scientist should no more interdict them than the poet does metaphors ; but he ought to know what they are worth. They may be useful to give a certain satisfaction to the mind, and they will not be injurious provided they are only indifferent hypotheses. These considerations explain to us why certain theories, sup- posed to be abandoned and finally condemned by experiment, suddenly arise from their ashes and recommence a new life. It is because they expressed true relations; and because they had not ceased to do so when, for one reason or another, we felt it necessary to enunciate the same relations in another language. So they retained a sort of latent life. THE THEORIES OF MODERN PHYSICS 143 Scarcely fifteen years ago was thpre anything more ridiculous, more naively antiquated, than Coulomb's fluids? And yet here they are reappearing under the name of electrons. Wherein do these permanently electrified molecules differ from Coulomb's electric molecules ? It is true that in the electrons the electricity is supported by a little, a very little matter ; in other words, they have a mass (and yet this is now contested) ; but Coulomb did not deny mass to his fluids, or, if he did, it was only with reluc- tance. It would be rash to affirm that the belief in electrons will not again suffer eclipse ; it ^as none the less curious to note this unexpected resurrection. But the most striking example is Carnot's principle. Camot set it up starting from false hypotheses ; when it was seen that heat is not indestructible, but may be transformed into work, his ideas were completely abandoned; afterwards Clausius returned to them and made them finally triumph. Carnot's theory, under its primitive form, expressed, aside from true relations, other inexact relations, debris of antiquated ideas ; but the presence of these latter did not change the reality of the others. Clausius had only to discard them as one lops off dead branches. The result was the second fundamental law of thermodynamics. There were always the same relations ; though these relations no longer subsisted, at least in appearance, between the same ob- jects. This was enough for the principle to retain its value. And even the reasonings of Carnot have not perished because of that ; they were applied to a material tainted with error ; but their form (that is to say, the essential) remained correct. What I have just said illuminates at the same time the role of general principles such as the principle of least action, or that of the conservation of energy. These principles have a very high value; they were obtained in seeking what there was in common in the enunciation of nu- merous physical laws; they represent therefore, as it were, the quintessence of innumerable observations. However, from their very generality a consequence results to which I have called attention in Chapter VIII., namely, that they can no longer be verified. As we can not give a general definition of energy, the principle of the conservation of energy 144 SCIENCE AND HYPOTHESIS signifies simply that there is something which remains constant. : Well, whatever be the new notions that future experiments shall give us about the world, we are sure in advance that there will be something there which will remain constant and which may be called energy. Is this to say that the principle has no meaning and vanishes in a tautology? Not at all; it signifies that the different things to which we give the name of energy are connected by a true kin- ship; it affirms a real relation between them. But then if this principle has a meaning, it may be false ; it may be that we have not the right to extend indefinitely its applications, and yet it is certain beforehand to be verified in the strict acceptation of the - term ; how then shall we know when it shall have attained all the extension which can legitimately be given it ? Just simply when it shall cease to be useful to us, that is, to make us correctly fore- see new phenomena. We shall be sure in such a case that the relation affirmed is no longer real; for otherwise it would be fruitful; experiment, without directly contradicting a new ex- tension of the principle, will yet have condemned it. Physics and Mechanism. — Most theorists have a constant predilection for explanations borrowed from mechanics or dy- namics. Some would be satisfied if they could explain all phe- nomena by motions of molecules attracting each other according to certain laws. Others are more exacting ; they would suppress attractions at a distance ; their molecules should follow rectilinear paths from which they could be made to deviate only by impacts. Others again, like Hertz, suppress forces also, but suppose their molecules subjected to geometric attachments analogous, for in- stance, to those of our linkages ; they try thus to reduce dynamics to a sort of kinematics. In a word, all would bend nature into a certain form outside of which their mind could not feel satisfied. Will nature be sufficiently fiexible for that ? We shall examine this question in Chapter XII., d propos of Maxwell's theory. Whenever the principles of energy and of ; least action are satisfied, we shall see not only that there is always , one possible mechanical explanation, but that there is always an : infinity of them. Thanks to a well-known theorem of Konig's on THE THEORIES OF MODERN PHYSICS 145 linkages, it could be shown that we can, in an infinity of ways, explain everything by attachments after the manner of Hertz, or also by central forces. "Without doubt it could be demonstrated just as easily that everything can always be explained by simple impacts. For that, of course, we need not be content with ordinary matter, with that which falls under our senses and whose motions we observe directly. Either we shall suppose that this common matter is formed of atoms whose internal motions elude us, the displacement of the totality alone remaining accessible to our senses. Or else we shall imagine some one of those subtile fluids which under the name of ether or under other names, have at all times played so great a role in physical theories. Often one goes further and regards the ether as the sole primitive matter or even as the only true matter. The more moderate consider common matter as condensed ether, which is nothing startling; but others reduce still further its importance and see in it nothing more than the geometric locus of the ether's singidarities. For instance, what we call matter is for Lord Kelvin only the locus of points where the ether is animated by vortex motions; for Riemann, it was the locus of points where ether is constantly destroyed; for other more recent authors, Wiechert or Larmor, it is the locus of points where the ether undergoes a sort of torsion of a very particular nature. If the attempt is made to occupy one of these points of view, I ask myself by what right shall we extend to the ether, under pretext that this is the true matter, mechanical properties observed in ordinary matter, which is only false matter. The ancient fluids, caloric, electricity, etc., were abandoned when it was perceived that heat is not indestructible. But they were abandoned for another reason also. In materializing them, their individuality was, so to speak, emphasized, a sort of abyss was opened between them. This had to be filled up on the coming of a more vivid feeling of the unity of nature, and the perception of the intimate relations which bind together aU its parts. Not only did the old physicists, in multiplying fluids, create entities unnecessarily, but they broke real ties. It is not sufficient for a theory to affirm no false relations, it must not hide true relations. 146 SCIENCE AND HYPOTHESIS And does our ether really exist? We know the origin of our belief in the ether. If light reaches us from a distant star, dur- ing several years it was no longer on the star and not yet on the earth ; it must then be somewhere and sustained, so to speak, by some material support. The same idea may be expressed under a more mathematical and more abstract form. "What we ascertain are the changes un- dergone by material molecules; we see, for instance, that our photographic plate feels the consequences of phenomena of which the incandescent mass of the star was the theater several years before. Now, in ordinary mechanics the state of the system studied depends only on its state at an instant immediately an- terior; therefore the system satisfies differential equations. On the contrary, if we should not believe in the ether, the state of the material universe would depend not only on the state immedi- ately preceding, but on states much older; the system would satisfy equations of finite differences. It is to escape this deroga- tion of the general laws of mechanics that we have invented the ether. That would still only oblige us to fill up, with the ether, the interplanetary void, but not to make it penetrate the bosom of the material media themselves. Fizeau's experiment goes fur- ther. By the interference of rays which have traversed air or water in motion, it seems to show us two different media inter- penetrating and yet changing place one with regard to the other. We seem to touch the ether with the finger. Yet experiments may be conceived which would make us touch it still more nearly. Suppose Newton's principle, of the equality of action and reaction, no longer true if applied to matter alone, aiid that we have established it. The geometric sum of all the forces applied to all the material molecules would no longer be null. It would be necessary then, if we did not wish to change all mechanics, to introduce the ether, in order that this action which matter appeared to experience should be counterbalanced by the reaction of matter on something. Or again, suppose we discover that optical and electrical phenomena are influenced by the motion of the earth. We should be led to conclude that these phenomena might reveal to us not THE THEORIES OF MODEBN PHYSICS 147 only the relative motions of material bodies, but what would seem to be their absolute motions. Again, an ether would be necessary, that these so-called absolute motions should not be their displacements with regard to a void space, but their dis- placements with regard to something concrete. Shall we ever arrive at that? I have not this hope, I shall soon say why, and yet it is not so absurd, since others have had it. For instance, if the theory of Lorentz, of which I shall speak in detail further on in Chapter XIII., were true, Newton's prin- ciple would not apply to matter alone, and the difference would not be very far from being accessible to experiment. On the other hand, many researches have been made on the influence of the earth's motion. The results have always been negative. But these experiments were undertaken because the outcome was not sure in advance, and, indeed, according to the ruling theories, the compensation would be only approximate, and one might expect to see precise methods give positive results. I believe that such a hope is iUusory; it was none the less interesting to show that a success of this sort would open to us, in some sort, a new world. And now I must be permitted a digression ; I must explain, in fact, why I do not believe, despite Lorentz, that more precise observations can ever put in evidence anything else than the rela- tive displacements of material bodies. Experiments have been made which should have disclosed the terms of the first order; the results have been negative; could that be by chance? No one has assumed that ; a general explanation has been sought, and Lorentz has found it; he has shown that the terms of the first order must destroy each other, but not those of the second. Then more precise experiments were made; they also were negative; neither could this be the effect of chance; an explanation was necessary; it was found; they always are found; of hypotheses there is never lack. But this is not enough; who does not feel that this is still to leave to chance too great a role? Would not that also be a chance, this singular coincidence which brought it about that a certain circumstance should come just in the nick of time to 148 SCIENCE AND HYPOTHESIS destroy the terms of the first order, and that another circum- stance, wholly different, but just as opportune, should take upon itself to destroy those of the second order 1 No, it is necessary to find an explanation the same for the one as for the other, and then everything leads us to think that this explanation will hold good equally well for the terms of higher order, and that the mutual destruction of these terms will be rigorous and absolute. Present State of the Science. — In the history of the de- velopment of physics we distinguish two inverse tendencies. On the one hand, new bonds are continually being discovered between objects which had seemed destined to remain forever unconnected ; scattered facts cease to be strangers to one another ; they tend to arrange themselves in an imposing synthesis. Science advances toward unity and simplicity. On the other hand, observation reveals to us every day new phenomena ; they must long await their place and sometimes, to make one for them, a corner of the edifice must be demolished. In the known phenomena themselves, where our crude senses showed us uniformity, we perceive details from day to day more varied; what we believed simple becomes complex, and science appears to advance toward variety and complexity. Of these two inverse tendencies, which seem to triumph turn about, which wiU win? If it be the first, science is possible; but nothing proves this a priori, and it may well be feared that after having made vain efforts to bend nature in spite of herself to our ideal of unity, submerged by the ever-rising flood of our new riches, we must renounce classifying them, abandon our ideal, and reduce science to the registration of innumerable recipes. To this question we can not reply. All we can do is to ob- serve the science of to-day and compare it with that of yesterday. From this examination we may doubtless draw some encourage- ment. Half a century ago, hope ran high. The discovery of the conservation of energy and of its transformations had revealed to us the unity of force. Thus it showed that the phenomena of heat could be explained by molecular motions. What was the nature of these motions was not exactly known, but no one ..'mL THE THEORIES OF MODERN PHYSICS 149 doubted that it soon would be. For light, the task seemed com- pletely accomplished. In what concerns electricity, things were less advanced. Electricity had just annexed magnetism. This was a considerable step toward unity, and a decisive step. But how should electricity in its turn enter into the general unity, how should it be reduced to the universal mechanism? Of that no one had any idea. Yet the possibility of this reduc- tion was doubted by none, there was faith. Finally, in what concerns the molecular properties of material bodies, the reduc- tion seemed still easier, but all the detail remained hazy. In a word, the hopes were vast and animated, but vague. To-day, what do we see? First of all, a prime progress, immense prog- ress. The relations of electricity and light are now known; the three realms, of light, of electricity and of magnetism, previously separated, form now but one ; and this annexation seems final. This conquest, however, has cost us some sacrifices. The optical phenomena subordinate themselves as particular cases under the electrical phenomena; so long as they remained isolated, it was easy to explain them by motions that were supposed to be known in all their details, that was a matter of course; but now an explanation, to be acceptable, must be easily capable of extension to the entire electric domain. Now that is a matter not without difficulties. The most satisfactory theory we have is that of Lorentz, which, as we shall see in the last chapter, explains electric currents by the motions of little electrified particles; it is unquestioflbly the one which best explains the known facts, the one which illumi- nates the greatest number of true relations, the one of which most traces will be found in the final construction. Nevertheless, it still has a serious defect, which I have indicated above; it is contrary to Newton's law of the equality of action and reaction; or rather, this principle, in the eyes of Lorentz, would not be applicable to matter alone ; for it to be true, it would be necessary to take account of the action of the ether on matter and of the reaction of matter on the ether. Now, from what we know at present, it seems probable that things do not happen in this way. However that may be, thanks to Lorentz, Fizeau's results on 150 SCIENCE AND HYPOTHESIS the optics of moving bodies, the laws of normal and anomalous dis- persion 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 broken. 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 de- fied Maxwell's efforts; this facility abundantly proves that the theory of Lorentz is not an artificial assemblage destined to fall asunder. It will probably have to be modified, but not destroyed. But Lorentz had no aim beyond that of embracing in one totality all the optics and electrodynamics of moving bodies ; he never pretended to give a mechanical explanation of them. Lar- mor goes further; retaining the theory of Lorentz in essentials, he grafts upon it, so to speak, MacCullagh's ideas on the direction of the motions of the ether. According to him, the velocity of the ether would have the same direction and the same magnitude as the magnetic force. However ingenious this attempt may be, the defect of the theory of Lorentz remains and is even aggravated. With Lorentz, we do not" know what are the motions of the ether; thanks to this igno- rance, we may suppose them such that, compensating those of matter, they reestablish the equality of action and reaction. With Larmor, we know the motions of the ether, and we can ascertain that the compensation does not take place. If Larmor has failed, as it seems to nie he has, does that mean that a mechanical explanation is impossible ? Par from it : I ' have said above that when a phenomenon obeys the two principles of energy and of least action, it admits of an infinity of mechan- ) ical explanations ; so it is, therefore, with the optical and electrical phenomena. But this is not enough: for a mechanical explanation to be good, it must be simple ; for choosing it among all which are pos- sible, there should be other reasons besides the necessity of mak- ing a choice. Well, we have not as yet a theory satisfying this condition and consequently good for something. Must we lament this? That would be to forget what is the goal sought; this is not mechanism ; the true, the sole aim is unity. We must therefore set bounds to our ambition ; let us not try THE THEORIES OF MODERN PHYSICS 101 to formulate a mechanical explanation; let us be content with showing that we could always find one if we wished to. In this J regard we have been successful ; the principle of the conservation of energy has received only confirmations ; a second principle has come to join it, that of least action, put under the form which is suitable for physics. It also has always been verified, at least in so far as concerns reversible phenomena which thus obey the equations of Lagrange, that is to say, the most general laws of mechanics. Irreversible phenomena are much more rebellious. Yet these also are being coordinated, and tend to come into unity ; the light which has illuminated them has come to us from Carnot's prin- ciple. Long did thermodynamics confine itself to the study of the dilatation of bodies and their changes of state. For some time peist it has been growing bolder and has considerably extended its domain. "We owe to it the theory of the galvanic battery, and that of the thermoelectric phenomena ; there is not in all physics a corner that it has not explored, and it has attacked chemistry itself. Everywhere the same laws reign ; everywhere, under the diver- sity of appearances, is found again Carnot's principle; every- where also is found that concept so prodigiously abstract of entropy, which is as universal as that of energy and seems like it to cover a reality. Radiant heat seemed destined to escape it ; but recently we have seen that submit to the same laws. In this way fresh analogies are revealed to us, which may often be followed into detail; ohmic resistance resembles the viscosity of liquids ; hysteresis would resemble rather the friction of solids. In all cases, friction would a,ppear to be the type which the most various irreversible phenomena copy, and this kinship is real and profound. Of these phenomena a mechanical explanation, properly so called, has also been sought. They hardly lent themselves to it. To find it, it was necessary to suppose that the irreversibility is only apparent, that the elementary phenomena are reversible and obey the known laws of dynamics. But the elements are extremely numerous and blend more and more, so that to our crude sight all appears to tend toward uniformity, that is, everything seems to 152 SCIENCE AND HYPOTHESIS go forward in the same sense without hope of return. The ap- parent irreversibility is thus only an effect of the law of great numbers. But, only a being with infinitely subtile senses, like Maxwell's imaginary demon, could disentangle this inextricable skein and turn back the course of the universe. This conception, which attaches itself to the kinetic theory of gases, has cost great efforts and has not, on the whole, been fruitful ; but it may become so. 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 things. "We signalize, however, M. Gouy 's original ideas on the Brownian movement. According to this scientist, this singular motion should escape Camot's principle. The particles which it puts in swing would be smaller than the links of that so compacted skein ; they would therefore be fitted to disentangle them and hence to make the world go backward. "We should almost see Maxwell's demon at work. To summarize, the previously known phenomena are better and better classified, but new phenomena come to claim their place; most of these, like the Zeeman effect, have at once found it. But we have the cathode rays, the X-rays, those of uranium and of radium. Herein is a whole world which no one suspected. How many unexpected guests must be stowed away ! No one can yet foresee the place they will occupy. But I do not believe they will destroy the general unity ; I think they will rather complete it. On the one hand, in fact, the new radiations seem connected with the phenomena of luminescence; not only do they excite fluorescence, but they sometimes take birth in the same conditions as it. • Nor are they without kinship with the causes which produce the electric spark under the action of the ultra-violet light. Finally, and above all, it is believed that in all these phenomena are found true ions, animated, it is true, by velocities incom- parably greater than in the electrolytes. That is all very vague, but it will all become more precise. Phosphorescence, the action of light on the spark, these were regions rather isolated, and consequently somewhat neglected by investigators. One may now hope that a new path will be con- THE THEOBIES OF MODERN PHYSICS 153 structed which will facilitate their communications with the rest of science. Not only do we discover new phenomena, but in those we thought we knew, unforeseen aspects reveal themselves. In the free ether, the laws retain their majestic simplicity ; but matter, properly so called, seems more and more complex; all that is said of it is never more than approximate, and at each instant our formulas require new terms. Nevertheless the frames are not broken ; the relations that we have recognized between objects we thought simple still subsist between these same objects when we know their complexity, and it is that alone which is of importance. Our equations become, it is true, more and more complicated, in order to embrace more closely the complexity of nature ; but nothing is changed in the relations which permit the deducing of these equations one from another. In a word, the form of these equations has persisted. Take, for example, the laws of reflection: Fresnel had estab- lished them by a simple and seductive theory which experiment seemed to confirm. Since then more precise researches have proved that this verification was only approximate; they have shown everywhere traces of elliptic polarization. • But, thanks to the help that the first approximation gave us, we found forthwith the cause of these anomalies, which is the presence of a transition layer; and Fresnel's theory has subsisted in its essentials. But there is a reflection we can not help making: All these relations would have remained unperceived if one had at first suspected the complexity of the objects they connect. It has long been said: If Tycho had had instruments ten times more pre- cise neither Kepler, nor Newton, nor astronomy would ever have been. It is a misfortune for a science to be born too late, when the means of observation have become too perfect. This is to-day the case with physical chemistry; its founders are embarrassed in their general grasp by third and fourth decimals ; happily they are men of a robust faith. The better one knows the properties of matter the more one sees continuity reign. Since the labors of Andrews and of van der Waals, we get an idea of how the passage is made from the liquid to the gaseous state and that this passage is not abrupt. Similarly, 154 SCIENCE AND HYPOTHESIS there is no gap between the liquid and solid states, and in the proceedings of a recent congress is to be seen, alongside of a work on the rigidity of liquids, a memoir on the flow of solids. By this tendency no doubt simplicity loses ; some phenomenon was formerly represented by several straight lines, now these straights must be joined by curves more or less complicated. In compensation unity gains notably. Those cut-off categories quieted the mind, but they did not satisfy it. Finally the methods of physics have invaded a new domain, that of chemistry; physical chemistry is born. It is still very young, but we already see that it will enable us to connect such phenomena as electrolysis, osmosis and the motions of ions. From this rapid exposition, what shall we conclude ? Everything considered, we have approached unity; we have not been as quick as was hoped fifty years ago, we have not always taken the predicted way; but, finally, we have gained ever so much ground. CHAPTER XI The Calculus of Probabilities Doubtless it will be astonisliing to find here thoughts about the calculus of probabilities. "What has it to do with the method of the physical sciences ? And yet the questions I shall raise with- out solving present themselves naturally to the philosopher who is thinking about physics. So far is this the case that in the two preceding chapters I have often been led to use the words 'probability' and 'chance.' 