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+The Project Gutenberg EBook of The Foundations of Science: Science and
+Hypothesis, The Value of Science, Science and Method, by Henri Poincaré
+
+This eBook is for the use of anyone anywhere at no cost and with
+almost no restrictions whatsoever. You may copy it, give it away or
+re-use it under the terms of the Project Gutenberg License included
+with this eBook or online at www.gutenberg.org
+
+
+Title: The Foundations of Science: Science and Hypothesis, The Value of Science, Science and Method
+
+Author: Henri Poincaré
+
+Translator: George Bruce Halsted
+
+Release Date: May 17, 2012 [EBook #39713]
+
+Language: English
+
+Character set encoding: ISO-8859-1
+
+*** START OF THIS PROJECT GUTENBERG EBOOK THE FOUNDATIONS OF SCIENCE: ***
+
+
+
+
+Produced by Bryan Ness and the Online Distributed
+Proofreading Team at http://www.pgdp.net (This book was
+produced from scanned images of public domain material
+from the Google Print project.)
+
+
+
+
+
+
+
+
+
+ SCIENCE AND EDUCATION
+
+ A SERIES OF 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. POINCARÉ. Containing
+ the authorised 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.
+ Councilman Theobald Smith, G. N. Stewart, C. M. Jackson,
+ E. P. Lyon, James B. Herrick, John M. Dodson, C. R. Bardeen,
+ W. Ophuls, S. J. Meltzer, James Ewing, W. W. Keen, Henry H.
+ Donaldson, Christian A. Herter, and Henry P. Bowditch.
+
+ Volume III. University Control. By J. MCKEEN CATTELL and other
+ authors.
+
+ AMERICAN MEN OF SCIENCE. A Biographical Directory.
+
+ SCIENCE. A weekly journal devoted to the advancement of science.
+ The official organ of the American Association for the
+ Advancement of Science.
+
+ THE POPULAR SCIENCE MONTHLY. A monthly magazine devoted to the
+ diffusion of science.
+
+ THE AMERICAN NATURALIST. A monthly journal devoted to the
+ biological sciences, with special reference to the factors
+ of evolution.
+
+ THE SCIENCE PRESS
+
+ NEW YORK GARRISON, N. Y.
+
+
+
+
+ THE FOUNDATIONS
+ OF SCIENCE
+
+ SCIENCE AND HYPOTHESIS
+ THE VALUE OF SCIENCE
+ SCIENCE AND METHOD
+
+
+ BY
+ H. POINCARÉ
+
+
+ AUTHORIZED TRANSLATION BY
+ GEORGE BRUCE HALSTED
+
+
+ WITH A SPECIAL PREFACE BY POINCARÉ, AND AN INTRODUCTION
+ BY JOSIAH ROYCE, HARVARD UNIVERSITY
+
+
+ THE SCIENCE PRESS
+ NEW YORK AND GARRISON, N. Y.
+ 1913
+
+
+
+
+ Copyright, 1913
+ BY THE SCIENCE PRESS
+
+
+ PRESS OF
+ THE NEW ERA PRINTING COMPANY
+ LANCASTER, PA.
+
+
+
+
+CONTENTS
+
+
+ PAGE
+ Henri Poincaré ix
+ Author's Preface to the Translation 3
+
+
+SCIENCE AND HYPOTHESIS
+
+ Introduction by Royce 9
+ Introduction 27
+
+ PART I. _Number and Magnitude_
+
+ CHAPTER I.--On the Nature of Mathematical Reasoning 31
+ Syllogistic Deduction 31
+ Verification and Proof 32
+ Elements of Arithmetic 33
+ Reasoning 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 Remarks (Curves without Tangents) 50
+ The Physical Continuum of Several Dimensions 52
+ The Mathematical Continuum of Several Dimensions 53
+
+ PART II. _Space_
+
+ CHAPTER III.--The Non-Euclidean Geometries 55
+ The Bolyai-Lobachevski Geometry 56
+ Riemann'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
+ Riemann'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
+ Solid Bodies and Geometry 72
+ Law of Homogeneity 74
+ The Non-Euclidean World 75
+ The World of Four Dimensions 78
+ Conclusions 79
+
+ CHAPTER 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
+
+ PART III. _Force_
+
+ CHAPTER VI.--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.--Relative Motion and Absolute Motion 107
+ The Principle of Relative Motion 107
+ Newton's Argument 108
+
+ CHAPTER VIII.--Energy and Thermodynamics 115
+ Energetics 115
+ Thermodynamics 119
+ General Conclusions on Part III 123
+
+ PART IV. _Nature_
+
+ CHAPTER IX.--Hypotheses in Physics 127
+ The Rôle of Experiment and Generalization 127
+ The Unity of Nature 130
+ The Rôle 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 164
+ Rouge et noir 167
+ The Probability of Causes 169
+ The Theory of Errors 170
+ Conclusions 172
+
+ CHAPTER XII.--Optics and Electricity 174
+ Fresnel's Theory 174
+ Maxwell's Theory 175
+ The Mechanical Explanation of Physical Phenomena 177
+
+ CHAPTER XIII.--Electrodynamics 184
+ Ampère's Theory 184
+ Closed Currents 185
+ Action of a Closed Current on a Portion of Current 186
+ Continuous Rotations 187
+ 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
+ Does 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
+
+ PART 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
+ Rôle 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 Central Forces 297
+ The Physics of the Principles 299
+
+ CHAPTER VIII.--The Present Crisis in Physics 303
+ The New Crisis 303
+ Carnot's Principle 303
+ The Principle of Relativity 305
+ Newton's Principle 308
+ Lavoisier's Principle 310
+ Mayer's Principle 312
+
+ CHAPTER IX.--The Future of Mathematical Physics 314
+ The Principles and Experiment 314
+ The Rôle of the Analyst 314
+ Aberration and Astronomy 315
+ Electrons and Spectra 316
+ Conventions preceding Experiment 317
+ Future Mathematical Physics 319
+
+ PART III. _The Objective Value of Science_
+
+ CHAPTER X.--Is Science Artificial? 321
+ The Philosophy of LeRoy 321
+ Science, Rule of Action 323
+ The Crude Fact and the Scientific Fact 325
+ Nominalism and the Universal Invariant 333
+
+ CHAPTER XI.--Science and Reality 340
+ Contingence 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 Reasoning_
+
+ CHAPTER I.--The Relativity of Space 413
+
+ CHAPTER II.--Mathematical Definitions and Teaching 430
+
+ 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 Mechanics_
+
+ 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 547
+
+
+
+
+HENRI POINCARÉ
+
+
+SIR GEORGE DARWIN, worthy son of an immortal father, said, referring to
+what Poincaré was to him and to his work: "He must be regarded as the
+presiding genius--or, shall I say, my patron saint?"
+
+Henri Poincaré 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 École Polytechnique, where, like John Bolyai at
+Maros Vásárhely, 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 _Nommé_, March 26, 1879. But he won his doctorate in the
+University of Paris, August 1, 1879, and was appointed to teach in the
+Faculté des Sciences de Caen, December 1, 1879, whence he was quickly
+called to the University 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
+l'Académie des Sciences, and, March 5, 1908, was chosen Membre de
+l'Académie Française. 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 Poincaré tells us the very
+minute it happened. Mounting 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 perspective new and
+immense. Moreover, the magic wand of his whole 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 commenced 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
+possibility of alternative laws of nature, which in the Introduction to
+the _Électricité 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_, convenient; but perhaps another
+may be vastly more advantageous. The old conception of _true_ has been
+revised. The first expression of the new idea occurs on the title page
+of John Bolyai's marvelous _Science Absolute of Space_, in the phrase
+"haud unquam 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,
+Poincaré proved the existence of a previously unknown pear-shaped
+figure, with the possibility that the progressive deformation of this
+figure with increasing angular velocity might result in the breaking up
+of the rotating body into two detached masses. Of his treatise _Les
+Méthodes nouvelles de la Méchanique céleste_, Sir George Darwin says:
+"It is probable that for half a century to come it will be the mine from
+which humbler investigators will excavate their materials." Brilliant
+was his appreciation of Poincaré 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
+Lobachevski medal of the Physico-Mathematical Society of Kazan, and the
+Bolyai prize of the Hungarian Academy of Sciences. His work must be
+reckoned with the greatest mathematical achievements of mankind.
+
+The kernel of Poincaré's power lies in an oracle Sylvester often quoted
+to me as from Hesiod: The whole is less than its part.
+
+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 immense 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 understanding physics; that this
+way seems to them superior to all others; that they feel no need of
+changing it, nor even of knowing 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 deplorable if there were no longer
+Anglo-Saxons.
+
+But continentals have on their part done things an Englishman 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 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 and 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 savoring of metaphysic.
+
+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 appreciating 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 uncompromising
+realists as has been said? Are they absolutely refractory, I do not say
+to metaphysic, but at least to everything metaphysical?
+
+Recall 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 ascensions
+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 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 all 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 mathematical 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 foreknowledge as only
+provisional, and they hasten to ask its definitive 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 ordinarily choose is not that preferred by the
+Anglo-Saxons.
+
+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 engrossed 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 studying the work of
+minds so unlike their own, they will immediately 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 will 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 geometric 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 perceived 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 perceiving them,
+have the advantage of showing them the hypothetical 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.
+
+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 formulas 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 construct the ether
+by material models, or in France to apply to it the laws of dynamic.
+
+The ether it is, the unknown, which explains matter, the known; matter
+is incapable of explaining the ether.
+
+ POINCARÉ.
+
+
+
+
+INTRODUCTION
+
+BY PROFESSOR JOSIAH ROYCE
+
+HARVARD 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.
+Poincaré'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. Poincaré has so notably contributed.
+
+
+I
+
+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. Poincaré 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
+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. Poincaré refers in his
+opening paragraphs, or whether, like Spencer 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 scientific orthodoxy was
+shocked amongst many of our American readers 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 presumption 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 details more assured. Phenomena men
+know and predict better than ever. But regarding the most general
+theories, and the 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 regarding 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 inconceivable' and the doctrine of energy beyond the reach of
+novel formulations. 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 interpretations. Certain properties of
+the irrational number [pi] 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 Dedekind, of Cantor, of Weierstrass and of various others,
+the general 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 several very
+notable respects, unconquered territory, despite the work of Hilbert and
+of Pieri, 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 radioactive bodies will occupy in that theory. The
+alchemists would not 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 invariant, than the theoretical
+orthodoxy of a recent period supposed. Again, regarding the foundations
+of biology, a theoretical orthodoxy grows less possible, less definable,
+less conceivable (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 objects, 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
+discoveries 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 dream, and that the
+results of science are coextensive with the range of actual empirical
+observation and of successful prediction? 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
+reasonable. 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 observed; 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 resignation? Why not substitute, for the
+old scientific orthodoxy, simply a confession of ignorance, and a
+resolution to devote ourselves to the business of enlarging the bounds
+of actual empirical knowledge?
+
+Such comments upon the situation just characterized are frequently 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
+these very comments. Never has there been better reason than there is
+to-day to be content, if rational man could be content, with a pure
+positivism. The splendid triumphs of special research in the most
+various fields, the constant increase in our practical control over
+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 'unification' 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 orthodoxy 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 regarding 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. Poincaré, in his
+book, sketches with the hand of the master, is another illustration of
+the consciousness 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 oppression 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, 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
+knowledge 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 different special issues. A student
+interested in the problems of the philosophy of mathematics, or in the
+theory of probabilities, or in the nature and office 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 construction--an
+abandonment of dogma, but a vindication of the rights of the
+constructive reason.
+
+
+III
+
+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 suggests 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 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 explicitly
+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 distinct 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. Poincaré'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 concepts 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.
+Poincaré's logical theory of this second class of hypotheses undertakes
+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
+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. Poincaré'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. Poincaré'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. Poincaré, on the contrary, all this
+adjustment of our interpretations 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 necessity. Unverifiable
+and irrefutable hypotheses in science are indeed, in general,
+indispensable aids to the organization and to the guidance of our
+interpretation of experience. But it is experience 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. Poincaré'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
+organization is due to hypotheses of the kind here in question, thus
+resembles that of a constitutional government--neither absolutely
+necessary, nor yet determined apart from the will of the subjects, nor
+yet accidental--a free, yet not a capricious establishment of good
+order, in conformity with empirical needs.
+
+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 described 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
+confirmed 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 existence of continuous
+extensive _quanta_ in nature; (2) The principles 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 intelligent beings whose experience could be best
+interpreted without using these hypotheses. Yet these hypotheses are
+_not_ subject 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 hypotheses 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 appreciation of the significance
+of empirical investigation. But the central problem of the logic of
+science thus becomes the problem of the relation between the two
+fundamentally distinct types of hypotheses, _i. e._, between those which
+can not be verified or refuted through experience, and those which can
+be empirically tested.
+
+
+IV
+
+The detailed treatment which M. Poincaré 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 observing that, while M.
+Poincaré 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. Poincaré'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, undertake theoretical
+constructions which relate to past time. Hypotheses 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 predictions of coming experience. These latter will be subject to
+control. 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 predictions proved
+verifiable. The result has been a considerable 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, regarding 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 hypotheses 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
+anthropology, or Old Testament history, the principal theoretical
+constructions 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 confrères of the sciences 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 criticism'--all these are matters
+which the men of the laboratory often regard with a general incredulity
+as belonging not at all to the domain of true science. Yet no one can
+doubt the importance 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
+verifiable 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 theoretical
+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 constructions in fact resemble
+in a measure those which M. Poincaré 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 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 well 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, internally rational, as well as (when that is possible)
+subject to external 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. Poincaré speaks when he characterizes the principles of mechanics and
+of the theory of energy. I venture to call attention here to this
+conception, which seems to me to illustrate M. Poincaré'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 Pathology' was the
+outcome of a very careful and elaborate induction. Virchow, himself,
+felt a strong aversion to mere speculation. 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 researches 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 diseases 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 the vital
+processes under altered conditions_" ('dasz sie nur Ablauf der
+Lebenserscheinungen unter veränderten Bedingungen 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 untouched 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 themselves, 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 _veränderte 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
+involves 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 case
+the disease is to be understood.
+
+For this very reason, however, Virchow's theoretical principle in its
+most general form _could be neither confirmed nor refuted by
+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
+_veränderte Bedingungen_ to which the diseased organism would be
+reacting. Let bullets or bacteria, poisons or compressed air, or the
+devil be the _Bedingungen_ to which a diseased organism reacts, the
+postulate that Virchow 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 Virchow's is no trial
+supposition, 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 interpretation 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 processes are linked. Now without some such leading idea, the
+cellular pathology itself could never have been reached; because the
+empirical facts in question would never have been observed. Hence this
+principle of Virchow's was indispensable to the growth of his science.
+Yet it was not a verifiable and not a refutable 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
+control, 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 irrefutable character,
+suggested, but not to be finally tested, by experience, the hypotheses
+in the narrower sense would lack that guidance which, as M. Poincaré 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 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 philosophical 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.
+Poincaré 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 problems 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 students 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 Poincaré was born at Nancy, in 1854, the son of a professor
+in the Faculty of Medicine at Nancy. He studied at the École
+Polytechnique and at the École des Mines, and later received his
+doctorate in mathematics in 1879. In 1883 he began courses of
+instruction in mathematics at the École Polytechnique; in 1886 received
+a professorship of mathematical 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 various branches of his
+chosen sciences is long; and his original 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 Poincaré 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. Poincaré has long been active. When, in 1893,
+the admirable _Revue de Métaphysique et de Morale_ began to appear, M.
+Poincaré was soon found amongst the most satisfactory of the
+contributors to the work of that journal, whose office it has especially
+been to bring philosophy and the various special sciences (both natural
+and moral) into a closer mutual understanding. The discussions brought
+together in the present volume are in large part the outcome of M.
+Poincaré's contributions to the _Revue de Métaphysique et de Morale_.
+The reader of M. Poincaré'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 possibility
+of doubt; the logic of science is infallible, and if the scientists 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. From 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 getting their
+first notions of physics, the origin of scientific certitude. This is
+what they suppose to be the rôle of experimentation 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 are two
+equally convenient solutions; each saves us from thinking.
+
+Instead of pronouncing a summary condemnation, we ought therefore to
+examine with care the rôle of hypothesis; we shall then recognize, not
+only that it is necessary, but that usually it is legitimate. We shall
+also see that there are several sorts of hypotheses; that some are
+verifiable, and once confirmed by experiment 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. Experiment 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 convention
+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 definitions and if the world he thinks he discovers
+is not simply created by his own caprice.[1] Under these conditions
+science would be certain, but deprived of significance.
+
+ [1] See Le Roy, 'Science et Philosophie,' _Revue de Métaphysique
+ et de Morale_, 1901.
+
+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
+naïve 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.
+
+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 mathematics puts into
+the hands of the investigator, we had to analyze another fundamental
+notion, that of mathematical magnitude. Do we find it in nature, or do
+we ourselves introduce it there? And, in this latter case, do we not
+risk marring everything? Comparing 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 imposed 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 geometry? A deeper discussion will show us it is not. We
+therefore conclude that the first principles of geometry are only
+conventions; 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 directly 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 hypotheses 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,
+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 principle 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 rôle 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 Maxwell, and what unconscious hypotheses
+were made by Ampère and the other founders of electrodynamics.
+
+
+
+
+PART I
+
+
+NUMBER AND MAGNITUDE
+
+
+
+
+CHAPTER I
+
+ON THE NATURE OF MATHEMATICAL REASONING
+
+
+I
+
+The very possibility of the science of mathematics seems an insoluble
+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? 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 so many volumes
+are nothing but devious ways of saying _A_ 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 mathematical necessity, we have yet the
+resource of classing them among synthetic _a priori_ judgments. This is
+not to solve the difficulty, 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
+reasoning 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 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 intention of
+generalizing some proposition already known. Does the mathematical
+method proceed from the particular to the general, 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 sufficiently powerful could at a glance perceive all
+its truths; nay more, we might even hope that some day one would invent
+to express them a language sufficiently simple to have them appear
+self-evident to an ordinary 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 called, 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 will suppose the number 1 defined and also the operation _x_ + 1 which
+consists in adding unity to a given number _x_.
+
+These definitions, whatever they be, do not enter into the course of the
+reasoning.
+
+I define then the numbers 2, 3 and 4 by the equalities
+
+ (1) 1 + 1 = 2; (2) 2 + 1 = 3; (3) 3 + 1 = 4.
+
+In the same way, I define the operation _x_ + 2 by the relation:
+
+ (4) _x_ + 2 = (_x_ + 1) + 1.
+
+That presupposed, we have
+
+ 2 + 1 + 1 = 3 + 1 (Definition 2),
+ 3 + 1 = 4 (Definition 3),
+ 2 + 2 = (2 + 1) + 1 (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 properly so called,' he will
+say to you: 'that is a verification.' We have confined ourselves to
+comparing two purely conventional definitions and have ascertained their
+identity; we have learned nothing new. _Verification_ differs from true
+demonstration precisely because it is purely analytic and because it is
+sterile. It is sterile because the conclusion is nothing but the
+premises translated 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.
+
+
+III
+
+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 complicated by
+arduous problems relative to the rôle of the postulates, to the nature
+and the origin of the notion of space. For analogous reasons we can not
+turn to the infinitesimal analysis. 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 undergone 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 precisely 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 mathematical rigor; they would see in it only
+useless and irksome subtleties; 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 impress itself
+upon all good minds? This is a logical and psychological 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 fail 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.
+
+DEFINITION OF ADDITION.--I suppose already defined the operation
+_x_ + 1, which consists in adding the number 1 to a given number _x_.
+
+This definition, whatever it be, does not enter into our subsequent
+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
+
+ _x_ + (_a_ - 1),
+
+the operation _x_ + _a_ will be defined by the equality
+
+ (1) _x_ + _a_ = [_x_ + (_a_ - 1)] + 1.
+
+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_ + 1 is,
+we can define successively and 'by recurrence' the operations _x_ + 2,
+_x_ + 3, etc.
+
+This definition deserves a moment's attention; it is of a particular
+nature which already distinguishes it from the purely logical
+definition; the equality (1) contains an infinity of distinct
+definitions, each having a meaning only when one knows the preceding.
+
+PROPERTIES OF ADDITION.--_Associativity._--I say that
+
+ _a_ + (_b_ + _c_) = (_a_ + _b_) + _c_.
+
+In fact the theorem is true for _c_ = 1; it is then written
+
+ _a_ + (_b_ + 1) = (_a_ + _b_) + 1,
+
+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_ = [gamma], I say it will be true for
+_c_ = [gamma] + 1.
+
+In fact, supposing
+
+ (_a_ + _b_) + [gamma] = _a_ + (_b_ + [gamma]),
+
+it follows that
+
+ [(_a_ + _b_) + [gamma]] + 1 = [_a_ + (_b_ + [gamma])] + 1
+
+or by definition (1)
+
+ (_a_ + _b_) + ([gamma] + 1) = _a_ + (_b_ + [gamma] + 1)
+ = _a_ + [_b_ + ([gamma] + 1)],
+
+which shows, by a series of purely analytic deductions, that the
+theorem is true for [gamma] + 1.
+
+Being true for _c_ = 1, we thus see successively that so it is for
+_c_ = 2, for _c_ = 3, etc.
+
+_Commutativity._--1º I say that
+
+ _a_ + 1 = 1 + _a_.
+
+The theorem is evidently true for _a_ = 1; we can _verify_ by purely
+analytic reasoning that if it is true for _a_ = [gamma] it will be true
+for _a_ = [gamma] + 1; for then
+
+ ([gamma] + 1) + 1 = (1 + [gamma]) + 1 = 1 + ([gamma] + 1);
+
+now it is true for _a_ = 1, therefore it will be true for _a_ = 2, for
+_a_ = 3, etc., which is expressed by saying that the enunciated
+proposition is demonstrated by recurrence.
+
+2º I say that
+
+ _a_ + _b_ = _b_ + _a_.
+
+The theorem has just been demonstrated for _b_ = 1; it can be verified
+analytically that if it is true for _b_ = [beta], it will be true for
+_b_ = [beta] + 1.
+
+The proposition is therefore established by recurrence.
+
+DEFINITION OF MULTIPLICATION.--We shall define multiplication by the
+equalities.
+
+ (1) _a_ × 1 = _a_.
+
+ (2) _a_ × _b_ = [_a_ × (_b_ - 1)] + _a_.
+
+Like equality (1), equality (2) contains an infinity of definitions;
+having defined a × 1, it enables us to define successively: _a_ × 2,
+_a_ × 3, etc.
+
+PROPERTIES OF MULTIPLICATION.--_Distributivity._--I say that
+
+ (_a_ + _b_) × _c_ = (_a_ × _c_) + (_b_ × _c_).
+
+We verify analytically that the equality is true for _c_ = 1; then
+that if the theorem is true for _c_ = [gamma], it will be true
+for _c_ = [gamma] + 1.
+
+The proposition is, therefore, demonstrated by recurrence.
+
+_Commutativity._--1º I say that
+
+ _a_ × 1 = 1 × _a_.
+
+The theorem is evident for _a_ = 1.
+
+We verify analytically that if it is true for _a_ = [alpha], it will be
+true for _a_ = [alpha] + 1.
+
+2º I say that
+
+ _a_ × _b_ = _b_ × _a_.
+
+The theorem has just been proven for _b_ = 1. We could verify
+analytically that if it is true for _b_ = [beta], it will be true
+for _b_ = [beta] + 1.
+
+
+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 recurrence. We first establish a
+theorem for _n_ = 1; then we show that if it is true of _n_ - 1, it is
+true of _n_, and thence conclude that it is true for all the whole
+numbers.
+
+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 transformation, 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 mathematics had no other instrument, it
+would therefore be forthwith 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 _n_ - 1, so it is of _n_.
+
+We see, then, that in reasoning by recurrence we confine ourselves 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.
+
+It is now easy to comprehend why every particular consequence 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 numbers, 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 cascade; 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 syllogisms 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 will 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 impracticable. 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 preponderant rôle, and without it
+there would be no science, because there would be nothing general.
+
+
+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 reasoning 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 contradiction.
+
+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 series, more or less long but always limited.
+
+Now if it were only a question of that, the principle of contradiction
+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, experience 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 induction? 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, demonstration 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_ + 1 = 1 + _a_
+
+and afterwards used it to establish the equality
+
+ _a_ + _b_ = _b_ + _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 mysterious to us, since we
+have ascertained the analogies between demonstration by recurrence and
+ordinary induction.
+
+Without doubt recurrent reasoning in mathematics and inductive 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_ + 1 = 1 + _a_
+
+and write
+
+ (2) _a_ + 2 = _a_ + 1 + 1 = 1 + _a_ + 1 = 1 + 1 + _a_ = 2 + _a_.
+
+The equality (2) thus deduced in purely analytic way from the equality
+(1) is, however, not simply a particular ease 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 'construct'
+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 relations 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 evidently the
+aggregates can not be regarded as more particular than their elements.
+
+Great importance, and justly, has been attached to this procedure 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 decomposable 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 prolonged
+exertions that they could be found by studying directly the relations of
+the elementary triangles. The knowledge of the general theorem spares us
+these efforts.
+
+A construction, therefore, becomes interesting only when it can be
+ranged beside other analogous constructions, forming species 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 importance than the whiteness of the chalk.
+
+The pure analyst has not this rock to fear. He has disengaged the
+science of mathematics from all 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 à la théorie 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
+conception, wherein is supposed between the elements of the continuum 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 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 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 necessary 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 property of these steps which is used
+in their reasonings[2] is that of being before or after such and such
+steps; therefore also this alone should occur in the definition.
+
+ [2] With those contained in the special conventions which serve to
+ define addition and of which we shall speak later.
+
+So how the intermediary terms should be intercalated need not concern
+us; on the other hand, no one will doubt the possibility 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 OF INCOMMENSURABLES.--The mathematicians of the Berlin
+school, Kronecker in particular, have devoted themselves to constructing
+this continuous scale of fractional and irrational 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 incommensurable
+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 objects; the
+replacement of these objects by others is therefore indifferent to them,
+provided the relations do not change. The 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 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 first 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 example, 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. Evidently 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 Dedekind's view, the incommensurable number
+
+ sqrt(2) or (2)^{1/2}
+
+is nothing but the symbol of this particular mode of partition of
+commensurable numbers; and to each mode of partition corresponds 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, 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 PHYSICAL 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 example, that a weight
+_A_ of 10 grams and a weight _B_ of 11 grams produce identical
+sensations, that the weight _B_ is just as indistinguishable 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_, _B_ = _C_, _A_ < _C_,
+
+which may be regarded as the formula of the physical continuum.
+
+But here is an intolerable discord with the principle of contradiction,
+and the need of stopping this has compelled us to invent the
+mathematical continuum.
+
+We are, therefore, forced to conclude that this notion has been created
+entirely by the mind, but that experience has given the occasion.
+
+We can not believe that two quantities equal to a third are not equal to
+one another, and so we are led to suppose that _A_ is different from _B_
+and _B_ from _C_, but that the imperfection of our senses has not
+permitted of our distinguishing them.
+
+CREATION OF THE MATHEMATICAL CONTINUUM.--_First Stage._ So far it would
+suffice, in accounting for the facts, to intercalate between _A_ and _B_
+a few terms, which would remain discrete. What happens now if we have
+recourse to some instrument to supplement the feebleness of our senses,
+if, for example, we make use of a microscope? Terms such as _A_ and _B_,
+before indistinguishable, appear now distinct; but between _A_ and _B_,
+now become distinct, will be intercalated a new term, _D_, that we can
+distinguish neither from _A_ nor from _B_. Despite the employment of the
+most highly perfected methods, the raw results of our experience will
+always present the characteristics of the physical continuum with the
+contradiction which is inherent in it.
+
+We shall escape it only by incessantly intercalating new terms between
+the terms already distinguished, and this operation must be continued
+indefinitely. We might conceive the stopping of this operation if we
+could imagine some instrument sufficiently powerful to decompose the
+physical continuum into discrete elements, as the telescope resolves the
+milky way into stars. But this we can not imagine; in fact, it is with
+the eye we observe the image magnified by the microscope, and
+consequently this image must always retain the characteristics of visual
+sensation and consequently those of the physical continuum.
+
+Nothing distinguishes a length observed directly from the half of this
+length doubled by the microscope. The whole is homogeneous with the
+part; this is a new contradiction, or rather it would be if the number
+of terms were supposed finite; in fact, it is clear that the part
+containing fewer terms than the whole could not be similar to the whole.
+
+The contradiction ceases when the number of terms is regarded as
+infinite; nothing hinders, for example, considering the aggregate of
+whole numbers as similar to the aggregate of even numbers, which,
+however, is only a part of it; and, in fact, to each whole number
+corresponds an even number, its double.
+
+But it is not only to escape this contradiction contained in the
+empirical data that the mind is led to create the concept of a
+continuum, formed of an indefinite number of terms.
+
+All happens as in the sequence of whole numbers. We have the faculty of
+conceiving that a unit can be added to a collection of units; thanks to
+experience, we have occasion to exercise this faculty and we become
+conscious of it; but from this moment we feel that our power has no
+limit and that we can count indefinitely, though we have never had to
+count more than a finite number of objects.
+
+Just so, as soon as we have been led to intercalate means between two
+consecutive terms of a series, we feel that this operation can be
+continued beyond all limit, and that there is, so to speak, no intrinsic
+reason for stopping.
+
+As an abbreviation, let me call a mathematical continuum of the first
+order every aggregate of terms formed according to the same law as the
+scale of commensurable numbers. If we afterwards intercalate new steps
+according to the law of formation of incommensurable numbers, we shall
+obtain what we will call a continuum of the second order.
+
+_Second Stage._--We have made hitherto only the first stride; we have
+explained the origin of continua of the first order; but it is necessary
+to see why even they are not sufficient and why the incommensurable
+numbers had to be invented.
+
+If we try to imagine a line, it must have the characteristics of the
+physical continuum, that is to say, we shall not be able to represent it
+except with a certain breadth. Two lines then will appear to us under
+the form of two narrow bands, and, if we are content with this rough
+image, it is evident that if the two lines cross they will have a common
+part.
+
+But the pure geometer makes a further effort; without entirely
+renouncing the aid of the senses, he tries to reach the concept of the
+line without breadth, of the point without extension. This he can only
+attain to by regarding the line as the limit toward which tends an ever
+narrowing band, and the point as the limit toward which tends an ever
+lessening area. And then, our two bands, however narrow they may be,
+will always have a common area, the smaller as they are the narrower,
+and whose limit will be what the pure geometer calls a point.
+
+This is why it is said two lines which cross have a point in common, and
+this truth seems intuitive.
+
+But it would imply contradiction if lines were conceived as continua of
+the first order, that is to say, if on the lines traced by the geometer
+should be found only points having for coordinates rational numbers. The
+contradiction would be manifest as soon as one affirmed, for example,
+the existence of straights and circles.
+
+It is clear, in fact, that if the points whose coordinates are
+commensurable were alone regarded as real, the circle inscribed in a
+square and the diagonal of this square would not intersect, since the
+coordinates of the point of intersection are incommensurable.
+
+That would not yet be sufficient, because we should get in this way only
+certain incommensurable numbers and not all those numbers.
+
+But conceive of a straight line divided into two rays. Each of these
+rays will appear to our imagination as a band of a certain breadth;
+these bands moreover will encroach one on the other, since there must be
+no interval between them. The common part will appear to us as a point
+which will always remain when we try to imagine our bands narrower and
+narrower, so that we admit as an intuitive truth that if a straight is
+cut into two rays their common frontier is a point; we recognize here
+the conception of Dedekind, in which an incommensurable number was
+regarded as the common frontier of two classes of rational numbers.
+
+Such is the origin of the continuum of the second order, which is the
+mathematical continuum properly so called.
+
+_Résumé._--In recapitulation, the mind has the faculty of creating
+symbols, and it is thus that it has constructed the mathematical
+continuum, which is only a particular system of symbols. Its power is
+limited only by the necessity of avoiding all contradiction; but the
+mind only makes use of this faculty if experience furnishes it a
+stimulus thereto.
+
+In the case considered, this stimulus was the notion of the physical
+continuum, drawn from the rough data of the senses. But this notion
+leads to a series of contradictions from which it is necessary
+successively to free ourselves. So we are forced to imagine a more and
+more complicated system of symbols. That at which we stop is not only
+exempt from internal contradiction (it was so already at all the stages
+we have traversed), but neither is it in contradiction with various
+propositions called intuitive, which are derived from empirical notions
+more or less elaborated.
+
+MEASURABLE MAGNITUDE.--The magnitudes we have studied hitherto are not
+_measurable_; we can indeed say whether a given one of these magnitudes
+is greater than another, but not whether it is twice or thrice as great.
+
+So far, I have only considered the order in which our terms are ranged.
+But for most applications that does not suffice. We must learn to
+compare the interval which separates any two terms. Only on this
+condition does the continuum become a measurable magnitude and the
+operations of arithmetic applicable.
+
+This can only be done by the aid of a new and special _convention_. We
+will _agree_ that in such and such a case the interval comprised between
+the terms _A_ and _B_ is equal to the interval which separates _C_ and
+_D_. For example, at the beginning of our work we have set out from the
+scale of the whole numbers and we have supposed intercalated between two
+consecutive steps _n_ intermediary steps; well, these new steps will be
+by convention regarded as equidistant.
+
+This is a way of defining the addition of two magnitudes, because if the
+interval _AB_ is by definition equal to the interval _CD_, the interval
+_AD_ will be by definition the sum of the intervals _AB_ and _AC_.
+
+This definition is arbitrary in a very large measure. It is not
+completely so, however. It is subjected to certain conditions and, for
+example, to the rules of commutativity and associativity of addition.
+But provided the definition chosen satisfies these rules, the choice is
+indifferent, and it is useless to particularize it.
+
+VARIOUS REMARKS.--We can now discuss several important questions:
+
+1º Is the creative power of the mind exhausted by the creation of the
+mathematical continuum?
+
+No: the works of Du Bois-Reymond demonstrate it in a striking way.
+
+We know that mathematicians distinguish between infinitesimals of
+different orders and that those of the second order are infinitesimal,
+not only in an absolute way, but also in relation to those of the first
+order. It is not difficult to imagine infinitesimals of fractional or
+even of irrational order, and thus we find again that scale of the
+mathematical continuum which has been dealt with in the preceding
+pages.
+
+Further, there are infinitesimals which are infinitely small in relation
+to those of the first order, and, on the contrary, infinitely great in
+relation to those of order 1 + [epsilon], and that however small
+[epsilon] may be. Here, then, are new terms intercalated in our series,
+and if I may be permitted to revert to the phraseology lately employed
+which is very convenient though not consecrated by usage, I shall say
+that thus has been created a sort of continuum of the third order.
+
+It would be easy to go further, but that would be idle; one would only
+be imagining symbols without possible application, and no one will think
+of doing that. The continuum of the third order, to which the
+consideration of the different orders of infinitesimals leads, is itself
+not useful enough to have won citizenship, and geometers regard it only
+as a mere curiosity. The mind uses its creative faculty only when
+experience requires it.
+
+2º Once in possession of the concept of the mathematical continuum, is
+one safe from contradictions analogous to those which gave birth to it?
+
+No, and I will give an example.
+
+One must be very wise not to regard it as evident that every curve has a
+tangent; and in fact if we picture this curve and a straight as two
+narrow bands we can always so dispose them that they have a part in
+common without crossing. If we imagine then the breadth of these two
+bands to diminish indefinitely, this common part will always subsist
+and, at the limit, so to speak, the two lines will have a point in
+common without crossing, that is to say, they will be tangent.
+
+The geometer who reasons in this way, consciously or not, is only doing
+what we have done above to prove two lines which cut have a point in
+common, and his intuition might seem just as legitimate.
+
+It would deceive him however. We can demonstrate that there are curves
+which have no tangent, if such a curve is defined as an analytic
+continuum of the second order.
+
+Without doubt some artifice analogous to those we have discussed above
+would have sufficed to remove the contradiction; but, as this is met
+with only in very exceptional cases, it has received no further
+attention.
+
+Instead of seeking to reconcile intuition with analysis, we have been
+content to sacrifice one of the two, and as analysis must remain
+impeccable, we have decided against intuition.
+
+THE PHYSICAL CONTINUUM OF SEVERAL DIMENSIONS.--We have discussed above
+the physical continuum as derived from the immediate data of our senses,
+or, if you wish, from the rough results of Fechner's experiments; I have
+shown that these results are summed up in the contradictory formulas
+
+ _A_ = _B_, _B_ = _C_, _A_ < _C_.
+
+Let us now see how this notion has been generalized and how from it has
+come the concept of many-dimensional continua.
+
+Consider any two aggregates of sensations. Either we can discriminate
+them one from another, or we can not, just as in Fechner's experiments a
+weight of 10 grams can be distinguished from a weight of 12 grams, but
+not from a weight of 11 grams. This is all that is required to construct
+the continuum of several dimensions.
+
+Let us call one of these aggregates of sensations an _element_. That
+will be something analogous to the _point_ of the mathematicians; it
+will not be altogether the same thing however. We can not say our
+element is without extension, since we can not distinguish it from
+neighboring elements and it is thus surrounded by a sort of haze. If the
+astronomical comparison may be allowed, our 'elements' would be like
+nebulae, whereas the mathematical points would be like stars.
+
+That being granted, a system of elements will form a _continuum_ if we
+can pass from any one of them to any other, by a series of consecutive
+elements such that each is indistinguishable from the preceding. This
+_linear_ series is to the _line_ of the mathematician what an isolated
+_element_ was to the point.
+
+Before going farther, I must explain what is meant by a _cut_. Consider
+a continuum _C_ and remove from it certain of its elements which for an
+instant we shall regard as no longer belonging to this continuum. The
+aggregate of the elements so removed will be called a cut. It may happen
+that, thanks to this cut, _C_ may be _subdivided_ into several distinct
+continua, the aggregate of the remaining elements ceasing to form a
+unique continuum.
+
+There will then be on _C_ two elements, _A_ and _B_, that must be
+regarded as belonging to two distinct continua, and this will be
+recognized because it will be impossible to find a linear series of
+consecutive elements of _C_, each of these elements indistinguishable
+from the preceding, the first being _A_ and the last _B_, _without one
+of the elements of this series being indistinguishable from one of the
+elements of the cut_.
+
+On the contrary, it may happen that the cut made is insufficient to
+subdivide the continuum _C_. To classify the physical continua, we will
+examine precisely what are the cuts which must be made to subdivide
+them.
+
+If a physical continuum _C_ can be subdivided by a cut reducing to a
+finite number of elements all distinguishable from one another (and
+consequently forming neither a continuum, nor several continua), we
+shall say _C_ is a _one-dimensional_ continuum.
+
+If, on the contrary, _C_ can be subdivided only by cuts which are
+themselves continua, we shall say _C_ has several dimensions. If cuts
+which are continua of one dimension suffice, we shall say _C_ has two
+dimensions; if cuts of two dimensions suffice, we shall say _C_ has
+three dimensions, and so on.
+
+Thus is defined the notion of the physical continuum of several
+dimensions, thanks to this very simple fact that two aggregates of
+sensations are distinguishable or indistinguishable.
+
+THE MATHEMATICAL CONTINUUM OF SEVERAL DIMENSIONS.--Thence the notion of
+the mathematical continuum of _n_ dimensions has sprung quite naturally
+by a process very like that we discussed at the beginning of this
+chapter. A point of such a continuum, you know, appears to us as defined
+by a system of _n_ distinct magnitudes called its coordinates.
+
+These magnitudes need not always be measurable; there is, for instance,
+a branch of geometry independent of the measurement of these magnitudes,
+in which it is only a question of knowing, for example, whether on a
+curve _ABC_, the point _B_ is between the points _A_ and _C_, and not of
+knowing whether the arc _AB_ is equal to the arc _BC_ or twice as great.
+This is what is called _Analysis Situs_.
+
+This is a whole body of doctrine which has attracted the attention of
+the greatest geometers and where we see flow one from another a series
+of remarkable theorems. What distinguishes these theorems from those of
+ordinary geometry is that they are purely qualitative and that they
+would remain true if the figures were copied by a draughtsman so awkward
+as to grossly distort the proportions and replace straights by strokes
+more or less curved.
+
+Through the wish to introduce measure next into the continuum just
+defined this continuum becomes space, and geometry is born. But the
+discussion of this is reserved for Part Second.
+
+
+
+
+PART II
+
+
+SPACE
+
+
+
+
+CHAPTER III
+
+THE NON-EUCLIDEAN GEOMETRIES
+
+
+Every conclusion supposes premises; these premises themselves either are
+self-evident and need no demonstration, or can be established only by
+relying upon other propositions, and since we can not go back thus to
+infinity, every deductive science, and in particular geometry, must rest
+on a certain number of undemonstrable axioms. All treatises on geometry
+begin, therefore, by the enunciation of these axioms. But among these
+there is a distinction to be made: Some, for example, 'Things which are
+equal to the same thing are equal to one another,' are not propositions
+of geometry, but propositions of analysis. I regard them as analytic
+judgments _a priori_, and shall not concern myself with them.
+
+But I must lay stress upon other axioms which are peculiar to geometry.
+Most treatises enunciate three of these explicitly:
+
+1º Through two points can pass only one straight;
+
+2º The straight line is the shortest path from one point to another;
+
+3º Through a given point there is not more than one parallel to a given
+straight.
+
+Although generally a proof of the second of these axioms is omitted, it
+would be possible to deduce it from the other two and from those, much
+more numerous, which are implicitly admitted without enunciating them,
+as I shall explain further on.
+
+It was long sought in vain to demonstrate likewise the third axiom,
+known as _Euclid's Postulate_. What vast effort has been wasted in this
+chimeric hope is truly unimaginable. Finally, in the first quarter of
+the nineteenth century, and almost at the same time, a Hungarian and a
+Russian, Bolyai and Lobachevski, established irrefutably that this
+demonstration is impossible; they have almost rid us of inventors of
+geometries 'sans postulatum'; since then the Académie des Sciences
+receives only about one or two new demonstrations a year.
+
+The question was not exhausted; it soon made a great stride by the
+publication of Riemann's celebrated memoir entitled: _Ueber die
+Hypothesen welche der Geometrie zu Grunde liegen_. This paper has
+inspired most of the recent works of which I shall speak further on, and
+among which it is proper to cite those of Beltrami and of Helmholtz.
+
+THE BOLYAI-LOBACHEVSKI GEOMETRY.--If it were possible to deduce Euclid's
+postulate from the other axioms, it is evident that in denying the
+postulate and admitting the other axioms, we should be led to
+contradictory consequences; it would therefore be impossible to base on
+such premises a coherent geometry.
+
+Now this is precisely what Lobachevski did.
+
+He assumes at the start that: _Through a given point can be drawn two
+parallels to a given straight_.
+
+And he retains besides all Euclid's other axioms. From these hypotheses
+he deduces a series of theorems among which it is impossible to find any
+contradiction, and he constructs a geometry whose faultless logic is
+inferior in nothing to that of the Euclidean geometry.
+
+The theorems are, of course, very different from those to which we are
+accustomed, and they can not fail to be at first a little disconcerting.
+
+Thus the sum of the angles of a triangle is always less than two right
+angles, and the difference between this sum and two right angles is
+proportional to the surface of the triangle.
+
+It is impossible to construct a figure similar to a given figure but of
+different dimensions.
+
+If we divide a circumference into _n_ equal parts, and draw tangents at
+the points of division, these _n_ tangents will form a polygon if the
+radius of the circle is small enough; but if this radius is sufficiently
+great they will not meet.
+
+It is useless to multiply these examples; Lobachevski's propositions
+have no relation to those of Euclid, but they are not less logically
+bound one to another.
+
+RIEMANN'S GEOMETRY.--Imagine a world uniquely peopled by beings of no
+thickness (height); and suppose these 'infinitely flat' animals are all
+in the same plane and can not get out. Admit besides that this world is
+sufficiently far from others to be free from their influence. While we
+are making hypotheses, it costs us no more to endow these beings with
+reason and believe them capable of creating a geometry. In that case,
+they will certainly attribute to space only two dimensions.
+
+But suppose now that these imaginary animals, while remaining without
+thickness, have the form of a spherical, and not of a plane, figure, and
+are all on the same sphere without power to get off. What geometry will
+they construct? First it is clear they will attribute to space only two
+dimensions; what will play for them the rôle of the straight line will
+be the shortest path from one point to another on the sphere, that is to
+say, an arc of a great circle; in a word, their geometry will be the
+spherical geometry.
+
+What they will call space will be this sphere on which they must stay,
+and on which happen all the phenomena they can know. Their space will
+therefore be _unbounded_ since on a sphere one can always go forward
+without ever being stopped, and yet it will be _finite_; one can never
+find the end of it, but one can make a tour of it.
+
+Well, Riemann's geometry is spherical geometry extended to three
+dimensions. To construct it, the German mathematician had to throw
+overboard, not only Euclid's postulate, but also the first axiom: _Only
+one straight can pass through two points_.
+
+On a sphere, through two given points we can draw _in general_ only one
+great circle (which, as we have just seen, would play the rôle of the
+straight for our imaginary beings); but there is an exception: if the
+two given points are diametrically opposite, an infinity of great
+circles can be drawn through them.
+
+In the same way, in Riemann's geometry (at least in one of its forms),
+through two points will pass in general only a single straight; but
+there are exceptional cases where through two points an infinity of
+straights can pass.
+
+There is a sort of opposition between Riemann's geometry and that of
+Lobachevski.
+
+Thus the sum of the angles of a triangle is:
+
+Equal to two right angles in Euclid's geometry;
+
+Less than two right angles in that of Lobachevski;
+
+Greater than two right angles in that of Riemann.
+
+The number of straights through a given point that can be drawn coplanar
+to a given straight, but nowhere meeting it, is equal:
+
+To one in Euclid's geometry;
+
+To zero in that of Riemann;
+
+To infinity in that of Lobachevski.
+
+Add that Riemann's space is finite, although unbounded, in the sense
+given above to these two words.
+
+THE SURFACES OF CONSTANT CURVATURE.--One objection still remained
+possible. The theorems of Lobachevski and of Riemann present no
+contradiction; but however numerous the consequences these two geometers
+have drawn from their hypotheses, they must have stopped before
+exhausting them, since their number would be infinite; who can say then
+that if they had pushed their deductions farther they would not have
+eventually reached some contradiction?
+
+This difficulty does not exist for Riemann's geometry, provided it is
+limited to two dimensions; in fact, as we have seen, two-dimensional
+Riemannian geometry does not differ from spherical geometry, which is
+only a branch of ordinary geometry, and consequently is beyond all
+discussion.
+
+Beltrami, in correlating likewise Lobachevski's two-dimensional geometry
+with a branch of ordinary geometry, has equally refuted the objection so
+far as it is concerned.
+
+Here is how he accomplished it. Consider any figure on a surface.
+Imagine this figure traced on a flexible and inextensible canvas applied
+over this surface in such a way that when the canvas is displaced and
+deformed, the various lines of this figure can change their form without
+changing their length. In general, this flexible and inextensible figure
+can not be displaced without leaving the surface; but there are certain
+particular surfaces for which such a movement would be possible; these
+are the surfaces of constant curvature.
+
+If we resume the comparison made above and imagine beings without
+thickness living on one of these surfaces, they will regard as possible
+the motion of a figure all of whose lines remain constant in length. On
+the contrary, such a movement would appear absurd to animals without
+thickness living on a surface of variable curvature.
+
+These surfaces of constant curvature are of two sorts: Some are of
+_positive curvature_, and can be deformed so as to be applied over a
+sphere. The geometry of these surfaces reduces itself therefore to the
+spherical geometry, which is that of Riemann.
+
+The others are of _negative curvature_. Beltrami has shown that the
+geometry of these surfaces is none other than that of Lobachevski. The
+two-dimensional geometries of Riemann and Lobachevski are thus
+correlated to the Euclidean geometry.
+
+INTERPRETATION OF NON-EUCLIDEAN GEOMETRIES.--So vanishes the objection
+so far as two-dimensional geometries are concerned.
+
+It would be easy to extend Beltrami's reasoning to three-dimensional
+geometries. The minds that space of four dimensions does not repel will
+see no difficulty in it, but they are few. I prefer therefore to proceed
+otherwise.
+
+Consider a certain plane, which I shall call the fundamental plane, and
+construct a sort of dictionary, by making correspond each to each a
+double series of terms written in two columns, just as correspond in the
+ordinary dictionaries the words of two languages whose significance is
+the same:
+
+_Space_: Portion of space situated above the fundamental plane.
+
+_Plane_: Sphere cutting the fundamental plane orthogonally.
+
+_Straight_: Circle cutting the fundamental plane orthogonally.
+
+_Sphere_: Sphere.
+
+_Circle_: Circle.
+
+_Angle_: Angle.
+
+_Distance between two points_: Logarithm of the cross ratio of these two
+points and the intersections of the fundamental plane with a circle
+passing through these two points and cutting it orthogonally. Etc.,
+Etc.
+
+Now take Lobachevski's theorems and translate them with the aid of this
+dictionary as we translate a German text with the aid of a
+German-English dictionary. _We shall thus obtain theorems of the
+ordinary geometry._ For example, that theorem of Lobachevski: 'the sum
+of the angles of a triangle is less than two right angles' is translated
+thus: "If a curvilinear triangle has for sides circle-arcs which
+prolonged would cut orthogonally the fundamental plane, the sum of the
+angles of this curvilinear triangle will be less than two right angles."
+Thus, however far the consequences of Lobachevski's hypotheses are
+pushed, they will never lead to a contradiction. In fact, if two of
+Lobachevski's theorems were contradictory, it would be the same with the
+translations of these two theorems, made by the aid of our dictionary,
+but these translations are theorems of ordinary geometry and no one
+doubts that the ordinary geometry is free from contradiction. Whence
+comes this certainty and is it justified? That is a question I can not
+treat here because it would require to be enlarged upon, but which is
+very interesting and I think not insoluble.
+
+Nothing remains then of the objection above formulated. This is not all.
+Lobachevski's geometry, susceptible of a concrete interpretation, ceases
+to be a vain logical exercise and is capable of applications; I have not
+the time to speak here of these applications, nor of the aid that Klein
+and I have gotten from them for the integration of linear differential
+equations.
+
+This interpretation moreover is not unique, and several dictionaries
+analogous to the preceding could be constructed, which would enable us
+by a simple 'translation' to transform Lobachevski's theorems into
+theorems of ordinary geometry.
+
+THE IMPLICIT AXIOMS.--Are the axioms explicitly enunciated in our
+treatises the sole foundations of geometry? We may be assured of the
+contrary by noticing that after they are successively abandoned there
+are still left over some propositions common to the theories of Euclid,
+Lobachevski and Riemann. These propositions must rest on premises the
+geometers admit without enunciation. It is interesting to try to
+disentangle them from the classic demonstrations.
+
+Stuart Mill has claimed that every definition contains an axiom,
+because in defining one affirms implicitly the existence of the object
+defined. This is going much too far; it is rare that in mathematics a
+definition is given without its being followed by the demonstration of
+the existence of the object defined, and when this is dispensed with it
+is generally because the reader can easily supply it. It must not be
+forgotten that the word existence has not the same sense when it refers
+to a mathematical entity and when it is a question of a material object.
+A mathematical entity exists, provided its definition implies no
+contradiction, either in itself, or with the propositions already
+admitted.
+
+But if Stuart Mill's observation can not be applied to all definitions,
+it is none the less just for some of them. The plane is sometimes
+defined as follows:
+
+The plane is a surface such that the straight which joins any two of its
+points is wholly on this surface.
+
+This definition manifestly hides a new axiom; it is true we might change
+it, and that would be preferable, but then we should have to enunciate
+the axiom explicitly.
+
+Other definitions would suggest reflections not less important.
+
+Such, for example, is that of the equality of two figures; two figures
+are equal when they can be superposed; to superpose them one must be
+displaced until it coincides with the other; but how shall it be
+displaced? If we should ask this, no doubt we should be told that it
+must be done without altering the shape and as a rigid solid. The
+vicious circle would then be evident.
+
+In fact this definition defines nothing; it would have no meaning for a
+being living in a world where there were only fluids. If it seems clear
+to us, that is because we are used to the properties of natural solids
+which do not differ much from those of the ideal solids, all of whose
+dimensions are invariable.
+
+Yet, imperfect as it may be, this definition implies an axiom.
+
+The possibility of the motion of a rigid figure is not a self-evident
+truth, or at least it is so only in the fashion of Euclid's postulate
+and not as an analytic judgment _a priori_ would be.
+
+Moreover, in studying the definitions and the demonstrations of
+geometry, we see that one is obliged to admit without proof not only the
+possibility of this motion, but some of its properties besides.
+
+This is at once seen from the definition of the straight line. Many
+defective definitions have been given, but the true one is that which is
+implied in all the demonstrations where the straight line enters:
+
+"It may happen that the motion of a rigid figure is such that all the
+points of a line belonging to this figure remain motionless while all
+the points situated outside of this line move. Such a line will be
+called a straight line." We have designedly, in this enunciation,
+separated the definition from the axiom it implies.
+
+Many demonstrations, such as those of the cases of the equality of
+triangles, of the possibility of dropping a perpendicular from a point
+to a straight, presume propositions which are not enunciated, for they
+require the admission that it is possible to transport a figure in a
+certain way in space.
+
+THE FOURTH GEOMETRY.--Among these implicit axioms, there is one which
+seems to me to merit some attention, because when it is abandoned a
+fourth geometry can be constructed as coherent as those of Euclid,
+Lobachevski and Riemann.
+
+To prove that a perpendicular may always be erected at a point _A_ to a
+straight _AB_, we consider a straight _AC_ movable around the point _A_
+and initially coincident with the fixed straight _AB_; and we make it
+turn about the point _A_ until it comes into the prolongation of _AB_.
+
+Thus two propositions are presupposed: First, that such a rotation is
+possible, and next that it may be continued until the two straights come
+into the prolongation one of the other.
+
+If the first point is admitted and the second rejected, we are led to a
+series of theorems even stranger than those of Lobachevski and Riemann,
+but equally exempt from contradiction.
+
+I shall cite only one of these theorems and that not the most singular:
+_A real straight may be perpendicular to itself_.
+
+LIE'S THEOREM.--The number of axioms implicitly introduced in the
+classic demonstrations is greater than necessary, and it would be
+interesting to reduce it to a minimum. It may first be asked whether
+this reduction is possible, whether the number of necessary axioms and
+that of imaginable geometries are not infinite.
+
+A theorem of Sophus Lie dominates this whole discussion. It may be thus
+enunciated:
+
+Suppose the following premises are admitted:
+
+1º Space has _n_ dimensions;
+
+2º The motion of a rigid figure is possible;
+
+3º It requires _p_ conditions to determine the position of this figure
+in space.
+
+_The number of geometries compatible with these premises will be
+limited._
+
+I may even add that if _n_ is given, a superior limit can be assigned to
+_p_.
+
+If therefore the possibility of motion is admitted, there can be
+invented only a finite (and even a rather small) number of
+three-dimensional geometries.
+
+RIEMANN'S GEOMETRIES.--Yet this result seems contradicted by Riemann,
+for this savant constructs an infinity of different geometries, and that
+to which his name is ordinarily given is only a particular case.
+
+All depends, he says, on how the length of a curve is defined. Now,
+there is an infinity of ways of defining this length, and each of them
+may be the starting point of a new geometry.
+
+That is perfectly true, but most of these definitions are incompatible
+with the motion of a rigid figure, which in the theorem of Lie is
+supposed possible. These geometries of Riemann, in many ways so
+interesting, could never therefore be other than purely analytic and
+would not lend themselves to demonstrations analogous to those of
+Euclid.
+
+ON THE NATURE OF AXIOMS.--Most mathematicians regard Lobachevski's
+geometry only as a mere logical curiosity; some of them, however, have
+gone farther. Since several geometries are possible, is it certain ours
+is the true one? Experience no doubt teaches us that the sum of the
+angles of a triangle is equal to two right angles; but this is because
+the triangles we deal with are too little; the difference, according to
+Lobachevski, is proportional to the surface of the triangle; will it not
+perhaps become sensible when we shall operate on larger triangles or
+when our measurements shall become more precise? The Euclidean geometry
+would thus be only a provisional geometry.
+
+To discuss this opinion, we should first ask ourselves what is the
+nature of the geometric axioms.
+
+Are they synthetic _a priori_ judgments, as Kant said?
+
+They would then impose themselves upon us with such force that we could
+not conceive the contrary proposition, nor build upon it a theoretic
+edifice. There would be no non-Euclidean geometry.
+
+To be convinced of it take a veritable synthetic _a priori_ judgment,
+the following, for instance, of which we have seen the preponderant rôle
+in the first chapter:
+
+_If a theorem is true for the number 1, and if it has been proved that
+it is true of n + 1 provided it is true of n, it will be true of all the
+positive whole numbers._
+
+Then try to escape from that and, denying this proposition, try to found
+a false arithmetic analogous to non-Euclidean geometry--it can not be
+done; one would even be tempted at first blush to regard these judgments
+as analytic.
+
+Moreover, resuming our fiction of animals without thickness, we can
+hardly admit that these beings, if their minds are like ours, would
+adopt the Euclidean geometry which would be contradicted by all their
+experience.
+
+Should we therefore conclude that the axioms of geometry are
+experimental verities? But we do not experiment on ideal straights or
+circles; it can only be done on material objects. On what then could be
+based experiments which should serve as foundation for geometry? The
+answer is easy.
+
+We have seen above that we constantly reason as if the geometric figures
+behaved like solids. What geometry would borrow from experience would
+therefore be the properties of these bodies. The properties of light and
+its rectilinear propagation have also given rise to some of the
+propositions of geometry, and in particular those of projective
+geometry, so that from this point of view one would be tempted to say
+that metric geometry is the study of solids, and projective, that of
+light.
+
+But a difficulty remains, and it is insurmountable. If geometry were an
+experimental science, it would not be an exact science, it would be
+subject to a continual revision. Nay, it would from this very day be
+convicted of error, since we know that there is no rigorously rigid
+solid.
+
+The _axioms of geometry therefore are neither synthetic_ a priori
+_judgments nor experimental facts_.
+
+They are _conventions_; our choice among all possible conventions is
+_guided_ by experimental facts; but it remains _free_ and is limited
+only by the necessity of avoiding all contradiction. Thus it is that the
+postulates can remain _rigorously_ true even though the experimental
+laws which have determined their adoption are only approximative.
+
+In other words, _the axioms of geometry_ (I do not speak of those of
+arithmetic) _are merely disguised definitions_.
+
+Then what are we to think of that question: Is the Euclidean geometry
+true?
+
+It has no meaning.
+
+As well ask whether the metric system is true and the old measures
+false; whether Cartesian coordinates are true and polar coordinates
+false. One geometry can not be more true than another; it can only be
+_more convenient_.
+
+Now, Euclidean geometry is, and will remain, the most convenient:
+
+1º Because it is the simplest; and it is so not only in consequence of
+our mental habits, or of I know not what direct intuition that we may
+have of Euclidean space; it is the simplest in itself, just as a
+polynomial of the first degree is simpler than one of the second; the
+formulas of spherical trigonometry are more complicated than those of
+plane trigonometry, and they would still appear so to an analyst
+ignorant of their geometric signification.
+
+2º Because it accords sufficiently well with the properties of natural
+solids, those bodies which our hands and our eyes compare and with which
+we make our instruments of measure.
+
+
+
+
+CHAPTER IV
+
+SPACE AND GEOMETRY
+
+
+Let us begin by a little paradox.
+
+Beings with minds like ours, and having the same senses as we, but
+without previous education, would receive from a suitably chosen
+external world impressions such that they would be led to construct a
+geometry other than that of Euclid and to localize the phenomena of that
+external world in a non-Euclidean space, or even in a space of four
+dimensions.
+
+As for us, whose education has been accomplished by our actual world, if
+we were suddenly transported into this new world, we should have no
+difficulty in referring its phenomena to our Euclidean space.
+Conversely, if these beings were transported into our environment, they
+would be led to relate our phenomena to non-Euclidean space.
+
+Nay more; with a little effort we likewise could do it. A person who
+should devote his existence to it might perhaps attain to a realization
+of the fourth dimension.
+
+GEOMETRIC SPACE AND PERCEPTUAL SPACE.--It is often said the images of
+external objects are localized in space, even that they can not be
+formed except on this condition. It is also said that this space, which
+serves thus as a ready prepared _frame_ for our sensations and our
+representations, is identical with that of the geometers, of which it
+possesses all the properties.
+
+To all the good minds who think thus, the preceding statement must have
+appeared quite extraordinary. But let us see whether they are not
+subject to an illusion that a more profound analysis would dissipate.
+
+What, first of all, are the properties of space, properly so called? I
+mean of that space which is the object of geometry and which I shall
+call _geometric space_.
+
+The following are some of the most essential:
+
+1º It is continuous;
+
+2º It is infinite;
+
+3º It has three dimensions;
+
+4º It is homogeneous, that is to say, all its points are identical one
+with another;
+
+5º It is isotropic, that is to say, all the straights which pass through
+the same point are identical one with another.
+
+Compare it now to the frame of our representations and our sensations,
+which I may call _perceptual space_.
+
+VISUAL SPACE.--Consider first a purely visual impression, due to an
+image formed on the bottom of the retina.
+
+A cursory analysis shows us this image as continuous, but as possessing
+only two dimensions; this already distinguishes from geometric space
+what we may call _pure visual space_.
+
+Besides, this image is enclosed in a limited frame.
+
+Finally, there is another difference not less important: _this pure
+visual space is not homogeneous_. All the points of the retina, aside
+from the images which may there be formed, do not play the same rôle.
+The yellow spot can in no way be regarded as identical with a point on
+the border of the retina. In fact, not only does the same object produce
+there much more vivid impressions, but in every _limited_ frame the
+point occupying the center of the frame will never appear as equivalent
+to a point near one of the borders.
+
+No doubt a more profound analysis would show us that this continuity of
+visual space and its two dimensions are only an illusion; it would
+separate it therefore still more from geometric space, but we shall not
+dwell on this remark.
+
+Sight, however, enables us to judge of distances and consequently to
+perceive a third dimension. But every one knows that this perception of
+the third dimension reduces itself to the sensation of the effort at
+accommodation it is necessary to make, and to that of the convergence
+which must be given to the two eyes, to perceive an object distinctly.
+
+These are muscular sensations altogether different from the visual
+sensations which have given us the notion of the first two dimensions.
+The third dimension therefore will not appear to us as playing the same
+rôle as the other two. What may be called _complete visual space_ is
+therefore not an isotropic space.
+
+It has, it is true, precisely three dimensions, which means that the
+elements of our visual sensations (those at least which combine to form
+the notion of extension) will be completely defined when three of them
+are known; to use the language of mathematics, they will be functions of
+three independent variables.
+
+But examine the matter a little more closely. The third dimension is
+revealed to us in two different ways: by the effort of accommodation and
+by the convergence of the eyes.
+
+No doubt these two indications are always concordant, there is a
+constant relation between them, or, in mathematical terms, the two
+variables which measure these two muscular sensations do not appear to
+us as independent; or again, to avoid an appeal to mathematical notions
+already rather refined, we may go back to the language of the preceding
+chapter and enunciate the same fact as follows: If two sensations of
+convergence, _A_ and _B_, are indistinguishable, the two sensations of
+accommodation, _A'_ and _B'_, which respectively accompany them, will be
+equally indistinguishable.
+
+But here we have, so to speak, an experimental fact; _a priori_ nothing
+prevents our supposing the contrary, and if the contrary takes place, if
+these two muscular sensations vary independently of one another, we
+shall have to take account of one more independent variable, and
+'complete visual space' will appear to us as a physical continuum of
+four dimensions.
+
+We have here even, I will add, a fact of _external_ experience. Nothing
+prevents our supposing that a being with a mind like ours, having the
+same sense organs that we have, may be placed in a world where light
+would only reach him after having traversed reflecting media of
+complicated form. The two indications which serve us in judging
+distances would cease to be connected by a constant relation. A being
+who should achieve in such a world the education of his senses would no
+doubt attribute four dimensions to complete visual space.
+
+TACTILE SPACE AND MOTOR SPACE.--'Tactile space' is still more
+complicated than visual space and farther removed from geometric space.
+It is superfluous to repeat for touch the discussion I have given for
+sight.
+
+But apart from the data of sight and touch, there are other sensations
+which contribute as much and more than they to the genesis of the notion
+of space. These are known to every one; they accompany all our
+movements, and are usually called muscular sensations.
+
+The corresponding frame constitutes what may be called _motor space_.
+
+Each muscle gives rise to a special sensation capable of augmenting or
+of diminishing, so that the totality of our muscular sensations will
+depend upon as many variables as we have muscles. From this point of
+view, _motor space would have as many dimensions as we have muscles_.
+
+I know it will be said that if the muscular sensations contribute to
+form the notion of space, it is because we have the sense of the
+_direction_ of each movement and that it makes an integrant part of the
+sensation. If this were so, if a muscular sensation could not arise
+except accompanied by this geometric sense of direction, geometric space
+would indeed be a form imposed upon our sensibility.
+
+But I perceive nothing at all of this when I analyze my sensations.
+
+What I do see is that the sensations which correspond to movements in
+the same direction are connected in my mind by a mere _association of
+ideas_. It is to this association that what we call 'the sense of
+direction' is reducible. This feeling therefore can not be found in a
+single sensation.
+
+This association is extremely complex, for the contraction of the same
+muscle may correspond, according to the position of the limbs, to
+movements of very different direction.
+
+Besides, it is evidently acquired; it is, like all associations of
+ideas, the result of a _habit_; this habit itself results from very
+numerous _experiences_; without any doubt, if the education of our
+senses had been accomplished in a different environment, where we should
+have been subjected to different impressions, contrary habits would have
+arisen and our muscular sensations would have been associated according
+to other laws.
+
+CHARACTERISTICS OF PERCEPTUAL SPACE.--Thus perceptual space, under its
+triple form, visual, tactile and motor, is essentially different from
+geometric space.
+
+It is neither homogeneous, nor isotropic; one can not even say that it
+has three dimensions.
+
+It is often said that we 'project' into geometric space the objects of
+our external perception; that we 'localize' them.
+
+Has this a meaning, and if so what?
+
+Does it mean that we _represent_ to ourselves external objects in
+geometric space?
+
+Our representations are only the reproduction of our sensations; they
+can therefore be ranged only in the same frame as these, that is to say,
+in perceptual space.
+
+It is as impossible for us to represent to ourselves external bodies in
+geometric space, as it is for a painter to paint on a plane canvas
+objects with their three dimensions.
+
+Perceptual space is only an image of geometric space, an image altered
+in shape by a sort of perspective, and we can represent to ourselves
+objects only by bringing them under the laws of this perspective.
+
+Therefore we do not _represent_ to ourselves external bodies in
+geometric space, but we _reason_ on these bodies as if they were
+situated in geometric space.
+
+When it is said then that we 'localize' such and such an object at such
+and such a point of space, what does it mean?
+
+_It simply means that we represent to ourselves the movements it would
+be necessary to make to reach that object_; and one may not say that to
+represent to oneself these movements, it is necessary to project the
+movements themselves in space and that the notion of space must,
+consequently, pre-exist.
+
+When I say that we represent to ourselves these movements, I mean only
+that we represent to ourselves the muscular sensations which accompany
+them and which have no geometric character whatever, which consequently
+do not at all imply the preexistence of the notion of space.
+
+CHANGE OF STATE AND CHANGE OF POSITION.--But, it will be said, if the
+idea of geometric space is not imposed upon our mind, and if, on the
+other hand, none of our sensations can furnish it, how could it have
+come into existence?
+
+This is what we have now to examine, and it will take some time, but I
+can summarize in a few words the attempt at explanation that I am about
+to develop.
+
+_None of our sensations, isolated, could have conducted us to the idea
+of space; we are led to it only in studying the laws, according to which
+these sensations succeed each other._
+
+We see first that our impressions are subject to change; but among the
+changes we ascertain we are soon led to make a distinction.
+
+At one time we say that the objects which cause these impressions have
+changed state, at another time that they have changed position, that
+they have only been displaced.
+
+Whether an object changes its state or merely its position, this is
+always translated for us in the same manner: _by a modification in an
+aggregate of impressions_.
+
+How then could we have been led to distinguish between the two? It is
+easy to account for. If there has only been a change of position, we can
+restore the primitive aggregate of impressions by making movements which
+replace us opposite the mobile object in the same _relative_ situation.
+We thus _correct_ the modification that happened and we reestablish the
+initial state by an inverse modification.
+
+If it is a question of sight, for example, and if an object changes its
+place before our eye, we can 'follow it with the eye' and maintain its
+image on the same point of the retina by appropriate movements of the
+eyeball.
+
+These movements we are conscious of because they are voluntary and
+because they are accompanied by muscular sensations, but that does not
+mean that we represent them to ourselves in geometric space.
+
+So what characterizes change of position, what distinguishes it from
+change of state, is that it can always be corrected in this way.
+
+It may therefore happen that we pass from the totality of impressions
+_A_ to the totality _B_ in two different ways:
+
+1º Involuntarily and without experiencing muscular sensations; this
+happens when it is the object which changes place;
+
+2° Voluntarily and with muscular sensations; this happens when the
+object is motionless, but we move so that the object has relative motion
+with reference to us.
+
+If this be so, the passage from the totality _A_ to the totality _B_ is
+only a change of position.
+
+It follows from this that sight and touch could not have given us the
+notion of space without the aid of the 'muscular sense.'
+
+Not only could this notion not be derived from a single sensation or
+even _from a series of sensations_, but what is more, an _immobile_
+being could never have acquired it, since, not being able to _correct_
+by his movements the effects of the changes of position of exterior
+objects, he would have had no reason whatever to distinguish them from
+changes of state. Just as little could he have acquired it if his
+motions had not been voluntary or were unaccompanied by any sensations.
+
+CONDITIONS OF COMPENSATION.--How is a like compensation possible, of
+such sort that two changes, otherwise independent of each other,
+reciprocally correct each other?
+
+A mind already familiar with geometry would reason as follows:
+Evidently, if there is to be compensation, the various parts of the
+external object, on the one hand, and the various sense organs, on the
+other hand, must be in the same _relative_ position after the double
+change. And, for that to be the case, the various parts of the external
+object must likewise have retained in reference to each other the same
+relative position, and the same must be true of the various parts of our
+body in regard to each other.
+
+In other words, the external object, in the first change, must be
+displaced as is a rigid solid, and so must it be with the whole of our
+body in the second change which corrects the first.
+
+Under these conditions, compensation may take place.
+
+But we who as yet know nothing of geometry, since for us the notion of
+space is not yet formed, we can not reason thus, we can not foresee _a
+priori_ whether compensation is possible. But experience teaches us that
+it sometimes happens, and it is from this experimental fact that we
+start to distinguish changes of state from changes of position.
+
+SOLID BODIES AND GEOMETRY.--Among surrounding objects there are some
+which frequently undergo displacements susceptible of being thus
+corrected by a correlative movement of our own body; these are the
+_solid bodies_. The other objects, whose form is variable, only
+exceptionally undergo like displacements (change of position without
+change of form). When a body changes its place _and its shape_, we can
+no longer, by appropriate movements, bring back our sense-organs into
+the same _relative_ situation with regard to this body; consequently we
+can no longer reestablish the primitive totality of impressions.
+
+It is only later, and as a consequence of new experiences, that we learn
+how to decompose the bodies of variable form into smaller elements, such
+that each is displaced almost in accordance with the same laws as solid
+bodies. Thus we distinguish 'deformations' from other changes of state;
+in these deformations, each element undergoes a mere change of position,
+which can be corrected, but the modification undergone by the aggregate
+is more profound and is no longer susceptible of correction by a
+correlative movement.
+
+Such a notion is already very complex and must have been relatively late
+in appearing; moreover it could not have arisen if the observation of
+solid bodies had not already taught us to distinguish changes of
+position.
+
+_Therefore, if there were no solid bodies in nature, there would be no
+geometry._
+
+Another remark also deserves a moment's attention. Suppose a solid body
+to occupy successively the positions [alpha] and [beta]; in its first
+position, it will produce on us the totality of impressions _A_, and in
+its second position the totality of impressions _B_. Let there be now a
+second solid body, having qualities entirely different from the first,
+for example, a different color. Suppose it to pass from the position
+[alpha], where it gives us the totality of impressions _A'_, to the
+position [beta], where it gives the totality of impressions _B'_.
+
+In general, the totality _A_ will have nothing in common with the
+totality _A'_, nor the totality _B_ with the totality _B'_. The
+transition from the totality _A_ to the totality _B_ and that from the
+totality _A'_ to the totality _B'_ are therefore two changes which _in
+themselves_ have in general nothing in common.
+
+And yet we regard these two changes both as displacements and,
+furthermore, we consider them as the _same_ displacement. How can that
+be?
+
+It is simply because they can both be corrected by the _same_
+correlative movement of our body.
+
+'Correlative movement' therefore constitutes the _sole connection_
+between two phenomena which otherwise we never should have dreamt of
+likening.
+
+On the other hand, our body, thanks to the number of its articulations
+and muscles, may make a multitude of different movements; but all are
+not capable of 'correcting' a modification of external objects; only
+those will be capable of it in which our whole body, or at least all
+those of our sense-organs which come into play, are displaced as a
+whole, that is, without their relative positions varying, or in the
+fashion of a solid body.
+
+To summarize:
+
+1º We are led at first to distinguish two categories of phenomena:
+
+Some, involuntary, unaccompanied by muscular sensations, are attributed
+by us to external objects; these are external changes;
+
+Others, opposite in character and attributed by us to the movements of
+our own body, are internal changes;
+
+2º We notice that certain changes of each of these categories may be
+corrected by a correlative change of the other category;
+
+3º We distinguish among external changes those which have thus a
+correlative in the other category; these we call displacements; and just
+so among the internal changes, we distinguish those which have a
+correlative in the first category.
+
+Thus are defined, thanks to this reciprocity, a particular class of
+phenomena which we call displacements.
+
+_The laws of these phenomena constitute the object of geometry._
+
+LAW OF HOMOGENEITY.--The first of these laws is the law of homogeneity.
+
+Suppose that, by an external change [alpha], we pass from the totality
+of impressions _A_ to the totality _B_, then that this change [alpha] is
+corrected by a correlative voluntary movement [beta], so that we are
+brought back to the totality _A_.
+
+Suppose now that another external change [alpha]' makes us pass anew
+from the totality _A_ to the totality _B_.
+
+Experience teaches us that this change [alpha]' is, like [alpha],
+susceptible of being corrected by a correlative voluntary movement
+[beta]' and that this movement [beta]' corresponds to the same muscular
+sensations as the movement [beta] which corrected [alpha].
+
+This fact is usually enunciated by saying that _space is homogeneous and
+isotropic_.
+
+It may also be said that a movement which has once been produced may be
+repeated a second and a third time, and so on, without its properties
+varying.
+
+In the first chapter, where we discussed the nature of mathematical
+reasoning, we saw the importance which must be attributed to the
+possibility of repeating indefinitely the same operation.
+
+It is from this repetition that mathematical reasoning gets its power;
+it is, therefore, thanks to the law of homogeneity, that it has a hold
+on the geometric facts.
+
+For completeness, to the law of homogeneity should be added a multitude
+of other analogous laws, into the details of which I do not wish to
+enter, but which mathematicians sum up in a word by saying that
+displacements form 'a group.'
+
+THE NON-EUCLIDEAN WORLD.--If geometric space were a frame imposed on
+_each_ of our representations, considered individually, it would be
+impossible to represent to ourselves an image stripped of this frame,
+and we could change nothing of our geometry.
+
+But this is not the case; geometry is only the résumé of the laws
+according to which these images succeed each other. Nothing then
+prevents us from imagining a series of representations, similar in all
+points to our ordinary representations, but succeeding one another
+according to laws different from those to which we are accustomed.
+
+We can conceive then that beings who received their education in an
+environment where these laws were thus upset might have a geometry very
+different from ours.
+
+Suppose, for example, a world enclosed in a great sphere and subject to
+the following laws:
+
+The temperature is not uniform; it is greatest at the center, and
+diminishes in proportion to the distance from the center, to sink to
+absolute zero when the sphere is reached in which this world is
+enclosed.
+
+To specify still more precisely the law in accordance with which this
+temperature varies: Let _R_ be the radius of the limiting sphere;
+let _r_ be the distance of the point considered from the center
+of this sphere. The absolute temperature shall be proportional
+to _R_^{2} - _r_^{2}.
+
+I shall further suppose that, in this world, all bodies have the same
+coefficient of dilatation, so that the length of any rule is
+proportional to its absolute temperature.
+
+Finally, I shall suppose that a body transported from one point to
+another of different temperature is put immediately into thermal
+equilibrium with its new environment.
+
+Nothing in these hypotheses is contradictory or unimaginable.
+
+A movable object will then become smaller and smaller in proportion as
+it approaches the limit-sphere.
+
+Note first that, though this world is limited from the point of view of
+our ordinary geometry, it will appear infinite to its inhabitants.
+
+In fact, when these try to approach the limit-sphere, they cool off and
+become smaller and smaller. Therefore the steps they take are also
+smaller and smaller, so that they can never reach the limiting sphere.
+
+If, for us, geometry is only the study of the laws according to which
+rigid solids move, for these imaginary beings it will be the study of
+the laws of motion of solids _distorted by the differences of
+temperature_ just spoken of.
+
+No doubt, in our world, natural solids likewise undergo variations of
+form and volume due to warming or cooling. But we neglect these
+variations in laying the foundations of geometry, because, besides their
+being very slight, they are irregular and consequently seem to us
+accidental.
+
+In our hypothetical world, this would no longer be the case, and these
+variations would follow regular and very simple laws.
+
+Moreover, the various solid pieces of which the bodies of its
+inhabitants would be composed would undergo the same variations of form
+and volume.
+
+I will make still another hypothesis; I will suppose light traverses
+media diversely refractive and such that the index of refraction is
+inversely proportional to _R_^{2} - _r_^{2}. It is easy to see that,
+under these conditions, the rays of light would not be rectilinear, but
+circular.
+
+To justify what precedes, it remains for me to show that certain changes
+in the position of external objects can be _corrected_ by correlative
+movements of the sentient beings inhabiting this imaginary world, and
+that in such a way as to restore the primitive aggregate of impressions
+experienced by these sentient beings.
+
+Suppose in fact that an object is displaced, undergoing deformation, not
+as a rigid solid, but as a solid subjected to unequal dilatations in
+exact conformity to the law of temperature above supposed. Permit me for
+brevity to call such a movement a _non-Euclidean displacement_.
+
+If a sentient being happens to be in the neighborhood, his impressions
+will be modified by the displacement of the object, but he can
+reestablish them by moving in a suitable manner. It suffices if finally
+the aggregate of the object and the sentient being, considered as
+forming a single body, has undergone one of those particular
+displacements I have just called non-Euclidean. This is possible if it
+be supposed that the limbs of these beings dilate according to the same
+law as the other bodies of the world they inhabit.
+
+Although from the point of view of our ordinary geometry there is a
+deformation of the bodies in this displacement and their various parts
+are no longer in the same relative position, nevertheless we shall see
+that the impressions of the sentient being have once more become the
+same.
+
+In fact, though the mutual distances of the various parts may have
+varied, yet the parts originally in contact are again in contact.
+Therefore the tactile impressions have not changed.
+
+On the other hand, taking into account the hypothesis made above in
+regard to the refraction and the curvature of the rays of light, the
+visual impressions will also have remained the same.
+
+These imaginary beings will therefore like ourselves be led to classify
+the phenomena they witness and to distinguish among them the 'changes of
+position' susceptible of correction by a correlative voluntary movement.
+
+If they construct a geometry, it will not be, as ours is, the study of
+the movements of our rigid solids; it will be the study of the changes
+of position which they will thus have distinguished and which are none
+other than the 'non-Euclidean displacements'; _it will be non-Euclidean
+geometry_.
+
+Thus beings like ourselves, educated in such a world, would not have the
+same geometry as ours.
+
+THE WORLD OF FOUR DIMENSIONS.--We can represent to ourselves a
+four-dimensional world just as well as a non-Euclidean.
+
+The sense of sight, even with a single eye, together with the muscular
+sensations relative to the movements of the eyeball, would suffice to
+teach us space of three dimensions.
+
+The images of external objects are painted on the retina, which is a
+two-dimensional canvas; they are _perspectives_.
+
+But, as eye and objects are movable, we see in succession various
+perspectives of the same body, taken from different points of view.
+
+At the same time, we find that the transition from one perspective to
+another is often accompanied by muscular sensations.
+
+If the transition from the perspective _A_ to the perspective _B_, and
+that from the perspective _A'_ to the perspective _B'_ are accompanied
+by the same muscular sensations, we liken them one to the other as
+operations of the same nature.
+
+Studying then the laws according to which these operations combine, we
+recognize that they form a group, which has the same structure as that
+of the movements of rigid solids.
+
+Now, we have seen that it is from the properties of this group we have
+derived the notion of geometric space and that of three dimensions.
+
+We understand thus how the idea of a space of three dimensions could
+take birth from the pageant of these perspectives, though each of them
+is of only two dimensions, since _they follow one another according to
+certain laws_.
+
+Well, just as the perspective of a three-dimensional figure can be made
+on a plane, we can make that of a four-dimensional figure on a picture
+of three (or of two) dimensions. To a geometer this is only child's
+play.
+
+We can even take of the same figure several perspectives from several
+different points of view.
+
+We can easily represent to ourselves these perspectives, since they are
+of only three dimensions.
+
+Imagine that the various perspectives of the same object succeed one
+another, and that the transition from one to the other is accompanied by
+muscular sensations.
+
+We shall of course consider two of these transitions as two operations
+of the same nature when they are associated with the same muscular
+sensations.
+
+Nothing then prevents us from imagining that these operations combine
+according to any law we choose, for example, so as to form a group with
+the same structure as that of the movements of a rigid solid of four
+dimensions.
+
+Here there is nothing unpicturable, and yet these sensations are
+precisely those which would be felt by a being possessed of a
+two-dimensional retina who could move in space of four dimensions. In
+this sense we may say the fourth dimension is imaginable.
+
+CONCLUSIONS.--We see that experience plays an indispensable rôle in the
+genesis of geometry; but it would be an error thence to conclude that
+geometry is, even in part, an experimental science.
+
+If it were experimental, it would be only approximative and provisional.
+And what rough approximation!
+
+Geometry would be only the study of the movements of solids; but in
+reality it is not occupied with natural solids, it has for object
+certain ideal solids, absolutely rigid, which are only a simplified and
+very remote image of natural solids.
+
+The notion of these ideal solids is drawn from all parts of our mind,
+and experience is only an occasion which induces us to bring it forth
+from them.
+
+The object of geometry is the study of a particular 'group'; but the
+general group concept pre-exists, at least potentially, in our minds. It
+is imposed on us, not as form of our sense, but as form of our
+understanding.
+
+Only, from among all the possible groups, that must be chosen which will
+be, so to speak, the _standard_ to which we shall refer natural
+phenomena.
+
+Experience guides us in this choice without forcing it upon us; it
+tells us not which is the truest geometry, but which is the most
+_convenient_.
+
+Notice that I have been able to describe the fantastic worlds above
+imagined _without ceasing to employ the language of ordinary geometry_.
+
+And, in fact, we should not have to change it if transported thither.
+
+Beings educated there would doubtless find it more convenient to create
+a geometry different from ours, and better adapted to their impressions.
+As for us, in face of the _same_ impressions, it is certain we should
+find it more convenient not to change our habits.
+
+
+
+
+CHAPTER V
+
+EXPERIENCE AND GEOMETRY
+
+
+1. Already in the preceding pages I have several times tried to show
+that the principles of geometry are not experimental facts and that in
+particular Euclid's postulate can not be proven experimentally.
+
+However decisive appear to me the reasons already given, I believe I
+should emphasize this point because here a false idea is profoundly
+rooted in many minds.
+
+2. If we construct a material circle, measure its radius and
+circumference, and see if the ratio of these two lengths is equal to
+[pi], what shall we have done? We shall have made an experiment on the
+properties of the matter with which we constructed this _round thing_,
+and of that of which the measure used was made.
+
+3. GEOMETRY AND ASTRONOMY.--The question has also been put in another
+way. If Lobachevski's geometry is true, the parallax of a very distant
+star will be finite; if Riemann's is true, it will be negative. These
+are results which seem within the reach of experiment, and there have
+been hopes that astronomical observations might enable us to decide
+between the three geometries.
+
+But in astronomy 'straight line' means simply 'path of a ray of light.'
+
+If therefore negative parallaxes were found, or if it were demonstrated
+that all parallaxes are superior to a certain limit, two courses would
+be open to us; we might either renounce Euclidean geometry, or else
+modify the laws of optics and suppose that light does not travel
+rigorously in a straight line.
+
+It is needless to add that all the world would regard the latter
+solution as the more advantageous.
+
+The Euclidean geometry has, therefore, nothing to fear from fresh
+experiments.
+
+4. Is the position tenable, that certain phenomena, possible in
+Euclidean space, would be impossible in non-Euclidean space, so that
+experience, in establishing these phenomena, would directly contradict
+the non-Euclidean hypothesis? For my part I think no such question can
+be put. To my mind it is precisely equivalent to the following, whose
+absurdity is patent to all eyes: are there lengths expressible in meters
+and centimeters, but which can not be measured in fathoms, feet and
+inches, so that experience, in ascertaining the existence of these
+lengths, would directly contradict the hypothesis that there are fathoms
+divided into six feet?
+
+Examine the question more closely. I suppose that the straight line
+possesses in Euclidean space any two properties which I shall call _A_
+and _B_; that in non-Euclidean space it still possesses the property
+_A_, but no longer has the property _B_; finally I suppose that in both
+Euclidean and non-Euclidean space the straight line is the only line
+having the property _A_.
+
+If this were so, experience would be capable of deciding between the
+hypothesis of Euclid and that of Lobachevski. It would be ascertained
+that a definite concrete object, accessible to experiment, for example,
+a pencil of rays of light, possesses the property _A_; we should
+conclude that it is rectilinear, and then investigate whether or not it
+has the property _B_.
+
+But _this is not so_; no property exists which, like this property _A_,
+can be an absolute criterion enabling us to recognize the straight line
+and to distinguish it from every other line.
+
+Shall we say, for instance: "the following is such a property: the
+straight line is a line such that a figure of which this line forms a
+part can be moved without the mutual distances of its points varying and
+so that all points of this line remain fixed"?
+
+This, in fact, is a property which, in Euclidean or non-Euclidean space,
+belongs to the straight and belongs only to it. But how shall we
+ascertain experimentally whether it belongs to this or that concrete
+object? It will be necessary to measure distances, and how shall one
+know that any concrete magnitude which I have measured with my material
+instrument really represents the abstract distance?
+
+We have only pushed back the difficulty.
+
+In reality the property just enunciated is not a property of the
+straight line alone, it is a property of the straight line and
+distance. For it to serve as absolute criterion, we should have to be
+able to establish not only that it does not also belong to a line other
+than the straight and to distance, but in addition that it does not
+belong to a line other than the straight and to a magnitude other than
+distance. Now this is not true.
+
+It is therefore impossible to imagine a concrete experiment which can be
+interpreted in the Euclidean system and not in the Lobachevskian system,
+so that I may conclude:
+
+No experience will ever be in contradiction to Euclid's postulate; nor,
+on the other hand, will any experience ever contradict the postulate of
+Lobachevski.
+
+5. But it is not enough that the Euclidean (or non-Euclidean) geometry
+can never be directly contradicted by experience. Might it not happen
+that it can accord with experience only by violating the principle of
+sufficient reason or that of the relativity of space?
+
+I will explain myself: consider any material system; we shall have to
+regard, on the one hand, 'the state' of the various bodies of this
+system (for instance, their temperature, their electric potential,
+etc.), and, on the other hand, their position in space; and among the
+data which enable us to define this position we shall, moreover,
+distinguish the mutual distances of these bodies, which define their
+relative positions, from the conditions which define the absolute
+position of the system and its absolute orientation in space.
+
+The laws of the phenomena which will happen in this system will depend
+on the state of these bodies and their mutual distances; but, because of
+the relativity and passivity of space, they will not depend on the
+absolute position and orientation of the system.
+
+In other words, the state of the bodies and their mutual distances at
+any instant will depend solely on the state of these same bodies and on
+their mutual distances at the initial instant, but will not at all
+depend on the absolute initial position of the system or on its absolute
+initial orientation. This is what for brevity I shall call the _law of
+relativity_.
+
+Hitherto I have spoken as a Euclidean geometer. As I have said, an
+experience, whatever it be, admits of an interpretation on the Euclidean
+hypothesis; but it admits of one equally on the non-Euclidean
+hypothesis. Well, we have made a series of experiments; we have
+interpreted them on the Euclidean hypothesis, and we have recognized
+that these experiments thus interpreted do not violate this 'law of
+relativity.'
+
+We now interpret them on the non-Euclidean hypothesis: this is always
+possible; only the non-Euclidean distances of our different bodies in
+this new interpretation will not generally be the same as the Euclidean
+distances in the primitive interpretation.
+
+Will our experiments, interpreted in this new manner, still be in accord
+with our 'law of relativity'? And if there were not this accord, should
+we not have also the right to say experience had proven the falsity of
+the non-Euclidean geometry?
+
+It is easy to see that this is an idle fear; in fact, to apply the law
+of relativity in all rigor, it must be applied to the entire universe.
+For if only a part of this universe were considered, and if the absolute
+position of this part happened to vary, the distances to the other
+bodies of the universe would likewise vary, their influence on the part
+of the universe considered would consequently augment or diminish, which
+might modify the laws of the phenomena happening there.
+
+But if our system is the entire universe, experience is powerless to
+give information about its absolute position and orientation in space.
+All that our instruments, however perfected they may be, can tell us
+will be the state of the various parts of the universe and their mutual
+distances.
+
+So our law of relativity may be thus enunciated:
+
+The readings we shall be able to make on our instruments at any instant
+will depend only on the readings we could have made on these same
+instruments at the initial instant.
+
+Now such an enunciation is independent of every interpretation of
+experimental facts. If the law is true in the Euclidean interpretation,
+it will also be true in the non-Euclidean interpretation.
+
+Allow me here a short digression. I have spoken above of the data which
+define the position of the various bodies of the system; I should
+likewise have spoken of those which define their velocities; I should
+then have had to distinguish the velocities with which the mutual
+distances of the different bodies vary; and, on the other hand, the
+velocities of translation and rotation of the system, that is to say,
+the velocities with which its absolute position and orientation vary.
+
+To fully satisfy the mind, the law of relativity should be expressible
+thus:
+
+The state of bodies and their mutual distances at any instant, as well
+as the velocities with which these distances vary at this same instant,
+will depend only on the state of those bodies and their mutual distances
+at the initial instant, and the velocities with which these distances
+vary at this initial instant, but they will not depend either upon the
+absolute initial position of the system, or upon its absolute
+orientation, or upon the velocities with which this absolute position
+and orientation varied at the initial instant.
+
+Unhappily the law thus enunciated is not in accord with experiments, at
+least as they are ordinarily interpreted.
+
+Suppose a man be transported to a planet whose heavens were always
+covered with a thick curtain of clouds, so that he could never see the
+other stars; on that planet he would live as if it were isolated in
+space. Yet this man could become aware that it turned, either by
+measuring its oblateness (done ordinarily by the aid of astronomic
+observations, but capable of being done by purely geodetic means), or by
+repeating the experiment of Foucault's pendulum. The absolute rotation
+of this planet could therefore be made evident.
+
+That is a fact which shocks the philosopher, but which the physicist is
+compelled to accept.
+
+We know that from this fact Newton inferred the existence of absolute
+space; I myself am quite unable to adopt this view. I shall explain why
+in Part III. For the moment it is not my intention to enter upon this
+difficulty.
+
+Therefore I must resign myself, in the enunciation of the law of
+relativity, to including velocities of every kind among the data which
+define the state of the bodies.
+
+However that may be, this difficulty is the same for Euclid's geometry
+as for Lobachevski's; I therefore need not trouble myself with it, and
+have only mentioned it incidentally.
+
+What is important is the conclusion: experiment can not decide between
+Euclid and Lobachevski.
+
+To sum up, whichever way we look at it, it is impossible to discover in
+geometric empiricism a rational meaning.
+
+6. Experiments only teach us the relations of bodies to one another;
+none of them bears or can bear on the relations of bodies with space, or
+on the mutual relations of different parts of space.
+
+"Yes," you reply, "a single experiment is insufficient, because it gives
+me only a single equation with several unknowns; but when I shall have
+made enough experiments I shall have equations enough to calculate all
+my unknowns."
+
+To know the height of the mainmast does not suffice for calculating the
+age of the captain. When you have measured every bit of wood in the ship
+you will have many equations, but you will know his age no better. All
+your measurements bearing only on your bits of wood can reveal to you
+nothing except concerning these bits of wood. Just so your experiments,
+however numerous they may be, bearing only on the relations of bodies to
+one another, will reveal to us nothing about the mutual relations of the
+various parts of space.
+
+7. Will you say that if the experiments bear on the bodies, they bear at
+least upon the geometric properties of the bodies? But, first, what do
+you understand by geometric properties of the bodies? I assume that it
+is a question of the relations of the bodies with space; these
+properties are therefore inaccessible to experiments which bear only on
+the relations of the bodies to one another. This alone would suffice to
+show that there can be no question of these properties.
+
+Still let us begin by coming to an understanding about the sense of the
+phrase: geometric properties of bodies. When I say a body is composed of
+several parts, I assume that I do not enunciate therein a geometric
+property, and this would remain true even if I agreed to give the
+improper name of points to the smallest parts I consider.
+
+When I say that such a part of such a body is in contact with such a
+part of such another body, I enunciate a proposition which concerns the
+mutual relations of these two bodies and not their relations with
+space.
+
+I suppose you will grant me these are not geometric properties; at least
+I am sure you will grant me these properties are independent of all
+knowledge of metric geometry.
+
+This presupposed, I imagine that we have a solid body formed of eight
+slender iron rods, _OA_, _OB_, _OC_, _OD_, _OE_, _OF_, _OG_, _OH_,
+united at one of their extremities _O_. Let us besides have a second
+solid body, for example a bit of wood, to be marked with three little
+flecks of ink which I shall call [alpha], [beta], [gamma]. I further
+suppose it ascertained that [alpha][beta][gamma] may be brought into
+contact with _AGO_ (I mean [alpha] with _A_, and at the same time [beta]
+with _G_ and [gamma] with _O_), then that we may bring successively into
+contact [alpha][beta][gamma] with _BGO_, _CGO_, _DGO_, _EGO_, _FGO_,
+then with _AHO_, _BHO_, _CHO_, _DHO_, _EHO_, _FHO_, then [alpha][gamma]
+successively with _AB_, _BC_, _CD_, _DE_, _EF_, _FA_.
+
+These are determinations we may make without having in advance any
+notion about form or about the metric properties of space. They in no
+wise bear on the 'geometric properties of bodies.' And these
+determinations will not be possible if the bodies experimented upon move
+in accordance with a group having the same structure as the
+Lobachevskian group (I mean according to the same laws as solid bodies
+in Lobachevski's geometry). They suffice therefore to prove that these
+bodies move in accordance with the Euclidean group, or at least that
+they do not move according to the Lobachevskian group.
+
+That they are compatible with the Euclidean group is easy to see. For
+they could be made if the body [alpha][beta][gamma] was a rigid solid of
+our ordinary geometry presenting the form of a right-angled triangle,
+and if the points _ABCDEFGH_ were the summits of a polyhedron formed of
+two regular hexagonal pyramids of our ordinary geometry, having for
+common base _ABCDEF_ and for apices the one _G_ and the other _H_.
+
+Suppose now that in place of the preceding determination it is observed
+that as above [alpha][beta][gamma] can be successively applied to _AGO_,
+_BGO_, _CGO_, _DGO_, _EGO_, _AHO_, _BHO_, _CHO_, _DHO_, _EHO_, _FHO_,
+then that [alpha][beta] (and no longer [alpha][gamma]) can be
+successively applied to _AB_, _BC_, _CD_, _DE_, _EF_ and _FA_.
+
+These are determinations which could be made if non-Euclidean geometry
+were true, if the bodies [alpha][beta][gamma] and _OABCDEFGH_ were rigid
+solids, and if the first were a right-angled triangle and the second a
+double regular hexagonal pyramid of suitable dimensions.
+
+Therefore these new determinations are not possible if the bodies move
+according to the Euclidean group; but they become so if it be supposed
+that the bodies move according to the Lobachevskian group. They would
+suffice, therefore (if one made them), to prove that the bodies in
+question do not move according to the Euclidean group.