'Predicted facts,' as I have said above, 'can only be probable.' "However solidly founded a prediction may seem to us to be, we are never absolutely sure that experiment will not prove it false. But the probability is often so great that practically we may be satisfied with it. ' ' And a little further on I have added : ' ' See what a role the belief in simplicity plays in our generaliza- tions. We have verified a simple law in a great number of par- ticular eases ; we refuse to admit that this coincidence, so often repeated, can be a mere effect of chance. ..." Thus in a multitude of circumstances the physicist is in the same position as the gambler who reckons up his chances. As often as he reasons by induction, he requires more or less con- sciously the calculus of probabilities, and this is why I am obliged to introduce a parenthesis, and interrupt our study of method in the physical sciences in order to examine a little more closely the value of this calculus, and what confidence it merits. The very name calculus of probabilities is a paradox. Prob- ability opposed to certainty is what we do not know, and how can we calculate what we do not know ? Yet many eminent savants have occupied themselves with this calculus, and it can not be denied that science has drawn therefrom no small advantage. How can we explain this apparent contradiction ? Has probability been defined? Can it even be defined? And if it can not, how dare we reason about it ? The definition, it will 155 156 SCIENCE AND HYPOTHESIS be said, is very simple : the probability of an event is the ratio of the number of cases favorable to this event to the total number of possible cases. A simple example will show how incomplete this definition is. I throw two dice. What is the probability that one of the two at least turns up a six? Bach die can turn up in six different ways ; the number of possible cases is 6X6 = 36; the number of favorable cases is 11 ; the probability is 11/36. That is the correct solution. But could I not just as well say : The points which turn up on the two dice can form 6X7/2 = 21 different combinations? Among these combinations 6 are favor- able ; the probability is 6/21. Now why is the first method of enumerating the possible cases more legitimate than the second? In any ease it is not our definition that tells us. "We are therefore obliged to complete this definition by saying : ' ... to the total number of possible cases provided these cases are equally probable. ' So, therefore, we are reduced to defining the probable by the probable. How can we know that two possible cases are equally probable ? Will it be by a convention ? If we place at the beginning of each problem an explicit convention, well and good. We shall then have nothing to do but apply the rules of arithmetic and of algebra, and we shall complete our calculation without our result leaving room for doubt. But if we wish to make the slightest application of this result, we must prove our convention was legitimate, and we shall find ourselves in the presence of the very difficulty we thought to escape. Will it be said that good sense suffices to show us what con- vention should be adopted ? Alas ! M. Bertrand has amused him- self by discussing the following simple problem: "What is the probability that a chord of a circle may be greater than the side of the inscribed equilateral triangle ? ' ' The illustrious geometer successively adopted two conventions which good sense seemed equally to dictate and with one he found 1/2, with the other 1/3. The conclusion which seems to follow from all this is that the calculus of probabilities is a useless science, and that the obscure THE CALCULUS OF PROBABILITIES 157 instinct which we may call good sense, and to which we are wont to appeal to legitimatize our conventions, must be distrusted. But neither can we subscribe to this conclusion; we can not do without this obscure instinct. "Without it science would be impossible, without it we could neither discover a law nor apply it. Have we the right, for instance, to enunciate Newton's law? Without doubt, numerous observations are in accord with it ; but is not this a simple effect of chance? Besides how do we know whether this law, true for so many centuries, will still be true next year? To this objection, you will find nothing to reply, except: 'That is very improbable.' But grant the law. Thanks to it, I believe myself able to calculate the position of Jupiter a year from now. Have I the right to believe this? "Who can tell if a gigantic mass of enor- mous velocity will not between now and that time pass near the solar system, and produce unforeseen perturbations ? Here again the only answer is: 'It is very improbable.' From this point of view, all the sciences would be only uncon- scious applications of the calculus of probabilities. To condemn this calculus would be to condemn the whole of science. I shall dwell lightly on the scientific problems in which the intervention of the calculus of probabilities is more evident. In the forefront of these is the problem of interpolation, in which, knowing a certain number of values of a function, we seek to divine the intermediate values. I shall likewise mention: the celebrated theory of errors of observation, to which I shall return later; the kinetic theory of gases, a well-known hypothesis, wherein each gaseous molecule is supposed to describe an extremely complicated trajectory ; but in which, through the effect of great numbers, the mean phenomena, alone observable, obey the simple laws of Mariotte and Gay- Lussac. An these theories are based on the laws of great numbers, and the calculus of probabilities would evidently involve them in its ruin. It is true that they have only a particular interest, and that, save as far as interpolation is concerned, these are sacrifices to which we might readily be resigned. But, as I have said above, it would not be only these partial 12 158 SCIENCE AND HYPOTHESIS sacrifices that would be in question ; it would be the legitimacy of the whole of science that would be challenged. I quite see that it might be said: "We are ignorant, and yet we must act. For action, we have not time to devote ourselves to an inquiry sufficient to dispel our ignorance. Besides, such an inquiry would demand an infinite time. We must therefore decide without knowing ; we are obliged to do so, hit or miss, and we must follow rules without quite believing them. What I know is not that such and such a thing is true, but that the best course for me is to act as if it were true." The calculus of probabilities, and consequently science itself, would thenceforth have merely a prac- tical value. Unfortunately the difficulty does not thus disappear. A gam- bler wants to try a coup; he asks my advice. If I give it to him, I shall use the calculus of probabilities, but I shall not guarantee success. This is what I shall call subjective probability. In this case, we might be content with the explanation of which I have just given a sketch. But suppose that an observer is present at the game, that he notes all its coups, and that the game goes on a long time. When he makes a summary of his book, he will find that events have taken place in conformity with the laws of the calculus of probabilities. This is what I shall call objective probability, and it is this phenomenon which has to be explained. There are numerous insurance companies which apply the rules of the calculus of probabilities, and they distribute to their share- holders dividends whose objective reality can not be contested. To invoke our ignorance and the necessity to act does not suffice to explain them. Thus absolute skepticism is not admissible. We may distrust, but we can not condemn en bloc. Discussion is necessary. I. Classification of the Problems of Probability. — In order to classify the problems which present themselves a propos of probabilities, we may look at them from many different points of view, and, first, from the point of view of generality. I have said above that probability is the ratio of the number of favorable cases to the number of possible cases. What for want of a better term I call the generality will increase with the number of pos- TSE CALCULUS OF PROBABILITIES 159 sible cases. This number may be finite, as, for instance, if we take a throw of the dice in which the number of possible cases is 36. That is the first degree of generality. But if we ask, for example, what is the probability that a point within a circle is within the inscribed square, there are as many possible cases as there are points in the circle, that is to say, an infinity. This is the second degree of generality. Gener- ality can be pushed further still. We may ask the probability that a function will satisfy a given condition. There are then as many possible cases as one can imagine different functions. This is the third degree of generality, to which we rise, for instance, when we seek to find the most probable law in conformity with a finite number of observations. "We may place ourselves at a point of view wholly different. If we were not ignorant, there would be no probability, there would be room for nothing but certainty. But our ignorance can not be absolute, for then there would no longer be any probability at all, since a little light is necessary to attain even this uncertain science. Thus the problems of probability may be classed accord- ing to the greater or less depth of this ignorance. In mathematics even we may set ourselves problems of prob- ability. "What is the probability that the fifth decimal of a log- arithm taken at random from a table is a '9'? There is no hesitation in answering that this probability is 1/10; here we possess all the data of the problem. "We can calculate our loga- rithm without recourse to the table, but we do not wish to give ourselves the trouble. This is the first degree of ignorance. In the physical sciences our ignorance becomes greater. The state of a system at a given instant depends on two things: Its initial state, and the law according to which that state varies. If we know both this law and this initial state, we shall have then only a mathematical problem to solve, and we fall back upon the first degree of ignorance. But it often happens that we know the law, and do not know the initial state. It may be asked, for instance, what is the present distribution of the minor planets ? "We know that from all time they have obeyed the laws of Kepler, but we do not know what was their initial distribution. 160 SCIENCE AND HYPOTHESIS In the kinetic theory of gases, we assume that the gaseous molecules follow rectilinear trajectories, and ohey the laws of impact of elastic bodies. But, as we know nothing of their initial velocities, we know nothing of their present velocities. The calculus of probabilities only enables us to predict the mean phenomena which will result from the combination of these velocities. This is the second degree of ignorance. Finally it is possible that not only the initial conditions but the laws themselves are unknown. 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 phenomenon. It often happens that instead of trying to guess an event, by means of a more or less imperfect knowledge of the law, the events may be known and we want to find the law ; or that instead of deducing effects from causes, we wish to deduce the causes from the effects. These are the problems called probability of causes, the most interesting from the point of view of their sci- entific applications. I play ecarte with a gentleman I know to be perfectly honest. He is about to deal. "What is the probability of his turning up the king? It is 1/8. This is a problem of the probability of effects. I play with a gentleman whom I do not know. He has dealt ten times, and he has turned up the king six times. What is the probability that he is a sharper? This is a problem in the probability of causes. It may be said that this is the essential problem of the experi- mental method. I have observed n values of x and the corres- ponding values of y. I have found that the ratio of the latter to the former is practically constant. There is the event, what is the cause? Is it probable that there is a general law according to which y would be proportional to x, and that the small divergencies are due to errors of observation ? This is a type of question that one is ever asking, and which we unconsciously solve whenever we are engaged in scientific work. I am now going to pass in review these different categories of THE CALCULUS OF PROBABILITIES 161 problems, discussing in succession what I have called above sub- jective and objective probability. II. Probability in Mathematics. — The impossibility of squar- ing the circle has been proved since 1882; but even before that date all geometers considered that impossibility as so 'probable,' that the Academy of Sciences rejected without exami- nation the alas ! too numerous memoirs on this subject, that some unhappy madmen sent in every year. Was the Academy wrong? Evidently not, and it knew well that in acting thus it did not run the least risk of stifling a dis- covery of moment. The Academy could not have proved that it was right ; but it knew quite well that its instinct was not mis- taken. If you had asked the Academicians, they would have answered: "We have compared the probability that an unknown savant should have found out what has been vainly sought for so long, with the probability that there is one madman the more on the earth ; the second appears to us the greater. ' ' These are very good reasons, but there is nothing mathematical about them ; they are purely psychological. And if you had pressed them further they would have added : "Why do you suppose a particular value of a transcendental function to be an algebraic number ; and if ir were a root of an algebraic equation, why do you suppose this root to be a period of the function sin 2x, and not the same about the other roots of this same equation?" To sum up, they would have invoked the prin- ciple of sufficient reason in its vaguest form. But what could they deduce from it? At most a rule of con- duct for the employment of their time, more usefully spent at their ordinary work than in reading a lucubration that inspired in them a legitimate distrust. But what I call above objective probability has nothing in common with this first problem. It is otherwise with the second problem. Consider the first 10,000 logarithms that we find in a table. Among these 10,000 logarithms I take one at random. What is the probability that its third decimal is an even number? You will not hesitate to answer 1/2 ; and in fact if you pick out in a table the third decimals of these 10,000 numbers, you will find nearly as many even digits as odd. 162 SCIENCE AND HYPOTHESIS Or if you prefer, let us write 10,000 numbers corresponding to our 10,000 logarithms, each of these numbers being + 1 if the third decimal of the corresponding logarithm is even, and — 1 if odd. Then take the mean of these 10,000 numbers. I do not hesitate to say that the mean of these 10,000 numbers is probably 0, and if I were actually to calculate it I should verify that it is extremely small. But even this verification is needless. I might have rigorously proved that this mean is less than 0.003. To prove this result, I should have had to make a rather long calculation for which there is no room here, and for which I confine myself to citing an article I published in the Revue generate des Sciences, April 15, 1899. The only point to which I wish to call attention is the following : in this calculation, I should have needed only to rest my case on two facts, to wit, that the first and second derivatives of the log- arithm remain, in the interval considered, between certain limits. Hence this important consequence that the property is true not only of the logarithm, but of any continuous function whatever, since the derivatives of every continuous function are limited. If I was certain beforehand of the result, it is first, because I had often observed analogous facts for other continuous func- tions; and next, because I made in my mind, in a more or less unconscious and imperfect manner, the reasoning which led me to the preceding inequalities, just as a skilled calculator before finishing his multiplication takes into account what it should come to approximately. And besides, since what I call my intuition was only an in- complete summary of a piece of true reasoning, it is clear why observation has confirmed my predictions, and why the objective probability has been in agreement with the subjective probability. As a third example I shall choose the following problem: A number u is taken at random, and w is a given very large integer. What is the probable value of sin nu ? This problem has no mean- ing by itself. To give it one a convention is needed. We shall agree that the probability for the number u to lie between a and a + da is equal to (fy (a) da ; that it is therefore proportional to the infinitely small interval da, and equal to this multiplied by a function <^(a) depending only on a. As for this function, I TRE CALCULUS OF PEOBABILITIES 163 choose it arbitrarily, but I must assume it to be continuous. The value of sin nu remaining the same when u increases by 27r, I may without loss of generality assume that u lies between and 2tz, and I shall thus be led to suppose that <^(a) is a periodic function whose period is 2ir. The probable value sought is readily expressed by a simple integral, and it is easy to show that this integral is less than 27rM:s/nfc, M)fc being the maximum value of the &*" derivative of <^(w). We see then that if the fc* derivative is finite, our probable value wiU tend toward when w increases indefinitely, and that more rapidly than l/m*-i. The probable value of sin nu when n is very large is therefore naught. To define this value I required a convention; but the result remains the same whatever that convention may he. I have imposed upon myself only slight restrictions in assuming that the function ^(a) is continuous and periodic, and these hy- potheses are so natural that we may ask ourselves how they can be escaped. Examination of the three preceding examples, so different in all respects, has already given us a glimpse, on the one hand, of the role of what philosophers call the principle of sufficient reason, and, on the other hand, of the importance of the fact that certain properties are common to all continuous functions. The study of probability in the physical sciences will lead us to the same result. III. Probability in the Physical Sciences. — ^We come now to the problems connected with what I have called the second degree of ignorance, those, namely, in which we know the law, but do not know the initial state of the system. I could multiply examples, but will take only one. What is the probable present distribution of the minor planets on the zodiac ? We know they obey the laws of Kepler. We may even, with- out at all changing the nature of the problem, suppose that their orbits are all circular, and situated in the same plane, and that we know this plane. On the other hand, we are in absolute ignorance as to what was their initial distribution. However, we do not 164 SCIENCE AND HYPOTHESIS hesitate to affirm that their distribution is now nearly uniform. Why? Let 6 be the longitude of a minor planet in the initial epoch, that is to say, the epoch zero. Let a be its mean motion. Its longi- tude at the present epoch, that is to say, at the epoch t, will be at -\- i. To say that the present distribution is uniform is to say that the mean value of the sines and cosines of multiples oiat-\-i is zero. Why do we assert this ? Let us represent each minor planet by a point in a plane, to wit, by a point whose coordinates are precisely a and 6. All these representative points will be contained in a certain region of the plane, but as they are very numerous, this region will appear dotted with points. We know nothing else about the dis- tribution of these points. What do we do when we wish to apply the calculus of proba- bilities to such a question ? What is the probability that one or more representative points may be found in a certain portion of the plane ? In our ignorance, we are reduced to making an arbi- trary hypothesis. To explain the nature of this hypothesis, allow me to use, in lieu of a mathematical formula, a crude but con- crete image. Let us suppose that over the surface of our plane has been spread an imaginary substance, whose density is vari- able, but varies continuously. 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 there. If we have then two regions of the plane of the same extent, 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 region. Here then are two distributions, one real, in which the repre- sentative points are very numerous, very close together, but dis- crete like the molecules of matter in the atomic hypothesis; the other remote from reality, in which our representative points are replaced by continuous fictitious matter. We know that the latter can not be real, but our ignorance forces us to adopt it. If again we had some idea of the real distribution of the representative points, we could arrange it so that in a region THE CALCULUS OF PROBABILITIES 165 of some extent the density of this imaginary eontimious matter would he nearly proportional to the number of the representative points, or, if you wish, to the number of atoms which are con- tained in that region. Even that is impossible, and our ignorance is so great that we are forced to choose arbitrarily the function which defines the density of our imaginary matter. Only we shall be forced to a hypothesis from which we can hardly get away, we shall suppose that this function is continuous. That is suf- ficient, as we shall see, to enable us to reach a conclusion. What is at the instant t the probable distribution of the minor planets? Or rather what is the probable value of the sine of the longitude at the instant t, that is to say of sin (a#-f 6) ? "We made at the outset an arbitrary convention, but if we adopt it, this probable value is entirely defined. Divide the plane into ele- ments of surface. Consider the value of sin (at -{-i) at the cen- ter of each of these elements ; multiply this value by the surface of the element, and by the corresponding density of the imaginary matter. Take then the sum for all the elements of the plane. This sum, by definition, will be the probable mean value we seek, which will thus be expressed by a double integral. It may be thought at first that this mean value depends on the choice of the function which defines the density of the imaginary matter, and that, as this function <^ is arbitrary, we can, according to the arbitrary choice which we make, obtain any mean value. This is not so. A simple calculation shows that our double integral decreases very rapidly when t increases. Thus I could not quite tell what hypothesis to make as to the probability of this or that initial distribution; but whatever the hypothesis made, the result will be the same, and this gets me out of my difficulty. "Whatever be the function <^, the mean value tends toward zero as t increases, and as the minor planets have certainly accom- plished a very great number of revolutions, I may assert that this mean value is very small. I may choose (^ as I wish, save always one restriction: this function must be continuous ; and, in fact, from the point of view of subjective probability, the choice of a discontinuous function would have been unreasonable. For instance, what reason could 166 SCIENCE AND HYPOTHESIS I have for supposing that the initial longitude might be exactly 0°, but that it could not lie between 0° and 1° ? But the difficulty reappears if we take the point of view of objective probability, if we pass from our imaginary distribution in which the fictitious matter was supposed continuous, to the real distribution in which our representative points form, as it were, discrete atoms. The mean value of sin {at-\-'b) will be represented quite simply by 1 n Ssm{6)d0. As for the function {0), I can choose it in an entirely arbitrary manner. There is nothing that can guide me in my choice, but I am naturally led to suppose this function continuous. Let £ be the length (measured on the circumference of radius 1) of each red and black subdivision. We have to calculate the integral of ^(^)di9, extending it, on the one hand, to all the red divisions, and, on the other hand, to all the black divisions, and to compare the results. Consider an interval 2e, comprising a red division and a black division which follows it. Let M and m be the greatest and least values of the function ^(6) in this interval. The integral extended to the red divisions will be smaller than SMe ; the integral extended to the black divisions vtdll be greater than Sme; the difference will therefore be less than 2(M — m)£. But, if the function 8 is supposed continuous; if, besides, the interval e is very 168 SCIENCE AND HYPOTHESIS small with respect to the total angle described by the needle, the difference M — m will be very small. The difference of the two integrals will therefore be very small, and the probability will be very nearly 1/2. We see that without knowing anything of the function 6, I must act as if the probability were 1/2. "We understand, on the other hand, why, if, placing myself at the objective point of view, I observe a certain number of coups, observation will give me about as many black coups as red. All players know this objective law; but it leads them into a remarkable error, which has been often exposed, but into which they always fall again. When the red has won, for instance, six times running, they bet on the black, thinking they are playing a safe game ; because, say they, it is very rare that red wins seven times running. In reality their probability of winning remains 1/2. Observa- tion shows, it is true, that series of seven consecutive reds are very rare, but series of six reds followed by a black are just as rare. They have noticed the rarity of the series of seven reds; if they have not remarked the rarity of six reds and a black, it is only because such series strike the attention less. V. The Pkobability of Causes. — We now come to the prob- lems of the probability of causes, the most important from the point of view of scientific applications. Two stars, for instance, are very close together on the celestial sphere. Is this apparent contiguity a mere effect of chance ? Are these stars, although on almost the same visual ray, situated at very different distances from the earth, and consequently very far from one another? Or, perhaps, does the apparent correspond to a real contiguity? This is a problem on the probability of causes. I recall first that at the outset of all problems of the proba- bility of effects that have hitherto occupied us, we have always had to make a convention, more or less justified. And if in most cases the result was, in a certain measure, independent of this convention, this was only because of certain hypotheses which permitted us to reject a priori discontinuous functions, for ex- ample, or certain absurd conventions. We shall find something analogous when we deal with the THE CALCULUS OF PROBABILITIES 169 probability of causes. An effect may be produced by the cause A or by the cause B. The effect has just been observed. We ask the probability that it is due to the cause A. This is an a posteriori probability of cause. But I could' not calculate it, 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 play ; I mean the probability of this event for some one who had not observed the effect. The better to explain myself I go back to the example of the game of ecarte mentioned above. My adversary deals for the first time and he turns up a king. What is the probability that he is a sharper ? The formulas ordinarily taught give 8/9, a result evidently rather surprising. If we look at it closer, we see that the calculation is made as if, before sitting down at the table, I had considered that there was one chance in two that my adver- sary was not honest. An absurd hypothesis, because in that case I should have certainly not played with him, and this explains the absurdity of the conclusion. The convention about the a priori probability was unjustified, and that is why the calculation of the a posteriori probability led me to an inadmissible result. We see the importance of this pre- liminary convention. I shall even add that if none were made, the problem of the a posteriori probability would have no mean- ing. It must always be made either explicitly or tacitly. Pass to an example of a more scientific character. I wish to determine an experimental law. This law, when I know it, can be represented by a curve. I make a certain number of isolated observations ; each of these will be represented by a point. When I have obtained these different points, I draw a curve between them, striving to pass as near to them as possible and yet preserve for my curve a regular form, without angular points, or inflec- tions too accentuated, or brusque variation of the radius of curva- ture. This curve will represent for me the probable law, and I assume not only that it will tell me the values of the function intermediate between those which have been observed, but also that it will give me the observed values themselves more exactly than direct observation. This is why I make it pass near the points, and not through the points themselves. 170 SCIENCE AND HYPOTHESIS Here is a problem in the probability of causes. The effects are the measurements I have recorded ; they depend on a combina- tion of two causes: the true law of the phenomenon and the errors of observation. Knowing the effects, we have to seek the probability that the phenomenon obeys this law or that, and that the observations have been affected by this or that error. The most probable law then corresponds to the curve traced, and the most probable error of an observation is represented by the dis; tance of the corresponding point from this curve. But the problem would have no meaning if, before any obser- vation, I had not fashioned an a priori idea of the probability of this or that law, and of the chances of error to which I am exposed. If my instruments are good (and that I knew before making the observations), I shall not permit my curve to depart much from the points which represent the rough measurements. If they are bad, I may go a little further away from them in order to obtain a less sinuous curve ; I shall sacrifice more to regularity. "Why then is it that I seek to trace a curve without sinuosities ? It is because I consider a priori a law represented by a continu- ous function (or by a function whose derivatives of high order are small), as more probable than a law not satisfying these con- ditions. "Without this belief, the problem of which we speak would have no meaning; interpolation would be impossible; no law could be deduced from a finite number of observations; science would not exist. Fifty years ago physicists considered, other things being equal, a simple law as more probable than a complicated law. They even invoked this principle in favor of Mariotte's law as against the experiments of Regnault. To-day they have repudiated this belief; and yet, how many times are they compelled to act as though they still held it ! However that may be, what remains of this tendency is the belief in continuity, and we have just seen that if this belief were to disappear in its turn, experimental science would become impossible. VI. The Theoey of Errors.— "We are thus led to speak of the theory of errors, which is directly connected with the problem of the probability of causes. Here again we find effects, to wit, a certain number of discordant observations, and we seek to THE CALCULUS OF PROBABILITIES 171 divine the causes, which are, on the one hand, the real value of the quantity to be measured; on the other hand, the error made in each isolated observation. It is necessary to calculate what is a posteriori the probable magnitude of each error, and conse- quently the probable value of the quantity to be measured. But as I have just explained, we should not know how to un- dertake this calculation if we did not admit a priori, that is to say, before all observation, a law of probability of errors. Is there a law of errors? The law of errors admitted by all calculators is Gauss's law, which is represented by a certain transcendental curve known under the name of ' the beU. ' But first it is proper to recall the classic distinction between systematic and accidental errors. If we measure a length with too long a meter, we shall always find too small a number, and it wiU be of no use to measure several times ; this is a systematic error. If we measure with an accurate meter, we may, however, make a mistake ; but we go wrong, now too much, now too little, and when we take the mean of a great number of measurements, the error will tend to grow small. These are accidental errors. It is evident from the first that systematic errors can not satisfy Gauss's law; but do the accidental errors satisfy it? A great number of demonstrations have been attempted ; almost all are crude paralogisms. Nevertheless, we may demonstrate Gauss 's law by starting from the following hypotheses : the error committed is the result of a great number of partial and inde- pendent errors; each of the partial errors is very little and besides, obeys any law of probability, provided that the prob- ability of a positive error is the same as that of an equal negative error. It is evident that these conditions will be often but not always fulfilled, and we may reserve the name of accidental for errors which satisfy them. We see that the method of least squares is not legitimate in every case; in general the physicists are more distrustful of it than the astronomers. This is, no doubt, because the latter, be- sides the systematic errors to which they and the physicists are subject alike, have to contend with an extremely important source of error which is wholly accidental ; I mean atmospheric undula- 172 SCIENCE AND HYPOTHESIS tions. So it is very curious to hear a physicist discuss with .an astronomer about a method of observation. The physicist, per- suaded that one good measurement is worth more than many bad ones, is before all concerned with eliminating by dint of precautions the least fystematie errors, and the astronomer says to him: 'But thus you can observe only a small number of stars; the accidental errors will not disappear. ' What should we conclude? Must we continue to use the method of least squares ? We must distinguish. We have elimi- nated all the systematic errors we could suspect; we know well there are still others, but we can not detect them; yet it is necessary to make up our mind and adopt a definitive value which will be regarded as the probable value ; and for that it is evident the best thing to do is to apply Gauss's method. We have only applied a practical rule referring to subjective prob- ability. There is nothing more to be said. But we wish to go farther and affirm that not only is the probable value so much, but that the probable error in the re- sult is so much. This is absolutely illegitimate; it would be true only if we were sure that all the systematic errors were elimi- nated, and of that we know absolutely nothing. We have two series of observations ; by applying the rule of least squares, we find that the probable error in the first series is twice as small as in the second. The second series may, however, be better than the first, because the first perhaps is affected by a large system- atic error. All we can say is that the first series is probably better than the second, since its accidental error is smaller, and we have no reason to affirm that the systematic error is greater for one of the series •'han for the other, our ignorance on this point being absolute. VII. Conclusions. — In the lines which precede, I have set many problems without solving any of them. Yet I do not regret having written them, because they will perhaps invite the reader to reflect on these delicate questions. However that may be, there are certain points which seem well established. To undertake any calculation of probability, and even for that calculation to have any meaning, it is neces- THE CALCULUS OF PROBABILITIES 173 sary to admit, as point of departure, a hypothesis or convention which has always something arbitrary about it. In the choice of this convention, we can be guided only by the principle of sufScient reason. Unfortunately this principle is very vague and very elastic, and in the cursory examination we have just made, we have seen it take many different forms. The form un- der which we have met it most often is the belief in continuity, a belief which it would be difScult to justify by apodeictic reason- ing, but without which all science would be impossible. 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 outset, provided only that this hypothesis satisfies the condition of continuity. 13 CHAPTER XII Optics and Electkicitt Feesnel's Theoet. — The best example^ that can be chosen of physics in the making is the theory of light and its relations to the theory of electricity. Thanks to Fresnel, optics is the best developed part of physics; the so-caUed wave-theory forms a whole truly satisfying to the mind. We must not, however, ask of it what it can not give us. The object of mathematical theories is not to reveal to us the true nature of things ; this would be an unreasonable pretension. Their sole aim is to coordinate the physical laws which experi- ment reveals to us, but which, without the help of mathematics, we should not be able even to state. It matters little whether the ether really exists; that is the affair of metaphysicians. The essential thing for us is that everything happens as if it existed, and that this hypothesis is convenient for the explanation of phenomena. After all, have we any other reason to believe in the existence of material objects? That, too, is only a convenient hypothesis; only this will never cease to be so, whereas, no doubt, some day the ether will be thrown aside as useless. But even at that day, the laws of optics and the equations which translate them analytically will remain true, at least as a first approximation. It wiU always be useful, then, to study a doctrine that unites all these equations. The undulatory theory rests on a molecular hjrpothesis. For those who think they have thus discovered the cause under the law, this is an advantage. For the others it is a reason for dis- trust. But this distrust seems to me as little justified as the illusion of the former. These hypotheses play only a secondary part. They might be sacrificed. They usually are not, because then the explanation would lose in clearness ; but that is the only reason. 1 This chapter is a partial reproduction of the prefaces of two of my works: TMorie matMrnatique de la lumi^re (Paris, Naud, 1889), and JBiec- triciU et optique (Paris, Naud, 1901). 174 OPTICS AND ELECTRICITY 175 In fact, if we looked closer we should see that only two things are borrowed from the molecular hypotheses : the principle of the conservation of energy, and the linear form of the equations, which is the general law of small movements, as of all small variations. This explains why most of Fresnel's conclusions remain un- changed when we adopt the electromagnetic theory of light. Maxwell's Theoey. — ^Maxwell, we know, connected by a close bond two parts of physics until then entirely foreign to one another, optics and electricity. By blending thus in a vaster whole, in a higher harmony, the optics of Fresnel has not ceased to be alive. Its various parts subsist, and their mutual relations are still the same. Only the language we used to express them has changed ; and, on the other hand. Maxwell has revealed to us other relations, before unsuspected, between the different parts of optics and the domain of electricity. When a French reader first opens Maxwell's book, a feeling of uneasiness and often even of mistrust mingles at first with his admiration. Only after a prolonged acquaintance and at the cost of many efforts does this feeling disappear. There are even some eminent minds that never lose it. Why are the English scientist's ideas with such difficulty acclimatized among us? It is, no doubt, because the education received by the majority of enlightened Frenchmen predisposes them to appreciate precision and logic above every other quality. The old theories of mathematical physics gave us in this re- spect complete satisfaction. All our masters, from Laplace to Cauchy, have proceeded in the same way. Starting from clearly stated hypotheses, they deduced all "their consequences with mathematical rigor, and then compared them with experiment. It seemed their aim to give every branch of physics the same pre- cision as celestial mechanics. A mind accustomed to admire such models is hard to suit with a theory. Not only will it not tolerate the least appearance of contradiction, but it will demand that the various parts be logically connected with one another, and that the number of distinct hypotheses be reduced to minimum. This is not all ; it will have still other demands, which seem to 176 SCIENCE AND HYPOTHESIS me less reasonable. Behind the matter which our senses can reach, and which experiment tells us of, it will desire to see another, and in its eyes the only real, matter, which wiH have only purely geometric properties, and whose atoms wiU he noth- ing but mathematical points, subject to the laws of dynamics alone. And yet these atoms, invisible and without color, it wUl seek by an unconscious contradiction to represent to itself and consequently to identify as closely as possible with common matter. Then only will it be fully satisfied and imagine that it has penetrated the secret of the universe. If this satisfaction is de- ceitful, it is none the less difficult to renounce. Thus, on opening Maxwell, a Frenchman expects to find a theoretical whole as logical and precise as the physical optics based on the hypothesis of the ether ; he thus prepares for him- self a disappointment which I should like to spare the reader by informing him immediately of what he must look for in Maxwell, and what he can not find there. Maxwell does not give a mechanical explanation of electricity and magnetism; he confines himself to demonstrating that such an explanation is possible. He shows also that optical phenomena are only a special case of electromagnetic phenomena. From every theory of electri- city, one can therefore deduce immediately a theory of light. The converse unfortunately is not true; from a complete ex- planation of light, it is not always easy to derive a complete ex- planation of electric phenomena. This is not easy, in particular, if we wish to start from Fresnel's theory. Doubtless it would not be impossible ; but nevertheless we must ask whether we are not going to be forced to renounce admirable results that we thought definitely acquired. That seems a step backward; and many good minds are not willing to submit to it. When the reader shall have consented to limit his hopes, he will still encounter other difficulties. The English scientist does not try to construct a single edifice, final and well ordered; he seems rather to erect a great number of provisional and inde- pendent constructions, between which communication is difficult and sometimes impossible. OPTICS AND ELECTRICITY 177 Take as example the chapter in which he explains electrostatic attractions by pressures and tensions in the dielectric medium. This chapter might be omitted without making thereby the rest of the book less clear or complete ; and, on the other hand, it con- tains a theory complete in itself which one could understand with- out having read a single line that precedes or follows. But it is not only independent of the rest of the work; it is difficult to reconcile with the fundamental ideas of the book. Maxwell does not even attempt this reconciliation; he merely says: "I have not been able to make the next step, namely, to account by mechanical considerations for these stresses in the dielectric." This example will suffice to make my thought understood; I could cite many others. Thus who would suspect, in reading the pages devoted to magnetic rotary polarization, that there is an identity between optical and magnetic phenomena ? One must not then flatter himself that he can avoid all con- tradiction; to that it is necessary to be resigned. In fact, two contradictory theories, provided one does not mingle them, and if one does not seek in them the basis of things, may both be useful instruments of research; and perhaps the reading of Maxwell would be less suggestive if he had not opened up to us so many new and divergent paths. The fundamental idea, however, is thus a little obscured. So far is this the case that in the majority of popularized versions it is the only point completely left aside. I feel, then, that the better to make its importance stand out, I' ought to explain in what this fundamental idea consists. But for that a short digression is necessary. The Mechanical Explanation of Physical Phenomena. — There is in every physical phenomenon a certain number of parameters which experiment reaches directly and allows us to measure. I shall call these the parameters q. Observation then teaches us the laws of the variations of these parameters; and these laws can generally be put in the form of differential equations, which connect the parameters q with the time. What is it necessary to do to give a mechanical interpretation of such a phenomenon? 178 SCIENCE AND ETPOTHESIS One will try to explain it either by the motions of ordinary matter, or by those of one or more hypothetical fluids. These fluids wiU be considered as formed of a very great num- ber of isolated molecules m. When shall we say, then, that we have a complete mechanical explanation of the phenomenon? It will be, on the one hand, when we know the differential equations satisfied by the coordi- nates of these hypothetical molecules m, equations which, more- over, must conform to the principles of dynamics; and, on the other hand, when we know the relations that define the coordi- nates of the molecules m as functions of the parameters q acces- sible to experiment. These equations, as I have said, must conform to the prin- ciples of dynamics, and, in particular, to the principle of the conservation of energy and the principle of least action. The first of these two principles teaches us that the total energy is constant and that this energy is divided into two parts : 1° The kinetic energy, or vis viva, which depends on the masses of the hypothetical molecules m, and their velocities, and which I shall call T. 2° The potential energy, which depends only on the coordi- nates of these molecules and which I shall call U. It is the sum of the two energies T and U which is constant. What now does the principle of least action tell us? It tells us that to pass from the initial position occupied at the instant t^ to the final position occupied at the instant t^, the system must take such a path that, in the interval of time that elapses be- tween the two instants ^o and t^^, the average value of 'the action' (that is to say, of the difference between the two energies T and U) shall be as small as possible. If the two functions T and U are known, this principle suffices to determine the equations of motion. Among all the possible ways of passing from one position to another, there is evidently one for which the average value of the action is less than for any other. There is, moreover, only one; and it results from this that the principle of least action suffices to determine the path followed and consequently the equations of motion. OPTICS AND ELECTRICITY 179 Thus we obtain what are called the equations of Lagrange. In these equations, the independent variables are the coordi- nates of the hypothetical molecules m ; but I now suppose that one takes as variables the parameters q directly accessible to ex- periment. The two parts of the energy must then be expressed as func- tions of the parameters q and of their derivatives. They will evidently appear under this form to the experimenter. The latter will naturally try to define the potential and the kinetic energy by the aid of quantities that he can directly observe.^ That granted, the system will always go from one position to another by a path such that the average action shall be a mini- mum. It matters little that T and V are now expressed by the aid of the parameters q and their derivatives ; it matters little that it is also by means of these parameters that we define the initial and final positions ; the principle of least action remains always true. Now here again, of all the paths that lead from one position to another, there is one for which the average action is a mini- mum, and there is only one. The principle of least action sufSces, then, to determine the differential equations which de- fine the variations of the parameters q. The equations thus obtained are another form of the equa- tions of Lagrange. To form these equations we need to know neither the relations that connect the parameters q with the coordinates of the hypothetical molecules, nor the masses of these molecules, nor the expression of ?7 as a function of the coordinates of these molecules. All we need to know is the expression of Z7 as a function of the parameters, and that of T as a function of the parameters q and their derivatives, that is, the expressions of the kinetic and of the potential energy as functions of the experimental data. Then we shall have one of two things: either for a suitable 2 We add that TJ will depend only on the parameters g, that T will depend on the parameters g and their derivatives with respect to the time and will be a homogeneous polynomial of the second degree with respect to these derivatives. 180 SCIENCE AND HTPOTHESIS choice of the functions T and U, the equations of Lagrange, con- structed as we have just said, will be identical with the differ- ential equations deduced from experiments; or else there will exist no functions T and U, for which this agreement takes place. In the latter case it is clear that no mechanical explanation is possible. The necessary condition for a mechanical explanation to be possible is therefore that we can choose the functions T and TJ in such a way as to satisfy the principle of least action, which in- volves that of the conservation of energy. This condition, moreover, is sufficient. Suppose, in fact, that we have found a function U of the parameters q, which- repre- sents one of the parts of the energy; that another part of the energy, which we shall represent by T, is a function of the parameters q and their derivatives, and that it is a homogeneous polynomial of the second degree with respect to these derivatives ; and finally that the equations of Lagrange,' formed by means of these two functions, T and U, conform to the data of the experiment. What is necessary in order to deduce from this a mechanical explanation? It is necessary that U can be regarded as the po- tential energy of a system and T as the vis viva of the same system. There is no difficulty as to U, but can T be regarded as the vis viva of a material system? It is easy to show that this is always possible, and even in an infinity of ways. I wiU confine myself to referring for more details to the preface of my work, 'Electricite et optique.' Thus if the principle of least action can not be satisfied, no mechanical explanation is possible ; if it can be satisfied, there is not only one, but an infinity, whence it follows that as soon as there is one there is an infinity of others. One more observation. Among the quantities that experiment gives us directly, we shall regard some as functions of the coordinates of our hypo- thetical molecules; these are our parameters q. We shall look upon the others as dependent not only on the coordinates, but on the velocities, or, what comes to the same thing, on the derivatives OPTICS AND ELECTRICITY 181 of the parameters q, or as combinations of these parameters and their derivatives. And then a question presents itself : among all these quantities measured experimentally, which shall we choose to represent the parameters g? Which shall we prefer to regard as the deriva- tives of these parameters? This choice remains arbitrary to a very large extent ; but, for a mechanical explanation to be possi- ble, it suffices if we can make the choice in such a way as to accord with the principle of least action. And then Maxwell asked himself whether he could make this choice and that of the two energies T and V, in such a way that the electrical phenomena would satisfy this principle. Ex- periment shows us that the energy of an electromagnetic field is decomposed into two parts, the electrostatic energy and the elec- trodynamic energy. Maxwell observed that if we regard the first as representing the potential energy U, the second as repre- senting the kinetic energy T; if, moreover, the electrostatic charges of -the conductors are considered as parameters q and the intensities of the currents as the derivatives of other para- meters q ; under these conditions, I say. Maxwell observed that the electric phenomena satisfy the principle of least action. Thence- forth he was certain of the possibility of a mechanical ex- planation. If he had explained this idea at the beginning of his book instead of relegating it to an obscure part of the second volume, it would not have escaped the majority of readers. If, then, a phenomenon admits of a complete mechanical ex- planation, it wiU admit of an infinity of others, that will render an account equally well of all the particulars revealed by ex- periment. And this is confirmed by the history of every branch of physics ; in optics, for instance, Fresnel believed vibration to be perpendicular to the plane of polarization; Neumann regarded it as parallel to this plane. An ' experimentum crucis' has long been sought which would enable us to decide between these two theories, but it has not been found. In the same way, without leaving the domain of electricity, we may ascertain that the theory of two fluids and that of the 182 SCIENCE AND HYPOTHESIS single fluid both account in a fashion equally satisfactory for all the observed laws of electrostatics. All these facts are easily explicable, thanks to the properties of the equations of Lagrange which I have just recalled. It is easy now to comprehend what is Maxwell's fundamental idea. To demonstrate the possibility of a mechanical explanation of electricity, we need not preoccupy ourselves with finding this explanation itself; it suffices us to know the expression of the two functions T and U, which are the two parts of energy, to form with these two functions the equations of Lagrange and then to compare these equations with the experimental laws. Among all these possible explanations, how make a choice for which the aid of experiment fails us ? A day will come perhaps when physicists will not interest themselves in these questions, inaccessible to positive methods, and will abandon them to the metaphysicians. This day has not yet arrived; man does not resign himself so easily to be forever ignorant of the foundation of things. Our choice can therefore be further guided only by considera- tions where the part of personal appreciation is very great ; there are, however, solutions that all the world will reject because of their whimsicality, and others that all the world will prefer be- cause of their simplicity. In what concerns electricity and magnetism. Maxwell abstains from making any choice. It is not that he systematically dis- dains all that is unattainable by positive methods; the time he has devoted to the kinetic theory of gases sufficiently proves that. I will add that if, in his great work, he develops no complete explanation, he had previously attempted to give one in an article in the Philosophical Magazine. The strangeness and the com- plexity of the hypotheses he had been obliged to make had led him afterwards to give this up. The same spirit is found throughout the whole work. What is essential, that is to say what must remain common to all theories, is made prominent; all that would only be suitable to a particular theory is nearly always passed over in silence. Thus the reader finds himself in the presence of a form almost devoid OPTICS AND ELECTBICITT 183 of matter, which he is at first tempted to take for a fugitive shadow not to be grasped. 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 at. CHAPTER XIII Electrodynamics The history of electrodynamics is particularly instructive from our point of view. Ampere entitled his immortal work, 'Theorie des phenomenes electrodynamiques, uniquement fondee sur 1 'experience.' He therefore imagined that he had made no hypothesis, but he had made them, as we shall soon see; only he made them without being conscious of it. His successors, on the other hand, perceived them, since their attention was attracted by the weak points in Ampere 's solution. They made new hypotheses, of which this time they were fully conscious; but how many times it was necessary to change them before arriving at the classic system of to-day which is perhaps not yet final; this we shall see. I. Ampere's Theory. — When Ampere studied experimentally the mutual actions of currents, he operated and he only could operate with closed currents. It was not that he denied the possibility of open currents. If two conductors are charged with positive and negative elec- tricity and brought into communication by a wire, a current is established going from one to the other, which continues until the two potentials are equal. According to the ideas of Ampere's time this was an open current; the current was known to go from the first conductor to the socond, it was not seen to return from the second to the first. So Ampere considered as open currents of this nature, for ex- ample, the currents of discharge of condensers; but he could not make them the objects of his experiments because their duration is too short. Another sort of open current may also be imagined. I sup- pose two conductors, A and B, connected by a wire AMB. Small conducting masses in motion first come in contact with the 184 ELECTRODYNAMICS 185 conductor B, take from it an electric charge, leave contact with B and move along the path BNA, and, transporting with them their charge, come into contact with A and give to it their charge, which returns then to B along the wire AMB. Now there we have in a sense a closed circuit, since the elec- tricity describes the closed circuit BNAMB; but the two parts of this current are very different. In the wire AMB, the elec- tricity is displaced through a fixed conductor, like a voltaic cur- rent, overcoming an ohmic resistance and developing heat; we say that it is displaced by conduction. In the part BNA, the electricity is carried by a moving conductor ; it is said to be dis- placed by convection. If then the current of convection is considered as altogether analogous to the current of conduction, the circuit BNAMB is closed ; if, on the contrary, the convection current is not ' a true current,' and, for example, does not act on the magnet, there remains only the conduction current AMB, which is open. For example, if we connect by a wire the two poles of a Holtz machine, the charged rotating disc transfers the electricity by convection from one pole to the other, and it returns to the first pole by conduction through the wire. But currents of this sort are very dififlcult to produce with ap- preciable intensity. With the means at Ampere's disposal, we may say that this was impossible. To sum up. Ampere could conceive of the existence of two kinds of open currents, but he could operate on neither because they were not strong enough or because their duration was too short. Experiment therefore could only show him the action of a closed current on a closed current, or, more accurately, the action of a closed current on a portion of a current, because a current can be made to describe a closed circuit composed of a moving part and a fixed part. It is possible then to study the displace- ments of the moving part under the action of another closed current. On the other hand, Ampere had no means of studying the action of an open current, either on a closed current or another open current. 186 SCIENCE AND HYPOTHESIS 1. The Case of Closed Currents. — In the case of the mutual action of two closed currents, experiment revealed to Ampere re- markably simple laws. I recall rapidly here those which will be useful to us in the sequel : 1° 7/ the intensity of the currents is kept constant, and if the two circuits, after having undergone any deformations and displacements whatsoever, return finally to their initial positions, the total work of the electrodynamic actions will be null. In other words, there is an electrodynamic potential of the two circuits, proportional to the product of the intensities, and depending on the form and relative position of the circuits ; the work of the electrodynamic actions is equal to the variation of this potential: 2° The action of a closed solenoid is null. 3° The action of a circuit C on another voltaic circuit C de- pends only on the 'magnetic field' developed by this circuit. At each point in space we can in fact define in magnitude and direc- tion a certain force called magnetic force, which enjoys the fol- lowing properties : (a) 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 pole ; (&) A very short magnetic needle tends to take the direction of the magnetic force, and the couple to which it tends to reduce is proportional to the magnetic force, the magnetic moment of the needle and the sine of the dip of the needle ; (c) If the circuit C is displaced, the work of the electrody- namic action exercised by C on C" will be equal to the increment of the 'flow of magnetic force' which passes through the circuit. 