+
+Thus, without making any hypothesis about form, about the nature of
+space, about the relations of bodies to space, and without attributing
+to bodies any geometric property, I have made observations which have
+enabled me to show in one case that the bodies experimented upon move
+according to a group whose structure is Euclidean, in the other case
+that they move according to a group whose structure is Lobachevskian.
+
+And one may not say that the first aggregate of determinations would
+constitute an experiment proving that space is Euclidean, and the second
+an experiment proving that space is non-Euclidean.
+
+In fact one could imagine (I say imagine) bodies moving so as to render
+possible the second series of determinations. And the proof is that the
+first mechanician met could construct such bodies if he cared to take
+the pains and make the outlay. You will not conclude from that, however,
+that space is non-Euclidean.
+
+Nay, since the ordinary solid bodies would continue to exist when the
+mechanician had constructed the strange bodies of which I have just
+spoken, it would be necessary to conclude that space is at the same time
+Euclidean and non-Euclidean.
+
+Suppose, for example, that we have a great sphere of radius _R_ and that
+the temperature decreases from the center to the surface of this sphere
+according to the law of which I have spoken in describing the
+non-Euclidean world.
+
+We might have bodies whose expansion would be negligible and which would
+act like ordinary rigid solids; and, on the other hand, bodies very
+dilatable and which would act like non-Euclidean solids. We might have
+two double pyramids _OABCDEFGH_ and _O'A'B'C'D'E'F'G'H'_ and two
+triangles [alpha][beta][gamma] and [alpha]'[beta]'[gamma]'. The first
+double pyramid might be rectilinear and the second curvilinear; the
+triangle [alpha][beta][gamma] might be made of inexpansible matter and
+the other of a very dilatable matter.
+
+It would then be possible to make the first observations with the double
+pyramid _OAH_ and the triangle [alpha][beta][gamma], and the second with
+the double pyramid _O'A'H'_ and the triangle [alpha]'[beta]'[gamma]'.
+And then experiment would seem to prove first that the Euclidean
+geometry is true and then that it is false.
+
+_Experiments therefore have a bearing, not on space, but on bodies._
+
+
+SUPPLEMENT
+
+8. To complete the matter, I ought to speak of a very delicate question,
+which would require long development; I shall confine myself to
+summarizing here what I have expounded in the _Revue de Métaphysique et
+de Morale_ and in _The Monist_. When we say space has three dimensions,
+what do we mean?
+
+We have seen the importance of those 'internal changes' revealed to us
+by our muscular sensations. They may serve to characterize the various
+_attitudes_ of our body. Take arbitrarily as origin one of these
+attitudes _A_. When we pass from this initial attitude to any other
+attitude _B_, we feel a series of muscular sensations, and this series
+_S_ will define _B_. Observe, however, that we shall often regard two
+series _S_ and _S'_ as defining the same attitude _B_ (since the initial
+and final attitudes _A_ and _B_ remaining the same, the intermediary
+attitudes and the corresponding sensations may differ). How then shall
+we recognize the equivalence of these two series? Because they may serve
+to compensate the same external change, or more generally because, when
+it is a question of compensating an external change, one of the series
+can be replaced by the other. Among these series, we have distinguished
+those which of themselves alone can compensate an external change, and
+which we have called 'displacements.' As we can not discriminate between
+two displacements which are too close together, the totality of these
+displacements presents the characteristics of a physical continuum;
+experience teaches us that they are those of a physical continuum of six
+dimensions; but we do not yet know how many dimensions space itself has,
+we must first solve another question.
+
+What is a point of space? Everybody thinks he knows, but that is an
+illusion. What we see when we try to represent to ourselves a point of
+space is a black speck on white paper, a speck of chalk on a blackboard,
+always an object. The question should therefore be understood as
+follows:
+
+What do I mean when I say the object _B_ is at the same point that the
+object _A_ occupied just now? Or further, what criterion will enable me
+to apprehend this?
+
+I mean that, _although I have not budged_ (which my muscular sense tells
+me), my first finger which just now touched the object _A_ touches at
+present the object _B_. I could have used other criteria; for instance
+another finger or the sense of sight. But the first criterion is
+sufficient; I know that if it answers yes, all the other criteria will
+give the same response. I know it _by experience_, I can not know it _a
+priori_. For the same reason I say that touch can not be exercised at a
+distance; this is another way of enunciating the same experimental fact.
+And if, on the contrary, I say that sight acts at a distance, it means
+that the criterion furnished by sight may respond yes while the others
+reply no.
+
+And in fact, the object, although moved away, may form its image at the
+same point of the retina. Sight responds yes, the object has remained at
+the same point and touch answers no, because my finger which just now
+touched the object, at present touches it no longer. If experience had
+shown us that one finger may respond no when the other says yes, we
+should likewise say that touch acts at a distance.
+
+In short, for each attitude of my body, my first finger determines a
+point, and this it is, and this alone, which defines a point of space.
+
+To each attitude corresponds thus a point; but it often happens that the
+same point corresponds to several different attitudes (in this case we
+say our finger has not budged, but the rest of the body has moved). We
+distinguish, therefore, among the changes of attitude those where the
+finger does not budge. How are we led thereto? It is because often we
+notice that in these changes the object which is in contact with the
+finger remains in contact with it.
+
+Range, therefore, in the same class all the attitudes obtainable from
+each other by one of the changes we have thus distinguished. To all the
+attitudes of the class will correspond the same point of space.
+Therefore to each class will correspond a point and to each point a
+class. But one may say that what experience arrives at is not the point,
+it is this class of changes or, better, the corresponding class of
+muscular sensations.
+
+And when we say space has three dimensions, we simply mean that the
+totality of these classes appears to us with the characteristics of a
+physical continuum of three dimensions.
+
+One might be tempted to conclude that it is experience which has taught
+us how many dimensions space has. But in reality here also our
+experiences have bearing, not on space, but on our body and its
+relations with the neighboring objects. Moreover they are excessively
+crude.
+
+In our mind pre-existed the latent idea of a certain number of
+groups--those whose theory Lie has developed. Which group shall we
+choose, to make of it a sort of standard with which to compare natural
+phenomena? And, this group chosen, which of its sub-groups shall we take
+to characterize a point of space? Experience has guided us by showing us
+which choice best adapts itself to the properties of our body. But its
+rôle is limited to that.
+
+
+ANCESTRAL EXPERIENCE
+
+It has often been said that if individual experience could not create
+geometry the same is not true of ancestral experience. But what does
+that mean? Is it meant that we could not experimentally demonstrate
+Euclid's postulate, but that our ancestors have been able to do it? Not
+in the least. It is meant that by natural selection our mind has
+_adapted_ itself to the conditions of the external world, that it has
+adopted the geometry _most advantageous_ to the species: or in other
+words _the most convenient_. This is entirely in conformity with our
+conclusions; geometry is not true, it is advantageous.
+
+
+
+
+PART III
+
+
+FORCE
+
+
+
+
+CHAPTER VI
+
+THE CLASSIC MECHANICS
+
+
+The English teach mechanics as an experimental science; on the continent
+it is always expounded as more or less a deductive and _a priori_
+science. The English are right, that goes without saying; but how could
+the other method have been persisted in so long? Why have the
+continental savants who have sought to get out of the ruts of their
+predecessors been usually unable to free themselves completely?
+
+On the other hand, if the principles of mechanics are only of
+experimental origin, are they not therefore only approximate and
+provisional? Might not new experiments some day lead us to modify or
+even to abandon them?
+
+Such are the questions which naturally obtrude themselves, and the
+difficulty of solution comes principally from the fact that the
+treatises on mechanics do not clearly distinguish between what is
+experiment, what is mathematical reasoning, what is convention, what is
+hypothesis.
+
+That is not all:
+
+1º There is no absolute space and we can conceive only of relative
+motions; yet usually the mechanical facts are enunciated as if there
+were an absolute space to which to refer them.
+
+2º There is no absolute time; to say two durations are equal is an
+assertion which has by itself no meaning and which can acquire one only
+by convention.
+
+3º Not only have we no direct intuition of the equality of two
+durations, but we have not even direct intuition of the simultaneity of
+two events occurring in different places: this I have explained in an
+article entitled _La mesure du temps_.[3]
+
+ [3] _Revue de Métaphysique et de Morale_, t. VI., pp. 1-13
+ (January, 1898).
+
+4º Finally, our Euclidean geometry is itself only a sort of convention
+of language; mechanical facts might be enunciated with reference to a
+non-Euclidean space which would be a guide less convenient than, but
+just as legitimate as, our ordinary space; the enunciation would thus
+become much more complicated, but it would remain possible.
+
+Thus absolute space, absolute time, geometry itself, are not conditions
+which impose themselves on mechanics; all these things are no more
+antecedent to mechanics than the French language is logically antecedent
+to the verities one expresses in French.
+
+We might try to enunciate the fundamental laws of mechanics in a
+language independent of all these conventions; we should thus without
+doubt get a better idea of what these laws are in themselves; this is
+what M. Andrade has attempted to do, at least in part, in his _Leçons de
+mécanique physique_.
+
+The enunciation of these laws would become of course much more
+complicated, because all these conventions have been devised expressly
+to abridge and simplify this enunciation.
+
+As for me, save in what concerns absolute space, I shall ignore all
+these difficulties; not that I fail to appreciate them, far from that;
+but we have sufficiently examined them in the first two parts of the
+book.
+
+I shall therefore admit, _provisionally_, absolute time and Euclidean
+geometry.
+
+THE PRINCIPLE OF INERTIA.--A body acted on by no force can only move
+uniformly in a straight line.
+
+Is this a truth imposed _a priori_ upon the mind? If it were so, how
+should the Greeks have failed to recognize it? How could they have
+believed that motion stops when the cause which gave birth to it ceases?
+Or again that every body if nothing prevents, will move in a circle, the
+noblest of motions?
+
+If it is said that the velocity of a body can not change if there is no
+reason for it to change, could it not be maintained just as well that
+the position of this body can not change, or that the curvature of its
+trajectory can not change, if no external cause intervenes to modify
+them?
+
+Is the principle of inertia, which is not an _a priori_ truth, therefore
+an experimental fact? But has any one ever experimented on bodies
+withdrawn from the action of every force? and, if so, how was it known
+that these bodies were subjected to no force? The example ordinarily
+cited is that of a ball rolling a very long time on a marble table; but
+why do we say it is subjected to no force? Is this because it is too
+remote from all other bodies to experience any appreciable action from
+them? Yet it is not farther from the earth than if it were thrown freely
+into the air; and every one knows that in this case it would experience
+the influence of gravity due to the attraction of the earth.
+
+Teachers of mechanics usually pass rapidly over the example of the ball;
+but they add that the principle of inertia is verified indirectly by its
+consequences. They express themselves badly; they evidently mean it is
+possible to verify various consequences of a more general principle, of
+which that of inertia is only a particular case.
+
+I shall propose for this general principle the following enunciation:
+
+The acceleration of a body depends only upon the position of this body
+and of the neighboring bodies and upon their velocities.
+
+Mathematicians would say the movements of all the material molecules of
+the universe depend on differential equations of the second order.
+
+To make it clear that this is really the natural generalization of the
+law of inertia, I shall beg you to permit me a bit of fiction. The law
+of inertia, as I have said above, is not imposed upon us _a priori_;
+other laws would be quite as compatible with the principle of sufficient
+reason. If a body is subjected to no force, in lieu of supposing its
+velocity not to change, it might be supposed that it is its position or
+else its acceleration which is not to change.
+
+Well, imagine for an instant that one of these two hypothetical laws is
+a law of nature and replaces our law of inertia. What would be its
+natural generalization? A moment's thought will show us.
+
+In the first case, we must suppose that the velocity of a body depends
+only upon its position and upon that of the neighboring bodies; in the
+second case that the change of acceleration of a body depends only upon
+the position of this body and of the neighboring bodies, upon their
+velocities and upon their accelerations.
+
+Or to speak the language of mathematics, the differential equations of
+motion would be of the first order in the first case, and of the third
+order in the second case.
+
+Let us slightly modify our fiction. Suppose a world analogous to our
+solar system, but where, by a strange chance, the orbits of all the
+planets are without eccentricity and without inclination. Suppose
+further that the masses of these planets are too slight for their mutual
+perturbations to be sensible. Astronomers inhabiting one of these
+planets could not fail to conclude that the orbit of a star can only be
+circular and parallel to a certain plane; the position of a star at a
+given instant would then suffice to determine its velocity and its whole
+path. The law of inertia which they would adopt would be the first of
+the two hypothetical laws I have mentioned.
+
+Imagine now that this system is some day traversed with great velocity
+by a body of vast mass, coming from distant constellations. All the
+orbits would be profoundly disturbed. Still our astronomers would not be
+too greatly astonished; they would very well divine that this new star
+was alone to blame for all the mischief. "But," they would say, "when it
+is gone, order will of itself be reestablished; no doubt the distances
+of the planets from the sun will not revert to what they were before the
+cataclysm, but when the perturbing star is gone, the orbits will again
+become circular."
+
+It would only be when the disturbing body was gone and when nevertheless
+the orbits, in lieu of again becoming circular, became elliptic, that
+these astronomers would become conscious of their error and the
+necessity of remaking all their mechanics.
+
+I have dwelt somewhat upon these hypotheses because it seems to me one
+can clearly comprehend what our generalized law of inertia really is
+only in contrasting it with a contrary hypothesis.
+
+Well, now, has this generalized law of inertia been verified by
+experiment, or can it be? When Newton wrote the _Principia_ he quite
+regarded this truth as experimentally acquired and demonstrated. It was
+so in his eyes, not only through the anthropomorphism of which we shall
+speak further on, but through the work of Galileo. It was so even from
+Kepler's laws themselves; in accordance with these laws, in fact, the
+path of a planet is completely determined by its initial position and
+initial velocity; this is just what our generalized law of inertia
+requires.
+
+For this principle to be only in appearance true, for one to have cause
+to dread having some day to replace it by one of the analogous
+principles I have just now contrasted with it, would be necessary our
+having been misled by some amazing chance, like that which, in the
+fiction above developed, led into error our imaginary astronomers.
+
+Such a hypothesis is too unlikely to delay over. No one will believe
+that such coincidences can happen; no doubt the probability of two
+eccentricities being both precisely null, to within errors of
+observation, is not less than the probability of one being precisely
+equal to 0.1, for instance, and the other to 0.2, to within errors of
+observation. The probability of a simple event is not less than that of
+a complicated event; and yet, if the first happens, we shall not consent
+to attribute it to chance; we should not believe that nature had acted
+expressly to deceive us. The hypothesis of an error of this sort being
+discarded, it may therefore be admitted that in so far as astronomy is
+concerned, our law has been verified by experiment.
+
+But astronomy is not the whole of physics.
+
+May we not fear lest some day a new experiment should come to falsify
+the law in some domain of physics? An experimental law is always subject
+to revision; one should always expect to see it replaced by a more
+precise law.
+
+Yet no one seriously thinks that the law we are speaking of will ever be
+abandoned or amended. Why? Precisely because it can never be subjected
+to a decisive test.
+
+First of all, in order that this trial should be complete, it would be
+necessary that after a certain time all the bodies in the universe
+should revert to their initial positions with their initial velocities.
+It might then be seen whether, starting from this moment, they would
+resume their original paths.
+
+But this test is impossible, it can be only partially applied, and,
+however well it is made, there will always be some bodies which will not
+revert to their initial positions; thus every derogation of the law will
+easily find its explanation.
+
+This is not all; in astronomy we _see_ the bodies whose motions we study
+and we usually assume that they are not subjected to the action of other
+invisible bodies. Under these conditions our law must indeed be either
+verified or not verified.
+
+But it is not the same in physics; if the physical phenomena are due to
+motions, it is to the motions of molecules which we do not see. If then
+the acceleration of one of the bodies we see appears to us to depend on
+_something else_ besides the positions or velocities of other visible
+bodies or of invisible molecules whose existence we have been previously
+led to admit, nothing prevents our supposing that this _something else_
+is the position or the velocity of other molecules whose presence we
+have not before suspected. The law will find itself safeguarded.
+
+Permit me to employ mathematical language a moment to express the same
+thought under another form. Suppose we observe _n_ molecules and
+ascertain that their 3_n_ coordinates satisfy a system of 3_n_
+differential equations of the fourth order (and not of the second order
+as the law of inertia would require). We know that by introducing 3_n_
+auxiliary variables, a system of 3_n_ equations of the fourth order can
+be reduced to a system of 6_n_ equations of the second order. If then we
+suppose these 3_n_ auxiliary variables represent the coordinates of _n_
+invisible molecules, the result is again in conformity with the law of
+inertia.
+
+To sum up, this law, verified experimentally in some particular cases,
+may unhesitatingly be extended to the most general cases, since we know
+that in these general cases experiment no longer is able either to
+confirm or to contradict it.
+
+THE LAW OF ACCELERATION.--The acceleration of a body is equal to the
+force acting on it divided by its mass. Can this law be verified by
+experiment? For that it would be necessary to measure the three
+magnitudes which figure in the enunciation: acceleration, force and
+mass.
+
+I assume that acceleration can be measured, for I pass over the
+difficulty arising from the measurement of time. But how measure force,
+or mass? We do not even know what they are.
+
+What is _mass_? According to Newton, it is the product of the volume by
+the density. According to Thomson and Tait, it would be better to say
+that density is the quotient of the mass by the volume. What is _force_?
+It is, replies Lagrange, that which moves or tends to move a body. It
+is, Kirchhoff will say, the product of the mass by the _acceleration_.
+But then, why not say the mass is the quotient of the force by the
+acceleration?
+
+These difficulties are inextricable.
+
+When we say force is the cause of motion, we talk metaphysics, and this
+definition, if one were content with it, would be absolutely sterile.
+For a definition to be of any use, it must teach us to _measure_ force;
+moreover that suffices; it is not at all necessary that it teach us what
+force is _in itself_, nor whether it is the cause or the effect of
+motion.
+
+We must therefore first define the equality of two forces. When shall we
+say two forces are equal? It is, we are told, when, applied to the same
+mass, they impress upon it the same acceleration, or when, opposed
+directly one to the other, they produce equilibrium. This definition is
+only a sham. A force applied to a body can not be uncoupled to hook it
+up to another body, as one uncouples a locomotive to attach it to
+another train. It is therefore impossible to know what acceleration such
+a force, applied to such a body, would impress upon such another body,
+_if_ it were applied to it. It is impossible to know how two forces
+which are not directly opposed would act, _if_ they were directly
+opposed.
+
+It is this definition we try to materialize, so to speak, when we
+measure a force with a dynamometer, or in balancing it with a weight.
+Two forces _F_ and _F'_, which for simplicity I will suppose vertical
+and directed upward, are applied respectively to two bodies _C_ and
+_C'_; I suspend the same heavy body _P_ first to the body _C_, then to
+the body _C'_; if equilibrium is produced in both cases, I shall
+conclude that the two forces _F_ and _F'_ are equal to one another,
+since they are each equal to the weight of the body _P_.
+
+But am I sure the body _P_ has retained the same weight when I have
+transported it from the first body to the second? Far from it; _I am
+sure of the contrary_; I know the intensity of gravity varies from one
+point to another, and that it is stronger, for instance, at the pole
+than at the equator. No doubt the difference is very slight and, in
+practise, I shall take no account of it; but a properly constructed
+definition should have mathematical rigor; this rigor is lacking. What I
+say of weight would evidently apply to the force of the resiliency of a
+dynamometer, which the temperature and a multitude of circumstances may
+cause to vary.
+
+This is not all; we can not say the weight of the body _P_ may be
+applied to the body _C_ and directly balance the force _F_. What is
+applied to the body _C_ is the action _A_ of the body _P_ on the body
+_C_; the body _P_ is submitted on its part, on the one hand, to its
+weight; on the other hand, to the reaction _R_ of the body _C_ on _P_.
+Finally, the force _F_ is equal to the force _A_, since it balances it;
+the force _A_ is equal to _R_, in virtue of the principle of the
+equality of action and reaction; lastly, the force _R_ is equal to the
+weight of _P_, since it balances it. It is from these three equalities
+we deduce as consequence the equality of _F_ and the weight of _P_.
+
+We are therefore obliged in the definition of the equality of the two
+forces to bring in the principle of the equality of action and reaction;
+_on this account, this principle must no longer be regarded as an
+experimental law, but as a definition_.
+
+For recognizing the equality of two forces here, we are then in
+possession of two rules: equality of two forces which balance; equality
+of action and reaction. But, as we have seen above, these two rules are
+insufficient; we are obliged to have recourse to a third rule and to
+assume that certain forces, as, for instance, the weight of a body, are
+constant in magnitude and direction. But this third rule, as I have
+said, is an experimental law; it is only approximately true; _it is a
+bad definition_.
+
+We are therefore reduced to Kirchhoff's definition; _force is equal to
+the mass multiplied by the acceleration_. This 'law of Newton' in its
+turn ceases to be regarded as an experimental law, it is now only a
+definition. But this definition is still insufficient, for we do not
+know what mass is. It enables us doubtless to calculate the relation of
+two forces applied to the same body at different instants; it teaches us
+nothing about the relation of two forces applied to two different
+bodies.
+
+To complete it, it is necessary to go back anew to Newton's third law
+(equality of action and reaction), regarded again, not as an
+experimental law, but as a definition. Two bodies _A_ and _B_ act one
+upon the other; the acceleration of _A_ multiplied by the mass of _A_ is
+equal to the action of _B_ upon _A_; in the same way, the product of the
+acceleration of _B_ by its mass is equal to the reaction of _A_ upon
+_B_. As, by definition, action is equal to reaction, the masses of _A_
+and _B_ are in the inverse ratio of their accelerations. Here we have
+the ratio of these two masses defined, and it is for experiment to
+verify that this ratio is constant.
+
+That would be all very well if the two bodies _A_ and _B_ alone were
+present and removed from the action of the rest of the world. This is
+not at all the case; the acceleration of _A_ is not due merely to the
+action of _B_, but to that of a multitude of other bodies _C_, _D_,...
+To apply the preceding rule, it is therefore necessary to separate the
+acceleration of _A_ into many components, and discern which of these
+components is due to the action of _B_.
+
+This separation would still be possible, if we _should assume_ that the
+action of _C_ upon _A_ is simply adjoined to that of _B_ upon _A_,
+without the presence of the body _C_ modifying the action of _B_ upon
+_A_; or the presence of _B_ modifying the action of _C_ upon _A_; if we
+should assume, consequently, that any two bodies attract each other,
+that their mutual action is along their join and depends only upon their
+distance apart; if, in a word, we assume _the hypothesis of central
+forces_.
+
+You know that to determine the masses of the celestial bodies we use a
+wholly different principle. The law of gravitation teaches us that the
+attraction of two bodies is proportional to their masses; if _r_ is
+their distance apart, _m_ and _m'_ their masses, _k_ a constant, their
+attraction will be _kmm'_/_r_^{2}.
+
+What we are measuring then is not mass, the ratio of force to
+acceleration, but the attracting mass; it is not the inertia of the
+body, but its attracting force.
+
+This is an indirect procedure, whose employment is not theoretically
+indispensable. It might very well have been that attraction was
+inversely proportional to the square of the distance without being
+proportional to the product of the masses, that it was equal
+to _f_/_r_^{2}, but without our having _f_ = _kmm'_.
+
+If it were so, we could nevertheless, by observation of the _relative_
+motions of the heavenly bodies, measure the masses of these bodies.
+
+But have we the right to admit the hypothesis of central forces? Is this
+hypothesis rigorously exact? Is it certain it will never be contradicted
+by experiment? Who would dare affirm that? And if we must abandon this
+hypothesis, the whole edifice so laboriously erected will crumble.
+
+We have no longer the right to speak of the component of the
+acceleration of _A_ due to the action of _B_. We have no means of
+distinguishing it from that due to the action of _C_ or of another body.
+The rule for the measurement of masses becomes inapplicable.
+
+What remains then of the principle of the equality of action and
+reaction? If the hypothesis of central forces is rejected, this
+principle should evidently be enunciated thus: the geometric resultant
+of all the forces applied to the various bodies of a system isolated
+from all external action will be null. Or, in other words, _the motion
+of the center of gravity of this system will be rectilinear and
+uniform_.
+
+There it seems we have a means of defining mass; the position of the
+center of gravity evidently depends on the values attributed to the
+masses; it will be necessary to dispose of these values in such a way
+that the motion of the center of gravity may be rectilinear and uniform;
+this will always be possible if Newton's third law is true, and possible
+in general only in a single way.
+
+But there exists no system isolated from all external action; all the
+parts of the universe are subject more or less to the action of all the
+other parts. _The law of the motion of the center of gravity is
+rigorously true only if applied to the entire universe._
+
+But then, to get from it the values of the masses, it would be necessary
+to observe the motion of the center of gravity of the universe. The
+absurdity of this consequence is manifest; we know only relative
+motions; the motion of the center of gravity of the universe will remain
+for us eternally unknown.
+
+Therefore nothing remains and our efforts have been fruitless; we are
+driven to the following definition, which is only an avowal of
+powerlessness: _masses are coefficients it is convenient to introduce
+into calculations_.
+
+We could reconstruct all mechanics by attributing different values to
+all the masses. This new mechanics would not be in contradiction either
+with experience or with the general principles of dynamics (principle of
+inertia, proportionality of forces to masses and to accelerations,
+equality of action and reaction, rectilinear and uniform motion of the
+center of gravity, principle of areas).
+
+Only the equations of this new mechanics would be _less simple_. Let us
+understand clearly: it would only be the first terms which would be less
+simple, that is those experience has already made us acquainted with;
+perhaps one could alter the masses by small quantities without the
+_complete_ equations gaining or losing in simplicity.
+
+Hertz has raised the question whether the principles of mechanics are
+rigorously true. "In the opinion of many physicists," he says, "it is
+inconceivable that the remotest experience should ever change anything
+in the immovable principles of mechanics; and yet, what comes from
+experience may always be rectified by experience." After what we have
+just said, these fears will appear groundless.
+
+The principles of dynamics at first appeared to us as experimental
+truths; but we have been obliged to use them as definitions. It is _by
+definition_ that force is equal to the product of mass by acceleration;
+here, then, is a principle which is henceforth beyond the reach of any
+further experiment. It is in the same way by definition that action is
+equal to reaction.
+
+But then, it will be said, these unverifiable principles are absolutely
+devoid of any significance; experiment can not contradict them; but they
+can teach us nothing useful; then what is the use of studying dynamics?
+
+This over-hasty condemnation would be unjust. There is not in nature any
+system _perfectly_ isolated, perfectly removed from all external action;
+but there are systems _almost_ isolated.
+
+If such a system be observed, one may study not only the relative
+motion of its various parts one in reference to another, but also the
+motion of its center of gravity in reference to the other parts of the
+universe. We ascertain then that the motion of this center of gravity is
+_almost_ rectilinear and uniform, in conformity with Newton's third law.
+
+That is an experimental truth, but it can not be invalidated by
+experience; in fact, what would a more precise experiment teach us? It
+would teach us that the law was only almost true; but that we knew
+already.
+
+_We can now understand how experience has been able to serve as basis
+for the principles of mechanics and yet will never be able to contradict
+them._
+
+ANTHROPOMORPHIC MECHANICS.--"Kirchhoff," it will be said, "has only
+acted in obedience to the general tendency of mathematicians toward
+nominalism; from this his ability as a physicist has not saved him. He
+wanted a definition of force, and he took for it the first proposition
+that presented itself; but we need no definition of force: the idea of
+force is primitive, irreducible, indefinable; we all know what it is, we
+have a direct intuition of it. This direct intuition comes from the
+notion of effort, which is familiar to us from infancy."
+
+But first, even though this direct intuition made known to us the real
+nature of force in itself, it would be insufficient as a foundation for
+mechanics; it would besides be wholly useless. What is of importance is
+not to know what force is, but to know how to measure it.
+
+Whatever does not teach us to measure it is as useless to mechanics as
+is, for instance, the subjective notion of warmth and cold to the
+physicist who is studying heat. This subjective notion can not be
+translated into numbers, therefore it is of no use; a scientist whose
+skin was an absolutely bad conductor of heat and who, consequently,
+would never have felt either sensations of cold or sensations of warmth,
+could read a thermometer just as well as any one else, and that would
+suffice him for constructing the whole theory of heat.
+
+Now this immediate notion of effort is of no use to us for measuring
+force; it is clear, for instance, that I should feel more fatigue in
+lifting a weight of fifty kilos than a man accustomed to carry burdens.
+
+But more than that: this notion of effort does not teach us the real
+nature of force; it reduces itself finally to a remembrance of muscular
+sensations, and it will hardly be maintained that the sun feels a
+muscular sensation when it draws the earth.
+
+All that can there be sought is a symbol, less precise and less
+convenient than the arrows the geometers use, but just as remote from
+the reality.
+
+Anthropomorphism has played a considerable historic rôle in the genesis
+of mechanics; perhaps it will still at times furnish a symbol which will
+appear convenient to some minds; but it can not serve as foundation for
+anything of a truly scientific or philosophic character.
+
+'THE SCHOOL OF THE THREAD.'--M. Andrade, in his _Leçons de mécanique
+physique_, has rejuvenated anthropomorphic mechanics. To the school of
+mechanics to which Kirchhoff belongs, he opposes that which he bizarrely
+calls the school of the thread.
+
+This school tries to reduce everything to "the consideration of certain
+material systems of negligible mass, envisaged in the state of tension
+and capable of transmitting considerable efforts to distant bodies,
+systems of which the ideal type is the _thread_."
+
+A thread which transmits any force is slightly elongated under the
+action of this force; the direction of the thread tells us the direction
+of the force, whose magnitude is measured by the elongation of the
+thread.
+
+One may then conceive an experiment such as this. A body _A_ is attached
+to a thread; at the other extremity of the thread any force acts which
+varies until the thread takes an elongation [alpha]; the acceleration of
+the body _A_ is noted; _A_ is detached and the body _B_ attached to the
+same thread; the same force or another force acts anew, and is made to
+vary until the thread takes again the elongation [alpha]; the
+acceleration of the body _B_ is noted. The experiment is then renewed
+with both _A_ and _B_, but so that the thread takes the elongation
+[beta]. The four observed accelerations should be proportional. We have
+thus an experimental verification of the law of acceleration above
+enunciated.
+
+Or still better, a body is submitted to the simultaneous action of
+several identical threads in equal tension, and by experiment it is
+sought what must be the orientations of all these threads that the body
+may remain in equilibrium. We have then an experimental verification of
+the law of the composition of forces.
+
+But, after all, what have we done? We have defined the force to which
+the thread is subjected by the deformation undergone by this thread,
+which is reasonable enough; we have further assumed that if a body is
+attached to this thread, the effort transmitted to it by the thread is
+equal to the action this body exercises on this thread; after all, we
+have therefore used the principle of the equality of action and
+reaction, in considering it, not as an experimental truth, but as the
+very definition of force.
+
+This definition is just as conventional as Kirchhoff's, but far less
+general.
+
+All forces are not transmitted by threads (besides, to be able to
+compare them, they would all have to be transmitted by identical
+threads). Even if it should be conceded that the earth is attached to
+the sun by some invisible thread, at least it would be admitted that we
+have no means of measuring its elongation.
+
+Nine times out of ten, consequently, our definition would be at fault;
+no sort of sense could be attributed to it, and it would be necessary to
+fall back on Kirchhoff's.
+
+Why then take this détour? You admit a certain definition of force which
+has a meaning only in certain particular cases. In these cases you
+verify by experiment that it leads to the law of acceleration. On the
+strength of this experiment, you then take the law of acceleration as a
+definition of force in all the other cases.
+
+Would it not be simpler to consider the law of acceleration as a
+definition in all cases, and to regard the experiments in question, not
+as verifications of this law, but as verifications of the principle of
+reaction, or as demonstrating that the deformations of an elastic body
+depend only on the forces to which this body is subjected?
+
+And this is without taking into account that the conditions under which
+your definition could be accepted are never fulfilled except
+imperfectly, that a thread is never without mass, that it is never
+removed from every force except the reaction of the bodies attached to
+its extremities.
+
+Andrade's ideas are nevertheless very interesting; if they do not
+satisfy our logical craving, they make us understand better the historic
+genesis of the fundamental ideas of mechanics. The reflections they
+suggest show us how the human mind has raised itself from a naïve
+anthropomorphism to the present conceptions of science.
+
+We see at the start a very particular and in sum rather crude
+experiment; at the finish, a law perfectly general, perfectly precise,
+the certainty of which we regard as absolute. This certainty we
+ourselves have bestowed upon it voluntarily, so to speak, by looking
+upon it as a convention.
+
+Are the law of acceleration, the rule of the composition of forces then
+only arbitrary conventions? Conventions, yes; arbitrary, no; they would
+be if we lost sight of the experiments which led the creators of the
+science to adopt them, and which, imperfect as they may be, suffice to
+justify them. It is well that from time to time our attention is carried
+back to the experimental origin of these conventions.
+
+
+
+
+CHAPTER VII
+
+RELATIVE MOTION AND ABSOLUTE MOTION
+
+
+THE PRINCIPLE OF RELATIVE MOTION.--The attempt has sometimes been made
+to attach the law of acceleration to a more general principle. The
+motion of any system must obey the same laws, whether it be referred to
+fixed axes, or to movable axes carried along in a rectilinear and
+uniform motion. This is the principle of relative motion, which forces
+itself upon us for two reasons: first, the commonest experience confirms
+it, and second, the contrary hypothesis is singularly repugnant to the
+mind.
+
+Assume it then, and consider a body subjected to a force; the relative
+motion of this body, in reference to an observer moving with a uniform
+velocity equal to the initial velocity of the body, must be identical to
+what its absolute motion would be if it started from rest. We conclude
+hence that its acceleration can not depend upon its absolute velocity;
+the attempt has even been made to derive from this a demonstration of
+the law of acceleration.
+
+There long were traces of this demonstration in the regulations for the
+degree B. ès Sc. It is evident that this attempt is idle. The obstacle
+which prevented our demonstrating the law of acceleration is that we had
+no definition of force; this obstacle subsists in its entirety, since
+the principle invoked has not furnished us the definition we lacked.
+
+The principle of relative motion is none the less highly interesting and
+deserves study for its own sake. Let us first try to enunciate it in a
+precise manner.
+
+We have said above that the accelerations of the different bodies
+forming part of an isolated system depend only on their relative
+velocities and positions, and not on their absolute velocities and
+positions, provided the movable axes to which the relative motion is
+referred move uniformly in a straight line. Or, if we prefer, their
+accelerations depend only on the differences of their velocities and the
+differences of their coordinates, and not on the absolute values of
+these velocities and coordinates.
+
+If this principle is true for relative accelerations, or rather for
+differences of acceleration, in combining it with the law of reaction we
+shall thence deduce that it is still true of absolute accelerations.
+
+It then remains to be seen how we may demonstrate that the differences
+of the accelerations depend only on the differences of the velocities
+and of the coordinates, or, to speak in mathematical language, that
+these differences of coordinates satisfy differential equations of the
+second order.
+
+Can this demonstration be deduced from experiments or from _a priori_
+considerations?
+
+Recalling what we have said above, the reader can answer for himself.
+
+Thus enunciated, in fact, the principle of relative motion singularly
+resembles what I called above the generalized principle of inertia; it
+is not altogether the same thing, since it is a question of the
+differences of coordinates and not of the coordinates themselves. The
+new principle teaches us therefore something more than the old, but the
+same discussion is applicable and would lead to the same conclusions; it
+is unnecessary to return to it.
+
+NEWTON'S ARGUMENT.--Here we encounter a very important and even somewhat
+disconcerting question. I have said the principle of relative motion was
+for us not solely a result of experiment and that _a priori_ every
+contrary hypothesis would be repugnant to the mind.
+
+But then, why is the principle true only if the motion of the movable
+axes is rectilinear and uniform? It seems that it ought to impose itself
+upon us with the same force, if this motion is varied, or at any rate if
+it reduces to a uniform rotation. Now, in these two cases, the principle
+is not true. I will not dwell long on the case where the motion of the
+axes is rectilinear without being uniform; the paradox does not bear a
+moment's examination. If I am on board, and if the train, striking any
+obstacle, stops suddenly, I shall be thrown against the seat in front of
+me, although I have not been directly subjected to any force. There is
+nothing mysterious in that; if I have undergone the action of no
+external force, the train itself has experienced an external impact.
+There can be nothing paradoxical in the relative motion of two bodies
+being disturbed when the motion of one or the other is modified by an
+external cause.
+
+I will pause longer on the case of relative motions referred to axes
+which rotate uniformly. If the heavens were always covered with clouds,
+if we had no means of observing the stars, we nevertheless might
+conclude that the earth turns round; we could learn this from its
+flattening or again by the Foucault pendulum experiment.
+
+And yet, in this case, would it have any meaning, to say the earth turns
+round? If there is no absolute space, can one turn without turning in
+reference to something else? and, on the other hand, how could we admit
+Newton's conclusion and believe in absolute space?
+
+But it does not suffice to ascertain that all possible solutions are
+equally repugnant to us; we must analyze, in each case, the reasons for
+our repugnance, so as to make our choice intelligently. The long
+discussion which follows will therefore be excused.
+
+Let us resume our fiction: thick clouds hide the stars from men, who can
+not observe them and are ignorant even of their existence; how shall
+these men know the earth turns round?
+
+Even more than our ancestors, no doubt, they will regard the ground
+which bears them as fixed and immovable; they will await much longer the
+advent of a Copernicus. But in the end the Copernicus would come--how?
+
+The students of mechanics in this world would not at first be confronted
+with an absolute contradiction. In the theory of relative motion,
+besides real forces, two fictitious forces are met which are called
+ordinary and compound centrifugal force. Our imaginary scientists could
+therefore explain everything by regarding these two forces as real, and
+they would not see therein any contradiction of the generalized
+principle of inertia, for these forces would depend, the one on the
+relative positions of the various parts of the system, as real
+attractions do, the other on their relative velocities, as real
+frictions do.
+
+Many difficulties, however, would soon awaken their attention; if they
+succeeded in realizing an isolated system, the center of gravity of this
+system would not have an almost rectilinear path. They would invoke, to
+explain this fact, the centrifugal forces which they would regard as
+real, and which they would attribute no doubt to the mutual actions of
+the bodies. Only they would not see these forces become null at great
+distances, that is to say in proportion as the isolation was better
+realized; far from it; centrifugal force increases indefinitely with the
+distance.
+
+This difficulty would seem to them already sufficiently great; and yet
+it would not stop them long; they would soon imagine some very subtile
+medium, analogous to our ether, in which all bodies would be immersed
+and which would exert a repellent action upon them.
+
+But this is not all. Space is symmetric, and yet the laws of motion
+would not show any symmetry; they would have to distinguish between
+right and left. It would be seen for instance that cyclones turn always
+in the same sense, whereas by reason of symmetry these winds should turn
+indifferently in one sense and in the other. If our scientists by their
+labor had succeeded in rendering their universe perfectly symmetric,
+this symmetry would not remain, even though there was no apparent reason
+why it should be disturbed in one sense rather than in the other.
+
+They would get themselves out of the difficulty doubtless, they would
+invent something which would be no more extraordinary than the glass
+spheres of Ptolemy, and so it would go on, complications accumulating,
+until the long-expected Copernicus sweeps them all away at a single
+stroke, saying: It is much simpler to assume the earth turns round.
+
+And just as our Copernicus said to us: It is more convenient to suppose
+the earth turns round, since thus the laws of astronomy are expressible
+in a much simpler language; this one would say: It is more convenient to
+suppose the earth turns round, since thus the laws of mechanics are
+expressible in a much simpler language.
+
+This does not preclude maintaining that absolute space, that is to say
+the mark to which it would be necessary to refer the earth to know
+whether it really moves, has no objective existence. Hence, this
+affirmation: 'the earth turns round' has no meaning, since it can be
+verified by no experiment; since such an experiment, not only could not
+be either realized or dreamed by the boldest Jules Verne, but can not be
+conceived of without contradiction; or rather these two propositions:
+'the earth turns round,' and, 'it is more convenient to suppose the
+earth turns round' have the same meaning; there is nothing more in the
+one than in the other.
+
+Perhaps one will not be content even with that, and will find it already
+shocking that among all the hypotheses, or rather all the conventions we
+can make on this subject, there is one more convenient than the others.
+
+But if it has been admitted without difficulty when it was a question of
+the laws of astronomy, why should it be shocking in that which concerns
+mechanics?
+
+We have seen that the coordinates of bodies are determined by
+differential equations of the second order, and that so are the
+differences of these coordinates. This is what we have called the
+generalized principle of inertia and the principle of relative motion.
+If the distances of these bodies were determined likewise by equations
+of the second order, it seems that the mind ought to be entirely
+satisfied. In what measure does the mind get this satisfaction and why
+is it not content with it?
+
+To account for this, we had better take a simple example. I suppose a
+system analogous to our solar system, but where one can not perceive
+fixed stars foreign to this system, so that astronomers can observe only
+the mutual distances of the planets and the sun, and not the absolute
+longitudes of the planets. If we deduce directly from Newton's law the
+differential equations which define the variation of these distances,
+these equations will not be of the second order. I mean that if, besides
+Newton's law, one knew the initial values of these distances and of
+their derivatives with respect to the time, that would not suffice to
+determine the values of these same distances at a subsequent instant.
+There would still be lacking one datum, and this datum might be for
+instance what astronomers call the area-constant.
+
+But here two different points of view may be taken; we may distinguish
+two sorts of constants. To the eyes of the physicist the world reduces
+to a series of phenomena, depending, on the one hand, solely upon the
+initial phenomena; on the other hand, upon the laws which bind the
+consequents to the antecedents. If then observation teaches us that a
+certain quantity is a constant, we shall have the choice between two
+conceptions.
+
+Either we shall assume that there is a law requiring this quantity not
+to vary, but that by chance, at the beginning of the ages, it had,
+rather than another, this value it has been forced to keep ever since.
+This quantity might then be called an _accidental_ constant.
+
+Or else we shall assume, on the contrary, that there is a law of nature
+which imposes upon this quantity such a value and not such another.
+
+We shall then have what we may call an _essential_ constant.
+
+For example, in virtue of Newton's laws, the duration of the revolution
+of the earth must be constant. But if it is 366 sidereal days and
+something over, and not 300 or 400, this is in consequence of I know not
+what initial chance. This is an accidental constant. If, on the
+contrary, the exponent of the distance which figures in the expression
+of the attractive force is equal to -2 and not to -3, this is not by
+chance, but because Newton's law requires it. This is an essential
+constant.
+
+I know not whether this way of giving chance its part is legitimate in
+itself, and whether this distinction is not somewhat artificial; it is
+certain at least that, so long as nature shall have secrets, this
+distinction will be in application extremely arbitrary and always
+precarious.
+
+As to the area-constant, we are accustomed to regard it as accidental.
+Is it certain our imaginary astronomers would do the same? If they could
+have compared two different solar systems, they would have the idea that
+this constant may have several different values; but my very supposition
+in the beginning was that their system should appear as isolated, and
+that they should observe no star foreign to it. Under these conditions,
+they would see only one single constant which would have a single value
+absolutely invariable; they would be led without any doubt to regard it
+as an essential constant.
+
+A word in passing to forestall an objection: the inhabitants of this
+imaginary world could neither observe nor define the area-constant as we
+do, since the absolute longitudes escape them; that would not preclude
+their being quickly led to notice a certain constant which would
+introduce itself naturally into their equations and which would be
+nothing but what we call the area-constant.
+
+But then see what would happen. If the area-constant is regarded as
+essential, as depending upon a law of nature, to calculate the distances
+of the planets at any instant it will suffice to know the initial values
+of these distances and those of their first derivatives. From this new
+point of view, the distances will be determined by differential
+equations of the second order.
+
+Yet would the mind of these astronomers be completely satisfied? I do
+not believe so; first, they would soon perceive that in differentiating
+their equations and thus raising their order, these equations became
+much simpler. And above all they would be struck by the difficulty which
+comes from symmetry. It would be necessary to assume different laws,
+according as the aggregate of the planets presented the figure of a
+certain polyhedron or of the symmetric polyhedron, and one would escape
+from this consequence only by regarding the area-constant as accidental.
+
+I have taken a very special example, since I have supposed astronomers
+who did not at all consider terrestrial mechanics, and whose view was
+limited to the solar system. Our universe is more extended than theirs,
+as we have fixed stars, but still it too is limited, and so we might
+reason on the totality of our universe as the astronomers on their solar
+system.
+
+Thus we see that finally we should be led to conclude that the equations
+which define distances are of an order superior to the second. Why
+should we be shocked at that, why do we find it perfectly natural for
+the series of phenomena to depend upon the initial values of the first
+derivatives of these distances, while we hesitate to admit that they may
+depend on the initial values of the second derivatives? This can only be
+because of the habits of mind created in us by the constant study of the
+generalized principle of inertia and its consequences.
+
+The values of the distances at any instant depend upon their initial
+values, upon those of their first derivatives and also upon something
+else. What is this _something else_?
+
+If we will not admit that this may be simply one of the second
+derivatives, we have only the choice of hypotheses. Either it may be
+supposed, as is ordinarily done, that this something else is the
+absolute orientation of the universe in space, or the rapidity with
+which this orientation varies; and this supposition may be correct; it
+is certainly the most convenient solution for geometry; it is not the
+most satisfactory for the philosopher, because this orientation does not
+exist.
+
+Or it may be supposed that this something else is the position or the
+velocity of some invisible body; this has been done by certain persons
+who have even called it the body alpha, although we are doomed never to
+know anything of this body but its name. This is an artifice entirely
+analogous to that of which I spoke at the end of the paragraph devoted
+to my reflections on the principle of inertia.
+
+But, after all, the difficulty is artificial. Provided the future
+indications of our instruments can depend only on the indications they
+have given us or would have given us formerly, this is all that is
+necessary. Now as to this we may rest easy.
+
+
+
+
+CHAPTER VIII
+
+ENERGY AND THERMODYNAMICS
+
+
+ENERGETICS.--The difficulties inherent in the classic mechanics have led
+certain minds to prefer a new system they call _energetics_.
+
+Energetics took its rise as an outcome of the discovery of the principle
+of the conservation of energy. Helmholtz gave it its final form.
+
+It begins by defining two quantities which play the fundamental rôle in
+this theory. They are _kinetic energy_, or _vis viva_, and _potential
+energy_.
+
+All the changes which bodies in nature can undergo are regulated by two
+experimental laws:
+
+1º The sum of kinetic energy and potential energy is constant. This is
+the principle of the conservation of energy.
+
+2º If a system of bodies is at _A_ at the time t_{0} and at _B_ at the
+time t_{1}, it always goes from the first situation to the second in
+such a way that the _mean_ value of the difference between the two sorts
+of energy, in the interval of time which separates the two epochs t_{0}
+and t_{1}, may be as small as possible.
+
+This is Hamilton's principle, which is one of the forms of the principle
+of least action.
+
+The energetic theory has the following advantages over the classic
+theory:
+
+1º It is less incomplete; that is to say, Hamilton's principle and that
+of the conservation of energy teach us more than the fundamental
+principles of the classic theory, and exclude certain motions not
+realized in nature and which would be compatible with the classic
+theory:
+
+2º It saves us the hypothesis of atoms, which it was almost impossible
+to avoid with the classic theory.
+
+But it raises in its turn new difficulties:
+
+The definitions of the two sorts of energy would raise difficulties
+almost as great as those of force and mass in the first system. Yet
+they may be gotten over more easily, at least in the simplest cases.
+
+Suppose an isolated system formed of a certain number of material
+points; suppose these points subjected to forces depending only on their
+relative position and their mutual distances, and independent of their
+velocities. In virtue of the principle of the conservation of energy, a
+function of forces must exist.
+
+In this simple case the enunciation of the principle of the conservation
+of energy is of extreme simplicity. A certain quantity, accessible to
+experiment, must remain constant. This quantity is the sum of two terms;
+the first depends only on the position of the material points and is
+independent of their velocities; the second is proportional to the
+square of these velocities. This resolution can take place only in a
+single way.
+
+The first of these terms, which I shall call _U_, will be the potential
+energy; the second, which I shall call _T_, will be the kinetic energy.
+
+It is true that if _T_ + _U_ is a constant, so is any function of _T_ +
+_U_,
+
+ {Phi}(_T_ + _U_).
+
+But this function {Phi}(_T_ + _U_) will not be the sum of two terms the
+one independent of the velocities, the other proportional to the square
+of these velocities. Among the functions which remain constant there is
+only one which enjoys this property, that is _T_ + _U_ (or a linear
+function of _T_ + _U_, which comes to the same thing, since this linear
+function may always be reduced to _T_ + _U_ by change of unit and of
+origin). This then is what we shall call energy; the first term we shall
+call potential energy and the second kinetic energy. The definition of
+the two sorts of energy can therefore be carried through without any
+ambiguity.
+
+It is the same with the definition of the masses. Kinetic energy, or
+_vis viva_, is expressed very simply by the aid of the masses and the
+relative velocities of all the material points with reference to one of
+them. These relative velocities are accessible to observation, and, when
+we know the expression of the kinetic energy as function of these
+relative velocities, the coefficients of this expression will give us
+the masses.
+
+Thus, in this simple case, the fundamental ideas may be defined without
+difficulty. But the difficulties reappear in the more complicated cases
+and, for instance, if the forces, in lieu of depending only on the
+distances, depend also on the velocities. For example, Weber supposes
+the mutual action of two electric molecules to depend not only on their
+distance, but on their velocity and their acceleration. If material
+points should attract each other according to an analogous law, _U_
+would depend on the velocity, and might contain a term proportional to
+the square of the velocity.
+
+Among the terms proportional to the squares of the velocities, how
+distinguish those which come from _T_ or from _U_? Consequently, how
+distinguish the two parts of energy?
+
+But still more; how define energy itself? We no longer have any reason
+to take as definition _T_ + _U_ rather than any other function of _T_ +
+_U_, when the property which characterized _T_ + _U_ has disappeared,
+that, namely, of being the sum of two terms of a particular form.
+
+But this is not all; it is necessary to take account, not only of
+mechanical energy properly so called, but of the other forms of energy,
+heat, chemical energy, electric energy, etc. The principle of the
+conservation of energy should be written:
+
+ _T_ + _U_ + _Q_ = const.
+
+where _T_ would represent the sensible kinetic energy, _U_ the potential
+energy of position, depending only on the position of the bodies, _Q_
+the internal molecular energy, under the thermal, chemic or electric
+form.
+
+All would go well if these three terms were absolutely distinct, if _T_
+were proportional to the square of the velocities, _U_ independent of
+these velocities and of the state of the bodies, _Q_ independent of the
+velocities and of the positions of the bodies and dependent only on
+their internal state.
+
+The expression for the energy could be resolved only in one single way
+into three terms of this form.
+
+But this is not the case; consider electrified bodies; the electrostatic
+energy due to their mutual action will evidently depend upon their
+charge, that is to say, on their state; but it will equally depend upon
+their position. If these bodies are in motion, they will act one upon
+another electrodynamically and the electrodynamic energy will depend not
+only upon their state and their position, but upon their velocities.
+
+We therefore no longer have any means of making the separation of the
+terms which should make part of _T_, of _U_ and of _Q_, and of
+separating the three parts of energy.
+
+If (_T_ + _U_ + _Q_) is constant so is any function [phi](_T_ + _U_ +
+_Q_).
+
+If _T_ + _U_ + _Q_ were of the particular form I have above considered,
+no ambiguity would result; among the functions [phi](_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 principle 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 principle 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.
+
+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 statement presents the molecule to
+us, so to speak, as a living and free being. Clearly it would be better
+to replace it by an enunciation less objectionable, and where, as the
+philosophers would say, final causes would not seem to be substituted
+for efficient causes.
+
+THERMODYNAMICS.[4]--The rôle of the two fundamental principles of
+thermodynamics in all branches of natural philosophy becomes daily more
+important. Abandoning the ambitious theories 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?
+
+ [4] The following lines are a partial reproduction of the preface
+ of my book _Thermodynamique_.
+
+An eminent physicist said to me one day _à propos_ of the law of errors:
+"All the world believes it firmly, because the mathematicians imagine
+that it is a fact of observation, and the observers 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 question, 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 generalization 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 Newton'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 perpetual
+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 perpetual
+motion implies the conservation of energy only for reversible phenomena.
+
+The imposing simplicity of Mayer's principle likewise contributes to
+strengthen our faith. In a law deduced immediately from experiment, like
+Mariotte's, this simplicity would rather seem to us a reason for
+distrust; but here this is no longer the case; we see elements, at first
+sight disparate, arrange themselves 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 impossible 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 hypothesis, the state
+of the universe is determined by an extremely great number _n_ of
+parameters which I shall call x_{1}, x_{2},... x_{_n_}. 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 _n_ - 1 integrals and consequently there are
+_n_ - 1 functions of x_{1}, x_{2},... x_{_n_}, which remain constant.
+_If 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.
+
+To simplify the enunciation, suppose that the sum of the work of the
+external forces is null, as well as that of the quantities 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 vanishing 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 influence 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 restriction, 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 conservation of
+energy can have only one signification, which is that there is a
+property common 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 I 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 approximations, and it is hoped to replace them little
+by little by laws more and more precise. If, on the other hand, the
+principle of Clausius reduces to an inequality, this is not caused by
+the imperfection of our means of observation, but by the very nature of
+the question.
+
+
+GENERAL CONCLUSIONS ON PART THIRD
+
+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 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 convention 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 experiments 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 system is true or false.
+
+Only, these conventions are convenient, and it is certain experiments
+which have taught us that.
+
+At first blush, the analogy is complete; the rôle of experiment seems
+the same. One will therefore be tempted to say: Either mechanics must be
+regarded as an experimental science, and then the same must hold for
+geometry; or else, on the contrary, 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 fundamental experiments
+are preeminently physiological experiments, which bear, not on space
+which is the object the geometer must 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 mechanics 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 compared 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 objectivity. 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 attributes an absolute 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.
+
+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 intermediary term _C_ more or
+less fictitious, and _C_ is _by definition_ that which has with _A_
+_exactly_ the relation expressed by the law.
+
+Then our law is separated into an absolute and rigorous principle 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 RÔLE OF 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 unquestionable
+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 naïvely. Descartes pitied the Ionians;
+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.
+
+And above all the scientist must foresee. Carlyle has somewhere said
+something like this: "Nothing but facts are of importance. 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
+Lackland 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 understood 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? 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 circumstances 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 interpolate.
+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; 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 corrections 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 preconceived 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 verifications that we can make will
+never be anything but a negligible quantity.
+
+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 continually. The
+librarian has at his disposal for his purchases only insufficient 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 rôle 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 difficulty. 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?
+
+There was a time when the simplicity of Mariotte's law was an argument
+invoked in favor of its accuracy; when Fresnel himself, 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.
+
+To-day ideas have greatly changed; and yet, those who do not believe
+that natural laws have to be simple, are still often obliged to act as
+if they did. They could 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 commonplace 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 movements 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 complex appearances;
+sometimes it is the simplicity which is apparent, and which disguises
+extremely complicated realities.
+
+What is more complicated than the confused movements of the planets?
+What simpler than Newton's law? Here nature, making sport, as Fresnel
+said, of analytical difficulties, employs only simple means, and by
+combining 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 capricious forms and traverse
+space in every direction. The observable result is Mariotte's simple
+law. Every individual fact was complicated. The law of great numbers has
+reestablished simplicity 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 analytical 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 superposition of small 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 numbers? In any case, 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 modifying 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.
+
+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, will the ground be firm enough? This is what must be
+investigated.
+
+For that purpose, let us see what part is played in our generalizations
+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 differences, 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 legitimately suppose that it will still be true
+in analogous cases. To refuse to do this would be to attribute to chance
+an inadmissible rôle.
+
+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 RÔLE OF HYPOTHESIS.--All generalization is a hypothesis. Hypothesis,
+then, has a necessary rôle that no one has ever contested. 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 experiment would have been made by chance, so
+that nothing would have been derived from it. One would have seen
+nothing extraordinary; 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
+still dangerous hypotheses; first, and above all, those which are tacit
+and unconscious. Since we make them without knowing 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 all the hypotheses that we should make without
+it, but unconsciously.
+
+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 necessary to change? It will be
+impossible to know. And inversely, if the experiment succeeds, shall we
+believe that we have demonstrated 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 mathematical 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 occasion to exclude them.
+
+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 will only be fruitful 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 number 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 substitute Newton's law.
+
+Next we try to analyze the phenomenon in space. What experiment gives us
+is a confused mass of facts presented on a stage of considerable extent.
+We must try to discover the elementary phenomenon, which will be, on the
+contrary, localized in a very small region of space.
+
+Some examples will perhaps make my thought better understood. If we
+wished to study in all its complexity the distribution 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 neighborhood, 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.
+
+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 verification. 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 elementary 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 superposition. 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 necessary 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 symmetry of the medium, all the differences are leveled
+down, and whatever hypothesis 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, and 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 phenomenon; 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 contrivance 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 decomposed 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 sufficient, 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 spectrum, but however small it may be, it will retain a certain
+breadth. Likewise the natural lights, called _monochromatic_, 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 perpendicular planes, then
+bring them back to the same plane of polarization, 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 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.
+
+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 simple 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 combination 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 explained this above, in the chapter on mathematical
+reasoning.
+
+But for this, all the operations must be alike. 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 OF 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 conclude that these are absolutely idle. This
+is what they call the _bankruptcy of science_.
+
+Their skepticism is superficial; they give no account to themselves of
+the aim and the rôle 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 rôle of
+mere practical recipes; these equations express relations, and if the
+equations remain true it is because these relations 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 the same relations exist between these objects as
+between the images by which we are forced to replace them. If these
+relations 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, acting 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 reality; that this relationship, this
+similitude, or rather this parallelism 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 will
+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 various ways, and its author
+himself imagined another founded on the principles of Maxwell. But, what
+is remarkable, all the scientists 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 the 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 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 regarded 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 all 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 billiards. 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 assertion, if neither
+do we wish the restricted sense I have just explained, 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, supposed 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.
+
+Scarcely fifteen years ago was there anything more ridiculous, more
+naïvely 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 reluctance. It would be rash to affirm that the
+belief in electrons will not again suffer eclipse; it was none the less
+curious to note this unexpected resurrection.
+
+But the most striking example is Carnot's principle. Carnot 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, _débris_ 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 objects. 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 rôle 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 numerous 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 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 kinship; 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 foresee 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
+extension of the principle, will yet have condemned it.
+
+PHYSICS AND MECHANISM.--Most theorists have a constant predilection for
+explanations borrowed from mechanics or dynamics. Some would be
+satisfied if they could explain all phenomena 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 instance, 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
+flexible for that?
+
+We shall examine this question in Chapter XII, _à 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 König's on 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 rôle 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 singularities. 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 all 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.
+
+And does our ether really exist? We know the origin of our belief in the
+ether. If light reaches us from a distant star, during 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 undergone 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 anterior; 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
+immediately preceding, but on states much older; the system would
+satisfy equations of finite differences. It is to escape this derogation
+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 further. By the
+interference of rays which have traversed air or water in motion, it
+seems to show us two different media interpenetrating 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_, and 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 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 displacements 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 principle 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 illusory; 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 relative 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 rôle? 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 destroy the terms of the first order,
+and that another circumstance, wholly different, but just as opportune,
+should take upon itself to destroy those of the second order? 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 development 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
+will 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 observe the
+science of to-day and compare it with that of yesterday. From this
+examination we may doubtless draw some encouragement.
+
+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 doubted that it soon would be. For light, the task
+seemed completely 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 reduction was
+doubted by none, there was faith. Finally, in what concerns the
+molecular properties of material bodies, the reduction 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 progress. 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 unquestionably the one which best
+explains the known facts, the one which illuminates 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 the optics
+of moving bodies, the laws of normal and anomalous dispersion and of
+absorption find themselves linked to one another and to the other
+properties of the ether by bonds which beyond any doubt will never more
+be 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 defied 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. Larmor 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 ignorance, 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 me he has, does that mean that a
+mechanical explanation is impossible? Far 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 mechanical 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 possible, there should be
+other reasons besides the necessity of making 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 to
+formulate a mechanical explanation; let us be content with showing that
+we could always find one if we wished to. In this 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 principle. Long did
+thermodynamics confine itself to the study of the dilatation of bodies
+and their changes of state. For some time past 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 diversity of
+appearances, is found again Carnot's principle; everywhere 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 appear 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 go forward in the same sense without hope of
+return. The apparent 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 Carnot'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 incomparably
+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 constructed 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 established 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 precise 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, 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 astonishing 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 without 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 rôle the belief
+in simplicity plays in our generalizations. We have verified a simple
+law in a great number of particular cases; 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 consciously 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. Probability
+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 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? Each die can turn up in six different ways; the number of
+possible cases is 6 × 6 = 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 6 × 7/2 = 21 different
+combinations? Among these combinations 6 are favorable; the probability
+is 6/21.
+
+Now why is the first method of enumerating the possible cases more
+legitimate than the second? In any case 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 convention
+should be adopted? Alas! M. Bertrand has amused himself 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 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 enormous 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 unconscious
+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.
+
+All 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
+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 practical value.
+
+Unfortunately the difficulty does not thus disappear. A gambler 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 shareholders
+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 _à 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 possible 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. Generality 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 according to the greater or less depth of
+this ignorance.
+
+In mathematics even we may set ourselves problems of probability. What
+is the probability that the fifth decimal of a logarithm 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 logarithm 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.
+
+In the kinetic theory of gases, we assume that the gaseous molecules
+follow rectilinear trajectories, and obey 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 scientific applications.
+
+I play écarté 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 experimental
+method. I have observed _n_ values of _x_ and the corresponding 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
+problems, discussing in succession what I have called above subjective
+and objective probability.
+
+II. PROBABILITY IN MATHEMATICS.--The impossibility of squaring 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 examination 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 discovery of moment.
+The Academy could not have proved that it was right; but it knew quite
+well that its instinct was not mistaken. 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 [pi] were a root of an algebraic equation, why
+do you suppose this root to be a period of the function sin 2_x_, and
+not the same about the other roots of this same equation?" To sum up,
+they would have invoked the principle of sufficient reason in its
+vaguest form.
+
+But what could they deduce from it? At most a rule of conduct 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.
+
+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 générale 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 logarithm 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 functions; 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 incomplete
+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 _n_ is a given very large integer. What is
+the probable value of sin _nu_? This problem has no meaning 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_+ is equal to
+[phi](_a_)_da_; that it is therefore proportional to the infinitely
+small interval _da_, and equal to this multiplied by _a_ function
+[phi](_a_) depending only on _a_. As for this function, I choose it
+arbitrarily, but I must assume it to be continuous. The value of sin
+_nu_ remaining the same when _u_ increases by 2[pi], I may without loss
+of generality assume that _u_ lies between 0 and 2[pi], and I shall thus
+be led to suppose that [phi](_a_) is a periodic function whose period is
+2[pi].
+
+The probable value sought is readily expressed by a simple integral, and
+it is easy to show that this integral is less than
+
+ 2[pi]M_{_k_}/_n_^{_k_},
+
+M_{_k_} being the maximum value of the _k_th derivative of [phi](_u_).
+We see then that if the _k_th derivative is finite, our probable value
+will tend toward 0 when _n_ increases indefinitely, and that more
+rapidly than 1/_n_^{_k_ - 1}.
+
+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 be_. I have imposed upon
+myself only slight restrictions in assuming that the function [phi](_a_)
+is continuous and periodic, and these hypotheses 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 rôle
+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, without 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 hesitate to affirm that their
+distribution is now nearly uniform. Why?
+
+Let _b_ 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 longitude at the
+present epoch, that is to say at the epoch _t_, will be _at_ + _b_. To
+say that the present distribution is uniform is to say that the mean
+value of the sines and cosines of multiples of _at_ + _b_ is 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 _b_. 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 distribution of these points.
+
+What do we do when we wish to apply the calculus of probabilities 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 arbitrary hypothesis. To explain the nature
+of this hypothesis, allow me to use, in lieu of a mathematical formula,
+a crude but concrete image. Let us suppose that over the surface of our
+plane has been spread an imaginary substance, whose density is variable,
+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 representative
+points are very numerous, very close together, but discrete 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 of some extent the
+density of this imaginary continuous matter would be nearly proportional
+to the number of the representative points, or, if you wish, to the
+number of atoms which are contained 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
+sufficient, 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 (_at_ + _b_)? We
+made at the outset an arbitrary convention, but if we adopt it, this
+probable value is entirely defined. Divide the plane into elements of
+surface. Consider the value of sin (_at_ + _b_) at the center 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 [phi] 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 [phi], the mean value tends toward zero as _t_
+increases, and as the minor planets have certainly accomplished a very
+great number of revolutions, I may assert that this mean value is very
+small.
+
+I may choose [phi] 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 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_){[Sigma] sin (_at_ + _b_)},
+
+_n_ being the number of minor planets. In lieu of a double integral
+referring to a continuous function, we shall have a sum of discrete
+terms. And yet no one will seriously doubt that this mean value is
+practically very small.
+
+Our representative points being very close together, our discrete sum
+will in general differ very little from an integral.
+
+An integral is the limit toward which a sum of terms tends when the
+number of these terms is indefinitely increased. If the terms are very
+numerous, the sum will differ very little from its limit, that is to say
+from the integral, and what I said of this latter will still be true of
+the sum itself.
+
+Nevertheless, there are exceptions. If, for instance, for all the minor
+planets,
+
+ _b_ = [pi]/2 - _at_,
+
+the longitude for all the planets at the time t would be [pi]/2, and the
+mean value would evidently be equal to unity. For this to be the case,
+it would be necessary that at the epoch 0, the minor planets must have
+all been lying on a spiral of peculiar form, with its spires very close
+together. Every one will admit that such an initial distribution is
+extremely improbable (and, even supposing it realized, the distribution
+would not be uniform at the present time, for example, on January 1,
+1913, but it would become so a few years later).
+
+Why then do we think this initial distribution improbable? This must be
+explained, because if we had no reason for rejecting as improbable this
+absurd hypothesis everything would break down, and we could no longer
+make any affirmation about the probability of this or that present
+distribution.
+
+Once more we shall invoke the principle of sufficient reason to which we
+must always recur. We might admit that at the beginning the planets were
+distributed almost in a straight line. We might admit that they were
+irregularly distributed. But it seems to us that there is no sufficient
+reason for the unknown cause that gave them birth to have acted along a
+curve so regular and yet so complicated, which would appear to have been
+expressly chosen so that the present distribution would not be uniform.
+
+IV. ROUGE ET NOIR.--The questions raised by games of chance, such as
+roulette, are, fundamentally, entirely analogous to those we have just
+treated. For example, a wheel is partitioned into a great number of
+equal subdivisions, alternately red and black. A needle is whirled with
+force, and after having made a great number of revolutions, it stops
+before one of these subdivisions. The probability that this division is
+red is evidently 1/2. The needle describes an angle [theta], including
+several complete revolutions. I do not know what is the probability that
+the needle may be whirled with a force such that this angle should lie
+between [theta] and [theta]+_d_[theta]; but I can make a convention. I
+can suppose that this probability is [phi]([theta])_d_[theta]. As for
+the function [phi]([theta]), 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 [epsilon] be the length (measured on the circumference of radius 1)
+of each red and black subdivision. We have to calculate the integral of
+[phi]([theta])_d_[theta], 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 2[epsilon], comprising a red division and a black
+division which follows it. Let M and _m_ be the greatest and least
+values of the function [phi]([theta]) in this interval. The integral
+extended to the red divisions will be smaller than [Sigma]M[epsilon];
+the integral extended to the black divisions will be greater than
+[Sigma]_m_[epsilon]; the difference will therefore be less than
+[Sigma](M - _m_)[epsilon]. But, if the function [theta] is supposed
+continuous; if, besides, the interval [epsilon] is very 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 [theta], 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. Observation 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 PROBABILITY OF CAUSES.--We now come to the problems 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 probability 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 example, or certain absurd conventions.
+
+We shall find something analogous when we deal with the 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
+écarté 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 adversary 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 preliminary
+convention. I shall even add that if none were made, the problem of the
+_a posteriori_ probability would have no meaning. 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 inflections too accentuated, or brusque variation of the
+radius of curvature. 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.
+
+Here is a problem in the probability of causes. The effects are the
+measurements I have recorded; they depend on a combination 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 distance of the corresponding point from this curve.
+
+But the problem would have no meaning if, before any observation, 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 continuous function
+(or by a function whose derivatives of high order are small), as more
+probable than a law not satisfying these conditions. 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 THEORY 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 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 consequently the probable value of the quantity to be
+measured.
+
+But as I have just explained, we should not know how to undertake 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 bell.'
+
+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 will 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 independent errors; each of the partial errors is
+very little and besides, obeys any law of probability, provided that the
+probability 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, besides the
+systematic errors to which they and the physicists are subject alike,
+have to control with an extremely important source of error which is
+wholly accidental; I mean atmospheric undulations. So it is very
+curious to hear a physicist discuss with an astronomer about a method of
+observation. The physicist, persuaded that one good measurement is worth
+more than many bad ones, is before all concerned with eliminating by
+dint of precautions the least systematic 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 eliminated 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 probability.
+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 result is so much. _This is
+absolutely illegitimate_; it would be true only if we were sure that all
+the systematic errors were eliminated, 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
+systematic 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 than 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 necessary 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 sufficient 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
+under which we have met it most often is the belief in continuity, a
+belief which it would be difficult to justify by apodeictic reasoning,
+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.
+
+
+
+
+CHAPTER XII
+
+OPTICS AND ELECTRICITY
+
+
+FRESNEL'S THEORY.--The best example[5] 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-called wave-theory forms a whole truly satisfying to the
+mind. We must not, however, ask of it what it can not give us.
+
+ [5] This chapter is a partial reproduction of the prefaces of two
+ of my works: _Théorie mathématique de la lumière_ (Paris, Naud,
+ 1889), and _Électricité et optique_ (Paris, Naud, 1901).
+
+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 experiment 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 will always be
+useful, then, to study a doctrine that unites all these equations.
+
+The undulatory theory rests on a molecular hypothesis. 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 distrust. 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.
+
+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 unchanged when we
+adopt the electromagnetic theory of light.
+
+MAXWELL'S THEORY.--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 respect
+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 precision 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 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 will have only purely geometric properties,
+and whose atoms will be nothing but mathematical points, subject to the
+laws of dynamics alone. And yet these atoms, invisible and without
+color, it will 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 deceitful, 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 himself 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 electricity, one can
+therefore deduce immediately a theory of light.
+
+The converse unfortunately is not true; from a complete explanation of
+light, it is not always easy to derive a complete explanation 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 independent constructions,
+between which communication is difficult and sometimes impossible.
+
+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 contains a theory
+complete in itself which one could understand without 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 contradiction;
+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?
+
+One will try to explain it either by the motions of ordinary matter, or
+by those of one or more hypothetical fluids.
+
+These fluids will be considered as formed of a very great number 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 coordinates of these
+hypothetical molecules _m_, equations which, moreover, must conform to
+the principles of dynamics; and, on the other hand, when we know the
+relations that define the coordinates of the molecules _m_ as functions
+of the parameters _q_ accessible to experiment.
+
+These equations, as I have said, must conform to the principles 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:
+
+1º 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_.
+
+2º The potential energy, which depends only on the coordinates 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_{0} to the
+final position occupied at the instant t_{1}, the system must take such
+a path that, in the interval of time that elapses between the two
+instants t_{0} and t_{1}, 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.
+
+Thus we obtain what are called the equations of Lagrange.
+
+In these equations, the independent variables are the coordinates of the
+hypothetical molecules _m_; but I now suppose that one takes as
+variables the parameters _q_ directly accessible to experiment.
+
+The two parts of the energy must then be expressed as functions 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.[6]
+
+ [6] We add that _U_ will depend only on the parameters _q_, that _T_
+ will depend on the parameters _q_ and their derivatives with
+ respect to the time and will be a homogeneous polynomial of the
+ second degree with respect to these derivatives.
+
+That granted, the system will always go from one position to another by
+a path such that the average action shall be a minimum.
+
+It matters little that _T_ and _U_ 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 minimum, and there is
+only one. The principle of least action suffices, then, to determine the
+differential equations which define the variations of the parameters
+_q_.
+
+The equations thus obtained are another form of the equations 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 _U_
+as a function of the coordinates of these molecules.
+
+All we need to know is the expression of _U_ 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 choice of
+the functions _T_ and _U_, the equations of Lagrange, constructed as we
+have just said, will be identical with the differential 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 _U_ in such a way as
+to satisfy the principle of least action, which involves 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 represents 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 potential 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 will confine myself to referring for more details to the
+preface of my work, 'Électricité 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 hypothetical 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 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 _q_?
+Which shall we prefer to regard as the derivatives of these parameters?
+This choice remains arbitrary to a very large extent; but, for a
+mechanical explanation to be possible, 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 _U_, in such a way that the electrical
+phenomena would satisfy this principle. Experiment shows us that the
+energy of an electromagnetic field is decomposed into two parts, the
+electrostatic energy and the electrodynamic energy. Maxwell observed
+that if we regard the first as representing the potential energy _U_,
+the second as representing 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
+parameters _q_; under these conditions, I say, Maxwell observed that the
+electric phenomena satisfy the principle of least action. Thenceforth he
+was certain of the possibility of a mechanical explanation.
+
+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 explanation, it
+will admit of an infinity of others, that will render an account equally
+well of all the particulars revealed by experiment.
+
+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 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 considerations 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 because of their simplicity.
+
+In what concerns electricity and magnetism, Maxwell abstains from making
+any choice. It is not that he systematically disdains 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 complexity 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 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.
+
+Ampère entitled his immortal work, 'Théorie des phénomènes
+électrodynamiques, _uniquement_ fondée sur l'expérience.' 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 Ampère'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 Ampère 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 electricity 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 Ampère's time this was an open current;
+the current was known to go from the first conductor to the second, it
+was not seen to return from the second to the first.
+
+So Ampère considered as open currents of this nature, for example, 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 suppose two
+conductors, _A_ and _B_, connected by a wire _AMB_. Small conducting
+masses in motion first come in contact with the 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 electricity
+describes the closed circuit _BNAMB_; but the two parts of this current
+are very different. In the wire _AMB_, the electricity is displaced
+through a fixed conductor, like a voltaic current, 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 displaced 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 difficult to produce with appreciable
+intensity. With the means at Ampère's disposal, we may say that this was
+impossible.
+
+To sum up, Ampère 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 displacements of the moving part under the
+action of another closed current.
+
+On the other hand, Ampère had no means of studying the action of an open
+current, either on a closed current or another open current.
+
+1. _The Case of Closed Currents._--In the case of the mutual action of
+two closed currents, experiment revealed to Ampère remarkably simple
+laws.
+
+I recall rapidly here those which will be useful to us in the sequel:
+
+1º _If 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.
+
+2º The action of a closed solenoid is null.
+
+3º The action of a circuit _C_ on another voltaic circuit _C'_ depends
+only on the 'magnetic field' developed by this circuit. At each point in
+space we can in fact define in magnitude and direction a certain force
+called _magnetic force_, which enjoys the following 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;
+
+(_b_) 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 electrodynamic
+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._--Ampère not
+having been able to produce an open current, properly 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 [alpha][beta] whose extremities [alpha] and [beta] could slide
+along a fixed wire. In one of the positions of the movable wire, the end
+[alpha] rested on the _A_ of the fixed wire and the extremity [beta] on
+the point _B_ of the fixed wire. The current circulated from [alpha] to
+[beta], that is to say, from _A_ to _B_ along the movable wire, and then
+it returned from _B_ to _A_ along the fixed wire. _This current was
+therefore closed._
+
+In a second position, the movable wire having slipped, the extremity
+[alpha] rested on another point _A'_ of the fixed wire, and the
+extremity [beta] on another point _B'_ of the fixed wire. The current
+circulated then from [alpha] to [beta], 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. Ampère _assumes_ that the apparent force to which this movable
+part _AB_ seems thus subjected, representing the action of the _C_ on
+the portion [alpha][beta] of the current, is the same as if
+[alpha][beta] were traversed by an open current, stopping at [alpha] and
+[beta], in place of being traversed by a closed current which after
+arriving at [beta] returns to [alpha] through the fixed part of the
+circuit.
+
+This hypothesis seems natural enough, and Ampère made it unconsciously;
+nevertheless _it is not necessary_, since we shall see further on that
+Helmholtz rejected it. However that may be, it permitted Ampère, though
+he had never been able to produce an open current, to enunciate the laws
+of the action of a closed current on an open current, or even on an
+element of current.
+
+The laws are simple:
+
+1º 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 magnetic force which is normal to
+the element.
+
+2º 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 intensities have been
+maintained constant, return to their initial positions, the total work
+is not null.
+
+3. _Continuous Rotations._--Among electrodynamic experiments, the most
+remarkable are those in which continuous rotations are produced and
+which are sometimes called _unipolar induction_ experiments. 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 _C_ 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 circuit _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 point 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 rotation.
+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 communication with a new point
+of the fixed part; it changes the connections, 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 indefinitely.
+
+Ampère 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.
+
+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 enunciated 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 down. Ampère has recourse to hypothesis.
+He supposes:
+
+1º That the mutual action of two elements reduces to a force acting
+along their join;
+
+2º 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 Ampère 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:
+
+1º By the action on a magnetic pole;
+
+2º By the director couple which orientates the magnetic needle;
+
+3º 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 meaning, and in fact:
+
+1º 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;
+
+2º 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;
+
+3º 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 disappeared_.
+
+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 contained 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, exercise 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 Ampère.
+
+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 Ampère. 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 Ampère'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.
+
+But, further, if a circuit _C_ is subjected to the induction of a
+variable voltaic system _S_, if this system _S_ be displaced and
+deformed 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 situation, 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_ contains
+only closed currents. This would no longer be true, if one accepts the
+theory of Ampère, 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 OF HELMHOLTZ.--I have dwelt upon the consequences of Ampère's
+theory, and of his method of explaining open currents.
+
+It is difficult to overlook the paradoxical and artificial character 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 something else.
+
+Helmholtz rejects Ampère'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 Bertrand and Helmholtz.
+
+Helmholtz replaces Ampère's hypothesis by the following: two elements
+always admit of an electrodynamic potential depending solely on their
+position and orientation; and the work of the forces that they exercise,
+one on the other, is equal to the variation of this potential. Thus
+Helmholtz can no more do without hypothesis than Ampère; but at least he
+does not make one without 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 Ampère supposed, the force which
+seems to act on the movable portion of a closed current is not the same
+as would act upon this movable portion if it were isolated and
+constituted an open current.
+
+Let us return to the circuit _C'_, of which we spoke above, and which
+was formed of a movable wire [alpha][beta] sliding on a fixed wire. In
+the only experiment that can be made, the movable portion [alpha][beta]
+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:
+
+1º 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_;
+
+2º 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, [alpha][beta] 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 _à 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 continuous 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 Ampère; it would
+not be null according to Helmholtz. From this follows an important
+consequence. We have given above three definitions of magnetic force.
+The third has 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 Ampère; it is necessary, however, that all the
+difficulties should be smoothed away. In the one as in the other, the
+phrase 'magnetic field' has no meaning, 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 distance, electrodynamic phenomena
+must be explained by a modification 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 galvanometer like a current, and which he calls _current
+of displacement_.
+
+If then two conductors bearing contrary charges are put in communication
+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 electrical
+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 confirmation of
+experiment. Hertz succeeded in producing systems of electric
+oscillations which reproduce all 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 displacement 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 induction 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. ROWLAND'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 whatever.
+
+There are also the cases in which electric discharges describe a closed
+contour, being displaced by conduction in one part of the circuit and by
+convection in the other part.
+
+For open currents of the first sort, the question might be considered as
+solved; they were closed by the currents of displacement.
+
+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 difficult in fact
+to obtain a sufficient intensity even by augmenting as much as possible
+the charge and the velocity of the conductors. It was Rowland, an
+extremely skillful experimenter, who first triumphed over these
+difficulties. A disc received a strong electrostatic charge and a very
+great speed of rotation. An astatic magnetic 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 making 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 electrode 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 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,
+electricity 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 farther. According 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 _electrons_. The circulation of these
+electrons through bodies would produce voltaic currents. And what would
+distinguish conductors 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 theories, even
+Maxwell's in its primitive form, could not explain in a satisfactory
+way; for example, the aberration of light, the partial carrying away of
+luminous waves, magnetic polarization and the Zeeman effect.
+
+Some objections still remained. The phenomena of an electric system
+seemed to depend on the absolute velocity of translation of the center
+of gravity of this system, which is contrary to the idea we have of the
+relativity of space. Supported by M. Crémieu, M. Lippmann 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.
+
+"No!" replied the partisans of Lorentz. "What we could measure in that
+way is not their absolute velocity, but their relative 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 satisfactory
+aspect. The theories of Ampère 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 experiments of M.
+Crémieu, which for a moment seemed to contradict the result previously
+obtained by Rowland.
+
+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 instructive; it
+will 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?_--Professor Rados of Budapest in
+his report to the Hungarian Academy of Science on the award to Poincaré
+of the Bolyai prize of ten thousand crowns, speaking of him as
+unquestionably the most powerful investigator 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 inspiration 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 geometry are conventions, disguised definitions or unprovable
+hypotheses precreated by auto-active 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 discovered from data." Then does
+the scientist create science? This is a question Poincaré here dissects
+with a master hand.
+
+The physiologic-psychologic investigation of the space problem must
+give the meaning of the words _geometric fact_, _geometric reality_.
+Poincaré 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 corresponding 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 illusions, 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 perception
+itself successively occur. What is called accuracy of movement 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 perception in the
+individual helping to make that cerebral organization and accurate motor
+adjustment on which its possibility and permanence seem in so far to
+depend. So-called correct perception 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 confess that the Kantian account of space
+is hopelessly and demonstrably antiquated. As Royce says in 'Kant's
+Doctrine of the Basis of Mathematics,' "That very use of intuition which
+Kant regarded as geometrically ideal, the modern geometer regards as
+scientifically defective, because surreptitious. No mathematical
+exactness without explicit proof from assumed principles--such is the
+motto of the modern geometer. But suppose the reasoning of Euclid
+purified of this comparatively surreptitious appeal to intuition.
+Suppose that the principles of geometry are made quite explicit at the
+outset of the treatise, as Pieri and Hilbert or Professor Halsted or Dr.
+Veblen makes his principles explicit in his recent treatment of
+geometry. Then, indeed, geometry 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 his 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 old
+hypotheses are still sufficiently 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, beating 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?_--What now is the definite, the permanent outcome?
+What new islets raise their fronded 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 Erdély'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 Damascus blade the spikes driven into his door-post, and strewing
+over the sod the thirteen Austrian cavalry officers? This book by the
+greatest mathematician of our time gives weightiest and most charming
+answer.
+
+ GEORGE BRUCE 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 well as to find
+the other, it is necessary to free the soul completely from prejudice
+and from passion; it is necessary to attain absolute sincerity. These
+two sorts of 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 conflict 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 in 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. Readers of my little book 'Science and
+Hypothesis' already know what I think about the question. The truth we
+are permitted to glimpse is not altogether 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 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 geometer's mind is not like the
+physicist's or the naturalist's, all the world would agree; but
+mathematicians themselves do not resemble 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 enclosed 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 quantitative 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
+'qualitative 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 without; 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 intimate 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 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 astonished 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 necessary 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 established
+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 needful now to set things to
+rights.
+
+Some people have exaggerated the rôle 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 artificial
+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 common 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 SCIENCES
+
+
+
+
+CHAPTER I
+
+INTUITION AND LOGIC IN 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 intuitionalists, 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 born a geometer or an analyst. I should like to cite
+examples and there are surely plenty; but to accentuate the contrast I
+shall begin with an extreme example, taking the liberty of seeking it in
+two living mathematicians.
+
+M. Méray wants to prove that a binomial equation always has a root, or,
+in ordinary words, that an angle may always be subdivided. 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? M. Méray 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 determine whether
+on a given Riemann surface there always exists a function admitting of
+given singularities. What does the celebrated 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 distribution 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 demonstration, 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 influences; 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 all their pupils these
+two faces are stamped in deathless lines; for all 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 eager to paint, this is
+why he calls gesture to his aid. With M. Hermite, 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. Weierstrass 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
+conceptions 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 venture 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
+rôles. But it is interesting 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 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 conclude
+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 establish 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 all the
+singularities functions can present when the word is understood in the
+most general sense.
+
+But it would not be the same had we used concrete images, had we, for
+example, considered this function as an electric potential; it would
+have been thought legitimate to affirm that electrostatic equilibrium
+can be attained. Yet perhaps a physical comparison 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 introduced in
+the reasoning unless first made to enter into the definitions. For the
+most part the objects treated of by mathematicians were long ill
+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.
+
+
+III
+
+A first question presents itself. Is this evolution ended? Have we
+finally attained absolute rigor? At each stage of the evolution 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 create nothing new; not
+from it alone can any science issue. In one sense these philosophers 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_ + 1 if true for _n_, then will it be true of all
+whole numbers; (3) if on a straight the point _C_ is between _A_ and _B_
+and the point _D_ between _A_ and _C_, then the point _D_ 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 imagination; 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 continuity_. 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 sciences; 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, because it is intuitive.
+At first I fail or must content myself with approximate proofs; finally
+I decide to give to my object a precise 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 definition. Only then could you affirm
+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 established. The second was
+an experimental verity. Experience alone can teach us that some real and
+concrete object corresponds or 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 rôle
+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. Without it young minds
+could not make a beginning in the understanding of mathematics; they
+could not learn to love it and would see in it only a vain logomachy;
+above all, without intuition they would never become capable of applying
+mathematics. But now I wish before all to speak of the rôle of intuition
+in science itself. If it is useful to the student it is still more so to
+the creative scientist.
+
+
+V
+
+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 interesting than that of the isolated
+elements, and should a naturalist who had never studied the elephant
+except by means of the microscope 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 operations one after
+the other and ascertained that each is correct, are we to think we have
+grasped the real meaning of the demonstration? 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 different 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 tell us which to choose? We need a faculty which
+makes us see 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
+successive moves a sort of organized whole. This faculty is still 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 function. 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 reproduced, so to speak, all the lines of the
+original image; this construction finished, the centering of the arch,
+so to say, was removed, that crude representation which had temporarily
+served 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 disappeared 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 still 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 _division_, _dissection_. It can have, therefore, no tool other
+than the scalpel and the microscope.
+
+Thus logic and intuition have each their necessary rôle. 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 mathematical 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 explanation.
+And first, do you think these logicians have always proceeded from the
+general to the particular, as the rules of formal 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 employment of dominant functions. We
+know it has already served to solve a multitude of problems; in what
+consists then the rôle 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 analogies 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 perceived 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 intuitionalists, 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 mathematical 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 essential 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 intervene. 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 intuition alone,
+would soon feel ourselves seized with vertigo. Our weakness has need of
+a staff more solid, and, despite the exceptions 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 not the
+time either to solve or even to enunciate with the developments 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 intuition; 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.
+
+
+
+
+CHAPTER 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 anterior 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 classification 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?
+
+
+II
+
+But that is not all; into this form we wish to put not only the
+phenomena of our own consciousness, but 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 because 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 impenetrable 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 measuring? 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?
+
+
+III
+
+The first difficulty has long been noticed; it has been the subject 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 affirmation?
+
+The least reflection shows that by itself it has none at all. It will
+only have that which I choose to give it, by a definition which will
+certainly possess a certain degree of arbitrariness. Psychologists 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 all the beats of this pendulum are of equal duration. But this is
+only a first approximation; the temperature, the resistance of the air,
+the barometric pressure, make the pace 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, electric, magnetic or others, 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 definition.
+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 phenomena 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 [alpha] happens, causing as consequence at the end of a
+certain time the effect [alpha]'. At another place in the world very
+far away from the first, happens the phenomenon [beta], which causes as
+consequence the effect [beta]'. The phenomena [alpha] and [beta] are
+simultaneous, as are also the effects [alpha]' and [beta]'.
+
+Later, the phenomenon [alpha] is reproduced under approximately the same
+conditions as before, and _simultaneously_ the phenomenon [beta] is also
+reproduced at a very distant place in the world and almost under the
+same circumstances. The effects [alpha]' and [beta]' also take place.
+Let us suppose that the effect [alpha]' happens perceptibly before the
+effect [beta]'.
+
+If experience made us witness such a sight, our postulate would be
+contradicted. For experience would tell us that the first duration
+[alpha][alpha]' is equal to the first duration [beta][beta]' and that
+the second duration [alpha][alpha]' is less than the second duration
+[beta][beta]'. On the other hand, our postulate would require that the
+two durations [alpha][alpha]' should be equal to each other, as likewise
+the two durations [beta][beta]'. 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 contradiction. Doubtless
+they could not happen without the principle of sufficient reason seeming
+violated. But to justify a definition so fundamental I should prefer
+some other guarantee.
+
+
+V
+
+But that is not all. In physical reality one cause does not produce 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 approximately. 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 pendulum.
+
+Under these conditions, it is clear that the causes which have produced
+a certain effect will never be reproduced except approximately. Then we
+should modify our postulate and our 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 approximate. Besides,
+as M. Calinon very justly remarks in a recent memoir:[7]
+
+ 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 measure time.
+
+ [7] _Etude sur les diverses grandeurs_, Paris, Gauthier-Villars, 1897.
+
+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 producing 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, calculated
+according to Newton's law, would be less than that deduced 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 enunciation of the law would be different,
+because it would be translated into another language; it would evidently
+be much less simple. So that the definition implicitly adopted by the
+astronomers may be summed up thus: Time should be so defined that 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 advantageous to conform to the indications of the
+first.
+
+The difficulty 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 consciousnesses;
+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 phenomenon, what do I mean?
+
+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 discovery 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
+phenomenon is anterior to the formation of the visual image of the isle
+of Española 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 one another. We
+should like to represent to ourselves the external universe, and only by
+so doing could we feel that we understood 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 all
+recollections would be equally present to it and for it there would be
+no time. And yet when we speak of time, for all which happens outside of
+us, do we not unconsciously 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?
+
+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 hypothetical intelligence, even
+if it existed, would be for us impenetrable. 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 without 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.
+
+
+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 discharge; 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 follow, 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 antecedent, 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 _D_,
+which I regard as a consequence of the act _A_; on the other hand, for
+whatever reason, I infer that this consequence is not immediate, but
+that outside my consciousness two facts _B_ and _C_, which I have not
+witnessed, have happened, and in such a way that _B_ is the effect of
+_A_, that _C_ is the effect of _B_, and _D_ of _C_.
+
+But why? If I think I have reason to regard the four facts _A_, _B_,
+_C_, _D_, as bound to one another by a causal connection, why range
+them in the causal order _A B C D_, and at the same time in the
+chronologic order _A B C 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 _C_ 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 _C_ because we always see _B_ happen before _C_.
+These two phenomena, when witnessed, happen in a certain order; when
+analogous phenomena happen without witness, there is no reason to invert
+this order.
+
+Doubtless, but take care; we never know directly the physical phenomena
+_B_ and _C_. What we know are sensations _B'_ and _C'_ produced
+respectively by _B_ and _C_. Our consciousness tells us immediately that
+_B'_ precedes _C'_ and we suppose that _B_ and _C_ 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 only 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?
+
+
+XI
+
+Another difficulty; have we really the right to speak of the cause of a
+phenomenon? If all the parts of the universe are interchained 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 circumstances 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 reduced 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 by Jupiter at the instant _t_ + _e_
+and Saturn at the instant _t_ + _a_ + _e_ is bound to the aggregate of
+positions occupied by Jupiter at the instant _t_ and Saturn at the
+instant _t_ + _a_, by laws as precise as that of Newton, though more
+complicated. Then why not regard one of these aggregates as the cause of
+the other, which would lead to considering as simultaneous the instant
+_t_ of Jupiter and the instant _t_ + _a_ of 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 simultaneity.
+
+I will 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 velocity 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 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 all events, resembling the principle of sufficient
+reason, has been accepted by everybody; what I wish to emphasize is that
+it furnishes us with a new rule for the investigation of simultaneity,
+entirely different 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 attributed
+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 complicated. 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 discussing; 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 depend 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 instantaneous; 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 sending 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 regarded as
+simultaneous. But, to be rigorous, a little correction 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 separate the qualitative problem of simultaneity from the
+quantitative 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 particular case.
+
+These rules are not imposed upon us and we might amuse ourselves in
+inventing others; but they could not be cast aside without 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 recapitulate them as follows:
+"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 OF 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 difficult of approach, such
+as those which pertain to the number of dimensions. All 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 amorphous 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 call _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 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 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 chiefly
+because 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 satisfy 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
+conceivable, 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 motions, the Euclidean
+straights remain Euclidean 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 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 triangles;
+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 Euclidean triangles if the non-Euclidean motions
+were called motions 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 natural 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 noteworthy natural objects
+affecting almost the form of non-Euclidean straights, and noteworthy
+natural bodies frequently undergoing 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 themselves which they serve to measure.
+The world, after this dilatation, would continue on its course without
+anything apprising 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 indistinguishable. 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 coordinates
+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 relativity 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 inhabitants
+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, therefore, be Euclidean geometry, but
+their straight will not be our Euclidean straight. It will be its
+transform by the point-transformation which carries over from our world
+to theirs. The straights of these men will not be our straights, but
+they will 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 themselves, 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_
+
+All that is relatively easy to understand, and I have already so often
+repeated it that I think it needless to expatiate further on the 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 Lobachevskian space, just as we can, by tracing upon it a
+proper graduation, transform an ungraduated thermometer into a
+Fahrenheit or a Réaumur thermometer.
+
+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
+Riemann and Betti and which has received the name of analysis situs. In
+this science abstraction 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 proportions and replace the
+straights by lines more or less sinuous. In mathematical terms, they are
+not altered by any 'point-transformation' whatsoever. It has often been
+said that metric geometry was quantitative, while projective geometry
+was purely qualitative. That is not altogether true. The straight is
+still distinguished from other lines by properties which remain
+quantitative in some respects. The real qualitative geometry is,
+therefore, 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 verities? 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 impossible 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 it without imagining the two terms of the
+alternative. If it were impossible to imagine one of these terms, it
+would be futile and besides impossible to consult experience. There is
+no need of observation 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 disembarrassed
+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 experiments 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 all the theorems of analysis situs, the most important is that which
+is expressed in saying that space has three dimensions. 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_
+
+I have explained in 'Science and Hypothesis' whence we derive the notion
+of physical continuity and how that of mathematical 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 readily 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
+symbols, may be written:
+
+ _A_ = _B_, _B_ = _C_, _A_ < _C_.
+
+This would be the formula of the physical continuum, as crude experience
+gives it to us, whence arises an intolerable contradiction that has
+been obviated by the introduction of the mathematical continuum. This is
+a scale of which the steps (commensurable or incommensurable numbers)
+are infinite in number but are exterior to one another, instead of
+encroaching on one another as do the elements of the physical continuum,
+in conformity with the preceding formula.
+
+The physical continuum is, so to speak, a nebula not resolved; the most
+perfect instruments could not attain to its resolution. Doubtless if we
+measured the weights with a good balance instead of judging them by the
+hand, we could distinguish the weight of 11 grams from those of 10 and
+12 grams, and our formula would become:
+
+ _A_ < _B_, _B_ < _C_, _A_ < _C_.
+
+But we should always find between _A_ and _B_ and between _B_ and _C_
+new elements _D_ and _E_, such that
+
+ _A_ = _D_, _D_ = _B_, _A_ < _B_;
+ _B_ = _E_, _E_ = _C_, _B_ < _C_,
+
+and the difficulty would only have receded and the nebula would always
+remain unresolved; the mind alone can resolve it and the mathematical
+continuum it is which is the nebula resolved into stars.
+
+Yet up to this point we have not introduced the notion of the number of
+dimensions. What is meant when we say that a mathematical continuum or
+that a physical continuum has two or three dimensions?
+
+First we must introduce the notion of cut, studying first physical
+continua. We have seen what characterizes the physical continuum. Each
+of the elements of this continuum consists of a manifold of impressions;
+and it may happen either that an element can not be discriminated from
+another element of the same continuum, if this new element corresponds
+to a manifold of impressions not sufficiently different, or, on the
+contrary, that the discrimination is possible; finally it may happen
+that two elements indistinguishable from a third may, nevertheless, be
+distinguished one from the other.
+
+That postulated, if _A_ and _B_ are two distinguishable elements of a
+continuum _C_, a series of elements may be found, E_{1}, E_{2}, ...,
+E_{_n_}, all belonging to this same continuum _C_ and such that
+each of them is indistinguishable from the preceding, that E_{1} is
+indistinguishable from _A_, and E_{_n_} indistinguishable from _B_.