2. Action of a Closed Current on a Portion of Current. — Ampere not having been able to produce an open current, prop- erly so called, had only one way of studying the action of a closed current on a portion of current. This was by operating on a circuit C composed of two parts, the one fixed, the other movable. The movable part was, for instance, a movable wire ajS whose extremities a and p could ELECTRODYNAMICS 187 slide along a fixed wire. In one of the positions of the movable wire, the end a rested on the A of the fixed wire and the extrem- ity p on the point B of the fixed wire. The current circulated from a to p, that is to say, from Aio B along the movable wire, and then it returned from B to A along the fixed wire. TMs current was therefore closed. In a second position, the movable wire having slipped, the ex- tremity a rested on another point A' of the fixed wire, and the extremity p on another point B' of the fixed wire. The current circulated then from a to p, that is to say from A' to B' along the movable wire, and it afterwards returned from B' to B, then from B to A, then finally from A to A', always following the fixed wire. The current was therefore also closed. If a like current is subjected to the action of a closed current C, the movable part will be displaced just as if it were acted upon by a force. Ampere assumes that the apparent force to which this movable part AB seems thus subjected, representing the action of the C on the portion ap of the current, is the same as if ap were traversed by an open current, stopping at a and p, in place of being traversed by a closed current which after arriv- ing at p returns to a through the fixed part of the circuit. This hypothesis seems natural enough, and Ampere made it unconsciously ; nevertheless it is not necessary, since we shall see further on that Helmholtz rejected it. However that may be, it permitted Ampere, though he had never been able to produce an open current, to enunciate the laws of the action of a closed cur- rent on an open current, or even on an element of current. The laws are simple : 1* The force which acts on an element of current is applied to this element ; it is normal to the element and to the magnetic force, and proportional to the component of this magne1;ie force which is normal to the element. '2° The action of a closed solenoid on an element of current is null. But the electrodynamic potential has disappeared, that is to say that, when a closed current and an open current, whose in- tensities have been maintained constant, return to their initial positions, the total work is not null. 188 SCIENCE AND HYPOTHESIS 3. Continuous Botations. — Among electrodynamic experiments, the most remarkable are those in which continuous rotations are produced and which are sometimes called unipolar induction ex- periments. A magnet may turn about its axis ; a current passes first through a fixed wire, enters the magnet by the pole N, for example, passes through half the magnet, emerges by a sliding contact and reenters the fixed wire. The magnet then begins to rotate continuously without being able ever to attain equilibrium; this is Faraday's experiment. How is it possible? If it were a question of two circuits of invariable form, the one G fixed, the other C movable about an axis, this latter could never take on continuous rotation ; in fact there is an electrodynamic potential; there must therefore be necessarily a position of equilibrium when this potential is a maximum. Continuous rotations are therefore possible only when the cir- cuit C is composed of two parts: one fixed, the other movable about an axis, as is the case in Faraday's experiment. Here again it is convenient to draw a distinction. The passage from the fixed to the movable part, or inversely, may take place either by simple contact (the same point of the movable part remaining constantly in contact with the same point of the fixed part) , or by a sliding contact (the same poiat of the movable part coming successively in contact with diverse points of the fixed part). It is only in the second case that there can be continuous rota- tion. This is what then happens: The system tends to take a position of equilibrium; but, when at the point of reaching that position, the sliding contact puts the movable part in communi- cation with a new point of the fixed part; it changes the con- nections, it changes therefore the conditions of equilibrium, so that the position of equilibrium fleeing, so to say, before the system which seeks to attain it, rotation may take place indefi- nitely. Ampere 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 exist, and therefore as if the current passing through the movable part were open. ELECTBODYNAMICS 189 He concludes therefore that the action of a closed on an open current, or inversely that of an open current on a closed current, may give rise to a continuous rotation. But this conclusion depends on the hypothesis I have enun- ciated and which, as I said above, is not admitted by Helmholtz. 4. Mutual Action of Two Open Currents. — In what concerns the mutual actions of two open currents, and in particular that of two elements of current, all experiment breaks dowii. Am- pere has recourse to hypothesis. He supposes : 1° That the mutual action of two elements reduces to a force acting along their join; 2° That the action of two closed currents is the resultant of the mutual actions of their diverse elements, which are besides the same as if these elements were isolated. What is remarkable is that here again Ampere makes these hypotheses unconsciously. However that may be, these two hypotheses, together with the experiments on closed currents, suffice to determine completely the law of the mutual action of two elements. But then most of the simple laws we have met in the case of closed currents are no longer true. In the first place, there is no electrodynamic potential ; nor was there any, as we have seen, in the case of a closed current acting on an open current. Next there is, properly speaking, no magnetic force. And, in fact, we have given above three different definitions of this force: 1° By the action on a magnetic pole; 2° By the director couple which orientates the magnetic needle ; 3° By the action on an element of current. But in the case which now occupies us, not only these three definitions are no longer in harmony, but each has lost its mean- ing, and in fact: 1° A magnetic pole is no longer acted upon simply by a single force applied to this pole. 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 pole, but to the element ; it may moreover be replaced by a force applied to the pole and by a couple ; 14 190 SCIENCE AND HYPOTHESIS 2° The couple which acts on the magnetic needle is no longer a simple director couple, for its moment with respect to the axis of the needle is not null. It breaks up into a director couple, properly so called, and a supplementary couple which tends to produce the continuous rotation of which we have above spoken ; 3° Finally the force acting on an element of current is not normal to this element. In other words, the unity of the magnetic force has disap peared. Let us see in what this unity consists. Two systems which exercise the same action on a magnetic pole will exert also the same action on an indefinitely small magnetic needle, or on an element of current placed at the same point of space as this pole. Well, this is true if these two systems contain only closed currents ; this would no longer be true if these two systems con- tained open currents. It suffices to remark, for instance, that, if a magnetic pole is placed at A and an element at B, the direction of the element being along the prolongation of the sect AB, this element which will exercise no action on this pole will, on the other hand, exer- cise an action either on a magnetic needle placed at the point A, or on an element of current placed at the point A. 5. Induction. — ^We know that the discovery of electrodynamic induction soon followed the immortal work of Ampere. As long as it is only a question of closed currents there is no difficulty, 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 Ampere. But always on one condition, as Bertrand has well shown; that we make besides a certain number of hypotheses. The same principle again permits this deduction in the case of open currents, although of course we can not submit the result to the test of experiment, since we can not produce such currents. If we try to apply this mode of analysis to Ampere's theory of open currents, we reach results calculated to surprise us. In the first place, induction can not be deduced from the variation of the magnetic field by the formula well known to savants and practicians, and, in fact, as we have said, properly speaking there is no longer a magnetic field. ELECTRODYNAMICS 191 But, further; if a circuit G is subjected to the induction of a variable voltaic system S, if this system S be displaced and de- formed in any way whatever, so that the intensity of the currents of this system varies according to any law whatever, but that after these variations the system finally returns to its initial sit- uation, it seems natural to suppose that the mean electromotive force induced in the circuit C is null. This is true if the circuit C is closed and if the system S con- tains only closed currents. This would no longer be true, if one accepts the theory of Ampere, if there were open currents. So that not only induction will no longer be the variation of the flow of magnetic force, in any of the usual senses of the word, but it can not be represented by the variation of anything whatever. II. Theory op Helmholtz. — I have dwelt upon the conse- quences of Ampere's theory, and of his method of explaining open currents. It is difficult to overlook the paradoxical and artificial char- acter of the propositions to which we are thus led. One can not help thinking 'that can not be so.' We understand therefore why Helmholtz was led to seek some- thing else. Helmholtz rejects Ampere's fundamental hypothesis, to wit, that the mutual action of two elements of current reduces to a force along their join. He assumes that an element of current is not subjected to a single force, but to a force and a couple. It is just this which gave rise to the celebrated polemic between Ber- trand and Helmholtz. Helmholtz replaces Ampere's hypothesis by the following: two elements always admit of an electrodynamic potential depend- ing solely on their position and orientation ; and the work of the forces that they exercise, one on the other, is equal to the varia- tion of this potential. Thus Helmholtz can no more do without hypothesis than Ampere ; but at least he does not make one with- out explicitly announcing it. In the case of closed currents, which are alone accessible to experiment, the two theories agree. In all other cases they differ. In the first place, contrary to what Ampere supposed, the force 192 SCIENCE AND HYPOTHESIS 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 current. Let us return to the circuit C, of which we spoke above, and which was formed of a movable wire a^ sliding on a fixed wire. In the only experiment that can be made, the movable portion ap is not isolated, but is part of a closed circuit. "When it passes from AB to A'B', the total electrodynamic potential varies for two reasons: 1° It undergoes a first increase because the potential of A'B' with respect to the circuit C is not the same as that of AB; 2° It takes a second increment because it must be increased by the potentials of the elements AA', BB' with respect to C. It is this double increment which represents the work of the force to which the portion AB seems subjected. If, on the contrary, ap were isolated, the potential would undergo only the first increase, and this first increment alone would measure the work of the force which acts on AB. In the second place, there could be no continuous rotation without sliding contact, and, in fact, that, as we have seen d propos of closed currents, is an immediate consequence of the existence of an electrodynamic potential. In Faraday's experiment, if the magnet is fixed and if the part of the current exterior to the magnet runs along a movable wire, that movable part may undergo a continuous rotation. But this does not mean to say that if the contacts of the wire with the magnet were suppressed, and an open current were to run along the wire, the wire would still take a movement of con- tinuous rotation. 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 circuit. Another difference: The action of a closed solenoid on a closed current is null according to experiment and according to the two theories. Its action on an open current would be null according to Ampere; it would not be null according to Helm- holtz. From this follows an important consequence. We have given above three definitions of magnetic force. The third has ELECTBODYNAMICS 193 no meaning here since an element of current is no longer acted upon by a single force. No more has the first any meaning. What, in fact, is a magnetic pole? It is the extremity of an indefinite linear magnet. This magnet may be replaced by an indefinite solenoid. For the definition of magnetic force to have any meaning, 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 solenoid, that is to say, that the action on a closed solenoid should be null. Now we have just seen that such is not the case. On the other hand, nothing prevents our adopting the second definition, which is founded on the measurement of the director couple which tends to orientate the magnetic needle. But if it is adopted, neither the effects of induction nor the electrodynamic effects will depend solely on the distribution of the lines of force in this magnetic field. III. Difficulties Raised by These Theories. — The theory of Helmholtz is in advance of that of Ampere; it is necessary, however, that all the difiSculties should be smoothed away. In the one as in the other, the phrase 'magnetic field' has no mean- ing, or, if we give it one, by a more or less artificial convention, the ordinary laws so familiar to all electricians no longer apply ; thus the electromotive force induced in a wire is no longer measured by the number of lines of force met by this wire. And our repugnance does not come alone from the difficulty of renouncing inveterate habits of language and of thought. There is something more. If we do not believe in action at a dis- tance, electrodynamic phenomena must be explained by a modi- fication of the medium. It is precisely this modification that we call 'magnetic field.' And then the electrodynamic effects must depend only on this field. All these difficulties arise from the hypothesis of open currents. IV. Maxwell's Theory. — Such were the difficulties raised by the dominant theories when Maxwell appeared, who with a stroke of the pen made them all vanish. To his mind, in fact, all currents are closed currents. Maxwell assumes that if in a dielectric the electric field happens to vary, this dielectric becomes the seat of a particular phenomenon, acting on the gal- 194 SCIENCE AND HYPOTHESIS vanometer like a current, and which he calls current of dds- placement. If then two conductors bearing contrary charges are put in conununication by a wire, in this wire during the discharge there is an open current of conduction ; but there are produced at the same time in the surrounding dielectric, currents of displacement which close this current of conduction. "We know that Maxwell's theory leads to the explanation of optical phenomena, which would be due to extremely rapid elec- trical oscillations. At that epoch such a conception was only a bold hypothesis, which could be supported by no experiment. At the end of twenty years, Maxwell's ideas received the con- firmation of experiment. Hertz succeeded in producing sys- tems of electric oscillations which reproduce aU the properties of light, and only differ from it by the length of their wave ; that is to say as violet differs from red. In some measure he made the synthesis of light. It might be said that Hertz has not demonstrated directly Maxwell's fundamental idea, the action of the current of dis- placement on the galvanometer. This is true in a sense. What he has shown in sum is that electromagnetic induction is not propagated instantaneously as was supposed ; but with the speed of light. But to suppose there is no current of displacement, and induc- tion is propagated with the speed of light ; or to suppose that the currents of displacement produce effects of induction, and that the induction is propagated instantaneously, comes to the same thing. This can not be seen at the first glance, but it is proved by an analysis of which I must not think of giving even a summary here. V. Eowland's Experiment. — But as I have said above, there are two kinds of open conduction currents. There are first the currents of discharge of a condenser or of any conductor what- ever. There are also the cases in which electric discharges describe ELECTRODYNAMICS 195 a 'closed contour, being displaced by conduction in one part of the circuit and by convection in tbe other part. For open currents of the first sort, the question might be con- sidered as solved; they were closed by the currents of displace- ment. For open currents of the second sort, the solution appeared still more simple. It seemed that if the current were closed, it could only be by the current of convection itself. For that it sufficed to assume that a 'convection current,' that is to say a charged conductor in motion, could act on the galvanometer. But experimental confirmation was lacking. It appeared diffi- cult in fact to obtain a sufficient intensity even by augmenting aa much as possible the charge and the velocity of the conductors. It was Rowland, an extremely skillful experimenter, who first tri- umphed over these difficulties. A disc received a strong electro- static charge and a very great speed of rotation. An astatic mag- netic system placed beside the disc underwent deviations. The experiment was made twice by Rowland, once in Berlin, once in Baltimore. It was afterwards repeated by Himstedt. These physicists even announced that they had succeeded in mak- ing quantitative measurements. In fact, for twenty years Rowland's law was admitted without objection by all physicists. Besides everything seemed to confirm it. The spark certainly does produce a magnetic effect. 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 elec- trode with their charge ? Is not the very spectrum of the spark, in which we recognize the lines of the metal of the electrode, a proof of it? The spark would then be a veritable current of convection. On the other hand, it is also admitted that in an electrolyte the electricity is carried by the ions in motion. The current in an electrolyte would therefore be also a current of convection; now, it acts on the magnetic needle. The same for cathode rays. Crookes attributed these rays to a very subtile matter charged with electricity and moving with a very great velocity. He regarded them, in other words, as currents of convection. Now these cathode rays are 196 SCIENCE AND HYPOTHESIS deviated by the magnet. In virtue of the principle of action and reaction, they should in turn deviate the magnetic needle. It is true that Hertz believed he had demonstrated that the cathode rays do not carry electricity, and that they do not act on the magnetic needle. But Hertz was mistaken. First of all, Perrin succeeded in collecting the electricity carried by these rays, elec- tricity of which Hertz denied the existence ; the German (scientist appears to have been deceived by effects due to the action of X-rays, which were not yet discovered. Afterwards, and quite recently, the action of the cathode rays on the magnetic needle has been put in evidence. Thus all these phenomena regarded as currents of convection, sparks, electrolytic currents, cathode rays, act in the same manner on the galvanometer and in conformity with Rowland's law. VI. Theory of Lorentz. — "We soon went further. Accord- ing to the theory of Lorentz, currents of conduction themselves would be true currents of convection. Electricity would remain inseparably connected with certain material particles called elec- trons. The circulation of these electrons through bodies would produce voltaic currents. And what would distinguish con- ductors from insulators would be that the one could be traversed by these electrons while the others would arrest their movements. The theory of Lorentz is very attractive. It gives a very simple explanation of certain phenomena which the earlier the- ories, even Maxwell 's in its primitive form, could not explain in a satisfactory way; for example, the aberration of light, the par- tial carrying away of luminous waves, magnetic polarization and the Zeeman effect. Some objections stiU remained. The phenomena of an elec- tric system seemed to depend on the absolute velocity of transla- tion of the center of gravity of this system, which is contrary to the idea we have of the relativity of space. Supported by M. Cremieu, M. Lippman has presented this objection in a striking form. Imagine two charged conductors with the same velocity of translation; they are relatively at rest. However, each of them being equivalent to a current of convection, they ought to attract one another, and by measuring this attraction we could measure their absolute velocity. ELECTRODYNAMICS 197 "No!" replied the partisans of Lorentz, "What we could measure in that way is not their absolute velocity, but their rela- tive velocity with respect to the ether, so that the principle of relativity is safe. ' ' "Whatever there may be in these latter objections, the edifice of electrodynamics, at least in its broad lines, seemed definitively constructed. Everything was presented under the most satis- factory aspect. The theories of Ampere and of Helmholtz, made for open currents which no longer existed, seemed to have no longer anything but a purely historic interest, and the inextricable complications to which these theories led were almost forgotten. This quiescence has been recently disturbed by the experi- ments of M. Cremieu, which for a moment seemed to contradict the result previously obtained by Eowland. But fresh researches have not confirmed them, and the theory of Lorentz has victoriously stood the test. The history of these variations will be none the less instruct- ive ; it wiU teach us to what pitfalls the scientist is exposed, and how he may hope to escape them. THE VALUE OF SCIENCE. TRANSLATOR'S INTRODUCTION 1. Does the Scientist create Science f — ^Professor Rados of Buda- pest in his report to the Hungarian Academy of Science on the award to Poincare of the Bolyai prize of ten thousand crowns, speaking of him as unquestionably the most powerful investiga- tor in the domain of mathematics and mathematical physics, characterized him as the intuitive genius drawing the inspiration for his wide-reaching researches from the exhaustless fountain of geometric and physical intuition, yet working this inspira- tion out in detail with marvelous logical keenness. With his brilliant creative genius was combined the capacity for sharp and successful generalization, pushing far out the boundaries of thought in the most widely different domains, so that his works must be ranked with the greatest mathematical achievements of all time. "Finally," says Rados, "permit me to make especial mention of his intensely interesting book, ' The Value of Science, * in which he in a way has laid down the scientist's creed." Now what is this creed ? Sense may act as stimulus, as suggestive, yet not to awaken a dormant depiction, or to educe the conception of an archetypal form, but rather to strike the hour for creation, to summon to work a sculptor capable of smoothing a Venus of Milo out of the formless clay. Knowledge is not a gift of bare experience, nor even made solely out of experience. The creative activity of mind is in mathematics particularly clear. The axioms of geom- etry are conventions, disguised definitions or unprovable hy- potheses precreated by auto-aetive animal and human minds. Bertrand Russell says of projective geometry: "It takes nothing from experience, and has, like arithmetic, a creature of the pure intellect for its object. It deals with an object whose properties are logically deduced from its definition, not empirically dis- covered from data." Then does the scientist create science? This is a question Poincare here dissects with a master hand. The physiologic-psychologic investigation of the space problem 201 202 THE VALUE OF SCIENCE must give the meaning of the words geometric fact, geometric reality. Poincare here subjects to the most successful analysis ever made the tridimensionality of our space. 2. The Mind Dispelling Optical Illusions. — ^Actual perception of spatial properties is accompanied by movements correspond- ing to its character. In the case of optical illusions, with the so- called false perceptions eye-movements are closely related. But though the perceived object and its environment remain constant, the sufficiently powerful mind can, as we say, dispel these illu- sions, the perception itself being creatively changed. Photo- graphs taken at intervals during the presence of these optical illusions, during the change, perhaps gradual and unconscious, in the perception, and after these illusions have, as the phrase is, finally disappeared, show quite clearly that changes in eye- movements corresponding to those internally created in percep- tion itself successively occur. What is called accuracy of move- ment is created by what is called correctness of perception. The higher creation in the perception is the determining cause of an improvement, a precision in the motion. Thus we see correct per- ception in the individual helping to make that cerebral organiza- tion and accurate motor adjustment on which its possibility and permanence seem in so far to depend. So-called correct percep- tion is connected with a long-continued process of perceptual education motived and initiated from within. How this may take place is here illustrated at length by our author. 3. Euclid not Necessary. — Geometry is a construction of the intellect, in application not certain but convenient. As Schiller says, when we see these facts as clearly as the development of metageometry has compelled us to see them, we must surely con- fess that the Kantian account of space is hopelessly and demon- strably antiquated. As Eoyce says in 'Kant's Doctrine of the Basis of Mathematics,' "That very use of intuition which Kant regarded as geometrically ideal, the modem geometer regards as scientifically defective, because surreptitious. No mathemat- ical exactness without explicit proof from assumed principles — I such is the motto of the modem geometer. But suppose the reasoning of Euclid purified of this comparatively surreptitious TRANSLATOR'S INTRODUCTION 203 appeal to intuition. Suppose that the principles of geometry are made quite explicit at the outset of the treatise, as Fieri and Hilbert or Professor Halsted or Dr. Veblen makes his principles explicit in his recent treatment of geometry. Then, indeed, geom- etry becomes for the modern mathematician a purely rational science. 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 necessity." Yet the environmental and perhaps hereditary premiums on Euclid still make even the scientist think Euclid most convenient. 4. Without Hypotheses, no Science. — Nobody ever observed an equidistantial, but also nobody ever observed a straight line. Emerson's Uriel "Gave Ms sentiment divine Against the being of a line. Line in Nature is not found." Clearly not, being an eject from man's mind. What is called 'a knowledge of facts' is usually merely a subjective realization that the oM hypotheses are still sufSciently elastic to serve in some domain; that is, with a sufficiency of conscious or unconscious omissions and doctorings and fudgings more or less wilful. In the present book we see the very foundation rocks of science, the conservation of energy and the indestructibility of matter, beat- ing against the bars of their cages, seemingly anxious to take wing away into the empyrean, to chase the once divine parallel postulate broken loose from Euclid and Kant. 5. What Outcome f — ^What now is the definite, the permanent outcome ? What new islets raise their f ronded palms in air within thought's musical domain? Over what age-gray barriers rise the fragrant floods of this new spring-tide, redolent of the wolf- haunted forest of Transylvania, of far Erdely's plunging river, Maros the bitter, or broad mother Volga at Kazan ? What victory heralded the great rocket for which young Lobachevski, the widow's son, was cast into prison? What severing of age-old mental fetters symbolized young Bolyai's cutting-off with his 204 THE VALVE OF SCIENCE Damascus blade the spikes driven into his door-post, and strew- ing over the sod the thirteen Austrian cavalry officers? This book by the greatest mathematician of our time gives vifeightiest and most charming answer. Geoege Beuce Halsted. INTRODUCTION The search for truth should be the goal of our activities; it is the sole end worthy of them. Doubtless we should first bend our efforts to assuage human suffering, but why ? Not to suffer is a negative ideal more surely attained by the annihilation of the world. If we wish more and more to free man from material cares, it is that he may be able to employ the liberty obtained in the study and contemplation of truth. But sometimes truth frightens us. And in fact we know that it is sometimes deceptive, that it is a phantom never showing itself for a moment except to ceaselessly flee, that it must be pursued further and ever further without ever being attained. Yet to work one must stop, as some Greek, Aristotle or another, has said. "We also know how cruel the truth often is, and we wonder whether illusion is not more consoling, yea, even more bracing, for illusion it is which gives confidence. "When it shall have vanished, will hope remain and shall we have the courage to achieve? Thus would not the horse harnessed to his treadmill refuse to go, were his eyes not bandaged? And then to seek truth it is necessary to be independent, wholly independent. If, on the contrary, we wish to act, to be strong, we should be united. This is why many of us fear truth; we consider it a cause of weakness. Yet truth should not be feared, for it alone is beautiful. "When I speak here of truth, assuredly I refer first to scientific truth ; but I also mean moral truth, of which what we call justice is only one aspect. It may seem that I am misusing words, that I combine thus under the same name two things having nothing in common ; that scientific truth, which is demonstrated, can in no way be likened to moral truth, which is felt. And yet I can not separate them, and whosoever loves the one can not help loving the other. To find the one, as weU as to find the other, it is neces- sary to free the soul completely from prejudice and from passion ; it is necessary to attain absolute sincerity. These two sorts of 15 205 206 TBE VALVE OF SCIENCE truth when discovered give the same joy; each when perceived beams with the same splendor, so that we must see it or close our eyes. Lastly, both attract us and flee from us; they are never fixed : when we think to have reached them, we find that we have still to advance, and he who pursues them is condemned never to know repose. It must be added that those who fear the one will also fear the other ; for they are the ones who in everything are concerned above all with consequences. In a word, I liken the two truths, because the same reasons make us love them and because the same reasons make us fear them. If we ought not to fear moral truth, still less should we dread scientific truth. In the first place it can not confiict with ethics. Ethics and science have their own domains, which touch but do not interpenetrate. The one shows us to what goal we should aspire, the other, given the goal, teaches us how to attain it. So they can never conflict since they can never meet. There can no more be immoral science than there can be scientific morals. But if science is feared, it is above all because it can not give us happiness. Of course it can not. We may even ask whether the beast does not suffer less than man. But can we regret that earthly paradise where man brute-like was really immortal m knowing not that he must die ? When we have tasted the apple, no suffering can make us forget its savor. We always come back to it. Could it be otherwise? As well ask if one who has seen and is blind will not long for the light. Man, then, can not be happy through science, but to-day he can much less be happy without it. But if truth be the sole aim worth pursuing, may we hope to attain it ? It may well be doubted. Eeaders of my little book 'Science and Hypothesis' already know what I think about the question. The truth we are permitted to glimpse is not alto- gether what most men call by that name. Does this mean that our most legitimate, most imperative aspiration is at the same time the most vain ? Or can we, despite all, approach truth on some side ? This it is which must be investigated. In the first place, what instrument have we at our disposal for this conquest? Is not human intelligence, more specifically the INTBODUCTION 207 intelligence of the scientist, susceptible of infinite variation? Volumes could be written without exhausting this subject ; I, in a few brief pages, have only touched it lightly. That the geom- eter's mind is not like the physicist's or the naturalist's, aU the world would agree; but mathematicians themselves do not re- semble each other; some recognize only implacable logic, others appeal to intuition and see in it the only source of discovery. And this would be a reason for distrust. To minds so unlike can the mathematical theorems themselves appear in the same light? Truth which is not the same for all, is it truth? But looking at things more closely, we see how these very different workers collaborate in a common task which could not be achieved without their cooperation. And that already reassures us. Next must be examined the frames in which nature seems en- closed and which are called time and space. In 'Science and Hypothesis' I have already shown how relative their value is; it is not nature which imposes them upon us, it is we who impose them upon nature because we find them convenient. But I have spoken of scarcely more than space, and particularly quanti- tative space, so to say, that is of the mathematical relations whose aggregate constitutes geometry. I should have shown that it is the same with time as with space and still the same with ' qualita- tive space'; in particular, I should have investigated why we attribute three dimensions to space. I may be pardoned then for taking up again these important questions. Is mathematical analysis, then, whose principal object is the study of these empty frames, only a vain play of the mind ? It can give to the physicist only a convenient language ; is this not a mediocre service, which, strictly speaking, could be done with- out ; and even is it not to be feared that this artificial language may be a veil interposed between reality and the eye of the physicist ? Far from it ; without this language most of the inti- mate analogies of things would have remained forever unknown to us ; and we should forever have been ignorant of the internal harmony of the world, which is, we shall see, the only true objective reality. The best expression of this harmony is law. Law is one of the 208 THE VALUE OF SCIENCE most recent conquests of the human mind; there still are people who live in the presence of a perpetual miracle and are not astonished at it. On the contrary, we it is who should be aston- ished at nature's regularity. Men demand of their gods to prove their existence by miracles ; but the eternal marvel is that there are not miracles without cease. The world is divine because it is a harmony. If it were ruled by caprice, what could prove to us it was not ruled by chance ? This conquest of law we owe to astronomy, and just this makes the grandeur of the science rather than the material grandeur of the objects it considers. It was altogether natural, then, that celestial mechanics should be the first model of mathematical physics; but since then this science has developed; it is still developing, even rapidly developing. And it is already neces- sary to modify in certain points the scheme from which I drew two chapters of ' Science and Hypothesis. ' In an address at the St. Louis exposition, I sought to survey the road traveled; the result of this investigation the reader shall see farther on. The progress of science has seemed to imperil the best estab- lished principles, those even which were regarded as fundamental. Yet nothing shows they will not be saved ; and if this comes about only imperfectly, they will still subsist even though they are modified. The advance of science is not comparable to the changes of a city, where old edifices are pitilessly torn down to give place to new, but to the continuous evolution of zoologic types which develop ceaselessly and end by becoming unrecognizable to the common sight, but where an expert eye finds always traces of the prior work of the centuries past. One must not think then that the old-fashioned theories have been sterile and vain. Were we to stop there, we should find in these pages some reasons for confidence in the value of science, but many more for distrusting it ; an impression of doubt would remain ; it is need- ful now to set things to rights. Some people have exaggerated the role of convention in science : they have even gone so far as to say that law, that scientific fact itself, was created by the scientist. This is going much too far in the direction of nominalism. No, scientific laws are not arti- INTRODUCTION 209 ficial creations ; we have no reason to regard them as accidental, though it be impossible to prove they are not. Does the harmony the human intelligence thinks it discovers in nature exist outside of this intelligence? No, beyond doubt a reality completely independent of the mind which conceives it, sees or feels it, is an impossibility. A world as exterior as that, even if it existed, would for us be forever inaccessible. But what we call objective reality is, in the last analysis, what is common to many thinking beings, and could be common to all ; this com- mon part, we shall see, can only be the harmony expressed by mathematical laws. It is this harmony then which is the sole objective reality, the only truth we can attain ; and when I add that the universal harmony of the world is the source of all beauty, it will be understood what price we should attach to the slow and difficult progress which little by little enables us to know it better. PART I THE MATHEMATICAL SCIEI^'CES CHAPTER I Intottion and Logic est Mathematics I It is impossible to study the works of the great mathematicians, or even those of the lesser, without noticing and distinguishing two opposite tendencies, or rather two entirely different kinds of minds. The one sort are above all preoccupied with logic; to read their works, one is tempted to believe they have advanced only step by step, after the manner of a Vauban who pushes on his trenches against the place besieged, leaving nothing to chance. The other sort are guided by intuition and at the first stroke make quick but sometimes precarious conquests, like bold cavalrymen of the advance guard. The method is not imposed by the matter treated. Though one often says of the first that they are analysts and calls the others geometers, that does not prevent the one sort from remaining analysts even when they work at geometry, while the others are still geometers even when they occupy themselves with pure analysis. It is the very nature of their mind which makes them logicians or intuiitionalists, and they can not lay it aside when they approach a new subject. Nor is it education which has developed in them one of the two tendencies and stifled the other. The mathematician is born, not made, and it seems he is bom a geometer or an analyst. I should like to cite examples and there are surely plenty ; but to accentu- ate the contrast I shall begin with an extreme example, taking the liberty of seeking it in two living mathematicians. 210 INTUITION AND LOGIC IN MATHEMATICS 211 M. Meray wants to prove that a binomial equation always has a root, or, in ordinary words, that an angle may always be sub- divided. If there is any truth that we think we know by direct intuition, it is this. Who could doubt that an angle may always be divided into any number of equal parts 1 M. Meray does not look at it that way; in his eyes this proposition is not at all evident and to prove it he needs several pages. On the other hand, look at Professor Klein : he is studying one of the most abstract questions of the theory of functions : to deter- mine whether on a given Riemann surface there always exists a function admitting of given singularities. What does the cele- brated German geometer do? He replaces his Riemann surface by a metallic surface whose electric conductivity varies according to certain laws. He connects two of its points with the two poles of a battery. The current, says he, must pass, and the distribu- tion of this current on the surface will define a function whose singularities will be precisely those called for by the enunciation. Doubtless Professor Klein well knows he has given here only a sketch; nevertheless he has not hesitated to publish it; and he would probably believe he finds in it, if not a rigorous demon- stration, at least a kind of moral certainty. A logician would have rejected with horror such a conception, or rather he would not have had to reject it, because in his mind it would never have originated. Again, permit me to compare two men, the honor of French science, who have recently been taken from us, but who both entered long ago into immortality. I speak of M. Bertrand and M. Hermite. They were scholars of the same school at the same time ; they had the same education, were under the same influ- ences; and yet what a difference! Not only does it blaze forth in their writings ; it is in their teaching, in their way of speaking, in their very look. In the memory of aU their pupils these two faces are stamped in deathless lines; for aU who have had the pleasure of following their teaching, this remembrance is still fresh ; it is easy for us to evoke it. While speaking, M. Bertrand is always in motion; now he seems in combat with some outside enemy, now he outlines with a gesture of the hand the figures he studies. Plainly he sees and he is 212 THE VALUE OF SCIENCE eager to paint, this is why he calls gesture to his aid. With M. Hennite, it is just the opposite; his eyes seem to shun contact with the world ; it is not without, it is within he seeks the vision of truth. Among the German geometers of this century, two names above all are illustrious, those of the two scientists who founded the general theory of functions, Weierstrass and Riemann. Weier- strass leads everything back to the consideration of series and their analytic transformations; to express it better, he reduces analysis to a sort of prolongation of arithmetic ; you may turn through all his books without finding a figure. Riemann, on the contrary, at once calls geometry to his aid; each of his concep- tions is an image that no one can forget, once he has caught its meaning. More recently, Lie was an intuitionalist ; this might have been doubted in reading his books, no one could doubt it after talking with him ; you saw at once that he thought in pictures. Madame Kovalevski was a logician. Among our students we notice the same differences ; some prefer to treat their problems 'by analysis,' others 'by geometry.' The first are incapable of 'seeing in space,' the others are quickly tired of long calculations and become perplexed. The two sorts of minds are equally necessary for the progress of science ; both the logicians and the intuitionalists have achieved great things that others could not have done. Who would ven- ture to say whether he preferred that Weierstrass had never written or that there had never been a Riemann ? Analysis and synthesis have then both their legitimate roles. But it is inter- esting to study more closely in the history of science the part which belongs to each. II Strange! If we read over the works of the ancients we are tempted to class them all among the intuitionalists. And yet nature is always the same ; it is hardly probable that it has begun in this century to create minds devoted to logic. If we could put ourselves into the flow of ideas which reigned in their time, we should recognize that many of the old geometers were in tendency INTUITION AND LOGIC IN MATHEMATICS 213 analysts. Euclid, for example, erected a scientific structure wherein his contemporaries could find no fault. In this vast construction, of which each piece however is due to intuition, we may still to-day, without much effort, recognize the work of a logician. It is not minds that have changed, it is ideas ; the intuitional minds have remained the same ; but their readers have required of them greater concessions. What is the cause of this evolution? It is not hard to find. Intuition can not give us rigor, nor even certainty ; this has been recognized more and more. Let us cite some examples. We know there exist continuous functions lacking derivatives. Nothing is more shocking to intuition than this proposition which is imposed upon us by logic. Our fathers would not have failed to say : " It is evident that every continuous function has a derivative, since every curve has a tangent." How can intuition deceive us on this point ? It is because when we seek to imagine a curve we can not represent it to ourselves without width ; just so, when we represent to ourselves a straight line, we see it under the form of a rectilinear band of a certain breadth. We well know these lines have no width; we try to imagine them narrower and narrower and thus to approach the limit; so we do in a certain measure, but we shall never attain this limit. And then it is clear we can always picture these two narrow bands, one straight, one curved, in a position such that they encroach slightly one upon the other without crossing. We shall thus be led, unless warned by a rigorous analysis, to con- clude that a curve always has a tangent. I shall take as second example Dirichlet's principle on which rest so many theorems of mathematical physics ; to-day we estab- lish it by reasoning very rigorous but very long ; heretofore, on the contrary, we were content with a very summary proof. A certain integral depending on an arbitrary function can never vanish. Hence it is concluded that it must have a minimum. The flaw in this reasoning strikes us immediately, since we use the abstract term function and are familiar with aU the singularities functions can present when the word is understood in the most general sense. 214 THE VALUE OF SCIENCE But it would not be the same had we used concrete images, had we, for example, considered this function as an electric poten- tial ; it would have been thought legitimate to affirm that electro- static equilibrium can be attained. Yet perhaps a physical com- parison would have awakened some vague distrust. But if care had been taken to translate the reasoning into the language of geometry, intermediate between that of analysis and that of physics, doubtless this distrust would not have been produced, and perhaps one might thus, even to-day, still deceive many readers not forewarned. Intuition, therefore, does not give us certainty. This is why the evolution had to happen ; let us now see how it happened. It was not slow in being noticed that rigor could not be intro- duced in the reasoning unless first made to enter into the defini- tions. For the most part the objects treated of by mathemati- cians were long iU defined; they were supposed to be known because represented by means of the senses or the imagination; but one had only a crude image of them and not a precise idea on which reasoning could take hold. It was there first that the logicians had to direct their efforts. So, in the case of incommensurable numbers. The vague idea of continuity, which we owe to intuition, resolved itself into a complicated system of inequalities referring to whole numbers. By that means the difficulties arising from passing to the limit, or from the consideration of infinitesimals, are finally removed. To-day in analysis only whole numbers are left or systems, finite or infinite, of whole numbers bound together by a net of equality or inequality relations. Mathematics, as they say, is arithmetized. Ill A first question presents itself. Is this evolution ended? Have we finally attained absolute rigor? At each stage of the evolu- tion our fathers also thought they had reached it. If they deceived themselves, do we not likewise cheat ourselves? We believe that in our reasonings we no longer appeal to intuition ; the philosophers will tell us this is an illusion. Pure logic could never lead us to anything but tautologies; it could INTUITION AND LOGIC IN MATHEMATICS 215 create nothing new ; not from it alone can any science issue. In one sense these philosopers are right; to make arithmetic, as to make geometry, or to make any science, something else than pure logic is necessary. To designate this something else we have no word other than intuition. But how many different ideas are hidden under this same word? Compare these four axioms: (1) Two quantities equal to a third are equal to one another; (2) if a theorem is true of the number 1 and if we prove that it is true of. n-{-l if true for n, then will it be true of all whole numbers; (3) if on a straight the point G is between A and B and the point D between A and C, then the point B will be between A and B ; (4) through a given point there is not more than one parallel to a given straight. All four are attributed to intuition, and yet the first is the enunciation of one of the rules of formal logic; the second is a real synthetic a priori judgment, it is the foundation of rigorous mathematical induction; the third is an appeal to the imagina- tion; the fourth is a disguised definition. Intuition is not necessarily founded on the evidence of the senses ; the senses would soon become powerless ; for example, we can not represent to ourselves a chiliagon, and yet we reason by intuition on polygons in general, which include the chiliagon as a particular case. You know what Poncelet understood by the principle of con- tinuity. "What is true of a real quantity, said Poncelet, should be true of an imaginary quantity ; what is true of the hyperbola whose asymptotes are real, should then be true of the ellipse whose asymptotes are imaginary. Poncelet was one of the most intuitive minds of this century; he was passionately, almost ostentatiously, so ; he regarded the principle of continuity as one of his boldest conceptions, and yet this principle did not rest on the evidence of the senses. To assimilate the hyperbola to the ellipse was rather to contradict this evidence. It was only a sort of precocious and instinctive generalization which, moreover, I have no desire to defend. We have then many kinds of intuition ; first, the appeal to the senses and the imagination; next generalization by induction, copied, so to speak, from the procedures of the experimental sci- 216 THE VALUE OF SCIENCE enees; finally, we have the intuition of pure number, whence arose the second of the axioms just enunciated, which is able to create the real mathematical reasoning. I have shown above by examples that the first two can not give us certainty ; but who will seriously doubt the third, who will doubt arithmetic ? Now in the analysis of to-day, when one cares to take the trouble to be rigorous, there can be nothing but syllogisms or appeals to this intuition of pure number, the only intuition which can not deceive us. It may be said that to-day absolute rigor is attained. IV The philosophers make still another objection : "What you gain in rigor, ' ' they say, ' ' you lose in objectivity. You can rise toward your logical ideal only by cutting the bonds which attach you to reality. Your science is infallible, but it can only remain so by imprisoning itself in an ivory tower and renouncing all relation with the external world. From this seclusion it must go out when it would attempt the slightest application." For example, I seek to show that some property pertains to some object whose concept seems to me at first indefinable, be- cause it is intuitive. At first I fail or must content myself with approximate proofs ; finally I decide to give to my object a pre- cise definition, and this enables me to establish this property in an irreproachable manner. "And then," say the philosophers, "it still remains to show that the object which corresponds to this definition is indeed the same made known to you by intuition ; or else that some real and concrete object whose conformity with your intuitive idea you believe you immediately recognize corresponds to your new defi- nition. Only then could you afiSrm that it has the property in question. You have only displaced the difficulty. ' ' That is not exactly so ; the difficulty has not been displaced, it has been divided. The proposition to be established was in reality composed of two different truths, at first not distinguished. The first was a mathematical truth, and it is now rigorously estab- lished. The second was an experimental verity. Experience alone can teach us that some real and concrete object corresponds or INTUITION AND LOGIC IN MATHEMATICS 217 does not correspond to some abstract definition. This second verity is not mathematically demonstrated, but neither can it be, no more than can the empirical laws of the physical and natural sciences. It would be unreasonable to ask more. "Well, is it not a great advance to have distinguished what long was wrongly confused? Does this mean that nothing is left of this objection of the philosophers? That I do not intend to say; in becoming rigorous, mathematical science takes a character so artificial as to strike every one ; it forgets its historical origins ; we see how the questions can be answered, we no longer see how and why they are put. This shows us that logic is not enough; that the science of demonstration is not all science and that intuition must retain its role as complement, I was about to say as counterpoise or as antidote of logic. I have already had occasion to insist on the place intuition should hold in the teaching of the mathematical sciences. With- out it young minds could not make a beginning in the under- standing of mathematics; they could not learn to love it and would see in it only a vain logomachy ; above all, without intui- tion they would never become capable of applying mathematics. But now I wish before all to speak of the role of intuition in science itself. If it is useful to the student it is still more so to the creative scientist. "We seek reality, but what is reality ? The physiologists tell us that organisms are formed of cells; the chemists add that cells themselves are formed of atoms. Does this mean that these atoms or these cells constitute reality, or rather the sole reality? The way in which these cells are arranged and from which results the unity of the individual, is not it also a reality much more inter- esting than that of the isolated elements, and should a naturalist who had never studied the elephant except by means of the micro- scope think himself sufficiently acquainted with that animal ? Well, there is something analogous to this in mathematics. The logician cuts up, so to speak, each demonstration into a very great number of elementary operations ; when we have examined these 218 THE VALUE OF SCIENCE operations one after tlie other and ascertained that each is correct, are we to think we have grasped the real meaning of the demon- stration ? Shall we have understood it even when, by an effort of memory, we have become able to repeat this proof by reproducing all these elementary operations in just the order in which the inventor had arranged them? Evidently not; we shall not yet possess the entire reality ; that I know not what, which makes the unity of the demonstration, will completely elude us. Pure analysis puts at our disposal a multitude of procedures whose infallibility it guarantees; it opens to us a thousand dif- ferent ways on which we can embark in all confidence; we are assured of meeting there no obstacles; but of all these ways, which will lead us most promptly to our goal? "Who shall teU us which to choose ? We need a faculty which makes us see the the end from afar, and intuition is this faculty. It is necessary to the explorer for choosing his route ; it is not less so to the one following his trail who wants to know why he chose it. If you are present at a game of chess, it will not suffice, for the understanding of the game, to know the rules for moving the pieces. That will only enable you to recognize that each move has been made conformably to these rules, and this knowledge will truly have very little value. Yet this is what the reader of a book on mathematics would do if he were a logician only. To understand the game is wholly another matter ; 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 game. It is to perceive the inward reason which makes of this series of succes- sive moves a sort of organized whole. This faculty is stiU more necessary for the player himself, that is, for the inventor. Let us drop this comparison and return to mathematics. For example, see what has happened to the idea of continuous func- tion. At the outset this was only a sensible image, for example, that of a continuous mark traced by the chalk on a blackboard. Then it became little by little more refined ; ere long it was used to construct a complicated system of inequalities, which repro- duced, so to speak, all the lines of the original image ; this eon- . struction finished, the centering of the arch, so to say, was removed, that crude representation which had temporarily served INTUITION AND LOGIC IN MATHEMATICS 219 as support and which was afterward useless was rejected; there remained only the construction itself, irreproachable in the eyes of the logician. And yet if the primitive image had totally dis- appeared from our recollection, how could we divine by what caprice all these inequalities were erected in this fashion one upon another? Perhaps you think I use too many comparisons ; yet pardon stiU another. You have doubtless seen those delicate assemblages of silicious needles which form the skeleton of certain sponges. When the organic matter has disappeared, there remains only a frail and elegant lace-work. True, nothing is there except silica, but what is interesting is the form this silica has taken, and we could not understand it if we did not know the living sponge which has given it precisely this form. Thus it is that the old intuitive notions of our fathers, even when we have abandoned them, still imprint their form upon the logical constructions we have put in their place. This view of the aggregate is necessary for the inventor ; it is equally necessary for whoever wishes really to comprehend the inventor. Can logic give it to us ? No ; the name mathematicians give it would suffice to prove this. In mathematics logic is called analysis and analysis means (Mvision, dissection. It can have, therefore, no tool other than the scalpel and the microscope. Thus logic and intuition have each their necessary role. Each is indispensable. Logic, which alone can give certainty, is the instrument of demonstration; intuition is the instrument of invention. VI But at the moment of formulating this conclusion I am seized with scruples. At the outset I distinguished two kinds of mathe- matical minds, the one sort logicians and analysts, the others intuitionalists and geometers. "Well, the analysts also have been inventors. The names I have just cited make my insistence on this unnecessary. Here is a contradiction, at least apparently, which needs expla- nation. And first, do you think these logicians have always pro- ceeded from the general to the particular, as the rules of formal 220 TBE VALUE OF SCIENCE logic would seem to require of them? Not thus could they have extended the boundaries of science; scientific conquest is to be made only by generalization. In one of the chapters of ' Science and Hypothesis, ' I have had occasion to study the nature of mathematical reasoning, and I have shown how this reasoning, without ceasing to be absolutely rigorous, could lift us from the particular to the general by a procedure I have called mathematical induction. It is by this procedure that the analysts have made science progress, and if we examine the detail itself of their demonstrations, we shall find it there at each instant beside the classic syllogism of Aristotle. We, therefore, see already that the analysts are not simply makers of syllogisms after