+Therefore we can go from _A_ to _B_ by a continuous route and without
+quitting _C_. If this condition is fulfilled for any two elements _A_
+and _B_ of the continuum _C_, we may say that this continuum _C_ is all
+in one piece. Now let us distinguish certain of the elements of _C_
+which may either be all distinguishable from one another, or themselves
+form one or several continua. The assemblage of the elements thus chosen
+arbitrarily among all those of _C_ will form what I shall call the _cut_
+or the _cuts_.
+
+Take on _C_ any two elements _A_ and _B_. Either we can also find a
+series of elements E_{1}, E_{2}, ..., E_{_n_}, such: (1) that they all
+belong to _C_; (2) that each of them is indistinguishable from the
+following, E_{1} indistinguishable from _A_ and E_{_n_} from _B_; (3)
+_and besides that none of the elements _E_ is indistinguishable from any
+element of the cut_. Or else, on the contrary, in each of the series
+E_{1}, E_{2}, ..., E_{_n_} satisfying the first two conditions, there
+will be an element _E_ indistinguishable from one of the elements of the
+cut. In the first case we can go from _A_ to _B_ by a continuous route
+without quitting _C_ and _without meeting the cuts_; in the second case
+that is impossible.
+
+If then for any two elements _A_ and _B_ of the continuum _C_, it is
+always the first case which presents itself, we shall say that _C_
+remains all in one piece despite the cuts.
+
+Thus, if we choose the cuts in a certain way, otherwise arbitrary, it
+may happen either that the continuum remains all in one piece or that it
+does not remain all in one piece; in this latter hypothesis we shall
+then say that it is _divided_ by the cuts.
+
+It will be noticed that all these definitions are constructed in setting
+out solely from this very simple fact, that two manifolds of impressions
+sometimes can be discriminated, sometimes can not be. That postulated,
+if, to _divide_ a continuum, it suffices to consider as cuts a certain
+number of elements all distinguishable from one another, we say that
+this continuum _is of one dimension_; if, on the contrary, to divide a
+continuum, it is necessary to consider as cuts a system of elements
+themselves forming one or several continua, we shall say that this
+continuum is _of several dimensions_.
+
+If to divide a continuum _C_, cuts forming one or several continua of
+one dimension suffice, we shall say that _C_ is a continuum _of two
+dimensions_; if cuts suffice which form one or several continua of two
+dimensions at most, we shall say that _C_ is a continuum _of three
+dimensions_; and so on.
+
+To justify this definition it is proper to see whether it is in this way
+that geometers introduce the notion of three dimensions at the beginning
+of their works. Now, what do we see? Usually they begin by defining
+surfaces as the boundaries of solids or pieces of space, lines as the
+boundaries of surfaces, points as the boundaries of lines, and they
+affirm that the same procedure can not be pushed further.
+
+This is just the idea given above: to divide space, cuts that are called
+surfaces are necessary; to divide surfaces, cuts that are called lines
+are necessary; to divide lines, cuts that are called points are
+necessary; we can go no further, the point can not be divided, so the
+point is not a continuum. Then lines which can be divided by cuts which
+are not continua will be continua of one dimension; surfaces which can
+be divided by continuous cuts of one dimension will be continua of two
+dimensions; finally, space which can be divided by continuous cuts of
+two dimensions will be a continuum of three dimensions.
+
+Thus the definition I have just given does not differ essentially from
+the usual definitions; I have only endeavored to give it a form
+applicable not to the mathematical continuum, but to the physical
+continuum, which alone is susceptible of representation, and yet to
+retain all its precision. Moreover, we see that this definition applies
+not alone to space; that in all which falls under our senses we find the
+characteristics of the physical continuum, which would allow of the same
+classification; that it would be easy to find there examples of continua
+of four, of five, dimensions, in the sense of the preceding definition;
+such examples occur of themselves to the mind.
+
+I should explain finally, if I had the time, that this science, of which
+I spoke above and to which Riemann gave the name of analysis situs,
+teaches us to make distinctions among continua of the same number of
+dimensions and that the classification of these continua rests also on
+the consideration of cuts.
+
+From this notion has arisen that of the mathematical continuum of
+several dimensions in the same way that the physical continuum of one
+dimension engendered the mathematical continuum of one dimension. The
+formula
+
+ _A_ > _C_, _A_ = _B_, _B_ = _C_,
+
+which summed up the data of crude experience, implied an intolerable
+contradiction. To get free from it, it was necessary to introduce a new
+notion while still respecting the essential characteristics 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 called 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 determination 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 satisfies the definition we
+have just given of the physical continuum of three dimensions. To be
+content with that would be to suppose 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.
+
+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 represent 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 [alpha] is the same
+as the point occupied by the object _B_ at the instant [beta], or still
+better, what this means?
+
+I am seated in my room; an object is placed on my table; during 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 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 position
+with regard to our body, we conclude that the relative position 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 coordinate 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 motion in
+absolute space, in reality I represent to myself my own self motionless
+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 ourselves? 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 reach it. I will
+explain myself. It is not a question of representing the movements
+themselves in space, but solely of representing 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 impressions 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 represent 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 everywhere with us
+a system of axes to which we refer all the points of space and that this
+system of axes seems to be invariably bound to our body. It should be
+noticed that rigorously we could not speak of axes invariably bound to
+the body unless the different parts of this body were themselves
+invariably bound to one 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_
+
+I have shown in 'Science and Hypothesis' the preponderant rôle played by
+the movements of our body in the genesis of the notion of space. For a
+being completely immovable 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 impossible). 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 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 because, 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 movements 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, qualitatively
+different, represent the same image, which has been displaced? It is
+because I _can follow the object with the eye_ and by a displacement of
+the eye, voluntary and accompanied by muscular 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 undergone the _same_ displacement.
+Why? Because in both cases I shall have been able to reestablish the
+primitive sensation, and 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 suppose that the
+sensation of red at the center of the retina is to the sensation of red
+at the border of the retina as that of blue at the center is to that of
+blue at the 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 distinction; before
+knowing geometry I ascertain the fact (experimentally, 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
+suffices me _to ascertain_ it.
+
+However that may be, the explanation is easy. Suppose that 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 produced 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 situation 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 correspond another series _S''_, of muscular sensations which
+generally will be completely different, since other muscles will be
+used.
+
+How am I led to regard these two series _S_ and _S''_ as corresponding
+to the same displacement _AB_? 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: [alpha] and [beta], 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. Consider, 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 [alpha] and [beta]
+correspond to the same displacement, and that _S_ and _S''_ correspond
+also to the same displacement. Why? Simply because _S_ can correct
+[alpha] as well as [beta] and because [alpha] can be corrected by _S''_
+as well as by _S_. And then a question suggests itself:
+
+If I have ascertained that _S_ corrects [alpha] and [beta] and that
+_S''_ corrects [alpha], am I certain that _S''_ likewise corrects
+[beta]? Experiment alone can teach us 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 classifying 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 rôle 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 satisfies the condition in
+question. I knew, in fact, before all experience, 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 rôle 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 movements 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
+common; we should not have been able to disengage from them what 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
+sensations 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 clear 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.
+
+If, therefore, the cut _C'_ has _n_ dimensions, the cut _C_ will have
+_n_ + 1 and the whole of visual space _n_ + 2.
+
+If all the red sensations affecting the same point of the retina were
+regarded as identical, the cut _C'_ reducing to a single element would
+have 0 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 recognize the distance
+of objects. When we seek to analyze this feeling, 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 sensation 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 sensation 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 preceding law they will all be indistinguishable and
+may be regarded as identical. Therefore _C''_ will not be a continuum
+and will have 0 dimension; and as _C''_ divides _C'_ it will thence
+result that _C'_ has one, _C_ two and _the whole visual space three
+dimensions_.
+
+But would it be the same if experience had taught us the contrary and if
+a certain sensation of convergence were not always accompanied by the
+same sensation of 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 _C''_, could nevertheless be distinguished since they would be
+accompanied by two different sensations of accommodation. Therefore
+_C''_ would be in its turn a continuum and would have one dimension (at
+least); then _C'_ would have two, _C_ three and _the whole visual 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 dimensions and that the optician who constructed
+them has given one more dimension to space? 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 explained 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 THREE 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 dimensions and we have
+asked first what is a physical continuum and when it may be said to have
+_n_ dimensions. If we consider different 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 saying 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 [Sigma] 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 [Sigma] 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 [Sigma] which join _A_ to _B_, we shall
+distinguish those of which an element is indistinguishable 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 [Sigma] which join _A_ to _B_
+cut the cut, we shall say that _A_ and _B_ are _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 continuum _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_, _C_ will have 2
+dimensions; if a cut forming a continuum 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
+element 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 different elements of space,
+which we call points, are all alike, and, to apply our definition, it is
+necessary that we know how to distinguish 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 physical
+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 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 ascertain that it may happen that an external change may be
+_corrected_ by an internal change which reestablishes the primitive
+sensations. The external changes, capable of being corrected by an
+internal change are called _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
+body_; the others are called _changes of attitude_.
+
+Now let [alpha] and [beta] be two external changes, [alpha]' and [beta]'
+two internal changes. Suppose that a may be corrected either by [alpha]'
+or by [beta]', and that [alpha]' can correct either [alpha] or [beta];
+experience tells us then that [beta]' can likewise correct [beta]. In
+this case we say that [alpha] and [beta] correspond to the _same_
+displacement and also that [alpha]' and [beta]' correspond to the _same_
+displacement. That postulated, we can imagine a physical continuum which
+we shall call _the continuum or group of displacements_ and which we
+shall define in the following manner. The elements of this continuum
+shall be the internal changes capable of correcting an external change.
+Two of these internal changes [alpha]' and [beta]' shall be regarded as
+indistinguishable: (1) if they are so naturally, that is, if they are
+too close to one another; (2) if [alpha]' is capable of correcting the
+same external change as a third internal change naturally
+indistinguishable from [beta]'. In this second case, they will be, so to
+speak, indistinguishable by convention, I mean by agreeing to disregard
+circumstances which might distinguish them.
+
+Our continuum is now entirely defined, since we know its elements and
+have fixed under what conditions they may be regarded 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 continuum 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 displacements has six dimensions? We know it _by
+experience_.
+
+It would be easy to describe the experiments by which we could arrive
+at this result. It would be seen that in this continuum cuts can be made
+which divide it and which are continua; that these cuts themselves can
+be divided by other cuts of the second order which yet are continua, and
+that this would stop only after cuts of the sixth order which would no
+longer be continua. 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 displacements, we
+have seen, is related to space, and space could be deduced 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 different? Or, in other words, when I say: The object _A_
+occupied at the instant [alpha] the point which the object _B_ occupies
+at the instant [beta], 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 replaced in the same attitude.
+
+I suppose that between the instants [alpha] and [beta] 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 ascertain that at the instant
+[alpha] 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 [beta] other impressions which I attribute to the object _B_ are
+transmitted to me, some by 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 sensations, a sensation purely luminous
+_a_ and a colored sensation _a'_, that _B_ produces in the same way
+simultaneously a luminous sensation _b_ and a colored sensation _b'_,
+that if these different sensations are transmitted to me by the same
+retinal fiber, _a_ is identical with _b_, but that in general the
+colored sensations _a'_ and _b'_ produced by different bodies are
+different? In that case it would be the identity of the sensation _a_
+which accompanies _a'_ with the sensation _b_ which accompanies _b'_,
+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 _b_ +_b'_, 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 I have
+just made: I suppose that I have ascertained that the impressions which
+I attribute to _B_ are transmitted to me at the instant [beta] by the
+same fibers, optic as well as tactile, which, at the instant [alpha],
+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 [beta] is identical with the point occupied by _A_ at the
+instant [alpha].
+
+I have just enunciated two conditions for these points being identical;
+one is relative to sight, the other to touch. Let us consider them
+separately. The first is necessary, but is not sufficient. The second is
+at once necessary and sufficient. A person knowing geometry could easily
+explain this in the following manner: Let _O_ be the point of the retina
+where is formed at the instant [alpha] the image of the body _A_; let
+_M_ be the point of space occupied at the instant [alpha] by this body
+_A_; let _M'_ be the point of space occupied at the instant [beta] by
+the body _B_. For this body _B_ to form its image in _O_, it is not
+necessary that the points _M_ and _M'_ coincide; since vision acts at a
+distance, it suffices for the three points _O_ _M_ _M'_ to be in a
+straight line. This condition that the two objects form their image on
+_O_ 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 [alpha], it
+is because _M_ and _P_ coincide; if _B_ touches my finger at the instant
+[beta], it is because _M'_ and _P_ coincide. Therefore _M_ and _M'_
+coincide. Thus this condition that if _A_ touches my finger at the
+instant [alpha], _B_ touches it at the instant [beta], 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 relative to touch may be
+fulfilled without that of sight being fulfilled and that, on the
+contrary, that of sight can not be fulfilled without 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 distance, 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 [alpha] 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 [alpha]', my _second_ finger receives a
+tactile impression that I attribute likewise to _A_. Afterward, at the
+instant [beta], without my having budged, as my muscular sense tells me,
+this same second finger transmits to me 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 movements. 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 compensate.
+That settled, should I expect that at the instant [beta]', 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 [alpha] and [alpha]', nor the object _B_ between the
+instants [beta] and [beta]'; assume this. At the instant [alpha], 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 _S_ of movements and at the end of this series, at the instant
+[alpha]', 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 [beta] 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 [beta]'. But between the
+instants [beta] and [beta]' I have made the movements _S'_; as these
+movements are the inverse of the movements _S_, they must have for
+effect bringing the first finger in the place of the second. At the
+instant [beta]' this first finger will, therefore, be at _M_; and as the
+object _B_ is likewise at _M_, this object _B_ will touch my first
+finger. To the question 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 [alpha] and [alpha]', or the object _B_ between the instants
+[beta] and [beta]'.
+
+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 impossible? 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 [beta] the point occupied
+by _A_ at the instant [alpha], 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 fingers 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 sensation 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 sensations, we should obtain 'simple
+visual space,' of only two dimensions. On the other hand, consider
+tactile space, limiting ourselves 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 impressions made on
+our retina by _A_ at the instant [alpha] and _B_ at the instant [beta]
+are transmitted by the same retinal fiber, although these impressions
+are qualitatively different? I have suggested a simple hypothesis, while
+adding that other hypotheses, decidedly 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 [alpha], and by the blue object _B_ at the
+instant [beta], 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 different though
+contiguous nervous fibers. What means have I then of knowing that these
+fibers are contiguous? It is probable that we should have none, if 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 sensation of blue at the point _B_ of the retina as
+between the sensation of red at the point _A_ and the sensation of red
+at the point _B_. They have shown us, in fact, that the same movements,
+corresponding 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 [alpha] and the point occupied by _B_ at
+the instant [beta], but only _on one condition_, namely, that I have not
+budged between the instants [alpha] and [beta]. That does not suffice
+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 [alpha] is identical with the point
+occupied by _B_ at the instant [beta]? I suppose that at the instant
+[alpha], the object _A_ was in contact with my first finger and that in
+the same way, at the instant [beta], 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 I have said it sometimes happens
+that we are led to consider two such series _S_ 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 [alpha] and [beta], but so as to feel successively the two
+series of muscular sensations _S_ and _S'_ that I consider inverses, I
+shall still conclude, just as if I had not budged, that the points
+occupied by _A_ at the instant [alpha] and by _B_ at the instant [beta]
+are identical, if I ascertain that my first finger touches _A_ at the
+instant [alpha], and _B_ at the instant [beta].
+
+This solution is not yet completely satisfactory, as one will see. Let
+us see, in fact, how many dimensions it would make us attribute to
+space. I wish to compare the two points occupied by _A_ and _B_ at the
+instants [alpha] and [beta], or (what amounts to the same thing since I
+suppose that my finger touches _A_ at the instant [alpha] and _B_ at the
+instant [beta]) I wish to compare the two points occupied by my finger
+at the two instants [alpha] and [beta]. The sole means I use for this
+comparison is the series [Sigma] of muscular sensations which have
+accompanied the movements of my body between these two instants. The
+different imaginable series [Sigma] form evidently 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 [Sigma] and [Sigma] +
+_S_ + _S'_, when _S_ and _S'_ are inverses one of the other in the sense
+above given to this word; in spite of this agreement, the aggregate of
+distinct series [Sigma] will still form a physical continuum and the
+number of dimensions will be less but still very great.
+
+To each of these series [Sigma] corresponds a point of space; to two
+series [Sigma] and [Sigma]' 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 [Sigma] is identical with [Sigma]';
+(2) if [Sigma]' = [Sigma] + _S_ + _S'_, _S_ and _S'_ being inverses one
+of the other. If in all the other cases we should regard _M_ and _M'_ as
+distinct, the manifold of points would have as many dimensions as the
+aggregate of distinct series [Sigma], that is, much more than three.
+
+For those who already know geometry, the following explanation would be
+easily comprehensible. Among the imaginable 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 [Sigma] and [Sigma] + [sigma], where the series
+[sigma] corresponds to movements where the finger does not budge, the
+aggregate of series will constitute a continuum of three dimensions, but
+that if one regards as distinct two series [Sigma] and [Sigma]' unless
+[Sigma]' = [Sigma] + _S_ + _S'_, _S_ and _S'_ being inverses, 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 C_{0} be the aggregate of all series
+[Sigma]. Let C_{1} be the aggregate of all the series [Sigma], such that
+at the end of corresponding movements the finger is found upon the
+surface _A_, and C_{2} or C_{3} the aggregate of series [Sigma] such
+that at the end the finger is found on _B_, or at _M_. It is clear,
+first that C_{1} will constitute a cut which will divide C_{0}, that
+C_{2} will be a cut which will divide C_{1}, and C_{3} a cut which will
+divide C_2. Thence it results, in accordance with our definitions, that
+if C_{3} is a continuum of _n_ dimensions, C_{0} will be a physical
+continuum of _n_ + 3 dimensions.
+
+Therefore, let [Sigma] and [Sigma]' = [Sigma] + [sigma] be two series
+forming part of C_{3}; 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 [sigma] the finger is at the same point _M_. This series
+[sigma] is therefore one of those which correspond to movements where
+the finger does not budge. If [Sigma] and [Sigma] + [sigma] are not
+regarded as distinct, all the series of C_{3} blend into one; therefore
+C_{3} will have 0 dimension, and C_{0} will have 3, as I wished to
+prove. If, on the contrary, I do not regard [Sigma] and [Sigma] +
+[sigma] as blending (unless [sigma] = _S_ + _S'_, _S_ and _S'_ being
+inverses), it is clear that C_{3} will contain a great number of series
+of distinct sensations; because, without the finger budging, the body
+may take a multitude of different attitudes. Then C_{3} will form a
+continuum and C_{0} will have more than three dimensions, and this also
+I wished to prove.
+
+We who do not yet know geometry can not reason in this way; we can only
+verify. But then a question arises; how, before knowing geometry, have
+we been led to distinguish from the others these series [sigma] where
+the finger does not budge? It is, in fact, only after having made this
+distinction that we could be led to regard [Sigma] and [Sigma] + [sigma]
+as identical, and it is on this condition alone, as we have just seen,
+that we can arrive at space of three dimensions.
+
+We are led to distinguish the series [sigma], because it often happens
+that when we have executed the movements which correspond to these
+series [sigma] of muscular sensations, the tactile sensations which are
+transmitted to us by the nerve of the finger that we have called the
+first finger, persist and are not altered by these movements. Experience
+alone tells us that and it alone could tell us.
+
+If we have distinguished the series of muscular sensations _S_ + _S'_
+formed by the union of two inverse series, it is because they preserve
+the totality of our impressions; if now we distinguish the series
+[sigma], it is because they preserve _certain_ of our impressions. (When
+I say that a series of muscular sensations _S_ 'preserves' one of our
+impressions _A_, I mean that we ascertain that if we feel the impression
+_A_, then the muscular sensations _S_, we _still_ feel the impression
+_A_ _after_ these sensations _S_.)
+
+I have said above it often happens that the series [sigma] do not alter
+the tactile impressions felt by our first finger; I said _often_, I did
+not say _always_. This it is that we express in our ordinary language by
+saying that the tactile impressions would not be altered if the finger
+has not moved, _on the condition_ that _neither has_ the object _A_,
+which was in contact with this finger, moved. Before knowing geometry,
+we could not give this explanation; all we could do is to ascertain that
+the impression often persists, but not always.
+
+But that the impression often continues is enough to make the series
+[sigma] appear remarkable to us, to lead us to put in the same class the
+series [Sigma] and [Sigma] + [sigma], and hence not regard them as
+distinct. Under these conditions we have seen that they will engender a
+physical continuum of three dimensions.
+
+Behold then a space of three dimensions engendered by my first finger.
+Each of my fingers will create one like it. It remains to consider how
+we are led to regard them as identical with visual space, as identical
+with geometric space.
+
+But one reflection before going further; according to the foregoing, we
+know the points of space, or more generally the final situation of our
+body, only by the series of muscular sensations revealing to us the
+movements which have carried us from a certain initial situation to this
+final situation. But it is clear that this final situation will depend,
+on the one hand, upon these movements and, _on the other hand, upon the
+initial situation_ from which we set out. Now these movements are
+revealed to us by our muscular sensations; but nothing tells us the
+initial situation; nothing can distinguish it for us from all the other
+possible situations. This puts well in evidence the essential relativity
+of space.
+
+
+4. _Identity of the Different Spaces_
+
+We are therefore led to compare the two continua _C_ and _C'_
+engendered, for instance, one by my first finger _D_, the other by my
+second finger _D'_. These two physical continua both have three
+dimensions. To each element of the continuum _C_, or, if you prefer, to
+each point of the first tactile space, corresponds a series of muscular
+sensations [Sigma], which carry me from a certain initial situation to a
+certain final situation.[8] Moreover, the same point of this first space
+will correspond to [Sigma] and [Sigma] + [sigma], if [sigma] is a series
+of which we know that it does not make the finger _D_ move.
+
+ [8] In place of saying that we refer space to axes rigidly bound to
+ our body, perhaps it would be better to say, in conformity to
+ what precedes, that we refer it to axes rigidly bound to the
+ initial situation of our body.
+
+Similarly to each element of the continuum _C'_, or to each point of the
+second tactile space, corresponds a series of sensations [Sigma]', and
+the same point will correspond to [Sigma]' and to [Sigma]' + [sigma]',
+if [sigma]' is a series which does not make the finger _D'_ move.
+
+What makes us distinguish the various series designated [sigma] from
+those called [sigma]' is that the first do not alter the tactile
+impressions felt by the finger _D_ and the second preserve those the
+finger _D'_ feels.
+
+Now see what we ascertain: in the beginning my finger _D'_ feels a
+sensation _A'_; I make movements which produce muscular sensations _S_;
+my finger _D_ feels the impression _A_; I make movements which produce a
+series of sensations [sigma]; my finger _D_ continues to feel the
+impression _A_, since this is the characteristic property of the series
+[sigma]; I then make movements which produce the series _S'_ of muscular
+sensations, _inverse_ to _S_ in the sense above given to this word. I
+ascertain then that my finger _D'_ feels anew the impression _A'_. (It
+is of course understood that _S_ has been suitably chosen.)
+
+This means that the series _S_ + [sigma] + _S'_, preserving the tactile
+impressions of the finger _D'_, is one of the series I have called
+[sigma]'. Inversely, if one takes any series [sigma]', _S'_ + [sigma]' +
+_S_ will be one of the series that we call [sigma]'.
+
+Thus if _S_ is suitably chosen, _S_ + [sigma] + _S'_ will be a series
+[sigma]', and by making [sigma] vary in all possible ways, we shall
+obtain all the possible series [sigma]'.
+
+Not yet knowing geometry, we limit ourselves to verifying all that, but
+here is how those who know geometry would explain the fact. In the
+beginning my finger _D'_ is at the point _M_, in contact with the object
+_a_, which makes it feel the impression _A'_. I make the movements
+corresponding to the series _S_; I have said that this series should be
+suitably chosen, I should so make this choice that these movements carry
+the finger _D_ to the point originally occupied by the finger _D'_, that
+is, to the point _M_; this finger _D_ will thus be in contact with the
+object _a_, which will make it feel the impression _A_.
+
+I then make the movements corresponding to the series [sigma]; in these
+movements, by hypothesis, the position of the finger _D_ does not
+change, this finger therefore remains in contact with the object a and
+continues to feel the impression _A_. Finally I make the movements
+corresponding to the series _S'_. As _S'_ is inverse to _S_, these
+movements carry the finger _D'_ to the point previously occupied by the
+finger _D_, that is, to the point _M_. If, as may be supposed, the
+object _a_ has not budged, this finger _D'_ will be in contact with this
+object and will feel anew the impression _A'_.... _Q.E.D._
+
+Let us see the consequences. I consider a series of muscular sensations
+[Sigma]. To this series will correspond a point _M_ of the first tactile
+space. Now take again the two series _S_ and _S'_, inverses of one
+another, of which we have just spoken. To the series _S_ + [Sigma] +
+_S'_ will correspond a point _N_ of the second tactile space, since to
+any series of muscular sensations corresponds, as we have said, a
+point, whether in the first space or in the second.
+
+I am going to consider the two points _N_ and _M_, thus defined, as
+corresponding. What authorizes me so to do? For this correspondence to
+be admissible, it is necessary that if two points _M_ and _M'_,
+corresponding in the first space to two series [Sigma] and [Sigma]', are
+identical, so also are the two corresponding points of the second space
+_N_ and _N'_, that is, the two points which correspond to the two series
+_S_ + [Sigma] + _S'_ and _S_ + [Sigma]' + _S'_. Now we shall see that
+this condition is fulfilled.
+
+First a remark. As _S_ and _S'_ are inverses of one another, we shall
+have _S_ + _S'_ = 0, and consequently _S_ + _S'_ + [Sigma] = [Sigma] +
+_S_ + _S'_ = [Sigma], or again [Sigma] + _S_ + _S'_ + [Sigma]' = [Sigma]
++ [Sigma]'; but it does not follow that we have _S_ + [Sigma] + _S'_ =
+[Sigma]; because, though we have used the addition sign to represent the
+succession of our sensations, it is clear that the order of this
+succession is not indifferent: we can not, therefore, as in ordinary
+addition, invert the order of the terms; to use abridged language, our
+operations are associative, but not commutative.
+
+That fixed, in order that [Sigma] and [Sigma]' should correspond to the
+same point _M_ = _M'_ of the first space, it is necessary and sufficient
+for us to have [Sigma]' = [Sigma] + [sigma]. We shall then have: _S_ +
+[Sigma]' + _S'_ = _S_ + [Sigma] + [sigma] + _S'_ = _S_ + [Sigma] + _S'_
++ _S_ + [sigma] + _S'_.
+
+But we have just ascertained that _S_ + [sigma] + _S'_ was one of the
+series [sigma]'. We shall therefore have: _S_ + [Sigma]' + _S'_ = _S_ +
+[Sigma] + _S'_ + [sigma]', which means that the series _S_ + [Sigma]' +
+_S'_ and _S_ + [Sigma] + _S'_ correspond to the same point _N_ = _N'_ of
+the second space. Q.E.D.
+
+Our two spaces therefore correspond point for point; they can be
+'transformed' one into the other; they are isomorphic. How are we led to
+conclude thence that they are identical?
+
+Consider the two series [sigma] and _S_ + [sigma] + _S'_ = [sigma]'. I
+have said that often, but not always, the series [sigma] preserves the
+tactile impression _A_ felt by the finger _D_; and similarly it often
+happens, but not always, that the series [sigma]' preserves the tactile
+impression _A'_ felt by the finger _D'_. Now I ascertain that it happens
+_very often_ (that is, much more often than what I have just called
+'often') that when the series [sigma] has preserved the impression _A_
+of the finger _D_, the series [sigma]' preserves at the same time the
+impression _A'_ of the finger _D'_; and, inversely, that if the first
+impression is altered, the second is likewise. That happens _very
+often_, but not always.
+
+We interpret this experimental fact by saying that the unknown object
+_a_ which gives the impression _A_ to the finger _D_ is identical with
+the unknown object _a'_ which gives the impression _A'_ to the finger
+_D'_. And in fact when the first object moves, which the disappearance
+of the impression _A_ tells us, the second likewise moves, since the
+impression _A'_ disappears likewise. When the first object remains
+motionless, the second remains motionless. If these two objects are
+identical, as the first is at the point _M_ of the first space and the
+second at the point _N_ of the second space, these two points are
+identical. This is how we are led to regard these two spaces as
+identical; or better, this is what we mean when we say that they are
+identical.
+
+What we have just said of the identity of the two tactile spaces makes
+unnecessary our discussing the question of the identity of tactile space
+and visual space, which could be treated in the same way.
+
+
+5. _Space and Empiricism_
+
+It seems that I am about to be led to conclusions in conformity with
+empiristic ideas. I have, in fact, sought to put in evidence the rôle of
+experience and to analyze the experimental facts which intervene in the
+genesis of space of three dimensions. But whatever may be the importance
+of these facts, there is one thing we must not forget and to which
+besides I have more than once called attention. These experimental facts
+are often verified but not always. That evidently does not mean that
+space has often three dimensions, but not always.
+
+I know well that it is easy to save oneself and that, if the facts do
+not verify, it will be easily explained by saying that the exterior
+objects have moved. If experience succeeds, we say that it teaches us
+about space; if it does not succeed, we hie to exterior objects which we
+accuse of having moved; in other words, if it does not succeed, it is
+given a fillip.
+
+These fillips are legitimate; I do not refuse to admit them; but they
+suffice to tell us that the properties of space are not experimental
+truths, properly so called. If we had wished to verify other laws, we
+could have succeeded also, by giving other analogous fillips. Should we
+not always have been able to justify these fillips by the same reasons?
+One could at most have said to us: 'Your fillips are doubtless
+legitimate, but you abuse them; why move the exterior objects so often?'
+
+To sum up, experience does not prove to us that space has three
+dimensions; it only proves to us that it is convenient to attribute
+three to it, because thus the number of fillips is reduced to a minimum.
+
+I will add that experience brings us into contact only with
+representative space, which is a physical continuum, never with
+geometric space, which is a mathematical continuum. At the very most it
+would appear to tell us that it is convenient to give to geometric space
+three dimensions, so that it may have as many as representative space.
+
+The empiric question may be put under another form. Is it impossible to
+conceive physical phenomena, the mechanical phenomena, for example,
+otherwise than in space of three dimensions? We should thus have an
+objective experimental proof, so to speak, independent of our
+physiology, of our modes of representation.
+
+But it is not so; I shall not here discuss the question completely, I
+shall confine myself to recalling the striking example given us by the
+mechanics of Hertz. You know that the great physicist did not believe in
+the existence of forces, properly so called; he supposed that visible
+material points are subjected to certain invisible bonds which join them
+to other invisible points and that it is the effect of these invisible
+bonds that we attribute to forces.
+
+But that is only a part of his ideas. Suppose a system formed of n
+material points, visible or not; that will give in all 3_n_ coordinates;
+let us regard them as the coordinates of a _single_ point in space of
+3_n_ dimensions. This single point would be constrained to remain upon a
+surface (of any number of dimensions < 3_n_) in virtue of the bonds of
+which we have just spoken; to go on this surface from one point to
+another, it would always take the shortest way; this would be the
+single principle which would sum up all mechanics.
+
+Whatever should be thought of this hypothesis, whether we be allured by
+its simplicity, or repelled by its artificial character, the simple fact
+that Hertz was able to conceive it, and to regard it as more convenient
+than our habitual hypotheses, suffices to prove that our ordinary ideas,
+and, in particular, the three dimensions of space, are in no wise
+imposed upon mechanics with an invincible force.
+
+
+6. _Mind and Space_
+
+Experience, therefore, has played only a single rôle, it has served as
+occasion. But this rôle was none the less very important; and I have
+thought it necessary to give it prominence. This rôle would have been
+useless if there existed an _a priori_ form imposing itself upon our
+sensitivity, and which was space of three dimensions.
+
+Does this form exist, or, if you choose, can we represent to ourselves
+space of more than three dimensions? And first what does this question
+mean? In the true sense of the word, it is clear that we can not
+represent to ourselves space of four, nor space of three, dimensions; we
+can not first represent them to ourselves empty, and no more can we
+represent to ourselves an object either in space of four, or in space of
+three, dimensions: (1) Because these spaces are both infinite and we can
+not represent to ourselves a figure _in_ space, that is, the part _in_
+the whole, without representing the whole, and that is impossible,
+because it is infinite; (2) because these spaces are both mathematical
+continua, and we can represent to ourselves only the physical continuum;
+(3) because these spaces are both homogeneous, and the frames in which
+we enclose our sensations, being limited, can not be homogeneous.
+
+Thus the question put can only be understood in one way; is it possible
+to imagine that, the results of the experiences related above having
+been different, we might have been led to attribute to space more than
+three dimensions; to imagine, for instance, that the sensation of
+accommodation might not be constantly in accord with the sensation of
+convergence of the eyes; or indeed that the experiences of which we
+have spoken in § 2, and of which we express the result by saying 'that
+touch does not operate at a distance,' might have led us to an inverse
+conclusion.
+
+And then yes evidently that is possible; from the moment one imagines an
+experience, one imagines just thereby the two contrary results it may
+give. That is possible, but that is difficult, because we have to
+overcome a multitude of associations of ideas, which are the fruit of a
+long personal experience and of the still longer experience of the race.
+Is it these associations (or at least those of them that we have
+inherited from our ancestors), which constitute this _a priori_ form of
+which it is said that we have pure intuition? Then I do not see why one
+should declare it refractory to analysis and should deny me the right of
+investigating its origin.
+
+When it is said that our sensations are 'extended' only one thing can be
+meant, that is that they are always associated with the idea of certain
+muscular sensations, corresponding to the movements which enable us to
+reach the object which causes them, which enable us, in other words, to
+defend ourselves against it. And it is just because this association is
+useful for the defense of the organism, that it is so old in the history
+of the species and that it seems to us indestructible. Nevertheless, it
+is only an association and we can conceive that it may be broken; so
+that we may not say that sensation can not enter consciousness without
+entering in space, but that in fact it does not enter consciousness
+without entering in space, which means, without being entangled in this
+association.
+
+No more can I understand one's saying that the idea of time is logically
+subsequent to space, since we can represent it to ourselves only under
+the form of a straight line; as well say that time is logically
+subsequent to the cultivation of the prairies, since it is usually
+represented armed with a scythe. That one can not represent to himself
+simultaneously the different parts of time, goes without saying, since
+the essential character of these parts is precisely not to be
+simultaneous. That does not mean that we have not the intuition of time.
+So far as that goes, no more should we have that of space, because
+neither can we represent it, in the proper sense of the word, for the
+reasons I have mentioned. What we represent to ourselves under the name
+of straight is a crude image which as ill resembles the geometric
+straight as it does time itself.
+
+Why has it been said that every attempt to give a fourth dimension to
+space always carries this one back to one of the other three? It is easy
+to understand. Consider our muscular sensations and the 'series' they
+may form. In consequence of numerous experiences, the ideas of these
+series are associated together in a very complex woof, our series are
+_classed_. Allow me, for convenience of language, to express my thought
+in a way altogether crude and even inexact by saying that our series of
+muscular sensations are classed in three classes corresponding to the
+three dimensions of space. Of course this classification is much more
+complicated than that, but that will suffice to make my reasoning
+understood. If I wish to imagine a fourth dimension, I shall suppose
+another series of muscular sensations, making part of a fourth class.
+But as _all_ my muscular sensations have already been classed in one of
+the three pre-existent classes, I can only represent to myself a series
+belonging to one of these three classes, so that my fourth dimension is
+carried back to one of the other three.
+
+What does that prove? This: that it would have been necessary first to
+destroy the old classification and replace it by a new one in which the
+series of muscular sensations should have been distributed into four
+classes. The difficulty would have disappeared.
+
+It is presented sometimes under a more striking form. Suppose I am
+enclosed in a chamber between the six impassable boundaries formed by
+the four walls, the floor and the ceiling; it will be impossible for me
+to get out and to imagine my getting out. Pardon, can you not imagine
+that the door opens, or that two of these walls separate? But of course,
+you answer, one must suppose that these walls remain immovable. Yes, but
+it is evident that I have the right to move; and then the walls that we
+suppose absolutely at rest will be in motion with regard to me. Yes, but
+such a relative motion can not be arbitrary; when objects are at rest,
+their relative motion with regard to any axes is that of a rigid solid;
+now, the apparent motions that you imagine are not in conformity with
+the laws of motion of a rigid solid. Yes, but it is experience which has
+taught us the laws of motion of a rigid solid; nothing would prevent our
+_imagining_ them different. To sum up, for me to imagine that I get out
+of my prison, I have only to imagine that the walls seem to open, when I
+move.
+
+I believe, therefore, that if by space is understood a mathematical
+continuum of three dimensions, were it otherwise amorphous, it is the
+mind which constructs it, but it does not construct it out of nothing;
+it needs materials and models. These materials, like these models,
+preexist within it. But there is not a single model which is imposed
+upon it; it has _choice_; it may choose, for instance, between space of
+four and space of three dimensions. What then is the rôle of experience?
+It gives the indications following which the choice is made.
+
+Another thing: whence does space get its quantitative character? It
+comes from the rôle which the series of muscular sensations play in its
+genesis. These are series which may _repeat themselves_, and it is from
+their repetition that number comes; it is because they can repeat
+themselves indefinitely that space is infinite. And finally we have
+seen, at the end of section 3, that it is also because of this that
+space is relative. So it is repetition which has given to space its
+essential characteristics; now, repetition supposes time; this is enough
+to tell that time is logically anterior to space.
+
+7. _Rôle of the Semicircular Canals_
+
+I have not hitherto spoken of the rôle of certain organs to which the
+physiologists attribute with reason a capital importance, I mean the
+semicircular canals. Numerous experiments have sufficiently shown that
+these canals are necessary to our sense of orientation; but the
+physiologists are not entirely in accord; two opposing theories have
+been proposed, that of Mach-Delage and that of M. de Cyon.
+
+M. de Cyon is a physiologist who has made his name illustrious by
+important discoveries on the innervation of the heart; I can not,
+however, agree with his ideas on the question before us. Not being a
+physiologist, I hesitate to criticize the experiments he has directed
+against the adverse theory of Mach-Delage; it seems to me, however, that
+they are not convincing, because in many of them the _total_ pressure
+was made to vary in one of the canals, while, physiologically, what
+varies is the _difference_ between the pressures on the two extremities
+of the canal; in others the organs were subjected to profound lesions,
+which must alter their functions.
+
+Besides, this is not important; the experiments, if they were
+irreproachable, might be convincing against the old theory. They would
+not be convincing _for_ the new theory. In fact, if I have rightly
+understood the theory, my explaining it will be enough for one to
+understand that it is impossible to conceive of an experiment confirming
+it.
+
+The three pairs of canals would have as sole function to tell us that
+space has three dimensions. Japanese mice have only two pairs of canals;
+they believe, it would seem, that space has only two dimensions, and
+they manifest this opinion in the strangest way; they put themselves in
+a circle, and, so ordered, they spin rapidly around. The lampreys,
+having only one pair of canals, believe that space has only one
+dimension, but their manifestations are less turbulent.
+
+It is evident that such a theory is inadmissible. The sense-organs are
+designed to tell us of _changes_ which happen in the exterior world. We
+could not understand why the Creator should have given us organs
+destined to cry without cease: Remember that space has three dimensions,
+since the number of these three dimensions is not subject to change.
+
+We must, therefore, come back to the theory of Mach-Delage. What the
+nerves of the canals can tell us is the difference of pressure on the
+two extremities of the same canal, and thereby: (1) the direction of the
+vertical with regard to three axes rigidly bound to the head; (2) the
+three components of the acceleration of translation of the center of
+gravity of the head; (3) the centrifugal forces developed by the
+rotation of the head; (4) the acceleration of the motion of rotation of
+the head.
+
+It follows from the experiments of M. Delage that it is this last
+indication which is much the most important; doubtless because the
+nerves are less sensible to the difference of pressure itself than to
+the brusque variations of this difference. The first three indications
+may thus be neglected.
+
+Knowing the acceleration of the motion of rotation of the head at each
+instant, we deduce from it, by an unconscious integration, the final
+orientation of the head, referred to a certain initial orientation taken
+as origin. The circular canals contribute, therefore, to inform us of
+the movements that we have executed, and that on the same ground as the
+muscular sensations. When, therefore, above we speak of the series _S_
+or of the series [Sigma], we should say, not that these were series of
+muscular sensations alone, but that they were series at the same time of
+muscular sensations and of sensations due to the semicircular canals.
+Apart from this addition, we should have nothing to change in what
+precedes.
+
+In the series _S_ and [Sigma], these sensations of the semicircular
+canals evidently hold a very important place. Yet alone they would not
+suffice, because they can tell us only of the movements of the head;
+they tell us nothing of the relative movements of the body or of the
+members in regard to the head. And more, it seems that they tell us only
+of the rotations of the head and not of the translations it may
+undergo.
+
+
+
+
+PART II
+
+
+THE PHYSICAL SCIENCES
+
+
+
+
+CHAPTER V
+
+ANALYSIS AND PHYSICS
+
+
+I
+
+You have doubtless often been asked of what good is mathematics and
+whether these delicate constructions entirely mind-made are not
+artificial and born of our caprice.
+
+Among those who put this question I should make a distinction; practical
+people ask of us only the means of money-making. These merit no reply;
+rather would it be proper to ask of them what is the good of
+accumulating so much wealth and whether, to get time to acquire it, we
+are to neglect art and science, which alone give us souls capable of
+enjoying it, 'and for life's sake to sacrifice all reasons for living.'
+
+Besides, a science made solely in view of applications is impossible;
+truths are fecund only if bound together. If we devote ourselves solely
+to those truths whence we expect an immediate result, the intermediary
+links are wanting and there will no longer be a chain.
+
+The men most disdainful of theory get from it, without suspecting it,
+their daily bread; deprived of this food, progress would quickly cease,
+and we should soon congeal into the immobility of old China.
+
+But enough of uncompromising practicians! Besides these, there are those
+who are only interested in nature and who ask us if we can enable them
+to know it better.
+
+To answer these, we have only to show them the two monuments already
+rough-hewn, Celestial Mechanics and Mathematical Physics.
+
+They would doubtless concede that these structures are well worth the
+trouble they have cost us. But this is not enough. Mathematics has a
+triple aim. It must furnish an instrument for the study of nature. But
+that is not all: it has a philosophic aim and, I dare maintain, an
+esthetic aim. It must aid the philosopher to fathom the notions of
+number, of space, of time. And above all, its adepts find therein
+delights analogous to those given by painting and music. They admire the
+delicate harmony of numbers and forms; they marvel when a new discovery
+opens to them an unexpected perspective; and has not the joy they thus
+feel the esthetic character, even though the senses take no part
+therein? Only a privileged few are called to enjoy it fully, it is true,
+but is not this the case for all the noblest arts?
+
+This is why I do not hesitate to say that mathematics deserves to be
+cultivated for its own sake, and the theories inapplicable to physics as
+well as the others. Even if the physical aim and the esthetic aim were
+not united, we ought not to sacrifice either.
+
+But more: these two aims are inseparable and the best means of attaining
+one is to aim at the other, or at least never to lose sight of it. This
+is what I am about to try to demonstrate in setting forth the nature of
+the relations between the pure science and its applications.
+
+The mathematician should not be for the physicist a mere purveyor of
+formulas; there should be between them a more intimate collaboration.
+Mathematical physics and pure analysis are not merely adjacent powers,
+maintaining good neighborly relations; they mutually interpenetrate and
+their spirit is the same. This will be better understood when I have
+shown what physics gets from mathematics and what mathematics, in
+return, borrows from physics.
+
+
+II
+
+The physicist can not ask of the analyst to reveal to him a new truth;
+the latter could at most only aid him to foresee it. It is a long time
+since one still dreamt of forestalling experiment, or of constructing
+the entire world on certain premature hypotheses. Since all those
+constructions in which one yet took a naïve delight it is an age, to-day
+only their ruins remain.
+
+All laws are therefore deduced from experiment; but to enunciate them, a
+special language is needful; ordinary language is too poor, it is
+besides too vague, to express relations so delicate, so rich, and so
+precise.
+
+This therefore is one reason why the physicist can not do without
+mathematics; it furnishes him the only language he can speak. And a
+well-made language is no indifferent thing; not to go beyond physics,
+the unknown man who invented the word _heat_ devoted many generations to
+error. Heat has been treated as a substance, simply because it was
+designated by a substantive, and it has been thought indestructible.
+
+On the other hand, he who invented the word _electricity_ had the
+unmerited good fortune to implicitly endow physics with a _new_ law,
+that of the conservation of electricity, which, by a pure chance, has
+been found exact, at least until now.
+
+Well, to continue the simile, the writers who embellish a language, who
+treat it as an object of art, make of it at the same time a more supple
+instrument, more apt for rendering shades of thought.
+
+We understand, then, how the analyst, who pursues a purely esthetic aim,
+helps create, just by that, a language more fit to satisfy the
+physicist.
+
+But this is not all: law springs from experiment, but not immediately.
+Experiment is individual, the law deduced from it is general; experiment
+is only approximate, the law is precise, or at least pretends to be.
+Experiment is made under conditions always complex, the enunciation of
+the law eliminates these complications. This is what is called
+'correcting the systematic errors.'
+
+In a word, to get the law from experiment, it is necessary to
+generalize; this is a necessity imposed upon the most circumspect
+observer. But how generalize? Every particular truth may evidently be
+extended in an infinity of ways. Among these thousand routes opening
+before us, it is necessary to make a choice, at least provisional; in
+this choice, what shall guide us?
+
+It can only be analogy. But how vague is this word! Primitive man knew
+only crude analogies, those which strike the senses, those of colors or
+of sounds. He never would have dreamt of likening light to radiant
+heat.
+
+What has taught us to know the true, profound analogies, those the eyes
+do not see but reason divines?
+
+It is the mathematical spirit, which disdains matter to cling only to
+pure form. This it is which has taught us to give the same name to
+things differing only in material, to call by the same name, for
+instance, the multiplication of quaternions and that of whole numbers.
+
+If quaternions, of which I have just spoken, had not been so promptly
+utilized by the English physicists, many persons would doubtless see in
+them only a useless fancy, and yet, in teaching us to liken what
+appearances separate, they would have already rendered us more apt to
+penetrate the secrets of nature.
+
+Such are the services the physicist should expect of analysis; but for
+this science to be able to render them, it must be cultivated in the
+broadest fashion without immediate expectation of utility--the
+mathematician must have worked as artist.
+
+What we ask of him is to help us to see, to discern our way in the
+labyrinth which opens before us. Now, he sees best who stands highest.
+Examples abound, and I limit myself to the most striking.
+
+The first will show us how to change the language suffices to reveal
+generalizations not before suspected.
+
+When Newton's law has been substituted for Kepler's we still know only
+elliptic motion. Now, in so far as concerns this motion, the two laws
+differ only in form; we pass from one to the other by a simple
+differentiation. And yet from Newton's law may be deduced by an
+immediate generalization all the effects of perturbations and the whole
+of celestial mechanics. If, on the other hand, Kepler's enunciation had
+been retained, no one would ever have regarded the orbits of the
+perturbed planets, those complicated curves of which no one has ever
+written the equation, as the natural generalizations of the ellipse. The
+progress of observations would only have served to create belief in
+chaos.
+
+The second example is equally deserving of consideration.
+
+When Maxwell began his work, the laws of electro-dynamics admitted up to
+his time accounted for all the known facts. It was not a new experiment
+which came to invalidate them. But in looking at them under a new bias,
+Maxwell saw that the equations became more symmetrical when a term was
+added, and besides, this term was too small to produce effects
+appreciable with the old methods.
+
+You know that Maxwell's _a priori_ views awaited for twenty years an
+experimental confirmation; or, if you prefer, Maxwell was twenty years
+ahead of experiment. How was this triumph obtained?
+
+It was because Maxwell was profoundly steeped in the sense of
+mathematical symmetry; would he have been so, if others before him had
+not studied this symmetry for its own beauty?
+
+It was because Maxwell was accustomed to 'think in vectors,' and yet it
+was through the theory of imaginaries (neomonics) that vectors were
+introduced into analysis. And those who invented imaginaries hardly
+suspected the advantage which would be obtained from them for the study
+of the real world, of this the name given them is proof sufficient.
+
+In a word, Maxwell was perhaps not an able analyst, but this ability
+would have been for him only a useless and bothersome baggage. On the
+other hand, he had in the highest degree the intimate sense of
+mathematical analogies. Therefore it is that he made good mathematical
+physics.
+
+Maxwell's example teaches us still another thing.
+
+How should the equations of mathematical physics be treated? Should we
+simply deduce all the consequences and regard them as intangible
+realities? Far from it; what they should teach us above all is what can
+and what should be changed. It is thus that we get from them something
+useful.
+
+The third example goes to show us how we may perceive mathematical
+analogies between phenomena which have physically no relation either
+apparent or real, so that the laws of one of these phenomena aid us to
+divine those of the other.
+
+The very same equation, that of Laplace, is met in the theory of
+Newtonian attraction, in that of the motion of liquids, in that of the
+electric potential, in that of magnetism, in that of the propagation of
+heat and in still many others. What is the result? These theories seem
+images copied one from the other; they are mutually illuminating,
+borrowing their language from each other; ask electricians if they do
+not felicitate themselves on having invented the phrase flow of force,
+suggested by hydrodynamics and the theory of heat.
+
+Thus mathematical analogies not only may make us foresee physical
+analogies, but besides do not cease to be useful when these latter fail.
+
+To sum up, the aim of mathematical physics is not only to facilitate for
+the physicist the numerical calculation of certain constants or the
+integration of certain differential equations. It is besides, it is
+above all, to reveal to him the hidden harmony of things in making him
+see them in a new way.
+
+Of all the parts of analysis, the most elevated, the purest, so to
+speak, will be the most fruitful in the hands of those who know how to
+use them.
+
+
+III
+
+Let us now see what analysis owes to physics.
+
+It would be necessary to have completely forgotten the history of
+science not to remember that the desire to understand nature has had on
+the development of mathematics the most constant and happiest influence.
+
+In the first place the physicist sets us problems whose solution he
+expects of us. But in proposing them to us, he has largely paid us in
+advance for the service we shall render him, if we solve them.
+
+If I may be allowed to continue my comparison with the fine arts, the
+pure mathematician who should forget the existence of the exterior world
+would be like a painter who knew how to harmoniously combine colors and
+forms, but who lacked models. His creative power would soon be
+exhausted.
+
+The combinations which numbers and symbols may form are an infinite
+multitude. In this multitude how shall we choose those which are worthy
+to fix our attention? Shall we let ourselves be guided solely by our
+caprice? This caprice, which itself would besides soon tire, would
+doubtless carry us very far apart and we should quickly cease to
+understand each other.
+
+But this is only the smaller side of the question. Physics will
+doubtless prevent our straying, but it will also preserve us from a
+danger much more formidable; it will prevent our ceaselessly going
+around in the same circle.
+
+History proves that physics has not only forced us to choose among
+problems which came in a crowd; it has imposed upon us such as we should
+without it never have dreamed of. However varied may be the imagination
+of man, nature is still a thousand times richer. To follow her we must
+take ways we have neglected, and these paths lead us often to summits
+whence we discover new countries. What could be more useful!
+
+It is with mathematical symbols as with physical realities; it is in
+comparing the different aspects of things that we are able to comprehend
+their inner harmony, which alone is beautiful and consequently worthy of
+our efforts.
+
+The first example I shall cite is so old we are tempted to forget it; it
+is nevertheless the most important of all.
+
+The sole natural object of mathematical thought is the whole number. It
+is the external world which has imposed the continuum upon us, which we
+doubtless have invented, but which it has forced us to invent. Without
+it there would be no infinitesimal analysis; all mathematical science
+would reduce itself to arithmetic or to the theory of substitutions.
+
+On the contrary, we have devoted to the study of the continuum almost
+all our time and all our strength. Who will regret it; who will think
+that this time and this strength have been wasted? Analysis unfolds
+before us infinite perspectives that arithmetic never suspects; it shows
+us at a glance a majestic assemblage whose array is simple and
+symmetric; on the contrary, in the theory of numbers, where reigns the
+unforeseen, the view is, so to speak, arrested at every step.
+
+Doubtless it will be said that outside of the whole number there is no
+rigor, and consequently no mathematical truth; that the whole number
+hides everywhere, and that we must strive to render transparent the
+screens which cloak it, even if to do so we must resign ourselves to
+interminable repetitions. Let us not be such purists and let us be
+grateful to the continuum, which, if _all_ springs from the whole
+number, was alone capable of making _so much_ proceed therefrom.
+
+Need I also recall that M. Hermite obtained a surprising advantage from
+the introduction of continuous variables into the theory of numbers?
+Thus the whole number's own domain is itself invaded, and this invasion
+has established order where disorder reigned.
+
+See what we owe to the continuum and consequently to physical nature.
+
+Fourier's series is a precious instrument of which analysis makes
+continual use, it is by this means that it has been able to represent
+discontinuous functions; Fourier invented it to solve a problem of
+physics relative to the propagation of heat. If this problem had not
+come up naturally, we should never have dared to give discontinuity its
+rights; we should still long have regarded continuous functions as the
+only true functions.
+
+The notion of function has been thereby considerably extended and has
+received from some logician-analysts an unforeseen development. These
+analysts have thus adventured into regions where reigns the purest
+abstraction and have gone as far away as possible from the real world.
+Yet it is a problem of physics which has furnished them the occasion.
+
+After Fourier's series, other analogous series have entered the domain
+of analysis; they have entered by the same door; they have been imagined
+in view of applications.
+
+The theory of partial differential equations of the second order has an
+analogous history. It has been developed chiefly by and for physics. But
+it may take many forms, because such an equation does not suffice to
+determine the unknown function, it is necessary to adjoin to it
+complementary conditions which are called conditions at the limits;
+whence many different problems.
+
+If the analysts had abandoned themselves to their natural tendencies,
+they would never have known but one, that which Madame Kovalevski has
+treated in her celebrated memoir. But there are a multitude of others
+which they would have ignored. Each of the theories of physics, that of
+electricity, that of heat, presents us these equations under a new
+aspect. It may, therefore, be said that without these theories we should
+not know partial differential equations.
+
+It is needless to multiply examples. I have given enough to be able to
+conclude: when physicists ask of us the solution of a problem, it is not
+a duty-service they impose upon us, it is on the contrary we who owe
+them thanks.
+
+
+IV
+
+But this is not all; physics not only gives us the occasion to solve
+problems; it aids us to find the means thereto, and that in two ways. It
+makes us foresee the solution; it suggests arguments to us.
+
+I have spoken above of Laplace's equation which is met in a multitude of
+diverse physical theories. It is found again in geometry, in the theory
+of conformal representation and in pure analysis, in that of
+imaginaries.
+
+In this way, in the study of functions of complex variables, the
+analyst, alongside of the geometric image, which is his usual
+instrument, finds many physical images which he may make use of with the
+same success. Thanks to these images, he can see at a glance what pure
+deduction would show him only successively. He masses thus the separate
+elements of the solution, and by a sort of intuition divines before
+being able to demonstrate.
+
+To divine before demonstrating! Need I recall that thus have been made
+all the important discoveries? How many are the truths that physical
+analogies permit us to present and that we are not in condition to
+establish by rigorous reasoning!
+
+For example, mathematical physics introduces a great number of
+developments in series. No one doubts that these developments converge;
+but the mathematical certitude is lacking. These are so many conquests
+assured for the investigators who shall come after us.
+
+On the other hand, physics furnishes us not alone solutions; it
+furnishes us besides, in a certain measure, arguments. It will suffice
+to recall how Felix Klein, in a question relative to Riemann surfaces,
+has had recourse to the properties of electric currents.
+
+It is true, the arguments of this species are not rigorous, in the sense
+the analyst attaches to this word. And here a question arises: How can a
+demonstration not sufficiently rigorous for the analyst suffice for the
+physicist? It seems there can not be two rigors, that rigor is or is
+not, and that, where it is not there can not be deduction.
+
+This apparent paradox will be better understood by recalling under what
+conditions number is applied to natural phenomena. Whence come in
+general the difficulties encountered in seeking rigor? We strike them
+almost always in seeking to establish that some quantity tends to some
+limit, or that some function is continuous, or that it has a derivative.
+
+Now the numbers the physicist measures by experiment are never known
+except approximately; and besides, any function always differs as little
+as you choose from a discontinuous function, and at the same time it
+differs as little as you choose from a continuous function. The
+physicist may, therefore, at will suppose that the function studied is
+continuous, or that it is discontinuous; that it has or has not a
+derivative; and may do so without fear of ever being contradicted,
+either by present experience or by any future experiment. We see that
+with such liberty he makes sport of difficulties which stop the analyst.
+He may always reason as if all the functions which occur in his
+calculations were entire polynomials.
+
+Thus the sketch which suffices for physics is not the deduction which
+analysis requires. It does not follow thence that one can not aid in
+finding the other. So many physical sketches have already been
+transformed into rigorous demonstrations that to-day this transformation
+is easy. There would be plenty of examples did I not fear in citing them
+to tire the reader.
+
+I hope I have said enough to show that pure analysis and mathematical
+physics may serve one another without making any sacrifice one to the
+other, and that each of these two sciences should rejoice in all which
+elevates its associate.
+
+
+
+
+CHAPTER VI
+
+ASTRONOMY
+
+
+Governments and parliaments must find that astronomy is one of the
+sciences which cost most dear: the least instrument costs hundreds of
+thousands of dollars, the least observatory costs millions; each eclipse
+carries with it supplementary appropriations. And all that for stars
+which are so far away, which are complete strangers to our electoral
+contests, and in all probability will never take any part in them. It
+must be that our politicians have retained a remnant of idealism, a
+vague instinct for what is grand; truly, I think they have been
+calumniated; they should be encouraged and shown that this instinct does
+not deceive them, that they are not dupes of that idealism.
+
+We might indeed speak to them of navigation, of which no one can
+underestimate the importance, and which has need of astronomy. But this
+would be to take the question by its smaller side.
+
+Astronomy is useful because it raises us above ourselves; it is useful
+because it is grand; that is what we should say. It shows us how small
+is man's body, how great his mind, since his intelligence can embrace
+the whole of this dazzling immensity, where his body is only an obscure
+point, and enjoy its silent harmony. Thus we attain the consciousness of
+our power, and this is something which can not cost too dear, since this
+consciousness makes us mightier.
+
+But what I should wish before all to show is, to what point astronomy
+has facilitated the work of the other sciences, more directly useful,
+since it has given us a soul capable of comprehending nature.
+
+Think how diminished humanity would be if, under heavens constantly
+overclouded, as Jupiter's must be, it had forever remained ignorant of
+the stars. Do you think that in such a world we should be what we are? I
+know well that under this somber vault we should have been deprived of
+the light of the sun, necessary to organisms like those which inhabit
+the earth. But if you please, we shall assume that these clouds are
+phosphorescent and emit a soft and constant light. Since we are making
+hypotheses, another will cost no more. Well! I repeat my question: Do
+you think that in such a world we should be what we are?
+
+The stars send us not only that visible and gross light which strikes
+our bodily eyes, but from them also comes to us a light far more subtle,
+which illuminates our minds and whose effects I shall try to show you.
+You know what man was on the earth some thousands of years ago, and what
+he is to-day. Isolated amidst a nature where everything was a mystery to
+him, terrified at each unexpected manifestation of incomprehensible
+forces, he was incapable of seeing in the conduct of the universe
+anything but caprice; he attributed all phenomena to the action of a
+multitude of little genii, fantastic and exacting, and to act on the
+world he sought to conciliate them by means analogous to those employed
+to gain the good graces of a minister or a deputy. Even his failures did
+not enlighten him, any more than to-day a beggar refused is discouraged
+to the point of ceasing to beg.
+
+To-day we no longer beg of nature; we command her, because we have
+discovered certain of her secrets and shall discover others each day. We
+command her in the name of laws she can not challenge, because they are
+hers; these laws we do not madly ask her to change, we are the first to
+submit to them. Nature can only be governed by obeying her.
+
+What a change must our souls have undergone to pass from the one state
+to the other! Does any one believe that, without the lessons of the
+stars, under the heavens perpetually overclouded that I have just
+supposed, they would have changed so quickly? Would the metamorphosis
+have been possible, or at least would it not have been much slower?
+
+And first of all, astronomy it is which taught that there are laws. The
+Chaldeans, who were the first to observe the heavens with some
+attention, saw that this multitude of luminous points is not a confused
+crowd wandering at random, but rather a disciplined army. Doubtless the
+rules of this discipline escaped them, but the harmonious spectacle of
+the starry night sufficed to give them the impression of regularity,
+and that was in itself already a great thing. Besides, these rules were
+discerned by Hipparchus, Ptolemy, Copernicus, Kepler, one after another,
+and finally, it is needless to recall that Newton it was who enunciated
+the oldest, the most precise, the most simple, the most general of all
+natural laws.
+
+And then, taught by this example, we have seen our little terrestrial
+world better and, under the apparent disorder, there also we have found
+again the harmony that the study of the heavens had revealed to us. It
+also is regular, it also obeys immutable laws, but they are more
+complicated, in apparent conflict one with another, and an eye untrained
+by other sights would have seen there only chaos and the reign of chance
+or caprice. If we had not known the stars, some bold spirits might
+perhaps have sought to foresee physical phenomena; but their failures
+would have been frequent, and they would have excited only the derision
+of the vulgar; do we not see, that even in our day the meteorologists
+sometimes deceive themselves, and that certain persons are inclined to
+laugh at them.
+
+How often would the physicists, disheartened by so many checks, have
+fallen into discouragement, if they had not had, to sustain their
+confidence, the brilliant example of the success of the astronomers!
+This success showed them that nature obeys laws; it only remained to
+know what laws; for that they only needed patience, and they had the
+right to demand that the sceptics should give them credit.
+
+This is not all: astronomy has not only taught us that there are laws,
+but that from these laws there is no escape, that with them there is no
+possible compromise. How much time should we have needed to comprehend
+that fact, if we had known only the terrestrial world, where each
+elemental force would always seem to us in conflict with other forces?
+Astronomy has taught us that the laws are infinitely precise, and that
+if those we enunciate are approximative, it is because we do not know
+them well. Aristotle, the most scientific mind of antiquity, still
+accorded a part to accident, to chance, and seemed to think that the
+laws of nature, at least here below, determine only the large features
+of phenomena. How much has the ever-increasing precision of
+astronomical predictions contributed to correct such an error, which
+would have rendered nature unintelligible!
+
+But are these laws not local, varying in different places, like those
+which men make; does not that which is truth in one corner of the
+universe, on our globe, for instance, or in our little solar system,
+become error a little farther away? And then could it not be asked
+whether laws depending on space do not also depend upon time, whether
+they are not simple habitudes, transitory, therefore, and ephemeral?
+Again it is astronomy that answers this question. Consider the double
+stars; all describe conics; thus, as far as the telescope carries, it
+does not reach the limits of the domain which obeys Newton's law.
+
+Even the simplicity of this law is a lesson for us; how many complicated
+phenomena are contained in the two lines of its enunciation; persons who
+do not understand celestial mechanics may form some idea of it at least
+from the size of the treatises devoted to this science; and then it may
+be hoped that the complication of physical phenomena likewise hides from
+us some simple cause still unknown.
+
+It is therefore astronomy which has shown us what are the general
+characteristics of natural laws; but among these characteristics there
+is one, the most subtle and the most important of all, which I shall ask
+leave to stress.
+
+How was the order of the universe understood by the ancients; for
+instance, by Pythagoras, Plato or Aristotle? It was either an immutable
+type fixed once for all, or an ideal to which the world sought to
+approach. Kepler himself still thought thus when, for instance, he
+sought whether the distances of the planets from the sun had not some
+relation to the five regular polyhedrons. This idea contained nothing
+absurd, but it was sterile, since nature is not so made. Newton has
+shown us that a law is only a necessary relation between the present
+state of the world and its immediately subsequent state. All the other
+laws since discovered are nothing else; they are in sum, differential
+equations; but it is astronomy which furnished the first model for them,
+without which we should doubtless long have erred.
+
+Astronomy has also taught us to set at naught appearances. The day
+Copernicus proved that what was thought the most stable was in motion,
+that what was thought moving was fixed, he showed us how deceptive could
+be the infantile reasonings which spring directly from the immediate
+data of our senses. True, his ideas did not easily triumph, but since
+this triumph there is no longer a prejudice so inveterate that we can
+not shake it off. How can we estimate the value of the new weapon thus
+won?
+
+The ancients thought everything was made for man, and this illusion must
+be very tenacious, since it must ever be combated. Yet it is necessary
+to divest oneself of it; or else one will be only an eternal myope,
+incapable of seeing the truth. To comprehend nature one must be able to
+get out of self, so to speak, and to contemplate her from many different
+points of view; otherwise we never shall know more than one side. Now,
+to get out of self is what he who refers everything to himself can not
+do. Who delivered us from this illusion? It was those who showed us that
+the earth is only one of the smallest planets of the solar system, and
+that the solar system itself is only an imperceptible point in the
+infinite spaces of the stellar universe.
+
+At the same time astronomy taught us not to be afraid of big numbers.
+This was needful, not only for knowing the heavens, but to know the
+earth itself; and was not so easy as it seems to us to-day. Let us try
+to go back and picture to ourselves what a Greek would have thought if
+told that red light vibrates four hundred millions of millions of times
+per second. Without any doubt, such an assertion would have appeared to
+him pure madness, and he never would have lowered himself to test it.
+To-day a hypothesis will no longer appear absurd to us because it
+obliges us to imagine objects much larger or smaller than those our
+senses are capable of showing us, and we no longer comprehend those
+scruples which arrested our predecessors and prevented them from
+discovering certain truths simply because they were afraid of them. But
+why? It is because we have seen the heavens enlarging and enlarging
+without cease; because we know that the sun is 150 millions of
+kilometers from the earth and that the distances of the nearest stars
+are hundreds of thousands of times greater yet. Habituated to the
+contemplation of the infinitely great, we have become apt to comprehend
+the infinitely small. Thanks to the education it has received, our
+imagination, like the eagle's eye that the sun does not dazzle, can look
+truth in the face.
+
+Was I wrong in saying that it is astronomy which has made us a soul
+capable of comprehending nature; that under heavens always overcast and
+starless, the earth itself would have been for us eternally
+unintelligible; that we should there have seen only caprice and
+disorder; and that, not knowing the world, we should never have been
+able to subdue it? What science could have been more useful? And in thus
+speaking I put myself at the point of view of those who only value
+practical applications. Certainly, this point of view is not mine; as
+for me, on the contrary, if I admire the conquests of industry, it is
+above all because if they free us from material cares, they will one day
+give to all the leisure to contemplate nature. I do not say: Science is
+useful, because it teaches us to construct machines. I say: Machines are
+useful, because in working for us, they will some day leave us more time
+to make science. But finally it is worth remarking that between the two
+points of view there is no antagonism, and that man having pursued a
+disinterested aim, all else has been added unto him.
+
+Auguste Comte has said somewhere, that it would be idle to seek to know
+the composition of the sun, since this knowledge would be of no use to
+sociology. How could he be so short-sighted? Have we not just seen that
+it is by astronomy that, to speak his language, humanity has passed from
+the theological to the positive state? He found an explanation for that
+because it had happened. But how has he not understood that what
+remained to do was not less considerable and would be not less
+profitable? Physical astronomy, which he seems to condemn, has already
+begun to bear fruit, and it will give us much more, for it only dates
+from yesterday.
+
+First was discovered the nature of the sun, what the founder of
+positivism wished to deny us, and there bodies were found which exist on
+the earth, but had here remained undiscovered; for example, helium, that
+gas almost as light as hydrogen. That already contradicted Comte. But to
+the spectroscope we owe a lesson precious in a quite different way; in
+the most distant stars, it shows us the same substances. It might have
+been asked whether the terrestrial elements were not due to some chance
+which had brought together more tenuous atoms to construct of them the
+more complex edifice that the chemists call atom; whether, in other
+regions of the universe, other fortuitous meetings had not engendered
+edifices entirely different. Now we know that this is not so, that the
+laws of our chemistry are the general laws of nature, and that they owe
+nothing to the chance which caused us to be born on the earth.
+
+But, it will be said, astronomy has given to the other sciences all it
+can give them, and now that the heavens have procured for us the
+instruments which enable us to study terrestrial nature, they could
+without danger veil themselves forever. After what we have just said, is
+there still need to answer this objection? One could have reasoned the
+same in Ptolemy's time; then also men thought they knew everything, and
+they still had almost everything to learn.
+
+The stars are majestic laboratories, gigantic crucibles, such as no
+chemist could dream. There reign temperatures impossible for us to
+realize. Their only defect is being a little far away; but the telescope
+will soon bring them near to us, and then we shall see how matter acts
+there. What good fortune for the physicist and the chemist!
+
+Matter will there exhibit itself to us under a thousand different
+states, from those rarefied gases which seem to form the nebulæ and
+which are luminous with I know not what glimmering of mysterious origin,
+even to the incandescent stars and to the planets so near and yet so
+different.
+
+Perchance even, the stars will some day teach us something about life;
+that seems an insensate dream and I do not at all see how it can be
+realized; but, a hundred years ago, would not the chemistry of the stars
+have also appeared a mad dream?
+
+But limiting our views to horizons less distant, there still will remain
+to us promises less contingent and yet sufficiently seductive. If the
+past has given us much, we may rest assured that the future will give us
+still more.
+
+In sum, it is incredible how useful belief in astrology has been to
+humanity. If Kepler and Tycho Brahe made a living, it was because they
+sold to naïve kings predictions founded on the conjunctions of the
+stars. If these princes had not been so credulous, we should perhaps
+still believe that nature obeys caprice, and we should still wallow in
+ignorance.
+
+
+
+
+CHAPTER VII
+
+THE HISTORY OF MATHEMATICAL PHYSICS
+
+
+_The Past and the Future of Physics._--What is the present state of
+mathematical physics? What are the problems it is led to set itself?
+What is its future? Is its orientation about to be modified?
+
+Ten years hence will the aim and the methods of this science appear to
+our immediate successors in the same light as to ourselves; or, on the
+contrary, are we about to witness a profound transformation? Such are
+the questions we are forced to raise in entering to-day upon our
+investigation.
+
+If it is easy to propound them: to answer is difficult. If we felt
+tempted to risk a prediction, we should easily resist this temptation,
+by thinking of all the stupidities the most eminent savants of a hundred
+years ago would have uttered, if some one had asked them what the
+science of the nineteenth century would be. They would have thought
+themselves bold in their predictions, and after the event, how very
+timid we should have found them. Do not, therefore, expect of me any
+prophecy.
+
+But if, like all prudent physicians, I shun giving a prognosis, yet I
+can not dispense with a little diagnostic; well, yes, there are
+indications of a serious crisis, as if we might expect an approaching
+transformation. Still, be not too anxious: we are sure the patient will
+not die of it, and we may even hope that this crisis will be salutary,
+for the history of the past seems to guarantee us this. This crisis, in
+fact, is not the first, and to understand it, it is important to recall
+those which have preceded. Pardon then a brief historical sketch.
+
+_The Physics of Central Forces._--Mathematical physics, as we know, was
+born of celestial mechanics, which gave birth to it at the end of the
+eighteenth century, at the moment when it itself attained its complete
+development. During its first years especially, the infant strikingly
+resembled its mother.
+
+The astronomic universe is formed of masses, very great, no doubt, but
+separated by intervals so immense that they appear to us only as
+material points. These points attract each other inversely as the square
+of the distance, and this attraction is the sole force which influences
+their movements. But if our senses were sufficiently keen to show us all
+the details of the bodies which the physicist studies, the spectacle
+thus disclosed would scarcely differ from the one the astronomer
+contemplates. There also we should see material points, separated from
+one another by intervals, enormous in comparison with their dimensions,
+and describing orbits according to regular laws. These infinitesimal
+stars are the atoms. Like the stars proper, they attract or repel each
+other, and this attraction or this repulsion, following the straight
+line which joins them, depends only on the distance. The law according
+to which this force varies as function of the distance is perhaps not
+the law of Newton, but it is an analogous law; in place of the exponent
+-2, we have probably a different exponent, and it is from this change of
+exponent that arises all the diversity of physical phenomena, the
+variety of qualities and of sensations, all the world, colored and
+sonorous, which surrounds us; in a word, all nature.
+
+Such is the primitive conception in all its purity. It only remains to
+seek in the different cases what value should be given to this exponent
+in order to explain all the facts. It is on this model that Laplace, for
+example, constructed his beautiful theory of capillarity; he regards it
+only as a particular case of attraction, or, as he says, of universal
+gravitation, and no one is astonished to find it in the middle of one of
+the five volumes of the 'Mécanique céleste.' More recently Briot
+believes he penetrated the final secret of optics in demonstrating that
+the atoms of ether attract each other in the inverse ratio of the sixth
+power of the distance; and Maxwell himself, does he not say somewhere
+that the atoms of gases repel each other in the inverse ratio of the
+fifth power of the distance? We have the exponent -6, or -5, in place of
+the exponent -2, but it is always an exponent.
+
+Among the theories of this epoch, one alone is an exception, that of
+Fourier; in it are indeed atoms acting at a distance one upon the other;
+they mutually transmit heat, but they do not attract, they never budge.
+From this point of view, Fourier's theory must have appeared to the eyes
+of his contemporaries, to those of Fourier himself, as imperfect and
+provisional.
+
+This conception was not without grandeur; it was seductive, and many
+among us have not finally renounced it; they know that one will attain
+the ultimate elements of things only by patiently disentangling the
+complicated skein that our senses give us; that it is necessary to
+advance step by step, neglecting no intermediary; that our fathers were
+wrong in wishing to skip stations; but they believe that when one shall
+have arrived at these ultimate elements, there again will be found the
+majestic simplicity of celestial mechanics.
+
+Neither has this conception been useless; it has rendered us an
+inestimable service, since it has contributed to make precise the
+fundamental notion of the physical law.
+
+I will explain myself; how did the ancients understand law? It was for
+them an internal harmony, static, so to say, and immutable; or else it
+was like a model that nature tried to imitate. For us a law is something
+quite different; it is a constant relation between the phenomenon of
+to-day and that of to-morrow; in a word, it is a differential equation.
+
+Behold the ideal form of physical law; well, it is Newton's law which
+first clothed it forth. If then one has acclimated this form in physics,
+it is precisely by copying as far as possible this law of Newton, that
+is by imitating celestial mechanics. This is, moreover, the idea I have
+tried to bring out in Chapter VI.
+
+_The Physics of the Principles._--Nevertheless, a day arrived when the
+conception of central forces no longer appeared sufficient, and this is
+the first of those crises of which I just now spoke.
+
+What was done then? The attempt to penetrate into the detail of the
+structure of the universe, to isolate the pieces of this vast mechanism,
+to analyze one by one the forces which put them in motion, was
+abandoned, and we were content to take as guides certain general
+principles the express object of which is to spare us this minute study.
+How so? Suppose we have before us any machine; the initial wheel work
+and the final wheel work alone are visible, but the transmission, the
+intermediary machinery by which the movement is communicated from one to
+the other, is hidden in the interior and escapes our view; we do not
+know whether the communication is made by gearing or by belts, by
+connecting-rods or by other contrivances. Do we say that it is
+impossible for us to understand anything about this machine so long as
+we are not permitted to take it to pieces? You know well we do not, and
+that the principle of the conservation of energy suffices to determine
+for us the most interesting point. We easily ascertain that the final
+wheel turns ten times less quickly than the initial wheel, since these
+two wheels are visible; we are able thence to conclude that a couple
+applied to the one will be balanced by a couple ten times greater
+applied to the other. For that there is no need to penetrate the
+mechanism of this equilibrium and to know how the forces compensate each
+other in the interior of the machine; it suffices to be assured that
+this compensation can not fail to occur.
+
+Well, in regard to the universe, the principle of the conservation of
+energy is able to render us the same service. The universe is also a
+machine, much more complicated than all those of industry, of which
+almost all the parts are profoundly hidden from us; but in observing the
+motion of those that we can see, we are able, by the aid of this
+principle, to draw conclusions which remain true whatever may be the
+details of the invisible mechanism which animates them.
+
+The principle of the conservation of energy, or Mayer's principle, is
+certainly the most important, but it is not the only one; there are
+others from which we can derive the same advantage. These are:
+
+Carnot's principle, or the principle of the degradation of energy.
+
+Newton's principle, or the principle of the equality of action and
+reaction.
+
+The principle of relativity, according to which the laws of physical
+phenomena must be the same for a stationary observer as for an observer
+carried along in a uniform motion of translation; so that we have not
+and can not have any means of discerning whether or not we are carried
+along in such a motion.
+
+The principle of the conservation of mass, or Lavoisier's principle.
+
+I will add the principle of least action.
+
+The application of these five or six general principles to the different
+physical phenomena is sufficient for our learning of them all that we
+could reasonably hope to know of them. The most remarkable example of
+this new mathematical physics is, beyond question, Maxwell's
+electromagnetic theory of light.
+
+We know nothing as to what the ether is, how its molecules are disposed,
+whether they attract or repel each other; but we know that this medium
+transmits at the same time the optical perturbations and the electrical
+perturbations; we know that this transmission must take place in
+conformity with the general principles of mechanics, and that suffices
+us for the establishment of the equations of the electromagnetic field.
+
+These principles are results of experiments boldly generalized; but they
+seem to derive from their very generality a high degree of certainty. In
+fact, the more general they are, the more frequent are the opportunities
+to check them, and the verifications multiplying, taking the most
+varied, the most unexpected forms, end by no longer leaving place for
+doubt.
+
+_Utility of the Old Physics._--Such is the second phase of the history
+of mathematical physics and we have not yet emerged from it. Shall we
+say that the first has been useless? that during fifty years science
+went the wrong way, and that there is nothing left but to forget so many
+accumulated efforts that a vicious conception condemned in advance to
+failure? Not the least in the world. Do you think the second phase could
+have come into existence without the first? The hypothesis of central
+forces contained all the principles; it involved them as necessary
+consequences; it involved both the conservation of energy and that of
+masses, and the equality of action and reaction, and the law of least
+action, which appeared, it is true, not as experimental truths, but as
+theorems; the enunciation of which had at the same time something more
+precise and less general than under their present form.
+
+It is the mathematical physics of our fathers which has familiarized us
+little by little with these various principles; which has habituated us
+to recognize them under the different vestments in which they disguise
+themselves. They have been compared with the data of experience, it has
+been seen how it was necessary to modify their enunciation to adapt them
+to these data; thereby they have been extended and consolidated. Thus
+they came to be regarded as experimental truths; the conception of
+central forces became then a useless support, or rather an
+embarrassment, since it made the principles partake of its hypothetical
+character.
+
+The frames then have not broken, because they are elastic; but they have
+enlarged; our fathers, who established them, did not labor in vain, and
+we recognize in the science of to-day the general traits of the sketch
+which they traced.
+
+
+
+
+CHAPTER VIII
+
+THE PRESENT CRISIS OF MATHEMATICAL PHYSICS
+
+
+_The New Crisis._--Are we now about to enter upon a third period? Are we
+on the eve of a second crisis? These principles on which we have built
+all, are they about to crumble away in their turn? This has been for
+some time a pertinent question.
+
+When I speak thus, you no doubt think of radium, that grand
+revolutionist of the present time, and in fact I shall come back to it
+presently; but there is something else. It is not alone the conservation
+of energy which is in question; all the other principles are equally in
+danger, as we shall see in passing them successively in review.
+
+_Carnot's Principle._--Let us commence with the principle of Carnot.
+This is the only one which does not present itself as an immediate
+consequence of the hypothesis of central forces; more than that, it
+seems, if not to directly contradict that hypothesis, at least not to be
+reconciled with it without a certain effort. If physical phenomena were
+due exclusively to the movements of atoms whose mutual attraction
+depended only on the distance, it seems that all these phenomena should
+be reversible; if all the initial velocities were reversed, these atoms,
+always subjected to the same forces, ought to go over their trajectories
+in the contrary sense, just as the earth would describe in the
+retrograde sense this same elliptic orbit which it describes in the
+direct sense, if the initial conditions of its motion had been reversed.
+On this account, if a physical phenomenon is possible, the inverse
+phenomenon should be equally so, and one should be able to reascend the
+course of time. Now, it is not so in nature, and this is precisely what
+the principle of Carnot teaches us; heat can pass from the warm body to
+the cold body; it is impossible afterward to make it take the inverse
+route and to reestablish differences of temperature which have been
+effaced. Motion can be wholly dissipated and transformed into heat by
+friction; the contrary transformation can never be made except
+partially.
+
+We have striven to reconcile this apparent contradiction. If the world
+tends toward uniformity, this is not because its ultimate parts, at
+first unlike, tend to become less and less different; it is because,
+shifting at random, they end by blending. For an eye which should
+distinguish all the elements, the variety would remain always as great;
+each grain of this dust preserves its originality and does not model
+itself on its neighbors; but as the blend becomes more and more
+intimate, our gross senses perceive only the uniformity. This is why,
+for example, temperatures tend to a level, without the possibility of
+going backwards.
+
+A drop of wine falls into a glass of water; whatever may be the law of
+the internal motion of the liquid, we shall soon see it colored of a
+uniform rosy tint, and however much from this moment one may shake it
+afterwards, the wine and the water do not seem capable of again
+separating. Here we have the type of the irreversible physical
+phenomenon: to hide a grain of barley in a heap of wheat, this is easy;
+afterwards to find it again and get it out, this is practically
+impossible. All this Maxwell and Boltzmann have explained; but the one
+who has seen it most clearly, in a book too little read because it is a
+little difficult to read, is Gibbs, in his `Elementary Principles of
+Statistical Mechanics.'
+
+For those who take this point of view, Carnot's principle is only an
+imperfect principle, a sort of concession to the infirmity of our
+senses; it is because our eyes are too gross that we do not distinguish
+the elements of the blend; it is because our hands are too gross that we
+can not force them to separate; the imaginary demon of Maxwell, who is
+able to sort the molecules one by one, could well constrain the world to
+return backward. Can it return of itself? That is not impossible; that
+is only infinitely improbable. The chances are that we should wait a
+long time for the concourse of circumstances which would permit a
+retrogradation; but sooner or later they will occur, after years whose
+number it would take millions of figures to write. These reservations,
+however, all remained theoretic; they were not very disquieting, and
+Carnot's principle retained all its practical value. But here the scene
+changes. The biologist, armed with his microscope, long ago noticed in
+his preparations irregular movements of little particles in suspension;
+this is the Brownian movement. He first thought this was a vital
+phenomenon, but soon he saw that the inanimate bodies danced with no
+less ardor than the others; then he turned the matter over to the
+physicists. Unhappily, the physicists remained long uninterested in this
+question; one concentrates the light to illuminate the microscopic
+preparation, thought they; with light goes heat; thence inequalities of
+temperature and in the liquid interior currents which produce the
+movements referred to. It occurred to M. Gouy to look more closely, and
+he saw, or thought he saw, that this explanation is untenable, that the
+movements become brisker as the particles are smaller, but that they are
+not influenced by the mode of illumination. If then these movements
+never cease, or rather are reborn without cease, without borrowing
+anything from an external source of energy, what ought we to believe? To
+be sure, we should not on this account renounce our belief in the
+conservation of energy, but we see under our eyes now motion transformed
+into heat by friction, now inversely heat changed into motion, and that
+without loss since the movement lasts forever. This is the contrary of
+Carnot's principle. If this be so, to see the world return backward, we
+no longer have need of the infinitely keen eye of Maxwell's demon; our
+microscope suffices. Bodies too large, those, for example, which are a
+tenth of a millimeter, are hit from all sides by moving atoms, but they
+do not budge, because these shocks are very numerous and the law of
+chance makes them compensate each other; but the smaller particles
+receive too few shocks for this compensation to take place with
+certainty and are incessantly knocked about. And behold already one of
+our principles in peril.
+
+_The Principle of Relativity._--Let us pass to the principle of
+relativity; this not only is confirmed by daily experience, not only is
+it a necessary consequence of the hypothesis of central forces, but it
+is irresistibly imposed upon our good sense, and yet it also is
+assailed. Consider two electrified bodies; though they seem to us at
+rest, they are both carried along by the motion of the earth; an
+electric charge in motion, Rowland has taught us, is equivalent to a
+current; these two charged bodies are, therefore, equivalent to two
+parallel currents of the same sense and these two currents should
+attract each other. In measuring this attraction, we shall measure the
+velocity of the earth; not its velocity in relation to the sun or the
+fixed stars, but its absolute velocity.
+
+I well know what will be said: It is not its absolute velocity that is
+measured, it is its velocity in relation to the ether. How
+unsatisfactory that is! Is it not evident that from the principle so
+understood we could no longer infer anything? It could no longer tell us
+anything just because it would no longer fear any contradiction. If we
+succeed in measuring anything, we shall always be free to say that this
+is not the absolute velocity, and if it is not the velocity in relation
+to the ether, it might always be the velocity in relation to some new
+unknown fluid with which we might fill space.
+
+Indeed, experiment has taken upon itself to ruin this interpretation of
+the principle of relativity; all attempts to measure the velocity of the
+earth in relation to the ether have led to negative results. This time
+experimental physics has been more faithful to the principle than
+mathematical physics; the theorists, to put in accord their other
+general views, would not have spared it; but experiment has been
+stubborn in confirming it. The means have been varied; finally Michelson
+pushed precision to its last limits; nothing came of it. It is precisely
+to explain this obstinacy that the mathematicians are forced to-day to
+employ all their ingenuity.
+
+Their task was not easy, and if Lorentz has got through it, it is only
+by accumulating hypotheses.
+
+The most ingenious idea was that of local time. Imagine two observers
+who wish to adjust their timepieces by optical signals; they exchange
+signals, but as they know that the transmission of light is not
+instantaneous, they are careful to cross them. When station B perceives
+the signal from station A, its clock should not mark the same hour as
+that of station A at the moment of sending the signal, but this hour
+augmented by a constant representing the duration of the transmission.
+Suppose, for example, that station A sends its signal when its clock
+marks the hour _O_, and that station B perceives it when its clock marks
+the hour _t_. The clocks are adjusted if the slowness equal to _t_
+represents the duration of the transmission, and to verify it, station B
+sends in its turn a signal when its clock marks _O_; then station A
+should perceive it when its clock marks _t_. The timepieces are then
+adjusted.
+
+And in fact they mark the same hour at the same physical instant, but on
+the one condition, that the two stations are fixed. Otherwise the
+duration of the transmission will not be the same in the two senses,
+since the station A, for example, moves forward to meet the optical
+perturbation emanating from B, whereas the station B flees before the
+perturbation emanating from A. The watches adjusted in that way will not
+mark, therefore, the true time; they will mark what may be called the
+_local time_, so that one of them will be slow of the other. It matters
+little, since we have no means of perceiving it. All the phenomena which
+happen at A, for example, will be late, but all will be equally so, and
+the observer will not perceive it, since his watch is slow; so, as the
+principle of relativity requires, he will have no means of knowing
+whether he is at rest or in absolute motion.
+
+Unhappily, that does not suffice, and complementary hypotheses are
+necessary; it is necessary to admit that bodies in motion undergo a
+uniform contraction in the sense of the motion. One of the diameters of
+the earth, for example, is shrunk by one two-hundred-millionth in
+consequence of our planet's motion, while the other diameter retains its
+normal length. Thus the last little differences are compensated. And
+then, there is still the hypothesis about forces. Forces, whatever be
+their origin, gravity as well as elasticity, would be reduced in a
+certain proportion in a world animated by a uniform translation; or,
+rather, this would happen for the components perpendicular to the
+translation; the components parallel would not change. Resume, then, our
+example of two electrified bodies; these bodies repel each other, but at
+the same time if all is carried along in a uniform translation, they are
+equivalent to two parallel currents of the same sense which attract each
+other. This electrodynamic attraction diminishes, therefore, the
+electrostatic repulsion, and the total repulsion is feebler than if the
+two bodies were at rest. But since to measure this repulsion we must
+balance it by another force, and all these other forces are reduced in
+the same proportion, we perceive nothing. Thus all seems arranged, but
+are all the doubts dissipated? What would happen if one could
+communicate by non-luminous signals whose velocity of propagation
+differed from that of light? If, after having adjusted the watches by
+the optical procedure, we wished to verify the adjustment by the aid of
+these new signals, we should observe discrepancies which would render
+evident the common translation of the two stations. And are such signals
+inconceivable, if we admit with Laplace that universal gravitation is
+transmitted a million times more rapidly than light?
+
+Thus, the principle of relativity has been valiantly defended in these
+latter times, but the very energy of the defense proves how serious was
+the attack.
+
+_Newton's Principle._--Let us speak now of the principle of Newton, on
+the equality of action and reaction. This is intimately bound up with
+the preceding, and it seems indeed that the fall of the one would
+involve that of the other. Thus we must not be astonished to find here
+the same difficulties.
+
+Electrical phenomena, according to the theory of Lorentz, are due to the
+displacements of little charged particles, called electrons, immersed in
+the medium we call ether. The movements of these electrons produce
+perturbations in the neighboring ether; these perturbations propagate
+themselves in every direction with the velocity of light, and in turn
+other electrons, originally at rest, are made to vibrate when the
+perturbation reaches the parts of the ether which touch them. The
+electrons, therefore, act on one another, but this action is not direct,
+it is accomplished through the ether as intermediary. Under these
+conditions can there be compensation between action and reaction, at
+least for an observer who should take account only of the movements of
+matter, that is, of the electrons, and who should be ignorant of those
+of the ether that he could not see? Evidently not. Even if the
+compensation should be exact, it could not be simultaneous. The
+perturbation is propagated with a finite velocity; it, therefore,
+reaches the second electron only when the first has long ago entered
+upon its rest. This second electron, therefore, will undergo, after a
+delay, the action of the first, but will certainly not at that moment
+react upon it, since around this first electron nothing any longer
+budges.
+
+The analysis of the facts permits us to be still more precise. Imagine,
+for example, a Hertzian oscillator, like those used in wireless
+telegraphy; it sends out energy in every direction; but we can provide
+it with a parabolic mirror, as Hertz did with his smallest oscillators,
+so as to send all the energy produced in a single direction. What
+happens then according to the theory? The apparatus recoils, as if it
+were a cannon and the projected energy a ball; and that is contrary to
+the principle of Newton, since our projectile here has no mass, it is
+not matter, it is energy. The case is still the same, moreover, with a
+beacon light provided with a reflector, since light is nothing but a
+perturbation of the electromagnetic field. This beacon light should
+recoil as if the light it sends out were a projectile. What is the force
+that should produce this recoil? It is what is called the
+Maxwell-Bartholi pressure. It is very minute, and it has been difficult
+to put it in evidence even with the most sensitive radiometers; but it
+suffices that it exists.
+
+If all the energy issuing from our oscillator falls on a receiver, this
+will act as if it had received a mechanical shock, which will represent
+in a sense the compensation of the oscillator's recoil; the reaction
+will be equal to the action, but it will not be simultaneous; the
+receiver will move on, but not at the moment when the oscillator
+recoils. If the energy propagates itself indefinitely without
+encountering a receiver, the compensation will never occur.
+
+Shall we say that the space which separates the oscillator from the
+receiver and which the perturbation must pass over in going from the one
+to the other is not void, that it is full not only of ether, but of air,
+or even in the interplanetary spaces of some fluid subtile but still
+ponderable; that this matter undergoes the shock like the receiver at
+the moment when the energy reaches it, and recoils in its turn when the
+perturbation quits it? That would save Newton's principle, but that is
+not true. If energy in its diffusion remained always attached to some
+material substratum, then matter in motion would carry along light with
+it, and Fizeau has demonstrated that it does nothing of the sort, at
+least for air. Michelson and Morley have since confirmed this. It might
+be supposed also that the movements of matter proper are exactly
+compensated by those of the ether; but that would lead us to the same
+reflections as before now. The principle so understood will explain
+everything, since, whatever might be the visible movements, we always
+could imagine hypothetical movements which compensate them. But if it is
+able to explain everything, this is because it does not enable us to
+foresee anything; it does not enable us to decide between the different
+possible hypotheses, since it explains everything beforehand. It
+therefore becomes useless.
+
+And then the suppositions that it would be necessary to make on the
+movements of the ether are not very satisfactory. If the electric
+charges double, it would be natural to imagine that the velocities of
+the diverse atoms of ether double also; but, for the compensation, it
+would be necessary that the mean velocity of the ether quadruple.
+
+This is why I have long thought that these consequences of theory,
+contrary to Newton's principle, would end some day by being abandoned,
+and yet the recent experiments on the movements of the electrons issuing
+from radium seem rather to confirm them.
+
+_Lavoisier's Principle._--I arrive at the principle of Lavoisier on the
+conservation of mass. Certainly, this is one not to be touched without
+unsettling all mechanics. And now certain persons think that it seems
+true to us only because in mechanics merely moderate velocities are
+considered, but that it would cease to be true for bodies animated by
+velocities comparable to that of light. Now these velocities are
+believed at present to have been realized; the cathode rays and those of
+radium may be formed of very minute particles or of electrons which are
+displaced with velocities smaller no doubt than that of light, but which
+might be its one tenth or one third.
+
+These rays can be deflected, whether by an electric field, or by a
+magnetic field, and we are able, by comparing these deflections, to
+measure at the same time the velocity of the electrons and their mass
+(or rather the relation of their mass to their charge). But when it was
+seen that these velocities approached that of light, it was decided that
+a correction was necessary. These molecules, being electrified, can not
+be displaced without agitating the ether; to put them in motion it is
+necessary to overcome a double inertia, that of the molecule itself and
+that of the ether. The total or apparent mass that one measures is
+composed, therefore, of two parts: the real or mechanical mass of the
+molecule and the electrodynamic mass representing the inertia of the
+ether.
+
+The calculations of Abraham and the experiments of Kaufmann have then
+shown that the mechanical mass, properly so called, is null, and that
+the mass of the electrons, or, at least, of the negative electrons, is
+of exclusively electrodynamic origin. This is what forces us to change
+the definition of mass; we can not any longer distinguish mechanical
+mass and electrodynamic mass, since then the first would vanish; there
+is no mass other than electrodynamic inertia. But in this case the mass
+can no longer be constant; it augments with the velocity, and it even
+depends on the direction, and a body animated by a notable velocity will
+not oppose the same inertia to the forces which tend to deflect it from
+its route, as to those which tend to accelerate or to retard its
+progress.
+
+There is still a resource; the ultimate elements of bodies are
+electrons, some charged negatively, the others charged positively. The
+negative electrons have no mass, this is understood; but the positive
+electrons, from the little we know of them, seem much greater. Perhaps
+they have, besides their electrodynamic mass, a true mechanical mass.
+The real mass of a body would, then, be the sum of the mechanical masses
+of its positive electrons, the negative electrons not counting; mass so
+defined might still be constant.
+
+Alas! this resource also evades us. Recall what we have said of the
+principle of relativity and of the efforts made to save it. And it is
+not merely a principle which it is a question of saving, it is the
+indubitable results of the experiments of Michelson.
+
+Well, as was above seen, Lorentz, to account for these results, was
+obliged to suppose that all forces, whatever their origin, were reduced
+in the same proportion in a medium animated by a uniform translation;
+this is not sufficient; it is not enough that this take place for the
+real forces, it must also be the same for the forces of inertia; it is
+therefore necessary, he says, that _the masses of all the particles be
+influenced by a translation to the same degree as the electromagnetic
+masses of the electrons_.
+
+So the mechanical masses must vary in accordance with the same laws as
+the electrodynamic masses; they can not, therefore, be constant.
+
+Need I point out that the fall of Lavoisier's principle involves that of
+Newton's? This latter signifies that the center of gravity of an
+isolated system moves in a straight line; but if there is no longer a
+constant mass, there is no longer a center of gravity, we no longer know
+even what this is. This is why I said above that the experiments on the
+cathode rays appeared to justify the doubts of Lorentz concerning
+Newton's principle.
+
+From all these results, if they were confirmed, would arise an entirely
+new mechanics, which would be, above all, characterized by this fact,
+that no velocity could surpass that of light,[9] any more than any
+temperature can fall below absolute zero.
+
+ [9] Because bodies would oppose an increasing inertia to the causes
+ which would tend to accelerate their motion; and this inertia
+ would become infinite when one approached the velocity of light.
+
+No more for an observer, carried along himself in a translation he does
+not suspect, could any apparent velocity surpass that of light; and this
+would be then a contradiction, if we did not recall that this observer
+would not use the same clocks as a fixed observer, but, indeed, clocks
+marking 'local time.'
+
+Here we are then facing a question I content myself with stating. If
+there is no longer any mass, what becomes of Newton's law? Mass has two
+aspects: it is at the same time a coefficient of inertia and an
+attracting mass entering as factor into Newtonian attraction. If the
+coefficient of inertia is not constant, can the attracting mass be? That
+is the question.