the fashion of the scholastics. Besides, do you think they have always marched step by step with no vision of the goal they wished to attain ? They must have divined the way leading thither, and for that they needed a guide. This guide is, first, analogy. For example, one of the methods of demonstration dear to analysts is that founded on the employ- ment of dominant functions. We know it has already served to solve a multitude of problems ; in what consists then the role of the inventor who wishes to apply it to a new problem? At the outset he must recognize the analogy of this question with those which have already been solved by this method; then he must perceive in what way this new question differs from the others, and thence deduce the modifications necessary to apply to the method. But how does one perceive these analogies and these differences? In the example just cited they are almost always evident, but I could have found others where they would have been much more deeply hidden ; often a very uncommon penetration is necessary for their discovery. The analysts, not to let these hidden analo- gies escape them, that is, in order to be inventors, must, without the aid of the senses and imagination, have a direct sense of what constitutes the unity of a piece of reasoning, of what makes, so to speak, its soul and inmost life. When one talked with M. Hermite, he never evoked a sensuous image, and yet you soon perceived that the most abstract entities were for him like living beings. He did not see them, but he per- INTUITION AND LOGIC IN MATHEMATICS 221 ceived that they are not an artificial assemblage, and that they have some principle of internal unity. But, one will say, that still is intuition. Shall we conclude that the distinction made at the outset was only apparent, that there is only one sort of mind and that all the mathematicians are intui- tionalists, at least those who are capable of inventing ? No, our distinction corresponds to something real. I have said above that there are many kinds of intuition. I have said how much the intuition of pure number, whence comes rigorous mathe- matical induction, differs from sensible intuition to which the imagination, properly so called, is the principal contributor. Is the abyss which separates them less profound than it at first appeared? Could we recognize with a little attention that this pure intuition itself could not do without the aid of the senses? This is the affair of the psychologist and the metaphysician and I shall not discuss the question. But the thing's being doubtful is enough to justify me in recognizing and affirming an essen- tial difference between the two kinds of intuition ; they have not the same object and seem to call into play two different faculties of our soul; one would think of two search-lights directed upon two worlds strangers to one another. It is the intuition of pure number, that of pure logical forms, which illumines and directs those we have called analysts. This it is which enables them not alone to demonstrate, but also to invent. By it they perceive at a glance the general plan of a logical edifice, and that too without the senses appearing to inter- vene. In rejecting the aid of the imagination, which, as we have seen, is not always infallible, they can advance without fear of deceiving themselves. Happy, therefore, are those who can do without this aid ! "We must admire them ; but how rare they are ! Among the analysts there will then be inventors, but they will be few. The majority of us, if we wished to see afar by pure intu- ition alone, would soon feel ourselves seized with vertigo. Our weakness has need of a staff more solid, and, despite the excep- tions of which we have just spoken, it is none the less true that sensible intuition is in mathematics the most usual instrument of invention. Apropos of these reflections, a question comes up that I have 16 222 TEE VALUE OF SCIENCE not the time either to solve or even to enunciate with the develop- ments it would admit of. Is there room for a new distinction, for distinguishing among the analysts those who above all use pure intuition and those who are first of all preoccupied with formal logic ? M. Hermite, for example, whom I have just cited, can not be classed among the geometers who make use of the sensible intui- tion ; but neither is he a logician, properly so called. He does not conceal his aversion to purely deductive procedures which start from the general and end in the particular. CHAPTBE II The Measure of Time I So long as we do not go outside the domain of consciousness, the notion of time is relatively clear. Not only do we distinguish without difficulty present sensation from the remembrance of past sensations or the anticipation of future sensations, but we know perfectly well what we mean when we say that of two conscious phenomena which we remember, one was anterior to the other; or that, of two foreseen conscious phenomena, one will be ante- rior to the other. When we say that two conscious facts are simultaneous, we mean that they profoundly interpenetrate, so that analysis can not separate them without mutilating them. The order in which we arrange conscious phenomena does not admit of any arbitrariness. It is imposed upon us and of it we can change nothing, I have only a single observation to add. For an aggregate of sensations to have become a remembrance capable of classifica- tion in time, it must have ceased to be actual, we must have lost the sense of its infinite complexity, otherwise it would have remained present. It must, so to speak, have crystallized around a center of associations of ideas which will be a sort of label. 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 herbarium. But these labels can only be finite in number. On that score, psychologic time should be discontinuous. Whence comes the feeling that between any two instants there are others? We arrange our recollections in time, but we know that there remain empty compartments. How could that be, if time were not a form pre-existent in our minds? How could we know there were empty compartments, if these compartments were revealed to us only by their content? 223 224 TEE VALUE OF SCIENCE II But that is not all ; into this form we wish to put not only the phenomena of our own consciousness, hut those of which other consciousnesses are the theater. But more, we wish to put there physical facts, these I know not what with which we people space and which no consciousness sees directly. This is necessary be- cause without it science could not exist. In a word, psychologic time is given to us and must needs create scientific and physical time. There the difficulty begins, or rather the difficulties, for there are two. Think of two consciousnesses, which are like two worlds im- penetrable one to the other. By what right do we strive to put them into the same mold, to measure them by the same standard? Is it not as if one strove to measure length with a gram or, weight with a meter ? And besides, why do we speak of measur- ing? We know perhaps that some fact is anterior to some other, but not by how much it is anterior. Therefore two difficulties: (1) Can we transform psychologic time, which is qualitative, into a quantitative time? (2) Can we reduce to one and the same measure facts which transpire in different worlds? Ill The first difficulty has long been noticed ; it has been the sub- ject of long discussions and one may say the question is settled. We have not a direct intuition of the equality of two intervals of time. The persons who believe they possess this intuition are dupes of an illusion. When I say, from noon to one the same time passes as from two to three, what meaning has this affir- mation ? The least reflection shows that by itself it has none at all. It wiU only have that which I choose to give it, by a definition which will certainly possess a certain degree of arbitrariness. Psy- chologists could have done without this definition ; physicists and astronomers could not ; let us see how they have managed. To measure time they use the pendulum and they suppose by definition that aU the beats of this pendulum are of equal dura- tion. But this is only a first approximation; the temperature, the resistance of the air, the barometric pressure, make the pace THE MEASURE OF TIME 225 of the pendulum vary. If we could escape these sources of error, we should obtain a much closer approximation, but it would still be only an approximation. New causes, hitherto neglected, elec- tric, magnetic or other;?, would introduce minute perturbations. In fact, the best chronometers must be corrected from time to time, and the corrections are made by the aid of astronomic observations; arrangements are made so that the sidereal clock marks the same hour when the same star passes the meridian. In other words, it is the sidereal day, that is, the duration of the rotation of the earth, which is the constant unit of time. It is supposed, by a new definition substituted for that based on the beats of the pendulum, that two complete rotations of the earth about its axis have the same duration. However, the astronomers are still not content with this defi- nition. Many of them think that the tides act as a check on our globe, and that the rotation of the earth is becoming slower and slower. Thus would be explained the apparent acceleration of the motion of the moon, which would seem to be going more rapidly than theory permits because our watch, which is the earth, is going slow. IV All this is unimportant, one will say; doubtless our instruments of measurement are imperfect, but it suffices that we can conceive a perfect instrument. This ideal can not be reached, but it is enough to have conceived it and so to have put rigor into the definition of the unit of time. The trouble is that there is no rigor in the definition. "When we use the pendulum to measure time, what postulate do we implicitly admit? It is that the duration of two identical phe- nomena is the same; or, if you prefer, that the same causes take the same time to produce the same effects. And at first blush, this is a good definition of the equality of two durations. But take care. Is it impossible that experiment may some day contradict our postulate ? Let me explain myself. I suppose that at a certain place in the world the phenomenon a happens, causing as consequence at the end of a certain time the effect a'. At another place in the world 226 THE VALVE OF SCIENCE very far away from the first, happens the phenomenon j8, which causes as consequence the effect /8'. The phenomena a and ^ are simultaneous, as are also the effects a' and /8'. Later, the phenomenon a is reproduced under approximately the same conditions as before, and simultaneously the phenom- enon p is also reproduced at a very distant place in the world and almost under the same circumstances. The effects a' and ^ also take place. Let us suppose that the effect a' happens per- ceptibly before the effect p'. If experience made us witness such a sight, our postulate would be contradicted. For experience would tell us that the first duration aa! is equal to the first duration j8j8' and that the second duration aa' is less than the second duration j8/8'. On the other hand, our postulate would require that the two durations aa' should be equal to each other, as likewise the two durations pp'. The equality and the inequality deduced from experience would be incompatible with the two equalities deduced from the postulate. Now can we affirm that the hypotheses I have just made are absurd? They are in no wise contrary to the principle of con- tradiction. Doubtless they could not happen without the prin- ciple of sufficient reason seeming violated. But to justify a definition so fundamental I should prefer some other guarantee. But that is not all. In physical reality one cause does not pro- duce a given effect, but a multitude of distinct causes contribute to produce it, without our having any means of discriminating the part of each of them. Physicists seek to make this distinction ; but they make it only approximately, and, however they progress, they never will make it except approsmately. It is approximately true that the motion of the pendulum is due solely to the earth's attraction; but in all rigor every attraction, even of Sirius, acts on the pen- dulum. Under these conditions, it is clear that the causes which have produced a certain effect will never be reproduced except ap- proximately. Then we should modify our postulate and our THE MEASURE OF TIME 227 definition. Instead of saying: 'The same causes take the same time to produce the same effects,' we should say: 'Causes almost identical take almost the same time to produce almost the same effects. ' Our definition therefore is no longer anything but approxi- mate. Besides, as M. Calinon very justly remarks in a recent memoir :^ One of the circumstances of any phenomenon is the velocity of the earth's rotation; if this velocity of rotation varies, it constitutes in the reproduction of this phenomenon a circumstance which no longer remains the same. But to suppose this velocity of rotation constant is to suppose that we know how to meEisure time. Our definition is therefore not yet satisfactory; it is certainly not that which the astronomers of whom I spoke above implicitly adopt, when they affirm that the terrestrial rotation is slowing down. What meaning according to them has this affirmation? "We can only understand it by analyzing the proofs they give of their proposition. They say first that the friction of the tides pro- ducing heat must destroy vis viva. They invoke therefore the principle of vis viva, or of the conservation of energy. They say next that the secular acceleration of the moon, cal- culated according to Newton's law, would be less than that de- duced from observations unless the correction relative to the slowing down of the terrestrial rotation were made. They invoke therefore Newton's law. In other words, they define duration in the following way: time should be so defined that Newton's law and that of vis viva may be verified. Newton's law is an experimental truth ; as such it is only approximate, which shows that we still have only a definition by approximation. If now it be supposed that another way of measuring time is adopted, the experiments on which Newton's law is founded would none the less have the same meaning. Only the enun- ciation of the law would be different, because it would be trans- lated into another language; it would evidently be much less simple. So that the definition implicitly adopted by the astron- omers may be summed up thus: Time should be so defined that 1 Etude sur les diverses grandeurs, Paris, Grauthier-Villars, 1897. 228 TBE VALVE OF SCIENCE the equations of mechanics may be as simple as possible. In other words, there is not one way of measuring time more true than another; that which is generally adopted is only more convenient. Of two watches, we have no right to say that the one goes true, the other wrong; we can only say that it is ad- vantageous to conform to the indications of the first. The diflBculty which has just occupied us has been, as I have said, often pointed out; among the most recent works in which it is considered, I may mention, besides M. Calinon's little book, the treatise on mechanics of Andrade. VI The second difficulty has up to the present attracted much less attention; yet it is altogether analogous to the preceding; and even, logically, I should have spoken of it first. Two psychological phenomena happen in two different con- sciousnesses ; when I say they are simultaneous, what do I mean ? When I say that a physical phenomenon, which happens outside of every consciousness, is before or after a psychological phenom- enon, what do I mean? j In 1572, Tycho Brahe noticed in the heavens a new star. An , immense conflagration had happened in some far distant heavenly body ; but it had happened long before ; at least two hundred years were necessary for the light from that star to reach our earth. This conflagration therefore happened before the discov- ery of America. "Well, when I say that ; when, considering this gigantic phenomenon, which perhaps had no witness, since the satellites of that star were perhaps uninhabited, I say this phe- nomenon is anterior to the formation of the visual image of the ' isle of Espanola in the consciousness of Christopher Columbus, what do I mean? A little reflection is sufficient to understand that all these affirmations have by themselves no meaning. They can have one only as the outcome of a convention. VII We should first ask ourselves how one could have had the idea of putting into the same frame so many worlds impenetrable to THE MEASURE OF TIME 229 one another. "We should like to represent to ourselves the ex- ternal universe, and only by so doing could we feel that we un- derstood it. We know we never can attain this representation : our weakness is too great. But at least we desire the ability to conceive an infinite intelligence for which this representation could be possible, a sort of great consciousness which should see ~ all, and which should classify all in its time, as we classify, in our time, the little we see. This hypothesis is indeed crude and incomplete, because this supreme intelligence would be only a demigod; infinite in one sense, it would be limited in another, since it would have only an imperfect recollection of the past; and it could have no other, since otherwise aU recollections would be equally present to it and for it there would be no time. And yet when we speak of time, for aU which happens outside of us, do we not uncon- sciously adopt this hypothesis; do we not put ourselves in the place of this imperfect god; and do not even the atheists put themselves in the place where god would be if he existed 1 What I have just said shows us, perhaps, why we have tried to put all physical phenomena into the same frame. But that can not pass for a definition of simultaneity, since this hypo- thetical intelligence, even if it existed, would be for us impene- trable. It is therefore necessary to seek something else. | VIII The ordinary definitions which are proper for psychologic time would suffice us no more. Two simultaneous psychologic facts are so closely bound together that analysis can not separate with- out mutilating them. Is it the same with two physical facts ? Is not my present nearer my past of yesterday than the present of Sirius? It has also been said that two facts should be regarded as simultaneous when the order of their succession may be inverted at will. It is evident that this definition would not suit two physical facts which happen far from one another, and that, in what concerns them, we no longer even understand what this reversibility would be; besides, succession itself must first be defined. 230 THE VALUE OF SCIENCE IX Let us then seek to give an account of what is understood by- simultaneity or antecedence, and for this let us analyze some examples. I write a letter ; it is afterward read by the friend to whom I have addressed it. There are two facts which have had for their theater two different consciousnesses. In writing this letter I have had the visual image of it, and my friend has had in his turn this same visual image in reading the letter. Though these two facts happen in impenetrable worlds, I do not hesitate to regard the first as anterior to the second, because I believe it is its cause. ,; I hear thunder, and I conclude there has been an electric dis- charge; I do not hesitate to consider the physical phenomenon as anterior to the auditory image perceived in my consciousness, because I believe it is its cause. Behold then the rule we foUow, and the only one we can follow : when a phenomenon appears to us as the cause of another, we /regard it as anterior. It is therefore by cause that we define time; but most often, when two facts appear to us bound by a constant relation, how do we recognize which is the cause and which the effect ? We assume that the anterior fact, the antece- dent, is the cause of the other, of the consequent. It is then by time that we define cause. How save ourselves from this petitio principii? We say now post hoc, ergo propter hoc; now propter hoc, ergo post hoc; shall we escape from this vicious circle ? X Let us see, not how we succeed in escaping, for we do not completely succeed, but how we try to escape. I execute a voluntary act A and I feel afterward a sensation B, which I regard as a consequence of the act A ; on the other hand, for whatever reason, I infer that this consequence is not imme- diate, but that outside my consciousness two facts B and G, which I have not witnessed, have happened, and in such a way that B is the effect of A, that G is the effect of B, and D of G. But why? If I think I have reason to regard the four facts A, B, G, D, as bound to one another by a causal connection, why THE MEASURE OF TIME 231 range them in the causal order A B C D, and at the same time in the chronologic order A B G D, rather than in any other order ? I clearly see that in the act A I have the feeling of having been active, while in undergoing the sensation D I have that of having been passive. This is why I regard A as the initial cause and D as the ultimate effect ; this is why I put A at the beginning of the chain and D at the end ; but why put B before G rather than C before B ? If this question is put, the reply ordinarily is : we know that it is B which is the cause of G because we always see B happen before G. These two phenomena, when witnessed, happen in a certain order; when analogous phenomena happen without wit- ness, there is no reason to invert this order. , Doubtless, but take care ; we never know directly the physical phenomena B and G. What we know are sensations B' and C produced respectively by B and G. Our consciousness tells us immediately that B' precedes C" and we suppose that B and G succeed one another in the same order. This rule appears in fact very natural, and yet we are often led to depart from it. We hear the sound of the thunder oidy some seconds after the electric discharge of the cloud. Of two flashes of lightning, the one distant, the other near, can not the first be anterior to the second, even though the sound of the second comes to us before that of the first 1 XI Another difficulty ; have we really the right to speak of the cause of a phenomenon ? If all the parts of the universe are inter- chained in a certain measure, any one phenomenon will not be the effect of a single cause, but the resultant of causes infinitely numerous; it is, one often says, the consequence of the state of the universe a moment before. How enunciate rules applicable to circiimstances so complex ? And yet it is only thus that these rules can be general and rigorous. Not to lose ourselves in this infinite complexity, let us make a simpler hypothesis. Consider three stars, for example, the sun, Jupiter and Saturn ; but, for greater simplicity, regard them as 232 TEE VALVE OF SCIENCE reduceij to material points and isolated from the rest of the world. The positions and the velocities of three bodies at a given instant suffice to determine their positions and velocities at the following instant, and consequently at any instant. Their positions at the instant t determine their positions at the instant t + h as well as their positions at the instant t — h. Even more ; the position of Jupiter at the instant t, together with that of Saturn at the instant t-\-a, determines the position of Jupiter at any instant and that of Saturn at any instant. The aggregate of positions occupied hy Jupiter at the instant t + e and Saturn at the instant i + a + e is bound to the aggre- gate of positions occupied by Jupiter at the instant t and Saturn at the instant * + a, by laws as precise as that of Newton, though more complicated. Then why not regard one of these aggre- gates as the cause of the other, which would lead to considering as simultaneous the instant t of Jupiter and the instant t-\-aot Saturn ? In answer there can only be reasons, very strong, it is true, of convenience and simplicity. XII But let us pass to examples less artificial; to understand the definition implicitly supposed by the savants, let us watch them at work and look for the rules by which they investigate simul- taneity. I win take two simple examples, the measurement of the velocity of light and the determination of longitude. When an astronomer tells me that some stellar phenomenon, which his telescope reveals to him at this moment, happened, nevertheless, fifty years ago, I seek his meaning, and to that end I shall ask him first how he knows it, that is, how he has measured the velocity of light. He has begun by supposing that light has a constant velocity, and in particular that its velocity is the same in all directions. That is a postulate without which no measurement of this veloc- ity could be attempted. This postulate could never be verified directly by experiment; it might be contradicted by it if the results of different measurements were not concordant. We THE MEASUBE OF TIME 233 should think ourselves fortunate that this contradiction has not happened and that the slight discordances which may happen can be readily explained. The postulate, at aU events, resembling the principle of suffi- cient reason, has been accepted by everybody ; what I wish to em- phasize is that it furnishes us with a new rule for the investi- gation of simultaneity, entirely dififerent from that which we have enunciated above. This postulate assumed, let us see how the velocity of light has been measured. You know that Roemer used eclipses of the satellites of Jupiter, and sought how much the event fell behind its prediction. But how is this prediction made? It is by the aid of astronomic laws; for instance Newton's law. Could not the observed facts be just as well explained if we at- tributed to the velocity of light a little different value from that adopted, and supposed Newton's law only approximate? Only this would lead to replacing Newton's law by another more com- plicated. So for the velocity of light a value is adopted, such that the astronomic laws compatible with this value may be as simple as possible. When navigators or geographers determine a longitude, they have to solve just the problem we are discuss- ing; they must, without being at Paris, calculate Paris time. How do they accomplish it? They carry a chronometer set for Paris. The qualitative problem of simultaneity is made to de- pend upon the quantitative problem of the measurement of time. I need not take up the difficulties relative to this latter problem, since above I have emphasized them at length. Or else they observe an astronomic phenomenon, such as an eclipse of the moon, and they suppose that this phenomenon is perceived simultaneously from all points of the earth. That is not altogether true, since the propagation of light is not instan- taneous; if absolute exactitude were desired, there would be a correction to make according to a complicated rule. Or else finally they use the telegraph. It is clear first that the reception of the signal at Berlin, for instance, is after the send- ing of this same signal from Paris. This is the rule of cause and effect analyzed above. But how much after? In general, the duration of the transmission is neglected and the two events are 234 THE VALVE OF SCIENCE regarded as simultaneous. But, to be rigorous, a little correc- tion would still have to be made by a complicated calculation; in practise it is not made, because it would be well within the errors of observation; its theoretic necessity is none the less from our point of view, which is that of a rigorous definition. From this discussion, I wish to emphasize two things: (1) The rules applied are exceedingly various. (2) It is difficult to sep- arate the qualitative problem of simultaneity from the quanti- tative problem of the measurement of time ; no matter whether a ': chronometer is used, or whether account must be taken of a velocity of transmission, as that of light, because such a velocity could not be measured without measuring a time. XIII To conclude : We have not a direct intuition of simultaneity, nor of the equality of two durations. If we think we have this intuition, this is an illusion. "We replace it by the aid of certain rules which we apply almost always without taking count of them. But what is the nature of these rules? No general rule, no rigorous rule ; a multitude of little rules applicable to each par- ticular case. These rules are not imposed upon us and we might amuse our- selves in inventing others ; but they could not be cast aside with- out greatly complicating the enunciation of the laws of physics, mechanics and astronomy. We therefore choose these rules, not because they are true, but because they are the most convenient, and we may recapitu- late them as foUows: " The simultaneity of two events, or the order of their succession, the equality of two durations, are to be so defined that the enunciation of the natural laws may be as simple as possible. In other words, all these rules, all these definitions are only the fruit of an unconscious opportunism." CHAPTER III The Notion op Space 1. Introduction In the articles I have heretofore devoted to space I have above all emphasized the problems raised by non-Euclidean geometry, while leaving almost completely aside other questions more diffi- cult of approach, such as those which pertain to the number of dimensions. AU the geometries I considered had thus a common basis, 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 it. In this continuum, primitively amorphous, we may imagine a network of lines and surfaces, we may then convene to regard the meshes of this net as equal to one another, and it is only after this convention that this continuum, become measurable, becomes Euclidean or non-Euclidean space. From this amor- phous continuum can therefore arise indifferently one or the other of the two spaces, just as on a blank sheet of paper may be traced indifferently a straight or a