+
+_Mayer's Principle._--At least, the principle of the conservation of
+energy yet remained to us, and this seemed more solid. Shall I recall to
+you how it was in its turn thrown into discredit? This event has made
+more noise than the preceding, and it is in all the memoirs. From the
+first words of Becquerel, and, above all, when the Curies had discovered
+radium, it was seen that every radioactive body was an inexhaustible
+source of radiation. Its activity seemed to subsist without alteration
+throughout the months and the years. This was in itself a strain on the
+principles; these radiations were in fact energy, and from the same
+morsel of radium this issued and forever issued. But these quantities of
+energy were too slight to be measured; at least that was the belief and
+we were not much disquieted.
+
+The scene changed when Curie bethought himself to put radium in a
+calorimeter; it was then seen that the quantity of heat incessantly
+created was very notable.
+
+The explanations proposed were numerous; but in such case we can not
+say, the more the better. In so far as no one of them has prevailed over
+the others, we can not be sure there is a good one among them. Since
+some time, however, one of these explanations seems to be getting the
+upper hand and we may reasonably hope that we hold the key to the
+mystery.
+
+Sir W. Ramsay has striven to show that radium is in process of
+transformation, that it contains a store of energy enormous but not
+inexhaustible. The transformation of radium then would produce a million
+times more heat than all known transformations; radium would wear itself
+out in 1,250 years; this is quite short, and you see that we are at
+least certain to have this point settled some hundreds of years from
+now. While waiting, our doubts remain.
+
+
+
+
+CHAPTER IX
+
+THE FUTURE OF MATHEMATICAL PHYSICS
+
+
+_The Principles and Experiment._--In the midst of so much ruin, what
+remains standing? The principle of least action is hitherto intact, and
+Larmor appears to believe that it will long survive the others; in
+reality, it is still more vague and more general.
+
+In presence of this general collapse of the principles, what attitude
+will mathematical physics take? And first, before too much excitement,
+it is proper to ask if all that is really true. All these derogations to
+the principles are encountered only among infinitesimals; the microscope
+is necessary to see the Brownian movement; electrons are very light;
+radium is very rare, and one never has more than some milligrams of it
+at a time. And, then, it may be asked whether, besides the infinitesimal
+seen, there was not another infinitesimal unseen counterpoise to the
+first.
+
+So there is an interlocutory question, and, as it seems, only experiment
+can solve it. We shall, therefore, only have to hand over the matter to
+the experimenters, and, while waiting for them to finally decide the
+debate, not to preoccupy ourselves with these disquieting problems, and
+to tranquilly continue our work as if the principles were still
+uncontested. Certes, we have much to do without leaving the domain where
+they may be applied in all security; we have enough to employ our
+activity during this period of doubts.
+
+_The Rôle of the Analyst._--And as to these doubts, is it indeed true
+that we can do nothing to disembarrass science of them? It must indeed
+be said, it is not alone experimental physics that has given birth to
+them; mathematical physics has well contributed. It is the experimenters
+who have seen radium throw out energy, but it is the theorists who have
+put in evidence all the difficulties raised by the propagation of light
+across a medium in motion; but for these it is probable we should not
+have become conscious of them. Well, then, if they have done their best
+to put us into this embarrassment, it is proper also that they help us
+to get out of it.
+
+They must subject to critical examination all these new views I have
+just outlined before you, and abandon the principles only after having
+made a loyal effort to save them. What can they do in this sense? That
+is what I will try to explain.
+
+It is a question before all of endeavoring to obtain a more satisfactory
+theory of the electrodynamics of bodies in motion. It is there
+especially, as I have sufficiently shown above, that difficulties
+accumulate. It is useless to heap up hypotheses, we can not satisfy all
+the principles at once; so far, one has succeeded in safeguarding some
+only on condition of sacrificing the others; but all hope of obtaining
+better results is not yet lost. Let us take, then, the theory of
+Lorentz, turn it in all senses, modify it little by little, and perhaps
+everything will arrange itself.
+
+Thus in place of supposing that bodies in motion undergo a contraction
+in the sense of the motion, and that this contraction is the same
+whatever be the nature of these bodies and the forces to which they are
+otherwise subjected, could we not make a more simple and natural
+hypothesis? We might imagine, for example, that it is the ether which is
+modified when it is in relative motion in reference to the material
+medium which penetrates it, that, when it is thus modified, it no longer
+transmits perturbations with the same velocity in every direction. It
+might transmit more rapidly those which are propagated parallel to the
+motion of the medium, whether in the same sense or in the opposite
+sense, and less rapidly those which are propagated perpendicularly. The
+wave surfaces would no longer be spheres, but ellipsoids, and we could
+dispense with that extraordinary contraction of all bodies.
+
+I cite this only as an example, since the modifications that might be
+essayed would be evidently susceptible of infinite variation.
+
+_Aberration and Astronomy._--It is possible also that astronomy may some
+day furnish us data on this point; she it was in the main who raised the
+question in making us acquainted with the phenomenon of the aberration
+of light. If we make crudely the theory of aberration, we reach a very
+curious result. The apparent positions of the stars differ from their
+real positions because of the earth's motion, and as this motion is
+variable, these apparent positions vary. The real position we can not
+ascertain, but we can observe the variations of the apparent position.
+The observations of the aberration show us, therefore, not the earth's
+motion, but the variations of this motion; they can not, therefore, give
+us information about the absolute motion of the earth.
+
+At least this is true in first approximation, but the case would be no
+longer the same if we could appreciate the thousandths of a second. Then
+it would be seen that the amplitude of the oscillation depends not alone
+on the variation of the motion, a variation which is well known, since
+it is the motion of our globe on its elliptic orbit, but on the mean
+value of this motion, so that the constant of aberration would not be
+quite the same for all the stars, and the differences would tell us the
+absolute motion of the earth in space.
+
+This, then, would be, under another form, the ruin of the principle of
+relativity. We are far, it is true, from appreciating the thousandth of
+a second, but, after all, say some, the earth's total absolute velocity
+is perhaps much greater than its relative velocity with respect to the
+sun. If, for example, it were 300 kilometers per second in place of 30,
+this would suffice to make the phenomenon observable.
+
+I believe that in reasoning thus one admits a too simple theory of
+aberration. Michelson has shown us, I have told you, that the physical
+procedures are powerless to put in evidence absolute motion; I am
+persuaded that the same will be true of the astronomic procedures,
+however far precision be carried.
+
+However that may be, the data astronomy will furnish us in this regard
+will some day be precious to the physicist. Meanwhile, I believe that
+the theorists, recalling the experience of Michelson, may anticipate a
+negative result, and that they would accomplish a useful work in
+constructing a theory of aberration which would explain this in advance.
+
+_Electrons and Spectra._--This dynamics of electrons can be approached
+from many sides, but among the ways leading thither is one which has
+been somewhat neglected, and yet this is one of those which promise us
+the most surprises. It is movements of electrons which produce the lines
+of the emission spectra; this is proved by the Zeeman effect; in an
+incandescent body what vibrates is sensitive to the magnet, therefore
+electrified. This is a very important first point, but no one has gone
+farther. Why are the lines of the spectrum distributed in accordance
+with a regular law? These laws have been studied by the experimenters in
+their least details; they are very precise and comparatively simple. A
+first study of these distributions recalls the harmonics encountered in
+acoustics; but the difference is great. Not only are the numbers of
+vibrations not the successive multiples of a single number, but we do
+not even find anything analogous to the roots of those transcendental
+equations to which we are led by so many problems of mathematical
+physics: that of the vibrations of an elastic body of any form, that of
+the Hertzian oscillations in a generator of any form, the problem of
+Fourier for the cooling of a solid body.
+
+The laws are simpler, but they are of wholly other nature, and to cite
+only one of these differences, for the harmonics of high order, the
+number of vibrations tends toward a finite limit, instead of increasing
+indefinitely.
+
+That has not yet been accounted for, and I believe that there we have
+one of the most important secrets of nature. A Japanese physicist, M.
+Nagaoka, has recently proposed an explanation; according to him, atoms
+are composed of a large positive electron surrounded by a ring formed of
+a great number of very small negative electrons. Such is the planet
+Saturn with its rings. This is a very interesting attempt, but not yet
+wholly satisfactory; this attempt should be renewed. We will penetrate,
+so to speak, into the inmost recess of matter. And from the particular
+point of view which we to-day occupy, when we know why the vibrations of
+incandescent bodies differ thus from ordinary elastic vibrations, why
+the electrons do not behave like the matter which is familiar to us, we
+shall better comprehend the dynamics of electrons and it will be perhaps
+more easy for us to reconcile it with the principles.
+
+_Conventions Preceding Experiment._--Suppose, now, that all these
+efforts fail, and, after all, I do not believe they will, what must be
+done? Will it be necessary to seek to mend the broken principles by
+giving what we French call a _coup de pouce_? That evidently is always
+possible, and I retract nothing of what I have said above.
+
+Have you not written, you might say if you wished to seek a quarrel with
+me--have you not written that the principles, though of experimental
+origin, are now unassailable by experiment because they have become
+conventions? And now you have just told us that the most recent
+conquests of experiment put these principles in danger.
+
+Well, formerly I was right and to-day I am not wrong. Formerly I was
+right, and what is now happening is a new proof of it. Take, for
+example, the calorimetric experiment of Curie on radium. Is it possible
+to reconcile it with the principle of the conservation of energy? This
+has been attempted in many ways. But there is among them one I should
+like you to notice; this is not the explanation which tends to-day to
+prevail, but it is one of those which have been proposed. It has been
+conjectured that radium was only an intermediary, that it only stored
+radiations of unknown nature which flashed through space in every
+direction, traversing all bodies, save radium, without being altered by
+this passage and without exercising any action upon them. Radium alone
+took from them a little of their energy and afterward gave it out to us
+in various forms.
+
+What an advantageous explanation, and how convenient! First, it is
+unverifiable and thus irrefutable. Then again it will serve to account
+for any derogation whatever to Mayer's principle; it answers in advance
+not only the objection of Curie, but all the objections that future
+experimenters might accumulate. This new and unknown energy would serve
+for everything.
+
+This is just what I said, and therewith we are shown that our principle
+is unassailable by experiment.
+
+But then, what have we gained by this stroke? The principle is intact,
+but thenceforth of what use is it? It enabled us to foresee that in such
+or such circumstance we could count on such a total quantity of energy;
+it limited us; but now that this indefinite provision of new energy is
+placed at our disposal, we are no longer limited by anything; and, as I
+have written in 'Science and Hypothesis,' if a principle ceases to be
+fecund, experiment without contradicting it directly will nevertheless
+have condemned it.
+
+_Future Mathematical Physics._--This, therefore, is not what would have
+to be done; it would be necessary to rebuild anew. If we were reduced to
+this necessity; we could moreover console ourselves. It would not be
+necessary thence to conclude that science can weave only a Penelope's
+web, that it can raise only ephemeral structures, which it is soon
+forced to demolish from top to bottom with its own hands.
+
+As I have said, we have already passed through a like crisis. I have
+shown you that in the second mathematical physics, that of the
+principles, we find traces of the first, that of central forces; it will
+be just the same if we must know a third. Just so with the animal that
+exuviates, that breaks its too narrow carapace and makes itself a fresh
+one; under the new envelope one will recognize the essential traits of
+the organism which have persisted.
+
+We can not foresee in what way we are about to expand; perhaps it is the
+kinetic theory of gases which is about to undergo development and serve
+as model to the others. Then the facts which first appeared to us as
+simple thereafter would be merely resultants of a very great number of
+elementary facts which only the laws of chance would make cooperate for
+a common end. Physical law would then assume an entirely new aspect; it
+would no longer be solely a differential equation, it would take the
+character of a statistical law.
+
+Perhaps, too, we shall have to construct an entirely new mechanics that
+we only succeed in catching a glimpse of, where, inertia increasing with
+the velocity, the velocity of light would become an impassable limit.
+The ordinary mechanics, more simple, would remain a first approximation,
+since it would be true for velocities not too great, so that the old
+dynamics would still be found under the new. We should not have to
+regret having believed in the principles, and even, since velocities too
+great for the old formulas would always be only exceptional, the surest
+way in practise would be still to act as if we continued to believe in
+them. They are so useful, it would be necessary to keep a place for
+them. To determine to exclude them altogether would be to deprive
+oneself of a precious weapon. I hasten to say in conclusion that we are
+not yet there, and as yet nothing proves that the principles will not
+come forth from out the fray victorious and intact.[10]
+
+ [10] These considerations on mathematical physics are borrowed from
+ my St. Louis address.
+
+
+
+
+PART III
+
+
+THE OBJECTIVE VALUE OF SCIENCE
+
+
+
+
+CHAPTER X
+
+IS SCIENCE ARTIFICIAL?
+
+
+1. _The Philosophy of M. LeRoy_
+
+There are many reasons for being sceptics; should we push this
+scepticism to the very end or stop on the way? To go to the end is the
+most tempting solution, the easiest and that which many have adopted,
+despairing of saving anything from the shipwreck.
+
+Among the writings inspired by this tendency it is proper to place in
+the first rank those of M. LeRoy. This thinker is not only a philosopher
+and a writer of the greatest merit, but he has acquired a deep knowledge
+of the exact and physical sciences, and even has shown rare powers of
+mathematical invention. Let us recapitulate in a few words his doctrine,
+which has given rise to numerous discussions.
+
+Science consists only of conventions, and to this circumstance solely
+does it owe its apparent certitude; the facts of science and, _a
+fortiori_, its laws are the artificial work of the scientist; science
+therefore can teach us nothing of the truth; it can only serve us as
+rule of action.
+
+Here we recognize the philosophic theory known under the name of
+nominalism; all is not false in this theory; its legitimate domain must
+be left it, but out of this it should not be allowed to go.
+
+This is not all; M. LeRoy's doctrine is not only nominalistic; it has
+besides another characteristic which it doubtless owes to M. Bergson, it
+is anti-intellectualistic. According to M. LeRoy, the intellect deforms
+all it touches, and that is still more true of its necessary instrument
+'discourse.' There is reality only in our fugitive and changing
+impressions, and even this reality, when touched, vanishes.
+
+And yet M. LeRoy is not a sceptic; if he regards the intellect as
+incurably powerless, it is only to give more scope to other sources of
+knowledge, to the heart, for instance, to sentiment, to instinct or to
+faith.
+
+However great my esteem for M. LeRoy's talent, whatever the ingenuity of
+this thesis, I can not wholly accept it. Certes, I am in accord on many
+points with M. LeRoy, and he has even cited, in support of his view,
+various passages of my writings which I am by no means disposed to
+reject. I think myself only the more bound to explain why I can not go
+with him all the way.
+
+M. LeRoy often complains of being accused of scepticism. He could not
+help being, though this accusation is probably unjust. Are not
+appearances against him? Nominalist in doctrine, but realist at heart,
+he seems to escape absolute nominalism only by a desperate act of faith.
+
+The fact is that anti-intellectualistic philosophy in rejecting analysis
+and 'discourse,' just by that condemns itself to being intransmissible;
+it is a philosophy essentially internal, or, at the very least, only its
+negations can be transmitted; what wonder then that for an external
+observer it takes the shape of scepticism?
+
+Therein lies the weak point of this philosophy; if it strives to remain
+faithful to itself, its energy is spent in a negation and a cry of
+enthusiasm. Each author may repeat this negation and this cry, may vary
+their form, but without adding anything.
+
+And, yet, would it not be more logical in remaining silent? See, you
+have written long articles; for that, it was necessary to use words. And
+therein have you not been much more 'discursive' and consequently much
+farther from life and truth than the animal who simply lives without
+philosophizing? Would not this animal be the true philosopher?
+
+However, because no painter has made a perfect portrait, should we
+conclude that the best painting is not to paint? When a zoologist
+dissects an animal, certainly he 'alters it.' Yes, in dissecting it, he
+condemns himself to never know all of it; but in not dissecting it, he
+would condemn himself to never know anything of it and consequently to
+never see anything of it.
+
+Certes, in man are other forces besides his intellect; no one has ever
+been mad enough to deny that. The first comer makes these blind forces
+act or lets them act; the philosopher must _speak_ of them; to speak of
+them, he must know of them the little that can be known, he should
+therefore _see_ them act. How? With what eyes, if not with his
+intellect? Heart, instinct, may guide it, but not render it useless;
+they may direct the look, but not replace the eye. It may be granted
+that the heart is the workman, and the intellect only the instrument.
+Yet is it an instrument not to be done without, if not for action, at
+least for philosophizing? Therefore a philosopher really
+anti-intellectualistic is impossible. Perhaps we shall have to declare
+for the supremacy of action; always it is our intellect which will thus
+conclude; in allowing precedence to action it will thus retain the
+superiority of the thinking reed. This also is a supremacy not to be
+disdained.
+
+Pardon these brief reflections and pardon also their brevity, scarcely
+skimming the question. The process of intellectualism is not the subject
+I wish to treat: I wish to speak of science, and about it there is no
+doubt; by definition, so to speak, it will be intellectualistic or it
+will not be at all. Precisely the question is, whether it will be.
+
+
+2. _Science, Rule of Action_
+
+For M. LeRoy, science is only a rule of action. We are powerless to know
+anything and yet we are launched, we must act, and at all hazards we
+have established rules. It is the aggregate of these rules that is
+called science.
+
+It is thus that men, desirous of diversion, have instituted rules of
+play, like those of tric-trac for instance, which, better than science
+itself, could rely upon the proof by universal consent. It is thus
+likewise that, unable to choose, but forced to choose, we toss up a
+coin, head or tail to win.
+
+The rule of tric-trac is indeed a rule of action like science, but does
+any one think the comparison just and not see the difference? The rules
+of the game are arbitrary conventions and the contrary convention might
+have been adopted, _which would have been none the less good_. On the
+contrary, science is a rule of action which is successful, generally at
+least, and I add, while the contrary rule would not have succeeded.
+
+If I say, to make hydrogen cause an acid to act on zinc, I formulate a
+rule which succeeds; I could have said, make distilled water act on
+gold; that also would have been a rule, only it would not have
+succeeded. If, therefore, scientific 'recipes' have a value, as rule of
+action, it is because we know they succeed, generally at least. But to
+know this is to know something and then why tell us we can know nothing?
+
+Science foresees, and it is because it foresees that it can be useful
+and serve as rule of action. I well know that its previsions are often
+contradicted by the event; that shows that science is imperfect, and if
+I add that it will always remain so, I am certain that this is a
+prevision which, at least, will never be contradicted. Always the
+scientist is less often mistaken than a prophet who should predict at
+random. Besides the progress though slow is continuous, so that
+scientists, though more and more bold, are less and less misled. This is
+little, but it is enough.
+
+I well know that M. LeRoy has somewhere said that science was mistaken
+oftener than one thought, that comets sometimes played tricks on
+astronomers, that scientists, who apparently are men, did not willingly
+speak of their failures, and that, if they should speak of them, they
+would have to count more defeats than victories.
+
+That day, M. LeRoy evidently overreached himself. If science did not
+succeed, it could not serve as rule of action; whence would it get its
+value? Because it is 'lived,' that is, because we love it and believe in
+it? The alchemists had recipes for making gold, they loved them and had
+faith in them, and yet our recipes are the good ones, although our faith
+be less lively, because they succeed.
+
+There is no escape from this dilemma; either science does not enable us
+to foresee, and then it is valueless as rule of action; or else it
+enables us to foresee, in a fashion more or less imperfect, and then it
+is not without value as means of knowledge.
+
+It should not even be said that action is the goal of science; should we
+condemn studies of the star Sirius, under pretext that we shall probably
+never exercise any influence on that star? To my eyes, on the contrary,
+it is the knowledge which is the end, and the action which is the means.
+If I felicitate myself on the industrial development, it is not alone
+because it furnishes a facile argument to the advocates of science; it
+is above all because it gives to the scientist faith in himself and also
+because it offers him an immense field of experience where he clashes
+against forces too colossal to be tampered with. Without this ballast,
+who knows whether he would not quit solid ground, seduced by the mirage
+of some scholastic novelty, or whether he would not despair, believing
+he had fashioned only a dream?
+
+
+3. _The Crude Fact and the Scientific Fact_
+
+What was most paradoxical in M. LeRoy's thesis was that affirmation that
+_the scientist creates the fact_; this was at the same time its
+essential point and it is one of those which have been most discussed.
+
+Perhaps, says he (I well believe that this was a concession), it is not
+the scientist that creates the fact in the rough; it is at least he who
+creates the scientific fact.
+
+This distinction between the fact in the rough and the scientific fact
+does not by itself appear to me illegitimate. But I complain first that
+the boundary has not been traced either exactly or precisely; and then
+that the author has seemed to suppose that the crude fact, not being
+scientific, is outside of science.
+
+Finally, I can not admit that the scientist creates without restraint
+the scientific fact, since it is the crude fact which imposes it upon
+him.
+
+The examples given by M. LeRoy have greatly astonished me. The first is
+taken from the notion of atom. The atom chosen as example of fact! I
+avow that this choice has so disconcerted me that I prefer to say
+nothing about it. I have evidently misunderstood the author's thought
+and I could not fruitfully discuss it.
+
+The second case taken as example is that of an eclipse where the crude
+phenomenon is a play of light and shadow, but where the astronomer can
+not intervene without introducing two foreign elements, to wit, a clock
+and Newton's law.
+
+Finally, M. LeRoy cites the rotation of the earth; it has been answered:
+but this is not a fact, and he has replied: it was one for Galileo, who
+affirmed it, as for the inquisitor, who denied it. It always remains
+that this is not a fact in the same sense as those just spoken of and
+that to give them the same name is to expose one's self to many
+confusions.
+
+Here then are four degrees:
+
+1º. It grows dark, says the clown.
+
+2º. The eclipse happened at nine o'clock, says the astronomer.
+
+3º. The eclipse happened at the time deducible from the tables
+constructed according to Newton's law, says he again.
+
+4º. That results from the earth's turning around the sun, says Galileo
+finally.
+
+Where then is the boundary between the fact in the rough and the
+scientific fact? To read M. LeRoy one would believe that it is between
+the first and the second stage, but who does not see that there is a
+greater distance from the second to the third, and still more from the
+third to the fourth.
+
+Allow me to cite two examples which perhaps will enlighten us a little.
+
+I observe the deviation of a galvanometer by the aid of a movable mirror
+which projects a luminous image or spot on a divided scale. The crude
+fact is this: I see the spot displace itself on the scale, and the
+scientific fact is this: a current passes in the circuit.
+
+Or again: when I make an experiment I should subject the result to
+certain corrections, because I know I must have made errors. These
+errors are of two kinds, some are accidental and these I shall correct
+by taking the mean; the others are systematic and I shall be able to
+correct those only by a thorough study of their causes. The first result
+obtained is then the fact in the rough, while the scientific fact is the
+final result after the finished corrections.
+
+Reflecting on this latter example, we are led to subdivide our second
+stage, and in place of saying:
+
+2. The eclipse happened at nine o'clock, we shall say:
+
+2_a_. The eclipse happened when my clock pointed to nine, and
+
+2_b_. My clock being ten minutes slow, the eclipse happened at ten
+minutes past nine.
+
+And this is not all: the first stage also should be subdivided, and not
+between these two subdivisions will be the least distance; it is
+necessary to distinguish between the impression of obscurity felt by one
+witnessing an eclipse, and the affirmation: It grows dark, which this
+impression extorts from him. In a sense it is the first which is the
+only true fact in the rough, and the second is already a sort of
+scientific fact.
+
+Now then our scale has six stages, and even though there is no reason
+for halting at this figure, there we shall stop.
+
+What strikes me at the start is this. At the first of our six stages,
+the fact, still completely in the rough, is, so to speak, individual, it
+is completely distinct from all other possible facts. From the second
+stage, already it is no longer the same. The enunciation of the fact
+would suit an infinity of other facts. So soon as language intervenes, I
+have at my command only a finite number of terms to express the shades,
+in number infinite, that my impressions might cover. When I say: It
+grows dark, that well expresses the impressions I feel in being present
+at an eclipse; but even in obscurity a multitude of shades could be
+imagined, and if, instead of that actually realized, had happened a
+slightly different shade, yet I should still have enunciated this
+_other_ fact by saying: It grows dark.
+
+Second remark: even at the second stage, the enunciation of a fact can
+only be _true or false_. This is not so of any proposition; if this
+proposition is the enunciation of a convention, it can not be said that
+this enunciation is _true_, in the proper sense of the word, since it
+could not be true apart from me and is true only because I wish it to
+be.
+
+When, for instance, I say the unit for length is the meter, this is a
+decree that I promulgate, it is not something ascertained which forces
+itself upon me. It is the same, as I think I have elsewhere shown, when
+it is a question, for example, of Euclid's postulate.
+
+When I am asked: Is it growing dark? I always know whether I ought to
+reply yes or no. Although an infinity of possible facts may be
+susceptible of this same enunciation, it grows dark, I shall always
+know whether the fact realized belongs or does not belong among those
+which answer to this enunciation. Facts are classed in categories, and
+if I am asked whether the fact that I ascertain belongs or does not
+belong in such a category, I shall not hesitate.
+
+Doubtless this classification is sufficiently arbitrary to leave a large
+part to man's freedom or caprice. In a word, this classification is a
+convention. _This convention being given_, if I am asked: Is such a fact
+true? I shall always know what to answer, and my reply will be imposed
+upon me by the witness of my senses.
+
+If therefore, during an eclipse, it is asked: Is it growing dark? all
+the world will answer yes. Doubtless those speaking a language where
+bright was called dark, and dark bright, would answer no. But of what
+importance is that?
+
+In the same way, in mathematics, _when I have laid down the definitions,
+and the postulates which are conventions_, a theorem henceforth can only
+be true or false. But to answer the question: Is this theorem true? it
+is no longer to the witness of my senses that I shall have recourse, but
+to reasoning.
+
+A statement of fact is always verifiable, and for the verification we
+have recourse either to the witness of our senses, or to the memory of
+this witness. This is properly what characterizes a fact. If you put the
+question to me: Is such a fact true? I shall begin by asking you, if
+there is occasion, to state precisely the conventions, by asking you, in
+other words, what language you have spoken; then once settled on this
+point, I shall interrogate my senses and shall answer yes or no. But it
+will be my senses that will have made answer, it will not be _you_ when
+you say to me: I have spoken to you in English or in French.
+
+Is there something to change in all that when we pass to the following
+stages? When I observe a galvanometer, as I have just said, if I ask an
+ignorant visitor: Is the current passing? he looks at the wire to try to
+see something pass; but if I put the same question to my assistant who
+understands my language, he will know I mean: Does the spot move? and he
+will look at the scale.
+
+What difference is there then between the statement of a fact in the
+rough and the statement of a scientific fact? The same difference as
+between the statement of the same crude fact in French and in German.
+The scientific statement is the translation of the crude statement into
+a language which is distinguished above all from the common German or
+French, because it is spoken by a very much smaller number of people.
+
+Yet let us not go too fast. To measure a current I may use a very great
+number of types of galvanometers or besides an electrodynamometer. And
+then when I shall say there is running in this circuit a current of so
+many amperes, that will mean: if I adapt to this circuit such a
+galvanometer I shall see the spot come to the division _a_;
+but that will mean equally: if I adapt to this circuit such an
+electrodynamometer, I shall see the spot go to the division _b_. And
+that will mean still many other things, because the current can manifest
+itself not only by mechanical effects, but by effects chemical, thermal,
+luminous, etc.
+
+Here then is one same statement which suits a very great number of facts
+absolutely different. Why? It is because I assume a law according to
+which, whenever such a mechanical effect shall happen, such a chemical
+effect will happen also. Previous experiments, very numerous, have never
+shown this law to fail, and then I have understood that I could express
+by the same statement two facts so invariably bound one to the other.
+
+When I am asked: Is the current passing? I can understand that that
+means: Will such a mechanical effect happen? But I can understand also:
+Will such a chemical effect happen? I shall then verify either the
+existence of the mechanical effect, or that of the chemical effect; that
+will be indifferent, since in both cases the answer must be the same.
+
+And if the law should one day be found false? If it was perceived that
+the concordance of the two effects, mechanical and chemical, is not
+constant? That day it would be necessary to change the scientific
+language to free it from a grave ambiguity.
+
+And after that? Is it thought that ordinary language by aid of which are
+expressed the facts of daily life is exempt from ambiguity?
+
+_Shall we thence conclude that the facts of daily life are the work of
+the grammarians?_
+
+You ask me: Is there a current? I try whether the mechanical effect
+exists, I ascertain it and I answer: Yes, there is a current. You
+understand at once that that means that the mechanical effect exists,
+and that the chemical effect, that I have not investigated, exists
+likewise. Imagine now, supposing an impossibility, the law we believe
+true, not to be, and the chemical effect not to exist. Under this
+hypothesis there will be two distinct facts, the one directly observed
+and which is true, the other inferred and which is false. It may
+strictly be said that we have created the second. So that error is the
+part of man's personal collaboration in the creation of the scientific
+fact.
+
+But if we can say that the fact in question is false, is this not just
+because it is not a free and arbitrary creation of our mind, a disguised
+convention, in which case it would be neither true nor false. And in
+fact it was verifiable; I had not made the verification, but I could
+have made it. If I answered amiss, it was because I chose to reply too
+quickly, without having asked nature, who alone knew the secret.
+
+When, after an experiment, I correct the accidental and systematic
+errors to bring out the scientific fact, the case is the same; the
+scientific fact will never be anything but the crude fact translated
+into another language. When I shall say: It is such an hour, that will
+be a short way of saying: There is such a relation between the hour
+indicated by my clock, and the hour it marked at the moment of the
+passing of such a star and such another star across the meridian. And
+this convention of language once adopted, when I shall be asked: Is it
+such an hour? it will not depend upon me to answer yes or no.
+
+Let us pass to the stage before the last: the eclipse happened at the
+hour given by the tables deduced from Newton's laws. This is still a
+convention of language which is perfectly clear for those who know
+celestial mechanics or simply for those who have the tables calculated
+by the astronomers. I am asked: Did the eclipse happen at the hour
+predicted? I look in the nautical almanac, I see that the eclipse was
+announced for nine o'clock and I understand that the question means: Did
+the eclipse happen at nine o'clock? There still we have nothing to
+change in our conclusions. _The scientific fact is only the crude fact
+translated into a convenient language._
+
+It is true that at the last stage things change. Does the earth rotate?
+Is this a verifiable fact? Could Galileo and the Grand Inquisitor, to
+settle the matter, appeal to the witness of their senses? On the
+contrary, they were in accord about the appearances, and whatever had
+been the accumulated experiences, they would have remained in accord
+with regard to the appearances without ever agreeing on their
+interpretation. It is just on that account that they were obliged to
+have recourse to procedures of discussion so unscientific.
+
+This is why I think they did not disagree about a _fact_: we have not
+the right to give the same name to the rotation of the earth, which was
+the object of their discussion, and to the facts crude or scientific we
+have hitherto passed in review.
+
+After what precedes, it seems superfluous to investigate whether the
+fact in the rough is outside of science, because there can neither be
+science without scientific fact, nor scientific fact without fact in the
+rough, since the first is only the translation of the second.
+
+And then, has one the right to say that the scientist creates the
+scientific fact? First of all, he does not create it from nothing, since
+he makes it with the fact in the rough. Consequently he does not make it
+freely and _as he chooses_. However able the worker may be, his freedom
+is always limited by the properties of the raw material on which he
+works.
+
+After all, what do you mean when you speak of this free creation of the
+scientific fact and when you take as example the astronomer who
+intervenes actively in the phenomenon of the eclipse by bringing his
+clock? Do you mean: The eclipse happened at nine o'clock; but if the
+astronomer had wished it to happen at ten, that depended only on him, he
+had only to advance his clock an hour?
+
+But the astronomer, in perpetrating that bad joke, would evidently have
+been guilty of an equivocation. When he tells me: The eclipse happened
+at nine, I understand that nine is the hour deduced from the crude
+indication of the pendulum by the usual series of corrections. If he has
+given me solely that crude indication, or if he has made corrections
+contrary to the habitual rules, he has changed the language agreed upon
+without forewarning me. If, on the contrary, he took care to forewarn
+me, I have nothing to complain of, but then it is always the same fact
+expressed in another language.
+
+In sum, _all the scientist creates in a fact is the language in which he
+enunciates it_. If he predicts a fact, he will employ this language, and
+for all those who can speak and understand it, his prediction is free
+from ambiguity. Moreover, this prediction once made, it evidently does
+not depend upon him whether it is fulfilled or not.
+
+What then remains of M. LeRoy's thesis? This remains: the scientist
+intervenes actively in choosing the facts worth observing. An isolated
+fact has by itself no interest; it becomes interesting if one has reason
+to think that it may aid in the prediction of other facts; or better,
+if, having been predicted, its verification is the confirmation of a
+law. Who shall choose the facts which, corresponding to these
+conditions, are worthy the freedom of the city in science? This is the
+free activity of the scientist.
+
+And that is not all. I have said that the scientific fact is the
+translation of a crude fact into a certain language; I should add that
+every scientific fact is formed of many crude facts. This is
+sufficiently shown by the examples cited above. For instance, for the
+hour of the eclipse my clock marked the hour [alpha] at the instant of
+the eclipse; it marked the hour [beta] at the moment of the last transit
+of the meridian of a certain star that we take as origin of right
+ascensions; it marked the hour [gamma] at the moment of the preceding
+transit of this same star. There are three distinct facts (still it will
+be noticed that each of them results itself from two simultaneous facts
+in the rough; but let us pass this over). In place of that I say: The
+eclipse happened at the hour 24 ([alpha]-[beta])/([beta]-[gamma]), and
+the three facts are combined in a single scientific fact. I have
+concluded that the three readings, [alpha], [beta], [gamma] made on my
+clock at three different moments lacked interest and that the only thing
+interesting was the combination ([alpha]-[beta])/([beta]-[gamma]) of
+the three. In this conclusion is found the free activity of my mind.
+
+But I have thus used up my power; I can not make this combination
+([alpha]-[beta])/([beta]-[gamma]) have such a value and not such
+another, since I can not influence either the value of [alpha], or that
+of [beta], or that of [gamma], which are imposed upon me as crude
+facts.
+
+In sum, facts are facts, and _if it happens that they satisfy a
+prediction, this is not an effect of our free activity_. There is no
+precise frontier between the fact in the rough and the scientific fact;
+it can only be said that such an enunciation of fact is _more crude_ or,
+on the contrary, _more scientific_ than such another.
+
+
+4. _'Nominalism' and 'the Universal Invariant'_
+
+If from facts we pass to laws, it is clear that the part of the free
+activity of the scientist will become much greater. But did not M. LeRoy
+make it still too great? This is what we are about to examine.
+
+Recall first the examples he has given. When I say: Phosphorus melts at
+44°, I think I am enunciating a law; in reality it is just the
+definition of phosphorus; if one should discover a body which,
+possessing otherwise all the properties of phosphorus, did not melt at
+44°, we should give it another name, that is all, and the law would
+remain true.
+
+Just so when I say: Heavy bodies falling freely pass over spaces
+proportional to the squares of the times, I only give the definition of
+free fall. Whenever the condition shall not be fulfilled, I shall say
+that the fall is not free, so that the law will never be wrong. It is
+clear that if laws were reduced to that, they could not serve in
+prediction; then they would be good for nothing, either as means of
+knowledge or as principle of action.
+
+When I say: Phosphorus melts at 44°, I mean by that: All bodies
+possessing such or such a property (to wit, all the properties of
+phosphorus, save fusing-point) fuse at 44°. So understood, my
+proposition is indeed a law, and this law may be useful to me, because
+if I meet a body possessing these properties I shall be able to predict
+that it will fuse at 44°.
+
+Doubtless the law may be found to be false. Then we shall read in the
+treatises on chemistry: "There are two bodies which chemists long
+confounded under the name of phosphorus; these two bodies differ only by
+their points of fusion." That would evidently not be the first time for
+chemists to attain to the separation of two bodies they were at first
+not able to distinguish; such, for example, are neodymium and
+praseodymium, long confounded under the name of didymium.
+
+I do not think the chemists much fear that a like mischance will ever
+happen to phosphorus. And if, to suppose the impossible, it should
+happen, the two bodies would probably not have _identically_ the same
+density, _identically_ the same specific heat, etc., so that after
+having determined with care the density, for instance, one could still
+foresee the fusion point.
+
+It is, moreover, unimportant; it suffices to remark that there is a law,
+and that this law, true or false, does not reduce to a tautology.
+
+Will it be said that if we do not know on the earth a body which does
+not fuse at 44° while having all the other properties of phosphorus, we
+can not know whether it does not exist on other planets? Doubtless that
+may be maintained, and it would then be inferred that the law in
+question, which may serve as a rule of action to us who inhabit the
+earth, has yet no general value from the point of view of knowledge, and
+owes its interest only to the chance which has placed us on this globe.
+This is possible, but, if it were so, the law would be valueless, not
+because it reduced to a convention, but because it would be false.
+
+The same is true in what concerns the fall of bodies. It would do me no
+good to have given the name of free fall to falls which happen in
+conformity with Galileo's law, if I did not know that elsewhere, in such
+circumstances, the fall will be _probably_ free or _approximately_ free.
+That then is a law which may be true or false, but which does not reduce
+to a convention.
+
+Suppose the astronomers discover that the stars do not exactly obey
+Newton's law. They will have the choice between two attitudes; they may
+say that gravitation does not vary exactly as the inverse of the square
+of the distance, or else they may say that gravitation is not the only
+force which acts on the stars and that there is in addition a different
+sort of force.
+
+In the second case, Newton's law will be considered as the definition of
+gravitation. This will be the nominalist attitude. The choice between
+the two attitudes is free, and is made from considerations of
+convenience, though these considerations are most often so strong that
+there remains practically little of this freedom.
+
+We can break up this proposition: (1) The stars obey Newton's law, into
+two others; (2) gravitation obeys Newton's law; (3) gravitation is the
+only force acting on the stars. In this case proposition (2) is no
+longer anything but a definition and is beyond the test of experiment;
+but then it will be on proposition (3) that this check can be exercised.
+This is indeed necessary, since the resulting proposition (1) predicts
+verifiable facts in the rough.
+
+It is thanks to these artifices that by an unconscious nominalism the
+scientists have elevated above the laws what they call principles. When
+a law has received a sufficient confirmation from experiment, we may
+adopt two attitudes: either we may leave this law in the fray; it will
+then remain subjected to an incessant revision, which without any doubt
+will end by demonstrating that it is only approximative. Or else we may
+elevate it into a _principle_ by adopting conventions such that the
+proposition may be certainly true. For that the procedure is always the
+same. The primitive law enunciated a relation between two facts in the
+rough, _A_ and _B_; between these two crude facts is introduced an
+abstract intermediary _C_, more or less fictitious (such was in the
+preceding example the impalpable entity, gravitation). And then we have
+a relation between _A_ and _C_ that we may suppose rigorous and which is
+the _principle_; and another between _C_ and _B_ which remains a _law_
+subject to revision.
+
+The principle, henceforth crystallized, so to speak, is no longer
+subject to the test of experiment. It is not true or false, it is
+convenient.
+
+Great advantages have often been found in proceeding in that way, but it
+is clear that if _all_ the laws had been transformed into principles
+_nothing_ would be left of science. Every law may be broken up into a
+principle and a law, but thereby it is very clear that, however far this
+partition be pushed, there will always remain laws.
+
+Nominalism has therefore limits, and this is what one might fail to
+recognize if one took to the very letter M. LeRoy's assertions.
+
+A rapid review of the sciences will make us comprehend better what are
+these limits. The nominalist attitude is justified only when it is
+convenient; when is it so?
+
+Experiment teaches us relations between bodies; this is the fact in the
+rough; these relations are extremely complicated. Instead of envisaging
+directly the relation of the body _A_ and the body _B_, we introduce
+between them an intermediary, which is space, and we envisage three
+distinct relations: that of the body _A_ with the figure _A'_ of space,
+that of the body _B_ with the figure _B'_ of space, that of the two
+figures _A'_ and _B'_ to each other. Why is this detour advantageous?
+Because the relation of _A_ and _B_ was complicated, but differed little
+from that of _A'_ and _B'_, which is simple; so that this complicated
+relation may be replaced by the simple relation between _A'_ and _B'_
+and by two other relations which tell us that the differences between
+_A_ and _A'_, on the one hand, between _B_ and _B'_, on the other hand,
+are _very small_. For example, if _A_ and _B_ are two natural solid
+bodies which are displaced with slight deformation, we envisage two
+movable _rigid_ figures _A'_ and _B'_. The laws of the relative
+displacement of these figures _A'_ and _B'_ will be very simple; they
+will be those of geometry. And we shall afterward add that the body _A_,
+which always differs very little from _A'_, dilates from the effect of
+heat and bends from the effect of elasticity. These dilatations and
+flexions, just because they are very small, will be for our mind
+relatively easy to study. Just imagine to what complexities of language
+it would have been necessary to be resigned if we had wished to
+comprehend in the same enunciation the displacement of the solid, its
+dilatation and its flexure?
+
+The relation between _A_ and _B_ was a rough law, and was broken up; we
+now have two laws which express the relations of _A_ and _A'_, of _B_
+and _B'_, and a principle which expresses that of _A'_ with _B'_. It is
+the aggregate of these principles that is called geometry.
+
+Two other remarks. We have a relation between two bodies _A_ and _B_,
+which we have replaced by a relation between two figures _A'_ and _B'_;
+but this same relation between the same two figures _A'_ and _B'_ could
+just as well have replaced advantageously a relation between two other
+bodies _A''_ and _B''_, entirely different from _A_ and _B_. And that in
+many ways. If the principles of geometry had not been invented, after
+having studied the relation of _A_ and _B_, it would be necessary to
+begin again _ab ovo_ the study of the relation of _A''_ and _B''_.
+That is why geometry is so precious. A geometrical relation can
+advantageously replace a relation which, considered in the rough state,
+should be regarded as mechanical, it can replace another which should be
+regarded as optical, etc.
+
+Yet let no one say: But that proves geometry an experimental science; in
+separating its principles from laws whence they have been drawn, you
+artificially separate it itself from the sciences which have given birth
+to it. The other sciences have likewise principles, but that does not
+preclude our having to call them experimental.
+
+It must be recognized that it would have been difficult not to make this
+separation that is pretended to be artificial. We know the rôle that the
+kinematics of solid bodies has played in the genesis of geometry; should
+it then be said that geometry is only a branch of experimental
+kinematics? But the laws of the rectilinear propagation of light have
+also contributed to the formation of its principles. Must geometry be
+regarded both as a branch of kinematics and as a branch of optics? I
+recall besides that our Euclidean space which is the proper object of
+geometry has been chosen, for reasons of convenience, from among a
+certain number of types which preexist in our mind and which are called
+groups.
+
+If we pass to mechanics, we still see great principles whose origin is
+analogous, and, as their 'radius of action,' so to speak, is smaller,
+there is no longer reason to separate them from mechanics proper and to
+regard this science as deductive.
+
+In physics, finally, the rôle of the principles is still more
+diminished. And in fact they are only introduced when it is of
+advantage. Now they are advantageous precisely because they are few,
+since each of them very nearly replaces a great number of laws.
+Therefore it is not of interest to multiply them. Besides an outcome is
+necessary, and for that it is needful to end by leaving abstraction to
+take hold of reality.
+
+Such are the limits of nominalism, and they are narrow.
+
+M. LeRoy has insisted, however, and he has put the question under
+another form.
+
+Since the enunciation of our laws may vary with the conventions that we
+adopt, since these conventions may modify even the natural relations of
+these laws, is there in the manifold of these laws something independent
+of these conventions and which may, so to speak, play the rôle of
+_universal invariant_? For instance, the fiction has been introduced of
+beings who, having been educated in a world different from ours, would
+have been led to create a non-Euclidean geometry. If these beings were
+afterward suddenly transported into our world, they would observe the
+same laws as we, but they would enunciate them in an entirely different
+way. In truth there would still be something in common between the two
+enunciations, but this is because these beings do not yet differ enough
+from us. Beings still more strange may be imagined, and the part common
+to the two systems of enunciations will shrink more and more. Will it
+thus shrink in convergence toward zero, or will there remain an
+irreducible residue which will then be the universal invariant sought?
+
+The question calls for precise statement. Is it desired that this common
+part of the enunciations be expressible in words? It is clear, then,
+that there are not words common to all languages, and we can not pretend
+to construct I know not what universal invariant which should be
+understood both by us and by the fictitious non-Euclidean geometers of
+whom I have just spoken; no more than we can construct a phrase which
+can be understood both by Germans who do not understand French and by
+French who do not understand German. But we have fixed rules which
+permit us to translate the French enunciations into German, and
+inversely. It is for that that grammars and dictionaries have been made.
+There are also fixed rules for translating the Euclidean language into
+the non-Euclidean language, or, if there are not, they could be made.
+
+And even if there were neither interpreter nor dictionary, if the
+Germans and the French, after having lived centuries in separate worlds,
+found themselves all at once in contact, do you think there would be
+nothing in common between the science of the German books and that of
+the French books? The French and the Germans would certainly end by
+understanding each other, as the American Indians ended by understanding
+the language of their conquerors after the arrival of the Spanish.
+
+But, it will be said, doubtless the French would be capable of
+understanding the Germans even without having learned German, but this
+is because there remains between the French and the Germans something in
+common, since both are men. We should still attain to an understanding
+with our hypothetical non-Euclideans, though they be not men, because
+they would still retain something human. But in any case a minimum of
+humanity is necessary.
+
+This is possible, but I shall observe first that this little humanness
+which would remain in the non-Euclideans would suffice not only to make
+possible the translation of _a little_ of their language, but to make
+possible the translation of _all_ their language.
+
+Now, that there must be a minimum is what I concede; suppose there
+exists I know not what fluid which penetrates between the molecules of
+our matter, without having any action on it and without being subject to
+any action coming from it. Suppose beings sensible to the influence of
+this fluid and insensible to that of our matter. It is clear that the
+science of these beings would differ absolutely from ours and that it
+would be idle to seek an 'invariant' common to these two sciences. Or
+again, if these beings rejected our logic and did not admit, for
+instance, the principle of contradiction.
+
+But truly I think it without interest to examine such hypotheses.
+
+And then, if we do not push whimsicality so far, if we introduce only
+fictitious beings having senses analogous to ours and sensible to the
+same impressions, and moreover admitting the principles of our logic, we
+shall then be able to conclude that their language, however different
+from ours it may be, would always be capable of translation. Now the
+possibility of translation implies the existence of an invariant. To
+translate is precisely to disengage this invariant. Thus, to decipher a
+cryptogram is to seek what in this document remains invariant, when the
+letters are permuted.
+
+What now is the nature of this invariant it is easy to understand, and a
+word will suffice us. The invariant laws are the relations between the
+crude facts, while the relations between the 'scientific facts' remain
+always dependent on certain conventions.
+
+
+
+
+CHAPTER XI
+
+SCIENCE AND REALITY
+
+
+5. _Contingence and Determinism_
+
+I do not intend to treat here the question of the contingence of the
+laws of nature, which is evidently insoluble, and on which so much has
+already been written. I only wish to call attention to what different
+meanings have been given to this word, contingence, and how advantageous
+it would be to distinguish them.
+
+If we look at any particular law, we may be certain in advance that it
+can only be approximate. It is, in fact, deduced from experimental
+verifications, and these verifications were and could be only
+approximate. We should always expect that more precise measurements will
+oblige us to add new terms to our formulas; this is what has happened,
+for instance, in the case of Mariotte's law.
+
+Moreover the statement of any law is necessarily incomplete. This
+enunciation should comprise the enumeration of _all_ the antecedents in
+virtue of which a given consequent can happen. I should first describe
+_all_ the conditions of the experiment to be made and the law would then
+be stated: If all the conditions are fulfilled, the phenomenon will
+happen.
+
+But we shall be sure of not having forgotten _any_ of these conditions
+only when we shall have described the state of the entire universe at
+the instant _t_; all the parts of this universe may, in fact, exercise
+an influence more or less great on the phenomenon which must happen at
+the instant _t_ + _dt_.
+
+Now it is clear that such a description could not be found in the
+enunciation of the law; besides, if it were made, the law would become
+incapable of application; if one required so many conditions, there
+would be very little chance of their ever being all realized at any
+moment.
+
+Then as one can never be certain of not having forgotten some essential
+condition, it can not be said: If such and such conditions are
+realized, such a phenomenon will occur; it can only be said: If such and
+such conditions are realized, it is probable that such a phenomenon will
+occur, very nearly.
+
+Take the law of gravitation, which is the least imperfect of all known
+laws. It enables us to foresee the motions of the planets. When I use
+it, for instance, to calculate the orbit of Saturn, I neglect the action
+of the stars, and in doing so I am certain of not deceiving myself,
+because I know that these stars are too far away for their action to be
+sensible.
+
+I announce, then, with a quasi-certitude that the coordinates of Saturn
+at such an hour will be comprised between such and such limits. Yet is
+that certitude absolute? Could there not exist in the universe some
+gigantic mass, much greater than that of all the known stars and whose
+action could make itself felt at great distances? That mass might be
+animated by a colossal velocity, and after having circulated from all
+time at such distances that its influence had remained hitherto
+insensible to us, it might come all at once to pass near us. Surely it
+would produce in our solar system enormous perturbations that we could
+not have foreseen. All that can be said is that such an event is wholly
+improbable, and then, instead of saying: Saturn will be near such a
+point of the heavens, we must limit ourselves to saying: Saturn will
+probably be near such a point of the heavens. Although this probability
+may be practically equivalent to certainty, it is only a probability.
+
+For all these reasons, no particular law will ever be more than
+approximate and probable. Scientists have never failed to recognize this
+truth; only they believe, right or wrong, that every law may be replaced
+by another closer and more probable, that this new law will itself be
+only provisional, but that the same movement can continue indefinitely,
+so that science in progressing will possess laws more and more probable,
+that the approximation will end by differing as little as you choose
+from exactitude and the probability from certitude.
+
+If the scientists who think thus are right, still could it be said that
+_the_ laws of nature are contingent, even though _each_ law, taken in
+particular, may be qualified as contingent? Or must one require, before
+concluding the contingence _of the_ natural laws, that this progress
+have an end, that the scientist finish some day by being arrested in his
+search for a closer and closer approximation, and that, beyond a certain
+limit, he thereafter meet in nature only caprice?
+
+In the conception of which I have just spoken (and which I shall call
+the scientific conception), every law is only a statement imperfect and
+provisional, but it must one day be replaced by another, a superior law,
+of which it is only a crude image. No place therefore remains for the
+intervention of a free will.
+
+It seems to me that the kinetic theory of gases will furnish us a
+striking example.
+
+You know that in this theory all the properties of gases are explained
+by a simple hypothesis; it is supposed that all the gaseous molecules
+move in every direction with great velocities and that they follow
+rectilineal paths which are disturbed only when one molecule passes very
+near the sides of the vessel or another molecule. The effects our crude
+senses enable us to observe are the mean effects, and in these means,
+the great deviations compensate, or at least it is very improbable that
+they do not compensate; so that the observable phenomena follow simple
+laws such as that of Mariotte or of Gay-Lussac. But this compensation of
+deviations is only probable. The molecules incessantly change place and
+in these continual displacements the figures they form pass successively
+through all possible combinations. Singly these combinations are very
+numerous; almost all are in conformity with Mariotte's law, only a few
+deviate from it. These also will happen, only it would be necessary to
+wait a long time for them. If a gas were observed during a sufficiently
+long time it would certainly be finally seen to deviate, for a very
+short time, from Mariotte's law. How long would it be necessary to wait?
+If it were desired to calculate the probable number of years, it would
+be found that this number is so great that to write only the number of
+places of figures employed would still require half a score places of
+figures. No matter; enough that it may be done.
+
+I do not care to discuss here the value of this theory. It is evident
+that if it be adopted, Mariotte's law will thereafter appear only as
+contingent, since a day will come when it will not be true. And yet,
+think you the partisans of the kinetic theory are adversaries of
+determinism? Far from it; they are the most ultra of mechanists. Their
+molecules follow rigid paths, from which they depart only under the
+influence of forces which vary with the distance, following a perfectly
+determinate law. There remains in their system not the smallest place
+either for freedom, or for an evolutionary factor, properly so-called,
+or for anything whatever that could be called contingence. I add, to
+avoid mistake, that neither is there any evolution of Mariotte's law
+itself; it ceases to be true after I know not how many centuries; but at
+the end of a fraction of a second it again becomes true and that for an
+incalculable number of centuries.
+
+And since I have pronounced the word evolution, let us clear away
+another mistake. It is often said: Who knows whether the laws do not
+evolve and whether we shall not one day discover that they were not at
+the Carboniferous epoch what they are to-day? What are we to understand
+by that? What we think we know about the past state of our globe, we
+deduce from its present state. And how is this deduction made? It is by
+means of laws supposed known. The law, being a relation between the
+antecedent and the consequent, enables us equally well to deduce the
+consequent from the antecedent, that is, to foresee the future, and to
+deduce the antecedent from the consequent, that is, to conclude from the
+present to the past. The astronomer who knows the present situation of
+the stars can from it deduce their future situation by Newton's law, and
+this is what he does when he constructs ephemerides; and he can equally
+deduce from it their past situation. The calculations he thus can make
+can not teach him that Newton's law will cease to be true in the future,
+since this law is precisely his point of departure; not more can they
+tell him it was not true in the past. Still, in what concerns the
+future, his ephemerides can one day be tested and our descendants will
+perhaps recognize that they were false. But in what concerns the past,
+the geologic past which had no witnesses, the results of his
+calculation, like those of all speculations where we seek to deduce the
+past from the present, escape by their very nature every species of
+test. So that if the laws of nature were not the same in the
+Carboniferous age as at the present epoch, we shall never be able to
+know it, since we can know nothing of this age, only what we deduce from
+the hypothesis of the permanence of these laws.
+
+Perhaps it will be said that this hypothesis might lead to contradictory
+results and that we shall be obliged to abandon it. Thus, in what
+concerns the origin of life, we may conclude that there have always been
+living beings, since the present world shows us always life springing
+from life; and we may also conclude that there have not always been,
+since the application of the existent laws of physics to the present
+state of our globe teaches us that there was a time when this globe was
+so warm that life on it was impossible. But contradictions of this sort
+can always be removed in two ways; it may be supposed that the actual
+laws of nature are not exactly what we have assumed; or else it may be
+supposed that the laws of nature actually are what we have assumed, but
+that it has not always been so.
+
+It is evident that the actual laws will never be sufficiently well known
+for us not to be able to adopt the first of these two solutions and for
+us to be constrained to infer the evolution of natural laws.
+
+On the other hand, suppose such an evolution; assume, if you wish, that
+humanity lasts sufficiently long for this evolution to have witnesses.
+The _same_ antecedent shall produce, for instance, different consequents
+at the Carboniferous epoch and at the Quaternary. That evidently means
+that the antecedents are closely alike; if all the circumstances were
+identical, the Carboniferous epoch would be indistinguishable from the
+Quaternary. Evidently this is not what is supposed. What remains is that
+such antecedent, accompanied by such accessory circumstance, produces
+such consequent; and that the same antecedent, accompanied by such other
+accessory circumstance, produces such other consequent. Time does not
+enter into the affair.
+
+The law, such as ill-informed science would have stated it, and which
+would have affirmed that this antecedent always produces this
+consequent, without taking account of the accessory circumstances, this
+law, which was only approximate and probable, must be replaced by
+another law more approximate and more probable, which brings in these
+accessory circumstances. We always come back, therefore, to that same
+process which we have analyzed above, and if humanity should discover
+something of this sort, it would not say that it is the laws which have
+evoluted, but the circumstances which have changed.
+
+Here, therefore, are several different senses of the word contingence.
+M. LeRoy retains them all and he does not sufficiently distinguish them,
+but he introduces a new one. Experimental laws are only approximate, and
+if some appear to us as exact, it is because we have artificially
+transformed them into what I have above called a principle. We have made
+this transformation freely, and as the caprice which has determined us
+to make it is something eminently contingent, we have communicated this
+contingence to the law itself. It is in this sense that we have the
+right to say that determinism supposes freedom, since it is freely that
+we become determinists. Perhaps it will be found that this is to give
+large scope to nominalism and that the introduction of this new sense of
+the word contingence will not help much to solve all those questions
+which naturally arise and of which we have just been speaking.
+
+I do not at all wish to investigate here the foundations of the
+principle of induction; I know very well that I should not succeed; it
+is as difficult to justify this principle as to get on without it. I
+only wish to show how scientists apply it and are forced to apply it.
+
+When the same antecedent recurs, the same consequent must likewise
+recur; such is the ordinary statement. But reduced to these terms this
+principle could be of no use. For one to be able to say that the same
+antecedent recurred, it would be necessary for the circumstances _all_
+to be reproduced, since no one is absolutely indifferent, and for them
+to be _exactly_ reproduced. And, as that will never happen, the
+principle can have no application.
+
+We should therefore modify the enunciation and say: If an antecedent _A_
+has once produced a consequent _B_, an antecedent _A'_, slightly
+different from _A_, will produce a consequent _B'_, slightly different
+from _B_. But how shall we recognize that the antecedents _A_ and _A'_
+are 'slightly different'? If some one of the circumstances can be
+expressed by a number, and this number has in the two cases values very
+near together, the sense of the phrase 'slightly different' is
+relatively clear; the principle then signifies that the consequent is a
+continuous function of the antecedent. And as a practical rule, we reach
+this conclusion that we have the right to interpolate. This is in fact
+what scientists do every day, and without interpolation all science
+would be impossible.
+
+Yet observe one thing. The law sought may be represented by a curve.
+Experiment has taught us certain points of this curve. In virtue of the
+principle we have just stated, we believe these points may be connected
+by a continuous graph. We trace this graph with the eye. New experiments
+will furnish us new points of the curve. If these points are outside of
+the graph traced in advance, we shall have to modify our curve, but not
+to abandon our principle. Through any points, however numerous they may
+be, a continuous curve may always be passed. Doubtless, if this curve is
+too capricious, we shall be shocked (and we shall even suspect errors of
+experiment), but the principle will not be directly put at fault.
+
+Furthermore, among the circumstances of a phenomenon, there are some
+that we regard as negligible, and we shall consider _A_ and _A'_ as
+slightly different if they differ only by these accessory circumstances.
+For instance, I have ascertained that hydrogen unites with oxygen under
+the influence of the electric spark, and I am certain that these two
+gases will unite anew, although the longitude of Jupiter may have
+changed considerably in the interval. We assume, for instance, that the
+state of distant bodies can have no sensible influence on terrestrial
+phenomena, and that seems in fact requisite, but there are cases where
+the choice of these practically indifferent circumstances admits of more
+arbitrariness or, if you choose, requires more tact.
+
+One more remark: The principle of induction would be inapplicable if
+there did not exist in nature a great quantity of bodies like one
+another, or almost alike, and if we could not infer, for instance, from
+one bit of phosphorus to another bit of phosphorus.
+
+If we reflect on these considerations, the problem of determinism and of
+contingence will appear to us in a new light.
+
+Suppose we were able to embrace the series of all phenomena of the
+universe in the whole sequence of time. We could envisage what might be
+called the _sequences_; I mean relations between antecedent and
+consequent. I do not wish to speak of constant relations or laws, I
+envisage separately (individually, so to speak) the different sequences
+realized.
+
+We should then recognize that among these sequences there are no two
+altogether alike. But, if the principle of induction, as we have just
+stated it, is true, there will be those almost alike and that can be
+classed alongside one another. In other words, it is possible to make a
+classification of sequences.
+
+It is to the possibility and the legitimacy of such a classification
+that determinism, in the end, reduces. This is all that the preceding
+analysis leaves of it. Perhaps under this modest form it will seem less
+appalling to the moralist.
+
+It will doubtless be said that this is to come back by a detour to M.
+LeRoy's conclusion which a moment ago we seemed to reject: we are
+determinists voluntarily. And in fact all classification supposes the
+active intervention of the classifier. I agree that this may be
+maintained, but it seems to me that this detour will not have been
+useless and will have contributed to enlighten us a little.
+
+
+6. _Objectivity of Science_
+
+I arrive at the question set by the title of this article: What is the
+objective value of science? And first what should we understand by
+objectivity?
+
+What guarantees the objectivity of the world in which we live is that
+this world is common to us with other thinking beings. Through the
+communications that we have with other men, we receive from them
+ready-made reasonings; we know that these reasonings do not come from us
+and at the same time we recognize in them the work of reasonable beings
+like ourselves. And as these reasonings appear to fit the world of our
+sensations, we think we may infer that these reasonable beings have seen
+the same thing as we; thus it is we know we have not been dreaming.
+
+Such, therefore, is the first condition of objectivity; what is
+objective must be common to many minds and consequently transmissible
+from one to the other, and as this transmission can only come about by
+that 'discourse' which inspires so much distrust in M. LeRoy, we are
+even forced to conclude: no discourse, no objectivity.
+
+The sensations of others will be for us a world eternally closed. We
+have no means of verifying that the sensation I call red is the same as
+that which my neighbor calls red.
+
+Suppose that a cherry and a red poppy produce on me the sensation _A_
+and on him the sensation _B_ and that, on the contrary, a leaf produces
+on me the sensation _B_ and on him the sensation _A_. It is clear we
+shall never know anything about it; since I shall call red the sensation
+_A_ and green the sensation _B_, while he will call the first green and
+the second red. In compensation, what we shall be able to ascertain is
+that, for him as for me, the cherry and the red poppy produce the _same_
+sensation, since he gives the same name to the sensations he feels and I
+do the same.
+
+Sensations are therefore intransmissible, or rather all that is pure
+quality in them is intransmissible and forever impenetrable. But it is
+not the same with relations between these sensations.
+
+From this point of view, all that is objective is devoid of all quality
+and is only pure relation. Certes, I shall not go so far as to say that
+objectivity is only pure quantity (this would be to particularize too
+far the nature of the relations in question), but we understand how some
+one could have been carried away into saying that the world is only a
+differential equation.
+
+With due reserve regarding this paradoxical proposition, we must
+nevertheless admit that nothing is objective which is not transmissible,
+and consequently that the relations between the sensations can alone
+have an objective value.
+
+Perhaps it will be said that the esthetic emotion, which is common to
+all mankind, is proof that the qualities of our sensations are also the
+same for all men and hence are objective. But if we think about this, we
+shall see that the proof is not complete; what is proved is that this
+emotion is aroused in John as in James by the sensations to which James
+and John give the same name or by the corresponding combinations of
+these sensations; either because this emotion is associated in John with
+the sensation _A_, which John calls red, while parallelly it is
+associated in James with the sensation _B_, which James calls red; or
+better because this emotion is aroused, not by the qualities themselves
+of the sensations, but by the harmonious combination of their relations
+of which we undergo the unconscious impression.
+
+Such a sensation is beautiful, not because it possesses such a quality,
+but because it occupies such a place in the woof of our associations of
+ideas, so that it can not be excited without putting in motion the
+'receiver' which is at the other end of the thread and which corresponds
+to the artistic emotion.
+
+Whether we take the moral, the esthetic or the scientific point of view,
+it is always the same thing. Nothing is objective except what is
+identical for all; now we can only speak of such an identity if a
+comparison is possible, and can be translated into a 'money of exchange'
+capable of transmission from one mind to another. Nothing, therefore,
+will have objective value except what is transmissible by 'discourse,'
+that is, intelligible.
+
+But this is only one side of the question. An absolutely disordered
+aggregate could not have objective value since it would be
+unintelligible, but no more can a well-ordered assemblage have it, if it
+does not correspond to sensations really experienced. It seems to me
+superfluous to recall this condition, and I should not have dreamed of
+it, if it had not lately been maintained that physics is not an
+experimental science. Although this opinion has no chance of being
+adopted either by physicists or by philosophers, it is well to be warned
+so as not to let oneself slip over the declivity which would lead
+thither. Two conditions are therefore to be fulfilled, and if the first
+separates reality[11] from the dream, the second distinguishes it from
+the romance.
+
+ [11] I here use the word real as a synonym of objective; I thus
+ conform to common usage; perhaps I am wrong, our dreams are
+ real, but they are not objective.
+
+Now what is science? I have explained in the preceding article, it is
+before all a classification, a manner of bringing together facts which
+appearances separate, though they were bound together by some natural
+and hidden kinship. Science, in other words, is a system of relations.
+Now we have just said, it is in the relations alone that objectivity
+must be sought; it would be vain to seek it in beings considered as
+isolated from one another.
+
+To say that science can not have objective value since it teaches us
+only relations, this is to reason backward, since, precisely, it is
+relations alone which can be regarded as objective.
+
+External objects, for instance, for which the word _object_ was
+invented, are really _objects_ and not fleeting and fugitive
+appearances, because they are not only groups of sensations, but groups
+cemented by a constant bond. It is this bond, and this bond alone, which
+is the object in itself, and this bond is a relation.
+
+Therefore, when we ask what is the objective value of science, that does
+not mean: Does science teach us the true nature of things? but it means:
+Does it teach us the true relations of things?
+
+To the first question, no one would hesitate to reply, no; but I think
+we may go farther; not only science can not teach us the nature of
+things; but nothing is capable of teaching it to us, and if any god knew
+it, he could not find words to express it. Not only can we not divine
+the response, but if it were given to us we could understand nothing of
+it; I ask myself even whether we really understand the question.
+
+When, therefore, a scientific theory pretends to teach us what heat is,
+or what is electricity, or life, it is condemned beforehand; all it can
+give us is only a crude image. It is, therefore, provisional and
+crumbling.
+
+The first question being out of reason, the second remains. Can science
+teach us the true relations of things? What it joins together should
+that be put asunder, what it puts asunder should that be joined
+together?
+
+To understand the meaning of this new question, it is needful to refer
+to what was said above on the conditions of objectivity. Have these
+relations an objective value? That means: Are these relations the same
+for all? Will they still be the same for those who shall come after us?
+
+It is clear that they are not the same for the scientist and the
+ignorant person. But that is unimportant, because if the ignorant person
+does not see them all at once, the scientist may succeed in making him
+see them by a series of experiments and reasonings. The thing essential
+is that there are points on which all those acquainted with the
+experiments made can reach accord.
+
+The question is to know whether this accord will be durable and whether
+it will persist for our successors. It may be asked whether the unions
+that the science of to-day makes will be confirmed by the science of
+to-morrow. To affirm that it will be so we can not invoke any _a priori_
+reason; but this is a question of fact, and science has already lived
+long enough for us to be able to find out by asking its history whether
+the edifices it builds stand the test of time, or whether they are only
+ephemeral constructions.
+
+Now what do we see? At the first blush, it seems to us that the theories
+last only a day and that ruins upon ruins accumulate. To-day the
+theories are born, to-morrow they are the fashion, the day after
+to-morrow they are classic, the fourth day they are superannuated, and
+the fifth they are forgotten. But if we look more closely, we see that
+what thus succumb are the theories properly so called, those which
+pretend to teach us what things are. But there is in them something
+which usually survives. If one of them taught us a true relation, this
+relation is definitively acquired, and it will be found again under a
+new disguise in the other theories which will successively come to reign
+in place of the old.
+
+Take only a single example: The theory of the undulations of the ether
+taught us that light is a motion; to-day fashion favors the
+electromagnetic theory which teaches us that light is a current. We do
+not consider whether we could reconcile them and say that light is a
+current, and that this current is a motion. As it is probable in any
+case that this motion would not be identical with that which the
+partisans of the old theory presume, we might think ourselves justified
+in saying that this old theory is dethroned. And yet something of it
+remains, since between the hypothetical currents which Maxwell supposes
+there are the same relations as between the hypothetical motions that
+Fresnel supposed. There is, therefore, something which remains over and
+this something is the essential. This it is which explains how we see
+the present physicists pass without any embarrassment from the language
+of Fresnel to that of Maxwell. Doubtless many connections that were
+believed well established have been abandoned, but the greatest number
+remain and it would seem must remain.
+
+And for these, then, what is the measure of their objectivity? Well, it
+is precisely the same as for our belief in external objects. These
+latter are real in this, that the sensations they make us feel appear to
+us as united to each other by I know not what indestructible cement and
+not by the hazard of a day. In the same way science reveals to us
+between phenomena other bonds finer but not less solid; these are
+threads so slender that they long remained unperceived, but once noticed
+there remains no way of not seeing them; they are therefore not less
+real than those which give their reality to external objects; small
+matter that they are more recently known, since neither can perish
+before the other.
+
+It may be said, for instance, that the ether is no less real than any
+external body; to say this body exists is to say there is between the
+color of this body, its taste, its smell, an intimate bond, solid and
+persistent; to say the ether exists is to say there is a natural kinship
+between all the optical phenomena, and neither of the two propositions
+has less value than the other.
+
+And the scientific syntheses have in a sense even more reality than
+those of the ordinary senses, since they embrace more terms and tend to
+absorb in them the partial syntheses.
+
+It will be said that science is only a classification and that a
+classification can not be true, but convenient. But it is true that it
+is convenient, it is true that it is so not only for me, but for all
+men; it is true that it will remain convenient for our descendants; it
+is true finally that this can not be by chance.
+
+In sum, the sole objective reality consists in the relations of things
+whence results the universal harmony. Doubtless these relations, this
+harmony, could not be conceived outside of a mind which conceives them.
+But they are nevertheless objective because they are, will become, or
+will remain, common to all thinking beings.
+
+This will permit us to revert to the question of the rotation of the
+earth which will give us at the same time a chance to make clear what
+precedes by an example.
+
+
+7. _The Rotation of the Earth_
+
+"... Therefore," have I said in _Science and Hypothesis_, "this
+affirmation, the earth turns round, has no meaning ... or rather these
+two propositions, the earth turns round, and, it is more convenient to
+suppose that the earth turns round, have one and the same meaning."
+
+These words have given rise to the strangest interpretations. Some have
+thought they saw in them the rehabilitation of Ptolemy's system, and
+perhaps the justification of Galileo's condemnation.
+
+Those who had read attentively the whole volume could not, however,
+delude themselves. This truth, the earth turns round, was put on the
+same footing as Euclid's postulate, for example. Was that to reject it?
+But better; in the same language it may very well be said: These two
+propositions, the external world exists, or, it is more convenient to
+suppose that it exists, have one and the same meaning. So the hypothesis
+of the rotation of the earth would have the same degree of certitude as
+the very existence of external objects.
+
+But after what we have just explained in the fourth part, we may go
+farther. A physical theory, we have said, is by so much the more true as
+it puts in evidence more true relations. In the light of this new
+principle, let us examine the question which occupies us.
+
+No, there is no absolute space; these two contradictory propositions:
+'The earth turns round' and 'The earth does not turn round' are,
+therefore, neither of them more true than the other. To affirm one while
+denying the other, _in the kinematic sense_, would be to admit the
+existence of absolute space.
+
+But if the one reveals true relations that the other hides from us, we
+can nevertheless regard it as physically more true than the other, since
+it has a richer content. Now in this regard no doubt is possible.
+
+Behold the apparent diurnal motion of the stars, and the diurnal motion
+of the other heavenly bodies, and besides, the flattening of the earth,
+the rotation of Foucault's pendulum, the gyration of cyclones, the
+trade-winds, what not else? For the Ptolemaist all these phenomena have
+no bond between them; for the Copernican they are produced by the one
+same cause. In saying, the earth turns round, I affirm that all these
+phenomena have an intimate relation, and _that is true_, and that
+remains true, although there is not and can not be absolute space.
+
+So much for the rotation of the earth upon itself; what shall we say of
+its revolution around the sun? Here again, we have three phenomena which
+for the Ptolemaist are absolutely independent and which for the
+Copernican are referred back to the same origin; they are the apparent
+displacements of the planets on the celestial sphere, the aberration of
+the fixed stars, the parallax of these same stars. Is it by chance that
+all the planets admit an inequality whose period is a year, and that
+this period is precisely equal to that of aberration, precisely equal
+besides to that of parallax? To adopt Ptolemy's system is to answer,
+yes; to adopt that of Copernicus is to answer, no; this is to affirm
+that there is a bond between the three phenomena, and that also is true,
+although there is no absolute space.
+
+In Ptolemy's system, the motions of the heavenly bodies can not be
+explained by the action of central forces, celestial mechanics is
+impossible. The intimate relations that celestial mechanics reveals to
+us between all the celestial phenomena are true relations; to affirm the
+immobility of the earth would be to deny these relations, that would be
+to fool ourselves.
+
+The truth for which Galileo suffered remains, therefore, the truth,
+although it has not altogether the same meaning as for the vulgar, and
+its true meaning is much more subtle, more profound and more rich.
+
+
+8. _Science for Its Own Sake_
+
+Not against M. LeRoy do I wish to defend science for its own sake; maybe
+this is what he condemns, but this is what he cultivates, since he loves
+and seeks truth and could not live without it. But I have some thoughts
+to express.
+
+We can not know all facts and it is necessary to choose those which are
+worthy of being known. According to Tolstoi, scientists make this choice
+at random, instead of making it, which would be reasonable, with a view
+to practical applications. On the contrary, scientists think that
+certain facts are more interesting than others, because they complete an
+unfinished harmony, or because they make one foresee a great number of
+other facts. If they are wrong, if this hierarchy of facts that they
+implicitly postulate is only an idle illusion, there could be no science
+for its own sake, and consequently there could be no science. As for me,
+I believe they are right, and, for example, I have shown above what is
+the high value of astronomical facts, not because they are capable of
+practical applications, but because they are the most instructive of
+all.
+
+It is only through science and art that civilization is of value. Some
+have wondered at the formula: science for its own sake; and yet it is as
+good as life for its own sake, if life is only misery; and even as
+happiness for its own sake, if we do not believe that all pleasures are
+of the same quality, if we do not wish to admit that the goal of
+civilization is to furnish alcohol to people who love to drink.
+
+Every act should have an aim. We must suffer, we must work, we must pay
+for our place at the game, but this is for seeing's sake; or at the very
+least that others may one day see.
+
+All that is not thought is pure nothingness; since we can think only
+thoughts and all the words we use to speak of things can express only
+thoughts, to say there is something other than thought, is therefore an
+affirmation which can have no meaning.
+
+And yet--strange contradiction for those who believe in time--geologic
+history shows us that life is only a short episode between two
+eternities of death, and that, even in this episode, conscious thought
+has lasted and will last only a moment. Thought is only a gleam in the
+midst of a long night.
+
+But it is this gleam which is everything.
+
+
+ * * * * *
+
+
+
+
+SCIENCE AND METHOD
+
+
+
+
+ * * * * *
+
+
+
+
+INTRODUCTION
+
+
+I bring together here different studies relating more or less directly
+to questions of scientific methodology. The scientific method consists
+in observing and experimenting; if the scientist had at his disposal
+infinite time, it would only be necessary to say to him: 'Look and
+notice well'; but, as there is not time to see everything, and as it is
+better not to see than to see wrongly, it is necessary for him to make
+choice. The first question, therefore, is how he should make this
+choice. This question presents itself as well to the physicist as to the
+historian; it presents itself equally to the mathematician, and the
+principles which should guide each are not without analogy. The
+scientist conforms to them instinctively, and one can, reflecting on
+these principles, foretell the future of mathematics.
+
+We shall understand them better yet if we observe the scientist at work,
+and first of all it is necessary to know the psychologic mechanism of
+invention and, in particular, that of mathematical creation. Observation
+of the processes of the work of the mathematician is particularly
+instructive for the psychologist.
+
+In all the sciences of observation account must be taken of the errors
+due to the imperfections of our senses and our instruments. Luckily, we
+may assume that, under certain conditions, these errors are in part
+self-compensating, so as to disappear in the average; this compensation
+is due to chance. But what is chance? This idea is difficult to justify
+or even to define; and yet what I have just said about the errors of
+observation, shows that the scientist can not neglect it. It therefore
+is necessary to give a definition as precise as possible of this
+concept, so indispensable yet so illusive.
+
+These are generalities applicable in sum to all the sciences; and for
+example the mechanism of mathematical invention does not differ sensibly
+from the mechanism of invention in general. Later I attack questions
+relating more particularly to certain special sciences and first to pure
+mathematics.
+
+In the chapters devoted to these, I have to treat subjects a little more
+abstract. I have first to speak of the notion of space; every one knows
+space is relative, or rather every one says so, but many think still as
+if they believed it absolute; it suffices to reflect a little however to
+perceive to what contradictions they are exposed.
+
+The questions of teaching have their importance, first in themselves,
+then because reflecting on the best way to make new ideas penetrate
+virgin minds is at the same time reflecting on how these notions were
+acquired by our ancestors, and consequently on their true origin, that
+is to say, in reality on their true nature. Why do children usually
+understand nothing of the definitions which satisfy scientists? Why is
+it necessary to give them others? This is the question I set myself in
+the succeeding chapter and whose solution should, I think, suggest
+useful reflections to the philosophers occupied with the logic of the
+sciences.
+
+On the other hand, many geometers believe we can reduce mathematics to
+the rules of formal logic. Unheard-of efforts have been made to do this;
+to accomplish it, some have not hesitated, for example, to reverse the
+historic order of the genesis of our conceptions and to try to explain
+the finite by the infinite. I believe I have succeeded in showing, for
+all those who attack the problem unprejudiced, that here there is a
+fallacious illusion. I hope the reader will understand the importance of
+the question and pardon me the aridity of the pages devoted to it.
+
+The concluding chapters relative to mechanics and astronomy will be
+easier to read.
+
+Mechanics seems on the point of undergoing a complete revolution. Ideas
+which appeared best established are assailed by bold innovators.
+Certainly it would be premature to decide in their favor at once simply
+because they are innovators.
+
+But it is of interest to make known their doctrines, and this is what I
+have tried to do. As far as possible I have followed the historic order;
+for the new ideas would seem too astonishing unless we saw how they
+arose.
+
+Astronomy offers us majestic spectacles and raises gigantic problems. We
+can not dream of applying to them directly the experimental method; our
+laboratories are too small. But analogy with phenomena these
+laboratories permit us to attain may nevertheless guide the astronomer.
+The Milky Way, for example, is an assemblage of suns whose movements
+seem at first capricious. But may not this assemblage be compared to
+that of the molecules of a gas, whose properties the kinetic theory of
+gases has made known to us? It is thus by a roundabout way that the
+method of the physicist may come to the aid of the astronomer.
+
+Finally I have endeavored to give in a few lines the history of the
+development of French geodesy; I have shown through what persevering
+efforts, and often what dangers, the geodesists have procured for us the
+knowledge we have of the figure of the earth. Is this then a question of
+method? Yes, without doubt, this history teaches us in fact by what
+precautions it is necessary to surround a serious scientific operation
+and how much time and pains it costs to conquer one new decimal.
+
+
+
+
+BOOK I
+
+
+SCIENCE AND THE SCIENTIST
+
+
+
+
+CHAPTER I
+
+THE CHOICE OF FACTS
+
+
+Tolstoi somewhere explains why 'science for its own sake' is in his eyes
+an absurd conception. We can not know _all_ facts, since their number is
+practically infinite. It is necessary to choose; then we may let this
+choice depend on the pure caprice of our curiosity; would it not be
+better to let ourselves be guided by utility, by our practical and above
+all by our moral needs; have we nothing better to do than to count the
+number of lady-bugs on our planet?
+
+It is clear the word utility has not for him the sense men of affairs
+give it, and following them most of our contemporaries. Little cares he
+for industrial applications, for the marvels of electricity or of
+automobilism, which he regards rather as obstacles to moral progress;
+utility for him is solely what can make man better.
+
+For my part, it need scarce be said, I could never be content with
+either the one or the other ideal; I want neither that plutocracy
+grasping and mean, nor that democracy goody and mediocre, occupied
+solely in turning the other cheek, where would dwell sages without
+curiosity, who, shunning excess, would not die of disease, but would
+surely die of ennui. But that is a matter of taste and is not what I
+wish to discuss.
+
+The question nevertheless remains and should fix our attention; if our
+choice can only be determined by caprice or by immediate utility, there
+can be no science for its own sake, and consequently no science. But is
+that true? That a choice must be made is incontestable; whatever be our
+activity, facts go quicker than we, and we can not catch them; while the
+scientist discovers one fact, there happen milliards of milliards in a
+cubic millimeter of his body. To wish to comprise nature in science
+would be to want to put the whole into the part.
+
+But scientists believe there is a hierarchy of facts and that among them
+may be made a judicious choice. They are right, since otherwise there
+would be no science, yet science exists. One need only open the eyes to
+see that the conquests of industry which have enriched so many practical
+men would never have seen the light, if these practical men alone had
+existed and if they had not been preceded by unselfish devotees who died
+poor, who never thought of utility, and yet had a guide far other than
+caprice.
+
+As Mach says, these devotees have spared their successors the trouble of
+thinking. Those who might have worked solely in view of an immediate
+application would have left nothing behind them, and, in face of a new
+need, all must have been begun over again. Now most men do not love to
+think, and this is perhaps fortunate when instinct guides them, for most
+often, when they pursue an aim which is immediate and ever the same,
+instinct guides them better than reason would guide a pure intelligence.
+But instinct is routine, and if thought did not fecundate it, it would
+no more progress in man than in the bee or ant. It is needful then to
+think for those who love not thinking, and, as they are numerous, it is
+needful that each of our thoughts be as often useful as possible, and
+this is why a law will be the more precious the more general it is.
+
+This shows us how we should choose: the most interesting facts are those
+which may serve many times; these are the facts which have a chance of
+coming up again. We have been so fortunate as to be born in a world
+where there are such. Suppose that instead of 60 chemical elements there
+were 60 milliards of them, that they were not some common, the others
+rare, but that they were uniformly distributed. Then, every time we
+picked up a new pebble there would be great probability of its being
+formed of some unknown substance; all that we knew of other pebbles
+would be worthless for it; before each new object we should be as the
+new-born babe; like it we could only obey our caprices or our needs.
+Biologists would be just as much at a loss if there were only
+individuals and no species and if heredity did not make sons like their
+fathers.
+
+In such a world there would be no science; perhaps thought and even life
+would be impossible, since evolution could not there develop the
+preservational instincts. Happily it is not so; like all good fortune to
+which we are accustomed, this is not appreciated at its true worth.
+
+Which then are the facts likely to reappear? They are first the simple
+facts. It is clear that in a complex fact a thousand circumstances are
+united by chance, and that only a chance still much less probable could
+reunite them anew. But are there any simple facts? And if there are, how
+recognize them? What assurance is there that a thing we think simple
+does not hide a dreadful complexity? All we can say is that we ought to
+prefer the facts which _seem_ simple to those where our crude eye
+discerns unlike elements. And then one of two things: either this
+simplicity is real, or else the elements are so intimately mingled as
+not to be distinguishable. In the first case there is chance of our
+meeting anew this same simple fact, either in all its purity or entering
+itself as element in a complex manifold. In the second case this
+intimate mixture has likewise more chances of recurring than a
+heterogeneous assemblage; chance knows how to mix, it knows not how to
+disentangle, and to make with multiple elements a well-ordered edifice
+in which something is distinguishable, it must be made expressly. The
+facts which appear simple, even if they are not so, will therefore be
+more easily revived by chance. This it is which justifies the method
+instinctively adopted by the scientist, and what justifies it still
+better, perhaps, is that oft-recurring facts appear to us simple,
+precisely because we are used to them.
+
+But where is the simple fact? Scientists have been seeking it in the two
+extremes, in the infinitely great and in the infinitely small. The
+astronomer has found it because the distances of the stars are immense,
+so great that each of them appears but as a point, so great that the
+qualitative differences are effaced, and because a point is simpler than
+a body which has form and qualities. The physicist on the other hand has
+sought the elementary phenomenon in fictively cutting up bodies into
+infinitesimal cubes, because the conditions of the problem, which
+undergo slow and continuous variation in passing from one point of the
+body to another, may be regarded as constant in the interior of each of
+these little cubes. In the same way the biologist has been instinctively
+led to regard the cell as more interesting than the whole animal, and
+the outcome has shown his wisdom, since cells belonging to organisms the
+most different are more alike, for the one who can recognize their
+resemblances, than are these organisms themselves. The sociologist is
+more embarrassed; the elements, which for him are men, are too unlike,
+too variable, too capricious, in a word, too complex; besides, history
+never begins over again. How then choose the interesting fact, which is
+that which begins again? Method is precisely the choice of facts; it is
+needful then to be occupied first with creating a method, and many have
+been imagined, since none imposes itself, so that sociology is the
+science which has the most methods and the fewest results.
+
+Therefore it is by the regular facts that it is proper to begin; but
+after the rule is well established, after it is beyond all doubt, the
+facts in full conformity with it are erelong without interest since they
+no longer teach us anything new. It is then the exception which becomes
+important. We cease to seek resemblances; we devote ourselves above all
+to the differences, and among the differences are chosen first the most
+accentuated, not only because they are the most striking, but because
+they will be the most instructive. A simple example will make my thought
+plainer: Suppose one wishes to determine a curve by observing some of
+its points. The practician who concerns himself only with immediate
+utility would observe only the points he might need for some special
+object. These points would be badly distributed on the curve; they would
+be crowded in certain regions, rare in others, so that it would be
+impossible to join them by a continuous line, and they would be
+unavailable for other applications. The scientist will proceed
+differently; as he wishes to study the curve for itself, he will
+distribute regularly the points to be observed, and when enough are
+known he will join them by a regular line and then he will have the
+entire curve. But for that how does he proceed? If he has determined an
+extreme point of the curve, he does not stay near this extremity, but
+goes first to the other end; after the two extremities the most
+instructive point will be the mid-point, and so on.
+
+So when a rule is established we should first seek the cases where this
+rule has the greatest chance of failing. Thence, among other reasons,
+come the interest of astronomic facts, and the interest of the geologic
+past; by going very far away in space or very far away in time, we may
+find our usual rules entirely overturned, and these grand overturnings
+aid us the better to see or the better to understand the little changes
+which may happen nearer to us, in the little corner of the world where
+we are called to live and act. We shall better know this corner for
+having traveled in distant countries with which we have nothing to do.
+
+But what we ought to aim at is less the ascertainment of resemblances
+and differences than the recognition of likenesses hidden under apparent
+divergences. Particular rules seem at first discordant, but looking more
+closely we see in general that they resemble each other; different as to
+matter, they are alike as to form, as to the order of their parts. When
+we look at them with this bias, we shall see them enlarge and tend to
+embrace everything. And this it is which makes the value of certain
+facts which come to complete an assemblage and to show that it is the
+faithful image of other known assemblages.
+
+I will not further insist, but these few words suffice to show that the
+scientist does not choose at random the facts he observes. He does not,
+as Tolstoi says, count the lady-bugs, because, however interesting
+lady-bugs may be, their number is subject to capricious variations. He
+seeks to condense much experience and much thought into a slender
+volume; and that is why a little book on physics contains so many past
+experiences and a thousand times as many possible experiences whose
+result is known beforehand.
+
+But we have as yet looked at only one side of the question. The
+scientist does not study nature because it is useful; he studies it
+because he delights in it, and he delights in it because it is
+beautiful. If nature were not beautiful, it would not be worth knowing,
+and if nature were not worth knowing, life would not be worth living. Of
+course I do not here speak of that beauty which strikes the senses, the
+beauty of qualities and of appearances; not that I undervalue such
+beauty, far from it, but it has nothing to do with science; I mean that
+profounder beauty which comes from the harmonious order of the parts
+and which a pure intelligence can grasp. This it is which gives body, a
+structure so to speak, to the iridescent appearances which flatter our
+senses, and without this support the beauty of these fugitive dreams
+would be only imperfect, because it would be vague and always fleeting.
+On the contrary, intellectual beauty is sufficient unto itself, and it
+is for its sake, more perhaps than for the future good of humanity, that
+the scientist devotes himself to long and difficult labors.
+
+It is, therefore, the quest of this especial beauty, the sense of the
+harmony of the cosmos, which makes us choose the facts most fitting to
+contribute to this harmony, just as the artist chooses from among the
+features of his model those which perfect the picture and give it
+character and life. And we need not fear that this instinctive and
+unavowed prepossession will turn the scientist aside from the search for
+the true. One may dream a harmonious world, but how far the real world
+will leave it behind! The greatest artists that ever lived, the Greeks,
+made their heavens; how shabby it is beside the true heavens, ours!
+
+And it is because simplicity, because grandeur, is beautiful, that we
+preferably seek simple facts, sublime facts, that we delight now to
+follow the majestic course of the stars, now to examine with the
+microscope that prodigious littleness which is also a grandeur, now to
+seek in geologic time the traces of a past which attracts because it is
+far away.
+
+We see too that the longing for the beautiful leads us to the same
+choice as the longing for the useful. And so it is that this economy of
+thought, this economy of effort, which is, according to Mach, the
+constant tendency of science, is at the same time a source of beauty and
+a practical advantage. The edifices that we admire are those where the
+architect has known how to proportion the means to the end, where the
+columns seem to carry gaily, without effort, the weight placed upon
+them, like the gracious caryatids of the Erechtheum.
+
+Whence comes this concordance? Is it simply that the things which seem
+to us beautiful are those which best adapt themselves to our
+intelligence, and that consequently they are at the same time the
+implement this intelligence knows best how to use? Or is there here a
+play of evolution and natural selection? Have the peoples whose ideal
+most conformed to their highest interest exterminated the others and
+taken their place? All pursued their ideals without reference to
+consequences, but while this quest led some to destruction, to others it
+gave empire. One is tempted to believe it. If the Greeks triumphed over
+the barbarians and if Europe, heir of Greek thought, dominates the
+world, it is because the savages loved loud colors and the clamorous
+tones of the drum which occupied only their senses, while the Greeks
+loved the intellectual beauty which hides beneath sensuous beauty, and
+this intellectual beauty it is which makes intelligence sure and strong.
+
+Doubtless such a triumph would horrify Tolstoi, and he would not like to
+acknowledge that it might be truly useful. But this disinterested quest
+of the true for its own beauty is sane also and able to make man better.
+I well know that there are mistakes, that the thinker does not always
+draw thence the serenity he should find therein, and even that there are
+scientists of bad character. Must we, therefore, abandon science and
+study only morals? What! Do you think the moralists themselves are
+irreproachable when they come down from their pedestal?
+
+
+
+
+CHAPTER II
+
+THE FUTURE OF MATHEMATICS
+
+
+To foresee the future of mathematics, the true method is to study its
+history and its present state.
+
+Is this not for us mathematicians in a way a professional procedure? We
+are accustomed to _extrapolate_, which is a means of deducing the future
+from the past and present, and as we well know what this amounts to, we
+run no risk of deceiving ourselves about the range of the results it
+gives us.
+
+We have had hitherto prophets of evil. They blithely reiterate that all
+problems capable of solution have already been solved, and that nothing
+is left but gleaning. Happily the case of the past reassures us. Often
+it was thought all problems were solved or at least an inventory was
+made of all admitting solution. And then the sense of the word solution
+enlarged, the insoluble problems became the most interesting of all, and
+others unforeseen presented themselves. For the Greeks a good solution
+was one employing only ruler and compasses; then it became one obtained
+by the extraction of roots, then one using only algebraic or logarithmic
+functions. The pessimists thus found themselves always outflanked,
+always forced to retreat, so that at present I think there are no more.
+
+My intention, therefore, is not to combat them, as they are dead; we
+well know that mathematics will continue to develop, but the question is
+how, in what direction? You will answer, 'in every direction,' and that
+is partly true; but if it were wholly true it would be a little
+appalling. Our riches would soon become encumbering and their
+accumulation would produce a medley as impenetrable as the unknown true
+was for the ignorant.
+
+The historian, the physicist, even, must make a choice among facts; the
+head of the scientist, which is only a corner of the universe, could
+never contain the universe entire; so that among the innumerable facts
+nature offers, some will be passed by, others retained.
+
+Just so, _a fortiori_, in mathematics; no more can the geometer hold
+fast pell-mell all the facts presenting themselves to him; all the more
+because he it is, almost I had said his caprice, that creates these
+facts. He constructs a wholly new combination by putting together its
+elements; nature does not in general give it to him ready made.
+
+Doubtless it sometimes happens that the mathematician undertakes a
+problem to satisfy a need in physics; that the physicist or engineer
+asks him to calculate a number for a certain application. Shall it be
+said that we geometers should limit ourselves to awaiting orders, and,
+in place of cultivating our science for our own delectation, try only to
+accommodate ourselves to the wants of our patrons? If mathematics has no
+other object besides aiding those who study nature, it is from these we
+should await orders. Is this way of looking at it legitimate? Certainly
+not; if we had not cultivated the exact sciences for themselves, we
+should not have created mathematics the instrument, and the day the call
+came from the physicist we should have been helpless.
+
+Nor do the physicists wait to study a phenomenon until some urgent need
+of material life has made it a necessity for them; and they are right.
+If the scientists of the eighteenth century had neglected electricity as
+being in their eyes only a curiosity without practical interest, we
+should have had in the twentieth century neither telegraphy, nor
+electro-chemistry, nor electro-technics. The physicists, compelled to
+choose, are therefore not guided in their choice solely by utility. How
+then do they choose between the facts of nature? We have explained it in
+the preceding chapter: the facts which interest them are those capable
+of leading to the discovery of a law, and so they are analogous to many
+other facts which do not seem to us isolated, but closely grouped with
+others. The isolated fact attracts all eyes, those of the layman as well
+as of the scientist. But what the genuine physicist alone knows how to
+see, is the bond which unites many facts whose analogy is profound but
+hidden. The story of Newton's apple is probably not true, but it is
+symbolic; let us speak of it then as if it were true. Well then, we must
+believe that before Newton plenty of men had seen apples fall; not one
+knew how to conclude anything therefrom. Facts would be sterile were
+there not minds capable of choosing among them, discerning those behind
+which something was hidden, and of recognizing what is hiding, minds
+which under the crude fact perceive the soul of the fact.
+
+We find just the same thing in mathematics. From the varied elements at
+our disposal we can get millions of different combinations; but one of
+these combinations, in so far as it is isolated, is absolutely void of
+value. Often we have taken great pains to construct it, but it serves no
+purpose, if not perhaps to furnish a task in secondary education. Quite
+otherwise will it be when this combination shall find place in a class
+of analogous combinations and we shall have noticed this analogy. We are
+no longer in the presence of a fact, but of a law. And upon that day the
+real discoverer will not be the workman who shall have patiently built
+up certain of these combinations; it will be he who brings to light
+their kinship. The first will have seen merely the crude fact, only the
+other will have perceived the soul of the fact. Often to fix this
+kinship it suffices him to make a new word, and this word is creative.
+The history of science furnishes us a crowd of examples familiar to all.
+
+The celebrated Vienna philosopher Mach has said that the rôle of science
+is to produce economy of thought, just as machines produce economy of
+effort. And that is very true. The savage reckons on his fingers or by
+heaping pebbles. In teaching children the multiplication table we spare
+them later innumerable pebble bunchings. Some one has already found out,
+with pebbles or otherwise, that 6 times 7 is 42 and has had the idea of
+noting the result, and so we need not do it over again. He did not waste
+his time even if he reckoned for pleasure: his operation took him only
+two minutes; it would have taken in all two milliards if a milliard men
+had had to do it over after him.
+
+The importance of a fact then is measured by its yield, that is to say,
+by the amount of thought it permits us to spare.
+
+In physics the facts of great yield are those entering into a very
+general law, since from it they enable us to foresee a great number of
+others, and just so it is in mathematics. Suppose I have undertaken a
+complicated calculation and laboriously reached a result: I shall not
+be compensated for my trouble if thereby I have not become capable of
+foreseeing the results of other analogous calculations and guiding them
+with a certainty that avoids the gropings to which one must be resigned
+in a first attempt. On the other hand, I shall not have wasted my time
+if these gropings themselves have ended by revealing to me the profound
+analogy of the problem just treated with a much more extended class of
+other problems; if they have shown me at once the resemblances and
+differences of these, if in a word they have made me perceive the
+possibility of a generalization. Then it is not a new result I have won,
+it is a new power.
+
+The simple example that comes first to mind is that of an algebraic
+formula which gives us the solution of a type of numeric problems when
+finally we replace the letters by numbers. Thanks to it, a single
+algebraic calculation saves us the pains of ceaselessly beginning over
+again new numeric calculations. But this is only a crude example; we all
+know there are analogies inexpressible by a formula and all the more
+precious.
+
+A new result is of value, if at all, when in unifying elements long
+known but hitherto separate and seeming strangers one to another it
+suddenly introduces order where apparently disorder reigned. It then
+permits us to see at a glance each of these elements and its place in
+the assemblage. This new fact is not merely precious by itself, but it
+alone gives value to all the old facts it combines. Our mind is weak as
+are the senses; it would lose itself in the world's complexity were this
+complexity not harmonious; like a near-sighted person, it would see only
+the details and would be forced to forget each of these details before
+examining the following, since it would be incapable of embracing all.
+The only facts worthy our attention are those which introduce order into
+this complexity and so make it accessible.