circle. In space we know rectilinear triangles the sum of whose angles is equal to two right angles; but equally we know curvilinear triangles the sum of whose angles is less than two right angles. The existence of the one sort is not more doubtful than that of the other. To give the name of straights to the sides of the first is to adopt Euclidean geometry ; to give the name of straights to the sides of the latter is to adopt the non-Euclidean geometry. So that to ask what geometry it is proper to adopt is to ask, to what line is it proper to give the name straight? It is evident that experiment can not settle such a question; one would not ask, for instance, experiment to decide whether I should caU AB or CD a straight. On the other hand, 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 235 236 TRE VALUE OF SCIENCE conformity with the eternal idea of straight which I have by intuition. I grant, indeed, that I have the intuitive idea of the side of the Euclidean triangle, but I have equally the intuitive idea of the side of the non-Euclidean triangle. Why should I have the right to apply the name of straight to the first of these ideas and not to the second? "Wherein does this syllable form an integrant part of this intuitive idea ? Evidently when we say that the Euclidean straight is a true straight and that the non- Euclidean straight is not a true straight, we simply mean that the first intuitive idea corresponds to a more noteworthy object than the second. But how do we decide that this object is more i noteworthy? This question I have investigated in 'Science and ; Hypothesis. ' It is here that we saw experience come in. If the Euclidean straight is more noteworthy than the non-Euclidean straight, it is so chiefiy hecause it differs little from certain noteworthy natural objects from which the non-Euclidean straight differs greatly. But, it will be said, the definition of the non-Euclidean straight is artificial; if we for a moment adopt it, we shall see that two circles of different radius both receive the name of non-Euclidean straights, while of two circles of the same radius one can satisfy the definition without the other being able to sat- isfy it, and then if we transport one of these so-called straights without deforming it, it will cease to be a straight. But by what right do we consider as equal these two figures which the Euclidean geometers call two circles with the same radius ? It is because by transporting one of them without deforming it we can make it coincide with the other. And why do we say this transportation is effected without deformation ? It is impossible to give a good reason for it. Among all the motions conceiv- able, there are some of which the Euclidean geometers say that they are not accompanied by deformation ; but there are others of which the non-Euclidean geometers would say that they are not accompanied by deformation. In the first, called Euclidean mo- tions, the Euclidean straights remain BucUdean straights and the non-Euclidean straights do not remain non-Euclidean straights; in the motions of the second sort, or non-Euclidean motions, the non-Euclidean straights remain non-Euclidean straights THE NOTION OF SPACE 237 and the Euclidean straights do not remain Euclidean straights. It has, therefore, not been demonstrated that it was unreasonable to call straights the sides of non-Euclidean tri- angles ; it has only been shown that that would be unreasonable if one continued to call the Euclidean motions motions without deformation; but it has at the same time been shown that it would be just as unreasonable to call straights the sides of Eu- clidean triangles if the non-Euclidean motions were called mo- tions without deformation. Now when we say that the Euclidean motions are the true motions without deformation, what do we mean? We simply mean that they are more noteworthy than the others. And why are they more noteworthy? It is because certain noteworthy najinyal bodies, the solid bodies, undergo motions almost similar. And then when we ask : Can one imagine non-Euclidean space ? That means : Can we imagine a world where there would be note- worthy natural objects affecting almost the form of non-Euclid- ean straights, and noteworthy natural bodies frequently under- going motions almost similar to the non-Euclidean motions? I have shown in 'Science and Hypothesis' that to this question we must answer yes. It has often been observed that if all the bodies in the universe were dilated simultaneously and in the same proportion, we should have no means of perceiving it, since all our measuring instruments would grow at the same time as the objects them- selves which they serve to measure. The world, after this dila- tation, would continue on its course without anything appris- ing us of so considerable an event. In other words, two worlds similar to one another (understanding the word similitude in the sense of Euclid, Book VI.) would be absolutely indistin- guMable. But more ; worlds will be indistinguishable not only if they are equal or similar, that is, if we can pass from one to the other by changing the axes of coordinates, or by changing the scale to which lengths are referred; but they will still be indistinguishable if we can pass from one to the other by any 'point-transformation' whatever. I will explain my meaning. I suppose that to each point of one corresponds one point of the other and only one, and inversely; and besides that the coordi- 17 238 THE VALUE OF SCIENCE nates of a point are continuous functions, otherwise altogether arbitrary, of the corresponding point. 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 point. I suppose finally that this correspondence fulfilled at the initial instant is maintained indefinitely. We should have no means ; of distinguishing these two worlds one from the other. The rela- tivity of space is not ordinarily understood in so broad a sense ; it is thus, however, that it would be proper to understand it. If one of these universes is our Euclidean world, what its in- habitants will call straight will be our Euclidean straight; 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 deformation. Their geometry will, there- fore, be Euclidean geometry, but their straight will not be our Euclidean straight. It will be its transform by the point-trans- formation which carries over from our world to theirs. The I straights of these men will not be our straights, but they will I have among themselves the same relations as our straights to one another. It is in this sense I say their geometry will be ours. If then we wish after all to proclaim that they deceive them- selves, that their straight is not the true straight, if we still are unwilling to admit that such an affirmation has no meaning, at least we must confess that these people have no means whatever of recognizing their error. 2. Qualitative Geometry AU that is relatively easy to understand, and I have already so often repeated it that I think it needless to expatiate further on rthe matter. Euclidean space is not a form imposed upon our ; sensibility, since we can imagine non-Euclidean space; but the two spaces, Euclidean and non-Euclidean, have a common basis, that amorphous continuum of which I spoke in the beginning. From this continuum we can get either Euclidean space or LobachevsMan space, just as we can, by tracing upon it a proper graduation, transform an ungraduated thermometer into a Fahr- enheit or a Eeaumur thermometer. TEE NOTION OF SPACE 239 And then comes a question: Is not this amorphous continuum, that our analysis has allowed to survive, a form imposed upon our sensibility? If so, we should have enlarged the prison in which this sensibility is confined, but it would always be a prison. This continuum has a certain number of properties, exempt, from all idea of measurement. The study of these properties is the object of a science which has been cultivated by many great geometers and in particular by Eiemann and Betti and which has received the name of analysis situs. In this science abstrac- tion is made of every quantitative idea and, for example, if we ascertain that on a line the point B is between the points A and C, we shall be content with this ascertainment and shall not trouble to know whether the line ABC is straight or curved, nor whether the length AB is equal to the length BC, or whether it is twice as great. The theorems of analysis situs have, therefore, this peculiarity, '. that they would remain true if the figures were copied by an inexpert draftsman who should grossly change all the propor- tions and replace the straights by lines more or less sinuous. In mathematical terms, they are not altered by any 'point-trans- formation' whatsoever. It has often been said that metric geom- etry was quantitative, while projective geometry was purely qual- itative. That is not altogether true. The straight is still dis- tinguished from other lines by properties which remain quanti- tative in some respects. The real qualitative geometry is, there- fore, analysis situs. The same questions which came up apropos of the truths of Euclidean geometry, come up anew apropos of the theorems of analysis situs. Are they obtainable by deductive reasoning? Are they disguised conventions? Are they experimental veri- ties? Are they the characteristics of a form imposed either, upon our sensibility or upon our understanding? I wish simply to observe that the last two solutions exclude each other. "We can not admit at the same time that it is impos- sible to imagine space of four dimensions and that experience proves to us that space has three dimensions. The experimenter puts to nature a question : Is it this or that ? and he can not put 240 THE VALVE OF SCIENCE it without imagining the two terms of the alternative. If it were impossible to imagine one of these terms, it would he futile and besides impossible to consult experience. There is no need of ob- servation to know that the hand of a watch is not marking the hour 15 on the dial, because we know beforehand that there are only 12, and we could not look at the mark 15 to see if the hand is there, because this mark does not exist.- Note likewise that in analysis situs the empiricists are disem- barrassed of one of the gravest objections that can be leveled against them, of that which renders absolutely vain in advance all their efforts to apply their thesis to the verities of Euclidean geometry. These verities are rigorous and all experimentation can only be approximate. In analysis situs approximate exper- iments may suffice to give a rigorous theorem and, for instance, if it is seen that space can not have either two or less than two dimensions, nor four or more than four, we are certain that it has exactly three, since it could not have two and a half or three and a half. Of aU the theorems of analysis situs, the most important is that which is expressed in saying that space has three dimen- sions. This it is that we are about to consider, and we shall put the question in these terms: "When we say that space has three dimensions, what do we mean? 3. The Physical Continuum of Several Dimensions 1 have explained in 'Science and Hypothesis' whence we derive the notion of physical continuity and how that of mathe- matical continuity has arisen from it. It happens that we are capable of distinguishing two impressions one from the other, while each is indistinguishable from a third. Thus we can read- ily distinguish a weight of 12 grams from a weight of 10 grams, while a weight of 11 grams could be distinguished from neither the one nor the other. Such a statement, translated into sym- bols, may be written : i A=B, B = C, Ac, A = B, B=:C, which summed up the data of crude experience, implied an in- tolerable contradiction. To get free from it, it was necessary to introduce a new notion while still respecting the essential char- acteristics of the physical continuum of several dimensions. The mathematical continuum of one dimension admitted of a scale whose divisions, infinite in number, corresponded to the different values, commensurable or not, of one same magnitude. To have the mathematical continuum of n dimensions, it will suffice to take n like scales whose divisions correspond to different values of n independent magnitudes caUed coordinates. We thus shall have an image of the physical continuum of n dimensions, and this image will be as faithful as it can be after the determina- tion not to allow the contradiction of which I spoke above. 4. The Notion of Point It seems now that the question we put to ourselves at the start is answered. When we say that space has three dimensions, it will be said, we mean that the manifold of points of space satis- fies the definition we have just given of the physical continuum of three dimensions. To be content vnth that would be to sup- pose that we know what is the manifold of points of space, or even one point of space. Now that is not as simple as one might think. Every one believes he knows what a point is, and it is just because we know it too well that we think there is no need of defining it. Surely we can not be required to know how to define it, because in going back from definition to definition a time must come when we must stop. But at what moment should we stop ? We shall stop first when we reach an object which falls under our senses or that we can represent to ourselves ; definition then will become useless; we do not define the sheep to a child; we say to him : See the sheep. TBE NOTION OF SPACE 245 So, then, we should ask ourselves if it is possible to represent to ourselves a point of space. 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 paper, and that they can represent to themselves only an object or rather the impressions that this object made on their senses. When they try to represent to themselves a point, they repre- sent the impressions that very little objects made them feel. It is needless to add that two different objects, though both very little, may produce extremely different impressions, but I shall not dwell on this difficulty, which would still require some discussion. But it is not a question of that ; it does not suffice to represent one point, it is necessary to represent a certain point and to have the means of distinguishing it from an other point. And in fact, 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 dimensions, we must rely upon the fact that two elements of this continuum sometimes can and sometimes can not be distinguished. 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 element. 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 myself, or whether it is a different point. In other words, how do we know whether the point occupied by the object A at the instant a is the same as the point occupied by the object B at the instant ^, or still better, what this means ? I am seated in my room ; an object is placed on my table ; dur- ing a second I do not move, no one touches the object. 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 end. Not at all ; from the point A to the point B is 30 kilometers, because the object has been carried along in the motion of the earth. We can not know whether an object, be it large or small, has not changed its absolute position in space, and not only can we not affirm it, but this affirmation has no 246 THE VALUE OF SCIENCE meaning and in any case can not correspond to any representation. But then we may ask ourselves if the relative position of an object with regard to other objects has changed or not, and first whether the relative position of this object with regard to our body has changed. If the impressions this object makes upon us have not changed, we shall be inclined to judge that neither has this relative position changed; if they have changed, we shall judge that this object has changed either in state or in relative position. It remains to decide which of the two. I have explained in 'Science and Hypothesis' how we have been led to distinguish the changes of position. Moreover, I shall return to that further on. We come to know, therefore, whether the relative position of an object with regard to our body has or has not remained the same. If now we see that two objects have retained their relative posi- tion with regard to our body, we conclude that the relative posi- 1 tion of these two objects with regard to one another has not \ changed ; but we reach this conclusion only by indirect reasoning. The only thing that we know directly is the relative position of the objects with regard to our body. A fortiori it is only by indirect reasoning that we think we know (and, moreover, this ! belief is delusive) whether the absolute position of the object has ' changed. In a word, the system of icoordinate axes to which we naturally refer all exterior objects is a system of axes invariably bound to our body, and carried around with us. It is impossible to represent to oneself absolute space ; when I try to represent to myself simultaneously objects and myself in niotion in absolute space, in reality I represent to myself my own self montionless and seeing move around me different objects and a man that is exterior to me, but that I convene to call me. Will the difficulty be solved if we agree to refer everything to these axes bound to our body? Shall we know then what is a point thus defined by its relative position with regard to our- selves? Many persons will answer yes and will say that they 'localize' exterior objects. What does this mean ? To localize an object simply means to represent to oneself the movements that would be necessary to THE NOTION OF SPACE 247 reach it. I will explain myself. It is not a question of repre- senting the movements themselves in space, but solely of repre- senting to oneself the muscular sensations which accompany these movements and which do not presuppose the preexistence of the notion of space. If we suppose two different objects which successively occupy the same relative position with regard to ourselves, the impres- sions that these two objects make upon us will be very different; if we localize them at the same point, this is simply because it is necessary to make the same movements to reach them ; apart from that, one can not just see what they could have in common. But, given an object, we can conceive many different series of movements which equally enable us to reach it. If then we repre- sent to ourselves a point by representing to ourselves the series of muscular sensations which accompany the movements which enable us to reach this point, there will be many ways entirely different of representing to oneself the same point. If one is not satisfied with this solution, but wishes, for instance, to bring in the visual sensations along with the muscular sensations, there will be one or two more ways of representing to oneself this same point and the difficulty will only be increased. In any case the following question comes up: Why do we think that all these representations so different from one another still represent the same point? Another remark: I have just said that it is to our own body that we naturally refer exterior objects ; that we carry about every- where 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 body. It should be noticed that rigorously we could not speak of axes invariably bound to the body unless the dif- ferent parts of this body were themselves invariably bound to one another. As this is not the case, we ought, before referring exterior objects to these fictitious axes, to suppose our body brought back to the initial attitude. 5. The Notion of Displacement 1 have shown in 'Science and Hypothesis' the preponderant role played by the movements of our body in the genesis of the 248 THE VALUE OF SCIENCE notion of space. For a being completely imnjovable there would be neither space nor geometry ; in vain would exterior objects be displaced about him, the variations which these displacements would make in his impressions would not be attributed by this being to changes of position, but to simple changes of state; this being would have no means of distinguishing these two sorts of changes, and this distinction, fundamental for us, would have no meaning for him. The movements that we impress upon our members have as effect the varying of the impressions produced on our senses by external objects ; other causes may likewise make them vary ; but we are led to distinguish the changes produced by our own motions and we easily discriminate them for two reasons: (1) because they are voluntary; (2) because they are accompanied by muscular sensations. So we naturally divide the changes that our impressions may undergo into two categories to which perhaps I have given an inappropriate designation: (1) the internal changes, which are voluntary and accompanied by muscular sensations; (2) the external changes, having the opposite characteristics. We then observe that among the external changes are some which can be corrected, thanks to an internal change which brings everything back to the primitive state ; others can not be corrected in this way (it is thus that, when an exterior object is displaced, we may then by changing our own position replace ourselves as regards this object in the same relative position as before, so as to reestablish the original aggregate of impressions; if this object was not displaced, but changed its state, that is impos- sible) . Thence comes a new distinction among external changes : those which may be so corrected we call changes of position; and the others, changes of state. Think, for example, of a sphere with one hemisphere blue and the other red ; it first presents to us the blue hemisphere, then it so revolves as to present the red hemisphere. Now think of a spherical vase containing a blue liquid which becomes red in consequence of a chemical reaction. In both cases the sensation of red has replaced that of blue ; our senses have experienced the same impressions which have succeeded each other in the same TRE NOTION OF SPACE 249 order, and yet these two changes are regarded by us as very different ; the first is a displacement, the second a change of state. Why ? 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 sensation. Still more; if the two hemispheres, in place of being red and blue, had been yellow and green, how should I have interpreted the revolution of the sphere ? Before, the red succeeded the blue, now the green succeeds the yellow; and yet I say that the two spheres have undergone the same revolution, that each has turned about its axis; yet I can not say that the green is to yellow as the red is to blue; how then am I led to decide that the two spheres have undergone the same displacement? Evidently be- cause, in one case as in the other, I am able to reestablish the original sensation by going around the sphere, by making the same movements, and I know that I have made the same move- ments because I have felt the same muscular sensations ; to know it, I do not need, therefore, to know geometry in advance and to represent to myself the movements of my body in geometric space. Another example: An object is displaced before my eye; its image was first formed at the center of the retina; then it is formed at the border; the old sensation was carried to me by a nerve fiber ending at the center of the retina ; the new sensation is carried to me by another nerve fiber starting from the border of the retina; these two sensations are qualitatively different; otherwise, how could I distinguish them ? Why then am I led to decide that these two sensations, quali- tatively different, represent the same image, which has been dis- placed? It is because I can follow the object with the eye and by a displacement of the eye, voluntary and accompanied by muscu- lar sensations, bring back the image to the center of the retina and reestablish the primitive sensation. I suppose that the image of a red object has gone from the center A to the border B of the retina, then that the image of a blue object goes in its turn from the center A to the border B of the retina ; I shall decide that these two objects have under- gone the same displacement. Why? Because in both cases I shall have been able to reestablish the primitive sensation, and 250 THE VALUE OF SCIENCE that to do it I shall have had to execute the same movement of the eye, and I shall know that my eye has executed the same movement because I shall have felt the same muscular sensations. If I could not move my eye, should I have any reason to sup- pose 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 border? I should only have four sensations qualitatively different, and if I were asked if they are connected by the proportion I have just stated, the question would seem to me ridiculous, just as if I were asked if there is an analogous proportion between an auditory sensation, a tactile sensation and an olfactory sensation. Let us now consider the internal changes, that is, those which are produced by the voluntary movements of our body and which are accompanied by muscular changes. They give rise to the two following observations, analogous to those we have just made on the subject of external changes. 1. I may suppose that my body has moved from one point to another, but that the same attitude is retained; all the parts of the body have therefore retained or resumed the same relative situation, although their absolute situation in space may have varied. I may suppose that not only has the position of my body changed, but that its attitude is no longer the same, that, for instance, my arms which before were folded are now stretched out. I should therefore distinguish the simple changes of position without change of attitude, and the changes of attitude. Both would appear to me under form of muscular sensations. How then am I led to distinguish them ? It is that the first may serve to correct an external change, and that the others can not, or at least can only give an imperfect correction. This fact I proceed to explain as I would explain it to some one who already knew geometry, but it need not thence be concluded that it is necessary already to know geometry to make this dis- tinction; before knowing geometry I ascertain the fact (experi- mentally, so to speak), without being able to explain it. But merely to make the distinction between the two kinds of change, I do not need to explain the fact, it sufiSces me to ascertain it. However that may be, the explanation is easy. Suppose that THE NOTION OF SPACE 261 an exterior object is displaced ; if we wish the different parts of our body to resume with regard to this object their initial relative position, it is necessary that these different parts should have resumed likewise their initial relative position with regard to one another. Only the internal changes which satisfy this latter condition will be capable of correcting the external change pro- duced by the displacement of that object. If, therefore, the relative position of my eye with regard to my finger has changed, I shall still be able to replace the eye in its initial relative situa- tion with regard to the object and reestablish thus the primitive visual sensations, but then the relative position of the finger with regard to the object will have changed and the tactile sensations will not be reestablished. 2. We ascertain likewise that the same external change may be corrected by two internal changes corresponding to different muscular sensations. Here again I can ascertain this without knowing geometry ; and I have no need of anything else ; but I proceed to give the explanation of the fact, employing geometrical language. To go from the position A to the position B I may take several routes. To the first of these routes will correspond a series S of muscular sensations ; to a second route will corre- spond another series 8", of muscular sensations which generally will be completely different, since other muscles will be used. How am I led to regaj*d these two series S and S" as corre- sponding to the same displacement AB 1 It is because these two series are capable of correcting the same external change. Apart from that, they have nothing in common. Let us now consider two external changes : a and p, which shall be, for instance, the rotation of a sphere half blue, half red, and that of a sphere half yellow, half green ; these two changes have nothing in common, since the one is for us the passing of blue into red and the other the passing of yellow into green. Con- sider, on the other hand, two series of internal changes S and S" ; like the others, they will have nothing in common. And yet I say that a and /8 correspond to the same displacement, and that 8 and 8" correspond also to the same displacement. Why? Simply because 8 can correct a as well as /3 and because a can be cor- rected by 8" as well as by 8. And then a question suggests itself : 252 THE VALUE OF SCIENCE If I have ascertained that S corrects a and ^ and that 8" corrects a, am I certain that S" likewise corrects pi Experiment alone can teach ns whether this law is verified. If it were not verified, at least approximately, there would be no geometry, there would be no space, because we should have no more interest in classi- fying the internal and external changes as I have just done, and, for instance, in distinguishing changes of state from changes of position. It is interesting to see what has been the role of experience in all this. It has shown me that a certain law is approximately verified. It has not told me how space is, and that it satis- fies the condition in question. I knew, in fact, before all experi- ence, that space satisfied this condition or that it would not be; nor have I any right to say that experience told me that geometry is possible ; I very well see that geometry is possible, since it does not imply contradiction ; experience only tells me that geometry is useful. 