+
+Mathematicians attach great importance to the elegance of their methods
+and their results. This is not pure dilettantism. What is it indeed that
+gives us the feeling of elegance in a solution, in a demonstration? It
+is the harmony of the diverse parts, their symmetry, their happy
+balance; in a word it is all that introduces order, all that gives
+unity, that permits us to see clearly and to comprehend at once both the
+_ensemble_ and the details. But this is exactly what yields great
+results; in fact the more we see this aggregate clearly and at a single
+glance, the better we perceive its analogies with other neighboring
+objects, consequently the more chances we have of divining the possible
+generalizations. Elegance may produce the feeling of the unforeseen by
+the unexpected meeting of objects we are not accustomed to bring
+together; there again it is fruitful, since it thus unveils for us
+kinships before unrecognized. It is fruitful even when it results only
+from the contrast between the simplicity of the means and the complexity
+of the problem set; it makes us then think of the reason for this
+contrast and very often makes us see that chance is not the reason; that
+it is to be found in some unexpected law. In a word, the feeling of
+mathematical elegance is only the satisfaction due to any adaptation of
+the solution to the needs of our mind, and it is because of this very
+adaptation that this solution can be for us an instrument. Consequently
+this esthetic satisfaction is bound up with the economy of thought.
+Again the comparison of the Erechtheum comes to my mind, but I must not
+use it too often.
+
+It is for the same reason that, when a rather long calculation has led
+to some simple and striking result, we are not satisfied until we have
+shown that we should have been _able to foresee_, if not this entire
+result, at least its most characteristic traits. Why? What prevents our
+being content with a calculation which has told us, it seems, all we
+wished to know? It is because, in analogous cases, the long calculation
+might not again avail, and that this is not so about the reasoning often
+half intuitive which would have enabled us to foresee. This reasoning
+being short, we see at a single glance all its parts, so that we
+immediately perceive what must be changed to adapt it to all the
+problems of the same nature which can occur. And then it enables us to
+foresee if the solution of these problems will be simple, it shows us at
+least if the calculation is worth undertaking.
+
+What we have just said suffices to show how vain it would be to seek to
+replace by any mechanical procedure the free initiative of the
+mathematician. To obtain a result of real value, it is not enough to
+grind out calculations, or to have a machine to put things in order; it
+is not order alone, it is unexpected order, which is worth while. The
+machine may gnaw on the crude fact, the soul of the fact will always
+escape it.
+
+Since the middle of the last century, mathematicians are more and more
+desirous of attaining absolute rigor; they are right, and this tendency
+will be more and more accentuated. In mathematics rigor is not
+everything, but without it there is nothing. A demonstration which is
+not rigorous is nothingness. I think no one will contest this truth. But
+if it were taken too literally, we should be led to conclude that before
+1820, for example, there was no mathematics; this would be manifestly
+excessive; the geometers of that time understood voluntarily what we
+explain by prolix discourse. This does not mean that they did not see it
+at all; but they passed over it too rapidly, and to see it well would
+have necessitated taking the pains to say it.
+
+But is it always needful to say it so many times? Those who were the
+first to emphasize exactness before all else have given us arguments
+that we may try to imitate; but if the demonstrations of the future are
+to be built on this model, mathematical treatises will be very long; and
+if I fear the lengthenings, it is not solely because I deprecate
+encumbering libraries, but because I fear that in being lengthened out,
+our demonstrations may lose that appearance of harmony whose usefulness
+I have just explained.
+
+The economy of thought is what we should aim at, so it is not enough to
+supply models for imitation. It is needful for those after us to be able
+to dispense with these models and, in place of repeating an argument
+already made, summarize it in a few words. And this has already been
+attained at times. For instance, there was a type of reasoning found
+everywhere, and everywhere alike. They were perfectly exact but long.
+Then all at once the phrase 'uniformity of convergence' was hit upon and
+this phrase made those arguments needless; we were no longer called upon
+to repeat them, since they could be understood. Those who conquer
+difficulties then do us a double service: first they teach us to do as
+they at need, but above all they enable us as often as possible to avoid
+doing as they, yet without sacrifice of exactness.
+
+We have just seen by one example the importance of words in mathematics,
+but many others could be cited. It is hard to believe how much a
+well-chosen word can economize thought, as Mach says. Perhaps I have
+already said somewhere that mathematics is the art of giving the same
+name to different things. It is proper that these things, differing in
+matter, be alike in form, that they may, so to speak, run in the same
+mold. When the language has been well chosen, we are astonished to see
+that all the proofs made for a certain object apply immediately to many
+new objects; there is nothing to change, not even the words, since the
+names have become the same.
+
+A well-chosen word usually suffices to do away with the exceptions from
+which the rules stated in the old way suffer; this is why we have
+created negative quantities, imaginaries, points at infinity, and what
+not. And exceptions, we must not forget, are pernicious because they
+hide the laws.
+
+Well, this is one of the characteristics by which we recognize the facts
+which yield great results. They are those which allow of these happy
+innovations of language. The crude fact then is often of no great
+interest; we may point it out many times without having rendered great
+service to science. It takes value only when a wiser thinker perceives
+the relation for which it stands, and symbolizes it by a word.
+
+Moreover the physicists do just the same. They have invented the word
+'energy,' and this word has been prodigiously fruitful, because it also
+made the law by eliminating the exceptions, since it gave the same name
+to things differing in matter and like in form.
+
+Among words that have had the most fortunate influence I would select
+'group' and 'invariant.' They have made us see the essence of many
+mathematical reasonings; they have shown us in how many cases the old
+mathematicians considered groups without knowing it, and how, believing
+themselves far from one another, they suddenly found themselves near
+without knowing why.
+
+To-day we should say that they had dealt with isomorphic groups. We now
+know that in a group the matter is of little interest, the form alone
+counts, and that when we know a group we thus know all the isomorphic
+groups; and thanks to these words 'group' and 'isomorphism,' which
+condense in a few syllables this subtile rule and quickly make it
+familiar to all minds, the transition is immediate and can be done with
+every economy of thought effort. The idea of group besides attaches to
+that of transformation. Why do we put such a value on the invention of a
+new transformation? Because from a single theorem it enables us to get
+ten or twenty; it has the same value as a zero adjoined to the right of
+a whole number.
+
+This then it is which has hitherto determined the direction of
+mathematical advance, and just as certainly will determine it in the
+future. But to this end the nature of the problems which come up
+contributes equally. We can not forget what must be our aim. In my
+opinion this aim is double. Our science borders upon both philosophy and
+physics, and we work for our two neighbors; so we have always seen and
+shall still see mathematicians advancing in two opposite directions.
+
+On the one hand, mathematical science must reflect upon itself, and that
+is useful since reflecting on itself is reflecting on the human mind
+which has created it, all the more because it is the very one of its
+creations for which it has borrowed least from without. This is why
+certain mathematical speculations are useful, such as those devoted to
+the study of the postulates, of unusual geometries, of peculiar
+functions. The more these speculations diverge from ordinary
+conceptions, and consequently from nature and applications, the better
+they show us what the human mind can create when it frees itself more
+and more from the tyranny of the external world, the better therefore
+they let us know it in itself.
+
+But it is toward the other side, the side of nature, that we must direct
+the bulk of our army. There we meet the physicist or the engineer, who
+says to us: "Please integrate this differential equation for me; I might
+need it in a week in view of a construction which should be finished by
+that time." "This equation," we answer, "does not come under one of the
+integrable types; you know there are not many." "Yes, I know; but then
+what good are you?" Usually to understand each other is enough; the
+engineer in reality does not need the integral in finite terms; he
+needs to know the general look of the integral function, or he simply
+wants a certain number which could readily be deduced from this integral
+if it were known. Usually it is not known, but the number can be
+calculated without it if we know exactly what number the engineer needs
+and with what approximation.
+
+Formerly an equation was considered solved only when its solution had
+been expressed by aid of a finite number of known functions; but that is
+possible scarcely once in a hundred times. What we always can do, or
+rather what we should always seek to do, is to solve the problem
+_qualitatively_ so to speak; that is to say, seek to know the general
+form of the curve which represents the unknown function.
+
+It remains to find the _quantitative_ solution of the problem; but if
+the unknown can not be determined by a finite calculation, it may always
+be represented by a convergent infinite series which enables us to
+calculate it. Can that be regarded as a true solution? We are told that
+Newton sent Leibnitz an anagram almost like this: aaaaabbbeeeeij, etc.
+Leibnitz naturally understood nothing at all of it; but we, who have the
+key, know that this anagram meant, translated into modern terms: "I can
+integrate all differential equations"; and we are tempted to say that
+Newton had either great luck or strange delusions. He merely wished to
+say he could form (by the method of indeterminate coefficients) a series
+of powers formally satisfying the proposed equation.
+
+Such a solution would not satisfy us to-day, and for two reasons:
+because the convergence is too slow and because the terms follow each
+other without obeying any law. On the contrary, the series [Theta] seems
+to us to leave nothing to be desired, first because it converges very
+quickly (this is for the practical man who wishes to get at a number as
+quickly as possible) and next because we see at a glance the law of the
+terms (this is to satisfy the esthetic need of the theorist).
+
+But then there are no longer solved problems and others which are not;
+there are only problems _more or less_ solved, according as they are
+solved by a series converging more or less rapidly, or ruled by a law
+more or less harmonious. It often happens however that an imperfect
+solution guides us toward a better one. Sometimes the series converges
+so slowly that the computation is impracticable and we have only
+succeeded in proving the possibility of the problem.
+
+And then the engineer finds this a mockery, and justly, since it will
+not aid him to complete his construction by the date fixed. He little
+cares to know if it will benefit engineers of the twenty-second century.
+But as for us, we think differently and we are sometimes happier to have
+spared our grandchildren a day's work than to have saved our
+contemporaries an hour.
+
+Sometimes by groping, empirically, so to speak, we reach a formula
+sufficiently convergent. "What more do you want?" says the engineer. And
+yet, in spite of all, we are not satisfied; we should have liked _to
+foresee_ that convergence. Why? Because if we had known how to foresee
+it once, we would know how to foresee it another time. We have
+succeeded; that is a small matter in our eyes if we can not validly
+expect to do so again.
+
+In proportion as science develops, its total comprehension becomes more
+difficult; then we seek to cut it in pieces and to be satisfied with one
+of these pieces: in a word, to specialize. If we went on in this way, it
+would be a grievous obstacle to the progress of science. As we have
+said, it is by unexpected union between its diverse parts that it
+progresses. To specialize too much would be to forbid these drawings
+together. It is to be hoped that congresses like those of Heidelberg and
+Rome, by putting us in touch with one another, will open for us vistas
+over neighboring domains and oblige us to compare them with our own, to
+range somewhat abroad from our own little village; thus they will be the
+best remedy for the danger just mentioned.
+
+But I have lingered too long over generalities; it is time to enter into
+detail.
+
+Let us pass in review the various special sciences which combined make
+mathematics; let us see what each has accomplished, whither it tends and
+what we may hope from it. If the preceding views are correct, we should
+see that the greatest advances in the past have happened when two of
+these sciences have united, when we have become conscious of the
+similarity of their form, despite the difference of their matter, when
+they have so modeled themselves upon each other that each could profit
+by the other's conquests. We should at the same time foresee in
+combinations of the same sort the progress of the future.
+
+
+ARITHMETIC
+
+Progress in arithmetic has been much slower than in algebra and
+analysis, and it is easy to see why. The feeling of continuity is a
+precious guide which the arithmetician lacks; each whole number is
+separated from the others--it has, so to speak, its own individuality.
+Each of them is a sort of exception and this is why general theorems are
+rarer in the theory of numbers; this is also why those which exist are
+more hidden and longer elude the searchers.
+
+If arithmetic is behind algebra and analysis, the best thing for it to
+do is to seek to model itself upon these sciences so as to profit by
+their advance. The arithmetician ought therefore to take as guide the
+analogies with algebra. These analogies are numerous and if, in many
+cases, they have not yet been studied sufficiently closely to become
+utilizable, they at least have long been foreseen, and even the language
+of the two sciences shows they have been recognized. Thus we speak of
+transcendent numbers and thus we account for the future classification
+of these numbers already having as model the classification of
+transcendent functions, and still we do not as yet very well see how to
+pass from one classification to the other; but had it been seen, it
+would already have been accomplished and would no longer be the work of
+the future.
+
+The first example that comes to my mind is the theory of congruences,
+where is found a perfect parallelism to the theory of algebraic
+equations. Surely we shall succeed in completing this parallelism, which
+must hold for instance between the theory of algebraic curves and that
+of congruences with two variables. And when the problems relative to
+congruences with several variables shall be solved, this will be a first
+step toward the solution of many questions of indeterminate analysis.
+
+
+ALGEBRA
+
+The theory of algebraic equations will still long hold the attention of
+geometers; numerous and very different are the sides whence it may be
+attacked.
+
+We need not think algebra is ended because it gives us rules to form all
+possible combinations; it remains to find the interesting combinations,
+those which satisfy such and such a condition. Thus will be formed a
+sort of indeterminate analysis where the unknowns will no longer be
+whole numbers, but polynomials. This time it is algebra which will model
+itself upon arithmetic, following the analogy of the whole number to the
+integral polynomial with any coefficients or to the integral polynomial
+with integral coefficients.
+
+
+GEOMETRY
+
+It looks as if geometry could contain nothing which is not already
+included in algebra or analysis; that geometric facts are only algebraic
+or analytic facts expressed in another language. It might then be
+thought that after our review there would remain nothing more for us to
+say relating specially to geometry. This would be to fail to recognize
+the importance of well-constructed language, not to comprehend what is
+added to the things themselves by the method of expressing these things
+and consequently of grouping them.
+
+First the geometric considerations lead us to set ourselves new
+problems; these may be, if you choose, analytic problems, but such as we
+never would have set ourselves in connection with analysis. Analysis
+profits by them however, as it profits by those it has to solve to
+satisfy the needs of physics.
+
+A great advantage of geometry lies in the fact that in it the senses can
+come to the aid of thought, and help find the path to follow, and many
+minds prefer to put the problems of analysis into geometric form.
+Unhappily our senses can not carry us very far, and they desert us when
+we wish to soar beyond the classical three dimensions. Does this mean
+that, beyond the restricted domain wherein they seem to wish to imprison
+us, we should rely only on pure analysis and that all geometry of more
+than three dimensions is vain and objectless? The greatest masters of a
+preceding generation would have answered 'yes'; to-day we are so
+familiarized with this notion that we can speak of it, even in a
+university course, without arousing too much astonishment.
+
+But what good is it? That is easy to see: First it gives us a very
+convenient terminology, which expresses concisely what the ordinary
+analytic language would say in prolix phrases. Moreover, this language
+makes us call like things by the same name and emphasize analogies it
+will never again let us forget. It enables us therefore still to find
+our way in this space which is too big for us and which we can not see,
+always recalling visible space, which is only an imperfect image of it
+doubtless, but which is nevertheless an image. Here again, as in all the
+preceding examples, it is analogy with the simple which enables us to
+comprehend the complex.
+
+This geometry of more than three dimensions is not a simple analytic
+geometry; it is not purely quantitative, but qualitative also, and it is
+in this respect above all that it becomes interesting. There is a
+science called _analysis situs_ and which has for its object the study
+of the positional relations of the different elements of a figure, apart
+from their sizes. This geometry is purely qualitative; its theorems
+would remain true if the figures, instead of being exact, were roughly
+imitated by a child. We may also make an _analysis situs_ of more than
+three dimensions. The importance of _analysis situs_ is enormous and can
+not be too much emphasized; the advantage obtained from it by Riemann,
+one of its chief creators, would suffice to prove this. We must achieve
+its complete construction in the higher spaces; then we shall have an
+instrument which will enable us really to see in hyperspace and
+supplement our senses.
+
+The problems of _analysis situs_ would perhaps not have suggested
+themselves if the analytic language alone had been spoken; or rather, I
+am mistaken, they would have occurred surely, since their solution is
+essential to a crowd of questions in analysis, but they would have come
+singly, one after another, and without our being able to perceive their
+common bond.
+
+
+CANTORISM
+
+I have spoken above of our need to go back continually to the first
+principles of our science, and of the advantage of this for the study of
+the human mind. This need has inspired two endeavors which have taken a
+very prominent place in the most recent annals of mathematics. The first
+is Cantorism, which has rendered our science such conspicuous service.
+Cantor introduced into science a new way of considering mathematical
+infinity. One of the characteristic traits of Cantorism is that in place
+of going up to the general by building up constructions more and more
+complicated and defining by construction, it starts from the _genus
+supremum_ and defines only, as the scholastics would have said, _per
+genus proximum et differentiam specificam_. Thence comes the horror it
+has sometimes inspired in certain minds, for instance in Hermite, whose
+favorite idea was to compare the mathematical to the natural sciences.
+With most of us these prejudices have been dissipated, but it has come
+to pass that we have encountered certain paradoxes, certain apparent
+contradictions that would have delighted Zeno, the Eleatic and the
+school of Megara. And then each must seek the remedy. For my part, I
+think, and I am not the only one, that the important thing is never to
+introduce entities not completely definable in a finite number of words.
+Whatever be the cure adopted, we may promise ourselves the joy of the
+doctor called in to follow a beautiful pathologic case.
+
+
+THE INVESTIGATION OF THE POSTULATES
+
+On the other hand, efforts have been made to enumerate the axioms and
+postulates, more or less hidden, which serve as foundation to the
+different theories of mathematics. Professor Hilbert has obtained the
+most brilliant results. It seems at first that this domain would be very
+restricted and there would be nothing more to do when the inventory
+should be ended, which could not take long. But when we shall have
+enumerated all, there will be many ways of classifying all; a good
+librarian always finds something to do, and each new classification will
+be instructive for the philosopher.
+
+Here I end this review which I could not dream of making complete. I
+think these examples will suffice to show by what mechanism the
+mathematical sciences have made their progress in the past and in what
+direction they must advance in the future.
+
+
+
+
+CHAPTER III
+
+MATHEMATICAL CREATION
+
+
+The genesis of mathematical creation is a problem which should intensely
+interest the psychologist. It is the activity in which the human mind
+seems to take least from the outside world, in which it acts or seems to
+act only of itself and on itself, so that in studying the procedure of
+geometric thought we may hope to reach what is most essential in man's
+mind.
+
+This has long been appreciated, and some time back the journal called
+_L'enseignement mathématique_, edited by Laisant and Fehr, began an
+investigation of the mental habits and methods of work of different
+mathematicians. I had finished the main outlines of this article when
+the results of that inquiry were published, so I have hardly been able
+to utilize them and shall confine myself to saying that the majority of
+witnesses confirm my conclusions; I do not say all, for when the appeal
+is to universal suffrage unanimity is not to be hoped.
+
+A first fact should surprise us, or rather would surprise us if we were
+not so used to it. How does it happen there are people who do not
+understand mathematics? If mathematics invokes only the rules of logic,
+such as are accepted by all normal minds; if its evidence is based on
+principles common to all men, and that none could deny without being
+mad, how does it come about that so many persons are here refractory?
+
+That not every one can invent is nowise mysterious. That not every one
+can retain a demonstration once learned may also pass. But that not
+every one can understand mathematical reasoning when explained appears
+very surprising when we think of it. And yet those who can follow this
+reasoning only with difficulty are in the majority: that is undeniable,
+and will surely not be gainsaid by the experience of secondary-school
+teachers.
+
+And further: how is error possible in mathematics? A sane mind should
+not be guilty of a logical fallacy, and yet there are very fine minds
+who do not trip in brief reasoning such as occurs in the ordinary doings
+of life, and who are incapable of following or repeating without error
+the mathematical demonstrations which are longer, but which after all
+are only an accumulation of brief reasonings wholly analogous to those
+they make so easily. Need we add that mathematicians themselves are not
+infallible?
+
+The answer seems to me evident. Imagine a long series of syllogisms, and
+that the conclusions of the first serve as premises of the following: we
+shall be able to catch each of these syllogisms, and it is not in
+passing from premises to conclusion that we are in danger of deceiving
+ourselves. But between the moment in which we first meet a proposition
+as conclusion of one syllogism, and that in which we reencounter it as
+premise of another syllogism occasionally some time will elapse, several
+links of the chain will have unrolled; so it may happen that we have
+forgotten it, or worse, that we have forgotten its meaning. So it may
+happen that we replace it by a slightly different proposition, or that,
+while retaining the same enunciation, we attribute to it a slightly
+different meaning, and thus it is that we are exposed to error.
+
+Often the mathematician uses a rule. Naturally he begins by
+demonstrating this rule; and at the time when this proof is fresh in his
+memory he understands perfectly its meaning and its bearing, and he is
+in no danger of changing it. But subsequently he trusts his memory and
+afterward only applies it in a mechanical way; and then if his memory
+fails him, he may apply it all wrong. Thus it is, to take a simple
+example, that we sometimes make slips in calculation because we have
+forgotten our multiplication table.
+
+According to this, the special aptitude for mathematics would be due
+only to a very sure memory or to a prodigious force of attention. It
+would be a power like that of the whist-player who remembers the cards
+played; or, to go up a step, like that of the chess-player who can
+visualize a great number of combinations and hold them in his memory.
+Every good mathematician ought to be a good chess-player, and inversely;
+likewise he should be a good computer. Of course that sometimes happens;
+thus Gauss was at the same time a geometer of genius and a very
+precocious and accurate computer.
+
+But there are exceptions; or rather I err; I can not call them
+exceptions without the exceptions being more than the rule. Gauss it is,
+on the contrary, who was an exception. As for myself, I must confess, I
+am absolutely incapable even of adding without mistakes. In the same way
+I should be but a poor chess-player; I would perceive that by a certain
+play I should expose myself to a certain danger; I would pass in review
+several other plays, rejecting them for other reasons, and then finally
+I should make the move first examined, having meantime forgotten the
+danger I had foreseen.
+
+In a word, my memory is not bad, but it would be insufficient to make me
+a good chess-player. Why then does it not fail me in a difficult piece
+of mathematical reasoning where most chess-players would lose
+themselves? Evidently because it is guided by the general march of the
+reasoning. A mathematical demonstration is not a simple juxtaposition of
+syllogisms, it is syllogisms _placed in a certain order_, and the order
+in which these elements are placed is much more important than the
+elements themselves. If I have the feeling, the intuition, so to speak,
+of this order, so as to perceive at a glance the reasoning as a whole, I
+need no longer fear lest I forget one of the elements, for each of them
+will take its allotted place in the array, and that without any effort
+of memory on my part.
+
+It seems to me then, in repeating a reasoning learned, that I could have
+invented it. This is often only an illusion; but even then, even if I am
+not so gifted as to create it by myself, I myself re-invent it in so far
+as I repeat it.
+
+We know that this feeling, this intuition of mathematical order, that
+makes us divine hidden harmonies and relations, can not be possessed by
+every one. Some will not have either this delicate feeling so difficult
+to define, or a strength of memory and attention beyond the ordinary,
+and then they will be absolutely incapable of understanding higher
+mathematics. Such are the majority. Others will have this feeling only
+in a slight degree, but they will be gifted with an uncommon memory and
+a great power of attention. They will learn by heart the details one
+after another; they can understand mathematics and sometimes make
+applications, but they cannot create. Others, finally, will possess in a
+less or greater degree the special intuition referred to, and then not
+only can they understand mathematics even if their memory is nothing
+extraordinary, but they may become creators and try to invent with more
+or less success according as this intuition is more or less developed in
+them.
+
+In fact, what is mathematical creation? It does not consist in making
+new combinations with mathematical entities already known. Any one could
+do that, but the combinations so made would be infinite in number and
+most of them absolutely without interest. To create consists precisely
+in not making useless combinations and in making those which are useful
+and which are only a small minority. Invention is discernment, choice.
+
+How to make this choice I have before explained; the mathematical facts
+worthy of being studied are those which, by their analogy with other
+facts, are capable of leading us to the knowledge of a mathematical law
+just as experimental facts lead us to the knowledge of a physical law.
+They are those which reveal to us unsuspected kinship between other
+facts, long known, but wrongly believed to be strangers to one another.
+
+Among chosen combinations the most fertile will often be those formed of
+elements drawn from domains which are far apart. Not that I mean as
+sufficing for invention the bringing together of objects as disparate as
+possible; most combinations so formed would be entirely sterile. But
+certain among them, very rare, are the most fruitful of all.
+
+To invent, I have said, is to choose; but the word is perhaps not wholly
+exact. It makes one think of a purchaser before whom are displayed a
+large number of samples, and who examines them, one after the other, to
+make a choice. Here the samples would be so numerous that a whole
+lifetime would not suffice to examine them. This is not the actual state
+of things. The sterile combinations do not even present themselves to
+the mind of the inventor. Never in the field of his consciousness do
+combinations appear that are not really useful, except some that he
+rejects but which have to some extent the characteristics of useful
+combinations. All goes on as if the inventor were an examiner for the
+second degree who would only have to question the candidates who had
+passed a previous examination.
+
+But what I have hitherto said is what may be observed or inferred in
+reading the writings of the geometers, reading reflectively.
+
+It is time to penetrate deeper and to see what goes on in the very soul
+of the mathematician. For this, I believe, I can do best by recalling
+memories of my own. But I shall limit myself to telling how I wrote my
+first memoir on Fuchsian functions. I beg the reader's pardon; I am
+about to use some technical expressions, but they need not frighten him,
+for he is not obliged to understand them. I shall say, for example, that
+I have found the demonstration of such a theorem under such
+circumstances. This theorem will have a barbarous name, unfamiliar to
+many, but that is unimportant; what is of interest for the psychologist
+is not the theorem but the circumstances.
+
+For fifteen days I strove to prove that there could not be any functions
+like those I have since called Fuchsian functions. I was then very
+ignorant; every day I seated myself at my work table, stayed an hour or
+two, tried a great number of combinations and reached no results. One
+evening, contrary to my custom, I drank black coffee and could not
+sleep. Ideas rose in crowds; I felt them collide until pairs
+interlocked, so to speak, making a stable combination. By the next
+morning I had established the existence of a class of Fuchsian
+functions, those which come from the hypergeometric series; I had only
+to write out the results, which took but a few hours.
+
+Then I wanted to represent these functions by the quotient of two
+series; this idea was perfectly conscious and deliberate, the analogy
+with elliptic functions guided me. I asked myself what properties these
+series must have if they existed, and I succeeded without difficulty in
+forming the series I have called theta-Fuchsian.
+
+Just at this time I left Caen, where I was then living, to go on a
+geologic excursion under the auspices of the school of mines. The
+changes of travel made me forget my mathematical work. Having reached
+Coutances, we entered an omnibus to go some place or other. At the
+moment when I put my foot on the step the idea came to me, without
+anything in my former thoughts seeming to have paved the way for it,
+that the transformations I had used to define the Fuchsian functions
+were identical with those of non-Euclidean geometry. I did not verify
+the idea; I should not have had time, as, upon taking my seat in the
+omnibus, I went on with a conversation already commenced, but I felt a
+perfect certainty. On my return to Caen, for conscience' sake I verified
+the result at my leisure.
+
+Then I turned my attention to the study of some arithmetical questions
+apparently without much success and without a suspicion of any
+connection with my preceding researches. Disgusted with my failure, I
+went to spend a few days at the seaside, and thought of something else.
+One morning, walking on the bluff, the idea came to me, with just the
+same characteristics of brevity, suddenness and immediate certainty,
+that the arithmetic transformations of indeterminate ternary quadratic
+forms were identical with those of non-Euclidean geometry.
+
+Returned to Caen, I meditated on this result and deduced the
+consequences. The example of quadratic forms showed me that there were
+Fuchsian groups other than those corresponding to the hypergeometric
+series; I saw that I could apply to them the theory of theta-Fuchsian
+series and that consequently there existed Fuchsian functions other than
+those from the hypergeometric series, the ones I then knew. Naturally I
+set myself to form all these functions. I made a systematic attack upon
+them and carried all the outworks, one after another. There was one
+however that still held out, whose fall would involve that of the whole
+place. But all my efforts only served at first the better to show me the
+difficulty, which indeed was something. All this work was perfectly
+conscious.
+
+Thereupon I left for Mont-Valérien, where I was to go through my
+military service; so I was very differently occupied. One day, going
+along the street, the solution of the difficulty which had stopped me
+suddenly appeared to me. I did not try to go deep into it immediately,
+and only after my service did I again take up the question. I had all
+the elements and had only to arrange them and put them together. So I
+wrote out my final memoir at a single stroke and without difficulty.
+
+I shall limit myself to this single example; it is useless to multiply
+them. In regard to my other researches I would have to say analogous
+things, and the observations of other mathematicians given in
+_L'enseignement mathématique_ would only confirm them.
+
+Most striking at first is this appearance of sudden illumination, a
+manifest sign of long, unconscious prior work. The rôle of this
+unconscious work in mathematical invention appears to me incontestable,
+and traces of it would be found in other cases where it is less evident.
+Often when one works at a hard question, nothing good is accomplished at
+the first attack. Then one takes a rest, longer or shorter, and sits
+down anew to the work. During the first half-hour, as before, nothing is
+found, and then all of a sudden the decisive idea presents itself to the
+mind. It might be said that the conscious work has been more fruitful
+because it has been interrupted and the rest has given back to the mind
+its force and freshness. But it is more probable that this rest has been
+filled out with unconscious work and that the result of this work has
+afterward revealed itself to the geometer just as in the cases I have
+cited; only the revelation, instead of coming during a walk or a
+journey, has happened during a period of conscious work, but
+independently of this work which plays at most a rôle of excitant, as if
+it were the goad stimulating the results already reached during rest,
+but remaining unconscious, to assume the conscious form.
+
+There is another remark to be made about the conditions of this
+unconscious work: it is possible, and of a certainty it is only
+fruitful, if it is on the one hand preceded and on the other hand
+followed by a period of conscious work. These sudden inspirations (and
+the examples already cited sufficiently prove this) never happen except
+after some days of voluntary effort which has appeared absolutely
+fruitless and whence nothing good seems to have come, where the way
+taken seems totally astray. These efforts then have not been as sterile
+as one thinks; they have set agoing the unconscious machine and without
+them it would not have moved and would have produced nothing.
+
+The need for the second period of conscious work, after the inspiration,
+is still easier to understand. It is necessary to put in shape the
+results of this inspiration, to deduce from them the immediate
+consequences, to arrange them, to word the demonstrations, but above all
+is verification necessary. I have spoken of the feeling of absolute
+certitude accompanying the inspiration; in the cases cited this feeling
+was no deceiver, nor is it usually. But do not think this a rule without
+exception; often this feeling deceives us without being any the less
+vivid, and we only find it out when we seek to put on foot the
+demonstration. I have especially noticed this fact in regard to ideas
+coming to me in the morning or evening in bed while in a semi-hypnagogic
+state.
+
+Such are the realities; now for the thoughts they force upon us. The
+unconscious, or, as we say, the subliminal self plays an important rôle
+in mathematical creation; this follows from what we have said. But
+usually the subliminal self is considered as purely automatic. Now we
+have seen that mathematical work is not simply mechanical, that it could
+not be done by a machine, however perfect. It is not merely a question
+of applying rules, of making the most combinations possible according to
+certain fixed laws. The combinations so obtained would be exceedingly
+numerous, useless and cumbersome. The true work of the inventor consists
+in choosing among these combinations so as to eliminate the useless ones
+or rather to avoid the trouble of making them, and the rules which must
+guide this choice are extremely fine and delicate. It is almost
+impossible to state them precisely; they are felt rather than
+formulated. Under these conditions, how imagine a sieve capable of
+applying them mechanically?
+
+A first hypothesis now presents itself: the subliminal self is in no way
+inferior to the conscious self; it is not purely automatic; it is
+capable of discernment; it has tact, delicacy; it knows how to choose,
+to divine. What do I say? It knows better how to divine than the
+conscious self, since it succeeds where that has failed. In a word, is
+not the subliminal self superior to the conscious self? You recognize
+the full importance of this question. Boutroux in a recent lecture has
+shown how it came up on a very different occasion, and what consequences
+would follow an affirmative answer. (See also, by the same author,
+_Science et Religion_, pp. 313 ff.)
+
+Is this affirmative answer forced upon us by the facts I have just
+given? I confess that, for my part, I should hate to accept it.
+Reexamine the facts then and see if they are not compatible with another
+explanation.
+
+It is certain that the combinations which present themselves to the mind
+in a sort of sudden illumination, after an unconscious working somewhat
+prolonged, are generally useful and fertile combinations, which seem the
+result of a first impression. Does it follow that the subliminal self,
+having divined by a delicate intuition that these combinations would be
+useful, has formed only these, or has it rather formed many others which
+were lacking in interest and have remained unconscious?
+
+In this second way of looking at it, all the combinations would be
+formed in consequence of the automatism of the subliminal self, but only
+the interesting ones would break into the domain of consciousness. And
+this is still very mysterious. What is the cause that, among the
+thousand products of our unconscious activity, some are called to pass
+the threshold, while others remain below? Is it a simple chance which
+confers this privilege? Evidently not; among all the stimuli of our
+senses, for example, only the most intense fix our attention, unless it
+has been drawn to them by other causes. More generally the privileged
+unconscious phenomena, those susceptible of becoming conscious, are
+those which, directly or indirectly, affect most profoundly our
+emotional sensibility.
+
+It may be surprising to see emotional sensibility invoked _à propos_ of
+mathematical demonstrations which, it would seem, can interest only the
+intellect. This would be to forget the feeling of mathematical beauty,
+of the harmony of numbers and forms, of geometric elegance. This is a
+true esthetic feeling that all real mathematicians know, and surely it
+belongs to emotional sensibility.
+
+Now, what are the mathematic entities to which we attribute this
+character of beauty and elegance, and which are capable of developing in
+us a sort of esthetic emotion? They are those whose elements are
+harmoniously disposed so that the mind without effort can embrace their
+totality while realizing the details. This harmony is at once a
+satisfaction of our esthetic needs and an aid to the mind, sustaining
+and guiding; And at the same time, in putting under our eyes a
+well-ordered whole, it makes us foresee a mathematical law. Now, as we
+have said above, the only mathematical facts worthy of fixing our
+attention and capable of being useful are those which can teach us a
+mathematical law. So that we reach the following conclusion: The useful
+combinations are precisely the most beautiful, I mean those best able to
+charm this special sensibility that all mathematicians know, but of
+which the profane are so ignorant as often to be tempted to smile at it.
+
+What happens then? Among the great numbers of combinations blindly
+formed by the subliminal self, almost all are without interest and
+without utility; but just for that reason they are also without effect
+upon the esthetic sensibility. Consciousness will never know them; only
+certain ones are harmonious, and, consequently, at once useful and
+beautiful. They will be capable of touching this special sensibility of
+the geometer of which I have just spoken, and which, once aroused, will
+call our attention to them, and thus give them occasion to become
+conscious.
+
+This is only a hypothesis, and yet here is an observation which may
+confirm it: when a sudden illumination seizes upon the mind of the
+mathematician, it usually happens that it does not deceive him, but it
+also sometimes happens, as I have said, that it does not stand the test
+of verification; well, we almost always notice that this false idea, had
+it been true, would have gratified our natural feeling for mathematical
+elegance.
+
+Thus it is this special esthetic sensibility which plays the rôle of the
+delicate sieve of which I spoke, and that sufficiently explains why the
+one lacking it will never be a real creator.
+
+Yet all the difficulties have not disappeared. The conscious self is
+narrowly limited, and as for the subliminal self we know not its
+limitations, and this is why we are not too reluctant in supposing that
+it has been able in a short time to make more different combinations
+than the whole life of a conscious being could encompass. Yet these
+limitations exist. Is it likely that it is able to form all the possible
+combinations, whose number would frighten the imagination? Nevertheless
+that would seem necessary, because if it produces only a small part of
+these combinations, and if it makes them at random, there would be
+small chance that the _good_, the one we should choose, would be found
+among them.
+
+Perhaps we ought to seek the explanation in that preliminary period of
+conscious work which always precedes all fruitful unconscious labor.
+Permit me a rough comparison. Figure the future elements of our
+combinations as something like the hooked atoms of Epicurus. During the
+complete repose of the mind, these atoms are motionless, they are, so to
+speak, hooked to the wall; so this complete rest may be indefinitely
+prolonged without the atoms meeting, and consequently without any
+combination between them.
+
+On the other hand, during a period of apparent rest and unconscious
+work, certain of them are detached from the wall and put in motion. They
+flash in every direction through the space (I was about to say the room)
+where they are enclosed, as would, for example, a swarm of gnats or, if
+you prefer a more learned comparison, like the molecules of gas in the
+kinematic theory of gases. Then their mutual impacts may produce new
+combinations.
+
+What is the rôle of the preliminary conscious work? It is evidently to
+mobilize certain of these atoms, to unhook them from the wall and put
+them in swing. We think we have done no good, because we have moved
+these elements a thousand different ways in seeking to assemble them,
+and have found no satisfactory aggregate. But, after this shaking up
+imposed upon them by our will, these atoms do not return to their
+primitive rest. They freely continue their dance.
+
+Now, our will did not choose them at random; it pursued a perfectly
+determined aim. The mobilized atoms are therefore not any atoms
+whatsoever; they are those from which we might reasonably expect the
+desired solution. Then the mobilized atoms undergo impacts which make
+them enter into combinations among themselves or with other atoms at
+rest which they struck against in their course. Again I beg pardon, my
+comparison is very rough, but I scarcely know how otherwise to make my
+thought understood.
+
+However it may be, the only combinations that have a chance of forming
+are those where at least one of the elements is one of those atoms
+freely chosen by our will. Now, it is evidently among these that is
+found what I called the _good combination_. Perhaps this is a way of
+lessening the paradoxical in the original hypothesis.
+
+Another observation. It never happens that the unconscious work gives us
+the result of a somewhat long calculation _all made_, where we have only
+to apply fixed rules. We might think the wholly automatic subliminal
+self particularly apt for this sort of work, which is in a way
+exclusively mechanical. It seems that thinking in the evening upon the
+factors of a multiplication we might hope to find the product ready made
+upon our awakening, or again that an algebraic calculation, for example
+a verification, would be made unconsciously. Nothing of the sort, as
+observation proves. All one may hope from these inspirations, fruits of
+unconscious work, is a point of departure for such calculations. As for
+the calculations themselves, they must be made in the second period of
+conscious work, that which follows the inspiration, that in which one
+verifies the results of this inspiration and deduces their consequences.
+The rules of these calculations are strict and complicated. They require
+discipline, attention, will, and therefore consciousness. In the
+subliminal self, on the contrary, reigns what I should call liberty, if
+we might give this name to the simple absence of discipline and to the
+disorder born of chance. Only, this disorder itself permits unexpected
+combinations.
+
+I shall make a last remark: when above I made certain personal
+observations, I spoke of a night of excitement when I worked in spite of
+myself. Such cases are frequent, and it is not necessary that the
+abnormal cerebral activity be caused by a physical excitant as in that I
+mentioned. It seems, in such cases, that one is present at his own
+unconscious work, made partially perceptible to the over-excited
+consciousness, yet without having changed its nature. Then we vaguely
+comprehend what distinguishes the two mechanisms or, if you wish, the
+working methods of the two egos. And the psychologic observations I have
+been able thus to make seem to me to confirm in their general outlines
+the views I have given.
+
+Surely they have need of it, for they are and remain in spite of all
+very hypothetical: the interest of the questions is so great that I do
+not repent of having submitted them to the reader.
+
+
+
+
+CHAPTER IV
+
+CHANCE
+
+
+I
+
+"How dare we speak of the laws of chance? Is not chance the antithesis
+of all law?" So says Bertrand at the beginning of his _Calcul des
+probabiltités_. Probability is opposed to certitude; so it is what we do
+not know and consequently it seems what we could not calculate. Here is
+at least apparently a contradiction, and about it much has already been
+written.
+
+And first, what is chance? The ancients distinguished between phenomena
+seemingly obeying harmonious laws, established once for all, and those
+which they attributed to chance; these were the ones unpredictable
+because rebellious to all law. In each domain the precise laws did not
+decide everything, they only drew limits between which chance might act.
+In this conception the word chance had a precise and objective meaning;
+what was chance for one was also chance for another and even for the
+gods.
+
+But this conception is not ours to-day. We have become absolute
+determinists, and even those who want to reserve the rights of human
+free will let determinism reign undividedly in the inorganic world at
+least. Every phenomenon, however minute, has a cause; and a mind
+infinitely powerful, infinitely well-informed about the laws of nature,
+could have foreseen it from the beginning of the centuries. If such a
+mind existed, we could not play with it at any game of chance; we should
+always lose.
+
+In fact for it the word chance would not have any meaning, or rather
+there would be no chance. It is because of our weakness and our
+ignorance that the word has a meaning for us. And, even without going
+beyond our feeble humanity, what is chance for the ignorant is not
+chance for the scientist. Chance is only the measure of our ignorance.
+Fortuitous phenomena are, by definition, those whose laws we do not
+know.
+
+But is this definition altogether satisfactory? When the first Chaldean
+shepherds followed with their eyes the movements of the stars, they knew
+not as yet the laws of astronomy; would they have dreamed of saying that
+the stars move at random? If a modern physicist studies a new
+phenomenon, and if he discovers its law Tuesday, would he have said
+Monday that this phenomenon was fortuitous? Moreover, do we not often
+invoke what Bertrand calls the laws of chance, to predict a phenomenon?
+For example, in the kinetic theory of gases we obtain the known laws of
+Mariotte and of Gay-Lussac by means of the hypothesis that the
+velocities of the molecules of gas vary irregularly, that is to say at
+random. All physicists will agree that the observable laws would be much
+less simple if the velocities were ruled by any simple elementary law
+whatsoever, if the molecules were, as we say, _organized_, if they were
+subject to some discipline. It is due to chance, that is to say, to our
+ignorance, that we can draw our conclusions; and then if the word chance
+is simply synonymous with ignorance what does that mean? Must we
+therefore translate as follows?
+
+"You ask me to predict for you the phenomena about to happen. If,
+unluckily, I knew the laws of these phenomena I could make the
+prediction only by inextricable calculations and would have to renounce
+attempting to answer you; but as I have the good fortune not to know
+them, I will answer you at once. And what is most surprising, my answer
+will be right."
+
+So it must well be that chance is something other than the name we give
+our ignorance, that among phenomena whose causes are unknown to us we
+must distinguish fortuitous phenomena about which the calculus of
+probabilities will provisionally give information, from those which are
+not fortuitous and of which we can say nothing so long as we shall not
+have determined the laws governing them. For the fortuitous phenomena
+themselves, it is clear that the information given us by the calculus of
+probabilities will not cease to be true upon the day when these
+phenomena shall be better known.
+
+The director of a life insurance company does not know when each of the
+insured will die, but he relies upon the calculus of probabilities and
+on the law of great numbers, and he is not deceived, since he
+distributes dividends to his stockholders. These dividends would not
+vanish if a very penetrating and very indiscreet physician should, after
+the policies were signed, reveal to the director the life chances of the
+insured. This doctor would dissipate the ignorance of the director, but
+he would have no influence on the dividends, which evidently are not an
+outcome of this ignorance.
+
+
+II
+
+To find a better definition of chance we must examine some of the facts
+which we agree to regard as fortuitous, and to which the calculus of
+probabilities seems to apply; we then shall investigate what are their
+common characteristics.
+
+The first example we select is that of unstable equilibrium; if a cone
+rests upon its apex, we know well that it will fall, but we do not know
+toward what side; it seems to us chance alone will decide. If the cone
+were perfectly symmetric, if its axis were perfectly vertical, if it
+were acted upon by no force other than gravity, it would not fall at
+all. But the least defect in symmetry will make it lean slightly toward
+one side or the other, and if it leans, however little, it will fall
+altogether toward that side. Even if the symmetry were perfect, a very
+slight tremor, a breath of air could make it incline some seconds of
+arc; this will be enough to determine its fall and even the sense of its
+fall which will be that of the initial inclination.
+
+A very slight cause, which escapes us, determines a considerable effect
+which we can not help seeing, and then we say this effect is due to
+chance. If we could know exactly the laws of nature and the situation of
+the universe at the initial instant, we should be able to predict
+exactly the situation of this same universe at a subsequent instant. But
+even when the natural laws should have no further secret for us, we
+could know the initial situation only _approximately_. If that permits
+us to foresee the subsequent situation _with the same degree of
+approximation_, this is all we require, we say the phenomenon has been
+predicted, that it is ruled by laws. But this is not always the case; it
+may happen that slight differences in the initial conditions produce
+very great differences in the final phenomena; a slight error in the
+former would make an enormous error in the latter. Prediction becomes
+impossible and we have the fortuitous phenomenon.
+
+Our second example will be very analogous to the first and we shall take
+it from meteorology. Why have the meteorologists such difficulty in
+predicting the weather with any certainty? Why do the rains, the
+tempests themselves seem to us to come by chance, so that many persons
+find it quite natural to pray for rain or shine, when they would think
+it ridiculous to pray for an eclipse? We see that great perturbations
+generally happen in regions where the atmosphere is in unstable
+equilibrium. The meteorologists are aware that this equilibrium is
+unstable, that a cyclone is arising somewhere; but where they can not
+tell; one-tenth of a degree more or less at any point, and the cyclone
+bursts here and not there, and spreads its ravages over countries it
+would have spared. This we could have foreseen if we had known that
+tenth of a degree, but the observations were neither sufficiently close
+nor sufficiently precise, and for this reason all seems due to the
+agency of chance. Here again we find the same contrast between a very
+slight cause, unappreciable to the observer, and important effects,
+which are sometimes tremendous disasters.
+
+Let us pass to another example, the distribution of the minor planets on
+the zodiac. Their initial longitudes may have been any longitudes
+whatever; but their mean motions were different and they have revolved
+for so long a time that we may say they are now distributed _at random_
+along the zodiac. Very slight initial differences between their
+distances from the sun, or, what comes to the same thing, between their
+mean motions, have ended by giving enormous differences between their
+present longitudes. An excess of the thousandth of a second in the daily
+mean motion will give in fact a second in three years, a degree in ten
+thousand years, an entire circumference in three or four million years,
+and what is that to the time which has passed since the minor planets
+detached themselves from the nebula of Laplace? Again therefore we see a
+slight cause and a great effect; or better, slight differences in the
+cause and great differences in the effect.
+
+The game of roulette does not take us as far as might seem from the
+preceding example. Assume a needle to be turned on a pivot over a dial
+divided into a hundred sectors alternately red and black. If it stops on
+a red sector I win; if not, I lose. Evidently all depends upon the
+initial impulse I give the needle. The needle will make, suppose, ten or
+twenty turns, but it will stop sooner or not so soon, according as I
+shall have pushed it more or less strongly. It suffices that the impulse
+vary only by a thousandth or a two thousandth to make the needle stop
+over a black sector or over the following red one. These are differences
+the muscular sense can not distinguish and which elude even the most
+delicate instruments. So it is impossible for me to foresee what the
+needle I have started will do, and this is why my heart throbs and I
+hope everything from luck. The difference in the cause is imperceptible,
+and the difference in the effect is for me of the highest importance,
+since it means my whole stake.
+
+
+III
+
+Permit me, in this connection, a thought somewhat foreign to my subject.
+Some years ago a philosopher said that the future is determined by the
+past, but not the past by the future; or, in other words, from knowledge
+of the present we could deduce the future, but not the past; because,
+said he, a cause can have only one effect, while the same effect might
+be produced by several different causes. It is clear no scientist can
+subscribe to this conclusion. The laws of nature bind the antecedent to
+the consequent in such a way that the antecedent is as well determined
+by the consequent as the consequent by the antecedent. But whence came
+the error of this philosopher? We know that in virtue of Carnot's
+principle physical phenomena are irreversible and the world tends toward
+uniformity. When two bodies of different temperature come in contact,
+the warmer gives up heat to the colder; so we may foresee that the
+temperature will equalize. But once equal, if asked about the anterior
+state, what can we answer? We might say that one was warm and the other
+cold, but not be able to divine which formerly was the warmer.
+
+And yet in reality the temperatures will never reach perfect equality.
+The difference of the temperatures only tends asymptotically toward
+zero. There comes a moment when our thermometers are powerless to make
+it known. But if we had thermometers a thousand times, a hundred
+thousand times as sensitive, we should recognize that there still is a
+slight difference, and that one of the bodies remains a little warmer
+than the other, and so we could say this it is which formerly was much
+the warmer.
+
+So then there are, contrary to what we found in the former examples,
+great differences in cause and slight differences in effect. Flammarion
+once imagined an observer going away from the earth with a velocity
+greater than that of light; for him time would have changed sign.
+History would be turned about, and Waterloo would precede Austerlitz.
+Well, for this observer, effects and causes would be inverted; unstable
+equilibrium would no longer be the exception. Because of the universal
+irreversibility, all would seem to him to come out of a sort of chaos in
+unstable equilibrium. All nature would appear to him delivered over to
+chance.
+
+
+IV
+
+Now for other examples where we shall see somewhat different
+characteristics. Take first the kinetic theory of gases. How should we
+picture a receptacle filled with gas? Innumerable molecules, moving at
+high speeds, flash through this receptacle in every direction. At every
+instant they strike against its walls or each other, and these
+collisions happen under the most diverse conditions. What above all
+impresses us here is not the littleness of the causes, but their
+complexity, and yet the former element is still found here and plays an
+important rôle. If a molecule deviated right or left from its
+trajectory, by a very small quantity, comparable to the radius of action
+of the gaseous molecules, it would avoid a collision or sustain it under
+different conditions, and that would vary the direction of its velocity
+after the impact, perhaps by ninety degrees or by a hundred and eighty
+degrees.
+
+And this is not all; we have just seen that it is necessary to deflect
+the molecule before the clash by only an infinitesimal, to produce its
+deviation after the collision by a finite quantity. If then the molecule
+undergoes two successive shocks, it will suffice to deflect it before
+the first by an infinitesimal of the second order, for it to deviate
+after the first encounter by an infinitesimal of the first order, and
+after the second hit, by a finite quantity. And the molecule will not
+undergo merely two shocks; it will undergo a very great number per
+second. So that if the first shock has multiplied the deviation by a
+very large number _A_, after _n_ shocks it will be multiplied by
+_A_^{_n_}. It will therefore become very great not merely because _A_
+is large, that is to say because little causes produce big effects, but
+because the exponent _n_ is large, that is to say because the shocks are
+very numerous and the causes very complex.
+
+Take a second example. Why do the drops of rain in a shower seem to be
+distributed at random? This is again because of the complexity of the
+causes which determine their formation. Ions are distributed in the
+atmosphere. For a long while they have been subjected to air-currents
+constantly changing, they have been caught in very small whirlwinds, so
+that their final distribution has no longer any relation to their
+initial distribution. Suddenly the temperature falls, vapor condenses,
+and each of these ions becomes the center of a drop of rain. To know
+what will be the distribution of these drops and how many will fall on
+each paving-stone, it would not be sufficient to know the initial
+situation of the ions, it would be necessary to compute the effect of a
+thousand little capricious air-currents.
+
+And again it is the same if we put grains of powder in suspension in
+water. The vase is ploughed by currents whose law we know not, we only
+know it is very complicated. At the end of a certain time the grains
+will be distributed at random, that is to say uniformly, in the vase;
+and this is due precisely to the complexity of these currents. If they
+obeyed some simple law, if for example the vase revolved and the
+currents circulated around the axis of the vase, describing circles, it
+would no longer be the same, since each grain would retain its initial
+altitude and its initial distance from the axis.
+
+We should reach the same result in considering the mixing of two liquids
+or of two fine-grained powders. And to take a grosser example, this is
+also what happens when we shuffle playing-cards. At each stroke the
+cards undergo a permutation (analogous to that studied in the theory of
+substitutions). What will happen? The probability of a particular
+permutation (for example, that bringing to the _n_th place the card
+occupying the [phi](_n_)th place before the permutation) depends upon
+the player's habits. But if this player shuffles the cards long enough,
+there will be a great number of successive permutations, and the
+resulting final order will no longer be governed by aught but chance; I
+mean to say that all possible orders will be equally probable. It is to
+the great number of successive permutations, that is to say to the
+complexity of the phenomenon, that this result is due.
+
+A final word about the theory of errors. Here it is that the causes are
+complex and multiple. To how many snares is not the observer exposed,
+even with the best instrument! He should apply himself to finding out
+the largest and avoiding them. These are the ones giving birth to
+systematic errors. But when he has eliminated those, admitting that he
+succeeds, there remain many small ones which, their effects
+accumulating, may become dangerous. Thence come the accidental errors;
+and we attribute them to chance because their causes are too complicated
+and too numerous. Here again we have only little causes, but each of
+them would produce only a slight effect; it is by their union and their
+number that their effects become formidable.
+
+
+V
+
+We may take still a third point of view, less important than the first
+two and upon which I shall lay less stress. When we seek to foresee an
+event and examine its antecedents, we strive to search into the anterior
+situation. This could not be done for all parts of the universe and we
+are content to know what is passing in the neighborhood of the point
+where the event should occur, or what would appear to have some relation
+to it. An examination can not be complete and we must know how to
+choose. But it may happen that we have passed by circumstances which at
+first sight seemed completely foreign to the foreseen happening, to
+which one would never have dreamed of attributing any influence and
+which nevertheless, contrary to all anticipation, come to play an
+important rôle.
+
+A man passes in the street going to his business; some one knowing the
+business could have told why he started at such a time and went by such
+a street. On the roof works a tiler. The contractor employing him could
+in a certain measure foresee what he would do. But the passer-by
+scarcely thinks of the tiler, nor the tiler of him; they seem to belong
+to two worlds completely foreign to one another. And yet the tiler drops
+a tile which kills the man, and we do not hesitate to say this is
+chance.
+
+Our weakness forbids our considering the entire universe and makes us
+cut it up into slices. We try to do this as little artificially as
+possible. And yet it happens from time to time that two of these slices
+react upon each other. The effects of this mutual action then seem to us
+to be due to chance.
+
+Is this a third way of conceiving chance? Not always; in fact most often
+we are carried back to the first or the second. Whenever two worlds
+usually foreign to one another come thus to react upon each other, the
+laws of this reaction must be very complex. On the other hand, a very
+slight change in the initial conditions of these two worlds would have
+been sufficient for the reaction not to have happened. How little was
+needed for the man to pass a second later or the tiler to drop his tile
+a second sooner.
+
+
+VI
+
+All we have said still does not explain why chance obeys laws. Does the
+fact that the causes are slight or complex suffice for our foreseeing,
+if not their effects _in each case_, at least what their effects will
+be, _on the average_? To answer this question we had better take up
+again some of the examples already cited.
+
+I shall begin with that of the roulette. I have said that the point
+where the needle will stop depends upon the initial push given it. What
+is the probability of this push having this or that value? I know
+nothing about it, but it is difficult for me not to suppose that this
+probability is represented by a continuous analytic function. The
+probability that the push is comprised between [alpha] and [alpha] +
+[epsilon] will then be sensibly equal to the probability of its being
+comprised between [alpha] + [epsilon] and [alpha] + 2[epsilon],
+_provided_ [epsilon] _be very small_. This is a property common to all
+analytic functions. Minute variations of the function are proportional
+to minute variations of the variable.
+
+But we have assumed that an exceedingly slight variation of the push
+suffices to change the color of the sector over which the needle finally
+stops. From [alpha] to [alpha]+[epsilon] it is red, from
+[alpha]+[epsilon] to [alpha]+2[epsilon] it is black; the probability of
+each red sector is therefore the same as of the following black, and
+consequently the total probability of red equals the total probability
+of black.
+
+The datum of the question is the analytic function representing the
+probability of a particular initial push. But the theorem remains true
+whatever be this datum, since it depends upon a property common to all
+analytic functions. From this it follows finally that we no longer need
+the datum.
+
+What we have just said for the case of the roulette applies also to the
+example of the minor planets. The zodiac may be regarded as an immense
+roulette on which have been tossed many little balls with different
+initial impulses varying according to some law. Their present
+distribution is uniform and independent of this law, for the same reason
+as in the preceding case. Thus we see why phenomena obey the laws of
+chance when slight differences in the causes suffice to bring on great
+differences in the effects. The probabilities of these slight
+differences may then be regarded as proportional to these differences
+themselves, just because these differences are minute, and the
+infinitesimal increments of a continuous function are proportional to
+those of the variable.
+
+Take an entirely different example, where intervenes especially the
+complexity of the causes. Suppose a player shuffles a pack of cards. At
+each shuffle he changes the order of the cards, and he may change them
+in many ways. To simplify the exposition, consider only three cards. The
+cards which before the shuffle occupied respectively the places 123, may
+after the shuffle occupy the places
+
+ 123, 231, 312, 321, 132, 213.
+
+Each of these six hypotheses is possible and they have respectively for
+probabilities:
+
+ p_1, p_2, p_3, p_4, p_5, p_6.
+
+The sum of these six numbers equals 1; but this is all we know of them;
+these six probabilities depend naturally upon the habits of the player
+which we do not know.
+
+At the second shuffle and the following, this will recommence, and under
+the same conditions; I mean that p_4 for example represents always the
+probability that the three cards which occupied after the _n_th shuffle
+and before the _n_ + 1th the places 123, occupy the places 321 after the
+_n_ + 1th shuffle. And this remains true whatever be the number _n_,
+since the habits of the player and his way of shuffling remain the same.
+
+But if the number of shuffles is very great, the cards which before the
+first shuffle occupied the places 123 may, after the last shuffle,
+occupy the places
+
+ 123, 231, 312, 321, 132, 213
+
+and the probability of these six hypotheses will be sensibly the same
+and equal to 1/6; and this will be true whatever be the numbers
+p_1 ... p_6 which we do not know. The great number of shuffles, that
+is to say the complexity of the causes, has produced uniformity.
+
+This would apply without change if there were more than three cards,
+but even with three cards the demonstration would be complicated;
+let it suffice to give it for only two cards. Then we have only two
+possibilities 12, 21 with the probabilities p_1 and p_2 = 1 - p_1.
+
+Suppose _n_ shuffles and suppose I win one franc if the cards are
+finally in the initial order and lose one if they are finally inverted.
+Then, my mathematical expectation will be (p_1 - p_2)^{_n_}.
+
+The difference p_1 - p_2 is certainly less than 1; so that if _n_ is
+very great my expectation will be zero; we need not learn p_1 and p_2
+to be aware that the game is equitable.
+
+There would always be an exception if one of the numbers p_1 and p_2
+was equal to 1 and the other naught. _Then it would not apply because
+our initial hypotheses would be too simple._
+
+What we have just seen applies not only to the mixing of cards, but to
+all mixings, to those of powders and of liquids; and even to those of
+the molecules of gases in the kinetic theory of gases.
+
+To return to this theory, suppose for a moment a gas whose molecules can
+not mutually clash, but may be deviated by hitting the insides of the
+vase wherein the gas is confined. If the form of the vase is
+sufficiently complex the distribution of the molecules and that of the
+velocities will not be long in becoming uniform. But this will not be so
+if the vase is spherical or if it has the shape of a cuboid. Why?
+Because in the first case the distance from the center to any trajectory
+will remain constant; in the second case this will be the absolute value
+of the angle of each trajectory with the faces of the cuboid.
+
+So we see what should be understood by conditions _too simple_; they are
+those which conserve something, which leave an invariant remaining. Are
+the differential equations of the problem too simple for us to apply the
+laws of chance? This question would seem at first view to lack precise
+meaning; now we know what it means. They are too simple if they conserve
+something, if they admit a uniform integral. If something in the initial
+conditions remains unchanged, it is clear the final situation can no
+longer be independent of the initial situation.
+
+We come finally to the theory of errors. We know not to what are due the
+accidental errors, and precisely because we do not know, we are aware
+they obey the law of Gauss. Such is the paradox. The explanation is
+nearly the same as in the preceding cases. We need know only one thing:
+that the errors are very numerous, that they are very slight, that each
+may be as well negative as positive. What is the curve of probability of
+each of them? We do not know; we only suppose it is symmetric. We prove
+then that the resultant error will follow Gauss's law, and this
+resulting law is independent of the particular laws which we do not
+know. Here again the simplicity of the result is born of the very
+complexity of the data.
+
+
+VII
+
+But we are not through with paradoxes. I have just recalled the figment
+of Flammarion, that of the man going quicker than light, for whom time
+changes sign. I said that for him all phenomena would seem due to
+chance. That is true from a certain point of view, and yet all these
+phenomena at a given moment would not be distributed in conformity with
+the laws of chance, since the distribution would be the same as for us,
+who, seeing them unfold harmoniously and without coming out of a primal
+chaos, do not regard them as ruled by chance.
+
+What does that mean? For Lumen, Flammarion's man, slight causes seem to
+produce great effects; why do not things go on as for us when we think
+we see grand effects due to little causes? Would not the same reasoning
+be applicable in his case?
+
+Let us return to the argument. When slight differences in the causes
+produce vast differences in the effects, why are these effects
+distributed according to the laws of chance? Suppose a difference of a
+millimeter in the cause produces a difference of a kilometer in the
+effect. If I win in case the effect corresponds to a kilometer bearing
+an even number, my probability of winning will be 1/2. Why? Because to
+make that, the cause must correspond to a millimeter with an even
+number. Now, according to all appearance, the probability of the cause
+varying between certain limits will be proportional to the distance
+apart of these limits, provided this distance be very small. If this
+hypothesis were not admitted there would no longer be any way of
+representing the probability by a continuous function.
+
+What now will happen when great causes produce small effects? This is
+the case where we should not attribute the phenomenon to chance and
+where on the contrary Lumen would attribute it to chance. To a
+difference of a kilometer in the cause would correspond a difference of
+a millimeter in the effect. Would the probability of the cause being
+comprised between two limits _n_ kilometers apart still be proportional
+to _n_? We have no reason to suppose so, since this distance, _n_
+kilometers, is great. But the probability that the effect lies between
+two limits _n_ millimeters apart will be precisely the same, so it will
+not be proportional to _n_, even though this distance, _n_ millimeters,
+be small. There is no way therefore of representing the law of
+probability of effects by a continuous curve. This curve, understand,
+may remain continuous in the _analytic_ sense of the word; to
+_infinitesimal_ variations of the abscissa will correspond infinitesimal
+variations of the ordinate. But _practically_ it will not be continuous,
+since _very small_ variations of the ordinate would not correspond to
+very small variations of the abscissa. It would become impossible to
+trace the curve with an ordinary pencil; that is what I mean.
+
+So what must we conclude? Lumen has no right to say that the
+probability of the cause (_his_ cause, our effect) should be represented
+necessarily by a continuous function. But then why have we this right?
+It is because this state of unstable equilibrium which we have been
+calling initial is itself only the final outcome of a long previous
+history. In the course of this history complex causes have worked a
+great while: they have contributed to produce the mixture of elements
+and they have tended to make everything uniform at least within a small
+region; they have rounded off the corners, smoothed down the hills and
+filled up the valleys. However capricious and irregular may have been
+the primitive curve given over to them, they have worked so much toward
+making it regular that finally they deliver over to us a continuous
+curve. And this is why we may in all confidence assume its continuity.
+
+Lumen would not have the same reasons for such a conclusion. For him
+complex causes would not seem agents of equalization and regularity, but
+on the contrary would create only inequality and differentiation. He
+would see a world more and more varied come forth from a sort of
+primitive chaos. The changes he could observe would be for him
+unforeseen and impossible to foresee. They would seem to him due to some
+caprice or another; but this caprice would be quite different from our
+chance, since it would be opposed to all law, while our chance still has
+its laws. All these points call for lengthy explications, which perhaps
+would aid in the better comprehension of the irreversibility of the
+universe.
+
+
+VIII
+
+We have sought to define chance, and now it is proper to put a question.
+Has chance thus defined, in so far as this is possible, objectivity?
+
+It may be questioned. I have spoken of very slight or very complex
+causes. But what is very little for one may be very big for another, and
+what seems very complex to one may seem simple to another. In part I
+have already answered by saying precisely in what cases differential
+equations become too simple for the laws of chance to remain applicable.
+But it is fitting to examine the matter a little more closely, because
+we may take still other points of view.
+
+What means the phrase 'very slight'? To understand it we need only go
+back to what has already been said. A difference is very slight, an
+interval is very small, when within the limits of this interval the
+probability remains sensibly constant. And why may this probability be
+regarded as constant within a small interval? It is because we assume
+that the law of probability is represented by a continuous curve,
+continuous not only in the analytic sense, but _practically_ continuous,
+as already explained. This means that it not only presents no absolute
+hiatus, but that it has neither salients nor reentrants too acute or too
+accentuated.
+
+And what gives us the right to make this hypothesis? We have already
+said it is because, since the beginning of the ages, there have always
+been complex causes ceaselessly acting in the same way and making the
+world tend toward uniformity without ever being able to turn back. These
+are the causes which little by little have flattened the salients and
+filled up the reentrants, and this is why our probability curves now
+show only gentle undulations. In milliards of milliards of ages another
+step will have been made toward uniformity, and these undulations will
+be ten times as gentle; the radius of mean curvature of our curve will
+have become ten times as great. And then such a length as seems to us
+to-day not very small, since on our curve an arc of this length can not
+be regarded as rectilineal, should on the contrary at that epoch be
+called very little, since the curvature will have become ten times less
+and an arc of this length may be sensibly identified with a sect.
+
+Thus the phrase 'very slight' remains relative; but it is not relative
+to such or such a man, it is relative to the actual state of the world.
+It will change its meaning when the world shall have become more
+uniform, when all things shall have blended still more. But then
+doubtless men can no longer live and must give place to other
+beings--should I say far smaller or far larger? So that our criterion,
+remaining true for all men, retains an objective sense.
+
+And on the other hand what means the phrase 'very complex'? I have
+already given one solution, but there are others. Complex causes we have
+said produce a blend more and more intimate, but after how long a time
+will this blend satisfy us? When will it have accumulated sufficient
+complexity? When shall we have sufficiently shuffled the cards? If we
+mix two powders, one blue, the other white, there comes a moment when
+the tint of the mixture seems to us uniform because of the feebleness of
+our senses; it will be uniform for the presbyte, forced to gaze from
+afar, before it will be so for the myope. And when it has become uniform
+for all eyes, we still could push back the limit by the use of
+instruments. There is no chance for any man ever to discern the infinite
+variety which, if the kinetic theory is true, hides under the uniform
+appearance of a gas. And yet if we accept Gouy's ideas on the Brownian
+movement, does not the microscope seem on the point of showing us
+something analogous?
+
+This new criterion is therefore relative like the first; and if it
+retains an objective character, it is because all men have approximately
+the same senses, the power of their instruments is limited, and besides
+they use them only exceptionally.
+
+
+IX
+
+It is just the same in the moral sciences and particularly in history.
+The historian is obliged to make a choice among the events of the epoch
+he studies; he recounts only those which seem to him the most important.
+He therefore contents himself with relating the most momentous events of
+the sixteenth century, for example, as likewise the most remarkable
+facts of the seventeenth century. If the first suffice to explain the
+second, we say these conform to the laws of history. But if a great
+event of the seventeenth century should have for cause a small fact of
+the sixteenth century which no history reports, which all the world has
+neglected, then we say this event is due to chance. This word has
+therefore the same sense as in the physical sciences; it means that
+slight causes have produced great effects.
+
+The greatest bit of chance is the birth of a great man. It is only by
+chance that meeting of two germinal cells, of different sex, containing
+precisely, each on its side, the mysterious elements whose mutual
+reaction must produce the genius. One will agree that these elements
+must be rare and that their meeting is still more rare. How slight a
+thing it would have required to deflect from its route the carrying
+spermatozoon. It would have sufficed to deflect it a tenth of a
+millimeter and Napoleon would not have been born and the destinies of a
+continent would have been changed. No example can better make us
+understand the veritable characteristics of chance.
+
+One more word about the paradoxes brought out by the application of the
+calculus of probabilities to the moral sciences. It has been proven that
+no Chamber of Deputies will ever fail to contain a member of the
+opposition, or at least such an event would be so improbable that we
+might without fear wager the contrary, and bet a million against a sou.
+
+Condorcet has striven to calculate how many jurors it would require to
+make a judicial error practically impossible. If we had used the results
+of this calculation, we should certainly have been exposed to the same
+disappointments as in betting, on the faith of the calculus, that the
+opposition would never be without a representative.
+
+The laws of chance do not apply to these questions. If justice be not
+always meted out to accord with the best reasons, it uses less than we
+think the method of Bridoye. This is perhaps to be regretted, for then
+the system of Condorcet would shield us from judicial errors.
+
+What is the meaning of this? We are tempted to attribute facts of this
+nature to chance because their causes are obscure; but this is not true
+chance. The causes are unknown to us, it is true, and they are even
+complex; but they are not sufficiently so, since they conserve
+something. We have seen that this it is which distinguishes causes 'too
+simple.' When men are brought together they no longer decide at random
+and independently one of another; they influence one another. Multiplex
+causes come into action. They worry men, dragging them to right or left,
+but one thing there is they can not destroy, this is their Panurge
+flock-of-sheep habits. And this is an invariant.
+
+
+X
+
+Difficulties are indeed involved in the application of the calculus of
+probabilities to the exact sciences. Why are the decimals of a table of
+logarithms, why are those of the number [pi] distributed in accordance
+with the laws of chance? Elsewhere I have already studied the question
+in so far as it concerns logarithms, and there it is easy. It is clear
+that a slight difference of argument will give a slight difference of
+logarithm, but a great difference in the sixth decimal of the logarithm.
+Always we find again the same criterion.
+
+But as for the number [pi], that presents more difficulties, and I have
+at the moment nothing worth while to say.
+
+There would be many other questions to resolve, had I wished to attack
+them before solving that which I more specially set myself. When we
+reach a simple result, when we find for example a round number, we say
+that such a result can not be due to chance, and we seek, for its
+explanation, a non-fortuitous cause. And in fact there is only a very
+slight probability that among 10,000 numbers chance will give a round
+number; for example, the number 10,000. This has only one chance in
+10,000. But there is only one chance in 10,000 for the occurrence of any
+other one number; and yet this result will not astonish us, nor will it
+be hard for us to attribute it to chance; and that simply because it
+will be less striking.
+
+Is this a simple illusion of ours, or are there cases where this way of
+thinking is legitimate? We must hope so, else were all science
+impossible. When we wish to check a hypothesis, what do we do? We can
+not verify all its consequences, since they would be infinite in number;
+we content ourselves with verifying certain ones and if we succeed we
+declare the hypothesis confirmed, because so much success could not be
+due to chance. And this is always at bottom the same reasoning.
+
+I can not completely justify it here, since it would take too much time;
+but I may at least say that we find ourselves confronted by two
+hypotheses, either a simple cause or that aggregate of complex causes we
+call chance. We find it natural to suppose that the first should produce
+a simple result, and then, if we find that simple result, the round
+number for example, it seems more likely to us to be attributable to the
+simple cause which must give it almost certainly, than to chance which
+could only give it once in 10,000 times. It will not be the same if we
+find a result which is not simple; chance, it is true, will not give
+this more than once in 10,000 times; but neither has the simple cause
+any more chance of producing it.
+
+
+
+
+BOOK II
+
+
+MATHEMATICAL REASONING
+
+
+
+
+CHAPTER I
+
+THE RELATIVITY OF SPACE
+
+
+I
+
+It is impossible to represent to oneself empty space; all our efforts to
+imagine a pure space, whence should be excluded the changing images of
+material objects, can result only in a representation where vividly
+colored surfaces, for example, are replaced by lines of faint
+coloration, and we can not go to the very end in this way without all
+vanishing and terminating in nothingness. Thence comes the irreducible
+relativity of space.
+
+Whoever speaks of absolute space uses a meaningless phrase. This is a
+truth long proclaimed by all who have reflected upon the matter, but
+which we are too often led to forget.
+
+I am at a determinate point in Paris, place du Panthéon for instance,
+and I say: I shall come back _here_ to-morrow. If I be asked: Do you
+mean you will return to the same point of space, I shall be tempted to
+answer: yes; and yet I shall be wrong, since by to-morrow the earth will
+have journeyed hence, carrying with it the place du Panthéon, which will
+have traveled over more than two million kilometers. And if I tried to
+speak more precisely, I should gain nothing, since our globe has run
+over these two million kilometers in its motion with relation to the
+sun, while the sun in its turn is displaced with reference to the Milky
+Way, while the Milky Way itself is doubtless in motion without our being
+able to perceive its velocity. So that we are completely ignorant, and
+always shall be, of how much the place du Panthéon is displaced in a
+day.
+
+In sum, I meant to say: To-morrow I shall see again the dome and the
+pediment of the Panthéon, and if there were no Panthéon my phrase would
+be meaningless and space would vanish.
+
+This is one of the most commonplace forms of the principle of the
+relativity of space; but there is another, upon which Delbeuf has
+particularly insisted. Suppose that in the night all the dimensions of
+the universe become a thousand times greater: the world will have
+remained _similar_ to itself, giving to the word _similitude_ the same
+meaning as in Euclid, Book VI. Only what was a meter long will measure
+thenceforth a kilometer, what was a millimeter long will become a meter.
+The bed whereon I lie and my body itself will be enlarged in the same
+proportion.
+
+When I awake to-morrow morning, what sensation shall I feel in presence
+of such an astounding transformation? Well, I shall perceive nothing at
+all. The most precise measurements will be incapable of revealing to me
+anything of this immense convulsion, since the measures I use will have
+varied precisely in the same proportion as the objects I seek to
+measure. In reality, this convulsion exists only for those who reason as
+if space were absolute. If I for a moment have reasoned as they do, it
+is the better to bring out that their way of seeing implies
+contradiction. In fact it would be better to say that, space being
+relative, nothing at all has happened, which is why we have perceived
+nothing.
+
+Has one the right, therefore, to say he knows the distance between two
+points? No, since this distance could undergo enormous variations
+without our being able to perceive them, provided the other distances
+have varied in the same proportion. We have just seen that when I say: I
+shall be here to-morrow, this does not mean: To-morrow I shall be at the
+same point of space where I am to-day, but rather: To-morrow I shall be
+at the same distance from the Panthéon as to-day. And we see that this
+statement is no longer sufficient and that I should say: To-morrow and
+to-day my distance from the Panthéon will be equal to the same number of
+times the height of my body.
+
+But this is not all; I have supposed the dimensions of the world to
+vary, but that at least the world remained always similar to itself. We
+might go much further, and one of the most astonishing theories of
+modern physics furnishes us the occasion.
+
+According to Lorentz and Fitzgerald, all the bodies borne along in the
+motion of the earth undergo a deformation.
+
+This deformation is, in reality, very slight, since all dimensions
+parallel to the movement of the earth diminish by a hundred millionth,
+while the dimensions perpendicular to this movement are unchanged. But
+it matters little that it is slight, that it exists suffices for the
+conclusion I am about to draw. And besides, I have said it was slight,
+but in reality I know nothing about it; I have myself been victim of the
+tenacious illusion which makes us believe we conceive an absolute space;
+I have thought of the motion of the earth in its elliptic orbit around
+the sun, and I have allowed thirty kilometers as its velocity. But its
+real velocity (I mean, this time, not its absolute velocity, which is
+meaningless, but its velocity with relation to the ether), I do not know
+that, and have no means of knowing it: it is perhaps, 10, 100 times
+greater, and then the deformation will be 100, 10,000 times more.
+
+Can we show this deformation? Evidently not; here is a cube with edge
+one meter; in consequence of the earth's displacement it is deformed,
+one of its edges, that parallel to the motion, becomes smaller, the
+others do not change. If I wish to assure myself of it by aid of a meter
+measure, I shall measure first one of the edges perpendicular to the
+motion and shall find that my standard meter fits this edge exactly; and
+in fact neither of these two lengths is changed, since both are
+perpendicular to the motion. Then I wish to measure the other edge, that
+parallel to the motion; to do this I displace my meter and turn it so as
+to apply it to the edge. But the meter, having changed orientation and
+become parallel to the motion, has undergone, in its turn, the
+deformation, so that though the edge be not a meter long, it will fit
+exactly, I shall find out nothing.
+
+You ask then of what use is the hypothesis of Lorentz and of Fitzgerald
+if no experiment can permit of its verification? It is my exposition
+that has been incomplete; I have spoken only of measurements that can be
+made with a meter; but we can also measure a length by the time it takes
+light to traverse it, on condition we suppose the velocity of light
+constant and independent of direction. Lorentz could have accounted for
+the facts by supposing the velocity of light greater in the direction
+of the earth's motion than in the perpendicular direction. He preferred
+to suppose that the velocity is the same in these different directions
+but that the bodies are smaller in the one than in the other. If the
+wave surfaces of light had undergone the same deformations as the
+material bodies we should never have perceived the Lorentz-Fitzgerald
+deformation.
+
+In either case, it is not a question of absolute magnitude, but of the
+measure of this magnitude by means of some instrument; this instrument
+may be a meter, or the path traversed by light; it is only the relation
+of the magnitude to the instrument that we measure; and if this relation
+is altered, we have no way of knowing whether it is the magnitude or the
+instrument which has changed.
+
+But what I wish to bring out is, that in this deformation the world has
+not remained similar to itself; squares have become rectangles, circles
+ellipses, spheres ellipsoids. And yet we have no way of knowing whether
+this deformation be real.
+
+Evidently one could go much further: in place of the Lorentz-Fitzgerald
+deformation, whose laws are particularly simple, we could imagine any
+deformation whatsoever. Bodies could be deformed according to any laws,
+as complicated as we might wish, we never should notice it provided all
+bodies without exception were deformed according to the same laws. In
+saying, all bodies without exception, I include of course our own body
+and the light rays emanating from different objects.
+
+If we look at the world in one of those mirrors of complicated shape
+which deform objects in a bizarre way, the mutual relations of the
+different parts of this world would not be altered; if, in fact two real
+objects touch, their images likewise seem to touch. Of course when we
+look in such a mirror we see indeed the deformation, but this is because
+the real world subsists alongside of its deformed image; and then even
+were this real world hidden from us, something there is could not be
+hidden, ourself; we could not cease to see, or at least to feel, our
+body and our limbs which have not been deformed and which continue to
+serve us as instruments of measure.
+
+But if we imagine our body itself deformed in the same way as if seen
+in the mirror, these instruments of measure in their turn will fail us
+and the deformation will no longer be ascertainable.
+
+Consider in the same way two worlds images of one another; to each
+object _P_ of the world _A_ corresponds in the world _B_ an object _P'_,
+its image; the coordinates of this image _P'_ are determinate functions
+of those of the object _P_; moreover these functions may be any
+whatsoever; I only suppose them chosen once for all. Between the
+position of _P_ and that of _P'_ there is a constant relation; what this
+relation is, matters not; enough that it be constant.
+
+Well, these two worlds will be indistinguishable one from the other. I
+mean the first will be for its inhabitants what the second is for its.
+And so it will be as long as the two worlds remain strangers to each
+other. Suppose we lived in world _A_, we shall have constructed our
+science and in particular our geometry; during this time the inhabitants
+of world _B_ will have constructed a science, and as their world is the
+image of ours, their geometry will also be the image of ours or, better,
+it will be the same. But if for us some day a window is opened upon
+world _B_, how we shall pity them: "Poor things," we shall say, "they
+think they have made a geometry, but what they call so is only a
+grotesque image of ours; their straights are all twisted, their circles
+are humped, their spheres have capricious inequalities." And we shall
+never suspect they say the same of us, and one never will know who is
+right.
+
+We see in how broad a sense should be understood the relativity of
+space; space is in reality amorphous and the things which are therein
+alone give it a form. What then should be thought of that direct
+intuition we should have of the straight or of distance? So little have
+we intuition of distance in itself that in the night, as we have said, a
+distance might become a thousand times greater without our being able to
+perceive it, if all other distances had undergone the same alteration.
+And even in a night the world _B_ might be substituted for the world _A_
+without our having any way of knowing it, and then the straight lines of
+yesterday would have ceased to be straight and we should never notice.
+
+One part of space is not by itself and in the absolute sense of the word
+equal to another part of space; because if so it is for us, it would not
+be for the dwellers in world _B_; and these have just as much right to
+reject our opinion as we to condemn theirs.
+
+I have elsewhere shown what are the consequences of these facts from the
+viewpoint of the idea we should form of non-Euclidean geometry and other
+analogous geometries; to that I do not care to return; and to-day I
+shall take a somewhat different point of view.
+
+
+II
+
+If this intuition of distance, of direction, of the straight line, if
+this direct intuition of space in a word does not exist, whence comes
+our belief that we have it? If this is only an illusion, why is this
+illusion so tenacious? It is proper to examine into this. We have said
+there is no direct intuition of size and we can only arrive at the
+relation of this magnitude to our instruments of measure. We should
+therefore not have been able to construct space if we had not had an
+instrument to measure it; well, this instrument to which we relate
+everything, which we use instinctively, it is our own body. It is in
+relation to our body that we place exterior objects, and the only
+spatial relations of these objects that we can represent are their
+relations to our body. It is our body which serves us, so to speak, as
+system of axes of coordinates.
+
+For example, at an instant [alpha], the presence of the object _A_ is
+revealed to me by the sense of sight; at another instant, [beta], the
+presence of another object, _B_, is revealed to me by another sense,
+that of hearing or of touch, for instance. I judge that this object _B_
+occupies the same place as the object _A_. What does that mean? First
+that does not signify that these two objects occupy, at two different
+moments, the same point of an absolute space, which even if it existed
+would escape our cognition, since, between the instants [alpha] and
+[beta], the solar system has moved and we can not know its displacement.
+That means these two objects occupy the same relative position with
+reference to our body.
+
+But even this, what does it mean? The impressions that have come to us
+from these objects have followed paths absolutely different, the optic
+nerve for the object _A_, the acoustic nerve for the object _B_.
+They have nothing in common from the qualitative point of view. The
+representations we are able to make of these two objects are absolutely
+heterogeneous, irreducible one to the other. Only I know that to reach
+the object _A_ I have just to extend the right arm in a certain way;
+even when I abstain from doing it, I represent to myself the muscular
+sensations and other analogous sensations which would accompany this
+extension, and this representation is associated with that of the
+object _A_.
+
+Now, I likewise know I can reach the object _B_ by extending my right
+arm in the same manner, an extension accompanied by the same train of
+muscular sensations. And when I say these two objects occupy the same
+place, I mean nothing more.
+
+I also know I could have reached the object _A_ by another appropriate
+motion of the left arm and I represent to myself the muscular sensations
+which would have accompanied this movement; and by this same motion of
+the left arm, accompanied by the same sensations, I likewise could have
+reached the object _B_.
+
+And that is very important, since thus I can defend myself against
+dangers menacing me from the object _A_ or the object _B_. With each of
+the blows we can be hit, nature has associated one or more parries which
+permit of our guarding ourselves. The same parry may respond to several
+strokes; and so it is, for instance, that the same motion of the right
+arm would have allowed us to guard at the instant [alpha] against the
+object _A_ and at the instant [beta] against the object _B_. Just so,
+the same stroke can be parried in several ways, and we have said, for
+instance, the object _A_ could be reached indifferently either by a
+certain movement of the right arm or by a certain movement of the left
+arm.
+
+All these parries have nothing in common except warding off the same
+blow, and this it is, and nothing else, which is meant when we say they
+are movements terminating at the same point of space. Just so, these
+objects, of which we say they occupy the same point of space, have
+nothing in common, except that the same parry guards against them.
+
+Or, if you choose, imagine innumerable telegraph wires, some
+centripetal, others centrifugal. The centripetal wires warn us of
+accidents happening without; the centrifugal wires carry the reparation.
+Connections are so established that when a centripetal wire is traversed
+by a current this acts on a relay and so starts a current in one of the
+centrifugal wires, and things are so arranged that several centripetal
+wires may act on the same centrifugal wire if the same remedy suits
+several ills, and that a centripetal wire may agitate different
+centrifugal wires, either simultaneously or in lieu one of the other
+when the same ill may be cured by several remedies.
+
+It is this complex system of associations, it is this table of
+distribution, so to speak, which is all our geometry or, if you wish,
+all in our geometry that is instinctive. What we call our intuition of
+the straight line or of distance is the consciousness we have of these
+associations and of their imperious character.
+
+And it is easy to understand whence comes this imperious character
+itself. An association will seem to us by so much the more
+indestructible as it is more ancient. But these associations are not,
+for the most part, conquests of the individual, since their trace is
+seen in the new-born babe: they are conquests of the race. Natural
+selection had to bring about these conquests by so much the more quickly
+as they were the more necessary.
+
+On this account, those of which we speak must have been of the earliest
+in date, since without them the defense of the organism would have been
+impossible. From the time when the cellules were no longer merely
+juxtaposed, but were called upon to give mutual aid, it was needful that
+a mechanism organize analogous to what we have described, so that this
+aid miss not its way, but forestall the peril.
+
+When a frog is decapitated, and a drop of acid is placed on a point of
+its skin, it seeks to wipe off the acid with the nearest foot, and, if
+this foot be amputated, it sweeps it off with the foot of the opposite
+side. There we have the double parry of which I have just spoken,
+allowing the combating of an ill by a second remedy, if the first fails.
+And it is this multiplicity of parries, and the resulting coordination,
+which is space.
+
+We see to what depths of the unconscious we must descend to find the
+first traces of these spatial associations, since only the inferior
+parts of the nervous system are involved. Why be astonished then at the
+resistance we oppose to every attempt made to dissociate what so long
+has been associated? Now, it is just this resistance that we call the
+evidence for the geometric truths; this evidence is nothing but the
+repugnance we feel toward breaking with very old habits which have
+always proved good.
+
+
+III
+
+The space so created is only a little space extending no farther than my
+arm can reach; the intervention of the memory is necessary to push back
+its limits. There are points which will remain out of my reach, whatever
+effort I make to stretch forth my hand; if I were fastened to the ground
+like a hydra polyp, for instance, which can only extend its tentacles,
+all these points would be outside of space, since the sensations we
+could experience from the action of bodies there situated, would be
+associated with the idea of no movement allowing us to reach them, of no
+appropriate parry. These sensations would not seem to us to have any
+spatial character and we should not seek to localize them.
+
+But we are not fixed to the ground like the lower animals; we can, if
+the enemy be too far away, advance toward him first and extend the hand
+when we are sufficiently near. This is still a parry, but a parry at
+long range. On the other hand, it is a complex parry, and into the
+representation we make of it enter the representation of the muscular
+sensations caused by the movements of the legs, that of the muscular
+sensations caused by the final movement of the arm, that of the
+sensations of the semicircular canals, etc. We must, besides, represent
+to ourselves, not a complex of simultaneous sensations, but a complex of
+successive sensations, following each other in a determinate order, and
+this is why I have just said the intervention of memory was necessary.
+Notice moreover that, to reach the same point, I may approach nearer the
+mark to be attained, so as to have to stretch my arm less. What more? It
+is not one, it is a thousand parries I can oppose to the same danger.
+All these parries are made of sensations which may have nothing in
+common and yet we regard them as defining the same point of space, since
+they may respond to the same danger and are all associated with the
+notion of this danger. It is the potentiality of warding off the same
+stroke which makes the unity of these different parries, as it is the
+possibility of being parried in the same way which makes the unity of
+the strokes so different in kind, which may menace us from the same
+point of space. It is this double unity which makes the individuality of
+each point of space, and, in the notion of point, there is nothing else.
+
+The space before considered, which might be called _restricted space_,
+was referred to coordinate axes bound to my body; these axes were fixed,
+since my body did not move and only my members were displaced. What are
+the axes to which we naturally refer the _extended space_? that is to
+say the new space just defined. We define a point by the sequence of
+movements to be made to reach it, starting from a certain initial
+position of the body. The axes are therefore fixed to this initial
+position of the body.
+
+But the position I call initial may be arbitrarily chosen among all the
+positions my body has successively occupied; if the memory more or less
+unconscious of these successive positions is necessary for the genesis
+of the notion of space, this memory may go back more or less far into
+the past. Thence results in the definition itself of space a certain
+indetermination, and it is precisely this indetermination which
+constitutes its relativity.
+
+There is no absolute space, there is only space relative to a certain
+initial position of the body. For a conscious being fixed to the ground
+like the lower animals, and consequently knowing only restricted space,
+space would still be relative (since it would have reference to his
+body), but this being would not be conscious of this relativity, because
+the axes of reference for this restricted space would be unchanging!
+Doubtless the rock to which this being would be fettered would not be
+motionless, since it would be carried along in the movement of our
+planet; for us consequently these axes would change at each instant; but
+for him they would be changeless. We have the faculty of referring our
+extended space now to the position _A_ of our body, considered as
+initial, again to the position _B_, which it had some moments afterward,
+and which we are free to regard in its turn as initial; we make
+therefore at each instant unconscious transformations of coordinates.
+This faculty would be lacking in our imaginary being, and from not
+having traveled, he would think space absolute. At every instant, his
+system of axes would be imposed upon him; this system would have to
+change greatly in reality, but for him it would be always the same,
+since it would be always the _only_ system. Quite otherwise is it with
+us, who at each instant have many systems between which we may choose at
+will, on condition of going back by memory more or less far into the
+past.
+
+This is not all; restricted space would not be homogeneous; the
+different points of this space could not be regarded as equivalent,
+since some could be reached only at the cost of the greatest efforts,
+while others could be easily attained. On the contrary, our extended
+space seems to us homogeneous, and we say all its points are equivalent.
+What does that mean?
+
+If we start from a certain place _A_, we can, from this position, make
+certain movements, _M_, characterized by a certain complex of muscular
+sensations. But, starting from another position, _B_, we make movements
+_M'_ characterized by the same muscular sensations. Let _a_, then, be
+the situation of a certain point of the body, the end of the index
+finger of the right hand for example, in the initial position _A_, and
+_b_ the situation of this same index when, starting from this position
+_A_, we have made the motions _M_. Afterwards, let _a'_ be the situation
+of this index in the position _B_, and _b'_ its situation when, starting
+from the position _B_, we have made the motions _M'_.
+
+Well, I am accustomed to say that the points of space _a_ and _b_ are
+related to each other just as the points _a'_ and _b'_, and this simply
+means that the two series of movements _M_ and _M'_ are accompanied by
+the same muscular sensations. And as I am conscious that, in passing
+from the position _A_ to the position _B_, my body has remained capable
+of the same movements, I know there is a point of space related to the
+point _a'_ just as any point _b_ is to the point _a_, so that the two
+points _a_ and _a'_ are equivalent. This is what is called the
+homogeneity of space. And, at the same time, this is why space is
+relative, since its properties remain the same whether it be referred to
+the axes _A_ or to the axes _B_. So that the relativity of space and its
+homogeneity are one sole and same thing.
+
+Now, if I wish to pass to the great space, which no longer serves only
+for me, but where I may lodge the universe, I get there by an act of
+imagination. I imagine how a giant would feel who could reach the
+planets in a few steps; or, if you choose, what I myself should feel in
+presence of a miniature world where these planets were replaced by
+little balls, while on one of these little balls moved a liliputian I
+should call myself. But this act of imagination would be impossible for
+me had I not previously constructed my restricted space and my extended
+space for my own use.
+
+
+IV
+
+Why now have all these spaces three dimensions? Go back to the "table of
+distribution" of which we have spoken. We have on the one side the list
+of the different possible dangers; designate them by _A1_, _A2_, etc.;
+and, on the other side, the list of the different remedies which I shall
+call in the same way _B1_, _B2_, etc. We have then connections between
+the contact studs or push buttons of the first list and those of the
+second, so that when, for instance, the announcer of danger _A3_
+functions, it will put or may put in action the relay corresponding to
+the parry _B4_.
+
+As I have spoken above of centripetal or centrifugal wires, I fear lest
+one see in all this, not a simple comparison, but a description of the
+nervous system. Such is not my thought, and that for several reasons:
+first I should not permit myself to put forth an opinion on the
+structure of the nervous system which I do not know, while those who
+have studied it speak only circumspectly; again because, despite my
+incompetence, I well know this scheme would be too simplistic; and
+finally because on my list of parries, some would figure very complex,
+which might even, in the case of extended space, as we have seen above,
+consist of many steps followed by a movement of the arm. It is not a
+question then of physical connection between two real conductors but of
+psychologic association between two series of sensations.
+
+If _A1_ and _A2_ for instance are both associated with the parry _B1_,
+and if _A1_ is likewise associated with the parry _B2_, it will
+generally happen that _A2_ and _B2_ will also themselves be associated.
+If this fundamental law were not generally true, there would exist only
+an immense confusion and there would be nothing resembling a conception
+of space or a geometry. How in fact have we defined a point of space. We
+have done it in two ways: it is on the one hand the aggregate of
+announcers _A_ in connection with the same parry _B_; it is on the other
+hand the aggregate of parries _B_ in connection with the same announcer
+_A_. If our law was not true, we should say _A1_ and _A2_ correspond to
+the same point since they are both in connection with _B1_; but we
+should likewise say they do not correspond to the same point, since _A1_
+would be in connection with _B2_ and the same would not be true of _A2_.
+This would be a contradiction.
+
+But, from another side, if the law were rigorously and always true,
+space would be very different from what it is. We should have categories
+strongly contrasted between which would be portioned out on the one hand
+the announcers _A_, on the other hand the parries _B_; these categories
+would be excessively numerous, but they would be entirely separated one
+from another. Space would be composed of points very numerous, but
+discrete; it would be _discontinuous_. There would be no reason for
+ranging these points in one order rather than another, nor consequently
+for attributing to space three dimensions.
+
+But it is not so; permit me to resume for a moment the language of those
+who already know geometry; this is quite proper since this is the
+language best understood by those I wish to make understand me.
+
+When I desire to parry the stroke, I seek to attain the point whence
+comes this blow, but it suffices that I approach quite near. Then the
+parry _B1_ may answer for _A1_ and for _A2_, if the point which
+corresponds to _B1_ is sufficiently near both to that corresponding to
+_A1_ and to that corresponding to _A2_. But it may happen that the point
+corresponding to another parry _B2_ may be sufficiently near to the
+point corresponding to A1 and not sufficiently near the point
+corresponding to _A2_; so that the parry _B2_ may answer for _A1_
+without answering for _A2_. For one who does not yet know geometry, this
+translates itself simply by a derogation of the law stated above. And
+then things will happen thus:
+
+Two parries _B1_ and _B2_ will be associated with the same warning _A1_
+and with a large number of warnings which we shall range in the same
+category as _A1_ and which we shall make correspond to the same point of
+space. But we may find warnings _A2_ which will be associated with _B2_
+without being associated with _B1_, and which in compensation will be
+associated with _B3_, which _B3_ was not associated with _A1_, and so
+forth, so that we may write the series
+
+ _B1_, _A1_, _B2_, _A2_, _B3_, _A3_, _B4_, _A4_,
+
+where each term is associated with the following and the preceding, but
+not with the terms several places away.
+
+Needless to add that each of the terms of these series is not isolated,
+but forms part of a very numerous category of other warnings or of other
+parries which have the same connections as it, and which may be regarded
+as belonging to the same point of space.
+
+The fundamental law, though admitting of exceptions, remains therefore
+almost always true. Only, in consequence of these exceptions, these
+categories, in place of being entirely separated, encroach partially one
+upon another and mutually penetrate in a certain measure, so that space
+becomes continuous.
+
+On the other hand, the order in which these categories are to be ranged
+is no longer arbitrary, and if we refer to the preceding series, we see
+it is necessary to put _B2_ between _A1_ and _A2_ and consequently
+between _B1_ and _B3_ and that we could not for instance put it between
+_B3_ and _B4_.
+
+There is therefore an order in which are naturally arranged our
+categories which correspond to the points of space, and experience
+teaches us that this order presents itself under the form of a table
+of triple entry, and this is why space has three dimensions.
+
+
+V
+
+So the characteristic property of space, that of having three
+dimensions, is only a property of our table of distribution, an internal
+property of the human intelligence, so to speak. It would suffice to
+destroy certain of these connections, that is to say of the associations
+of ideas to give a different table of distribution, and that might be
+enough for space to acquire a fourth dimension.
+
+Some persons will be astonished at such a result. The external world,
+they will think, should count for something. If the number of dimensions
+comes from the way we are made, there might be thinking beings living in
+our world, but who might be made differently from us and who would
+believe space has more or less than three dimensions. Has not M. de Cyon
+said that the Japanese mice, having only two pair of semicircular
+canals, believe that space is two-dimensional? And then this thinking
+being, if he is capable of constructing a physics, would he not make a
+physics of two or of four dimensions, and which in a sense would still
+be the same as ours, since it would be the description of the same world
+in another language?
+
+It seems in fact that it would be possible to translate our physics into
+the language of geometry of four dimensions; to attempt this translation
+would be to take great pains for little profit, and I shall confine
+myself to citing the mechanics of Hertz where we have something
+analogous. However, it seems that the translation would always be less
+simple than the text, and that it would always have the air of a
+translation, that the language of three dimensions seems the better
+fitted to the description of our world, although this description can be
+rigorously made in another idiom. Besides, our table of distribution was
+not made at random. There is connection between the warning _A1_ and the
+parry _B1_, this is an internal property of our intelligence; but why
+this connection? It is because the parry _B1_ affords means effectively
+to guard against the danger _A1_; and this is a fact exterior to us,
+this is a property of the exterior world. Our table of distribution is
+therefore only the translation of an aggregate of exterior facts; if it
+has three dimensions, this is because it has adapted itself to a world
+having certain properties; and the chief of these properties is that
+there exist natural solids whose displacements follow sensibly the laws
+we call laws of motion of rigid solids. If therefore the language of
+three dimensions is that which permits us most easily to describe our
+world, we should not be astonished; this language is copied from our
+table of distribution; and it is in order to be able to live in this
+world that this table has been established.
+
+I have said we could conceive, living in our world, thinking beings
+whose table of distribution would be four-dimensional and who
+consequently would think in hyperspace. It is not certain however that
+such beings, admitting they were born there, could live there and defend
+themselves against the thousand dangers by which they would there be
+assailed.
+
+
+VI
+
+A few remarks to end with. There is a striking contrast between the
+roughness of this primitive geometry, reducible to what I call a table
+of distribution, and the infinite precision of the geometers' geometry.
+And yet this is born of that; but not of that alone; it must be made
+fecund by the faculty we have of constructing mathematical concepts,
+such as that of group, for instance; it was needful to seek among the
+pure concepts that which best adapts itself to this rough space whose
+genesis I have sought to explain and which is common to us and the
+higher animals.
+
+The evidence for certain geometric postulates, we have said, is only our
+repugnance to renouncing very old habits. But these postulates are
+infinitely precise, while these habits have something about them
+essentially pliant. When we wish to think, we need postulates infinitely
+precise, since this is the only way to avoid contradiction; but among
+all the possible systems of postulates, there are some we dislike to
+choose because they are not sufficiently in accord with our habits;
+however pliant, however elastic they may be, these have a limit of
+elasticity.
+
+We see that if geometry is not an experimental science, it is a science
+born apropos of experience; that we have created the space it studies,
+but adapting it to the world wherein we live. We have selected the most
+convenient space, but experience has guided our choice; as this choice
+has been unconscious, we think it has been imposed upon us; some say
+experience imposes it, others that we are born with our space ready
+made; we see from the preceding considerations, what in these two
+opinions is the part of truth, what of error.
+
+In this progressive education whose outcome has been the construction of
+space, it is very difficult to determine what is the part of the
+individual, what the part of the race. How far could one of us,
+transported from birth to an entirely different world, where were
+dominant, for instance, bodies moving in conformity to the laws of
+motion of non-Euclidean solids, renounce the ancestral space to build a
+space completely new?
+
+The part of the race seems indeed preponderant; yet if to it we owe
+rough space, the soft space I have spoken of, the space of the higher
+animals, is it not to the unconscious experience of the individual we
+owe the infinitely precise space of the geometer? This is a question not
+easy to solve. Yet we cite a fact showing that the space our ancestors
+have bequeathed us still retains a certain plasticity. Some hunters
+learn to shoot fish under water, though the image of these fish be
+turned up by refraction. Besides they do it instinctively: they
+therefore have learned to modify their old instinct of direction; or, if
+you choose, to substitute for the association _A1_, _B1_, another
+association _A1_, _B2_, because experience showed them the first would
+not work.
+
+
+
+
+CHAPTER II
+
+MATHEMATICAL DEFINITIONS AND TEACHING
+
+
+1. I should speak here of general definitions in mathematics; at least
+that is the title, but it will be impossible to confine myself to the
+subject as strictly as the rule of unity of action would require; I
+shall not be able to treat it without touching upon a few other related
+questions, and if thus I am forced from time to time to walk on the
+bordering flower-beds on the right or left, I pray you bear with me.
+
+What is a good definition? For the philosopher or the scientist it is a
+definition which applies to all the objects defined, and only those; it
+is the one satisfying the rules of logic. But in teaching it is not
+that; a good definition is one understood by the scholars.
+
+How does it happen that so many refuse to understand mathematics? Is
+that not something of a paradox? Lo and behold! a science appealing only
+to the fundamental principles of logic, to the principle of
+contradiction, for instance, to that which is the skeleton, so to speak,
+of our intelligence, to that of which we can not divest ourselves
+without ceasing to think, and there are people who find it obscure! and
+they are even in the majority! That they are incapable of inventing may
+pass, but that they do not understand the demonstrations shown them,
+that they remain blind when we show them a light which seems to us
+flashing pure flame, this it is which is altogether prodigious.
+
+And yet there is no need of a wide experience with examinations to know
+that these blind men are in no wise exceptional beings. This is a
+problem not easy to solve, but which should engage the attention of all
+those wishing to devote themselves to teaching.
+
+What is it, to understand? Has this word the same meaning for all the
+world? To understand the demonstration of a theorem, is that to examine
+successively each of the syllogisms composing it and to ascertain its
+correctness, its conformity to the rules of the game? Likewise, to
+understand a definition, is this merely to recognize that one already
+knows the meaning of all the terms employed and to ascertain that it
+implies no contradiction?
+
+For some, yes; when they have done this, they will say: I understand.
+
+For the majority, no. Almost all are much more exacting; they wish to
+know not merely whether all the syllogisms of a demonstration are
+correct, but why they link together in this order rather than another.
+In so far as to them they seem engendered by caprice and not by an
+intelligence always conscious of the end to be attained, they do not
+believe they understand.
+
+Doubtless they are not themselves just conscious of what they crave and
+they could not formulate their desire, but if they do not get
+satisfaction, they vaguely feel that something is lacking. Then what
+happens? In the beginning they still perceive the proofs one puts under
+their eyes; but as these are connected only by too slender a thread to
+those which precede and those which follow, they pass without leaving
+any trace in their head; they are soon forgotten; a moment bright, they
+quickly vanish in night eternal. When they are farther on, they will no
+longer see even this ephemeral light, since the theorems lean one upon
+another and those they would need are forgotten; thus it is they become
+incapable of understanding mathematics.
+
+This is not always the fault of their teacher; often their mind, which
+needs to perceive the guiding thread, is too lazy to seek and find it.
+But to come to their aid, we first must know just what hinders them.
+
+Others will always ask of what use is it; they will not have understood
+if they do not find about them, in practise or in nature, the
+justification of such and such a mathematical concept. Under each word
+they wish to put a sensible image; the definition must evoke this image,
+so that at each stage of the demonstration they may see it transform and
+evolve. Only upon this condition do they comprehend and retain. Often
+these deceive themselves; they do not listen to the reasoning, they look
+at the figures; they think they have understood and they have only seen.
+
+2. How many different tendencies! Must we combat them? Must we use them?
+And if we wish to combat them, which should be favored? Must we show
+those content with the pure logic that they have seen only one side of
+the matter? Or need we say to those not so cheaply satisfied that what
+they demand is not necessary?
+
+In other words, should we constrain the young people to change the
+nature of their minds? Such an attempt would be vain; we do not possess
+the philosopher's stone which would enable us to transmute one into
+another the metals confided to us; all we can do is to work with them,
+adapting ourselves to their properties.
+
+Many children are incapable of becoming mathematicians, to whom however
+it is necessary to teach mathematics; and the mathematicians themselves
+are not all cast in the same mold. To read their works suffices to
+distinguish among them two sorts of minds, the logicians like
+Weierstrass for example, the intuitives like Riemann. There is the same
+difference among our students. The one sort prefer to treat their
+problems 'by analysis' as they say, the others 'by geometry.'
+
+It is useless to seek to change anything of that, and besides would it
+be desirable? It is well that there are logicians and that there are
+intuitives; who would dare say whether he preferred that Weierstrass had
+never written or that there never had been a Riemann? We must therefore
+resign ourselves to the diversity of minds, or better we must rejoice in
+it.
+
+3. Since the word understand has many meanings, the definitions which
+will be best understood by some will not be best suited to others. We
+have those which seek to produce an image, and those where we confine
+ourselves to combining empty forms, perfectly intelligible, but purely
+intelligible, which abstraction has deprived of all matter.
+
+I know not whether it be necessary to cite examples. Let us cite them,
+anyhow, and first the definition of fractions will furnish us an extreme
+case. In the primary schools, to define a fraction, one cuts up an apple
+or a pie; it is cut up mentally of course and not in reality, because I
+do not suppose the budget of the primary instruction allows of such
+prodigality. At the Normal School, on the other hand, or at the college,
+it is said: a fraction is the combination of two whole numbers separated
+by a horizontal bar; we define by conventions the operations to which
+these symbols may be submitted; it is proved that the rules of these
+operations are the same as in calculating with whole numbers, and we
+ascertain finally that multiplying the fraction, according to these
+rules, by the denominator gives the numerator. This is all very well
+because we are addressing young people long familiarized with the notion
+of fractions through having cut up apples or other objects, and whose
+mind, matured by a hard mathematical education, has come little by
+little to desire a purely logical definition. But the débutant to whom
+one should try to give it, how dumfounded!
+
+Such also are the definitions found in a book justly admired and greatly
+honored, the _Foundations of Geometry_ by Hilbert. See in fact how he
+begins: _We think three systems of_ THINGS _which we shall call points,
+straights and planes_. What are these 'things'?
+
+We know not, nor need we know; it would even be a pity to seek to know;
+all we have the right to know of them is what the assumptions tell us;
+this for example: _Two distinct points always determine a straight_,
+which is followed by this remark: _in place of determine, we may say the
+two points are on the straight, or the straight goes through these two
+points or joins the two points_.
+
+Thus 'to be on a straight' is simply defined as synonymous with
+'determine a straight.' Behold a book of which I think much good, but
+which I should not recommend to a school boy. Yet I could do so without
+fear, he would not read much of it. I have taken extreme examples and no
+teacher would dream of going that far. But even stopping short of such
+models, does he not already expose himself to the same danger?
+
+Suppose we are in a class; the professor dictates: the circle is the
+locus of points of the plane equidistant from an interior point called
+the center. The good scholar writes this phrase in his note-book; the
+bad scholar draws faces; but neither understands; then the professor
+takes the chalk and draws a circle on the board. "Ah!" think the
+scholars, "why did he not say at once: a circle is a ring, we should
+have understood." Doubtless the professor is right. The scholars'
+definition would have been of no avail, since it could serve for no
+demonstration, since besides it would not give them the salutary habit
+of analyzing their conceptions. But one should show them that they do
+not comprehend what they think they know, lead them to be conscious of
+the roughness of their primitive conception, and of themselves to wish
+it purified and made precise.
+
+4. I shall return to these examples; I only wished to show you the two
+opposed conceptions; they are in violent contrast. This contrast the
+history of science explains. If we read a book written fifty years ago,
+most of the reasoning we find there seems lacking in rigor. Then it was
+assumed a continuous function can change sign only by vanishing; to-day
+we prove it. It was assumed the ordinary rules of calculation are
+applicable to incommensurable numbers; to-day we prove it. Many other
+things were assumed which sometimes were false.
+
+We trusted to intuition; but intuition can not give rigor, nor even
+certainty; we see this more and more. It tells us for instance that
+every curve has a tangent, that is to say that every continuous function
+has a derivative, and that is false. And as we sought certainty, we had
+to make less and less the part of intuition.
+
+What has made necessary this evolution? We have not been slow to
+perceive that rigor could not be established in the reasonings, if it
+were not first put into the definitions.
+
+The objects occupying mathematicians were long ill defined; we thought
+we knew them because we represented them with the senses or the
+imagination; but we had of them only a rough image and not a precise
+concept upon which reasoning could take hold. It is there that the
+logicians would have done well to direct their efforts.
+
+So for the incommensurable number, the vague idea of continuity, which
+we owe to intuition, has resolved itself into a complicated system of
+inequalities bearing on whole numbers. Thus have finally vanished all
+those difficulties which frightened our fathers when they reflected upon
+the foundations of the infinitesimal calculus. To-day only whole numbers
+are left in analysis, or systems finite or infinite of whole numbers,
+bound by a plexus of equalities and inequalities. Mathematics we say is
+arithmetized.
+
+5. But do you think mathematics has attained absolute rigor without
+making any sacrifice? Not at all; what it has gained in rigor it has
+lost in objectivity. It is by separating itself from reality that it has
+acquired this perfect purity. We may freely run over its whole domain,
+formerly bristling with obstacles, but these obstacles have not
+disappeared. They have only been moved to the frontier, and it would be
+necessary to vanquish them anew if we wished to break over this frontier
+to enter the realm of the practical.
+
+We had a vague notion, formed of incongruous elements, some _a priori_,
+others coming from experiences more or less digested; we thought we
+knew, by intuition, its principal properties. To-day we reject the
+empiric elements, retaining only the _a priori_; one of the properties
+serves as definition and all the others are deduced from it by rigorous
+reasoning. This is all very well, but it remains to be proved that this
+property, which has become a definition, pertains to the real objects
+which experience had made known to us and whence we drew our vague
+intuitive notion. To prove that, it would be necessary to appeal to
+experience, or to make an effort of intuition, and if we could not prove
+it, our theorems would be perfectly rigorous, but perfectly useless.
+
+Logic sometimes makes monsters. Since half a century we have seen arise
+a crowd of bizarre functions which seem to try to resemble as little as
+possible the honest functions which serve some purpose. No longer
+continuity, or perhaps continuity, but no derivatives, etc. Nay more,
+from the logical point of view, it is these strange functions which are
+the most general, those one meets without seeking no longer appear
+except as particular case. There remains for them only a very small
+corner.
+
+Heretofore when a new function was invented, it was for some practical
+end; to-day they are invented expressly to put at fault the reasonings
+of our fathers, and one never will get from them anything more than
+that.
+
+If logic were the sole guide of the teacher, it would be necessary to
+begin with the most general functions, that is to say with the most
+bizarre. It is the beginner that would have to be set grappling with
+this teratologic museum. If you do not do it, the logicians might say,
+you will achieve rigor only by stages.
+
+6. Yes, perhaps, but we can not make so cheap of reality, and I mean not
+only the reality of the sensible world, which however has its worth,
+since it is to combat against it that nine tenths of your students ask
+of you weapons. There is a reality more subtile, which makes the very
+life of the mathematical beings, and which is quite other than logic.
+
+Our body is formed of cells, and the cells of atoms; are these cells and
+these atoms then all the reality of the human body? The way these cells
+are arranged, whence results the unity of the individual, is it not also
+a reality and much more interesting?
+
+A naturalist who never had studied the elephant except in the
+microscope, would he think he knew the animal adequately? It is the same
+in mathematics. When the logician shall have broken up each
+demonstration into a multitude of elementary operations, all correct, he
+still will not possess the whole reality; this I know not what which
+makes the unity of the demonstration will completely escape him.
+
+In the edifices built up by our masters, of what use to admire the work
+of the mason if we can not comprehend the plan of the architect? Now
+pure logic can not give us this appreciation of the total effect; this
+we must ask of intuition.
+
+Take for instance the idea of continuous function. This is at first only
+a sensible image, a mark traced by the chalk on the blackboard. Little
+by little it is refined; we use it to construct a complicated system of
+inequalities, which reproduces all the features of the primitive image;
+when all is done, we have _removed the centering_, as after the
+construction of an arch; this rough representation, support thenceforth
+useless, has disappeared and there remains only the edifice itself,
+irreproachable in the eyes of the logician. And yet, if the professor
+did not recall the primitive image, if he did not restore momentarily
+the _centering_, how could the student divine by what caprice all these
+inequalities have been scaffolded in this fashion one upon another? The
+definition would be logically correct, but it would not show him the
+veritable reality.
+
+7. So back we must return; doubtless it is hard for a master to teach
+what does not entirely satisfy him; but the satisfaction of the master
+is not the unique object of teaching; we should first give attention to
+what the mind of the pupil is and to what we wish it to become.
+
+Zoologists maintain that the embryonic development of an animal
+recapitulates in brief the whole history of its ancestors throughout
+geologic time. It seems it is the same in the development of minds. The
+teacher should make the child go over the path his fathers trod; more
+rapidly, but without skipping stations. For this reason, the history of
+science should be our first guide.
+
+Our fathers thought they knew what a fraction was, or continuity, or the
+area of a curved surface; we have found they did not know it. Just so
+our scholars think they know it when they begin the serious study of
+mathematics. If without warning I tell them: "No, you do not know it;
+what you think you understand, you do not understand; I must prove to
+you what seems to you evident," and if in the demonstration I support
+myself upon premises which to them seem less evident than the
+conclusion, what shall the unfortunates think? They will think that the
+science of mathematics is only an arbitrary mass of useless subtilities;
+either they will be disgusted with it, or they will play it as a game
+and will reach a state of mind like that of the Greek sophists.
+
+Later, on the contrary, when the mind of the scholar, familiarized with
+mathematical reasoning, has been matured by this long frequentation, the
+doubts will arise of themselves and then your demonstration will be
+welcome. It will awaken new doubts, and the questions will arise
+successively to the child, as they arose successively to our fathers,
+until perfect rigor alone can satisfy him. To doubt everything does not
+suffice, one must know why he doubts.
+
+8. The principal aim of mathematical teaching is to develop certain
+faculties of the mind, and among them intuition is not the least
+precious. It is through it that the mathematical world remains in
+contact with the real world, and if pure mathematics could do without
+it, it would always be necessary to have recourse to it to fill up the
+chasm which separates the symbol from reality. The practician will
+always have need of it, and for one pure geometer there should be a
+hundred practicians.
+
+
+The engineer should receive a complete mathematical education, but for
+what should it serve him?
+
+To see the different aspects of things and see them quickly; he has no
+time to hunt mice. It is necessary that, in the complex physical objects
+presented to him, he should promptly recognize the point where the
+mathematical tools we have put in his hands can take hold. How could he
+do it if we should leave between instruments and objects the deep chasm
+hollowed out by the logicians?
+
+9. Besides the engineers, other scholars, less numerous, are in their
+turn to become teachers; they therefore must go to the very bottom; a
+knowledge deep and rigorous of the first principles is for them before
+all indispensable. But this is no reason not to cultivate in them
+intuition; for they would get a false idea of the science if they never
+looked at it except from a single side, and besides they could not
+develop in their students a quality they did not themselves possess.
+
+For the pure geometer himself, this faculty is necessary; it is by logic
+one demonstrates, by intuition one invents. To know how to criticize is
+good, to know how to create is better. You know how to recognize if a
+combination is correct; what a predicament if you have not the art of
+choosing among all the possible combinations. Logic tells us that on
+such and such a way we are sure not to meet any obstacle; it does not
+say which way leads to the end. For that it is necessary to see the end
+from afar, and the faculty which teaches us to see is intuition. Without
+it the geometer would be like a writer who should be versed in grammar
+but had no ideas. Now how could this faculty develop if, as soon as it
+showed itself, we chase it away and proscribe it, if we learn to set it
+at naught before knowing the good of it.
+
+And here permit a parenthesis to insist upon the importance of written
+exercises. Written compositions are perhaps not sufficiently emphasized
+in certain examinations, at the polytechnic school, for instance. I am
+told they would close the door against very good scholars who have
+mastered the course, thoroughly understanding it, and who nevertheless
+are incapable of making the slightest application. I have just said the
+word understand has several meanings: such students only understand in
+the first way, and we have seen that suffices neither to make an
+engineer nor a geometer. Well, since choice must be made, I prefer those
+who understand completely.
+
+10. But is the art of sound reasoning not also a precious thing, which
+the professor of mathematics ought before all to cultivate? I take good
+care not to forget that. It should occupy our attention and from the
+very beginning. I should be distressed to see geometry degenerate into I
+know not what tachymetry of low grade and I by no means subscribe to the
+extreme doctrines of certain German Oberlehrer. But there are occasions
+enough to exercise the scholars in correct reasoning in the parts of
+mathematics where the inconveniences I have pointed out do not present
+themselves. There are long chains of theorems where absolute logic has
+reigned from the very first and, so to speak, quite naturally, where the
+first geometers have given us models we should constantly imitate and
+admire.
+
+It is in the exposition of first principles that it is necessary to
+avoid too much subtility; there it would be most discouraging and
+moreover useless. We can not prove everything and we can not define
+everything; and it will always be necessary to borrow from intuition;
+what does it matter whether it be done a little sooner or a little
+later, provided that in using correctly premises it has furnished us, we
+learn to reason soundly.
+
+11. Is it possible to fulfill so many opposing conditions? Is this
+possible in particular when it is a question of giving a definition? How
+find a concise statement satisfying at once the uncompromising rules of
+logic, our desire to grasp the place of the new notion in the totality
+of the science, our need of thinking with images? Usually it will not be
+found, and this is why it is not enough to state a definition; it must
+be prepared for and justified.
+
+What does that mean? You know it has often been said: every definition
+implies an assumption, since it affirms the existence of the object
+defined. The definition then will not be justified, from the purely
+logical point of view, until one shall have _proved_ that it involves no
+contradiction, neither in the terms, nor with the verities previously
+admitted.
+
+But this is not enough; the definition is stated to us as a convention;
+but most minds will revolt if we wish to impose it upon them as an
+_arbitrary_ convention. They will be satisfied only when you have
+answered numerous questions.
+
+Usually mathematical definitions, as M. Liard has shown, are veritable
+constructions built up wholly of more simple notions. But why assemble
+these elements in this way when a thousand other combinations were
+possible?
+
+Is it by caprice? If not, why had this combination more right to exist
+than all the others? To what need does it respond? How was it foreseen
+that it would play an important rôle in the development of the science,
+that it would abridge our reasonings and our calculations? Is there in
+nature some familiar object which is so to speak the rough and vague
+image of it?
+
+This is not all; if you answer all these questions in a satisfactory
+manner, we shall see indeed that the new-born had the right to be
+baptized; but neither is the choice of a name arbitrary; it is needful
+to explain by what analogies one has been guided and that if analogous
+names have been given to different things, these things at least differ
+only in material and are allied in form; that their properties are
+analogous and so to say parallel.
+
+At this cost we may satisfy all inclinations. If the statement is
+correct enough to please the logician, the justification will satisfy
+the intuitive. But there is still a better procedure; wherever possible,
+the justification should precede the statement and prepare for it; one
+should be led on to the general statement by the study of some
+particular examples.
+
+Still another thing: each of the parts of the statement of a definition
+has as aim to distinguish the thing to be defined from a class of other
+neighboring objects. The definition will be understood only when you
+have shown, not merely the object defined, but the neighboring objects
+from which it is proper to distinguish it, when you have given a grasp
+of the difference and when you have added explicitly: this is why in
+stating the definition I have said this or that.
+
+But it is time to leave generalities and examine how the somewhat
+abstract principles I have expounded may be applied in arithmetic,
+geometry, analysis and mechanics.
+
+
+ARITHMETIC
+
+12. The whole number is not to be defined; in return, one ordinarily
+defines the operations upon whole numbers; I believe the scholars learn
+these definitions by heart and attach no meaning to them. For that there
+are two reasons: first they are made to learn them too soon, when their
+mind as yet feels no need of them; then these definitions are not
+satisfactory from the logical point of view. A good definition for
+addition is not to be found just simply because we must stop and can not
+define everything. It is not defining addition to say it consists in
+adding. All that can be done is to start from a certain number of
+concrete examples and say: the operation we have performed is called
+addition.
+
+For subtraction it is quite otherwise; it may be logically defined as
+the operation inverse to addition; but should we begin in that way? Here
+also start with examples, show on these examples the reciprocity of the
+two operations; thus the definition will be prepared for and justified.
+
+Just so again for multiplication; take a particular problem; show that
+it may be solved by adding several equal numbers; then show that we
+reach the result more quickly by a multiplication, an operation the
+scholars already know how to do by routine and out of that the logical
+definition will issue naturally.
+
+Division is defined as the operation inverse to multiplication; but
+begin by an example taken from the familiar notion of partition and show
+on this example that multiplication reproduces the dividend.
+
+There still remain the operations on fractions. The only difficulty is
+for multiplication. It is best to expound first the theory of
+proportion; from it alone can come a logical definition; but to make
+acceptable the definitions met at the beginning of this theory, it is
+necessary to prepare for them by numerous examples taken from classic
+problems of the rule of three, taking pains to introduce fractional
+data.
+
+Neither should we fear to familiarize the scholars with the notion of
+proportion by geometric images, either by appealing to what they
+remember if they have already studied geometry, or in having recourse to
+direct intuition, if they have not studied it, which besides will
+prepare them to study it. Finally I shall add that after defining
+multiplication of fractions, it is needful to justify this definition by
+showing that it is commutative, associative and distributive, and
+calling to the attention of the auditors that this is established to
+justify the definition.
+
+One sees what a rôle geometric images play in all this; and this rôle is
+justified by the philosophy and the history of the science. If
+arithmetic had remained free from all admixture of geometry, it would
+have known only the whole number; it is to adapt itself to the needs of
+geometry that it invented anything else.
+
+
+GEOMETRY
+
+In geometry we meet forthwith the notion of the straight line. Can the
+straight line be defined? The well-known definition, the shortest path
+from one point to another, scarcely satisfies me. I should start simply
+with the _ruler_ and show at first to the scholar how one may verify a
+ruler by turning; this verification is the true definition of the
+straight line; the straight line is an axis of rotation. Next he should
+be shown how to verify the ruler by sliding and he would have one of the
+most important properties of the straight line.
+
+As to this other property of being the shortest path from one point to
+another, it is a theorem which can be demonstrated apodictically, but
+the demonstration is too delicate to find a place in secondary teaching.
+It will be worth more to show that a ruler previously verified fits on a
+stretched thread. In presence of difficulties like these one need not
+dread to multiply assumptions, justifying them by rough experiments.
+
+It is needful to grant these assumptions, and if one admits a few more
+of them than is strictly necessary, the evil is not very great; the
+essential thing is to learn to reason soundly on the assumptions
+admitted. Uncle Sarcey, who loved to repeat, often said that at the
+theater the spectator accepts willingly all the postulates imposed upon
+him at the beginning, but the curtain once raised, he becomes
+uncompromising on the logic. Well, it is just the same in mathematics.
+
+For the circle, we may start with the compasses; the scholars will
+recognize at the first glance the curve traced; then make them observe
+that the distance of the two points of the instrument remains constant,
+that one of these points is fixed and the other movable, and so we shall
+be led naturally to the logical definition.
+
+The definition of the plane implies an axiom and this need not be
+hidden. Take a drawing board and show that a moving ruler may be kept
+constantly in complete contact with this plane and yet retain three
+degrees of freedom. Compare with the cylinder and the cone, surfaces on
+which an applied straight retains only two degrees of freedom; next take
+three drawing boards; show first that they will glide while remaining
+applied to one another and this with three degrees of freedom; and
+finally to distinguish the plane from the sphere, show that two of these
+boards which fit a third will fit each other.
+
+Perhaps you are surprised at this incessant employment of moving things;
+this is not a rough artifice; it is much more philosophic than one would
+at first think. What is geometry for the philosopher? It is the study of
+a group. And what group? That of the motions of solid bodies. How define
+this group then without moving some solids?
+
+Should we retain the classic definition of parallels and say parallels
+are two coplanar straights which do not meet, however far they be
+prolonged? No, since this definition is negative, since it is
+unverifiable by experiment, and consequently can not be regarded as an
+immediate datum of intuition. No, above all because it is wholly strange
+to the notion of group, to the consideration of the motion of solid
+bodies which is, as I have said, the true source of geometry. Would it
+not be better to define first the rectilinear translation of an
+invariable figure, as a motion wherein all the points of this figure
+have rectilinear trajectories; to show that such a translation is
+possible by making a square glide on a ruler?
+
+From this experimental ascertainment, set up as an assumption, it would
+be easy to derive the notion of parallel and Euclid's postulate itself.
+
+
+
+MECHANICS
+
+I need not return to the definition of velocity, or acceleration, or
+other kinematic notions; they may be advantageously connected with that
+of the derivative.
+
+I shall insist, on the other hand, upon the dynamic notions of force and
+mass.
+
+I am struck by one thing: how very far the young people who have
+received a high-school education are from applying to the real world the
+mechanical laws they have been taught. It is not only that they are
+incapable of it; they do not even think of it. For them the world of
+science and the world of reality are separated by an impervious
+partition wall.
+
+If we try to analyze the state of mind of our scholars, this will
+astonish us less. What is for them the real definition of force? Not
+that which they recite, but that which, crouching in a nook of their
+mind, from there directs it wholly. Here is the definition: forces are
+arrows with which one makes parallelograms. These arrows are imaginary
+things which have nothing to do with anything existing in nature. This
+would not happen if they had been shown forces in reality before
+representing them by arrows.
+
+How shall we define force?
+
+I think I have elsewhere sufficiently shown there is no good logical
+definition. There is the anthropomorphic definition, the sensation of
+muscular effort; this is really too rough and nothing useful can be
+drawn from it.
+
+Here is how we should go: first, to make known the genus force, we must
+show one after the other all the species of this genus; they are very
+numerous and very different; there is the pressure of fluids on the
+insides of the vases wherein they are contained; the tension of threads;
+the elasticity of a spring; the gravity working on all the molecules of
+a body; friction; the normal mutual action and reaction of two solids in
+contact.
+
+This is only a qualitative definition; it is necessary to learn to
+measure force. For that begin by showing that one force may be replaced
+by another without destroying equilibrium; we may find the first example
+of this substitution in the balance and Borda's double weighing.
+
+Then show that a weight may be replaced, not only by another weight,
+but by force of a different nature; for instance, Prony's brake permits
+replacing weight by friction.
+
+From all this arises the notion of the equivalence of two forces.
+
+The direction of a force must be defined. If a force _F_ is equivalent
+to another force _F'_ applied to the body considered by means of a
+stretched string, so that _F_ may be replaced by _F'_ without affecting
+the equilibrium, then the point of attachment of the string will be by
+definition the point of application of the force _F'_, and that of the
+equivalent force _F_; the direction of the string will be the direction
+of the force _F'_ and that of the equivalent force _F_.
+
+From that, pass to the comparison of the magnitude of forces. If a force
+can replace two others with the same direction, it equals their sum;
+show for example that a weight of 20 grams may replace two 10-gram
+weights.
+
+Is this enough? Not yet. We now know how to compare the intensity of two
+forces which have the same direction and same point of application; we
+must learn to do it when the directions are different. For that, imagine
+a string stretched by a weight and passing over a pulley; we shall say
+that the tensor of the two legs of the string is the same and equal to
+the tension weight.
+
+This definition of ours enables us to compare the tensions of the two
+pieces of our string, and, using the preceding definitions, to compare
+any two forces having the same direction as these two pieces. It should
+be justified by showing that the tension of the last piece of the string
+remains the same for the same tensor weight, whatever be the number and
+the disposition of the reflecting pulleys. It has still to be completed
+by showing this is only true if the pulleys are frictionless.
+
+Once master of these definitions, it is to be shown that the point of
+application, the direction and the intensity suffice to determine a
+force; that two forces for which these three elements are the same are
+_always_ equivalent and may _always_ be replaced by one another, whether
+in equilibrium or in movement, and this whatever be the other forces
+acting.
+
+It must be shown that two concurrent forces may always be replaced by a
+unique resultant; and that _this resultant remains the same_, whether
+the body be at rest or in motion and whatever be the other forces
+applied to it.
+
+Finally it must be shown that forces thus defined satisfy the principle
+of the equality of action and reaction.
+
+Experiment it is, and experiment alone, which can teach us all that. It
+will suffice to cite certain common experiments, which the scholars make
+daily without suspecting it, and to perform before them a few
+experiments, simple and well chosen.
+
+It is after having passed through all these meanders that one may
+represent forces by arrows, and I should even wish that in the
+development of the reasonings return were made from time to time from
+the symbol to the reality. For instance it would not be difficult to
+illustrate the parallelogram of forces by aid of an apparatus formed of
+three strings, passing over pulleys, stretched by weights and in
+equilibrium while pulling on the same point.
+
+Knowing force, it is easy to define mass; this time the definition
+should be borrowed from dynamics; there is no way of doing otherwise,
+since the end to be attained is to give understanding of the distinction
+between mass and weight. Here again, the definition should be led up to
+by experiments; there is in fact a machine which seems made expressly to
+show what mass is, Atwood's machine; recall also the laws of the fall of
+bodies, that the acceleration of gravity is the same for heavy as for
+light bodies, and that it varies with the latitude, etc.
+
+Now, if you tell me that all the methods I extol have long been applied
+in the schools, I shall rejoice over it more than be surprised at it. I
+know that on the whole our mathematical teaching is good. I do not wish
+it overturned; that would even distress me. I only desire betterments
+slowly progressive. This teaching should not be subjected to brusque
+oscillations under the capricious blast of ephemeral fads. In such
+tempests its high educative value would soon founder. A good and sound
+logic should continue to be its basis. The definition by example is
+always necessary, but it should prepare the way for the logical
+definition, it should not replace it; it should at least make this
+wished for, in the cases where the true logical definition can be
+advantageously given only in advanced teaching.
+
+Understand that what I have here said does not imply giving up what I
+have written elsewhere. I have often had occasion to criticize certain
+definitions I extol to-day. These criticisms hold good completely. These
+definitions can only be provisory. But it is by way of them that we must
+pass.
+
+
+
+
+CHAPTER III
+
+MATHEMATICS AND LOGIC
+
+
+INTRODUCTION
+
+Can mathematics be reduced to logic without having to appeal to
+principles peculiar to mathematics? There is a whole school, abounding
+in ardor and full of faith, striving to prove it. They have their own
+special language, which is without words, using only signs. This
+language is understood only by the initiates, so that commoners are
+disposed to bow to the trenchant affirmations of the adepts. It is
+perhaps not unprofitable to examine these affirmations somewhat closely,
+to see if they justify the peremptory tone with which they are
+presented.
+
+But to make clear the nature of the question it is necessary to enter
+upon certain historical details and in particular to recall the
+character of the works of Cantor.
+
+Since long ago the notion of infinity had been introduced into
+mathematics; but this infinite was what philosophers call a _becoming_.
+The mathematical infinite was only a quantity capable of increasing
+beyond all limit: it was a variable quantity of which it could not be
+said that it _had passed_ all limits, but only that it _could pass_
+them.
+
+Cantor has undertaken to introduce into mathematics an _actual
+infinite_, that is to say a quantity which not only is capable of
+passing all limits, but which is regarded as having already passed them.
+He has set himself questions like these: Are there more points in space
+than whole numbers? Are there more points in space than points in a
+plane? etc.
+
+And then the number of whole numbers, that of the points of space, etc.,
+constitutes what he calls a _transfinite cardinal number_, that is to
+say a cardinal number greater than all the ordinary cardinal numbers.
+And he has occupied himself in comparing these transfinite cardinal
+numbers. In arranging in a proper order the elements of an aggregate
+containing an infinity of them, he has also imagined what he calls
+transfinite ordinal numbers upon which I shall not dwell.
+
+Many mathematicians followed his lead and set a series of questions of
+the sort. They so familiarized themselves with transfinite numbers that
+they have come to make the theory of finite numbers depend upon that of
+Cantor's cardinal numbers. In their eyes, to teach arithmetic in a way
+truly logical, one should begin by establishing the general properties
+of transfinite cardinal numbers, then distinguish among them a very
+small class, that of the ordinary whole numbers. Thanks to this détour,
+one might succeed in proving all the propositions relative to this
+little class (that is to say all our arithmetic and our algebra) without
+using any principle foreign to logic. This method is evidently contrary
+to all sane psychology; it is certainly not in this way that the human
+mind proceeded in constructing mathematics; so its authors do not dream,
+I think, of introducing it into secondary teaching. But is it at least
+logic, or, better, is it correct? It may be doubted.
+
+The geometers who have employed it are however very numerous. They have
+accumulated formulas and they have thought to free themselves from what
+was not pure logic by writing memoirs where the formulas no longer
+alternate with explanatory discourse as in the books of ordinary
+mathematics, but where this discourse has completely disappeared.
+
+Unfortunately they have reached contradictory results, what are called
+the _cantorian antinomies_, to which we shall have occasion to return.
+These contradictions have not discouraged them and they have tried to
+modify their rules so as to make those disappear which had already shown
+themselves, without being sure, for all that, that new ones would not
+manifest themselves.
+
+It is time to administer justice on these exaggerations. I do not hope
+to convince them; for they have lived too long in this atmosphere.
+Besides, when one of their demonstrations has been refuted, we are sure
+to see it resurrected with insignificant alterations, and some of them
+have already risen several times from their ashes. Such long ago was the
+Lernæan hydra with its famous heads which always grew again. Hercules
+got through, since his hydra had only nine heads, or eleven; but here
+there are too many, some in England, some in Germany, in Italy, in
+France, and he would have to give up the struggle. So I appeal only to
+men of good judgment unprejudiced.
+
+
+I
+
+In these latter years numerous works have been published on pure
+mathematics and the philosophy of mathematics, trying to separate and
+isolate the logical elements of mathematical reasoning. These works have
+been analyzed and expounded very clearly by M. Couturat in a book
+entitled: _The Principles of Mathematics_.
+
+For M. Couturat, the new works, and in particular those of Russell and
+Peano, have finally settled the controversy, so long pending between
+Leibnitz and Kant. They have shown that there are no synthetic judgments
+a priori (Kant's phrase to designate judgments which can neither be
+demonstrated analytically, nor reduced to identities, nor established
+experimentally), they have shown that mathematics is entirely reducible
+to logic and that intuition here plays no rôle.
+
+This is what M. Couturat has set forth in the work just cited; this he
+says still more explicitly in his Kant jubilee discourse, so that I
+heard my neighbor whisper: "I well see this is the centenary of Kant's
+_death_."
+
+Can we subscribe to this conclusive condemnation? I think not, and I
+shall try to show why.
+
+
+II
+
+What strikes us first in the new mathematics is its purely formal
+character: "We think," says Hilbert, "three sorts of _things_, which we
+shall call points, straights and planes. We convene that a straight
+shall be determined by two points, and that in place of saying this
+straight is determined by these two points, we may say it passes through
+these two points, or that these two points are situated on this
+straight." What these _things_ are, not only we do not know, but we
+should not seek to know. We have no need to, and one who never had seen
+either point or straight or plane could geometrize as well as we. That
+the phrase _to pass through_, or the phrase _to be situated upon_ may
+arouse in us no image, the first is simply a synonym of to _be
+determined_ and the second of _to determine_.
+
+Thus, be it understood, to demonstrate a theorem, it is neither
+necessary nor even advantageous to know what it means. The geometer
+might be replaced by the _logic piano_ imagined by Stanley Jevons; or,
+if you choose, a machine might be imagined where the assumptions were
+put in at one end, while the theorems came out at the other, like the
+legendary Chicago machine where the pigs go in alive and come out
+transformed into hams and sausages. No more than these machines need the
+mathematician know what he does.
+
+I do not make this formal character of his geometry a reproach to
+Hilbert. This is the way he should go, given the problem he set himself.
+He wished to reduce to a minimum the number of the fundamental
+assumptions of geometry and completely enumerate them; now, in
+reasonings where our mind remains active, in those where intuition still
+plays a part, in living reasonings, so to speak, it is difficult not to
+introduce an assumption or a postulate which passes unperceived. It is
+therefore only after having carried back all the geometric reasonings to
+a form purely mechanical that he could be sure of having accomplished
+his design and finished his work.
+
+What Hilbert did for geometry, others have tried to do for arithmetic
+and analysis. Even if they had entirely succeeded, would the Kantians be
+finally condemned to silence? Perhaps not, for in reducing mathematical
+thought to an empty form, it is certainly mutilated.
+
+Even admitting it were established that all the theorems could be
+deduced by procedures purely analytic, by simple logical combinations of
+a finite number of assumptions, and that these assumptions are only
+conventions; the philosopher would still have the right to investigate
+the origins of these conventions, to see why they have been judged
+preferable to the contrary conventions.
+
+And then the logical correctness of the reasonings leading from the
+assumptions to the theorems is not the only thing which should occupy
+us. The rules of perfect logic, are they the whole of mathematics? As
+well say the whole art of playing chess reduces to the rules of the
+moves of the pieces. Among all the constructs which can be built up of
+the materials furnished by logic, choice must be made; the true geometer
+makes this choice judiciously because he is guided by a sure instinct,
+or by some vague consciousness of I know not what more profound and more
+hidden geometry, which alone gives value to the edifice constructed.
+
+To seek the origin of this instinct, to study the laws of this deep
+geometry, felt, not stated, would also be a fine employment for the
+philosophers who do not want logic to be all. But it is not at this
+point of view I wish to put myself, it is not thus I wish to consider
+the question. The instinct mentioned is necessary for the inventor, but
+it would seem at first we might do without it in studying the science
+once created. Well, what I wish to investigate is if it be true that,
+the principles of logic once admitted, one can, I do not say discover,
+but demonstrate, all the mathematical verities without making a new
+appeal to intuition.
+
+
+III
+
+I once said no to this question:[12] should our reply be modified by the
+recent works? My saying no was because "the principle of complete
+induction" seemed to me at once necessary to the mathematician and
+irreducible to logic. The statement of this principle is: "If a property
+be true of the number 1, and if we establish that it is true of _n_ + 1
+provided it be of _n_, it will be true of all the whole numbers."
+Therein I see the mathematical reasoning par excellence. I did not mean
+to say, as has been supposed, that all mathematical reasonings can be
+reduced to an application of this principle. Examining these reasonings
+closely, we there should see applied many other analogous principles,
+presenting the same essential characteristics. In this category of
+principles, that of complete induction is only the simplest of all and
+this is why I have chosen it as type.
+
+ [12] See _Science and Hypothesis_, chapter I.
+
+The current name, principle of complete induction, is not justified.
+This mode of reasoning is none the less a true mathematical induction
+which differs from ordinary induction only by its certitude.
+
+
+IV
+
+DEFINITIONS AND ASSUMPTIONS
+
+The existence of such principles is a difficulty for the uncompromising
+logicians; how do they pretend to get out of it? The principle of
+complete induction, they say, is not an assumption properly so called or
+a synthetic judgment _a priori_; it is just simply the definition of
+whole number. It is therefore a simple convention. To discuss this way
+of looking at it, we must examine a little closely the relations between
+definitions and assumptions.
+
+Let us go back first to an article by M. Couturat on mathematical
+definitions which appeared in _l'Enseignement mathématique_, a magazine
+published by Gauthier-Villars and by Georg at Geneva. We shall see there
+a distinction between the _direct definition and the definition by
+postulates_.
+
+"The definition by postulates," says M. Couturat, "applies not to a
+single notion, but to a system of notions; it consists in enumerating
+the fundamental relations which unite them and which enable us to
+demonstrate all their other properties; these relations are postulates."
+
+If previously have been defined all these notions but one, then this
+last will be by definition the thing which verifies these postulates.
+Thus certain indemonstrable assumptions of mathematics would be only
+disguised definitions. This point of view is often legitimate; and I
+have myself admitted it in regard for instance to Euclid's postulate.
+
+The other assumptions of geometry do not suffice to completely define
+distance; the distance then will be, by definition, among all the
+magnitudes which satisfy these other assumptions, that which is such as
+to make Euclid's postulate true.
+
+Well the logicians suppose true for the principle of complete induction
+what I admit for Euclid's postulate; they want to see in it only a
+disguised definition.
+
+But to give them this right, two conditions must be fulfilled. Stuart
+Mill says every definition implies an assumption, that by which the
+existence of the defined object is affirmed. According to that, it
+would no longer be the assumption which might be a disguised definition,
+it would on the contrary be the definition which would be a disguised
+assumption. Stuart Mill meant the word existence in a material and
+empirical sense; he meant to say that in defining the circle we affirm
+there are round things in nature.
+
+Under this form, his opinion is inadmissible. Mathematics is independent
+of the existence of material objects; in mathematics the word exist can
+have only one meaning, it means free from contradiction. Thus rectified,
+Stuart Mill's thought becomes exact; in defining a thing, we affirm that
+the definition implies no contradiction.
+
+If therefore we have a system of postulates, and if we can demonstrate
+that these postulates imply no contradiction, we shall have the right to
+consider them as representing the definition of one of the notions
+entering therein. If we can not demonstrate that, it must be admitted
+without proof, and that then will be an assumption; so that, seeking the
+definition under the postulate, we should find the assumption under the
+definition.
+
+Usually, to show that a definition implies no contradiction, we proceed
+by _example_, we try to make an example of a thing satisfying the
+definition. Take the case of a definition by postulates; we wish to
+define a notion _A_, and we say that, by definition, an _A_ is anything
+for which certain postulates are true. If we can prove directly that all
+these postulates are true of a certain object _B_, the definition will
+be justified; the object _B_ will be an _example_ of an _A_. We shall be
+certain that the postulates are not contradictory, since there are cases
+where they are all true at the same time.
+
+But such a direct demonstration by example is not always possible.
+
+To establish that the postulates imply no contradiction, it is then
+necessary to consider all the propositions deducible from these
+postulates considered as premises, and to show that, among these
+propositions, no two are contradictory. If these propositions are finite
+in number, a direct verification is possible. This case is infrequent
+and uninteresting. If these propositions are infinite in number, this
+direct verification can no longer be made; recourse must be had to
+procedures where in general it is necessary to invoke just this
+principle of complete induction which is precisely the thing to be
+proved.
+
+This is an explanation of one of the conditions the logicians should
+satisfy, _and further on we shall see they have not done it_.
+
+
+V
+
+There is a second. When we give a definition, it is to use it.
+
+We therefore shall find in the sequel of the exposition the word
+defined; have we the right to affirm, of the thing represented by this
+word, the postulate which has served for definition? Yes, evidently, if
+the word has retained its meaning, if we do not attribute to it
+implicitly a different meaning. Now this is what sometimes happens and
+it is usually difficult to perceive it; it is needful to see how this
+word comes into our discourse, and if the gate by which it has entered
+does not imply in reality a definition other than that stated.
+
+This difficulty presents itself in all the applications of mathematics.
+The mathematical notion has been given a definition very refined and
+very rigorous; and for the pure mathematician all doubt has disappeared;
+but if one wishes to apply it to the physical sciences for instance, it
+is no longer a question of this pure notion, but of a concrete object
+which is often only a rough image of it. To say that this object
+satisfies, at least approximately, the definition, is to state a new
+truth, which experience alone can put beyond doubt, and which no longer
+has the character of a conventional postulate.
+
+But without going beyond pure mathematics, we also meet the same
+difficulty.
+
+You give a subtile definition of numbers; then, once this definition
+given, you think no more of it; because, in reality, it is not it which
+has taught you what number is; you long ago knew that, and when the word
+number further on is found under your pen, you give it the same sense as
+the first comer. To know what is this meaning and whether it is the same
+in this phrase or that, it is needful to see how you have been led to
+speak of number and to introduce this word into these two phrases. I
+shall not for the moment dilate upon this point, because we shall have
+occasion to return to it.
+
+Thus consider a word of which we have given explicitly a definition _A_;
+afterwards in the discourse we make a use of it which implicitly
+supposes another definition _B_. It is possible that these two
+definitions designate the same thing. But that this is so is a new truth
+which must either be demonstrated or admitted as an independent
+assumption.
+
+_We shall see farther on that the logicians have not fulfilled the
+second condition any better than the first._
+
+
+VI
+
+The definitions of number are very numerous and very different; I forego
+the enumeration even of the names of their authors. We should not be
+astonished that there are so many. If one among them was satisfactory,
+no new one would be given. If each new philosopher occupying himself
+with this question has thought he must invent another one, this was
+because he was not satisfied with those of his predecessors, and he was
+not satisfied with them because he thought he saw a petitio principii.
+
+I have always felt, in reading the writings devoted to this problem, a
+profound feeling of discomfort; I was always expecting to run against a
+petitio principii, and when I did not immediately perceive it, I feared
+I had overlooked it.
+
+This is because it is impossible to give a definition without using a
+sentence, and difficult to make a sentence without using a number word,
+or at least the word several, or at least a word in the plural. And then
+the declivity is slippery and at each instant there is risk of a fall
+into petitio principii.
+
+I shall devote my attention in what follows only to those of these
+definitions where the petitio principii is most ably concealed.
+
+
+VII
+
+PASIGRAPHY
+
+The symbolic language created by Peano plays a very grand rôle in these
+new researches. It is capable of rendering some service, but I think M.
+Couturat attaches to it an exaggerated importance which must astonish
+Peano himself.
+
+The essential element of this language is certain algebraic signs which
+represent the different conjunctions: if, and, or, therefore. That these
+signs may be convenient is possible; but that they are destined to
+revolutionize all philosophy is a different matter. It is difficult to
+admit that the word _if_ acquires, when written C, a virtue it had not
+when written if. This invention of Peano was first called _pasigraphy_,
+that is to say the art of writing a treatise on mathematics without
+using a single word of ordinary language. This name defined its range
+very exactly. Later, it was raised to a more eminent dignity by
+conferring on it the title of _logistic_. This word is, it appears,
+employed at the Military Academy, to designate the art of the
+quartermaster of cavalry, the art of marching and cantoning troops; but
+here no confusion need be feared, and it is at once seen that this new
+name implies the design of revolutionizing logic.
+
+We may see the new method at work in a mathematical memoir by
+Burali-Forti, entitled: _Una Questione sui numeri transfiniti_, inserted
+in Volume XI of the _Rendiconti del circolo matematico di Palermo_.
+
+I begin by saying this memoir is very interesting, and my taking it here
+as example is precisely because it is the most important of all those
+written in the new language. Besides, the uninitiated may read it,
+thanks to an Italian interlinear translation.
+
+What constitutes the importance of this memoir is that it has given the
+first example of those antinomies met in the study of transfinite
+numbers and making since some years the despair of mathematicians. The
+aim, says Burali-Forti, of this note is to show there may be two
+transfinite numbers (ordinals), _a_ and _b_, such that _a_ is neither
+equal to, greater than, nor less than _b_.
+
+To reassure the reader, to comprehend the considerations which follow,
+he has no need of knowing what a transfinite ordinal number is.
+
+Now, Cantor had precisely proved that between two transfinite numbers as
+between two finite, there can be no other relation than equality or
+inequality in one sense or the other. But it is not of the substance of
+this memoir that I wish to speak here; that would carry me much too far
+from my subject; I only wish to consider the form, and just to ask if
+this form makes it gain much in rigor and whether it thus compensates
+for the efforts it imposes upon the writer and the reader.
+
+First we see Burali-Forti define the number 1 as follows:
+
+ 1 = [iota]T'{Ko[(n_](u, h)[epsilon](u[epsilon]Un)},
+
+a definition eminently fitted to give an idea of the number 1 to persons
+who had never heard speak of it.
+
+I understand Peanian too ill to dare risk a critique, but still I fear
+this definition contains a petitio principii, considering that I see the
+figure 1 in the first member and Un in letters in the second.
+
+However that may be, Burali-Forti starts from this definition and, after
+a short calculation, reaches the equation:
+
+ (27) 1[epsilon]No,
+
+which tells us that One is a number.
+
+And since we are on these definitions of the first numbers, we recall
+that M. Couturat has also defined 0 and 1.
+
+What is zero? It is the number of elements of the null class. And what
+is the null class? It is that containing no element.
+
+To define zero by null, and null by no, is really to abuse the wealth of
+language; so M. Couturat has introduced an improvement in his
+definition, by writing:
+
+ 0 = [iota][Lambda]:[phi]x = [Lambda]·[inverted c]·[Lambda]
+ = (x[epsilon][phi]x),
+
+which means: zero is the number of things satisfying a condition never
+satisfied.
+
+But as never means _in no case_ I do not see that the progress is great.
+
+I hasten to add that the definition M. Couturat gives of the number 1 is
+more satisfactory.
+
+One, says he in substance, is the number of elements in a class in which
+any two elements are identical.
+
+It is more satisfactory, I have said, in this sense that to define 1, he
+does not use the word one; in compensation, he uses the word two. But I
+fear, if asked what is two, M. Couturat would have to use the word one.
+
+
+
+VIII
+
+But to return to the memoir of Burali-Forti; I have said his conclusions
+are in direct opposition to those of Cantor. Now, one day M. Hadamard
+came to see me and the talk fell upon this antinomy.
+
+"Burali-Forti's reasoning," I said, "does it not seem to you
+irreproachable?" "No, and on the contrary I find nothing to object to in
+that of Cantor. Besides, Burali-Forti had no right to speak of the
+aggregate of _all_ the ordinal numbers."
+
+"Pardon, he had the right, since he could always put
+
+ [Omega] = T'(No,[epsilon]>).
+
+I should like to know who was to prevent him, and can it be said a thing
+does not exist, when we have called it [Omega]?"
+
+It was in vain, I could not convince him (which besides would have been
+sad, since he was right). Was it merely because I do not speak the
+Peanian with enough eloquence? Perhaps; but between ourselves I do not
+think so.
+
+Thus, despite all this pasigraphic apparatus, the question was not
+solved. What does that prove? In so far as it is a question only of
+proving one a number, pasigraphy suffices, but if a difficulty presents
+itself, if there is an antinomy to solve, pasigraphy becomes impotent.
+
+
+
+
+CHAPTER IV
+
+THE NEW LOGICS
+
+
+I
+
+_The Russell Logic_
+
+To justify its pretensions, logic had to change. We have seen new logics
+arise of which the most interesting is that of Russell. It seems he has
+nothing new to write about formal logic, as if Aristotle there had
+touched bottom. But the domain Russell attributes to logic is infinitely
+more extended than that of the classic logic, and he has put forth on
+the subject views which are original and at times well warranted.
+
+First, Russell subordinates the logic of classes to that of
+propositions, while the logic of Aristotle was above all the logic of
+classes and took as its point of departure the relation of subject to
+predicate. The classic syllogism, "Socrates is a man," etc., gives place
+to the hypothetical syllogism: "If _A_ is true, _B_ is true; now if _B_
+is true, _C_ is true," etc. And this is, I think, a most happy idea,
+because the classic syllogism is easy to carry back to the hypothetical
+syllogism, while the inverse transformation is not without difficulty.
+
+And then this is not all. Russell's logic of propositions is the study
+of the laws of combination of the conjunctions _if_, _and_, _or_, and
+the negation _not_.
+
+In adding here two other conjunctions, _and_ and _or_, Russell opens to
+logic a new field. The symbols _and_, _or_ follow the same laws as the
+two signs × and +, that is to say the commutative associative and
+distributive laws. Thus _and_ represents logical multiplication, while
+_or_ represents logical addition. This also is very interesting.
+
+Russell reaches the conclusion that any false proposition implies all
+other propositions true or false. M. Couturat says this conclusion will
+at first seem paradoxical. It is sufficient however to have corrected a
+bad thesis in mathematics to recognize how right Russell is. The
+candidate often is at great pains to get the first false equation; but
+that once obtained, it is only sport then for him to accumulate the most
+surprising results, some of which even may be true.
+
+
+II
+
+We see how much richer the new logic is than the classic logic; the
+symbols are multiplied and allow of varied combinations _which are no
+longer limited in number_. Has one the right to give this extension to
+the meaning of the word _logic_? It would be useless to examine this
+question and to seek with Russell a mere quarrel about words. Grant him
+what he demands; but be not astonished if certain verities declared
+irreducible to logic in the old sense of the word find themselves now
+reducible to logic in the new sense--something very different.
+
+A great number of new notions have been introduced, and these are not
+simply combinations of the old. Russell knows this, and not only at the
+beginning of the first chapter, 'The Logic of Propositions,' but at the
+beginning of the second and third, 'The Logic of Classes' and 'The Logic
+of Relations,' he introduces new words that he declares indefinable.
+
+And this is not all; he likewise introduces principles he declares
+indemonstrable. But these indemonstrable principles are appeals to
+intuition, synthetic judgments _a priori_. We regard them as intuitive
+when we meet them more or less explicitly enunciated in mathematical
+treatises; have they changed character because the meaning of the word
+logic has been enlarged and we now find them in a book entitled
+_Treatise on Logic_? _They have not changed nature; they have only
+changed place._
+
+
+III
+
+Could these principles be considered as disguised definitions? It would
+then be necessary to have some way of proving that they imply no
+contradiction. It would be necessary to establish that, however far one
+followed the series of deductions, he would never be exposed to
+contradicting himself.
+
+We might attempt to reason as follows: We can verify that the
+operations of the new logic applied to premises exempt from
+contradiction can only give consequences equally exempt from
+contradiction. If therefore after _n_ operations we have not met
+contradiction, we shall not encounter it after _n_ + 1. Thus it is
+impossible that there should be a moment when contradiction _begins_,
+which shows we shall never meet it. Have we the right to reason in this
+way? No, for this would be to make use of complete induction; and
+_remember, we do not yet know the principle of complete induction_.
+
+We therefore have not the right to regard these assumptions as disguised
+definitions and only one resource remains for us, to admit a new act of
+intuition for each of them. Moreover I believe this is indeed the
+thought of Russell and M. Couturat.
+
+Thus each of the nine indefinable notions and of the twenty
+indemonstrable propositions (I believe if it were I that did the
+counting, I should have found some more) which are the foundation of the
+new logic, logic in the broad sense, presupposes a new and independent
+act of our intuition and (why not say it?) a veritable synthetic
+judgment _a priori_. On this point all seem agreed, but what Russell
+claims, and _what seems to me doubtful, is that after these appeals to
+intuition, that will be the end of it; we need make no others and can
+build all mathematics without the intervention of any new element_.
+
+
+IV
+
+M. Couturat often repeats that this new logic is altogether independent
+of the idea of number. I shall not amuse myself by counting how many
+numeral adjectives his exposition contains, both cardinal and ordinal,
+or indefinite adjectives such as several. We may cite, however, some
+examples:
+
+"The logical product of _two_ or _more_ propositions is....";
+
+"All propositions are capable only of _two_ values, true and false";
+
+"The relative product of _two_ relations is a relation";
+
+"A relation exists between two terms," etc., etc.
+
+Sometimes this inconvenience would not be unavoidable, but sometimes
+also it is essential. A relation is incomprehensible without two terms;
+it is impossible to have the intuition of the relation, without having
+at the same time that of its two terms, and without noticing they are
+two, because, if the relation is to be conceivable, it is necessary that
+there be two and only two.
+
+
+V
+
+_Arithmetic_
+
+I reach what M. Couturat calls the _ordinal theory_ which is the
+foundation of arithmetic properly so called. M. Couturat begins by
+stating Peano's five assumptions, which are independent, as has been
+proved by Peano and Padoa.
+
+1. Zero is an integer.
+
+2. Zero is not the successor of any integer.
+
+3. The successor of an integer is an integer. To this it would be proper
+to add,
+
+Every integer has a successor.
+
+4. Two integers are equal if their successors are.
+
+The fifth assumption is the principle of complete induction.
+
+M. Couturat considers these assumptions as disguised definitions; they
+constitute the definition by postulates of zero, of successor, and of
+integer.
+
+But we have seen that for a definition by postulates to be acceptable we
+must be able to prove that it implies no contradiction.
+
+Is this the case here? Not at all.
+
+The demonstration can not be made _by example_. We can not take a part
+of the integers, for instance the first three, and prove they satisfy
+the definition.
+
+If I take the series 0, 1, 2, I see it fulfils the assumptions 1, 2, 4
+and 5; but to satisfy assumption 3 it still is necessary that 3 be an
+integer, and consequently that the series 0, 1, 2, 3, fulfil the
+assumptions; we might prove that it satisfies assumptions 1, 2, 4, 5,
+but assumption 3 requires besides that 4 be an integer and that the
+series 0, 1, 2, 3, 4 fulfil the assumptions, and so on.
+
+It is therefore impossible to demonstrate the assumptions for certain
+integers without proving them for all; we must give up proof by
+example.
+
+It is necessary then to take all the consequences of our assumptions and
+see if they contain no contradiction.
+
+If these consequences were finite in number, this would be easy; but
+they are infinite in number; they are the whole of mathematics, or at
+least all arithmetic.
+
+What then is to be done? Perhaps strictly we could repeat the reasoning
+of number III.
+
+But as we have said, this reasoning is complete induction, and it is
+precisely the principle of complete induction whose justification would
+be the point in question.
+
+
+VI
+
+_The Logic of Hilbert_
+
+I come now to the capital work of Hilbert which he communicated to the
+Congress of Mathematicians at Heidelberg, and of which a French
+translation by M. Pierre Boutroux appeared in _l'Enseignement
+mathématique_, while an English translation due to Halsted appeared in
+_The Monist_.[13] In this work, which contains profound thoughts, the
+author's aim is analogous to that of Russell, but on many points he
+diverges from his predecessor.
+
+ [13] 'The Foundations of Logic and Arithmetic,' _Monist_, XV.,
+ 338-352.
+
+"But," he says (_Monist_, p. 340), "on attentive consideration we become
+aware that in the usual exposition of the laws of logic certain
+fundamental concepts of arithmetic are already employed; for example,
+the concept of the aggregate, in part also the concept of number.
+
+"We fall thus into a vicious circle and therefore to avoid paradoxes a
+partly simultaneous development of the laws of logic and arithmetic is
+requisite."
+
+We have seen above that what Hilbert says of the principles of logic _in
+the usual exposition_ applies likewise to the logic of Russell. So for
+Russell logic is prior to arithmetic; for Hilbert they are
+'simultaneous.' We shall find further on other differences still
+greater, but we shall point them out as we come to them. I prefer to
+follow step by step the development of Hilbert's thought, quoting
+textually the most important passages.
+
+"Let us take as the basis of our consideration first of all a
+thought-thing 1 (one)" (p. 341). Notice that in so doing we in no wise
+imply the notion of number, because it is understood that 1 is here only
+a symbol and that we do not at all seek to know its meaning. "The taking
+of this thing together with itself respectively two, three or more
+times...." Ah! this time it is no longer the same; if we introduce the
+words 'two,' 'three,' and above all 'more,' 'several,' we introduce the
+notion of number; and then the definition of finite whole number which
+we shall presently find, will come too late. Our author was too
+circumspect not to perceive this begging of the question. So at the end
+of his work he tries to proceed to a truly patching-up process.
+
+Hilbert then introduces two simple objects 1 and =, and considers all
+the combinations of these two objects, all the combinations of their
+combinations, etc. It goes without saying that we must forget the
+ordinary meaning of these two signs and not attribute any to them.
+
+Afterwards he separates these combinations into two classes, the class
+of the existent and the class of the non-existent, and till further
+orders this separation is entirely arbitrary. Every affirmative
+statement tells us that a certain combination belongs to the class of
+the existent; every negative statement tells us that a certain
+combination belongs to the class of the non-existent.
+
+
+VII
+
+Note now a difference of the highest importance. For Russell any object
+whatsoever, which he designates by _x_, is an object absolutely
+undetermined and about which he supposes nothing; for Hilbert it is one
+of the combinations formed with the symbols 1 and =; he could not
+conceive of the introduction of anything other than combinations of
+objects already defined. Moreover Hilbert formulates his thought in the
+neatest way, and I think I must reproduce _in extenso_ his statement (p.
+348):
+
+"In the assumptions the arbitraries (as equivalent for the concept
+'every' and 'all' in the customary logic) represent only those
+thought-things and their combinations with one another, which at this
+stage are laid down as fundamental or are to be newly defined.
+Therefore in the deduction of inferences from the assumptions, the
+arbitraries, which occur in the assumptions, can be replaced only by
+such thought-things and their combinations.
+
+"Also we must duly remember, that through the super-addition and making
+fundamental of a new thought-thing the preceding assumptions undergo an
+enlargement of their validity, and where necessary, are to be subjected
+to a change in conformity with the sense."
+
+The contrast with Russell's view-point is complete. For this philosopher
+we may substitute for _x_ not only objects already known, but anything.
+
+Russell is faithful to his point of view, which is that of
+comprehension. He starts from the general idea of being, and enriches it
+more and more while restricting it, by adding new qualities. Hilbert on
+the contrary recognizes as possible beings only combinations of objects
+already known; so that (looking at only one side of his thought) we
+might say he takes the view-point of extension.
+
+
+VIII
+
+Let us continue with the exposition of Hilbert's ideas. He introduces
+two assumptions which he states in his symbolic language but which
+signify, in the language of the uninitiated, that every quality is equal
+to itself and that every operation performed upon two identical
+quantities gives identical results.
+
+So stated, they are evident, but thus to present them would be to
+misrepresent Hilbert's thought. For him mathematics has to combine only
+pure symbols, and a true mathematician should reason upon them without
+preconceptions as to their meaning. So his assumptions are not for him
+what they are for the common people.
+
+He considers them as representing the definition by postulates of the
+symbol (=) heretofore void of all signification. But to justify this
+definition we must show that these two assumptions lead to no
+contradiction. For this Hilbert used the reasoning of our number III,
+without appearing to perceive that he is using complete induction.
+
+
+IX
+
+The end of Hilbert's memoir is altogether enigmatic and I shall not lay
+stress upon it. Contradictions accumulate; we feel that the author is
+dimly conscious of the _petitio principii_ he has committed, and that he
+seeks vainly to patch up the holes in his argument.
+
+What does this mean? At the point of proving that the definition of the
+whole number by the assumption of complete induction implies no
+contradiction, Hilbert withdraws as Russell and Couturat withdrew,
+because the difficulty is too great.
+
+
+X
+
+_Geometry_
+
+Geometry, says M. Couturat, is a vast body of doctrine wherein the
+principle of complete induction does not enter. That is true in a
+certain measure; we can not say it is entirely absent, but it enters
+very slightly. If we refer to the _Rational Geometry_ of Dr. Halsted
+(New York, John Wiley and Sons, 1904) built up in accordance with the
+principles of Hilbert, we see the principle of induction enter for the
+first time on page 114 (unless I have made an oversight, which is quite
+possible).[14]
+
+ [14] Second ed., 1907, p. 86; French ed., 1911, p. 97. G. B. H.
+
+So geometry, which only a few years ago seemed the domain where the
+reign of intuition was uncontested, is to-day the realm where the
+logicians seem to triumph. Nothing could better measure the importance
+of the geometric works of Hilbert and the profound impress they have
+left on our conceptions.
+
+But be not deceived. What is after all the fundamental theorem of
+geometry? It is that the assumptions of geometry imply no contradiction,
+and this we can not prove without the principle of induction.
+
+How does Hilbert demonstrate this essential point? By leaning upon
+analysis and through it upon arithmetic and through it upon the
+principle of induction.
+
+And if ever one invents another demonstration, it will still be
+necessary to lean upon this principle, since the possible consequences
+of the assumptions, of which it is necessary to show that they are not
+contradictory, are infinite in number.
+
+
+XI
+
+_Conclusion_
+
+Our conclusion straightway is that the principle of induction can not be
+regarded as the disguised definition of the entire world.
+
+Here are three truths: (1) The principle of complete induction; (2)
+Euclid's postulate; (3) the physical law according to which phosphorus
+melts at 44° (cited by M. Le Roy).
+
+These are said to be three disguised definitions: the first, that of the
+whole number; the second, that of the straight line; the third, that of
+phosphorus.
+
+I grant it for the second; I do not admit it for the other two. I must
+explain the reason for this apparent inconsistency.
+
+First, we have seen that a definition is acceptable only on condition
+that it implies no contradiction. We have shown likewise that for the
+first definition this demonstration is impossible; on the other hand, we
+have just recalled that for the second Hilbert has given a complete
+proof.
+
+As to the third, evidently it implies no contradiction. Does this mean
+that the definition guarantees, as it should, the existence of the
+object defined? We are here no longer in the mathematical sciences, but
+in the physical, and the word existence has no longer the same meaning.
+It no longer signifies absence of contradiction; it means objective
+existence.
+
+You already see a first reason for the distinction I made between the
+three cases; there is a second. In the applications we have to make of
+these three concepts, do they present themselves to us as defined by
+these three postulates?
+
+The possible applications of the principle of induction are innumerable;
+take, for example, one of those we have expounded above, and where it is
+sought to prove that an aggregate of assumptions can lead to no
+contradiction. For this we consider one of the series of syllogisms we
+may go on with in starting from these assumptions as premises. When we
+have finished the _n_th syllogism, we see we can make still another and
+this is the _n_ + 1th. Thus the number _n_ serves to count a series of
+successive operations; it is a number obtainable by successive
+additions. This therefore is a number from which we may go back to
+unity by _successive subtractions_. Evidently we could not do this if we
+had _n_ = _n_ - 1, since then by subtraction we should always obtain
+again the same number. So the way we have been led to consider this
+number _n_ implies a definition of the finite whole number and this
+definition is the following: A finite whole number is that which can be
+obtained by successive additions; it is such that _n_ is not equal to
+_n_ - 1.
+
+That granted, what do we do? We show that if there has been no
+contradiction up to the _n_th syllogism, no more will there be up to the
+_n_ + 1th, and we conclude there never will be. You say: I have the
+right to draw this conclusion, since the whole numbers are by definition
+those for which a like reasoning is legitimate. But that implies another
+definition of the whole number, which is as follows: A whole number is
+that on which we may reason by recurrence. In the particular case it is
+that of which we may say that, if the absence of contradiction up to the
+time of a syllogism of which the number is an integer carries with it
+the absence of contradiction up to the time of the syllogism whose
+number is the following integer, we need fear no contradiction for any
+of the syllogisms whose number is an integer.
+
+The two definitions are not identical; they are doubtless equivalent,
+but only in virtue of a synthetic judgment _a priori_; we can not pass
+from one to the other by a purely logical procedure. Consequently we
+have no right to adopt the second, after having introduced the whole
+number by a way that presupposes the first.
+
+On the other hand, what happens with regard to the straight line? I have
+already explained this so often that I hesitate to repeat it again, and
+shall confine myself to a brief recapitulation of my thought. We have
+not, as in the preceding case, two equivalent definitions logically
+irreducible one to the other. We have only one expressible in words.
+Will it be said there is another which we feel without being able to
+word it, since we have the intuition of the straight line or since we
+represent to ourselves the straight line? First of all, we can not
+represent it to ourselves in geometric space, but only in representative
+space, and then we can represent to ourselves just as well the objects
+which possess the other properties of the straight line, save that of
+satisfying Euclid's postulate. These objects are 'the non-Euclidean
+straights,' which from a certain point of view are not entities void of
+sense, but circles (true circles of true space) orthogonal to a certain
+sphere. If, among these objects equally capable of representation, it is
+the first (the Euclidean straights) which we call straights, and not the
+latter (the non-Euclidean straights), this is properly by definition.
+
+And arriving finally at the third example, the definition of phosphorus,
+we see the true definition would be: Phosphorus is the bit of matter I
+see in yonder flask.
+
+
+XII
+
+And since I am on this subject, still another word. Of the phosphorus
+example I said: "This proposition is a real verifiable physical law,
+because it means that all bodies having all the other properties of
+phosphorus, save its point of fusion, melt like it at 44°." And it was
+answered: "No, this law is not verifiable, because if it were shown that
+two bodies resembling phosphorus melt one at 44° and the other at 50°,
+it might always be said that doubtless, besides the point of fusion,
+there is some other unknown property by which they differ."
+
+That was not quite what I meant to say. I should have written, "All
+bodies possessing such and such properties finite in number (to wit, the
+properties of phosphorus stated in the books on chemistry, the
+fusion-point excepted) melt at 44°."
+
+And the better to make evident the difference between the case of the
+straight and that of phosphorus, one more remark. The straight has in
+nature many images more or less imperfect, of which the chief are the
+light rays and the rotation axis of the solid. Suppose we find the ray
+of light does not satisfy Euclid's postulate (for example by showing
+that a star has a negative parallax), what shall we do? Shall we
+conclude that the straight being by definition the trajectory of light
+does not satisfy the postulate, or, on the other hand, that the straight
+by definition satisfying the postulate, the ray of light is not
+straight?
+
+Assuredly we are free to adopt the one or the other definition and
+consequently the one or the other conclusion; but to adopt the first
+would be stupid, because the ray of light probably satisfies only
+imperfectly not merely Euclid's postulate, but the other properties of
+the straight line, so that if it deviates from the Euclidean straight,
+it deviates no less from the rotation axis of solids which is another
+imperfect image of the straight line; while finally it is doubtless
+subject to change, so that such a line which yesterday was straight will
+cease to be straight to-morrow if some physical circumstance has
+changed.
+
+Suppose now we find that phosphorus does not melt at 44°, but at 43.9°.
+Shall we conclude that phosphorus being by definition that which melts
+at 44°, this body that we did call phosphorus is not true phosphorus,
+or, on the other hand, that phosphorous melts at 43.9°? Here again we
+are free to adopt the one or the other definition and consequently the
+one or the other conclusion; but to adopt the first would be stupid
+because we can not be changing the name of a substance every time we
+determine a new decimal of its fusion-point.
+
+
+XIII
+
+To sum up, Russell and Hilbert have each made a vigorous effort; they
+have each written a work full of original views, profound and often well
+warranted. These two works give us much to think about and we have much
+to learn from them. Among their results, some, many even, are solid and
+destined to live.
+
+But to say that they have finally settled the debate between Kant and
+Leibnitz and ruined the Kantian theory of mathematics is evidently
+incorrect. I do not know whether they really believed they had done it,
+but if they believed so, they deceived themselves.
+
+
+
+
+CHAPTER V
+
+THE LATEST EFFORTS OF THE LOGISTICIANS
+
+
+I
+
+The logicians have attempted to answer the preceding considerations. For
+that, a transformation of logistic was necessary, and Russell in
+particular has modified on certain points his original views. Without
+entering into the details of the debate, I should like to return to the
+two questions to my mind most important: Have the rules of logistic
+demonstrated their fruitfulness and infallibility? Is it true they
+afford means of proving the principle of complete induction without any
+appeal to intuition?
+
+
+II
+
+_The Infallibility of Logistic_
+
+On the question of fertility, it seems M. Couturat has naïve illusions.
+Logistic, according to him, lends invention 'stilts and wings,' and on
+the next page: "_Ten years ago_, Peano published the first edition of
+his _Formulaire_." How is that, ten years of wings and not to have
+flown!
+
+I have the highest esteem for Peano, who has done very pretty things
+(for instance his 'space-filling curve,' a phrase now discarded); but
+after all he has not gone further nor higher nor quicker than the
+majority of wingless mathematicians, and would have done just as well
+with his legs.
+
+On the contrary I see in logistic only shackles for the inventor. It is
+no aid to conciseness--far from it, and if twenty-seven equations were
+necessary to establish that 1 is a number, how many would be needed to
+prove a real theorem? If we distinguish, with Whitehead, the individual
+_x_, the class of which the only member is _x_ and which shall be called
+[iota]_x_, then the class of which the only member is the class of which
+the only member is _x_ and which shall be called [mu]_x_, do you think
+these distinctions, useful as they may be, go far to quicken our pace?
+
+Logistic forces us to say all that is ordinarily left to be understood;
+it makes us advance step by step; this is perhaps surer but not quicker.
+
+It is not wings you logisticians give us, but leading-strings. And then
+we have the right to require that these leading-strings prevent our
+falling. This will be their only excuse. When a bond does not bear much
+interest, it should at least be an investment for a father of a family.
+
+Should your rules be followed blindly? Yes, else only intuition could
+enable us to distinguish among them; but then they must be infallible;
+for only in an infallible authority can one have a blind confidence.
+This, therefore, is for you a necessity. Infallible you shall be, or not
+at all.
+
+You have no right to say to us: "It is true we make mistakes, but so do
+you." For us to blunder is a misfortune, a very great misfortune; for
+you it is death.
+
+Nor may you ask: Does the infallibility of arithmetic prevent errors in
+addition? The rules of calculation are infallible, and yet we see those
+blunder _who do not apply these rules_; but in checking their
+calculation it is at once seen where they went wrong. Here it is not at
+all the case; the logicians _have applied_ their rules, and they have
+fallen into contradiction; and so true is this, that they are preparing
+to change these rules and to "sacrifice the notion of class." Why change
+them if they were infallible?
+
+"We are not obliged," you say, "to solve _hic et nunc_ all possible
+problems." Oh, we do not ask so much of you. If, in face of a problem,
+you would give _no_ solution, we should have nothing to say; but on the
+contrary you give us _two_ of them and those contradictory, and
+consequently at least one false; this it is which is failure.
+
+Russell seeks to reconcile these contradictions, which can only be done,
+according to him, "by restricting or even sacrificing the notion of
+class." And M. Couturat, discovering the success of his attempt, adds:
+"If the logicians succeed where others have failed, M. Poincaré will
+remember this phrase, and give the honor of the solution to logistic."
+
+But no! Logistic exists, it has its code which has already had four
+editions; or rather this code is logistic itself. Is Mr. Russell
+preparing to show that one at least of the two contradictory reasonings
+has transgressed the code? Not at all; he is preparing to change these
+laws and to abrogate a certain number of them. If he succeeds, I shall
+give the honor of it to Russell's intuition and not to the Peanian
+logistic which he will have destroyed.
+
+
+III
+
+_The Liberty of Contradiction_
+
+I made two principal objections to the definition of whole number
+adopted in logistic. What says M. Couturat to the first of these
+objections?
+
+What does the word _exist_ mean in mathematics? It means, I said, to be
+free from contradiction. This M. Couturat contests. "Logical existence,"
+says he, "is quite another thing from the absence of contradiction. It
+consists in the fact that a class is not empty." To say: _a_'s exist,
+is, by definition, to affirm that the class _a_ is not null.
+
+And doubtless to affirm that the class _a_ is not null, is, by
+definition, to affirm that _a_'s exist. But one of the two affirmations
+is as denuded of meaning as the other, if they do not both signify,
+either that one may see or touch _a_'s which is the meaning physicists
+or naturalists give them, or that one may conceive an _a_ without being
+drawn into contradictions, which is the meaning given them by logicians
+and mathematicians.
+
+For M. Couturat, "it is not non-contradiction that proves existence, but
+it is existence that proves non-contradiction." To establish the
+existence of a class, it is necessary therefore to establish, by an
+_example_, that there is an individual belonging to this class: "But, it
+will be said, how is the existence of this individual proved? Must not
+this existence be established, in order that the existence of the class
+of which it is a part may be deduced? Well, no; however paradoxical may
+appear the assertion, we never demonstrate the existence of an
+individual. Individuals, just because they are individuals, are always
+considered as existent.... We never have to express that an individual
+exists, absolutely speaking, but only that it exists in a class." M.
+Couturat finds his own assertion paradoxical, and he will certainly not
+be the only one. Yet it must have a meaning. It doubtless means that the
+existence of an individual, alone in the world, and of which nothing is
+affirmed, can not involve contradiction; in so far as it is all alone it
+evidently will not embarrass any one. Well, so let it be; we shall admit
+the existence of the individual, 'absolutely speaking,' but nothing
+more. It remains to prove the existence of the individual 'in a class,'
+and for that it will always be necessary to prove that the affirmation,
+"Such an individual belongs to such a class," is neither contradictory
+in itself, nor to the other postulates adopted.
+
+"It is then," continues M. Couturat, "arbitrary and misleading to
+maintain that a definition is valid only if we first prove it is not
+contradictory." One could not claim in prouder and more energetic terms
+the liberty of contradiction. "In any case, the _onus probandi_ rests
+upon those who believe that these principles are contradictory."
+Postulates are presumed to be compatible until the contrary is proved,
+just as the accused person is presumed innocent. Needless to add that I
+do not assent to this claim. But, you say, the demonstration you require
+of us is impossible, and you can not ask us to jump over the moon.
+Pardon me; that is impossible for you, but not for us, who admit the
+principle of induction as a synthetic judgment _a priori_. And that
+would be necessary for you, as for us.
+
+To demonstrate that a system of postulates implies no contradiction, it
+is necessary to apply the principle of complete induction; this mode of
+reasoning not only has nothing 'bizarre' about it, but it is the only
+correct one. It is not 'unlikely' that it has ever been employed; and it
+is not hard to find 'examples and precedents' of it. I have cited two
+such instances borrowed from Hilbert's article. He is not the only one
+to have used it, and those who have not done so have been wrong. What I
+have blamed Hilbert for is not his having recourse to it (a born
+mathematician such as he could not fail to see a demonstration was
+necessary and this the only one possible), but his having recourse
+without recognizing the reasoning by recurrence.
+
+
+IV
+
+_The Second Objection_
+
+I pointed out a second error of logistic in Hilbert's article. To-day
+Hilbert is excommunicated and M. Couturat no longer regards him as of
+the logistic cult; so he asks if I have found the same fault among the
+orthodox. No, I have not seen it in the pages I have read; I know not
+whether I should find it in the three hundred pages they have written
+which I have no desire to read.
+
+Only, they must commit it the day they wish to make any application of
+mathematics. This science has not as sole object the eternal
+contemplation of its own navel; it has to do with nature and some day it
+will touch it. Then it will be necessary to shake off purely verbal
+definitions and to stop paying oneself with words.
+
+To go back to the example of Hilbert: always the point at issue is
+reasoning by recurrence and the question of knowing whether a system of
+postulates is not contradictory. M. Couturat will doubtless say that
+then this does not touch him, but it perhaps will interest those who do
+not claim, as he does, the liberty of contradiction.
+
+We wish to establish, as above, that we shall never encounter
+contradiction after any number of deductions whatever, provided this
+number be finite. For that, it is necessary to apply the principle of
+induction. Should we here understand by finite number every number to
+which by definition the principle of induction applies? Evidently not,
+else we should be led to most embarrassing consequences. To have the
+right to lay down a system of postulates, we must be sure they are not
+contradictory. This is a truth admitted by _most_ scientists; I should
+have written _by all_ before reading M. Couturat's last article. But
+what does this signify? Does it mean that we must be sure of not meeting
+contradiction after a _finite_ number of propositions, the _finite_
+number being by definition that which has all properties of recurrent
+nature, so that if one of these properties fails--if, for instance, we
+come upon a contradiction--we shall agree to say that the number in
+question is not finite? In other words, do we mean that we must be sure
+not to meet contradictions, on condition of agreeing to stop just when
+we are about to encounter one? To state such a proposition is enough to
+condemn it.
+
+So, Hilbert's reasoning not only assumes the principle of induction, but
+it supposes that this principle is given us not as a simple definition,
+but as a synthetic judgment _a priori_.
+
+To sum up:
+
+A demonstration is necessary.
+
+The only demonstration possible is the proof by recurrence.
+
+This is legitimate only if we admit the principle of induction and if we
+regard it not as a definition but as a synthetic judgment.
+
+
+V
+
+_The Cantor Antinomies_
+
+Now to examine Russell's new memoir. This memoir was written with the
+view to conquer the difficulties raised by those Cantor antinomies to
+which frequent allusion has already been made. Cantor thought he could
+construct a science of the infinite; others went on in the way he
+opened, but they soon ran foul of strange contradictions. These
+antinomies are already numerous, but the most celebrated are:
+
+1. The Burali-Forti antinomy;
+
+2. The Zermelo-König antinomy;
+
+3. The Richard antinomy.
+
+Cantor proved that the ordinal numbers (the question is of transfinite
+ordinal numbers, a new notion introduced by him) can be ranged in a
+linear series; that is to say that of two unequal ordinals one is always
+less than the other. Burali-Forti proves the contrary; and in fact he
+says in substance that if one could range _all_ the ordinals in a linear
+series, this series would define an ordinal greater than _all_ the
+others; we could afterwards adjoin 1 and would obtain again an ordinal
+which would be _still greater_, and this is contradictory.
+
+We shall return later to the Zermelo-König antinomy which is of a
+slightly different nature. The Richard antinomy[15] is as follows:
+Consider all the decimal numbers definable by a finite number of words;
+these decimal numbers form an aggregate _E_, and it is easy to see that
+this aggregate is countable, that is to say we can _number_ the
+different decimal numbers of this assemblage from 1 to infinity. Suppose
+the numbering effected, and define a number _N_ as follows: If the _n_th
+decimal of the _n_th number of the assemblage _E_ is
+
+ 0, 1, 2, 3, 4, 5, 6, 7, 8, 9
+
+the _n_th decimal of _N_ shall be:
+
+ 1, 2, 3, 4, 5, 6, 7, 8, 1, 1
+
+ [15] _Revue générale des sciences_, June 30, 1905.
+
+As we see, _N_ is not equal to the _n_th number of _E_, and as _n_ is
+arbitrary, _N_ does not appertain to _E_ and yet _N_ should belong to
+this assemblage since we have defined it with a finite number of words.
+
+We shall later see that M. Richard has himself given with much sagacity
+the explanation of his paradox and that this extends, _mutatis
+mutandis_, to the other like paradoxes. Again, Russell cites another
+quite amusing paradox: _What is the least whole number which can not be
+defined by a phrase composed of less than a hundred English words_?
+
+This number exists; and in fact the numbers capable of being defined by
+a like phrase are evidently finite in number since the words of the
+English language are not infinite in number. Therefore among them will
+be one less than all the others. And, on the other hand, this number
+does not exist, because its definition implies contradiction. This
+number, in fact, is defined by the phrase in italics which is composed
+of less than a hundred English words; and by definition this number
+should not be capable of definition by a like phrase.
+
+
+VI
+
+_Zigzag Theory and No-class Theory_
+
+What is Mr. Russell's attitude in presence of these contradictions?
+After having analyzed those of which we have just spoken, and cited
+still others, after having given them a form recalling Epimenides, he
+does not hesitate to conclude: "A propositional function of one
+variable does not always determine a class." A propositional function
+(that is to say a definition) does not always determine a class. A
+'propositional function' or 'norm' may be 'non-predicative.' And this
+does not mean that these non-predicative propositions determine an empty
+class, a null class; this does not mean that there is no value of x
+satisfying the definition and capable of being one of the elements of
+the class. The elements exist, but they have no right to unite in a
+syndicate to form a class.
+
+But this is only the beginning and it is needful to know how to
+recognize whether a definition is or is not predicative. To solve this
+problem Russell hesitates between three theories which he calls
+
+A. The zigzag theory;
+
+B. The theory of limitation of size;
+
+C. The no-class theory.
+
+According to the zigzag theory "definitions (propositional functions)
+determine a class when they are very simple and cease to do so only when
+they are complicated and obscure." Who, now, is to decide whether a
+definition may be regarded as simple enough to be acceptable? To this
+question there is no answer, if it be not the loyal avowal of a complete
+inability: "The rules which enable us to recognize whether these
+definitions are predicative would be extremely complicated and can not
+commend themselves by any plausible reason. This is a fault which might
+be remedied by greater ingenuity or by using distinctions not yet
+pointed out. But hitherto in seeking these rules, I have not been able
+to find any other directing principle than the absence of
+contradiction."
+
+This theory therefore remains very obscure; in this night a single
+light--the word zigzag. What Russell calls the 'zigzaginess' is
+doubtless the particular characteristic which distinguishes the argument
+of Epimenides.
+
+According to the theory of limitation of size, a class would cease to
+have the right to exist if it were too extended. Perhaps it might be
+infinite, but it should not be too much so. But we always meet again the
+same difficulty; at what precise moment does it begin to be too much
+so? Of course this difficulty is not solved and Russell passes on to the
+third theory.
+
+In the no-classes theory it is forbidden to speak the word 'class' and
+this word must be replaced by various periphrases. What a change for
+logistic which talks only of classes and classes of classes! It becomes
+necessary to remake the whole of logistic. Imagine how a page of
+logistic would look upon suppressing all the propositions where it is a
+question of class. There would only be some scattered survivors in the
+midst of a blank page. _Apparent rari nantes in gurgite vasto._
+
+Be that as it may, we see how Russell hesitates and the modifications to
+which he submits the fundamental principles he has hitherto adopted.
+Criteria are needed to decide whether a definition is too complex or too
+extended, and these criteria can only be justified by an appeal to
+intuition.
+
+It is toward the no-classes theory that Russell finally inclines. Be
+that as it may, logistic is to be remade and it is not clear how much of
+it can be saved. Needless to add that Cantorism and logistic are alone
+under consideration; real mathematics, that which is good for something,
+may continue to develop in accordance with its own principles without
+bothering about the storms which rage outside it, and go on step by step
+with its usual conquests which are final and which it never has to
+abandon.
+
+
+VII
+
+_The True Solution_
+
+What choice ought we to make among these different theories? It seems to
+me that the solution is contained in a letter of M. Richard of which I
+have spoken above, to be found in the _Revue générale des sciences_ of
+June 30, 1905. After having set forth the antinomy we have called
+Richard's antinomy, he gives its explanation. Recall what has already
+been said of this antinomy. _E_ is the aggregate of _all_ the numbers
+definable by a finite number of words, _without introducing the notion
+of the aggregate E itself_. Else the definition of _E_ would contain a
+vicious circle; we must not define _E_ by the aggregate _E_ itself.
+
+Now we have defined _N_ with a finite number of words, it is true, but
+with the aid of the notion of the aggregate _E_. And this is why _N_ is
+not part of _E_. In the example selected by M. Richard, the conclusion
+presents itself with complete evidence and the evidence will appear
+still stronger on consulting the text of the letter itself. But the same
+explanation holds good for the other antinomies, as is easily verified.
+Thus _the definitions which should be regarded as not predicative are
+those which contain a vicious circle_. And the preceding examples
+sufficiently show what I mean by that. Is it this which Russell calls
+the 'zigzaginess'? I put the question without answering it.
+
+
+VIII
+
+_The Demonstrations of the Principle of Induction_
+
+Let us now examine the pretended demonstrations of the principle of
+induction and in particular those of Whitehead and of Burali-Forti.
+
+We shall speak of Whitehead's first, and take advantage of certain new
+terms happily introduced by Russell in his recent memoir. Call
+_recurrent class_ every class containing zero, and containing _n_ + 1 if
+it contains _n_. Call _inductive number_ every number which is a part of
+_all_ the recurrent classes. Upon what condition will this latter
+definition, which plays an essential rôle in Whitehead's proof, be
+'predicative' and consequently acceptable?
+
+In accordance with what has been said, it is necessary to understand by
+_all_ the recurrent classes, all those in whose definition the notion of
+inductive number does not enter. Else we fall again upon the vicious
+circle which has engendered the antinomies.
+
+Now _Whitehead has not taken this precaution_. Whitehead's reasoning is
+therefore fallacious; it is the same which led to the antinomies. It was
+illegitimate when it gave false results; it remains illegitimate when by
+chance it leads to a true result.
+
+A definition containing a vicious circle defines nothing. It is of no
+use to say, we are sure, whatever meaning we may give to our definition,
+zero at least belongs to the class of inductive numbers; it is not a
+question of knowing whether this class is void, but whether it can be
+rigorously deliminated. A 'non-predicative' class is not an empty
+class, it is a class whose boundary is undetermined. Needless to add
+that this particular objection leaves in force the general objections
+applicable to all the demonstrations.
+
+
+IX
+
+Burali-Forti has given another demonstration.[16] But he is obliged to
+assume two postulates: First, there always exists at least one infinite
+class. The second is thus expressed:
+
+ u[epsilon]K(K - [iota][Lambda]) · [inverted c] · u < v'u.
+
+The first postulate is not more evident than the principle to be proved.
+The second not only is not evident, but it is false, as Whitehead has
+shown; as moreover any recruit would see at the first glance, if the
+axiom had been stated in intelligible language, since it means that the
+number of combinations which can be formed with several objects is less
+than the number of these objects.
+
+ [16] In his article 'Le classi finite,' _Atti di Torino_, Vol. XXXII.
+
+
+X
+
+_Zermelo's Assumption_
+
+A famous demonstration by Zermelo rests upon the following assumption:
+In any aggregate (or the same in each aggregate of an assemblage of
+aggregates) we can always choose _at random_ an element (even if this
+assemblage of aggregates should contain an infinity of aggregates). This
+assumption had been applied a thousand times without being stated, but,
+once stated, it aroused doubts. Some mathematicians, for instance
+M. Borel, resolutely reject it; others admire it. Let us see what,
+according to his last article, Russell thinks of it. He does not speak
+out, but his reflections are very suggestive.
+
+And first a picturesque example: Suppose we have as many pairs of shoes
+as there are whole numbers, and so that we can number _the pairs_ from
+one to infinity, how many shoes shall we have? Will the number of shoes
+be equal to the number of pairs? Yes, if in each pair the right shoe is
+distinguishable from the left; it will in fact suffice to give the
+number 2_n_ - 1 to the right shoe of the _n_th pair, and the number 2_n_
+to the left shoe of the _n_th pair. No, if the right shoe is just like
+the left, because a similar operation would become impossible--unless we
+admit Zermelo's assumption, since then we could choose _at random_ in
+each pair the shoe to be regarded as the right.
+
+
+XI
+
+_Conclusions_
+
+A demonstration truly founded upon the principles of analytic logic will
+be composed of a series of propositions. Some, serving as premises, will
+be identities or definitions; the others will be deduced from the
+premises step by step. But though the bond between each proposition and
+the following is immediately evident, it will not at first sight appear
+how we get from the first to the last, which we may be tempted to regard
+as a new truth. But if we replace successively the different expressions
+therein by their definition and if this operation be carried as far as
+possible, there will finally remain only identities, so that all will
+reduce to an immense tautology. Logic therefore remains sterile unless
+made fruitful by intuition.
+
+This I wrote long ago; logistic professes the contrary and thinks it has
+proved it by actually proving new truths. By what mechanism? Why in
+applying to their reasonings the procedure just described--namely,
+replacing the terms defined by their definitions--do we not see them
+dissolve into identities like ordinary reasonings? It is because this
+procedure is not applicable to them. And why? Because their definitions
+are not predicative and present this sort of hidden vicious circle which
+I have pointed out above; non-predicative definitions can not be
+substituted for the terms defined. Under these conditions _logistic is
+not sterile, it engenders antinomies_.
+
+It is the belief in the existence of the actual infinite which has given
+birth to those non-predicative definitions. Let me explain. In these
+definitions the word 'all' figures, as is seen in the examples cited
+above. The word 'all' has a very precise meaning when it is a question
+of a finite number of objects; to have another one, when the objects are
+infinite in number, would require there being an actual (given complete)
+infinity. Otherwise _all_ these objects could not be conceived as
+postulated anteriorly to their definition, and then if the definition of
+a notion _N_ depends upon _all_ the objects _A_, it may be infected with
+a vicious circle, if among the objects _A_ are some indefinable without
+the intervention of the notion _N_ itself.
+
+The rules of formal logic express simply the properties of all possible
+classifications. But for them to be applicable it is necessary that
+these classifications be immutable and that we have no need to modify
+them in the course of the reasoning. If we have to classify only a
+finite number of objects, it is easy to keep our classifications without
+change. If the objects are _indefinite_ in number, that is to say if one
+is constantly exposed to seeing new and unforeseen objects arise, it may
+happen that the appearance of a new object may require the
+classification to be modified, and thus it is we are exposed to
+antinomies. _There is no actual (given complete) infinity._ The
+Cantorians have forgotten this, and they have fallen into contradiction.
+It is true that Cantorism has been of service, but this was when applied
+to a real problem whose terms were precisely defined, and then we could
+advance without fear.
+
+Logistic also forgot it, like the Cantorians, and encountered the same
+difficulties. But the question is to know whether they went this way by
+accident or whether it was a necessity for them. For me, the question is
+not doubtful; belief in an actual infinity is essential in the Russell
+logic. It is just this which distinguishes it from the Hilbert logic.
+Hilbert takes the view-point of extension, precisely in order to avoid
+the Cantorian antinomies. Russell takes the view-point of comprehension.
+Consequently for him the genus is anterior to the species, and the
+_summum genus_ is anterior to all. That would not be inconvenient if the
+_summum genus_ was finite; but if it is infinite, it is necessary to
+postulate the infinite, that is to say to regard the infinite as actual
+(given complete). And we have not only infinite classes; when we pass
+from the genus to the species in restricting the concept by new
+conditions, these conditions are still infinite in number. Because they
+express generally that the envisaged object presents such or such a
+relation with all the objects of an infinite class.
+
+But that is ancient history. Russell has perceived the peril and takes
+counsel. He is about to change everything, and, what is easily
+understood, he is preparing not only to introduce new principles which
+shall allow of operations formerly forbidden, but he is preparing to
+forbid operations he formerly thought legitimate. Not content to adore
+what he burned, he is about to burn what he adored, which is more
+serious. He does not add a new wing to the building, he saps its
+foundation.
+
+The old logistic is dead, so much so that already the zigzag theory and
+the no-classes theory are disputing over the succession. To judge of the
+new, we shall await its coming.
+
+
+
+
+BOOK III
+
+
+THE NEW MECHANICS
+
+
+
+
+CHAPTER I
+
+MECHANICS AND RADIUM
+
+
+I
+
+_Introduction_
+
+The general principles of Dynamics, which have, since Newton, served as
+foundation for physical science, and which appeared immovable, are they
+on the point of being abandoned or at least profoundly modified? This is
+what many people have been asking themselves for some years. According
+to them, the discovery of radium has overturned the scientific dogmas we
+believed the most solid: on the one hand, the impossibility of the
+transmutation of metals; on the other hand, the fundamental postulates
+of mechanics.