6. Visual Space Although motor impressions have had, as I have just explained, an altogether preponderant influence in the genesis of the notion of space, which never would have taken birth without them, it will not be without interest to examine also the role of visual impressions and to investigate how many dimensions 'visual space' has, and for that purpose to apply to these impressions the definition of § 3. A first difficulty presents itself : consider a red color sensation affecting a certain point of the retina ; and on the other hand a blue color sensation affecting the same point of the retina. It is necessary that we have some means of recognizing that these two sensations, qualitatively different, have something in common. Now, according to the considerations expounded in the preceding paragraph, we have been able to recognize this only by the move- ments of the eye and the observations to which they have given rise. If the eye were immovable, or if we were unconscious of its movements, we should not have been able to recognize that these two sensations, of different quality, had something in com- mon ; we should not have been able to disengage from them what THE NOTION OF SPACE 253 gives them a geometric character. The visual sensations, without the muscular sensations, would have nothing geometric, so that it may be said there is no pure visual space. To do away with this difficulty, consider only sensations of the same nature, red sensations, for instance, differing one from another only as regards the point of the retina that they affect. It is clear that I have no reason for making such an arbitrary choice among all the possible visual sensations, for the purpose of uniting in the same class all the sensations of the same color, whatever may be the point of the retina affected. I should never have dreamt of it, had I not before learned, by the means we have just seen, to distinguish changes of state from changes of position, that is, if my eye were immovable. Two sensations of the same color affecting two different parts of the retina would have appeared to me as qualitatively distinct, just as two sensa- tions of different color. In restricting myself to red sensations, I therefore impose upon myself an artificial limitation and I neglect systematically one whole side of the question ; but it is only by this artifice that I am able to analyze visual space without mingling any motor sensation. Imagine a line traced on the retina and dividing in two its surface ; and set apart the red sensations affecting a point of this line, or those differing from them too little to be distinguished from them. The aggregate of these sensations will form a sort of cut that I shall call C, and it is dear that this cut suffices to divide the manifold of possible red sensations, and that if I take two red sensations affecting two points situated on one side and the other of the line, 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 cut. If, therefore, the cut has n dimensions, the total manifold of my red sensations, or if you wish, the whole visual space, will have n + 1. Now, I distinguish the red sensations affecting a point of the cut C. The assemblage of these sensations will form a new cut C. It is clear that this will divide the cut C, always giving to the word divide the same meaning. 18 254 THE VALUE OF SCIENCE If, therefore, the cut C has n dimensions, the cut C will have w + 1 and the whole of visual space n -\- 2. If all the red sensations aifecting the same point of the retina were regarded as identical, the cut C reducing to a single ele- ment would have dimensions, and visual space would have 2. And yet most often it is said that the eye gives us the sense of a third dimension, and enables us in a certain measure to recog- nize the distance of objects. When we seek to analyze this feel- ing, we ascertain that it reduces either to the consciousness of the convergence of the eyes, or to that of the effort of accommodation which the ciliary muscle makes to focus the image. 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 sensa- tion of effort of accommodation or at least by sensations of convergence and accommodation so slightly different as to be indistinguishable. On this account the cut C" is itself a continuum and the cut C has more than one dimension. But it happens precisely that experience teaches us that when two visual sensations are accompanied by the same sensation of convergence, they are likewise accompanied by the same sensa- tion of accommodation. If then we form a new cut C" with all those of the sensations of the cut C, which are accompanied by a certain sensation of convergence, in accordance with the preced- ing law they will all be indistinguishable and may be regarded as identical. Therefore C" will not be a continuum and will have dimension ; and as C" divides C it will thence result that C has one, G two and the whole visual space three dimensions. But would it be the same if experience had taught us the con- trary and if a certain sensation of convergence were not always accompanied by the same sensation of accommodation? In this case two sensations affecting the same point of the retina and accompanied by the same sense of convergence, two sensations which consequently would both appertain to the cut G", could nevertheless be distinguished since they would be accompanied by two different sensations of accommodation. Therefore G" would be in its turn a continuum and would have one dimension (at THE NOTION OF SPACE 255 least) ; then C would have two, C three and the whole visiial space would have four dimensions. Will it then be said that it is experience which teaches us that space has three dimensions, since it is in setting out from an experimental law that we have come to attribute three to it ? But we have therein performed, so to speak, only an experiment in physiology ; 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 accommodation, are we to say that putting on spectacles is enough to make space have four dimen- sions and that the optician who constructed them has given one more dimension to space 1 Evidently not ; all we can say is that experience has taught us that it is convenient to attribute three dimensions to space. But visual space is only one part of space, and in even the notion of this space there is something artificial, as I have ex- plained at the beginning. The real space is motor space and this it is that we shall examine in the following chapter. CHAPTER IV Space and its Thkee Dimensions 1. The Group of Displacements Let us sum up briefly the results obtained. "We proposed to investigate what was meant in saying that space has three dimen- sions and we have asked first what is a physical continuum and when it may be said to have n dimensions. If we consider dif- ferent systems of impressions and compare them with one another, we often recognize that two of these systems of impressions are indistinguishable (which is ordinarily expressed in sajdng that they are too close to one another, and that our senses are too crude, for us to distinguish them) and we ascertain besides that two of these systems can sometimes be discriminated from one another though indistinguishable from a third system. In that case we say the manifold of these systems of impressions forms a physical continuum C. And each of these systems is called an element of the continuum C. How many dimensions has this continuum? Take first two elements A and B of C, and suppose there exists a series S of elements, all belonging to the continuum C, 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 preceding. If such a series S can be found, we say that A and B are joined to one another; and if any two elements of C are joined to one another, we say that C is all of one piece. Now take on the continuum C a certain number of elements in a way altogether arbitrary. The aggregate of these elements will be called a cut. Among the various series S which join A to B, we shall distinguish those of which an element is indistinguish- able from one of the elements of the cut (we shall say that these are they which cut the cut) and those of which all the elements are distinguishable from all those of the cut. If all the series S which join A to B cut the cut, we shall say that A and B are 256 J ^ SPACE AND ITS THREE DIMENSIONS 257 separated by the cut, and that the cut divides C. If we can not find on C two elements which are separated by the cut, we shall say that the cut does not divide C. These definitions laid down, if the continuum C can be divided by cuts which do not themselves form a continuum, this con- tinuum C has only one dimension; in the contrary case it has several. If a cut forming a continuum of 1 dimension suffices to divide C, G will have 2 dimensions; if a cut forming a con- tinuum of 2 dimensions suffices, C will have 3 dimensions, etc. Thanks to these definitions, we can always recognize how many dimensions any physical continuum has. It only remains to find a physical continuum which is, so to speak, equivalent to space, of such a sort that to every point of space corresponds an ele- ment of this continuum, and that to points of space very near one another correspond indistinguishable elements. Space will have then as many dimensions as this continuum. The intermediation of this physical continuum, capable of representation, is indispensable; because we can not represent space to ourselves, and that for a multitude of reasons. Space is a mathematical continuum, it is infinite, and we can represent to ourselves only physical continua and finite objects. The dif- ferent elements of space, which we call points, are all alike, and, to apply our definition, it is necessary that we know how to dis- tinguish the elements from one another, at least if they are not too close. Finally absolute space is nonsense, and it is necessary for us to begin by referring space to a system of axes invariably bound to our body (which we must always suppose put back in the initial attitude). Then I have sought to form with our visual sensations a phys- ical continuum equivalent to space ; that certainly is easy and this example is particularly appropriate for the discussion of the number of dimensions; this discussion has enabled us to see in what measure it is allowable to say that 'visual space' has three dimensions. Only this solution is incomplete and artificial. I have explained why, and it is not on visual space, but on motor space that it is necessary to bring our efforts to bear. I have then recalled what is the origin of the distinction we make between 258 THE VALUE OF SCIENCE changes of position and changes of state. Among the changes which occur in our impressions, we distinguish, first the internal changes, voluntary and accompanied by muscular sensations, and the external changes, having opposite characteristics. "We ascer- tain that it may happen that an external change may be corrected by an internal change which reestablishes the primitive sensa- tions. The external changes, capable of being corrected by an internal change are caUed changes of position, those not capable of it are called changes of state. The internal changes capable of correcting an external change are called displacements of the whole iody; the others are called changes of attitude. Now let a and j8 be two external changes, a' and /3' two internal changes. Suppose that a may be corrected either by a' or by p', and that a' can correct either a or j8 ; experience tells us then that P' can likewise correct p. In this case we say that a and p cor- respond to the same displacement and also that a' and p' cor- respond to the same displacement. That postulated, we can imagine a physical continuum which we shaU call the continuum or group of displacements and which we shall define in the fol- lowing manner. The elements of this continuum shall be the in- ternal changes capable of correcting an external change. Two of these internal changes a' and p' shall be regarded as indis- tinguishable: (1) if they are so naturally, that is, if they are too close to one another; (2) if a' is capable of correcting the same external change as a third internal change natu- rally indistinguishable from p'. In this second case, they will be, so to speak, indistinguishable by convention, I mean by agree- ing to disregard circumstances which might distinguish them. Our continuum is now entirely defined, since we know its ele- ments and have fixed under what conditions they may be re- garded as indistinguishable. "We thus have all that is necessary to apply our definition and determine how many dimensions this continuum has. "We shall recognize that it has six. The con- tinuum of displacements is, therefore, not equivalent to space, since the number of dimensions is not the same ; it is only related to space. Now how do we know that this continuum of displace- ments has six dimensions? "We know it hy experience. It would be easy to describe the experiments by which we SPACE AND ITS THREE DIMENSIONS 259 could arrive at this result. It would be seen that in this con- tinuum cuts can be made which divide it and which are con- tinua; that these cuts themselves can be divided by other cuts of the second order which yet are eontinua, and that this would stop only after cuts of the sixth order which would no longer be eontinua. From our definitions that would mean that the group of displacements has six dimensions. That would be easy, I have said, but that would be rather long ; and would it not be a little superficial ? This group of displace- ments, we have seen, is related to space, and space could be de- duced from it, but it is not equivalent to space, since it has not the same number of dimensions ; and when we shall have shown how the notion of this continuum can be formed and how that of space may be deduced from it, it might always be asked why space of three dimensions is much more familiar to us than this continuum of six dimensions, and consequently doubted whether it was by this detour that the notion of space was formed in the human mind. 2. Identity of Two Points What is a point? How do we know whether two points of space are identical or difiEerent ? Or, in other words, when I say : The object A occupied at the instant a the point which the object B occupies at the instant j8, what does that mean ? Such is the problem we set ourselves in the preceding chapter, §4. As I have explained it, it is not a question of comparing the positions of the objects A and B in absolute space ; the question then would manifestly have no meaning. It is a question of comparing the positions of these two objects with regard to axes invariably bound to my body, supposing always this body re- placed in the same attitude. I suppose that between the instants a and /8 I have moved neither my body nor my eye, as I know from my muscular sense. Nor have I moved either my head, my arm or my hand. I ascer- tain that at the instant a impressions that I attributed to the object A were transmitted to me, some by one of the fibers of my optic nerve, the others by one of the sensitive tactile nerves of my finger ; I ascertain that at the instant ^ other impressions which I attribute to the object B are transmitted to me, some by 260 TEE VALUE OF SCIENCE this same fiber of the optic nerve, the others by this same tactile nerve. Here I must pause for an explanation ; how am I told that this impression which I attribute to A, and that which I attribute to B, impressions which are qualitatively different, are transmitted to me by the same nerve ? Must we suppose, to take for example the visual sensations, that A produces two simultaneous sensa- tions, a sensation purely luminous a and a colored sensation a', that B produces in the same way simultaneously a luminous sen- sation 6 and a colored sensation V, that if these different sensa- tions are transmitted to me by the same retinal fiber, a is iden- tical with 6, but that in general the colored sensations a' and 6' produced by different bodies are different ? In that case it would be the identity of the sensation a which accompanies a' with the sensation ft which accompanies V, which would tell that all these sensations are transmitted to me by the same fiber. However it may be with this hypothesis and although I am led to prefer to it others considerably more complicated, it is certain that we are told in some way that there is something in common between these sensations a-\-a' and 6 -j- V, without which we should have no means of recognizing that the object B has taken the place of the object A. Therefore I do not further insist and I recall the hypothesis 1 have just made : I suppose that I have ascertained that the im- pressions which I attribute to B are transmitted to me at the instant p by the same fibers, optic as well as tactile, which, at the instant a, had transmitted to me the impressions that I attributed to A. If it is so, we shall not hesitate to declare that the point occupied by B at the instant y8 is identical with the point occu- pied by A at the instant a. I have just enunciated two conditions for these points being identical ; one is relative to sight, the other to touch. Let us con- sider them separately. The first is necessary, but is not sufiS- cient. The second is at once necessary and sufficient. A person knowing geometry could easily explain this in the following manner : Let be the point of the retina where is formed at the instant a the image of the body 4 ; let M be the point of space occupied at the instant a by this body A ; let M' be the point of SPACE AND ITS THREE DIMENSIONS 261 space occupied at the instant p by the body B. For this body B to form its image in 0, it is not necessary that the points M and M' coincide; since vision acts at a distance, it suffices for the three points M M' to be in a straight line. This condition that the two objects form their image on is therefore necessary, but not sufficient for the points M and M' to coincide. Let now P be the point occupied by my finger and where it remains, since it does not budge. As touch does not act at a distance, if the body A touches my finger at the instant a, it is because M and P cotaeide ; if B touches my finger at the instant )8, it is because M' and P coincide. Therefore M and M' coincide. Thus this condition that if A touches my finger at the instant a, B touches it at the instant /?, is at once necessary and sufficient for M and M' to coincide. But we who, as yet, do not know geometry can not reason thus; all that we can do is to ascertain experimentally that the first condition relative to sight may be fulfilled without the second, which is relative to touch, but that the second can not be fulfilled without the first. Suppose experience had taught us the contrary, as might well be ; this hypothesis contains nothing absurd. Suppose, therefore, that we had ascertained experimentally that the condition rela- tive to touch may be fulfilled without that of sight being fulfilled and that, on the contrary, that of sight can not be fulfilled with- out that of touch being also. It is clear that if this were so we should conclude that it is touch which may be exercised at a dis- tance, and that sight does not operate at a distance. But this is not all; 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 finger ; but I could just as well have employed other means, for example, all my other fingers. I suppose that my first finger receives at the instant a a tactile impression which I attribute to the object A. I make a series of movements, corresponding to a series S of muscular sensations. After these movements, at the instant a', my second finger re- ceives a tactile impression that I attribute likewise to A. After- ward, at the instant 13, without my having budged, as my mus- cular sense tells me, this same second finger transmits to me 262 THE VALUE OF SCIENCE anew a tactile impression which I attribute this time to the object B; I then make a series of movements, corresponding to a series S' of muscular sensations. I know that this series S' is the inverse of the series S and corresponds to contrary move- ments. 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', the primitive impressions would be reestablished, in other words, that the two series mutually com- pensate. That settled, should I expect that at the instant /S', when the second series of movements is ended, my first finger would feel a tactile impression attributable to the object B? To answer this question, those already knowing geometry would reason as follows : There are chances that the object A has not budged, between the instants a and a', nor the object B between the instants p and )8'; assume this. At the instant a, the object A occupied a certain point M of space. Now at this instant it touched my first finger, and as touch does not operate at a distance, my first finger was likewise at the point M. I afterward made the series 8 of movements and at the end of this series, at the instant a', I ascertained that the object A touched my second finger. I thence conclude that this second finger was then at M, that is, that the movements S had the result of bringing the second finger to the place of the first. At the instant ^ the object B has come in contact with my second finger: as I have not budged, this second finger has remained at M; therefore the object B has come to M; by hypothesis it does not budge up to the instant p'. But between the instants ^ and jS' I have made the movements S' ; as these movements are the in- verse of the movements 8, they must have for effect bringing the first finger in the place of the second. At the instant 13' this first finger will, therefore, be at M ; and as the object B is like- wise at M, this object B will touch my first finger. To the ques- tion put, the answer should therefore be yes. We who do not yet know geometry can not reason thus; but we ascertain that this anticipation is ordinarily realized ; and we can always explain the exceptions by saying that the object A has moved between the instants a and a', or the object B between the instants /8 and /S'. SPACE AND ITS THREE DIMENSIONS 263 But could not experience have given a contrary result ? Would this contrary result have been absurd in itself ? Evidently not. What should we have done then if experience had given this contrary result? Would all geometry thus have become impos- sible? Not the least in the world. We should have contented ourselves with concluding that touch can operate at a distance. When I say, touch does not operate at a distance, but sight operates at a distance, this assertion has only one meaning, which is as follows: To recognize whether B occupies at the instant j8 the point occupied by A at the instant a, I can use a multitude of different criteria. In one my eye intervenes, in another my first finger, in another my second finger, etc. Well, it is sufficient for the criterion relative to one of my fin- gers to be satisfied in order that all the others should be satisfied, but it is not sufficient that the criterion relative to the eye should be. This is the sense of my assertion, I content myself with affirming an experimental fact which is ordinarily verified. At the end of the preceding chapter we analyzed visual space ; we saw that to engender this space it is necessary to bring in the retinal sensations, the sensation of convergence and the sensa- tion of accommodation; that if these last two were not always in accord, visual space would have four dimensions in place of three ; we also saw that if we brought in only the retinal sensa- tions, we should obtain 'simple visual space,' of only two dimen- sions. On the other hand, consider tactile space, limiting our- selves to the sensations of a single finger, that is in sum to the assemblage of positions this finger can occupy. This tactile space that we shall analyze in the following section and which consequently I ask permission not to consider further for the moment, this tactile space, I say, has three dimensions. Why has space properly so called as many dimensions as tactile space and more than simple visual space ? It is because touch does not operate at a distance, while vision does operate at a distance. These two assertions have the same meaning and we have just seen what this is. Now I return to a point over which I passed rapidly in order not to interrupt the discussion. How do we know that the im- pressions made on our retina by A at the instant a and B at the 264 TEE VALVE OF SCIENCE instant j8 are transmitted by the same retinal fiber, although these impressions are qualitatively different? I have suggested a simple hypothesis, while adding that other hypotheses, decid- edly more complex, would seem to me more probably true. Here then are these hypotheses, of which I have already said a word. How do we know that the impressions produced by the red object A at the instant a, and by the blue object B at the instant jS, if these two objects have been imaged on the same point of the retina, have something in common? The simple hypothesis above made may be rejected and we may suppose that these two impressions, qualitatively different, are transmitted by two dif- ferent though contiguous nervous fibers. What means have I then of knowing that these fibers are contiguous ? It is probable that we should have none, tf the eye were immovable. 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 sen- sation 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 B. They have shown us, in fact, that the same movements, corre- sponding to the same muscular sensations, carry us from the first to the second, or from the third to the fourth. I do not emphasize these considerations, which belong, as one sees, to the question of local signs raised by Lotze. 3. Tactile Space Thus I know how to recognize the identity of two points, the point occupied by A at the instant a and the point occupied by B at the instant p, but only on one condition, namely, that I have not budged between the instants a and j8. That does not sufBce for our object. Suppose, therefore, that I have moved in any manner in the interval between these two instants, how shall I know whether the point occupied by A at the instant a is identi- cal with the point occupied by B at the instant j8? I suppose that at the instant a, the object A was in contact with my first finger and that in the same way, at the instant p, the object B touches this first finger ; but at the same time, my muscular sense has told me that in the interval my body has moved. I have considered above two series of muscular sensations S and S', and SPACE AND ITS THREE DIMENSIONS 265 I have said it sometimes happens that we are led to consider two such series 8 and S' as inverse one of the other, because we have often observed that when these two series succeed one another our primitive impressions are reestablished. If then my muscular sense tells me that I have moved between the two instants a and p, but so as to feel successively the two series of muscular sensations 8 and 8' that I consider inverses, I shall still conclude, just as if I had not budged, that the points occupied by A at the instant a and by B at the instant p are identical, if I ascertain that my first finger touches A at the instant a, and B at the instant p. This solution is not yet completely satisfactory, as one will see. Let us see, in fact, how many dimensions it would make us at- tribute to space. I wish to compare the two points occupied, by A and B at the instants a and 13, or (what amounts to the same thing since I suppose that my finger touches A at the instant a and B at the instant j8) I wish to compare the two points occu- pied by my finger at the two instants a and jS. The sole means I use for this comparison is the series 2 of muscular sensations which have accompanied the movements of my body between these two instants. The different imaginable series 2 form evi- dently a physical continuum of which the number of dimensions is very great. Let us agree, as I have done, not to consider as distinct the two series 2 and 2 + S + 8', when 8 and 8' are in- verses one of the other in the sense above given to this word; in spite of this agreement, the aggregate of distinct series 2 will still form a physical continuum and the number of dimensions will be less but still very great. To each of these series 2 corresponds a point of space ; to two series 2 and 2' thus correspond two points M and M'. The means we have hitherto used enable us to recognize that M and M' are not distinct in two cases : (1) if 2 is identical with 2' ; (2) if 2' = ^ ^ S -\- 8', 8 and 8' being inverses one of the other. If in all the other cases we should regard M and M' as distinct, the mani- fold of points would have as many dimensions as the aggregate of distinct series 2, that is, much more than three. For those who already know geometry, the following explana- tion would be easily comprehensible. Among the imaginable 266 TRE VALUE OF SCIENCE series of muscular sensations, there are those which correspond to series of movements where the finger does not budge. I say that if one does not consider as distinct the series S and S + (t, where the series o- corresponds to movements where the finger does not budge, the aggregate of series will constitute a con- tinuum of three dimensions, but that if one regards as distinct two series S and 2' unless 2' = S + /S + ;S', S and 8' being in- verses, the aggregate of series will constitute a continuum of more than three dimensions. In fact, let there be in space a surface A, on this surface a line B, on this line a point M. Let Co be the aggregate of all series 2. Let C^ be the aggregate of all the series 2, such that at the end of corresponding movements the finger is found upon the surface A, and Cj or Cg the aggregate of series 2 such that at the end the finger is found on B, or at M. It is clear, first that Ci will constitute a cut which will divide Co, that Cj will be a cut which will divide G^, and Cg a cut which will divide Cj. Thence it results, in accordance with our definitions, that if Cg is a con- tinuum of n dimensions, Cj will be a physical continuum of n-\-3 dimensions. Therefore, let 2 and 2' = 2 + a be two series forming part of Cj ; for both, at the end of the movements, the finger is found at M ; thence results that at the beginning and at the end of the series