+
+Perhaps one is too hasty in considering these novelties as finally
+established, and breaking our idols of yesterday; perhaps it would be
+proper, before taking sides, to await experiments more numerous and more
+convincing. None the less is it necessary, from to-day, to know the new
+doctrines and the arguments, already very weighty, upon which they rest.
+
+In few words let us first recall in what those principles consist:
+
+_A._ The motion of a material point isolated and apart from all exterior
+force is straight and uniform; this is the principle of inertia: without
+force no acceleration;
+
+_B._ The acceleration of a moving point has the same direction as the
+resultant of all the forces to which it is subjected; it is equal to the
+quotient of this resultant by a coefficient called _mass_ of the moving
+point.
+
+The mass of a moving point, so defined, is a constant; it does not
+depend upon the velocity acquired by this point; it is the same whether
+the force, being parallel to this velocity, tends only to accelerate or
+to retard the motion of the point, or whether, on the contrary, being
+perpendicular to this velocity, it tends to make this motion deviate
+toward the right, or the left, that is to say to _curve_ the trajectory;
+
+_C._ All the forces affecting a material point come from the action of
+other material points; they depend only upon the _relative_ positions
+and velocities of these different material points.
+
+Combining the two principles _B_ and _C_, we reach the _principle of
+relative motion_, in virtue of which the laws of the motion of a system
+are the same whether we refer this system to fixed axes, or to moving
+axes animated by a straight and uniform motion of translation, so that
+it is impossible to distinguish absolute motion from a relative motion
+with reference to such moving axes;
+
+_D._ If a material point _A_ acts upon another material point _B_, the
+body _B_ reacts upon _A_, and these two actions are two equal and
+directly opposite forces. This is _the principle of the equality of
+action and reaction_, or, more briefly, the _principle of reaction_.
+
+Astronomic observations and the most ordinary physical phenomena seem to
+have given of these principles a confirmation complete, constant and
+very precise. This is true, it is now said, but it is because we have
+never operated with any but very small velocities; Mercury, for example,
+the fastest of the planets, goes scarcely 100 kilometers a second. Would
+this planet act the same if it went a thousand times faster? We see
+there is yet no need to worry; whatever may be the progress of
+automobilism, it will be long before we must give up applying to our
+machines the classic principles of dynamics.
+
+How then have we come to make actual speeds a thousand times greater
+than that of Mercury, equal, for instance, to a tenth or a third of the
+velocity of light, or approaching still more closely to that velocity?
+It is by aid of the cathode rays and the rays from radium.
+
+We know that radium emits three kinds of rays, designated by the three
+Greek letters [alpha], [beta], [gamma]; in what follows, unless the
+contrary be expressly stated, it will always be a question of the [beta]
+rays, which are analogous to the cathode rays.
+
+After the discovery of the cathode rays two theories appeared. Crookes
+attributed the phenomena to a veritable molecular bombardment; Hertz, to
+special undulations of the ether. This was a renewal of the debate which
+divided physicists a century ago about light; Crookes took up the
+emission theory, abandoned for light; Hertz held to the undulatory
+theory. The facts seem to decide in favor of Crookes.
+
+It has been recognized, in the first place, that the cathode rays carry
+with them a negative electric charge; they are deviated by a magnetic
+field and by an electric field; and these deviations are precisely such
+as these same fields would produce upon projectiles animated by a very
+high velocity and strongly charged with electricity. These two
+deviations depend upon two quantities: one the velocity, the other the
+relation of the electric charge of the projectile to its mass; we cannot
+know the absolute value of this mass, nor that of the charge, but only
+their relation; in fact, it is clear that if we double at the same time
+the charge and the mass, without changing the velocity, we shall double
+the force which tends to deviate the projectile, but, as its mass is
+also doubled, the acceleration and deviation observable will not be
+changed. The observation of the two deviations will give us therefore
+two equations to determine these two unknowns. We find a velocity of
+from 10,000 to 30,000 kilometers a second; as to the ratio of the charge
+to the mass, it is very great. We may compare it to the corresponding
+ratio in regard to the hydrogen ion in electrolysis; we then find that a
+cathodic projectile carries about a thousand times more electricity than
+an equal mass of hydrogen would carry in an electrolyte.
+
+To confirm these views, we need a direct measurement of this velocity to
+compare with the velocity so calculated. Old experiments of J. J.
+Thomson had given results more than a hundred times too small; but they
+were exposed to certain causes of error. The question was taken up again
+by Wiechert in an arrangement where the Hertzian oscillations were
+utilized; results were found agreeing with the theory, at least as to
+order of magnitude; it would be of great interest to repeat these
+experiments. However that may be, the theory of undulations appears
+powerless to account for this complex of facts.
+
+The same calculations made with reference to the [beta] rays of radium
+have given velocities still greater: 100,000 or 200,000 kilometers or
+more yet. These velocities greatly surpass all those we know. It is true
+that light has long been known to go 300,000 kilometers a second; but it
+is not a carrying of matter, while, if we adopt the emission theory for
+the cathode rays, there would be material molecules really impelled at
+the velocities in question, and it is proper to investigate whether the
+ordinary laws of mechanics are still applicable to them.
+
+
+II
+
+_Mass Longitudinal and Mass Transversal_
+
+We know that electric currents produce the phenomena of induction, in
+particular _self-induction_. When a current increases, there develops an
+electromotive force of self-induction which tends to oppose the current;
+on the contrary, when the current decreases, the electromotive force of
+self-induction tends to maintain the current. The self-induction
+therefore opposes every variation of the intensity of the current, just
+as in mechanics the inertia of a body opposes every variation of its
+velocity.
+
+_Self-induction is a veritable inertia._ Everything happens as if the
+current could not establish itself without putting in motion the
+surrounding ether and as if the inertia of this ether tended, in
+consequence, to keep constant the intensity of this current. It would be
+requisite to overcome this inertia to establish the current, it would be
+necessary to overcome it again to make the current cease.
+
+A cathode ray, which is a rain of projectiles charged with negative
+electricity, may be likened to a current; doubtless this current
+differs, at first sight at least, from the currents of ordinary
+conduction, where the matter does not move and where the electricity
+circulates through the matter. This is a _current of convection_, where
+the electricity, attached to a material vehicle, is carried along by the
+motion of this vehicle. But Rowland has proved that currents of
+convection produce the same magnetic effects as currents of conduction;
+they should produce also the same effects of induction. First, if this
+were not so, the principle of the conservation of energy would be
+violated; besides, Crémieu and Pender have employed a method putting in
+evidence _directly_ these effects of induction.
+
+If the velocity of a cathode corpuscle varies, the intensity of the
+corresponding current will likewise vary; and there will develop effects
+of self-induction which will tend to oppose this variation. These
+corpuscles should therefore possess a double inertia: first their own
+proper inertia, and then the apparent inertia, due to self-induction,
+which produces the same effects. They will therefore have a total
+apparent mass, composed of their real mass and of a fictitious mass of
+electromagnetic origin. Calculation shows that this fictitious mass
+varies with the velocity, and that the force of inertia of
+self-induction is not the same when the velocity of the projectile
+accelerates or slackens, or when it is deviated; therefore so it is with
+the force of the total apparent inertia.
+
+The total apparent mass is therefore not the same when the real force
+applied to the corpuscle is parallel to its velocity and tends to
+accelerate the motion as when it is perpendicular to this velocity and
+tends to make the direction vary. It is necessary therefore to
+distinguish the _total longitudinal mass_ from the _total transversal
+mass_. These two total masses depend, moreover, upon the velocity. This
+follows from the theoretical work of Abraham.
+
+In the measurements of which we speak in the preceding section, what is
+it we determine in measuring the two deviations? It is the velocity on
+the one hand, and on the other hand the ratio of the charge to the
+_total transversal mass_. How, under these conditions, can we make out
+in this total mass the part of the real mass and that of the fictitious
+electromagnetic mass? If we had only the cathode rays properly so
+called, it could not be dreamed of; but happily we have the rays of
+radium which, as we have seen, are notably swifter. These rays are not
+all identical and do not behave in the same way under the action of an
+electric field and a magnetic field. It is found that the electric
+deviation is a function of the magnetic deviation, and we are able, by
+receiving on a sensitive plate radium rays which have been subjected to
+the action of the two fields, to photograph the curve which represents
+the relation between these two deviations. This is what Kaufmann has
+done, deducing from it the relation between the velocity and the ratio
+of the charge to the total apparent mass, a ratio we shall call
+[epsilon].
+
+One might suppose there are several species of rays, each characterized
+by a fixed velocity, by a fixed charge and by a fixed mass. But this
+hypothesis is improbable; why, in fact, would all the corpuscles of the
+same mass take always the same velocity? It is more natural to suppose
+that the charge as well as the _real_ mass are the same for all the
+projectiles, and that these differ only by their velocity. If the ratio
+[epsilon] is a function of the velocity, this is not because the real
+mass varies with this velocity; but, since the fictitious
+electromagnetic mass depends upon this velocity, the total apparent
+mass, alone observable, must depend upon it, though the real mass does
+not depend upon it and may be constant.
+
+The calculations of Abraham let us know the law according to which the
+_fictitious_ mass varies as a function of the velocity; Kaufmann's
+experiment lets us know the law of variation of the _total_ mass.
+
+The comparison of these two laws will enable us therefore to determine
+the ratio of the real mass to the total mass.
+
+Such is the method Kaufmann used to determine this ratio. The result is
+highly surprising: _the real mass is naught_.
+
+This has led to conceptions wholly unexpected. What had only been proved
+for cathode corpuscles was extended to all bodies. What we call mass
+would be only semblance; all inertia would be of electromagnetic origin.
+But then mass would no longer be constant, it would augment with the
+velocity; sensibly constant for velocities up to 1,000 kilometers a
+second, it then would increase and would become infinite for the
+velocity of light. The transversal mass would no longer be equal to the
+longitudinal: they would only be nearly equal if the velocity is not too
+great. The principle _B_ of mechanics would no longer be true.
+
+
+III
+
+_The Canal Rays_
+
+At the point where we now are, this conclusion might seem premature. Can
+one apply to all matter what has been proved only for such light
+corpuscles, which are a mere emanation of matter and perhaps not true
+matter? But before entering upon this question, a word must be said of
+another sort of rays. I refer to the _canal rays_, the _Kanalstrahlen_
+of Goldstein.
+
+The cathode, together with the cathode rays charged with negative
+electricity, emits canal rays charged with positive electricity. In
+general, these canal rays not being repelled by the cathode, are
+confined to the immediate neighborhood of this cathode, where they
+constitute the `chamois cushion,' not very easy to perceive; but, if the
+cathode is pierced with holes and if it almost completely blocks up the
+tube, the canal rays spread _back_ of the cathode, in the direction
+opposite to that of the cathode rays, and it becomes possible to study
+them. It is thus that it has been possible to show their positive charge
+and to show that the magnetic and electric deviations still exist, as
+for the cathode rays, but are much feebler.
+
+Radium likewise emits rays analogous to the canal rays, and relatively
+very absorbable, called [alpha] rays.
+
+We can, as for the cathode rays, measure the two deviations and thence
+deduce the velocity and the ratio [epsilon]. The results are less
+constant than for the cathode rays, but the velocity is less, as well as
+the ratio [epsilon]; the positive corpuscles are less charged than the
+negative; or if, which is more natural, we suppose the charges equal and
+of opposite sign, the positive corpuscles are much the larger. These
+corpuscles, charged the ones positively, the others negatively, have
+been called _electrons_.
+
+
+IV
+
+_The Theory of Lorentz_
+
+But the electrons do not merely show us their existence in these rays
+where they are endowed with enormous velocities. We shall see them in
+very different rôles, and it is they that account for the principal
+phenomena of optics and electricity. The brilliant synthesis about to be
+noticed is due to Lorentz.
+
+Matter is formed solely of electrons carrying enormous charges, and, if
+it seems to us neutral, this is because the charges of opposite sign of
+these electrons compensate each other. We may imagine, for example, a
+sort of solar system formed of a great positive electron, around which
+gravitate numerous little planets, the negative electrons, attracted by
+the electricity of opposite name which charges the central electron. The
+negative charges of these planets would balance the positive charge of
+this sun, so that the algebraic sum of all these charges would be
+naught.
+
+All these electrons swim in the ether. The ether is everywhere
+identically the same, and perturbations in it are propagated according
+to the same laws as light or the Hertzian oscillations _in vacuo_. There
+is nothing but electrons and ether. When a luminous wave enters a part
+of the ether where electrons are numerous, these electrons are put in
+motion under the influence of the perturbation of the ether, and they
+then react upon the ether. So would be explained refraction, dispersion,
+double refraction and absorption. Just so, if for any cause an electron
+be put in motion, it would trouble the ether around it and would give
+rise to luminous waves, and this would explain the emission of light by
+incandescent bodies.
+
+In certain bodies, the metals for example, we should have fixed
+electrons, between which would circulate moving electrons enjoying
+perfect liberty, save that of going out from the metallic body and
+breaking the surface which separates it from the exterior void or from
+the air, or from any other non-metallic body.
+
+These movable electrons behave then, within the metallic body, as do,
+according to the kinetic theory of gases, the molecules of a gas within
+the vase where this gas is confined. But, under the influence of a
+difference of potential, the negative movable electrons would tend to go
+all to one side, and the positive movable electrons to the other. This
+is what would produce electric currents, and _this is why these bodies
+would be conductors_. On the other hand, the velocities of our electrons
+would be the greater the higher the temperature, if we accept the
+assimilation with the kinetic theory of gases. When one of these movable
+electrons encounters the surface of the metallic body, whose boundary it
+can not pass, it is reflected like a billiard ball which has hit the
+cushion, and its velocity undergoes a sudden change of direction. But
+when an electron changes direction, as we shall see further on, it
+becomes the source of a luminous wave, and this is why hot metals are
+incandescent.
+
+In other bodies, the dielectrics and the transparent bodies, the movable
+electrons enjoy much less freedom. They remain as if attached to fixed
+electrons which attract them. The farther they go away from them the
+greater becomes this attraction and tends to pull them back. They
+therefore can make only small excursions; they can no longer circulate,
+but only oscillate about their mean position. This is why these bodies
+would not be conductors; moreover they would most often be transparent,
+and they would be refractive, since the luminous vibrations would be
+communicated to the movable electrons, susceptible of oscillation, and
+thence a perturbation would result.
+
+I can not here give the details of the calculations; I confine myself to
+saying that this theory accounts for all the known facts, and has
+predicted new ones, such as the Zeeman effect.
+
+
+V
+
+_Mechanical Consequences_
+
+We now may face two hypotheses:
+
+1º The positive electrons have a real mass, much greater than their
+fictitious electromagnetic mass; the negative electrons alone lack real
+mass. We might even suppose that apart from electrons of the two signs,
+there are neutral atoms which have only their real mass. In this case,
+mechanics is not affected; there is no need of touching its laws; the
+real mass is constant; simply, motions are deranged by the effects of
+self-induction, as has always been known; moreover, these perturbations
+are almost negligible, except for the negative electrons which, not
+having real mass, are not true matter.
+
+2º But there is another point of view; we may suppose there are no
+neutral atoms, and the positive electrons lack real mass just as the
+negative electrons. But then, real mass vanishing, either the word
+_mass_ will no longer have any meaning, or else it must designate the
+fictitious electromagnetic mass; in this case, mass will no longer be
+constant, the transversal _mass_ will no longer be equal to the
+longitudinal, the principles of mechanics will be overthrown.
+
+First a word of explanation. We have said that, for the same charge, the
+_total_ mass of a positive electron is much greater than that of a
+negative. And then it is natural to think that this difference is
+explained by the positive electron having, besides its fictitious mass,
+a considerable real mass; which takes us back to the first hypothesis.
+But we may just as well suppose that the real mass is null for these as
+for the others, but that the fictitious mass of the positive electron is
+much the greater since this electron is much the smaller. I say
+advisedly: much the smaller. And, in fact, in this hypothesis inertia is
+exclusively electromagnetic in origin; it reduces itself to the inertia
+of the ether; the electrons are no longer anything by themselves; they
+are solely holes in the ether and around which the ether moves; the
+smaller these holes are, the more will there be of ether, the greater,
+consequently, will be the inertia of the ether.
+
+How shall we decide between these two hypotheses? By operating upon the
+canal rays as Kaufmann did upon the [beta] rays? This is impossible; the
+velocity of these rays is much too slight. Should each therefore decide
+according to his temperament, the conservatives going to one side and
+the lovers of the new to the other? Perhaps, but, to fully understand
+the arguments of the innovators, other considerations must come in.
+
+
+
+
+CHAPTER II
+
+MECHANICS AND OPTICS
+
+
+I
+
+_Aberration_
+
+You know in what the phenomenon of aberration, discovered by Bradley,
+consists. The light issuing from a star takes a certain time to go
+through a telescope; during this time, the telescope, carried along by
+the motion of the earth, is displaced. If therefore the telescope were
+pointed in the _true_ direction of the star, the image would be formed
+at the point occupied by the crossing of the threads of the network when
+the light has reached the objective; and this crossing would no longer
+be at this same point when the light reached the plane of the network.
+We would therefore be led to mis-point the telescope to bring the image
+upon the crossing of the threads. Thence results that the astronomer
+will not point the telescope in the direction of the absolute velocity
+of the light, that is to say toward the true position of the star, but
+just in the direction of the relative velocity of the light with
+reference to the earth, that is to say toward what is called the
+apparent position of the star.
+
+The velocity of light is known; we might therefore suppose that we have
+the means of calculating the _absolute_ velocity of the earth. (I shall
+soon explain my use here of the word absolute.) Nothing of the sort; we
+indeed know the apparent position of the star we observe; but we do not
+know its true position; we know the velocity of the light only in
+magnitude and not in direction.
+
+If therefore the absolute velocity of the earth were straight and
+uniform, we should never have suspected the phenomenon of aberration;
+but it is variable; it is composed of two parts: the velocity of the
+solar system, which is straight and uniform; the velocity of the earth
+with reference to the sun, which is variable. If the velocity of the
+solar system, that is to say if the constant part existed alone, the
+observed direction would be invariable. This position that one would
+thus observe is called the _mean_ apparent position of the star.
+
+Taking account now at the same time of the two parts of the velocity of
+the earth, we shall have the actual apparent position, which describes a
+little ellipse around the mean apparent position, and it is this ellipse
+that we observe.
+
+Neglecting very small quantities, we shall see that the dimensions of
+this ellipse depend only upon the ratio of the velocity of the earth
+with reference to the sun to the velocity of light, so that the relative
+velocity of the earth with regard to the sun has alone come in.
+
+But wait! This result is not exact, it is only approximate; let us push
+the approximation a little farther. The dimensions of the ellipse will
+depend then upon the absolute velocity of the earth. Let us compare the
+major axes of the ellipse for the different stars: we shall have,
+theoretically at least, the means of determining this absolute velocity.
+
+That would be perhaps less shocking than it at first seems; it is a
+question, in fact, not of the velocity with reference to an absolute
+void, but of the velocity with regard to the ether, which is taken _by
+definition_ as being absolutely at rest.
+
+Besides, this method is purely theoretical. In fact, the aberration is
+very small; the possible variations of the ellipse of aberration are
+much smaller yet, and, if we consider the aberration as of the first
+order, they should therefore be regarded as of the second order: about a
+millionth of a second; they are absolutely inappreciable for our
+instruments. We shall finally see, further on, why the preceding theory
+should be rejected, and why we could not determine this absolute
+velocity even if our instruments were ten thousand times more precise!
+
+One might imagine some other means, and in fact, so one has. The
+velocity of light is not the same in water as in air; could we not
+compare the two apparent positions of a star seen through a telescope
+first full of air, then full of water? The results have been negative;
+the apparent laws of reflection and refraction are not altered by the
+motion of the earth. This phenomenon is capable of two explanations:
+
+1º It might be supposed that the ether is not at rest, but that it is
+carried along by the body in motion. It would then not be astonishing
+that the phenomena of refraction are not altered by the motion of the
+earth, since all, prisms, telescopes and ether, are carried along
+together in the same translation. As to the aberration itself, it would
+be explained by a sort of refraction happening at the surface of
+separation of the ether at rest in the interstellar spaces and the ether
+carried along by the motion of the earth. It is upon this hypothesis
+(bodily carrying along of the ether) that is founded the _theory of
+Hertz_ on the electrodynamics of moving bodies.
+
+2º Fresnel, on the contrary, supposes that the ether is at absolute rest
+in the void, at rest almost absolute in the air, whatever be the
+velocity of this air, and that it is partially carried along by
+refractive media. Lorentz has given to this theory a more satisfactory
+form. For him, the ether is at rest, only the electrons are in motion;
+in the void, where it is only a question of the ether, in the air, where
+this is almost the case, the carrying along is null or almost null; in
+refractive media, where perturbation is produced at the same time by
+vibrations of the ether and those of electrons put in swing by the
+agitation of the ether, the undulations are _partially_ carried along.
+
+To decide between the two hypotheses, we have Fizeau's experiment,
+comparing by measurements of the fringes of interference, the velocity
+of light in air at rest or in motion. These experiments have confirmed
+Fresnel's hypothesis of partial carrying along. They have been repeated
+with the same result by Michelson. _The theory of Hertz must therefore
+be rejected._
+
+
+II
+
+_The Principle of Relativity_
+
+But if the ether is not carried along by the motion of the earth, is it
+possible to show, by means of optical phenomena, the absolute velocity
+of the earth, or rather its velocity with respect to the unmoving ether?
+Experiment has answered negatively, and yet the experimental procedures
+have been varied in all possible ways. Whatever be the means employed
+there will never be disclosed anything but relative velocities; I mean
+the velocities of certain material bodies with reference to other
+material bodies. In fact, if the source of light and the apparatus of
+observation are on the earth and participate in its motion, the
+experimental results have always been the same, whatever be the
+orientation of the apparatus with reference to the orbital motion of the
+earth. If astronomic aberration happens, it is because the source, a
+star, is in motion with reference to the observer.
+
+The hypotheses so far made perfectly account for this general result,
+_if we neglect very small quantities of the order of the square of the
+aberration_. The explanation rests upon the notion of _local time_,
+introduced by Lorentz, which I shall try to make clear. Suppose two
+observers, placed one at _A_, the other at _B_, and wishing to set their
+watches by means of optical signals. They agree that _B_ shall send a
+signal to _A_ when his watch marks an hour determined upon, and _A_ is
+to put his watch to that hour the moment he sees the signal. If this
+alone were done, there would be a systematic error, because as the light
+takes a certain time _t_ to go from _B_ to _A_, _A_'s watch would be
+behind _B_'s the time _t_. This error is easily corrected. It suffices
+to cross the signals. _A_ in turn must signal _B_, and, after this new
+adjustment, _B_'s watch will be behind _A_'s the time _t_. Then it will
+be sufficient to take the arithmetic mean of the two adjustments.
+
+But this way of doing supposes that light takes the same time to go from
+_A_ to _B_ as to return from _B_ to _A_. That is true if the observers
+are motionless; it is no longer so if they are carried along in a common
+translation, since then _A_, for example, will go to meet the light
+coming from _B_, while _B_ will flee before the light coming from _A_.
+If therefore the observers are borne along in a common translation and
+if they do not suspect it, their adjustment will be defective; their
+watches will not indicate the same time; each will show the _local time_
+belonging to the point where it is.
+
+The two observers will have no way of perceiving this, if the unmoving
+ether can transmit to them only luminous signals all of the same
+velocity, and if the other signals they might send are transmitted by
+media carried along with them in their translation. The phenomenon each
+observes will be too soon or too late; it would be seen at the same
+instant only if the translation did not exist; but as it will be
+observed with a watch that is wrong, this will not be perceived and the
+appearances will not be altered.
+
+It results from this that the compensation is easy to explain so long as
+we neglect the square of the aberration, and for a long time the
+experiments were not sufficiently precise to warrant taking account of
+it. But the day came when Michelson imagined a much more delicate
+procedure: he made rays interfere which had traversed different courses,
+after being reflected by mirrors; each of the paths approximating a
+meter and the fringes of interference permitting the recognition of a
+fraction of a thousandth of a millimeter, the square of the aberration
+could no longer be neglected, and _yet the results were still negative_.
+Therefore the theory required to be completed, and it has been by the
+_Lorentz-Fitzgerald hypothesis_.
+
+These two physicists suppose that all bodies carried along in a
+translation undergo a contraction in the sense of this translation,
+while their dimensions perpendicular to this translation remain
+unchanged. _This contraction is the same for all bodies_; moreover, it
+is very slight, about one two-hundred-millionth for a velocity such as
+that of the earth. Furthermore our measuring instruments could not
+disclose it, even if they were much more precise; our measuring rods in
+fact undergo the same contraction as the objects to be measured. If the
+meter exactly fits when applied to a body, if we point the body and
+consequently the meter in the sense of the motion of the earth, it will
+not cease to exactly fit in another orientation, and that although the
+body and the meter have changed in length as well as orientation, and
+precisely because the change is the same for one as for the other. But
+it is quite different if we measure a length, not now with a meter, but
+by the time taken by light to pass along it, and this is just what
+Michelson has done.
+
+A body, spherical when at rest, will take thus the form of a flattened
+ellipsoid of revolution when in motion; but the observer will always
+think it spherical, since he himself has undergone an analogous
+deformation, as also all the objects serving as points of reference. On
+the contrary, the surfaces of the waves of light, remaining rigorously
+spherical, will seem to him elongated ellipsoids.
+
+What happens then? Suppose an observer and a source of light carried
+along together in the translation: the wave surfaces emanating from the
+source will be spheres having as centers the successive positions of the
+source; the distance from this center to the actual position of the
+source will be proportional to the time elapsed after the emission, that
+is to say to the radius of the sphere. All these spheres are therefore
+homothetic one to the other, with relation to the actual position _S_ of
+the source. But, for our observer, because of the contraction, all these
+spheres will seem elongated ellipsoids, and all these ellipsoids will
+moreover be homothetic, with reference to the point _S_; the
+excentricity of all these ellipsoids is the same and depends solely upon
+the velocity of the earth. _We shall so select the law of contraction
+that the point S may be at the focus of the meridian section of the
+ellipsoid._
+
+This time the compensation is _rigorous_, and this it is which explains
+Michelson's experiment.
+
+I have said above that, according to the ordinary theories, observations
+of the astronomic aberration would give us the absolute velocity of the
+earth, if our instruments were a thousand times more precise. I must
+modify this statement. Yes, the observed angles would be modified by the
+effect of this absolute velocity, but the graduated circles we use to
+measure the angles would be deformed by the translation: they would
+become ellipses; thence would result an error in regard to the angle
+measured, and _this second error would exactly compensate the first_.
+
+This Lorentz-Fitzgerald hypothesis seems at first very extraordinary;
+all we can say for the moment, in its favor, is that it is only the
+immediate translation of Michelson's experimental result, if we _define_
+lengths by the time taken by light to run along them.
+
+However that may be, it is impossible to escape the impression that the
+principle of relativity is a general law of nature, that one will never
+be able by any imaginable means to show any but relative velocities, and
+I mean by that not only the velocities of bodies with reference to the
+ether, but the velocities of bodies with regard to one another. Too many
+different experiments have given concordant results for us not to feel
+tempted to attribute to this principle of relativity a value comparable
+to that, for example, of the principle of equivalence. In any case, it
+is proper to see to what consequences this way of looking at things
+would lead us and then to submit these consequences to the control of
+experiment.
+
+
+III
+
+_The Principle of Reaction_
+
+Let us see what the principle of the equality of action and reaction
+becomes in the theory of Lorentz. Consider an electron _A_ which for any
+cause begins to move; it produces a perturbation in the ether; at the
+end of a certain time, this perturbation reaches another electron _B_,
+which will be disturbed from its position of equilibrium. In these
+conditions there can not be equality between action and reaction, at
+least if we do not consider the ether, but only the electrons, _which
+alone are observable_, since our matter is made of electrons.
+
+In fact it is the electron _A_ which has disturbed the electron _B_;
+even in case the electron _B_ should react upon _A_, this reaction could
+be equal to the action, but in no case simultaneous, since the electron
+_B_ can begin to move only after a certain time, necessary for the
+propagation. Submitting the problem to a more exact calculation, we
+reach the following result: Suppose a Hertz discharger placed at the
+focus of a parabolic mirror to which it is mechanically attached; this
+discharger emits electromagnetic waves, and the mirror reflects all
+these waves in the same direction; the discharger therefore will radiate
+energy in a determinate direction. Well, the calculation shows that _the
+discharger recoils_ like a cannon which has shot out a projectile. In
+the case of the cannon, the recoil is the natural result of the equality
+of action and reaction. The cannon recoils because the projectile upon
+which it has acted reacts upon it. But here it is no longer the same.
+What has been sent out is no longer a material projectile: it is energy,
+and energy has no mass: it has no counterpart. And, in place of a
+discharger, we could have considered just simply a lamp with a reflector
+concentrating its rays in a single direction.
+
+It is true that, if the energy sent out from the discharger or from the
+lamp meets a material object, this object receives a mechanical push as
+if it had been hit by a real projectile, and this push will be equal to
+the recoil of the discharger and of the lamp, if no energy has been lost
+on the way and if the object absorbs the whole of the energy. Therefore
+one is tempted to say that there still is compensation between the
+action and the reaction. But this compensation, even should it be
+complete, is always belated. It never happens if the light, after
+leaving its source, wanders through interstellar spaces without ever
+meeting a material body; it is incomplete, if the body it strikes is not
+perfectly absorbent.
+
+Are these mechanical actions too small to be measured, or are they
+accessible to experiment? These actions are nothing other than those due
+to the _Maxwell-Bartholi_ pressures; Maxwell had predicted these
+pressures from calculations relative to electrostatics and magnetism;
+Bartholi reached the same result by thermodynamic considerations.
+
+This is how the _tails of comets_ are explained. Little particles detach
+themselves from the nucleus of the comet; they are struck by the light
+of the sun, which pushes them back as would a rain of projectiles coming
+from the sun. The mass of these particles is so little that this
+repulsion sweeps it away against the Newtonian attraction; so in moving
+away from the sun they form the tails.
+
+The direct experimental verification was not easy to obtain. The first
+endeavor led to the construction of the _radiometer_. But this
+instrument _turns backward_, in the sense opposite to the theoretic
+sense, and the explanation of its rotation, since discovered, is wholly
+different. At last success came, by making the vacuum more complete, on
+the one hand, and on the other by not blackening one of the faces of the
+paddles and directing a pencil of luminous rays upon one of the faces.
+The radiometric effects and the other disturbing causes are eliminated
+by a series of pains-taking precautions, and one obtains a deviation
+which is very minute, but which is, it would seem, in conformity with
+the theory.
+
+The same effects of the Maxwell-Bartholi pressure are forecast likewise
+by the theory of Hertz of which we have before spoken, and by that of
+Lorentz. But there is a difference. Suppose that the energy, under the
+form of light, for example, proceeds from a luminous source to any body
+through a transparent medium. The Maxwell-Bartholi pressure will act,
+not alone upon the source at the departure, and on the body lit up at
+the arrival, but upon the matter of the transparent medium which it
+traverses. At the moment when the luminous wave reaches a new region of
+this medium, this pressure will push forward the matter there
+distributed and will put it back when the wave leaves this region. So
+that the recoil of the source has for counterpart the forward movement
+of the transparent matter which is in contact with this source; a little
+later, the recoil of this same matter has for counterpart the forward
+movement of the transparent matter which lies a little further on, and
+so on.
+
+Only, is the compensation perfect? Is the action of the Maxwell-Bartholi
+pressure upon the matter of the transparent medium equal to its reaction
+upon the source, and that whatever be this matter? Or is this action by
+so much the less as the medium is less refractive and more rarefied,
+becoming null in the void?
+
+If we admit the theory of Hertz, who regards matter as mechanically
+bound to the ether, so that the ether may be entirely carried along by
+matter, it would be necessary to answer yes to the first question and no
+to the second.
+
+There would then be perfect compensation, as required by the principle
+of the equality of action and reaction, even in the least refractive
+media, even in the air, even in the interplanetary void, where it would
+suffice to suppose a residue of matter, however subtile. If on the
+contrary we admit the theory of Lorentz, the compensation, always
+imperfect, is insensible in the air and becomes null in the void.
+
+But we have seen above that Fizeau's experiment does not permit of our
+retaining the theory of Hertz; it is necessary therefore to adopt the
+theory of Lorentz, and consequently _to renounce the principle of
+reaction_.
+
+
+IV
+
+_Consequences of the Principle of Relativity_
+
+We have seen above the reasons which impel us to regard the principle of
+relativity as a general law of nature. Let us see to what consequences
+this principle would lead, should it be regarded as finally
+demonstrated.
+
+First, it obliges us to generalize the hypothesis of Lorentz and
+Fitzgerald on the contraction of all bodies in the sense of the
+translation. In particular, we must extend this hypothesis to the
+electrons themselves. Abraham considered these electrons as spherical
+and indeformable; it will be necessary for us to admit that these
+electrons, spherical when in repose, undergo the Lorentz contraction
+when in motion and take then the form of flattened ellipsoids.
+
+This deformation of the electrons will influence their mechanical
+properties. In fact I have said that the displacement of these charged
+electrons is a veritable current of convection and that their apparent
+inertia is due to the self-induction of this current: exclusively as
+concerns the negative electrons; exclusively or not, we do not yet know,
+for the positive electrons. Well, the deformation of the electrons, a
+deformation which depends upon their velocity, will modify the
+distribution of the electricity upon their surface, consequently the
+intensity of the convection current they produce, consequently the laws
+according to which the self-induction of this current will vary as a
+function of the velocity.
+
+At this price, the compensation will be perfect and will conform to the
+requirements of the principle of relativity, but only upon two
+conditions:
+
+1º That the positive electrons have no real mass, but only a fictitious
+electromagnetic mass; or at least that their real mass, if it exists, is
+not constant and varies with the velocity according to the same laws as
+their fictitious mass;
+
+2º That all forces are of electromagnetic origin, or at least that they
+vary with the velocity according to the same laws as the forces of
+electromagnetic origin.
+
+It still is Lorentz who has made this remarkable synthesis; stop a
+moment and see what follows therefrom. First, there is no more matter,
+since the positive electrons no longer have real mass, or at least no
+constant real mass. The present principles of our mechanics, founded
+upon the constancy of mass, must therefore be modified. Again, an
+electromagnetic explanation must be sought of all the known forces, in
+particular of gravitation, or at least the law of gravitation must be so
+modified that this force is altered by velocity in the same way as the
+electromagnetic forces. We shall return to this point.
+
+All that appears, at first sight, a little artificial. In particular,
+this deformation of electrons seems quite hypothetical. But the thing
+may be presented otherwise, so as to avoid putting this hypothesis of
+deformation at the foundation of the reasoning. Consider the electrons
+as material points and ask how their mass should vary as function of the
+velocity not to contravene the principle of relativity. Or, still
+better, ask what should be their acceleration under the influence of an
+electric or magnetic field, that this principle be not violated and that
+we come back to the ordinary laws when we suppose the velocity very
+slight. We shall find that the variations of this mass, or of these
+accelerations, must be _as if_ the electron underwent the Lorentz
+deformation.
+
+
+V
+
+_Kaufmann's Experiment_
+
+We have before us, then, two theories: one where the electrons are
+indeformable, this is that of Abraham; the other where they undergo the
+Lorentz deformation. In both cases, their mass increases with the
+velocity, becoming infinite when this velocity becomes equal to that of
+light; but the law of the variation is not the same. The method employed
+by Kaufmann to bring to light the law of variation of the mass seems
+therefore to give us an experimental means of deciding between the two
+theories.
+
+Unhappily, his first experiments were not sufficiently precise for that;
+so he decided to repeat them with more precautions, and measuring with
+great care the intensity of the fields. Under their new form _they are
+in favor of the theory of Abraham_. Then the principle of relativity
+would not have the rigorous value we were tempted to attribute to it;
+there would no longer be reason for believing the positive electrons
+denuded of real mass like the negative electrons. However, before
+definitely adopting this conclusion, a little reflection is necessary.
+The question is of such importance that it is to be wished Kaufmann's
+experiment were repeated by another experimenter.[17] Unhappily, this
+experiment is very delicate and could be carried out successfully only
+by a physicist of the same ability as Kaufmann. All precautions have
+been properly taken and we hardly see what objection could be made.
+
+ [17] At the moment of going to press we learn that M. Bucherer has
+ repeated the experiment, taking new precautions, and that he
+ has obtained, contrary to Kaufmann, results confirming the
+ views of Lorentz.
+
+There is one point however to which I wish to draw attention: that is to
+the measurement of the electrostatic field, a measurement upon which all
+depends. This field was produced between the two armatures of a
+condenser; and, between these armatures, there was to be made an
+extremely perfect vacuum, in order to obtain a complete isolation. Then
+the difference of potential of the two armatures was measured, and the
+field obtained by dividing this difference by the distance apart of the
+armatures. That supposes the field uniform; is this certain? Might there
+not be an abrupt fall of potential in the neighborhood of one of the
+armatures, of the negative armature, for example? There may be a
+difference of potential at the meeting of the metal and the vacuum, and
+it may be that this difference is not the same on the positive side and
+on the negative side; what would lead me to think so is the electric
+valve effects between mercury and vacuum. However slight the probability
+that it is so, it seems that it should be considered.
+
+
+VI
+
+_The Principle of Inertia_
+
+In the new dynamics, the principle of inertia is still true, that is to
+say that an _isolated_ electron will have a straight and uniform motion.
+At least this is generally assumed; however, Lindemann has made
+objections to this view; I do not wish to take part in this discussion,
+which I can not here expound because of its too difficult character. In
+any case, slight modifications to the theory would suffice to shelter it
+from Lindemann's objections.
+
+We know that a body submerged in a fluid experiences, when in motion,
+considerable resistance, but this is because our fluids are viscous; in
+an ideal fluid, perfectly free from viscosity, the body would stir up
+behind it a liquid hill, a sort of wake; upon departure, a great effort
+would be necessary to put it in motion, since it would be necessary to
+move not only the body itself, but the liquid of its wake. But, the
+motion once acquired, it would perpetuate itself without resistance,
+since the body, in advancing, would simply carry with it the
+perturbation of the liquid, without the total vis viva of the liquid
+augmenting. Everything would happen therefore as if its inertia was
+augmented. An electron advancing in the ether would behave in the same
+way: around it, the ether would be stirred up, but this perturbation
+would accompany the body in its motion; so that, for an observer carried
+along with the electron, the electric and magnetic fields accompanying
+this electron would appear invariable, and would change only if the
+velocity of the electron varied. An effort would therefore be necessary
+to put the electron in motion, since it would be necessary to create the
+energy of these fields; on the contrary, once the movement acquired, no
+effort would be necessary to maintain it, since the created energy would
+only have to go along behind the electron as a wake. This energy,
+therefore, could only augment the inertia of the electron, as the
+agitation of the liquid augments that of the body submerged in a perfect
+fluid. And anyhow, the negative electrons at least have no other inertia
+except that.
+
+In the hypothesis of Lorentz, the vis viva, which is only the energy of
+the ether, is not proportional to _v_^{2}. Doubtless if _v_ is very
+slight, the vis viva is sensibly proportional to _v_^{2}, the quantity
+of motion sensibly proportional to _v_, the two masses sensibly constant
+and equal to each other. But _when the velocity tends toward the
+velocity of light, the vis viva, the quantity of motion and the two
+masses increase beyond all limit_.
+
+In the hypothesis of Abraham, the expressions are a little more
+complicated; but what we have just said remains true in essentials.
+
+So the mass, the quantity of motion, the vis viva become infinite when
+the velocity is equal to that of light.
+
+Thence results that _no body can attain in any way a velocity beyond
+that of light_. And in fact, in proportion as its velocity increases,
+its mass increases, so that its inertia opposes to any new increase of
+velocity a greater and greater obstacle.
+
+A question then suggests itself: let us admit the principle of
+relativity; an observer in motion would not have any means of perceiving
+his own motion. If therefore no body in its absolute motion can exceed
+the velocity of light, but may approach it as nearly as you choose, it
+should be the same concerning its relative motion with reference to our
+observer. And then we might be tempted to reason as follows: The
+observer may attain a velocity of 200,000 kilometers; the body in its
+relative motion with reference to the observer may attain the same
+velocity; its absolute velocity will then be 400,000 kilometers, which
+is impossible, since this is beyond the velocity of light. This is only
+a seeming, which vanishes when account is taken of how Lorentz evaluates
+local time.
+
+
+VII
+
+_The Wave of Acceleration_
+
+When an electron is in motion, it produces a perturbation in the ether
+surrounding it; if its motion is straight and uniform, this perturbation
+reduces to the wake of which we have spoken in the preceding section.
+But it is no longer the same, if the motion be curvilinear or varied.
+The perturbation may then be regarded as the superposition of two
+others, to which Langevin has given the names _wave of velocity_ and
+_wave of acceleration_. The wave of velocity is only the wave which
+happens in uniform motion.
+
+As to the wave of acceleration, this is a perturbation altogether
+analogous to light waves, which starts from the electron at the instant
+when it undergoes an acceleration, and which is then propagated by
+successive spherical waves with the velocity of light. Whence follows:
+in a straight and uniform motion, the energy is wholly conserved; but,
+when there is an acceleration, there is loss of energy, which is
+dissipated under the form of luminous waves and goes out to infinity
+across the ether.
+
+However, the effects of this wave of acceleration, in particular the
+corresponding loss of energy, are in most cases negligible, that is to
+say not only in ordinary mechanics and in the motions of the heavenly
+bodies, but even in the radium rays, where the velocity is very great
+without the acceleration being so. We may then confine ourselves to
+applying the laws of mechanics, putting the force equal to the product
+of acceleration by mass, this mass, however, varying with the velocity
+according to the laws explained above. We then say the motion is
+_quasi-stationary_.
+
+It would not be the same in all cases where the acceleration is great,
+of which the chief are the following:
+
+1º In incandescent gases certain electrons take an oscillatory motion of
+very high frequency; the displacements are very small, the velocities
+are finite, and the accelerations very great; energy is then
+communicated to the ether, and this is why these gases radiate light of
+the same period as the oscillations of the electron;
+
+2º Inversely, when a gas receives light, these same electrons are put in
+swing with strong accelerations and they absorb light;
+
+3º In the Hertz discharger, the electrons which circulate in the
+metallic mass undergo, at the instant of the discharge, an abrupt
+acceleration and take then an oscillatory motion of high frequency.
+Thence results that a part of the energy radiates under the form of
+Hertzian waves;
+
+4º In an incandescent metal, the electrons enclosed in this metal are
+impelled with great velocity; upon reaching the surface of the metal,
+which they can not get through, they are reflected and thus undergo a
+considerable acceleration. This is why the metal emits light. The
+details of the laws of the emission of light by dark bodies are
+perfectly explained by this hypothesis;
+
+5º Finally when the cathode rays strike the anticathode, the negative
+electrons, constituting these rays, which are impelled with very great
+velocity, are abruptly arrested. Because of the acceleration they thus
+undergo, they produce undulations in the ether. This, according to
+certain physicists, is the origin of the Röntgen rays, which would only
+be light rays of very short wave-length.
+
+
+
+
+CHAPTER III
+
+THE NEW MECHANICS AND ASTRONOMY
+
+
+I
+
+_Gravitation_
+
+Mass may be defined in two ways:
+
+1º By the quotient of the force by the acceleration; this is the true
+definition of the mass, which measures the inertia of the body.
+
+2° By the attraction the body exercises upon an exterior body, in virtue
+of Newton's law. We should therefore distinguish the mass coefficient of
+inertia and the mass coefficient of attraction. According to Newton's
+law, there is rigorous proportionality between these two coefficients.
+But that is demonstrated only for velocities to which the general
+principles of dynamics are applicable. Now, we have seen that the mass
+coefficient of inertia increases with the velocity; should we conclude
+that the mass coefficient of attraction increases likewise with the
+velocity and remains proportional to the coefficient of inertia, or, on
+the contrary, that this coefficient of attraction remains constant? This
+is a question we have no means of deciding.
+
+On the other hand, if the coefficient of attraction depends upon the
+velocity, since the velocities of two bodies which mutually attract are
+not in general the same, how will this coefficient depend upon these two
+velocities?
+
+Upon this subject we can only make hypotheses, but we are naturally led
+to investigate which of these hypotheses would be compatible with the
+principle of relativity. There are a great number of them; the only one
+of which I shall here speak is that of Lorentz, which I shall briefly
+expound.
+
+Consider first electrons at rest. Two electrons of the same sign repel
+each other and two electrons of contrary sign attract each other; in the
+ordinary theory, their mutual actions are proportional to their electric
+charges; if therefore we have four electrons, two positive _A_ and
+_A'_, and two negative _B_ and _B'_, the charges of these four being the
+same in absolute value, the repulsion of _A_ for _A'_ will be, at the
+same distance, equal to the repulsion of _B_ for _B'_ and equal also to
+the attraction of _A_ for _B'_, or of _A'_ for _B_. If therefore _A_ and
+_B_ are very near each other, as also _A'_ and _B'_, and we examine the
+action of the system _A_ + _B_ upon the system _A'_ + _B'_, we shall
+have two repulsions and two attractions which will exactly compensate
+each other and the resulting action will be null.
+
+Now, material molecules should just be regarded as species of solar
+systems where circulate the electrons, some positive, some negative, and
+_in such a way that the algebraic sum of all the charges is null_. A
+material molecule is therefore wholly analogous to the system _A_ + _B_
+of which we have spoken, so that the total electric action of two
+molecules one upon the other should be null.
+
+But experiment shows us that these molecules attract each other in
+consequence of Newtonian gravitation; and then we may make two
+hypotheses: we may suppose gravitation has no relation to the
+electrostatic attractions, that it is due to a cause entirely different,
+and is simply something additional; or else we may suppose the
+attractions are not proportional to the charges and that the attraction
+exercised by a charge +1 upon a charge -1 is greater than the mutual
+repulsion of two +1 charges, or two -1 charges.
+
+In other words, the electric field produced by the positive electrons
+and that which the negative electrons produce might be superposed and
+yet remain distinct. The positive electrons would be more sensitive to
+the field produced by the negative electrons than to the field produced
+by the positive electrons; the contrary would be the case for the
+negative electrons. It is clear that this hypothesis somewhat
+complicates electrostatics, but that it brings back into it gravitation.
+This was, in sum, Franklin's hypothesis.
+
+What happens now if the electrons are in motion? The positive electrons
+will cause a perturbation in the ether and produce there an electric and
+magnetic field. The same will be the case for the negative electrons.
+The electrons, positive as well as negative, undergo then a mechanical
+impulsion by the action of these different fields. In the ordinary
+theory, the electromagnetic field, due to the motion of the positive
+electrons, exercises, upon two electrons of contrary sign and of the
+same absolute charge, equal actions with contrary sign. We may then
+without inconvenience not distinguish the field due to the motion of the
+positive electrons and the field due to the motion of the negative
+electrons and consider only the algebraic sum of these two fields, that
+is to say the resulting field.
+
+In the new theory, on the contrary, the action upon the positive
+electrons of the electromagnetic field due to the positive electrons
+follows the ordinary laws; it is the same with the action upon the
+negative electrons of the field due to the negative electrons. Let us
+now consider the action of the field due to the positive electrons upon
+the negative electrons (or inversely); it will still follow the same
+laws, but _with a different coefficient_. Each electron is more
+sensitive to the field created by the electrons of contrary name than to
+the field created by the electrons of the same name.
+
+Such is the hypothesis of Lorentz, which reduces to Franklin's
+hypothesis for slight velocities; it will therefore explain, for these
+small velocities, Newton's law. Moreover, as gravitation goes back to
+forces of electrodynamic origin, the general theory of Lorentz will
+apply, and consequently the principle of relativity will not be
+violated.
+
+We see that Newton's law is no longer applicable to great velocities and
+that it must be modified, for bodies in motion, precisely in the same
+way as the laws of electrostatics for electricity in motion.
+
+We know that electromagnetic perturbations spread with the velocity of
+light. We may therefore be tempted to reject the preceding theory upon
+remembering that gravitation spreads, according to the calculations of
+Laplace, at least ten million times more quickly than light, and that
+consequently it can not be of electromagnetic origin. The result of
+Laplace is well known, but one is generally ignorant of its
+signification. Laplace supposed that, if the propagation of gravitation
+is not instantaneous, its velocity of spread combines with that of the
+body attracted, as happens for light in the phenomenon of astronomic
+aberration, so that the effective force is not directed along the
+straight joining the two bodies, but makes with this straight a small
+angle. This is a very special hypothesis, not well justified, and, in
+any case, entirely different from that of Lorentz. Laplace's result
+proves nothing against the theory of Lorentz.
+
+
+II
+
+_Comparison with Astronomic Observations_
+
+Can the preceding theories be reconciled with astronomic observations?
+
+First of all, if we adopt them, the energy of the planetary motions will
+be constantly dissipated by the effect of the _wave of acceleration_.
+From this would result that the mean motions of the stars would
+constantly accelerate, as if these stars were moving in a resistant
+medium. But this effect is exceedingly slight, far too much so to be
+discerned by the most precise observations. The acceleration of the
+heavenly bodies is relatively slight, so that the effects of the wave of
+acceleration are negligible and the motion may be regarded as _quasi
+stationary_. It is true that the effects of the wave of acceleration
+constantly accumulate, but this accumulation itself is so slow that
+thousands of years of observation would be necessary for it to become
+sensible. Let us therefore make the calculation considering the motion
+as quasi-stationary, and that under the three following hypotheses:
+
+A. Admit the hypothesis of Abraham (electrons indeformable) and retain
+Newton's law in its usual form;
+
+B. Admit the hypothesis of Lorentz about the deformation of electrons
+and retain the usual Newton's law;
+
+C. Admit the hypothesis of Lorentz about electrons and modify Newton's
+law as we have done in the preceding paragraph, so as to render it
+compatible with the principle of relativity.
+
+It is in the motion of Mercury that the effect will be most sensible,
+since this planet has the greatest velocity. Tisserand formerly made an
+analogous calculation, admitting Weber's law; I recall that Weber had
+sought to explain at the same time the electrostatic and electrodynamic
+phenomena in supposing that electrons (whose name was not yet invented)
+exercise, one upon another, attractions and repulsions directed along
+the straight joining them, and depending not only upon their distances,
+but upon the first and second derivatives of these distances,
+consequently upon their velocities and their accelerations. This law of
+Weber, different enough from those which to-day tend to prevail, none
+the less presents a certain analogy with them.
+
+Tisserand found that, if the Newtonian attraction conformed to Weber's
+law there resulted, for Mercury's perihelion, secular variation of 14",
+_of the same sense as that which has been observed and could not be
+explained_, but smaller, since this is 38".
+
+Let us recur to the hypotheses A, B and C, and study first the motion of
+a planet attracted by a fixed center. The hypotheses B and C are no
+longer distinguished, since, if the attracting point is fixed, the field
+it produces is a purely electrostatic field, where the attraction varies
+inversely as the square of the distance, in conformity with Coulomb's
+electrostatic law, identical with that of Newton.
+
+The vis viva equation holds good, taking for vis viva the new
+definition; in the same way, the equation of areas is replaced by
+another equivalent to it; the moment of the quantity of motion is a
+constant, but the quantity of motion must be defined as in the new
+dynamics.
+
+The only sensible effect will be a secular motion of the perihelion.
+With the theory of Lorentz, we shall find, for this motion, half of what
+Weber's law would give; with the theory of Abraham, two fifths.
+
+If now we suppose two moving bodies gravitating around their common
+center of gravity, the effects are very little different, though the
+calculations may be a little more complicated. The motion of Mercury's
+perihelion would therefore be 7" in the theory of Lorentz and 5".6 in
+that of Abraham.
+
+The effect moreover is proportional to (_n_^{3})(_a_^{2}), where _n_ is
+the star's mean motion and a the radius of its orbit. For the planets,
+in virtue of Kepler's law, the effect varies then inversely as
+sqrt(_a_^{5}); it is therefore insensible, save for Mercury.
+
+It is likewise insensible for the moon though _n_ is great, because _a_
+is extremely small; in sum, it is five times less for Venus, and six
+hundred times less for the moon than for Mercury. We may add that as to
+Venus and the earth, the motion of the perihelion (for the same angular
+velocity of this motion) would be much more difficult to discern by
+astronomic observations, because the excentricity of their orbits is
+much less than for Mercury.
+
+To sum up, _the only sensible effect upon astronomic observations would
+be a motion of Mercury's perihelion, in the same sense as that which has
+been observed without being explained, but notably slighter_.
+
+That can not be regarded as an argument in favor of the new dynamics,
+since it will always be necessary to seek another explanation for the
+greater part of Mercury's anomaly; but still less can it be regarded as
+an argument against it.
+
+
+III
+
+_The Theory of Lesage_
+
+It is interesting to compare these considerations with a theory long
+since proposed to explain universal gravitation.
+
+Suppose that, in the interplanetary spaces, circulate in every
+direction, with high velocities, very tenuous corpuscles. A body
+isolated in space will not be affected, apparently, by the impacts of
+these corpuscles, since these impacts are equally distributed in all
+directions. But if two bodies _A_ and _B_ are present, the body _B_ will
+play the rôle of screen and will intercept part of the corpuscles which,
+without it, would have struck _A_. Then, the impacts received by _A_ in
+the direction opposite that from _B_ will no longer have a counterpart,
+or will now be only partially compensated, and this will push _A_ toward
+_B_.
+
+Such is the theory of Lesage; and we shall discuss it, taking first the
+view-point of ordinary mechanics.
+
+First, how should the impacts postulated by this theory take place; is
+it according to the laws of perfectly elastic bodies, or according to
+those of bodies devoid of elasticity, or according to an intermediate
+law? The corpuscles of Lesage can not act as perfectly elastic bodies;
+otherwise the effect would be null, since the corpuscles intercepted by
+the body _B_ would be replaced by others which would have rebounded from
+_B_, and calculation proves that the compensation would be perfect. It
+is necessary then that the impact make the corpuscles lose energy, and
+this energy should appear under the form of heat. But how much heat
+would thus be produced? Note that attraction passes through bodies; it
+is necessary therefore to represent to ourselves the earth, for example,
+not as a solid screen, but as formed of a very great number of very
+small spherical molecules, which play individually the rôle of little
+screens, but between which the corpuscles of Lesage may freely
+circulate. So, not only the earth is not a solid screen, but it is not
+even a cullender, since the voids occupy much more space than the
+plenums. To realize this, recall that Laplace has demonstrated that
+attraction, in traversing the earth, is weakened at most by one
+ten-millionth part, and his proof is perfectly satisfactory: in fact, if
+attraction were absorbed by the body it traverses, it would no longer be
+proportional to the masses; it would be _relatively_ weaker for great
+bodies than for small, since it would have a greater thickness to
+traverse. The attraction of the sun for the earth would therefore be
+_relatively_ weaker than that of the sun for the moon, and thence would
+result, in the motion of the moon, a very sensible inequality. We should
+therefore conclude, if we adopt the theory of Lesage, that the total
+surface of the spherical molecules which compose the earth is at most
+the ten-millionth part of the total surface of the earth.
+
+Darwin has proved that the theory of Lesage only leads exactly to
+Newton's law when we postulate particles entirely devoid of elasticity.
+The attraction exerted by the earth on a mass 1 at a distance 1 will
+then be proportional, at the same time, to the total surface _S_ of the
+spherical molecules composing it, to the velocity _v_ of the corpuscles,
+to the square root of the density [rho] of the medium formed by the
+corpuscles. The heat produced will be proportional to _S_, to the
+density [rho], and to the cube of the velocity _v_.
+
+But it is necessary to take account of the resistance experienced by a
+body moving in such a medium; it can not move, in fact, without going
+against certain impacts, in fleeing, on the contrary, before those
+coming in the opposite direction, so that the compensation realized in
+the state of rest can no longer subsist. The calculated resistance is
+proportional to _S_, to [rho] and to _v_; now, we know that the heavenly
+bodies move as if they experienced no resistance, and the precision of
+observations permits us to fix a limit to the resistance of the medium.
+
+This resistance varying as _S_[rho]_v_, while the attraction varies as
+_S_{sqrt([rho]_v_)}, we see that the ratio of the resistance to the
+square of the attraction is inversely as the product _Sv_.
+
+We have therefore a lower limit of the product _Sv_. We have already an
+upper limit of _S_ (by the absorption of attraction by the body it
+traverses); we have therefore a lower limit of the velocity _v_, which
+must be at least 24·10^{17} times that of light.
+
+From this we are able to deduce [rho] and the quantity of heat produced;
+this quantity would suffice to raise the temperature 10^{26} degrees a
+second; the earth would receive in a given time 10^{20} times more heat
+than the sun emits in the same time; I am not speaking of the heat the
+sun sends to the earth, but of that it radiates in all directions.
+
+It is evident the earth could not long stand such a régime.
+
+We should not be led to results less fantastic if, contrary to Darwin's
+views, we endowed the corpuscles of Lesage with an elasticity imperfect
+without being null. In truth, the vis viva of these corpuscles would not
+be entirely converted into heat, but the attraction produced would
+likewise be less, so that it would be only the part of this vis viva
+converted into heat, which would contribute to produce the attraction
+and that would come to the same thing; a judicious employment of the
+theorem of the viriel would enable us to account for this.
+
+The theory of Lesage may be transformed; suppress the corpuscles and
+imagine the ether overrun in all senses by luminous waves coming from
+all points of space. When a material object receives a luminous wave,
+this wave exercises upon it a mechanical action due to the
+Maxwell-Bartholi pressure, just as if it had received the impact of a
+material projectile. The waves in question could therefore play the rôle
+of the corpuscles of Lesage. This is what is supposed, for example, by
+M. Tommasina.
+
+The difficulties are not removed for all that; the velocity of
+propagation can be only that of light, and we are thus led, for the
+resistance of the medium, to an inadmissible figure. Besides, if the
+light is all reflected, the effect is null, just as in the hypothesis of
+the perfectly elastic corpuscles.
+
+That there should be attraction, it is necessary that the light be
+partially absorbed; but then there is production of heat. The
+calculations do not differ essentially from those made in the ordinary
+theory of Lesage, and the result retains the same fantastic character.
+
+On the other hand, attraction is not absorbed by the body it traverses,
+or hardly at all; it is not so with the light we know. Light which would
+produce the Newtonian attraction would have to be considerably different
+from ordinary light and be, for example, of very short wave length. This
+does not count that, if our eyes were sensible of this light, the whole
+heavens should appear to us much more brilliant than the sun, so that
+the sun would seem to us to stand out in black, otherwise the sun would
+repel us instead of attracting us. For all these reasons, light which
+would permit of the explanation of attraction would be much more like
+Röntgen rays than like ordinary light.
+
+And besides, the X-rays would not suffice; however penetrating they may
+seem to us, they could not pass through the whole earth; it would be
+necessary therefore to imagine X'-rays much more penetrating than the
+ordinary X-rays. Moreover a part of the energy of these X'-rays would
+have to be destroyed, otherwise there would be no attraction. If you do
+not wish it transformed into heat, which would lead to an enormous heat
+production, you must suppose it radiated in every direction under the
+form of secondary rays, which might be called X'' and which would have
+to be much more penetrating still than the X'-rays, otherwise they would
+in their turn derange the phenomena of attraction.
+
+Such are the complicated hypotheses to which we are led when we try to
+give life to the theory of Lesage.
+
+But all we have said presupposes the ordinary laws of mechanics.
+
+Will things go better if we admit the new dynamics? And first, can we
+conserve the principles of relativity? Let us give at first to the
+theory of Lesage its primitive form, and suppose space ploughed by
+material corpuscles; if these corpuscles were perfectly elastic, the
+laws of their impact would conform to this principle of relativity, but
+we know that then their effect would be null. We must therefore suppose
+these corpuscles are not elastic, and then it is difficult to imagine a
+law of impact compatible with the principle of relativity. Besides, we
+should still find a production of considerable heat, and yet a very
+sensible resistance of the medium.
+
+If we suppress these corpuscles and revert to the hypothesis of the
+Maxwell-Bartholi pressure, the difficulties will not be less. This is
+what Lorentz himself has attempted in his Memoir to the Amsterdam
+Academy of Sciences of April 25, 1900.
+
+Consider a system of electrons immersed in an ether permeated in every
+sense by luminous waves; one of these electrons, struck by one of these
+waves, begins to vibrate; its vibration will be synchronous with that of
+light; but it may have a difference of phase, if the electron absorbs a
+part of the incident energy. In fact, if it absorbs energy, this is
+because the vibration of the ether _impels_ the electron; the electron
+must therefore be slower than the ether. An electron in motion is
+analogous to a convection current; therefore every magnetic field, in
+particular that due to the luminous perturbation itself, must exert a
+mechanical action upon this electron. This action is very slight;
+moreover, it changes sign in the current of the period; nevertheless,
+the mean action is not null if there is a difference of phase between
+the vibrations of the electron and those of the ether. The mean action
+is proportional to this difference, consequently to the energy absorbed
+by the electron. I can not here enter into the detail of the
+calculations; suffice it to say only that the final result is an
+attraction of any two electrons, varying inversely as the square of the
+distance and proportional to the energy absorbed by the two electrons.
+
+Therefore there can not be attraction without absorption of light and,
+consequently, without production of heat, and this it is which
+determined Lorentz to abandon this theory, which, at bottom, does not
+differ from that of Lesage-Maxwell-Bartholi. He would have been much
+more dismayed still if he had pushed the calculation to the end. He
+would have found that the temperature of the earth would have to
+increase 10^{12} degrees a second.
+
+
+IV
+
+_Conclusions_
+
+I have striven to give in few words an idea as complete as possible of
+these new doctrines; I have sought to explain how they took birth;
+otherwise the reader would have had ground to be frightened by their
+boldness. The new theories are not yet demonstrated; far from it; only
+they rest upon an aggregate of probabilities sufficiently weighty for us
+not to have the right to treat them with disregard.
+
+New experiments will doubtless teach us what we should finally think of
+them. The knotty point of the question lies in Kaufmann's experiment and
+those that may be undertaken to verify it.
+
+In conclusion, permit me a word of warning. Suppose that, after some
+years, these theories undergo new tests and triumph; then our secondary
+education will incur a great danger; certain professors will doubtless
+wish to make a place for the new theories.
+
+Novelties are so attractive, and it is so hard not to seem highly
+advanced! At least there will be the wish to open vistas to the pupils
+and, before teaching them the ordinary mechanics, to let them know it
+has had its day and was at best good enough for that old dolt Laplace.
+And then they will not form the habit of the ordinary mechanics.
+
+Is it well to let them know this is only approximative? Yes; but later,
+when it has penetrated to their very marrow, when they shall have taken
+the bent of thinking only through it, when there shall no longer be risk
+of their unlearning it, then one may, without inconvenience, show them
+its limits.
+
+It is with the ordinary mechanics that they must live; this alone will
+they ever have to apply. Whatever be the progress of automobilism, our
+vehicles will never attain speeds where it is not true. The other is
+only a luxury, and we should think of the luxury only when there is no
+longer any risk of harming the necessary.
+
+
+
+
+BOOK IV
+
+
+ASTRONOMIC SCIENCE
+
+
+
+
+CHAPTER I
+
+THE MILKY WAY AND THE THEORY OF GASES
+
+
+The considerations to be here developed have scarcely as yet drawn the
+attention of astronomers; there is hardly anything to cite except an
+ingenious idea of Lord Kelvin's, which has opened a new field of
+research, but still waits to be followed out. Nor have I original
+results to impart, and all I can do is to give an idea of the problems
+presented, but which no one hitherto has undertaken to solve. Every one
+knows how a large number of modern physicists represent the constitution
+of gases; gases are formed of an innumerable multitude of molecules
+which, at high speeds, cross and crisscross in every direction. These
+molecules probably act at a distance one upon another, but this action
+decreases very rapidly with distance, so that their trajectories remain
+sensibly straight; they cease to be so only when two molecules happen to
+pass very near to each other; in this case, their mutual attraction or
+repulsion makes them deviate to right or left. This is what is sometimes
+called an impact; but the word _impact_ is not to be understood in its
+usual sense; it is not necessary that the two molecules come into
+contact, it suffices that they approach sufficiently near each other for
+their mutual attractions to become sensible. The laws of the deviation
+they undergo are the same as for a veritable impact.
+
+It seems at first that the disorderly impacts of this innumerable dust
+can engender only an inextricable chaos before which analysis must
+recoil. But the law of great numbers, that supreme law of chance, comes
+to our aid; in presence of a semi-disorder, we must despair, but in
+extreme disorder, this statistical law reestablishes a sort of mean
+order where the mind can recover. It is the study of this mean order
+which constitutes the kinetic theory of gases; it shows us that the
+velocities of the molecules are equally distributed among all the
+directions, that the rapidity of these velocities varies from one
+molecule to another, but that even this variation is subject to a law
+called Maxwell's law. This law tells us how many of the molecules move
+with such and such a velocity. As soon as the gas departs from this law,
+the mutual impacts of the molecules, in modifying the rapidity and
+direction of their velocities, tend to bring it promptly back.
+Physicists have striven, not without success, to explain in this way the
+experimental properties of gases; for example Mariotte's law.
+
+Consider now the milky way; there also we see an innumerable dust; only
+the grains of this dust are not atoms, they are stars; these grains move
+also with high velocities; they act at a distance one upon another, but
+this action is so slight at great distance that their trajectories are
+straight; and yet, from time to time, two of them may approach near
+enough to be deviated from their path, like a comet which has passed too
+near Jupiter. In a word, to the eyes of a giant for whom our suns would
+be as for us our atoms, the milky way would seem only a bubble of gas.
+
+Such was Lord Kelvin's leading idea. What may be drawn from this
+comparison? In how far is it exact? This is what we are to investigate
+together; but before reaching a definite conclusion, and without wishing
+to prejudge it, we foresee that the kinetic theory of gases will be for
+the astronomer a model he should not follow blindly, but from which he
+may advantageously draw inspiration. Up to the present, celestial
+mechanics has attacked only the solar system or certain systems of
+double stars. Before the assemblage presented by the milky way, or the
+agglomeration of stars, or the resolvable nebulae it recoils, because it
+sees therein only chaos. But the milky way is not more complicated than
+a gas; the statistical methods founded upon the calculus of
+probabilities applicable to a gas are also applicable to it. Before all,
+it is important to grasp the resemblance of the two cases, and their
+difference.
+
+Lord Kelvin has striven to determine in this manner the dimensions of
+the milky way; for that we are reduced to counting the stars visible in
+our telescopes; but we are not sure that behind the stars we see, there
+are not others we do not see; so that what we should measure in this way
+would not be the size of the milky way, it would be the range of our
+instruments.
+
+The new theory comes to offer us other resources. In fact, we know the
+motions of the stars nearest us, and we can form an idea of the rapidity
+and direction of their velocities. If the ideas above set forth are
+exact, these velocities should follow Maxwell's law, and their mean
+value will tell us, so to speak, that which corresponds to the
+temperature of our fictitious gas. But this temperature depends itself
+upon the dimensions of our gas bubble. In fact, how will a gaseous mass
+let loose in the void act, if its elements attract one another according
+to Newton's law? It will take a spherical form; moreover, because of
+gravitation, the density will be greater at the center, the pressure
+also will increase from the surface to the center because of the weight
+of the outer parts drawn toward the center; finally, the temperature
+will increase toward the center: the temperature and the pressure being
+connected by the law called adiabatic, as happens in the successive
+layers of our atmosphere. At the surface itself, the pressure will be
+null, and it will be the same with the absolute temperature, that is to
+say with the velocity of the molecules.
+
+A question comes here: I have spoken of the adiabatic law, but this law
+is not the same for all gases, since it depends upon the ratio of their
+two specific heats; for the air and like gases, this ratio is 1.42; but
+is it to air that it is proper to liken the milky way? Evidently not, it
+should be regarded as a mono-atomic gas, like mercury vapor, like argon,
+like helium, that is to say that the ratio of the specific heats should
+be taken equal to 1.66. And, in fact, one of our molecules would be for
+example the solar system; but the planets are very small personages, the
+sun alone counts, so that our molecule is indeed mono-atomic. And even
+if we take a double star, it is probable that the action of a strange
+star which might approach it would become sufficiently sensible to
+deviate the motion of general translation of the system much before
+being able to trouble the relative orbits of the two components; the
+double star, in a word, would act like an indivisible atom.
+
+However that may be, the pressure, and consequently the temperature, at
+the center of the gaseous sphere would be by so much the greater as the
+sphere was larger since the pressure increases by the weight of all the
+superposed layers. We may suppose that we are nearly at the center of
+the milky way, and by observing the mean proper velocity of the stars,
+we shall know that which corresponds to the central temperature of our
+gaseous sphere and we shall determine its radius.
+
+We may get an idea of the result by the following considerations: make a
+simpler hypothesis: the milky way is spherical, and in it the masses are
+distributed in a homogeneous manner; thence results that the stars in it
+describe ellipses having the same center. If we suppose the velocity
+becomes nothing at the surface, we may calculate this velocity at the
+center by the equation of vis viva. Thus we find that this velocity is
+proportional to the radius of the sphere and to the square root of its
+density. If the mass of this sphere was that of the sun and its radius
+that of the terrestrial orbit, this velocity would be (it is easy to
+see) that of the earth in its orbit. But in the case we have supposed,
+the mass of the sun should be distributed in a sphere of radius
+1,000,000 times greater, this radius being the distance of the nearest
+stars; the density is therefore 10^{18} times less; now, the velocities
+are of the same order, therefore it is necessary that the radius be
+10^{9} times greater, be 1,000 times the distance of the nearest stars,
+which would give about a thousand millions of stars in the milky way.
+
+But you will say these hypothesis differ greatly from the reality;
+first, the milky way is not spherical and we shall soon return to this
+point, and again the kinetic theory of gases is not compatible with the
+hypothesis of a homogeneous sphere. But in making the exact calculation
+according to this theory, we should find a different result, doubtless,
+but of the same order of magnitude; now in such a problem the data are
+so uncertain that the order of magnitude is the sole end to be aimed at.
+
+And here a first remark presents itself; Lord Kelvin's result, which I
+have obtained again by an approximative calculation, agrees sensibly
+with the evaluations the observers have made with their telescopes; so
+that we must conclude we are very near to piercing through the milky
+way. But that enables us to answer another question. There are the stars
+we see because they shine; but may there not be dark stars circulating
+in the interstellar spaces whose existence might long remain unknown?
+But then, what Lord Kelvin's method would give us would be the total
+number of stars, including the dark stars; as his figure is comparable
+to that the telescope gives, this means there is no dark matter, or at
+least not so much as of shining matter.
+
+Before going further, we must look at the problem from another angle. Is
+the milky way thus constituted truly the image of a gas properly so
+called? You know Crookes has introduced the notion of a fourth state of
+matter, where gases having become too rarefied are no longer true gases
+and become what he calls radiant matter. Considering the slight density
+of the milky way, is it the image of gaseous matter or of radiant
+matter? The consideration of what is called the _free path_ will furnish
+us the answer.
+
+The trajectory of a gaseous molecule may be regarded as formed of
+straight segments united by very small arcs corresponding to the
+successive impacts. The length of each of these segments is what is
+called the free path; of course this length is not the same for all the
+segments and for all the molecules; but we may take a mean; this is what
+is called the _mean path_. This is the greater the less the density of
+the gas. The matter will be radiant if the mean path is greater than the
+dimensions of the receptacle wherein the gas is enclosed, so that a
+molecule has a chance to go across the whole receptacle without
+undergoing an impact; if the contrary be the case, it is gaseous. From
+this it follows that the same fluid may be radiant in a little
+receptacle and gaseous in a big one; this perhaps is why, in a Crookes
+tube, it is necessary to make the vacuum by so much the more complete as
+the tube is larger.
+
+How is it then for the milky way? This is a mass of gas of which the
+density is very slight, but whose dimensions are very great; has a star
+chances of traversing it without undergoing an impact, that is to say
+without passing sufficiently near another star to be sensibly deviated
+from its route! What do we mean by _sufficiently near_? That is perforce
+a little arbitrary; take it as the distance from the sun to Neptune,
+which would represent a deviation of a dozen degrees; suppose therefore
+each of our stars surrounded by a protective sphere of this radius;
+could a straight pass between these spheres? At the mean distance of the
+stars of the milky way, the radius of these spheres will be seen under
+an angle of about a tenth of a second; and we have a thousand millions
+of stars. Put upon the celestial sphere a thousand million little
+circles of a tenth of a second radius. Are the chances that these
+circles will cover a great number of times the celestial sphere? Far
+from it; they will cover only its sixteen thousandth part. So the milky
+way is not the image of gaseous matter, but of Crookes' radiant matter.
+Nevertheless, as our foregoing conclusions are happily not at all
+precise, we need not sensibly modify them.
+
+But there is another difficulty: the milky way is not spherical, and we
+have reasoned hitherto as if it were, since this is the form of
+equilibrium a gas isolated in space would take. To make amends,
+agglomerations of stars exist whose form is globular and to which would
+better apply what we have hitherto said. Herschel has already endeavored
+to explain their remarkable appearances. He supposed the stars of the
+aggregates uniformly distributed, so that an assemblage is a homogeneous
+sphere; each star would then describe an ellipse and all these orbits
+would be passed over in the same time, so that at the end of a period
+the aggregate would take again its primitive configuration and this
+configuration would be stable. Unluckily, the aggregates do not appear
+to be homogeneous; we see a condensation at the center, we should
+observe it even were the sphere homogeneous, since it is thicker at the
+center; but it would not be so accentuated. We may therefore rather
+compare an aggregate to a gas in adiabatic equilibrium, which takes the
+spherical form because this is the figure of equilibrium of a gaseous
+mass.
+
+But, you will say, these aggregates are much smaller than the milky way,
+of which they even in probability make part, and even though they be
+more dense, they will rather present something analogous to radiant
+matter; now, gases attain their adiabatic equilibrium only through
+innumerable impacts of the molecules. That might perhaps be adjusted.
+Suppose the stars of the aggregate have just enough energy for their
+velocity to become null when they reach the surface; then they may
+traverse the aggregate without impact, but arrived at the surface they
+will go back and will traverse it anew; after a great number of
+crossings, they will at last be deviated by an impact; under these
+conditions, we should still have a matter which might be regarded as
+gaseous; if perchance there had been in the aggregate stars whose
+velocity was greater, they have long gone away out of it, they have left
+it never to return. For all these reasons, it would be interesting to
+examine the known aggregates, to seek to account for the law of the
+densities, and to see if it is the adiabatic law of gases.
+
+But to return to the milky way; it is not spherical and would rather be
+represented as a flattened disc. It is clear then that a mass starting
+without velocity from the surface will reach the center with different
+velocities, according as it starts from the surface in the neighborhood
+of the middle of the disc or just on the border of the disc; the
+velocity would be notably greater in the latter case. Now, up to the
+present, we have supposed that the proper velocities of the stars, those
+we observe, must be comparable to those which like masses would attain;
+this involves a certain difficulty. We have given above a value for the
+dimensions of the milky way, and we have deduced it from the observed
+proper velocities which are of the same order of magnitude as that of
+the earth in its orbit; but which is the dimension we have thus
+measured? Is it the thickness? Is it the radius of the disc? It is
+doubtless something intermediate; but what can we say then of the
+thickness itself, or of the radius of the disc? Data are lacking to make
+the calculation; I shall confine myself to giving a glimpse of the
+possibility of basing an evaluation at least approximate upon a deeper
+discussion of the proper motions.
+
+And then we find ourselves facing two hypotheses: either the stars of
+the milky way are impelled by velocities for the most part parallel to
+the galactic plane, but otherwise distributed uniformly in all
+directions parallel to this plane. If this be so, observation of the
+proper motions should show a preponderance of components parallel to the
+milky way; this is to be determined, because I do not know whether a
+systematic discussion has ever been made from this view-point. On the
+other hand, such an equilibrium could only be provisory, since because
+of impacts the molecules, I mean the stars, would in the long run
+acquire notable velocities in the sense perpendicular to the milky way
+and would end by swerving from its plane, so that the system would tend
+toward the spherical form, the only figure of equilibrium of an isolated
+gaseous mass.
+
+Or else the whole system is impelled by a common rotation, and for that
+reason is flattened like the earth, like Jupiter, like all bodies that
+twirl. Only, as the flattening is considerable, the rotation must be
+rapid; rapid doubtless, but it must be understood in what sense this
+word is used. The density of the milky way is 10^{23} times less than
+that of the sun; a velocity of rotation sqrt(10^{25}) times less than
+that of the sun, for it would, therefore, be the equivalent so far as
+concerns flattening; a velocity 10^{12} times slower than that of the
+earth, say a thirtieth of a second of arc in a century, would be a very
+rapid rotation, almost too rapid for stable equilibrium to be possible.
+
+In this hypothesis, the observable proper motions would appear to us
+uniformly distributed, and there would no longer be a preponderance of
+components parallel to the galactic plane.
+
+They will tell us nothing about the rotation itself, since we belong to
+the turning system. If the spiral nebulæ are other milky ways, foreign
+to ours, they are not borne along in this rotation, and we might study
+their proper motions. It is true they are very far away; if a nebula has
+the dimensions of the milky way and if its apparent radius is for
+example 20", its distance is 10,000 times the radius of the milky way.
+
+But that makes no difference, since it is not about the translation of
+our system that we ask information from them, but about its rotation.
+The fixed stars, by their apparent motion, reveal to us the diurnal
+rotation of the earth, though their distance is immense. Unluckily, the
+possible rotation of the milky way, however rapid it may be relatively,
+is very slow viewed absolutely, and besides the pointings on nebulæ can
+not be very precise; therefore thousands of years of observations would
+be necessary to learn anything.
+
+However that may be, in this second hypothesis, the figure of the milky
+way would be a figure of final equilibrium.
+
+I shall not further discuss the relative value of these two hypotheses
+since there is a third which is perhaps more probable. We know that
+among the irresolvable nebulæ, several kinds may be distinguished: the
+irregular nebulæ like that of Orion, the planetary and annular nebulæ,
+the spiral nebulæ. The spectra of the first two families have been
+determined, they are discontinuous; these nebulæ are therefore not
+formed of stars; besides, their distribution on the heavens seems to
+depend upon the milky way; whether they have a tendency to go away from
+it, or on the contrary to approach it, they make therefore a part of the
+system. On the other hand, the spiral nebulæ are generally considered as
+independent of the milky way; it is supposed that they, like it, are
+formed of a multitude of stars, that they are, in a word, other milky
+ways very far away from ours. The recent investigations of Stratonoff
+tend to make us regard the milky way itself as a spiral nebula, and this
+is the third hypothesis of which I wish to speak.
+
+How can we explain the very singular appearances presented by the spiral
+nebulæ, which are too regular and too constant to be due to chance?
+First of all, to take a look at one of these representations is enough
+to see that the mass is in rotation; we may even see what the sense of
+the rotation is; all the spiral radii are curved in the same sense; it
+is evident that the _moving wing_ lags behind the pivot and that fixes
+the sense of the rotation. But this is not all; it is evident that these
+nebulæ can not be likened to a gas at rest, nor even to a gas in
+relative equilibrium under the sway of a uniform rotation; they are to
+be compared to a gas in permanent motion in which internal currents
+prevail.
+
+Suppose, for example, that the rotation of the central nucleus is rapid
+(you know what I mean by this word), too rapid for stable equilibrium;
+then at the equator the centrifugal force will drive it away over the
+attraction, and the stars will tend to break away at the equator and
+will form divergent currents; but in going away, as their moment of
+rotation remains constant, while the radius vector augments, their
+angular velocity will diminish, and this is why the moving wing seems to
+lag back.
+
+From this point of view, there would not be a real permanent motion, the
+central nucleus would constantly lose matter which would go out of it
+never to return, and would drain away progressively. But we may modify
+the hypothesis. In proportion as it goes away, the star loses its
+velocity and ends by stopping; at this moment attraction regains
+possession of it and leads it back toward the nucleus; so there will be
+centripetal currents. We must suppose the centripetal currents are the
+first rank and the centrifugal currents the second rank, if we adopt the
+comparison with a troop in battle executing a change of front; and, in
+fact, it is necessary that the composite centrifugal force be
+compensated by the attraction exercised by the central layers of the
+swarm upon the extreme layers.
+
+Besides, at the end of a certain time a permanent régime establishes
+itself; the swarm being curved, the attraction exercised upon the pivot
+by the moving wing tends to slow up the pivot and that of the pivot upon
+the moving wing tends to accelerate the advance of this wing which no
+longer augments its lag, so that finally all the radii end by turning
+with a uniform velocity. We may still suppose that the rotation of the
+nucleus is quicker than that of the radii.
+
+A question remains; why do these centripetal and centrifugal swarms tend
+to concentrate themselves in radii instead of disseminating themselves a
+little everywhere? Why do these rays distribute themselves regularly? If
+the swarms concentrate themselves, it is because of the attraction
+exercised by the already existing swarms upon the stars which go out
+from the nucleus in their neighborhood. After an inequality is produced,
+it tends to accentuate itself in this way.
+
+Why do the rays distribute themselves regularly? That is less obvious.
+Suppose there is no rotation, that all the stars are in two planes at
+right angles, in such a way that their distribution is symmetric with
+regard to these two planes.
+
+By symmetry, there would be no reason for their going out of these
+planes, nor for the symmetry changing. This configuration would give us
+therefore equilibrium, but _this would be an unstable equilibrium_.
+
+If on the contrary, there is rotation, we shall find an analogous
+configuration of equilibrium with four curved rays, equal to each other
+and intersecting at 90°, and if the rotation is sufficiently rapid, this
+equilibrium is stable.
+
+I am not in position to make this more precise: enough if you see that
+these spiral forms may perhaps some day be explained by only the law of
+gravitation and statistical consideration recalling those of the theory
+of gases.
+
+What has been said of internal currents shows it is of interest to
+discuss systematically the aggregate of proper motions; this may be done
+in a hundred years, when the second edition is issued of the chart of
+the heavens and compared with the first, that we now are making.
+
+But, in conclusion, I wish to call your attention to a question, that of
+the age of the milky way or the nebulæ. If what we think we see is
+confirmed, we can get an idea of it. That sort of statistical
+equilibrium of which gases give us the model is established only in
+consequence of a great number of impacts. If these impacts are rare, it
+can come about only after a very long time; if really the milky way (or
+at least the agglomerations which are contained in it), if the nebulæ
+have attained this equilibrium, this means they are very old, and we
+shall have an inferior limit of their age. Likewise we should have of it
+a superior limit; this equilibrium is not final and can not last always.
+Our spiral nebulæ would be comparable to gases impelled by permanent
+motions; but gases in motion are viscous and their velocities end by
+wearing out. What here corresponds to the viscosity (and which depends
+upon the chances of impact of the molecules) is excessively slight, so
+that the present régime may persist during an extremely long time, yet
+not forever, so that our milky ways can not live eternally nor become
+infinitely old.
+
+And this is not all. Consider our atmosphere: at the surface must reign
+a temperature infinitely small and the velocity of the molecules there
+is near zero. But this is a question only of the mean velocity; as a
+consequence of impacts, one of these molecules may acquire (rarely, it
+is true) an enormous velocity, and then it will rush out of the
+atmosphere, and once out, it will never return; therefore our atmosphere
+drains off thus with extreme slowness. The milky way also from time to
+time loses a star by the same mechanism, and that likewise limits its
+duration.
+
+Well, it is certain that if we compute in this manner the age of the
+milky way, we shall get enormous figures. But here a difficulty presents
+itself. Certain physicists, relying upon other considerations, reckon
+that suns can have only an ephemeral existence, about fifty million
+years; our minimum would be much greater than that. Must we believe that
+the evolution of the milky way began when the matter was still dark? But
+how have the stars composing it reached all at the same time adult age,
+an age so briefly to endure? Or must they reach there all successively,
+and are those we see only a feeble minority compared with those
+extinguished or which shall one day light up? But how reconcile that
+with what we have said above on the absence of a noteworthy proportion
+of dark matter? Should we abandon one of the two hypotheses, and which?
+I confine myself to pointing out the difficulty without pretending to
+solve it; I shall end therefore with a big interrogation point.
+
+However, it is interesting to set problems, even when their solution
+seems very far away.
+
+
+
+
+CHAPTER II
+
+FRENCH GEODESY
+
+
+Every one understands our interest in knowing the form and dimensions of
+our earth; but some persons will perhaps be surprised at the exactitude
+sought after. Is this a useless luxury? What good are the efforts so
+expended by the geodesist?
+
+Should this question be put to a congressman, I suppose he would say:
+"I am led to believe that geodesy is one of the most useful of the
+sciences; because it is one of those costing us most dear." I shall try
+to give you an answer a little more precise.
+
+The great works of art, those of peace as well as those of war, are not
+to be undertaken without long studies which save much groping,
+miscalculation and useless expense. These studies can only be based upon
+a good map. But a map will be only a valueless phantasy if constructed
+without basing it upon a solid framework. As well make stand a human
+body minus the skeleton.
+
+Now, this framework is given us by geodesic measurements; so, without
+geodesy, no good map; without a good map, no great public works.
+
+These reasons will doubtless suffice to justify much expense; but these
+are arguments for practical men. It is not upon these that it is proper
+to insist here; there are others higher and, everything considered, more
+important.
+
+So we shall put the question otherwise; can geodesy aid us the better to
+know nature? Does it make us understand its unity and harmony? In
+reality an isolated fact is of slight value, and the conquests of
+science are precious only if they prepare for new conquests.
+
+If therefore a little hump were discovered on the terrestrial ellipsoid,
+this discovery would be by itself of no great interest. On the other
+hand, it would become precious if, in seeking the cause of this hump, we
+hoped to penetrate new secrets.
+
+Well, when, in the eighteenth century, Maupertuis and La Condamine
+braved such opposite climates, it was not solely to learn the shape of
+our planet, it was a question of the whole world-system.
+
+If the earth was flattened, Newton triumphed and with him the doctrine
+of gravitation and the whole modern celestial mechanics.
+
+And to-day, a century and a half after the victory of the Newtonians,
+think you geodesy has nothing more to teach us?
+
+We know not what is within our globe. The shafts of mines and borings
+have let us know a layer of 1 or 2 kilometers thickness, that is to say,
+the millionth part of the total mass; but what is beneath?
+
+Of all the extraordinary journeys dreamed by Jules Verne, perhaps that
+to the center of the earth took us to regions least explored.
+
+But these deep-lying rocks we can not reach, exercise from afar their
+attraction which operates upon the pendulum and deforms the terrestrial
+spheroid. Geodesy can therefore weigh them from afar, so to speak, and
+tell us of their distribution. Thus will it make us really see those
+mysterious regions which Jules Verne only showed us in imagination.
+
+This is not an empty illusion. M. Faye, comparing all the measurements,
+has reached a result well calculated to surprise us. Under the oceans,
+in the depths, are rocks of very great density; under the continents, on
+the contrary, are empty spaces.
+
+New observations will modify perhaps the details of these conclusions.
+
+In any case, our venerated dean has shown us where to search and what
+the geodesist may teach the geologist, desirous of knowing the interior
+constitution of the earth, and even the thinker wishing to speculate
+upon the past and the origin of this planet.
+
+And now, why have I entitled this chapter _French Geodesy_? It is
+because, in each country, this science has taken, more than all others,
+perhaps, a national character. It is easy to see why.
+
+There must be rivalry. The scientific rivalries are always courteous, or
+at least almost always; in any case, they are necessary, because they
+are always fruitful. Well, in those enterprises which require such long
+efforts and so many collaborators, the individual is effaced, in spite
+of himself, of course; no one has the right to say: this is my work.
+Therefore it is not between men, but between nations that rivalries go
+on.
+
+So we are led to seek what has been the part of France. Her part I
+believe we are right to be proud of.
+
+At the beginning of the eighteenth century, long discussions arose
+between the Newtonians who believed the earth flattened, as the theory
+of gravitation requires, and Cassini, who, deceived by inexact
+measurements, believed our globe elongated. Only direct observation
+could settle the question. It was our Academy of Sciences that undertook
+this task, gigantic for the epoch.
+
+While Maupertuis and Clairaut measured a degree of meridian under the
+polar circle, Bouguer and La Condamine went toward the Andes Mountains,
+in regions then under Spain which to-day are the Republic of Ecuador.
+
+Our envoys were exposed to great hardships. Traveling was not as easy as
+at present.
+
+Truly, the country where Maupertuis operated was not a desert and he
+even enjoyed, it is said, among the Laplanders those sweet satisfactions
+of the heart that real arctic voyagers never know. It was almost the
+region where, in our days, comfortable steamers carry, each summer,
+hosts of tourists and young English people. But in those days Cook's
+agency did not exist and Maupertuis really believed he had made a polar
+expedition.
+
+Perhaps he was not altogether wrong. The Russians and the Swedes carry
+out to-day analogous measurements at Spitzbergen, in a country where
+there is real ice-cap. But they have quite other resources, and the
+difference of time makes up for that of latitude.
+
+The name of Maupertuis has reached us much scratched by the claws of
+Doctor Akakia; the scientist had the misfortune to displease Voltaire,
+who was then the king of mind. He was first praised beyond measure; but
+the flatteries of kings are as much to be dreaded as their displeasure,
+because the days after are terrible. Voltaire himself knew something of
+this.
+
+Voltaire called Maupertuis, my amiable master in thinking, marquis of
+the polar circle, dear flattener out of the world and Cassini, and even,
+flattery supreme, Sir Isaac Maupertuis; he wrote him: "Only the king of
+Prussia do I put on a level with you; he only lacks being a geometer."
+But soon the scene changes, he no longer speaks of deifying him, as in
+days of yore the Argonauts, or of calling down from Olympus the council
+of the gods to contemplate his works, but of chaining him up in a
+madhouse. He speaks no longer of his sublime mind, but of his despotic
+pride, plated with very little science and much absurdity.
+
+I care not to relate these comico-heroic combats; but permit me some
+reflections on two of Voltaire's verses. In his 'Discourse on
+Moderation' (no question of moderation in praise and criticism), the
+poet has written:
+
+ You have confirmed in regions drear
+ What Newton discerned without going abroad.
+
+These two verses (which replace the hyperbolic praises of the first
+period) are very unjust, and doubtless Voltaire was too enlightened not
+to know it.
+
+Then, only those discoveries were esteemed which could be made without
+leaving one's house.
+
+To-day, it would rather be theory that one would make light of.
+
+This is to misunderstand the aim of science.
+
+Is nature governed by caprice, or does harmony rule there? That is the
+question. It is when it discloses to us this harmony that science is
+beautiful and so worthy to be cultivated. But whence can come to us this
+revelation, if not from the accord of a theory with experiment? To seek
+whether this accord exists or if it fails, this therefore is our aim.
+Consequently these two terms, which we must compare, are as
+indispensable the one as the other. To neglect one for the other would
+be nonsense. Isolated, theory would be empty, experiment would be blind;
+each would be useless and without interest.
+
+Maupertuis therefore deserves his share of glory. Truly, it will not
+equal that of Newton, who had received the spark divine; nor even that
+of his collaborator Clairaut. Yet it is not to be despised, because his
+work was necessary, and if France, outstripped by England in the
+seventeenth century, has so well taken her revenge in the century
+following, it is not alone to the genius of Clairauts, d'Alemberts,
+Laplaces that she owes it; it is also to the long patience of the
+Maupertuis and the La Condamines.
+
+We reach what may be called the second heroic period of geodesy. France
+is torn within. All Europe is armed against her; it would seem that
+these gigantic combats might absorb all her forces. Far from it; she
+still has them for the service of science. The men of that time recoiled
+before no enterprise, they were men of faith.
+
+Delambre and Méchain were commissioned to measure an arc going from
+Dunkerque to Barcelona. This time there was no going to Lapland or to
+Peru; the hostile squadrons had closed to us the ways thither. But,
+though the expeditions are less distant, the epoch is so troubled that
+the obstacles, the perils even, are just as great.
+
+In France, Delambre had to fight against the ill-will of suspicious
+municipalities. One knows that the steeples, which are visible from so
+far, and can be aimed at with precision, often serve as signal points to
+geodesists. But in the region Delambre traversed there were no longer
+any steeples. A certain proconsul had passed there, and boasted of
+knocking down all the steeples rising proudly above the humble abode of
+the sans-culottes. Pyramids then were built of planks and covered with
+white cloth to make them more visible. That was quite another thing:
+with white cloth! What was this rash person who, upon our heights so
+recently set free, dared to raise the hateful standard of the
+counter-revolution? It was necessary to border the white cloth with blue
+and red bands.
+
+Méchain operated in Spain; the difficulties were other; but they were
+not less. The Spanish peasants were hostile. There steeples were not
+lacking: but to install oneself in them with mysterious and perhaps
+diabolic instruments, was it not sacrilege? The revolutionists were
+allies of Spain, but allies smelling a little of the stake.
+
+"Without cease," writes Méchain, "they threaten to butcher us."
+Fortunately, thanks to the exhortations of the priests, to the pastoral
+letters of the bishops, these ferocious Spaniards contented themselves
+with threatening.
+
+Some years after Méchain made a second expedition into Spain: he
+proposed to prolong the meridian from Barcelona to the Balearics. This
+was the first time it had been attempted to make the triangulations
+overpass a large arm of the sea by observing signals installed upon some
+high mountain of a far-away isle. The enterprise was well conceived and
+well prepared; it failed however.
+
+The French scientist encountered all sorts of difficulties of which he
+complains bitterly in his correspondence. "Hell," he writes, perhaps
+with some exaggeration--"hell and all the scourges it vomits upon the
+earth, tempests, war, the plague and black intrigues are therefore
+unchained against me!"
+
+The fact is that he encountered among his collaborators more of proud
+obstinacy than of good will and that a thousand accidents retarded his
+work. The plague was nothing, the fear of the plague was much more
+redoubtable; all these isles were on their guard against the neighboring
+isles and feared lest they should receive the scourge from them. Méchain
+obtained permission to disembark only after long weeks upon the
+condition of covering all his papers with vinegar; this was the
+antisepsis of that time.
+
+Disgusted and sick, he had just asked to be recalled, when he died.
+
+Arago and Biot it was who had the honor of taking up the unfinished work
+and carrying it on to completion.
+
+Thanks to the support of the Spanish government, to the protection of
+several bishops and, above all, to that of a famous brigand chief, the
+operations went rapidly forward. They were successfully completed, and
+Biot had returned to France when the storm burst.
+
+It was the moment when all Spain took up arms to defend her independence
+against France. Why did this stranger climb the mountains to make
+signals? It was evidently to call the French army. Arago was able to
+escape the populace only by becoming a prisoner. In his prison, his only
+distraction was reading in the Spanish papers the account of his own
+execution. The papers of that time sometimes gave out news prematurely.
+He had at least the consolation of learning that he died with courage
+and like a Christian.
+
+Even the prison was no longer safe; he had to escape and reach Algiers.
+There, he embarked for Marseilles on an Algerian vessel. This ship was
+captured by a Spanish corsair, and behold Arago carried back to Spain
+and dragged from dungeon to dungeon, in the midst of vermin and in the
+most shocking wretchedness.
+
+If it had only been a question of his subjects and his guests, the dey
+would have said nothing. But there were on board two lions, a present
+from the African sovereign to Napoleon. The dey threatened war.
+
+The vessel and the prisoners were released. The port should have been
+properly reached, since they had on board an astronomer; but the
+astronomer was seasick, and the Algerian seamen, who wished to make
+Marseilles, came out at Bougie. Thence Arago went to Algiers, traversing
+Kabylia on foot in the midst of a thousand perils. He was long detained
+in Africa and threatened with the convict prison. Finally he was able to
+get back to France; his observations, which he had preserved and
+safeguarded under his shirt, and, what is still more remarkable, his
+instruments had traversed unhurt these terrible adventures. Up to this
+point, not only did France hold the foremost place, but she occupied the
+stage almost alone.
+
+In the years which follow, she has not been inactive and our
+staff-office map is a model. However, the new methods of observation and
+calculation have come to us above all from Germany and England. It is
+only in the last forty years that France has regained her rank. She owes
+it to a scientific officer, General Perrier, who has successfully
+executed an enterprise truly audacious, the junction of Spain and
+Africa. Stations were installed on four peaks upon the two sides of the
+Mediterranean. For long months they awaited a calm and limpid
+atmosphere. At last was seen the little thread of light which had
+traversed 300 kilometers over the sea. The undertaking had succeeded.
+
+To-day have been conceived projects still more bold. From a mountain
+near Nice will be sent signals to Corsica, not now for geodesic
+determinations, but to measure the velocity of light. The distance is
+only 200 kilometers; but the ray of light is to make the journey there
+and return, after reflection by a mirror installed in Corsica. And it
+should not wander on the way, for it must return exactly to the point of
+departure.
+
+Ever since, the activity of French geodesy has never slackened. We have
+no more such astonishing adventures to tell; but the scientific work
+accomplished is immense. The territory of France beyond the sea, like
+that of the mother country, is covered by triangles measured with
+precision.
+
+We have become more and more exacting and what our fathers admired does
+not satisfy us to-day. But in proportion as we seek more exactitude, the
+difficulties greatly increase; we are surrounded by snares and must be
+on our guard against a thousand unsuspected causes of error. It is
+needful, therefore, to create instruments more and more faultless.
+
+Here again France has not let herself be distanced. Our appliances for
+the measurement of bases and angles leave nothing to desire, and, I may
+also mention the pendulum of Colonel Defforges, which enables us to
+determine gravity with a precision hitherto unknown.
+
+The future of French geodesy is at present in the hands of the
+Geographic Service of the army, successively directed by General Bassot
+and General Berthaut. We can not sufficiently congratulate ourselves
+upon it. For success in geodesy, scientific aptitudes are not enough; it
+is necessary to be capable of standing long fatigues in all sorts of
+climates; the chief must be able to win obedience from his collaborators
+and to make obedient his native auxiliaries. These are military
+qualities. Besides, one knows that, in our army, science has always
+marched shoulder to shoulder with courage.
+
+I add that a military organization assures the indispensable unity of
+action. It would be more difficult to reconcile the rival pretensions of
+scientists jealous of their independence, solicitous of what they call
+their fame, and who yet must work in concert, though separated by great
+distances. Among the geodesists of former times there were often
+discussions, of which some aroused long echoes. The Academy long
+resounded with the quarrel of Bouguer and La Condamine. I do not mean to
+say that soldiers are exempt from passion, but discipline imposes
+silence upon a too sensitive self-esteem.
+
+Several foreign governments have called upon our officers to organize
+their geodesic service: this is proof that the scientific influence of
+France abroad has not declined.
+
+Our hydrographic engineers contribute also to the common achievement a
+glorious contingent. The survey of our coasts, of our colonies, the
+study of the tides, offer them a vast domain of research. Finally I may
+mention the general leveling of France which is carried out by the
+ingenious and precise methods of M. Lallemand.
+
+With such men we are sure of the future. Moreover, work for them will
+not be lacking; our colonial empire opens for them immense expanses illy
+explored. That is not all: the International Geodetic Association has
+recognized the necessity of a new measurement of the arc of Quito,
+determined in days of yore by La Condamine. It is France that has been
+charged with this operation; she had every right to it, since our
+ancestors had made, so to speak, the scientific conquest of the
+Cordilleras. Besides, these rights have not been contested and our
+government has undertaken to exercise them.
+
+Captains Maurain and Lacombe completed a first reconnaissance, and the
+rapidity with which they accomplished their mission, crossing the
+roughest regions and climbing the most precipitous summits, is worthy of
+all praise. It won the admiration of General Alfaro, President of the
+Republic of Ecuador, who called them 'los hombres de hierro,' the men of
+iron.
+
+The final commission then set out under the command of
+Lieutenant-Colonel (then Major) Bourgeois. The results obtained have
+justified the hopes entertained. But our officers have encountered
+unforeseen difficulties due to the climate. More than once, one of them
+has been forced to remain several months at an altitude of 4,000 meters,
+in the clouds and the snow, without seeing anything of the signals he
+had to aim at and which refused to unmask themselves. But thanks to
+their perseverance and courage, there resulted from this only a delay
+and an increase of expense, without the exactitude of the measurements
+suffering therefrom.
+
+
+
+
+GENERAL CONCLUSIONS
+
+
+What I have sought to explain in the preceding pages is how the
+scientist should guide himself in choosing among the innumerable facts
+offered to his curiosity, since indeed the natural limitations of his
+mind compel him to make a choice, even though a choice be always a
+sacrifice. I have expounded it first by general considerations,
+recalling on the one hand the nature of the problem to be solved and on
+the other hand seeking to better comprehend that of the human mind,
+which is the principal instrument of the solution. I then have explained
+it by examples; I have not multiplied them indefinitely; I also have had
+to make a choice, and I have chosen naturally the questions I had
+studied most. Others would doubtless have made a different choice; but
+what difference, because I believe they would have reached the same
+conclusions.
+
+There is a hierarchy of facts; some have no reach; they teach us nothing
+but themselves. The scientist who has ascertained them has learned
+nothing but a fact, and has not become more capable of foreseeing new
+facts. Such facts, it seems, come once, but are not destined to
+reappear.
+
+There are, on the other hand, facts of great yield; each of them teaches
+us a new law. And since a choice must be made, it is to these that the
+scientist should devote himself.
+
+Doubtless this classification is relative and depends upon the weakness
+of our mind. The facts of slight outcome are the complex facts, upon
+which various circumstances may exercise a sensible influence,
+circumstances too numerous and too diverse for us to discern them all.
+But I should rather say that these are the facts we think complex, since
+the intricacy of these circumstances surpasses the range of our mind.
+Doubtless a mind vaster and finer than ours would think differently of
+them. But what matter; we can not use that superior mind, but only our
+own.
+
+The facts of great outcome are those we think simple; may be they really
+are so, because they are influenced only by a small number of
+well-defined circumstances, may be they take on an appearance of
+simplicity because the various circumstances upon which they depend obey
+the laws of chance and so come to mutually compensate. And this is what
+happens most often. And so we have been obliged to examine somewhat more
+closely what chance is.
+
+Facts where the laws of chance apply become easy of access to the
+scientist who would be discouraged before the extraordinary complication
+of the problems where these laws are not applicable. We have seen that
+these considerations apply not only to the physical sciences, but to the
+mathematical sciences. The method of demonstration is not the same for
+the physicist and the mathematician. But the methods of invention are
+very much alike. In both cases they consist in passing up from the fact
+to the law, and in finding the facts capable of leading to a law.
+
+To bring out this point, I have shown the mind of the mathematician at
+work, and under three forms: the mind of the mathematical inventor and
+creator; that of the unconscious geometer who among our far distant
+ancestors, or in the misty years of our infancy, has constructed for us
+our instinctive notion of space; that of the adolescent to whom the
+teachers of secondary education unveil the first principles of the
+science, seeking to give understanding of the fundamental definitions.
+Everywhere we have seen the rôle of intuition and of the spirit of
+generalization without which these three stages of mathematicians, if I
+may so express myself, would be reduced to an equal impotence.
+
+And in the demonstration itself, the logic is not all; the true
+mathematical reasoning is a veritable induction, different in many
+regards from the induction of physics, but proceeding like it from the
+particular to the general. All the efforts that have been made to
+reverse this order and to carry back mathematical induction to the rules
+of logic have eventuated only in failures, illy concealed by the
+employment of a language inaccessible to the uninitiated. The examples I
+have taken from the physical sciences have shown us very different cases
+of facts of great outcome. An experiment of Kaufmann on radium rays
+revolutionizes at the same time mechanics, optics and astronomy. Why?
+Because in proportion as these sciences have developed, we have the
+better recognized the bonds uniting them, and then we have perceived a
+species of general design of the chart of universal science. There are
+facts common to several sciences, which seem the common source of
+streams diverging in all directions and which are comparable to that
+knoll of Saint Gothard whence spring waters which fertilize four
+different valleys.
+
+And then we can make choice of facts with more discernment than our
+predecessors who regarded these valleys as distinct and separated by
+impassable barriers.
+
+It is always simple facts which must be chosen, but among these simple
+facts we must prefer those which are situated upon these sorts of knolls
+of Saint Gothard of which I have just spoken.
+
+And when sciences have no direct bond, they still mutually throw light
+upon one another by analogy. When we studied the laws obeyed by gases we
+knew we had attacked a fact of great outcome; and yet this outcome was
+still estimated beneath its value, since gases are, from a certain point
+of view, the image of the milky way, and those facts which seemed of
+interest only for the physicist, ere long opened new vistas to astronomy
+quite unexpected.
+
+And finally when the geodesist sees it is necessary to move his
+telescope some seconds to see a signal he has set up with great pains,
+this is a very small fact; but this is a fact of great outcome, not only
+because this reveals to him the existence of a small protuberance upon
+the terrestrial globe, that little hump would be by itself of no great
+interest, but because this protuberance gives him information about the
+distribution of matter in the interior of the globe, and through that
+about the past of our planet, about its future, about the laws of its
+development.
+
+
+ * * * * *
+
+
+
+
+INDEX
+
+
+ aberration of light, 315, 496
+
+ Abraham, 311, 490-1, 505-7, 509, 515-6
+
+ absolute motion, 107
+ orientation, 83
+ space, 85, 93, 246, 257, 353
+
+ acceleration, 94, 98, 486, 509
+
+ accidental constant, 112
+ errors, 171, 402
+
+ accommodation of the eyes, 67-8
+
+ action at a distance, 137
+
+ addition, 34
+
+ aim of mathematics, 280
+
+ alchemists, 11
+
+ Alfaro, 543
+
+ algebra, 379
+
+ analogy, 220
+
+ analysis, 218-9, 279
+
+ analysis situs, 53, 239, 381
+
+ analyst, 210, 221
+
+ ancestral experience, 91
+
+ Andrade, 93, 104, 228
+
+ Andrews, 153
+
+ angle sum of triangle, 58
+
+ Anglo-Saxons, 3
+
+ antinomies, 449, 457, 477
+
+ Arago, 540-1
+
+ Aristotle, 205, 292, 460
+
+ arithmetic, 34, 379, 441, 463
+
+ associativity, 35
+
+ assumptions, 451, 453
+
+ astronomy, 81, 289, 315, 512
+
+ Atwood, 446
+
+ axiom, 60, 63, 65, 215
+
+
+ Bacon, 128
+
+ Bartholi, 503
+
+ Bassot, 542
+
+ beauty, 349, 368
+
+ Becquerel, 312
+
+ Beltrami, 56, 58
+
+ Bergson, 321
+
+ Berkeley, 4
+
+ Berthaut, 542
+
+ Bertrand, 156, 190, 211, 395
+
+ Betti, 239
+
+ Biot, 540
+
+ bodies, solid, 72
+
+ Boltzmann, 304
+
+ Bolyai, 56, 201, 203
+
+ Borel, 482
+
+ Bouguer, 537, 542
+
+ Bourgeois, 543
+
+ Boutroux, 390, 464
+
+ Bradley, 496
+
+ Briot, 298
+
+ Brownian movement, 152, 410
+
+ Bucherer, 507
+
+ Burali-Forti, 457-9, 477, 481-2
+
+
+ Caen, 387-8
+
+ Calinon, 228
+
+ canal rays, 491-2
+
+ canals, semicircular, 276
+
+ Cantor, 11, 448-9, 457, 459, 477
+
+ Cantorism, 381, 382, 480, 484
+
+ capillarity, 298
+
+ Carlyle, 128
+
+ Carnot's principle, 143, 151, 300, 303-5, 399
+
+ Cassini, 537
+
+ cathode rays, 487-92
+
+ cells, 217
+
+ center of gravity, 103
+
+ central forces, 297
+
+ Chaldeans, 290
+
+ chance, 395, 408
+
+ change of position, 70
+ state, 70
+
+ chemistry of the stars, 295
+
+ circle-squarers, 11
+
+ Clairaut, 537-8
+
+ Clausius, 119, 123, 143
+
+ color sensation, 252
+
+ Columbus, 228
+
+ commutativity, 35-6
+
+ compensation, 72
+
+ complete induction, 40
+
+ Comte, 294
+
+ Condorcet, 411
+
+ contingence, 340
+
+ continuity, 173
+
+ continuum, 43
+ amorphous, 238
+ mathematical, 46
+ physical, 46, 240
+ tridimensional, 240
+
+ convention, 50, 93, 106, 125, 173, 208, 317, 440, 451
+
+ convergence, 67-8
+
+ coordinates, 244
+
+ Copernicus, 109, 291, 354
+
+ Coulomb, 143, 516
+
+ Couturat, 450, 453, 456, 460, 462-3, 467, 472-6
+
+ creation, mathematical, 383
+
+ creed, 1
+
+ Crémieu, 168-9, 490
+
+ crisis, 303
+
+ Crookes, 195, 488, 527-8
+
+ crude fact, 326, 330
+
+ Curie, 312-3, 318
+
+ current, 186
+
+ curvature, 58-9
+
+ curve, 213, 346
+
+ curves without tangents, 51
+
+ cut, 52, 256
+
+ cyclones, 353
+
+
+ d'Alembert, 538
+
+ Darwin, 518-9
+
+ De Cyon, 276, 427
+
+ Dedekind, 44-5
+
+ Defforges, 542
+
+ definitions, 430, 453
+
+ deformation, 73, 415
+
+ Delage, 277
+
+ Delambre, 539
+
+ Delbeuf, 414
+
+ Descartes, 127
+
+ determinism, 123, 340
+
+ dictionary, 59
+
+ didymium, 333
+
+ dilatation, 76
+
+ dimensions, 53, 68, 78, 241, 256, 426
+
+ direction, 69
+
+ Dirichlet, 213
+
+ dispersion, 141
+
+ displacement, 73, 77, 247, 256
+
+ distance, 59, 292
+
+ distributivity, 36
+
+ Du Bois-Reymond, 50
+
+
+ earth, rotation of, 326, 353
+
+ eclipse, 326
+
+ electricity, 174
+
+ electrified bodies, 117
+
+ electrodynamic attraction, 308
+ induction, 188
+ mass, 311
+
+ electrodynamics, 184, 282
+
+ electromagnetic theory of light, 301
+
+ electrons, 316, 492-4, 505-8, 510, 512-4
+
+ elephant, 217, 436
+
+ ellipse, 215
+
+ Emerson, 203
+
+ empiricism, 86, 271
+
+ Epimenides, 478-9
+
+ equation of Laplace, 283
+
+ Erdély, 203
+
+ errors, accidental, 171, 402
+ law of, 119
+ systematic, 171, 402
+ theory of, 402, 406
+
+ ether, 145, 351
+
+ ethics, 205
+
+ Euclid, 62, 86, 202-3, 213
+
+ Euclidean geometry, 65, 235-6, 337
+
+ Euclid's postulate, 83, 91, 124, 353, 443, 453, 468, 470-1
+
+ experience, 90-1
+
+ experiment, 127, 317, 336, 446
+
+
+ fact, crude, 326, 330
+ in the rough, 327
+ scientific, 326
+
+ facts, 362, 371
+
+ Fahrenheit, 238
+
+ Faraday, 150, 192
+
+ Faye, 536
+
+ Fechner, 46, 52
+
+ Fehr, 383
+
+ finite, 57
+
+ Fitzgerald, 415-6, 500-1, 505
+
+ Fizeau, 146, 149, 309, 498, 504
+
+ Flammarion, 400, 406-7
+
+ flattening of the earth, 353
+
+ force, 72, 98, 444
+ direction of, 445
+ -flow, 284
+
+ forces, central, 297
+ equivalence of, 445
+ magnitude of, 445
+
+ Foucault's pendulum, 85, 109, 353
+
+ four dimensions, 78
+
+ Fourier, 298-9
+
+ Fourier's problem, 317
+ series, 286
+
+ Franklin, 513-4
+
+ Fresnel, 132, 140, 153, 174, 176, 181, 351, 498
+
+ Fuchsian, 387-8
+
+ function, 213
+ continuous, 218, 288
+
+
+ Galileo, 97, 331, 353-4
+
+ gaseous pressure, 141
+
+ gases, theory of, 400, 405, 523
+
+ Gauss, 384-5, 406
+
+ Gay-Lussac, 157
+
+ generalize, 342
+
+ geodesy, 535
+
+ geometer, 83, 210, 438
+
+ geometric space, 66
+
+ geometry, 72, 81, 125, 207, 380, 428, 442, 467
+ Euclidean, 65, 93
+ fourth, 62
+ non-Euclidean, 55
+ projective, 201
+ qualitative, 238
+ rational, 5, 467
+ Riemann's, 57
+ spheric, 59
+
+ Gibbs, 304
+
+ Goldstein, 492
+
+ Gouy, 152, 305, 410
+
+ gravitation, 512
+
+ Greeks, 93, 368
+
+
+ Hadamard, 459
+
+ Halsted, 3, 203, 464, 467
+
+ Hamilton, 115
+
+ helium, 294
+
+ Helmholtz, 56, 115, 118, 141, 190, 196
+
+ Hercules, 449
+
+ Hermite, 211, 220, 222, 285
+
+ Herschel, 528
+
+ Hertz, 102, 145, 194-5, 427, 488, 498, 502, 504, 510
+
+ Hertzian oscillator, 309, 317
+
+ Hilbert, 5, 11, 203, 433, 450-1, 464-8, 471, 475-7, 484
+
+ Himstedt, 195
+
+ Hipparchus, 291
+
+ homogeneity, 74, 423
+
+ homogeneous, 67
+
+ hydrodynamics, 284
+
+ hyperbola, 215
+
+ hypotheses, 6, 15, 127, 133
+
+ hysteresis, 151
+
+
+ identity of spaces, 268
+ of two points, 259
+
+ illusions, optical, 202
+
+ incommensurable numbers, 44
+
+ induction, complete, 40, 452-3, 467-8
+ electromagnetic, 188
+ mathematical, 40, 220
+ principle of, 481
+
+ inertia, 93, 486, 489, 507
+
+ infinite, 448
+
+ infinitesimals, 50
+
+ inquisitor, 331
+
+ integration, 139
+
+ interpolation, 131
+
+ intuition, 210, 213, 215
+
+ invariant, 333
+
+ Ionians, 127
+
+ ions, 152
+
+ irrational number, 44
+
+ irreversible phenomena, 151
+
+ isotropic, 67
+
+
+ Japanese mice, 277, 427
+
+ Jevons, 451
+
+ John Lackland, 128
+
+ Jules Verne, 111, 536
+
+ Jupiter, 131, 157, 231, 289
+
+
+ Kant, 16, 64, 202-3, 450-1, 471
+
+ Kauffman, 311, 490-1, 495, 506-7, 522, 545
+
+ Kazan, 203
+
+ Kelvin, 145, 523-4, 526-7
+
+ Kepler, 120, 133, 153, 282, 291-2
+
+ Kepler's laws, 136, 516
+
+ kinematics, 337
+
+ kinetic energy, 116
+ theory of gases, 141
+
+ Kirchhoff, 98-9, 103-5
+
+ Klein, 60, 211, 287
+
+ knowledge, 201
+
+ König, 144, 477
+
+ Kovalevski, 212, 286
+
+ Kronecker, 44
+
+
+ Lacombe, 543
+
+ La Condamine, 535, 537-8, 542-3
+
+ Lagrange, 98, 151, 179
+
+ Laisant, 383
+
+ Lallamand, 543
+
+ Langevin, 509
+
+ Laplace, 298, 398, 514-5, 518, 522, 538
+
+ Laplace's equation, 283, 287
+
+ Larmor, 145, 150
+
+ Lavoisier's principle, 301, 310, 312
+
+ law, 207, 291, 395
+
+ Leibnitz, 32, 450, 471
+
+ Le Roy, 28, 321-6, 332, 335, 337, 347-8, 354, 468
+
+ Lesage, 517-21
+
+ Liard, 440
+
+ Lie, 62-3, 212
+
+ light sensations, 252
+ theory of, 351
+ velocity of, 232, 312
+
+ Lindemann, 508
+
+ line, 203, 243
+
+ linkages, 144
+
+ Lippmann, 196
+
+ Lobachevski, 29, 56, 60, 62, 83, 86, 203
+
+ Lobachevski's space, 239
+
+ local time, 306-7, 499
+
+ logic, 214, 435, 448, 460-2, 464
+
+ logistic, 457, 472-4
+
+ logisticians, 472
+
+ Lorentz, 147, 149, 196-7, 306, 308, 311, 315, 415-6, 492, 498-502,
+ 504-9, 512, 514-6, 521
+
+ Lotze, 264
+
+ luck, 399
+
+ Lumen, 407-8
+
+
+ MacCullagh, 150
+
+ Mach, 375
+
+ Mach-Delage, 276
+
+ magnetism, 149
+
+ magnitude, 49
+
+ Mariotte's law, 120, 132, 157, 342, 524
+
+ Maros, 203
+
+ mass, 98, 312, 446, 486, 489, 494, 515
+
+ mathematical analysis, 218
+ continuum, 46
+ creation, 383
+ induction, 40, 220
+ physics, 136, 297, 319
+
+ mathematics, 369, 448
+
+ matter, 492
+
+ Maupertuis, 535, 537-8
+
+ Maurain, 543
+
+ Maxwell, 140, 152, 175, 177, 181, 193, 282-3, 298, 301, 304-5, 351,
+ 503, 524-5
+
+ Maxwell-Bartholi, 309, 503-4, 519, 521
+
+ Mayer, 119, 123, 300, 312, 318
+
+ measurement, 49
+
+ Méchain, 539-40
+
+ mechanical explanation, 177
+ mass, 312
+
+ mechanics, 92, 444, 486, 496, 512
+ anthropomorphic, 103
+ celestial, 279
+ statistical, 304
+
+ Méray, 211
+
+ metaphysician, 221
+
+ meteorology, 398
+
+ mice, 277
+
+ Michelson, 306, 309, 311, 316, 498, 500-1
+
+ milky way, 523-30
+
+ Mill, Stuart, 60-1, 453-4
+
+ Monist, 4, 89, 464
+
+ moons of Jupiter, 233
+
+ Morley, 309
+
+ motion of liquids, 283
+ of moon, 28
+ of planets, 341
+ relative, 107, 487
+ without deformation, 236
+
+ multiplication, 36
+
+ muscular sensations, 69
+
+
+ Nagaoka, 317
+
+ nature, 127
+
+ navigation, 289
+
+ neodymium, 333
+
+ neomonics, 283
+
+ Neumann, 181
+
+ Newton, 85, 96, 98, 109, 153, 291, 370, 486, 516, 536, 538
+
+ Newton's argument, 108, 334, 343
+ law, 111, 118, 132, 136, 149, 157, 233, 282, 292, 512, 514-5,
+ 518, 525
+ principle, 146, 300, 308-9, 312
+
+ no-class theory, 478
+
+ nominalism, 28, 125, 321, 333, 335
+
+ non-Euclidean geometry, 55, 59, 388
+ language, 127
+ space, 55, 235, 237
+ straight, 236, 470
+ world, 75
+
+ number, 31
+ big, 88
+ imaginary, 283
+ incommensurable, 44
+ transfinite, 448-9
+ whole, 44, 469
+
+
+ objectivity, 209, 347, 349, 408
+
+ optical illusions, 202
+
+ optics, 174, 496
+
+ orbit of Saturn, 341
+
+ order, 385
+
+ orientation, 83
+
+ osmotic, 141
+
+
+ Padoa, 463
+
+ Panthéon, 414
+
+ parallax, 470
+
+ parallels, 56, 443
+
+ Paris time, 233
+
+ parry, 419-22, 427
+
+ partition, 45
+
+ pasigraphy, 456-7
+
+ Pasteur, 128
+
+ Peano, 450, 456-9, 463, 472
+
+ Pender, 490
+
+ pendulum, 224
+
+ Perrier, 541
+
+ Perrin, 195
+
+ phosphorus, 333, 468, 470-1
+
+ physical continuum, 46
+
+ physics, 127, 140, 144, 279, 297
+
+ physics of central forces, 297
+ of the principles, 299
+
+ Pieri, 11, 203
+
+ Plato, 292
+
+ Poincaré, 473
+
+ point, 89, 244
+
+ Poncelet, 215
+
+ postulates, 382
+
+ potential energy, 116
+
+ praseodymium, 333
+
+ principle, 125, 299
+ Carnot's, 143, 151, 300, 303-5, 399
+ Clausius', 119, 123, 143
+ Hamilton's, 115
+ Lavoisier's, 300, 310
+ Mayer's, 119, 121, 123, 300, 312, 318
+ Newton's, 146, 300, 308-9, 312
+ of action and reaction, 300, 487, 502
+ of conservation of energy, 300
+ of degradation of energy, 300
+ of inertia, 93, 486, 507
+ of least action, 118, 300
+ of relativity, 300, 305, 498, 505
+
+ Prony, 445
+
+ psychologist, 383
+
+ Ptolemy, 110, 291, 353-4
+
+ Pythagoras, 292
+
+
+ quadrature of the circle, 161
+
+ qualitative geometry, 238
+ space, 207
+ time, 224
+
+ quaternions, 282
+
+
+ radiometer, 503
+
+ radium, 312, 318, 486-7
+
+ Rados, 201
+
+ Ramsay, 313
+
+ rational geometry, 5, 467
+
+ reaction, 502
+
+ reality, 217, 340, 349
+
+ Réaumur, 238
+
+ recurrence, 37
+
+ Regnault, 170
+
+ relativity, 83, 305, 417, 423, 498, 505
+
+ Richard, 477-8, 480-1
+
+ Riemann, 56, 62, 145, 212, 239, 243, 381, 432
+ surface, 211, 287
+
+ Roemer, 233
+
+ Röntgen, 511, 520
+
+ rotation of earth, 225, 331, 353
+
+ roulette, 403
+
+ Rowland, 194-7, 305, 489
+
+ Royce, 202
+
+ Russell, 201, 450, 460-2, 464-7, 471-4, 477-82, 484-5
+
+
+ St. Louis exposition, 208, 320
+
+ Sarcey, 442
+
+ Saturn, 231, 317
+
+ Schiller, 202
+
+ Schliemann, 19
+
+ science, 205, 321, 323, 340, 354
+
+ Science and Hypothesis, 205-7, 220, 240, 246-7, 319, 353, 452
+
+ semicircular canals, 276
+
+ series, development in, 287
+ Fourier's, 286
+
+ Sirius, 226, 229
+
+ solid bodies, 72
+
+ space, 55, 66, 89, 235, 256
+ absolute, 85, 93
+ amorphous, 417
+ Bolyai, 56
+ Euclidean, 65
+ geometric, 66
+ Lobachevski's, 239
+ motor, 69
+ non-Euclidean, 55, 235, 237
+ of four dimensions, 78
+ perceptual, 66, 69
+ tactile, 68, 264
+ visual, 67, 252
+
+ spectra, 316
+
+ spectroscope, 294
+
+ Spencer, 9
+
+ sponge, 219
+
+ Stallo, 10
+
+ stars, 292
+
+ statistical mechanics, 304
+
+ straight, 62, 82, 236, 433, 450, 470
+
+ Stratonoff, 531
+
+ surfaces, 58
+
+ systematic errors, 171
+
+
+ tactile space, 68, 264
+
+ Tait, 98
+
+ tangent, 51
+
+ Tannery, 43
+
+ teaching, 430, 437
+
+ thermodynamics, 115, 119
+
+ Thomson, 98, 488
+
+ thread, 104
+
+ time, 223
+ equality, 225
+ local, 306, 307
+ measure of, 223-4
+
+ Tisserand, 515-6
+
+ Tolstoi, 354, 362, 368
+
+ Tommasina, 519
+
+ Transylvania, 203
+
+ triangle, 58
+ angle sum of, 58
+
+ truth, 205
+
+ Tycho Brahe, 133, 153, 228
+
+
+ unity of nature, 130
+
+ universal invariant, 333
+
+ Uriel, 203
+
+
+ van der Waals, 153
+
+ Vauban, 210
+
+ Veblen, 203
+
+ velocity of light, 232, 312
+
+ Venus of Milo, 201
+
+ verification, 33
+
+ Virchow, 21
+
+ visual impressions, 252
+ space, 67, 252
+
+ Volga, 203
+
+ Voltaire, 537-8
+
+
+ Weber, 117, 515-6
+
+ Weierstrass, 11, 212, 432
+
+ Whitehead, 472, 481-2
+
+ whole numbers, 44
+
+ Wiechert, 145, 488
+
+
+ x-rays, 152, 511, 520
+
+
+ Zeeman effect, 152, 196, 317, 494
+
+ Zeno, 382
+
+ Zermelo, 477, 482-3
+
+ zigzag theory, 478
+
+ zodiac, 398, 404
+
+
+ * * * * *
+
+
+Transcriber's Note: The Greek alphabets are represented within square
+brackets. For example, [alpha] stands for first Greek alphabet alpha.
+Square root of a number is represented using the symbol (sqrt). That
+is to say, sqrt(25) stands for square root of 25. The superscript is
+shown with carat (^) symbol, example, 10^{5} stands for 5th power of
+10. Similarly, subscript is represented by underscore (_) symbol. For
+instance, n_{3} stands for letter n with subscript 3.
+
+
+
+
+
+End of the Project Gutenberg EBook of The Foundations of Science: Science
+and Hypothesis, The Value of Science, Science and Method, by Henri Poincaré
+
+*** END OF THIS PROJECT GUTENBERG EBOOK THE FOUNDATIONS OF SCIENCE: ***
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+ The Project Gutenberg eBook of The Foundations of Science: Science and Hypothesis, The Value of Science, Science and Method, by Henri Poincaré.
+ </title>
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+
+The Project Gutenberg EBook of The Foundations of Science: Science and
+Hypothesis, The Value of Science, Science and Method, by Henri Poincaré
+
+This eBook is for the use of anyone anywhere at no cost and with
+almost no restrictions whatsoever. You may copy it, give it away or
+re-use it under the terms of the Project Gutenberg License included
+with this eBook or online at www.gutenberg.org
+
+
+Title: The Foundations of Science: Science and Hypothesis, The Value of Science, Science and Method
+
+Author: Henri Poincaré
+
+Translator: George Bruce Halsted
+
+Release Date: May 17, 2012 [EBook #39713]
+
+Language: English
+
+Character set encoding: ISO-8859-1
+
+*** START OF THIS PROJECT GUTENBERG EBOOK THE FOUNDATIONS OF SCIENCE: ***
+
+
+
+
+Produced by Bryan Ness and the Online Distributed
+Proofreading Team at http://www.pgdp.net (This book was
+produced from scanned images of public domain material
+from the Google Print project.)
+
+
+
+
+
+
+</pre>
+
+
+
+<p class="center">SCIENCE AND EDUCATION<br /><br />
+
+<small>A SERIES OF VOLUMES FOR THE PROMOTION OF<br />
+SCIENTIFIC RESEARCH AND EDUCATIONAL PROGRESS<br /><br />
+
+<span class="smcap">Edited by</span> J. McKEEN CATTELL</small></p>
+
+<p>&nbsp;</p>
+<p>&nbsp;</p>
+
+<p class="center">VOLUME I&mdash;THE FOUNDATIONS OF SCIENCE</p>
+<p>&nbsp;</p>
+<p>&nbsp;</p>
+
+
+<div class="bbox">
+<p class="center">UNDER THE SAME EDITORSHIP</p>
+
+<hr class="half" />
+<p class="negidt">SCIENCE AND EDUCATION. A series of volumes for
+the promotion of scientific research and educational
+progress.</p>
+
+<p class="negidt"> &nbsp; &nbsp; Volume I. The Foundations of Science. By <span class="smcap">H.
+Poincar&eacute;</span>. Containing the authorised English
+translation by George Bruce Halsted of "Science
+and Hypothesis," "The Value of Science," and
+"Science and Method."</p>
+<p class="negidt"> &nbsp; &nbsp; 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. Councilman
+Theobald Smith, G. N. Stewart, C. M. Jackson,
+E. P. Lyon, James B. Herrick, John M. Dodson,
+C. R. Bardeen, W. Ophuls, S. J. Meltzer, James
+Ewing, W. W. Keen, Henry H. Donaldson, Christian
+A. Herter, and Henry P. Bowditch.</p>
+
+<p class="negidt"> &nbsp; &nbsp; Volume III. University Control. By <span class="smcap">J. McKeen
+Cattell</span> and other authors.</p>
+
+<p class="negidt">AMERICAN MEN OF SCIENCE. A Biographical
+Directory.</p>
+
+<p class="negidt">SCIENCE. A weekly journal devoted to the advancement
+of science. The official organ of the American Association
+for the Advancement of Science.</p>
+
+<p class="negidt">THE POPULAR SCIENCE MONTHLY. A monthly
+magazine devoted to the diffusion of science.</p>
+
+<p class="negidt">THE AMERICAN NATURALIST. A monthly journal
+devoted to the biological sciences, with special reference
+to the factors of evolution.</p>
+
+<hr class="half" />
+<p class="center"><big>THE SCIENCE PRESS</big><br />
+
+NEW YORK &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; GARRISON, N. Y.</p>
+</div>
+
+
+<p>&nbsp;</p>
+
+<h1>THE FOUNDATIONS<br />
+OF SCIENCE</h1>
+
+<p class="center">SCIENCE AND HYPOTHESIS<br />
+THE VALUE OF SCIENCE<br />
+SCIENCE AND METHOD</p>
+
+<p>&nbsp;</p>
+
+<h3><small>BY</small><br />
+H. POINCAR&Eacute;</h3>
+
+<p>&nbsp;</p>
+
+<p class="center"><small>AUTHORIZED TRANSLATION BY</small><br />
+GEORGE BRUCE HALSTED</p>
+
+<p>&nbsp;</p>
+
+
+<p class="center"><small>WITH A SPECIAL PREFACE BY POINCAR&Eacute;, AND AN INTRODUCTION<br />
+BY JOSIAH ROYCE, HARVARD UNIVERSITY</small></p>
+
+<p>&nbsp;</p>
+
+<p class="center">THE SCIENCE PRESS<br />
+<small>NEW YORK AND GARRISON, N. Y.</small><br />
+1913</p>
+
+<p>&nbsp;</p>
+<p>&nbsp;</p>
+
+
+<p class="center">Copyright, 1913<br />
+
+<span class="smcap">BY The Science Press</span></p>
+
+<p>&nbsp;</p>
+<p>&nbsp;</p>
+
+
+
+<p class="center"><small>PRESS OF<br />
+THE NEW ERA PRINTING COMPANY<br />
+LANCASTER, PA.</small></p>
+
+
+<hr style="width: 100%;" />
+<p><span class='pagenum'><a name="Page_v" id="Page_v">[Pg v]</a></span></p>
+<h2>CONTENTS</h2>
+
+
+<div class='center'>
+<table border="0" cellpadding="4" cellspacing="0" summary="">
+<tr><td align='left'></td><td align='right'><small>PAGE</small></td></tr>
+<tr><td align='left'>Henri Poincar&eacute;</td><td align='right'><a href="#Page_ix">ix</a></td></tr>
+<tr><td align='left'>Author's Preface to the Translation</td><td align='right'><a href="#Page_3">3</a></td></tr>
+<tr><td></td></tr>
+<tr><td align='center' colspan='2'><a href="#SCIENCE_AND_HYPOTHESIS">SCIENCE AND HYPOTHESIS</a></td></tr>
+<tr><td align='left'>Introduction by Royce</td><td align='right'><a href="#Page_9">9</a></td></tr>
+<tr><td align='left'>Introduction</td><td align='right'><a href="#Page_27">27</a></td></tr>
+<tr><td></td></tr>
+<tr><td align='center' colspan='2'><span class="smcap">Part I.</span> <i>Number and Magnitude</i></td></tr>
+<tr><td align='left'><span class="smcap">Chapter I.</span>&mdash;On the Nature of Mathematical Reasoning</td><td align='right'><a href="#Page_31">31</a></td></tr>
+<tr><td class='tdl'>Syllogistic Deduction</td><td align='right'><a href="#Page_31">31</a></td></tr>
+<tr><td class='tdl'>Verification and Proof</td><td align='right'><a href="#Page_32">32</a></td></tr>
+<tr><td class='tdl'>Elements of Arithmetic</td><td align='right'><a href="#Page_33">33</a></td></tr>
+<tr><td class='tdl'>Reasoning by Recurrence</td><td align='right'><a href="#Page_37">37</a></td></tr>
+<tr><td class='tdl'>Induction</td><td align='right'><a href="#Page_40">40</a></td></tr>
+<tr><td class='tdl'>Mathematical Construction</td><td align='right'><a href="#Page_41">41</a></td></tr>
+<tr><td align='left'><span class="smcap">Chapter II.</span>&mdash;Mathematical Magnitude and Experience</td><td align='right'><a href="#Page_43">43</a></td></tr>
+<tr><td class='tdl'>Definition of Incommensurables</td><td align='right'><a href="#Page_44">44</a></td></tr>
+<tr><td class='tdl'>The Physical Continuum</td><td align='right'><a href="#Page_46">46</a></td></tr>
+<tr><td class='tdl'>Creation of the Mathematical Continuum</td><td align='right'><a href="#Page_46">46</a></td></tr>
+<tr><td class='tdl'>Measurable Magnitude</td><td align='right'><a href="#Page_49">49</a></td></tr>
+<tr><td class='tdl'>Various Remarks (Curves without Tangents)</td><td align='right'><a href="#Page_50">50</a></td></tr>
+<tr><td class='tdl'>The Physical Continuum of Several Dimensions</td><td align='right'><a href="#Page_52">52</a></td></tr>
+<tr><td class='tdl'>The Mathematical Continuum of Several Dimensions</td><td align='right'><a href="#Page_53">53</a></td></tr>
+<tr><td></td></tr>
+<tr><td align='center' colspan='2'><span class="smcap">Part II.</span> <i>Space</i></td></tr>
+<tr><td align='left'><span class="smcap">Chapter III.</span>&mdash;The Non-Euclidean Geometries</td><td align='right'><a href="#Page_55">55</a></td></tr>
+<tr><td class='tdl'>The Bolyai-Lobachevski Geometry</td><td align='right'><a href="#Page_56">56</a></td></tr>
+<tr><td class='tdl'>Riemann's Geometry</td><td align='right'><a href="#Page_57">57</a></td></tr>
+<tr><td class='tdl'>The Surfaces of Constant Curvature</td><td align='right'><a href="#Page_58">58</a></td></tr>
+<tr><td class='tdl'>Interpretation of Non-Euclidean Geometries</td><td align='right'><a href="#Page_59">59</a></td></tr>
+<tr><td class='tdl'>The Implicit Axioms</td><td align='right'><a href="#Page_60">60</a></td></tr>
+<tr><td class='tdl'>The Fourth Geometry</td><td align='right'><a href="#Page_62">62</a></td></tr>
+<tr><td class='tdl'>Lie's Theorem</td><td align='right'><a href="#Page_62">62</a></td></tr>
+<tr><td class='tdl'>Riemann's Geometries</td><td align='right'><a href="#Page_63">63</a></td></tr>
+<tr><td class='tdl'>On the Nature of Axioms</td><td align='right'><a href="#Page_63">63</a></td></tr>
+<tr><td align='left'><span class="smcap">Chapter IV.</span>&mdash;Space and Geometry</td><td align='right'><a href="#Page_66">66</a></td></tr>
+<tr><td class='tdl'>Geometric Space and Perceptual Space</td><td align='right'><a href="#Page_66">66</a></td></tr>
+<tr><td class='tdl'>Visual Space</td><td align='right'><a href="#Page_67">67</a></td></tr>
+<tr><td class='tdl'>Tactile Space and Motor Space</td><td align='right'><a href="#Page_68">68</a></td></tr>
+<tr><td class='tdl'>Characteristics of Perceptual Space</td><td align='right'><a href="#Page_69">69</a></td></tr>
+<tr><td class='tdl'>Change of State and Change of Position</td><td align='right'><a href="#Page_70">70</a></td></tr>
+<tr><td class='tdl'>Conditions of Compensation</td><td align='right'><a href="#Page_72">72</a></td></tr>
+<tr><td class='tdl'><span class='pagenum'><a name="Page_vi" id="Page_vi">[Pg vi]</a></span>Solid Bodies and Geometry</td><td align='right'><a href="#Page_72">72</a></td></tr>
+<tr><td class='tdl'>Law of Homogeneity</td><td align='right'><a href="#Page_74">74</a></td></tr>
+<tr><td class='tdl'>The Non-Euclidean World</td><td align='right'><a href="#Page_75">75</a></td></tr>
+<tr><td class='tdl'>The World of Four Dimensions</td><td align='right'><a href="#Page_78">78</a></td></tr>
+<tr><td class='tdl'>Conclusions</td><td align='right'><a href="#Page_79">79</a></td></tr>
+<tr><td align='left'><span class="smcap">Chapter V.</span>&mdash;Experience and Geometry</td><td align='right'><a href="#Page_81">81</a></td></tr>
+<tr><td class='tdl'>Geometry and Astronomy</td><td align='right'><a href="#Page_81">81</a></td></tr>
+<tr><td class='tdl'>The Law of Relativity</td><td align='right'><a href="#Page_83">83</a></td></tr>
+<tr><td class='tdl'>Bearing of Experiments</td><td align='right'><a href="#Page_86">86</a></td></tr>
+<tr><td class='tdl'>Supplement (What is a Point?)</td><td align='right'><a href="#Page_89">89</a></td></tr>
+<tr><td class='tdl'>Ancestral Experience</td><td align='right'><a href="#Page_91">91</a></td></tr>
+<tr><td></td></tr>
+<tr><td align='center' colspan='2'><span class="smcap">Part III.</span> <i>Force</i></td></tr>
+<tr><td align='left'><span class="smcap">Chapter VI.</span>&mdash;The Classic Mechanics</td><td align='right'><a href="#Page_92">92</a></td></tr>
+<tr><td class='tdl'>The Principle of Inertia</td><td align='right'><a href="#Page_93">93</a></td></tr>
+<tr><td class='tdl'>The Law of Acceleration</td><td align='right'><a href="#Page_97">97</a></td></tr>
+<tr><td class='tdl'>Anthropomorphic Mechanics</td><td align='right'><a href="#Page_103">103</a></td></tr>
+<tr><td class='tdl'>The School of the Thread</td><td align='right'><a href="#Page_104">104</a></td></tr>
+<tr><td align='left'><span class="smcap">Chapter VII.</span>&mdash;Relative Motion and Absolute Motion</td><td align='right'><a href="#Page_107">107</a></td></tr>
+<tr><td class='tdl'>The Principle of Relative Motion</td><td align='right'><a href="#Page_107">107</a></td></tr>
+<tr><td class='tdl'>Newton's Argument</td><td align='right'><a href="#Page_108">108</a></td></tr>
+<tr><td align='left'><span class="smcap">Chapter VIII.</span>&mdash;Energy and Thermodynamics</td><td align='right'><a href="#Page_115">115</a></td></tr>
+<tr><td class='tdl'>Energetics</td><td align='right'><a href="#Page_115">115</a></td></tr>
+<tr><td class='tdl'>Thermodynamics</td><td align='right'><a href="#Page_119">119</a></td></tr>
+<tr><td class='tdl'>General Conclusions on Part III</td><td align='right'><a href="#Page_123">123</a></td></tr>
+<tr><td></td></tr>
+<tr><td align='center' colspan='2'><span class="smcap">Part IV.</span> <i>Nature</i></td></tr>
+<tr><td align='left'><span class="smcap">Chapter IX.</span>&mdash;Hypotheses in Physics</td><td align='right'><a href="#Page_127">127</a></td></tr>
+<tr><td class='tdl'>The R&ocirc;le of Experiment and Generalization</td><td align='right'><a href="#Page_127">127</a></td></tr>
+<tr><td class='tdl'>The Unity of Nature</td><td align='right'><a href="#Page_130">130</a></td></tr>
+<tr><td class='tdl'>The R&ocirc;le of Hypothesis</td><td align='right'><a href="#Page_133">133</a></td></tr>
+<tr><td class='tdl'>Origin of Mathematical Physics</td><td align='right'><a href="#Page_136">136</a></td></tr>
+<tr><td align='left'><span class="smcap">Chapter X.</span>&mdash;The Theories of Modern Physics</td><td align='right'><a href="#Page_140">140</a></td></tr>
+<tr><td class='tdl'>Meaning of Physical Theories</td><td align='right'><a href="#Page_140">140</a></td></tr>
+<tr><td class='tdl'>Physics and Mechanism</td><td align='right'><a href="#Page_144">144</a></td></tr>
+<tr><td class='tdl'>Present State of the Science</td><td align='right'><a href="#Page_148">148</a></td></tr>
+<tr><td align='left'><span class="smcap">Chapter XI.</span>&mdash;The Calculus of Probabilities</td><td align='right'><a href="#Page_155">155</a></td></tr>
+<tr><td class='tdl'>Classification of the Problems of Probability</td><td align='right'><a href="#Page_158">158</a></td></tr>
+<tr><td class='tdl'>Probability in Mathematics</td><td align='right'><a href="#Page_161">161</a></td></tr>
+<tr><td class='tdl'>Probability in the Physical Sciences</td><td align='right'><a href="#Page_164">164</a></td></tr>
+<tr><td class='tdl'>Rouge et noir</td><td align='right'><a href="#Page_167">167</a></td></tr>
+<tr><td class='tdl'>The Probability of Causes</td><td align='right'><a href="#Page_169">169</a></td></tr>
+<tr><td class='tdl'>The Theory of Errors</td><td align='right'><a href="#Page_170">170</a></td></tr>
+<tr><td class='tdl'>Conclusions</td><td align='right'><a href="#Page_172">172</a></td></tr>
+<tr><td align='left'><span class="smcap">Chapter XII.</span>&mdash;Optics and Electricity</td><td align='right'><a href="#Page_174">174</a></td></tr>
+<tr><td class='tdl'>Fresnel's Theory</td><td align='right'><a href="#Page_174">174</a></td></tr>
+<tr><td class='tdl'>Maxwell's Theory</td><td align='right'><a href="#Page_175">175</a></td></tr>
+<tr><td class='tdl'>The Mechanical Explanation of Physical Phenomena</td><td align='right'><a href="#Page_177">177</a></td></tr>
+<tr><td align='left'><span class='pagenum'><a name="Page_vii" id="Page_vii">[Pg vii]</a></span><span class="smcap">Chapter XIII.</span>&mdash;Electrodynamics</td><td align='right'><a href="#Page_184">184</a></td></tr>
+<tr><td class='tdl'>Amp&egrave;re's Theory</td><td align='right'><a href="#Page_184">184</a></td></tr>
+<tr><td class='tdl'>Closed Currents</td><td align='right'><a href="#Page_185">185</a></td></tr>
+<tr><td class='tdl'>Action of a Closed Current on a Portion of Current</td><td align='right'><a href="#Page_186">186</a></td></tr>
+<tr><td class='tdl'>Continuous Rotations</td><td align='right'><a href="#Page_187">187</a></td></tr>
+<tr><td class='tdl'>Mutual Action of Two Open Currents</td><td align='right'><a href="#Page_189">189</a></td></tr>
+<tr><td class='tdl'>Induction</td><td align='right'><a href="#Page_190">190</a></td></tr>
+<tr><td class='tdl'>Theory of Helmholtz</td><td align='right'><a href="#Page_191">191</a></td></tr>
+<tr><td class='tdl'>Difficulties Raised by these Theories</td><td align='right'><a href="#Page_193">193</a></td></tr>
+<tr><td class='tdl'>Maxwell's Theory</td><td align='right'><a href="#Page_193">193</a></td></tr>
+<tr><td class='tdl'>Rowland's Experiment</td><td align='right'><a href="#Page_194">194</a></td></tr>
+<tr><td class='tdl'>The Theory of Lorentz</td><td align='right'><a href="#Page_196">196</a></td></tr>
+<tr><td></td></tr>
+<tr><td align='center' colspan='2'><a href="#THE_VALUE_OF_SCIENCE">THE VALUE OF SCIENCE</a></td></tr>
+<tr><td align='left'>Translator's Introduction</td><td align='right'><a href="#Page_201">201</a></td></tr>
+<tr><td align='left'>Does the Scientist Create Science?</td><td align='right'><a href="#Page_201">201</a></td></tr>
+<tr><td align='left'>The Mind Dispelling Optical Illusions</td><td align='right'><a href="#Page_202">202</a></td></tr>
+<tr><td align='left'>Euclid not Necessary</td><td align='right'><a href="#Page_202">202</a></td></tr>
+<tr><td align='left'>Without Hypotheses, no Science</td><td align='right'><a href="#Page_203">203</a></td></tr>
+<tr><td align='left'>What Outcome?</td><td align='right'><a href="#Page_203">203</a></td></tr>
+<tr><td align='left'>Introduction</td><td align='right'><a href="#Page_205">205</a></td></tr>
+<tr><td></td></tr>
+<tr><td align='center' colspan='2'><span class="smcap">Part I.</span> <i>The Mathematical Sciences</i></td></tr>
+<tr><td align='left'><span class="smcap">Chapter I.</span>&mdash;Intuition and Logic in Mathematics</td><td align='right'><a href="#Page_210">210</a></td></tr>
+<tr><td align='left'><span class="smcap">Chapter II.</span>&mdash;The Measure of Time</td><td align='right'><a href="#Page_223">223</a></td></tr>
+<tr><td align='left'><span class="smcap">Chapter III.</span>&mdash;The Notion of Space</td><td align='right'><a href="#Page_235">235</a></td></tr>
+<tr><td class='tdl'>Qualitative Geometry</td><td align='right'><a href="#Page_238">238</a></td></tr>
+<tr><td class='tdl'>The Physical Continuum of Several Dimensions</td><td align='right'><a href="#Page_240">240</a></td></tr>
+<tr><td class='tdl'>The Notion of Point</td><td align='right'><a href="#Page_244">244</a></td></tr>
+<tr><td class='tdl'>The Notion of Displacement</td><td align='right'><a href="#Page_247">247</a></td></tr>
+<tr><td class='tdl'>Visual Space</td><td align='right'><a href="#Page_252">252</a></td></tr>
+<tr><td align='left'><span class="smcap">Chapter IV.</span>&mdash;Space and its Three Dimensions</td><td align='right'><a href="#Page_256">256</a></td></tr>
+<tr><td class='tdl'>The Group of Displacements</td><td align='right'><a href="#Page_256">256</a></td></tr>
+<tr><td class='tdl'>Identity of Two Points</td><td align='right'><a href="#Page_259">259</a></td></tr>
+<tr><td class='tdl'>Tactile Space</td><td align='right'><a href="#Page_264">264</a></td></tr>
+<tr><td class='tdl'>Identity of the Different Spaces</td><td align='right'><a href="#Page_268">268</a></td></tr>
+<tr><td class='tdl'>Space and Empiricism</td><td align='right'><a href="#Page_271">271</a></td></tr>
+<tr><td class='tdl'>R&ocirc;le of the Semicircular Canals</td><td align='right'><a href="#Page_276">276</a></td></tr>
+<tr><td></td></tr>
+<tr><td align='center' colspan='2'><span class="smcap">Part II.</span> <i>The Physical Sciences</i></td></tr>
+<tr><td align='left'><span class="smcap">Chapter V.</span>&mdash;Analysis and Physics</td><td align='right'><a href="#Page_279">279</a></td></tr>
+<tr><td align='left'><span class="smcap">Chapter VI.</span>&mdash;Astronomy</td><td align='right'><a href="#Page_289">289</a></td></tr>
+<tr><td align='left'><span class="smcap">Chapter VII.</span>&mdash;The History of Mathematical Physics</td><td align='right'><a href="#Page_297">297</a></td></tr>
+<tr><td class='tdl'>The Physics of Central Forces</td><td align='right'><a href="#Page_297">297</a></td></tr>
+<tr><td class='tdl'>The Physics of the Principles</td><td align='right'><a href="#Page_299">299</a></td></tr>
+<tr><td align='left'><span class="smcap">Chapter VIII.</span>&mdash;The Present Crisis in Physics</td><td align='right'><a href="#Page_303">303</a></td></tr>
+<tr><td class='tdl'>The New Crisis</td><td align='right'><a href="#Page_303">303</a></td></tr>
+<tr><td class='tdl'>Carnot's Principle</td><td align='right'><a href="#Page_303">303</a></td></tr>
+<tr><td class='tdl'><span class='pagenum'><a name="Page_viii" id="Page_viii">[Pg viii]</a></span>The Principle of Relativity</td><td align='right'><a href="#Page_305">305</a></td></tr>
+<tr><td class='tdl'>Newton's Principle</td><td align='right'><a href="#Page_308">308</a></td></tr>
+<tr><td class='tdl'>Lavoisier's Principle</td><td align='right'><a href="#Page_310">310</a></td></tr>
+<tr><td class='tdl'>Mayer's Principle</td><td align='right'><a href="#Page_312">312</a></td></tr>
+<tr><td align='left'><span class="smcap">Chapter IX.</span>&mdash;The Future of Mathematical Physics</td><td align='right'><a href="#Page_314">314</a></td></tr>
+<tr><td class='tdl'>The Principles and Experiment</td><td align='right'><a href="#Page_314">314</a></td></tr>
+<tr><td class='tdl'>The R&ocirc;le of the Analyst</td><td align='right'><a href="#Page_314">314</a></td></tr>
+<tr><td class='tdl'>Aberration and Astronomy</td><td align='right'><a href="#Page_315">315</a></td></tr>
+<tr><td class='tdl'>Electrons and Spectra</td><td align='right'><a href="#Page_316">316</a></td></tr>
+<tr><td class='tdl'>Conventions preceding Experiment</td><td align='right'><a href="#Page_317">317</a></td></tr>
+<tr><td class='tdl'>Future Mathematical Physics</td><td align='right'><a href="#Page_319">319</a></td></tr>
+<tr><td></td></tr>
+<tr><td align='center' colspan='2'><span class="smcap">Part III.</span> <i>The Objective Value of Science</i></td></tr>
+<tr><td align='left'><span class="smcap">Chapter X.</span>&mdash;Is Science Artificial?</td><td align='right'><a href="#Page_321">321</a></td></tr>
+<tr><td class='tdl'>The Philosophy of LeRoy</td><td align='right'><a href="#Page_321">321</a></td></tr>
+<tr><td class='tdl'>Science, Rule of Action</td><td align='right'><a href="#Page_323">323</a></td></tr>
+<tr><td class='tdl'>The Crude Fact and the Scientific Fact</td><td align='right'><a href="#Page_325">325</a></td></tr>
+<tr><td class='tdl'>Nominalism and the Universal Invariant</td><td align='right'><a href="#Page_333">333</a></td></tr>
+<tr><td align='left'><span class="smcap">Chapter XI.</span>&mdash;Science and Reality</td><td align='right'><a href="#Page_340">340</a></td></tr>
+<tr><td class='tdl'>Contingence and Determinism</td><td align='right'><a href="#Page_340">340</a></td></tr>
+<tr><td class='tdl'>Objectivity of Science</td><td align='right'><a href="#Page_347">347</a></td></tr>
+<tr><td class='tdl'>The Rotation of the Earth</td><td align='right'><a href="#Page_353">353</a></td></tr>
+<tr><td class='tdl'>Science for Its Own Sake</td><td align='right'><a href="#Page_354">354</a></td></tr>
+<tr><td></td></tr>
+<tr><td align='center' colspan='2'><a href="#SCIENCE_AND_METHOD">SCIENCE AND METHOD</a></td></tr>
+<tr><td align='left'>Introduction</td><td align='right'><a href="#Page_359">359</a></td></tr>
+<tr><td align='center' colspan='2'><span class="smcap">Book I.</span> <i>Science and the Scientist</i></td></tr>
+<tr><td align='left'><span class="smcap">Chapter I.</span>&mdash;The Choice of Facts</td><td align='right'><a href="#Page_362">362</a></td></tr>
+<tr><td align='left'><span class="smcap">Chapter II.</span>&mdash;The Future of Mathematics</td><td align='right'><a href="#Page_369">369</a></td></tr>
+<tr><td align='left'><span class="smcap">Chapter III.</span>&mdash;Mathematical Creation</td><td align='right'><a href="#Page_383">383</a></td></tr>
+<tr><td align='left'><span class="smcap">Chapter IV.</span>&mdash;Chance</td><td align='right'><a href="#Page_395">395</a></td></tr>
+<tr><td></td></tr>
+<tr><td align='center' colspan='2'><span class="smcap">Book II.</span> <i>Mathematical Reasoning</i></td></tr>
+<tr><td align='left'><span class="smcap">Chapter I.</span>&mdash;The Relativity of Space</td><td align='right'><a href="#Page_413">413</a></td></tr>
+<tr><td align='left'><span class="smcap">Chapter II.</span>&mdash;Mathematical Definitions and Teaching</td><td align='right'><a href="#Page_430">430</a></td></tr>
+<tr><td align='left'><span class="smcap">Chapter III.</span>&mdash;Mathematics and Logic</td><td align='right'><a href="#Page_448">448</a></td></tr>
+<tr><td align='left'><span class="smcap">Chapter IV.</span>&mdash;The New Logics</td><td align='right'><a href="#Page_460">460</a></td></tr>
+<tr><td align='left'><span class="smcap">Chapter V.</span>&mdash;The Latest Efforts of the Logisticians</td><td align='right'><a href="#Page_472">472</a></td></tr>
+<tr><td></td></tr>
+<tr><td align='center' colspan='2'><span class="smcap">Book III.</span> <i>The New Mechanics</i></td></tr>
+<tr><td align='left'><span class="smcap">Chapter I.</span>&mdash;Mechanics and Radium</td><td align='right'><a href="#Page_486">486</a></td></tr>
+<tr><td align='left'><span class="smcap">Chapter II.</span>&mdash;Mechanics and Optics</td><td align='right'><a href="#Page_496">496</a></td></tr>
+<tr><td align='left'><span class="smcap">Chapter III.</span>&mdash;The New Mechanics and Astronomy</td><td align='right'><a href="#Page_512">512</a></td></tr>
+<tr><td></td></tr>
+<tr><td align='center' colspan='2'><span class="smcap">Book IV.</span> <i>Astronomic Science</i></td></tr>
+<tr><td align='left'><span class="smcap">Chapter I.</span>&mdash;The Milky Way and the Theory of Gases</td><td align='right'><a href="#Page_523">523</a></td></tr>
+<tr><td align='left'><span class="smcap">Chapter II.</span>&mdash;French Geodesy</td><td align='right'><a href="#Page_535">535</a></td></tr>
+<tr><td align='left'>General Conclusions</td><td align='right'><a href="#Page_544">544</a></td></tr>
+<tr><td align='left'>Index</td><td align='right'><a href="#Page_547">547</a></td></tr>
+</table></div>
+
+<hr style="width: 100%;" />
+<p><span class='pagenum'><a name="Page_ix" id="Page_ix">[Pg ix]</a></span></p>
+<h2>HENRI POINCAR&Eacute;</h2>
+
+
+<p><span class="smcap">Sir George Darwin</span>, worthy son of an immortal father, said,
+referring to what Poincar&eacute; was to him and to his work: "He
+must be regarded as the presiding genius&mdash;or, shall I say, my
+patron saint?"</p>
+
+<p>Henri Poincar&eacute; 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 &Eacute;cole Polytechnique, where, like John Bolyai at Maros
+V&aacute;s&aacute;rhely, 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 <i>Nomm&eacute;</i>, March 26, 1879. But he won
+his doctorate in the University of Paris, August 1, 1879, and
+was appointed to teach in the Facult&eacute; des Sciences de Caen,
+December 1, 1879, whence he was quickly called to the University
+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 l'Acad&eacute;mie des Sciences, and, March 5, 1908, was chosen
+Membre de l'Acad&eacute;mie Fran&ccedil;aise. July 1, 1909, the number of
+his writings was 436.</p>
+
+<p>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 Poincar&eacute; tells us the very minute it happened. Mounting
+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 perspective
+new and immense. Moreover, the magic wand of his whole
+<span class='pagenum'><a name="Page_x" id="Page_x">[Pg x]</a></span>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 commenced
+in his book <i>Science and Hypothesis</i>, 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 possibility
+of alternative laws of nature, which in the Introduction
+to the <i>&Eacute;lectricit&eacute; et Optique</i>, 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."</p>
+
+<p>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 <i>commode</i>, convenient;
+but perhaps another may be vastly more advantageous.
+The old conception of <i>true</i> has been revised. The first expression
+of the new idea occurs on the title page of John Bolyai's
+marvelous <i>Science Absolute of Space</i>, in the phrase "haud unquam
+a priori decidenda."</p>
+
+<p>With bearing on the history of the earth and moon system and
+the origin of double stars, in formulating the geometric criterion
+of stability, Poincar&eacute; proved the existence of a previously unknown
+pear-shaped figure, with the possibility that the progressive
+deformation of this figure with increasing angular velocity
+might result in the breaking up of the rotating body into two
+detached masses. Of his treatise <i>Les M&eacute;thodes nouvelles de la
+M&eacute;chanique c&eacute;leste</i>, Sir George Darwin says: "It is probable that
+for half a century to come it will be the mine from which humbler
+investigators will excavate their materials." Brilliant was his
+appreciation of Poincar&eacute; 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 Lobachevski medal of the Physico-Mathematical
+Society of Kazan, and the Bolyai prize of the Hungarian Academy
+of Sciences. His work must be reckoned with the greatest
+mathematical achievements of mankind.</p>
+
+<p>The kernel of Poincar&eacute;'s power lies in an oracle Sylvester often
+quoted to me as from Hesiod: The whole is less than its part.</p>
+<p><span class='pagenum'><a name="Page_xi" id="Page_xi">[Pg xi]</a></span></p>
+<p>He penetrates at once the divine simplicity of the perfectly
+general case, and thence descends, as from Olympus, to the
+special concrete earthly particulars.</p>
+
+<p>A combination of seemingly extremely simple analytic and
+geometric concepts gave necessary general conclusions of immense
+scope from which sprang a disconcerting wilderness of
+possible deductions. And so he leaves a noble, fruitful heritage.</p>
+
+<p>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."</p>
+
+<p class="ralign"><span class="smcap">George Bruce Halsted.</span></p>
+<p><span class='pagenum'><a name="Page_xii" id="Page_xii">[Pg xii]</a></span></p>
+
+
+<hr style="width: 100%;" />
+<p><span class='pagenum'><a name="Page_1" id="Page_1">[Pg 1]</a></span></p>
+<p>&nbsp;</p>
+<h1><a name="SCIENCE_AND_HYPOTHESIS" id="SCIENCE_AND_HYPOTHESIS"></a><b>SCIENCE AND HYPOTHESIS</b></h1>
+<p>&nbsp;</p>
+<p><span class='pagenum'><a name="Page_2" id="Page_2">[Pg 2]</a></span></p>
+<hr style="width: 100%;" />
+<p><span class='pagenum'><a name="Page_3" id="Page_3">[Pg 3]</a></span></p>
+
+
+<h3>AUTHOR'S PREFACE TO THE<br />
+TRANSLATION</h3>
+
+
+<p>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.</p>
+
+<p>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.</p>
+
+<p>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 understanding
+physics; that this way seems to them superior to all
+others; that they feel no need of changing it, nor even of knowing
+the ways of other peoples.</p>
+
+<p>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.</p>
+
+<p>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.</p>
+
+<p>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 deplorable
+if there were no longer Anglo-Saxons.</p>
+
+<p>But continentals have on their part done things an Englishman
+could not have done, so that there is no need either for
+wishing all the world Anglo-Saxon.</p>
+
+<p>Each has his characteristic aptitudes, and these aptitudes<span class='pagenum'><a name="Page_4" id="Page_4">[Pg 4]</a></span>
+should be diverse, else would the scientific concert resemble a
+quartet where every one wanted to play the violin.</p>
+
+<p>And yet it is not bad for the violin to know what the violon-cello
+is playing, and <i>vice versa</i>.</p>
+
+<p>This it is that the English and Americans are comprehending
+more and more; and from this point of view the translations
+undertaken by Dr. Halsted are most opportune and timely.</p>
+
+<p>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 savoring
+of metaphysic.</p>
+
+<p>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 appreciating
+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.</p>
+
+<p>But that is not true. And first of all, are they such uncompromising
+realists as has been said? Are they absolutely refractory,
+I do not say to metaphysic, but at least to everything
+metaphysical?</p>
+
+<p>Recall 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.</p>
+
+<p>Is this not enough to show they are capable of making ascensions
+otherwise than in a captive balloon?</p>
+
+<p>And to return to America, is not the <i>Monist</i> published at
+Chicago, that review which even to us seems bold and yet which
+finds readers?</p>
+
+<p>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<span class='pagenum'><a name="Page_5" id="Page_5">[Pg 5]</a></span>
+groups of substitutions, and under its most abstract form, the
+farthest removed from the practical.</p>
+
+<p>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.</p>
+
+<p>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.</p>
+
+<p>The American public is therefore much better prepared than
+has been thought for investigating the origin of the notion of
+space.</p>
+
+<p>Moreover, to analyze this concept is not to sacrifice reality to
+I know not what phantom. The geometric language is after all
+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.</p>
+
+<p>In the study of nature, the contrast between the Anglo-Saxon
+spirit and the Latin spirit is still greater.</p>
+
+<p>The Latins seek in general to put their thought in mathematical
+form; the English prefer to express it by a material
+representation.</p>
+
+<p>Both doubtless rely only on experience for knowing the world;
+when they happen to go beyond this, they consider their foreknowledge
+as only provisional, and they hasten to ask its definitive
+confirmation from nature herself.</p>
+
+<p>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&mdash;the instrument the Latins ordinarily
+choose is not that preferred by the Anglo-Saxons.<span class='pagenum'><a name="Page_6" id="Page_6">[Pg 6]</a></span></p>
+
+<p>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.</p>
+
+<p>The Anglo-Saxon to depict a phenomenon will first be engrossed
+in making a <i>model</i>, 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.</p>
+
+<p>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.</p>
+
+<p>From this point of view again, it would be well for these two
+sorts of physicists to know something of each other; in studying
+the work of minds so unlike their own, they will immediately
+recognize that in this work there has been an accumulation
+of hypotheses.</p>
+
+<p>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 will warn
+their rivals, and it may be supposed these will not fail to render
+them the same service.</p>
+
+<p>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 geometric
+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 perceived
+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.</p>
+
+<p>And then these very contradictions, when they end by perceiving
+them, have the advantage of showing them the hypothetical
+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.<span class='pagenum'><a name="Page_7" id="Page_7">[Pg 7]</a></span></p>
+
+<p>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.</p>
+
+<p>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.</p>
+
+<p>The Latin wants to make it out of formulas, but these formulas
+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.</p>
+
+<p>And yet is this legitimate, if the unknown be the simple and
+the known the complex?</p>
+
+<p>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?</p>
+
+<p>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&mdash;as
+when Newton replaced Kepler's three laws by the single law of
+gravitation, which was something simpler, equivalent, yet unlike?</p>
+
+<p>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.</p>
+
+<p>So it was in vain the attempt was made in England to construct
+the ether by material models, or in France to apply to
+it the laws of dynamic.</p>
+
+<p>The ether it is, the unknown, which explains matter, the
+known; matter is incapable of explaining the ether.</p>
+
+<p class="ralign"><span class="smcap">Poincar&eacute;.</span></p>
+<p><span class='pagenum'><a name="Page_8" id="Page_8">[Pg 8]</a></span></p>
+
+
+<hr style="width: 65%;" />
+<p><span class='pagenum'><a name="Page_9" id="Page_9">[Pg 9]</a></span></p>
+<h3>INTRODUCTION</h3>
+
+<h4>BY PROFESSOR JOSIAH ROYCE<br />
+<br />
+<span class="smcap">Harvard University</span></h4>
+
+
+<p>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. Poincar&eacute;'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. Poincar&eacute; has so notably contributed.</p>
+
+
+<h4>I</h4>
+
+<p>The branches of inquiry collectively known as the Philosophy
+of Science have undergone great changes since the appearance of
+Herbert Spencer's <i>First Principles</i>, 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.
+Poincar&eacute; 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<span class='pagenum'><a name="Page_10" id="Page_10">[Pg 10]</a></span>
+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.
+Poincar&eacute; refers in his opening paragraphs, or whether, like Spencer
+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.'</p>
+
+<p>Some of us well remember how, when Stallo's 'Principles and
+Theories of Modern Physics' first appeared, this sense of scientific
+orthodoxy was shocked amongst many of our American readers
+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 presumption
+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.</p>
+
+<p>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 details
+more assured. Phenomena men know and predict better
+than ever. But regarding the most general theories, and the<span class='pagenum'><a name="Page_11" id="Page_11">[Pg 11]</a></span>
+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&mdash;yes, the field of the logic of
+science&mdash;this whole region is to-day an open one. Whoever will
+work there must indeed accept the verdict of experience regarding
+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 inconceivable'
+and the doctrine of energy beyond the reach of novel formulations.
+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 interpretations.
+Certain properties of the irrational number &#960; 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 Dedekind,
+of Cantor, of Weierstrass and of various others, the general
+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 several
+very notable respects, unconquered territory, despite the
+work of Hilbert and of Pieri, 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 radioactive
+bodies will occupy in that theory. The alchemists would not<span class='pagenum'><a name="Page_12" id="Page_12">[Pg 12]</a></span>
+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 invariant,
+than the theoretical orthodoxy of a recent period supposed.
+Again, regarding the foundations of biology, a theoretical
+orthodoxy grows less possible, less definable, less conceivable
+(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 objects, 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 discoveries
+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.</p>
+
+<p>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.</p>
+
+
+<h4>II</h4>
+
+<p>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<span class='pagenum'><a name="Page_13" id="Page_13">[Pg 13]</a></span>
+dream, and that the results of science are coextensive with the
+range of actual empirical observation and of successful prediction?
+If this is indeed the lesson, then the decline of theoretical
+orthodoxy in science is&mdash;like the eclipse of dogma in religion&mdash;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 reasonable.
+What we are free to do as we please&mdash;is it any longer a
+serious business? What we are free to think as we please&mdash;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 observed;
+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&mdash;or else is
+some matter to be tested by another experiment?" Why not
+limit our philosophy of science strictly to such a counsel of resignation?
+Why not substitute, for the old scientific orthodoxy,
+simply a confession of ignorance, and a resolution to devote ourselves
+to the business of enlarging the bounds of actual empirical
+knowledge?</p>
+
+<p>Such comments upon the situation just characterized are frequently
+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 these very comments.
+Never has there been better reason than there is to-day
+to be content, if rational man could be content, with a pure positivism.
+The splendid triumphs of special research in the most
+various fields, the constant increase in our practical control over<span class='pagenum'><a name="Page_14" id="Page_14">[Pg 14]</a></span>
+nature&mdash;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 'unification'
+any further, when what we daily get from our sciences is
+an increasing wealth of detailed information and of practical
+guidance?</p>
+
+<p>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 orthodoxy
+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 regarding
+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. Poincar&eacute;, in his book, sketches
+with the hand of the master, is another illustration of the consciousness
+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&mdash;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 oppression
+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.</p>
+
+<p>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,<span class='pagenum'><a name="Page_15" id="Page_15">[Pg 15]</a></span>
+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 knowledge
+in more or less completely unified systems of doctrine?</p>
+
+<p>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 different
+special issues. A student interested in the problems of
+the philosophy of mathematics, or in the theory of probabilities,
+or in the nature and office 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&mdash;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 construction&mdash;an
+abandonment of dogma, but a vindication of the
+rights of the constructive reason.</p>
+
+
+<h4>III</h4>
+
+<p>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:</p>
+
+<p>1. The hypotheses which are valuable <i>precisely</i> because they
+are either verifiable or else refutable through a definite appeal
+to the tests furnished by experience; and</p>
+
+<p>2. The hypotheses which, despite the fact that experience suggests
+them, are valuable <i>despite</i>, or even <i>because</i>, of the fact that
+experience can <i>neither</i> confirm nor refute them. The contrast
+between these two kinds of hypotheses is a prominent topic of
+our author's discussion.</p>
+
+<p>Hypotheses of the general type which I have here placed first<span class='pagenum'><a name="Page_16" id="Page_16">[Pg 16]</a></span>
+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 explicitly
+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 distinct
+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.
+Poincar&eacute;'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 concepts
+of science) as principles regarding the structure of the
+world in its wholeness; <i>i. e.</i>, 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.
+Poincar&eacute;'s logical theory of this second class of hypotheses undertakes
+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<span class='pagenum'><a name="Page_17" id="Page_17">[Pg 17]</a></span>
+adjustment of our conceptions of things to the internal needs
+of our intelligence, rather than a grasping of things as they are
+in themselves.</p>
+
+<p>To be sure, M. Poincar&eacute;'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. Poincar&eacute;'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.
+Poincar&eacute;, on the contrary, all this adjustment of our interpretations
+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 necessity.
+Unverifiable and irrefutable hypotheses in science are indeed,
+in general, indispensable aids to the organization and to the
+guidance of our interpretation of experience. But it is experience
+itself which points out to us what lines of interpretation
+will prove most convenient. Instead of Kant's rigid list of
+<i>a priori</i> 'forms,' we consequently have in M. Poincar&eacute;'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 organization
+is due to hypotheses of the kind here in question, thus
+resembles that of a constitutional government&mdash;neither absolutely
+necessary, nor yet determined apart from the will of the
+subjects, nor yet accidental&mdash;a free, yet not a capricious establishment
+of good order, in conformity with empirical needs.<span class='pagenum'><a name="Page_18" id="Page_18">[Pg 18]</a></span></p>
+
+<p>Characteristic remains, however, for our author, as, in his
+decidedly contrasting way, for Kant, the thought that <i>without
+principles which at every stage transcend precise confirmation
+through such experience as is then accessible the organization of
+experience is impossible</i>. Whether one views these principles as
+conventions or as <i>a priori</i> 'forms,' they may therefore be described
+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 confirmed
+nor refuted by the experiences which we possess or which
+we can hope to attain.</p>
+
+<p>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 existence
+of continuous extensive <i>quanta</i> in nature; (2) The principles
+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 <i>find</i> 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 <i>a priori</i> necessities; and we can easily conceive intelligent
+beings whose experience could be best interpreted without
+using these hypotheses. Yet these hypotheses are <i>not</i> subject
+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>i. e.</i>, to the hypotheses
+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 appreciation
+of the significance of empirical investigation. But the central<span class='pagenum'><a name="Page_19" id="Page_19">[Pg 19]</a></span>
+problem of the logic of science thus becomes the problem of
+the relation between the two fundamentally distinct types of
+hypotheses, <i>i. e.</i>, between those which can not be verified or refuted
+through experience, and those which can be empirically
+tested.</p>
+
+
+<h4>IV</h4>
+
+<p>The detailed treatment which M. Poincar&eacute; 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 observing
+that, while M. Poincar&eacute; 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. Poincar&eacute;'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.</p>
+
+<p>The historical sciences, and in fact all those sciences such as
+geology, and such as the evolutionary sciences in general, undertake
+theoretical constructions which relate to past time. Hypotheses
+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 predictions
+of coming experience. These latter will be subject to control.
+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 predictions
+proved verifiable. The result has been a considerable<span class='pagenum'><a name="Page_20" id="Page_20">[Pg 20]</a></span>
+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, regarding
+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 hypotheses
+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 anthropology,
+or Old Testament history, the principal theoretical constructions
+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 confr&egrave;res of the sciences
+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 criticism'&mdash;all
+these are matters which the men of the laboratory
+often regard with a general incredulity as belonging not at all
+to the domain of true science. Yet no one can doubt the importance
+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 verifiable
+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 theoretical
+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 constructions
+in fact resemble in a measure those which M. Poincar&eacute;
+in this book has analyzed in the case of geometry. They are
+constructions molded, but <i>not</i> predetermined in their details, by
+experience. We report facts; we let the facts speak; but we, as<span class='pagenum'><a name="Page_21" id="Page_21">[Pg 21]</a></span>
+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 well 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, internally
+rational, as well as (when that is possible) subject to external
+tests.</p>
+
+<p>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. Poincar&eacute; speaks
+when he characterizes the principles of mechanics and of the
+theory of energy. I venture to call attention here to this conception,
+which seems to me to illustrate M. Poincar&eacute;'s view of the
+functions of hypothesis in scientific work.</p>
+
+<p>The modern science of pathology is usually regarded as dating
+from the earlier researches of Virchow, whose 'Cellular Pathology'
+was the outcome of a very careful and elaborate induction.
+Virchow, himself, felt a strong aversion to mere speculation.
+He endeavored to keep close to observation, and to relieve
+medical science from the control of fantastic theories, such as
+those of the <i>Naturphilosophen</i> had been. Yet Virchow's researches
+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 diseases
+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 <i>that they merely show us the course of<span class='pagenum'><a name="Page_22" id="Page_22">[Pg 22]</a></span>
+the vital processes under altered conditions</i>" ('dasz sie nur
+Ablauf der Lebenserscheinungen unter ver&auml;nderten Bedingungen
+darstellen').</p>
+
+<p>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 untouched
+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 themselves,
+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 <i>ver&auml;nderte Bedingungen</i> 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 involves
+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 case the disease is to be understood.</p>
+
+<p>For this very reason, however, Virchow's theoretical principle
+in its most general form <i>could be neither confirmed nor refuted
+by experience</i>. 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 <i>ver&auml;nderte Bedingungen</i> to which the
+diseased organism would be reacting. Let bullets or bacteria,
+poisons or compressed air, or the devil be the <i>Bedingungen</i> to
+which a diseased organism reacts, the postulate that Virchow<span class='pagenum'><a name="Page_23" id="Page_23">[Pg 23]</a></span>
+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.</p>
+
+<p>I insist, then, that this principle of Virchow's is no trial supposition,
+no scientific hypothesis in the narrower sense&mdash;capable
+of being submitted to precise empirical tests. It is, on the
+contrary, a very precious <i>leading idea</i>, a theoretical interpretation
+of phenomena, in the light of which observations are to be
+made&mdash;'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 processes
+are linked. Now without some such leading idea, the cellular
+pathology itself could never have been reached; because the
+empirical facts in question would never have been observed.
+Hence this principle of Virchow's was indispensable to the
+growth of his science. Yet it was not a verifiable and not a refutable
+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 control,
+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 irrefutable
+character, suggested, but not to be finally tested, by
+experience, the hypotheses in the narrower sense would lack that
+guidance which, as M. Poincar&eacute; has shown, the larger ideas of
+science give to empirical investigation.</p>
+
+
+<h4>V</h4>
+
+<p>I have dwelt, no doubt, at too great length upon one aspect
+only of our author's varied and well-balanced discussion of the<span class='pagenum'><a name="Page_24" id="Page_24">[Pg 24]</a></span>
+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 philosophical
+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.
+Poincar&eacute; defends a thesis regarding the office of 'demonstration
+by recurrence'&mdash;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 problems
+that lie at the basis of recurrent proofs and processes are,
+as I have elsewhere argued, of the most fundamental importance.</p>
+
+<p>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.</p>
+
+<p>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 students
+of his own science, to whom his position is well known, but
+to the general reader who may seek guidance in these pages.</p>
+
+<p>Jules Henri Poincar&eacute; was born at Nancy, in 1854, the son
+of a professor in the Faculty of Medicine at Nancy. He
+studied at the &Eacute;cole Polytechnique and at the &Eacute;cole des Mines,
+and later received his doctorate in mathematics in 1879. In
+1883 he began courses of instruction in mathematics at the
+&Eacute;cole Polytechnique; in 1886 received a professorship of mathematical
+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<span class='pagenum'><a name="Page_25" id="Page_25">[Pg 25]</a></span>
+various branches of his chosen sciences is long; and his original
+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 Poincar&eacute; 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&mdash;a highly recondite region of pure mathematics&mdash;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.</p>
+
+<p>Meanwhile, as contributor to the philosophical discussion of
+the bases and methods of science, M. Poincar&eacute; has long been
+active. When, in 1893, the admirable <i>Revue de M&eacute;taphysique et
+de Morale</i> began to appear, M. Poincar&eacute; was soon found amongst
+the most satisfactory of the contributors to the work of that
+journal, whose office it has especially been to bring philosophy
+and the various special sciences (both natural and moral) into
+a closer mutual understanding. The discussions brought together
+in the present volume are in large part the outcome of
+M. Poincar&eacute;'s contributions to the <i>Revue de M&eacute;taphysique et de
+Morale</i>. The reader of M. Poincar&eacute;'s book is in presence, then,
+of a great special investigator who is also a philosopher.</p>
+<p><span class='pagenum'><a name="Page_26" id="Page_26">[Pg 26]</a></span></p>
+
+
+<hr style="width: 65%;" />
+<p><span class='pagenum'><a name="Page_27" id="Page_27">[Pg 27]</a></span></p>
+<h2>SCIENCE AND HYPOTHESIS</h2>
+
+<h3>INTRODUCTION</h3>
+
+
+<p>For a superficial observer, scientific truth is beyond the possibility
+of doubt; the logic of science is infallible, and if the scientists
+are sometimes mistaken, this is only from their mistaking
+its rules.</p>
+
+<p>"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. From 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."</p>
+
+<p>Behold what is for many people in the world, for scholars getting
+their first notions of physics, the origin of scientific certitude.
+This is what they suppose to be the r&ocirc;le of experimentation
+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.</p>
+
+<p>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
+are two equally convenient solutions; each saves us from
+thinking.</p>
+
+<p>Instead of pronouncing a summary condemnation, we ought
+therefore to examine with care the r&ocirc;le of hypothesis; we shall
+then recognize, not only that it is necessary, but that usually it is<span class='pagenum'><a name="Page_28" id="Page_28">[Pg 28]</a></span>
+legitimate. We shall also see that there are several sorts of hypotheses;
+that some are verifiable, and once confirmed by experiment
+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.</p>
+
+<p>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 <i>our</i> science, which, without
+them, would be impossible, they are not imposed upon nature.
+Are they then arbitrary? No, else were they sterile. Experiment
+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.</p>
+
+<p>Some people have been struck by this character of free convention
+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 <i>nominalism</i>, and have
+asked themselves if the savant is not the dupe of his own definitions
+and if the world he thinks he discovers is not simply
+created by his own caprice.<a name="FNanchor_1_1" id="FNanchor_1_1"></a><a href="#Footnote_1_1" class="fnanchor">[1]</a> Under these conditions science
+would be certain, but deprived of significance.</p>
+
+<p>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 na&iuml;ve dogmatists think, but only
+the relations between things. Outside of these relations there
+is no knowable reality.</p>
+
+<p>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.<span class='pagenum'><a name="Page_29" id="Page_29">[Pg 29]</a></span></p>
+
+<p>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.</p>
+
+<p>Knowing better now one of the instruments which mathematics
+puts into the hands of the investigator, we had to analyze another
+fundamental notion, that of mathematical magnitude. Do
+we find it in nature, or do we ourselves introduce it there? And,
+in this latter case, do we not risk marring everything? Comparing
+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.</p>
+
+<p>Another frame which we impose on the world is space.
+Whence come the first principles of geometry? Are they imposed
+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 geometry?
+A deeper discussion will show us it is not. We therefore
+conclude that the first principles of geometry are only conventions;
+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.</p>
+
+<p>In mechanics we should be led to analogous conclusions, and
+should see that the principles of this science, though more directly
+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 hypotheses
+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,<span class='pagenum'><a name="Page_30" id="Page_30">[Pg 30]</a></span>
+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.</p>
+
+<p>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
+<i>all</i> these circumstances could be reproduced at once, this principle
+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 r&ocirc;le 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.</p>
+
+<p>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 Maxwell,
+and what unconscious hypotheses were made by Amp&egrave;re
+and the other founders of electrodynamics.</p>
+
+
+
+<hr style="width: 65%;" />
+<p><span class='pagenum'><a name="Page_31" id="Page_31">[Pg 31]</a></span></p>
+<h2><b>PART I<br />
+
+<br />
+
+<small>NUMBER AND MAGNITUDE</small></b></h2>
+
+
+
+<hr style="width: 65%;" />
+<h3>CHAPTER I</h3>
+
+<h3><span class="smcap">On the Nature of Mathematical Reasoning</span></h3>
+
+
+<h4>I</h4>
+
+<p>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? 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
+so many volumes are nothing but devious ways of saying <i>A</i> is <i>A</i>?</p>
+
+<p>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 mathematical
+necessity, we have yet the resource of classing them
+among synthetic <i>a priori</i> judgments. This is not to solve the difficulty,
+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 reasoning
+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.</p>
+
+<p>No theorem could be new if no new axiom intervened in its
+demonstration; reasoning could give us only the immediately<span class='pagenum'><a name="Page_32" id="Page_32">[Pg 32]</a></span>
+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?</p>
+
+<p>The contradiction will strike us the more if we open any book
+on mathematics; on every page the author will announce his intention
+of generalizing some proposition already known. Does
+the mathematical method proceed from the particular to the general,
+and, if so, how then can it be called deductive?</p>
+
+<p>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 sufficiently powerful could at
+a glance perceive all its truths; nay more, we might even hope
+that some day one would invent to express them a language sufficiently
+simple to have them appear self-evident to an ordinary
+intelligence.</p>
+
+<p>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.</p>
+
+<p>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.</p>
+
+<p>All these modes of reasoning, whether or not they be reducible
+to the syllogism properly so called, retain the analytic character,
+and just because of that are powerless.</p>
+
+
+<h4>II</h4>
+
+<p>The discussion is old; Leibnitz tried to prove 2 and 2 make 4;
+let us look a moment at his demonstration.</p>
+
+<p>I will suppose the number 1 defined and also the operation
+<i>x</i> + 1 which consists in adding unity to a given number <i>x</i>.</p>
+
+<p>These definitions, whatever they be, do not enter into the
+course of the reasoning.</p>
+
+<p>I define then the numbers 2, 3 and 4 by the equalities</p>
+
+<p class="center">(1) 1 + 1 = 2; &nbsp;&nbsp;&nbsp;&nbsp;&nbsp; (2) 2 + 1 = 3; &nbsp;&nbsp;&nbsp;&nbsp;&nbsp; (3) 3 + 1 = 4.</p>
+
+<p>In the same way, I define the operation <i>x</i> + 2 by the relation:<span class='pagenum'><a name="Page_33" id="Page_33">[Pg 33]</a></span></p>
+
+<p class="center">(4) <i>x</i> + 2 = (<i>x</i> + 1) + 1.</p>
+
+<p>That presupposed, we have</p>
+
+
+<div class='center'>
+<table border="0" cellpadding="1" cellspacing="0" summary="">
+<tr><td align='left'>2 + 1 + 1 = 3 + 1</td><td align='left'>(Definition 2),</td></tr>
+<tr><td align='left'>3 + 1 = 4</td><td align='left'>(Definition 3),</td></tr>
+<tr><td align='left'>2 + 2 = (2 + 1) + 1 &nbsp; &nbsp;</td><td align='left'>(Definition 4),</td></tr>
+</table></div>
+
+<p class="noidt">whence</p>
+
+<p class="center">2 + 2 = 4 Q.E.D.</p>
+
+<p>It can not be denied that this reasoning is purely analytic.
+But ask any mathematician: 'That is not a demonstration properly
+so called,' he will say to you: 'that is a verification.' We
+have confined ourselves to comparing two purely conventional
+definitions and have ascertained their identity; we have learned
+nothing new. <i>Verification</i> differs from true demonstration precisely
+because it is purely analytic and because it is sterile. It is
+sterile because the conclusion is nothing but the premises translated
+into another language. On the contrary, true demonstration
+is fruitful because the conclusion here is in a sense more general
+than the premises.</p>
+
+<p>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.</p>
+
+<p>It may even be said the very object of the exact sciences is to
+spare us these direct verifications.</p>
+
+
+<h4>III</h4>
+
+<p>Let us, therefore, see the geometer at work and seek to catch
+his process.</p>
+
+<p>The task is not without difficulty; it does not suffice to open
+a work at random and analyze any demonstration in it.</p>
+
+<p>We must first exclude geometry, where the question is complicated
+by arduous problems relative to the r&ocirc;le of the postulates,
+to the nature and the origin of the notion of space. For
+analogous reasons we can not turn to the infinitesimal analysis.<span class='pagenum'><a name="Page_34" id="Page_34">[Pg 34]</a></span>
+We must seek mathematical thought where it has remained pure,
+that is, in arithmetic.</p>
+
+<p>A choice still is necessary; in the higher parts of the theory
+of numbers, the primitive mathematical notions have already undergone
+an elaboration so profound that it becomes difficult to
+analyze them.</p>
+
+<p>It is, therefore, at the beginning of arithmetic that we must
+expect to find the explanation we seek, but it happens that precisely
+in the demonstration of the most elementary theorems the
+authors of the classic treatises have shown the least precision and
+rigor. We must not impute this to them as a crime; they have
+yielded to a necessity; beginners are not prepared for real mathematical
+rigor; they would see in it only useless and irksome subtleties;
+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.</p>
+
+<p>Why is so long a preparation necessary to become habituated
+to this perfect rigor, which, it would seem, should naturally impress
+itself upon all good minds? This is a logical and psychological
+problem well worthy of study.</p>
+
+<p>But we shall not take it up; it is foreign to our purpose; all
+I wish to insist on is that, not to fail 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.</p>
+
+<p><span class="smcap">Definition of Addition.</span>&mdash;I suppose already defined the
+operation <i>x</i> + 1, which consists in adding the number 1 to a
+given number <i>x</i>.</p>
+
+<p>This definition, whatever it be, does not enter into our subsequent
+reasoning.</p>
+
+<p>We now have to define the operation <i>x</i> + <i>a</i>, which consists in
+adding the number <i>a</i> to a given number <i>x</i>.</p>
+
+<p>Supposing we have defined the operation</p>
+
+<p class="center"><i>x</i> + (<i>a</i> &minus; 1),</p>
+
+<p class="noidt">the operation <i>x</i> + <i>a</i> will be defined by the equality</p>
+
+<p class="center"><span class="linenum">(1)</span> <i>x</i> + <i>a</i> = [<i>x</i> + (<i>a</i> &minus; 1)] + 1.</p>
+
+<p><span class='pagenum'><a name="Page_35" id="Page_35">[Pg 35]</a></span></p>
+
+<p>We shall know then what <i>x + a</i> is when we know what
+<i>x</i> + (<i>a</i> &minus; 1) is, and as I have supposed that to start with we
+knew what <i>x</i> + 1 is, we can define successively and 'by recurrence'
+the operations <i>x</i> + 2, <i>x</i> + 3, etc.</p>
+
+<p>This definition deserves a moment's attention; it is of a particular
+nature which already distinguishes it from the purely
+logical definition; the equality (1) contains an infinity of distinct
+definitions, each having a meaning only when one knows the
+preceding.</p>
+
+<p><span class="smcap">Properties of Addition.</span>&mdash;<i>Associativity.</i>&mdash;I say that</p>
+
+<p class="center">
+<i>a</i> + (<i>b + c</i>) = (<i>a + b</i>) + <i>c</i>.<br />
+</p>
+
+<p>In fact the theorem is true for <i>c</i> = 1; it is then written</p>
+
+<p class="center">
+<i>a</i> + (<i>b</i> + 1) = (<i>a + b</i>) + 1,<br />
+</p>
+
+<p class="noidt">which, apart from the difference of notation, is nothing but the
+equality (1), by which I have just defined addition.</p>
+
+<p>Supposing the theorem true for <i>c</i> = &#947;, I say it will be true for
+<i>c</i> = &#947; + 1.</p>
+
+<p>In fact, supposing</p>
+
+<p class="center">
+(<i>a + b</i>) + &#947; = <i>a</i> + (<i>b</i> + &#947;),<br />
+</p>
+
+<p class="noidt">it follows that</p>
+
+<p class="center">
+[(<i>a + b</i>) + &#947;] + 1 = [<i>a</i> + (<i>b</i> + &#947;)] + 1<br />
+</p>
+
+<p class="noidt">or by definition (1)</p>
+
+<p class="center">
+(<i>a + b</i>) + (&#947; + 1) = <i>a</i> + (<i>b</i> + &#947; + 1) = <i>a</i> + [<i>b</i> + (&#947; + 1)],<br />
+</p>
+
+<p class="noidt">which shows, by a series of purely analytic deductions, that the
+theorem is true for &#947; + 1.</p>
+
+<p>Being true for <i>c</i> = 1, we thus see successively that so it is for
+<i>c</i> = 2, for <i>c</i> = 3, etc.</p>
+
+<p><i>Commutativity.</i>&mdash;1&ordm; I say that</p>
+
+<p class="center">
+<i>a</i> + 1 = 1 + <i>a</i>.<br />
+</p>
+
+<p>The theorem is evidently true for <i>a</i> = 1; we can <i>verify</i> by
+purely analytic reasoning that if it is true for <i>a</i> = &#947; it will be
+true for <i>a</i> = &#947; + 1; for then</p>
+
+<p class="center">
+(&#947; + 1) + 1 = (1 + &#947;) + 1 = 1 + (&#947; + 1);<br />
+</p>
+
+<p class="noidt">now it is true for <i>a</i> = 1, therefore it will be true for <i>a</i> = 2, for
+<i>a</i> = 3, etc., which is expressed by saying that the enunciated
+proposition is demonstrated by recurrence.<span class='pagenum'><a name="Page_36" id="Page_36">[Pg 36]</a></span></p>
+
+<p>2&ordm; I say that</p>
+
+<p class="center"><i>a</i> + <i>b</i> = <i>b</i> + <i>a</i>.</p>
+
+<p>The theorem has just been demonstrated for <i>b</i> = 1; it can be
+verified analytically that if it is true for <i>b</i> = &#946;, it will be true for
+<i>b</i> = &#946; + 1.</p>
+
+<p>The proposition is therefore established by recurrence.</p>
+
+<p><span class="smcap">Definition of Multiplication.</span>&mdash;We shall define multiplication
+by the equalities.</p>
+
+<p class="center"><span class="linenum">(1)</span> <i>a</i> &times; 1 = <i>a</i>.</p>
+
+<p class="center"><span class="linenum">(2)</span> <i>a</i> &times; <i>b</i> = [<i>a</i> &times; (<i>b</i> &minus; 1)] + <i>a</i>.</p>
+
+<p>Like equality (1), equality (2) contains an infinity of definitions;
+having defined a &times; 1, it enables us to define successively:
+<i>a</i> &times; 2, <i>a</i> &times; 3, etc.</p>
+
+<p><span class="smcap">Properties of Multiplication.</span>&mdash;<i>Distributivity.</i>&mdash;I say that</p>
+
+<p class="center">(<i>a</i> + <i>b</i>) &times; <i>c</i> = (<i>a</i> &times; <i>c</i>) + (<i>b</i> &times; <i>c</i>).</p>
+
+<p>We verify analytically that the equality is true for <i>c</i> = 1; then
+that if the theorem is true for <i>c</i> = &#947;, it will be true for <i>c</i> = &#947; + 1.</p>
+
+<p>The proposition is, therefore, demonstrated by recurrence.</p>
+
+<p><i>Commutativity.</i>&mdash;1&ordm; I say that</p>
+
+<p class="center"><i>a</i> &times; 1 = 1 &times; <i>a</i>.</p>
+
+<p>The theorem is evident for <i>a</i> = 1.</p>
+
+<p>We verify analytically that if it is true for <i>a</i> = &#945;, it will be
+true for <i>a</i> = &#945; + 1.</p>
+
+<p>2&ordm; I say that</p>
+
+<p class="center"><i>a</i> &times; <i>b</i> = <i>b</i> &times; <i>a</i>.</p>
+
+<p>The theorem has just been proven for <i>b</i> = 1. We could verify
+analytically that if it is true for <i>b</i> = &#946;, it will be true for
+<i>b</i> = &#946; + 1.</p>
+
+<h4>IV</h4>
+
+<p>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.</p>
+
+<p>This procedure is the demonstration by recurrence. We first
+establish a theorem for <i>n</i> = 1; then we show that if it is true of
+<i>n</i> &minus; 1, it is true of <i>n</i>, and thence conclude that it is true for all
+the whole numbers.<span class='pagenum'><a name="Page_37" id="Page_37">[Pg 37]</a></span></p>
+
+<p>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 transformation,
+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 mathematics
+had no other instrument, it would therefore be forthwith
+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.</p>
+
+<p>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.</p>
+
+<p>Here then we have the mathematical reasoning <i>par excellence</i>,
+and we must examine it more closely.</p>
+
+
+<h4>V</h4>
+
+<p>The essential characteristic of reasoning by recurrence is that
+it contains, condensed, so to speak, in a single formula, an
+infinity of syllogisms.</p>
+
+<p>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.'</p>
+
+<p>These are of course hypothetical syllogisms.</p>
+
+<p>The theorem is true of the number 1.</p>
+
+<p>Now, if it is true of 1, it is true of 2.</p>
+
+<p>Therefore it is true of 2.</p>
+
+<p>Now, if it is true of 2, it is true of 3.</p>
+
+<p>Therefore it is true of 3, and so on.</p>
+
+<p>We see that the conclusion of each syllogism serves as minor to
+the following.</p>
+
+<p>Furthermore the majors of all our syllogisms can be reduced
+to a single formula.</p>
+
+<p>If the theorem is true of <i>n</i> &minus; 1, so it is of <i>n</i>.</p>
+
+<p>We see, then, that in reasoning by recurrence we confine ourselves
+to stating the minor of the first syllogism, and the general
+formula which contains as particular cases all the majors.</p>
+
+<p>This never-ending series of syllogisms is thus reduced to a
+phrase of a few lines.<span class='pagenum'><a name="Page_38" id="Page_38">[Pg 38]</a></span></p>
+
+<p>It is now easy to comprehend why every particular consequence
+of a theorem can, as I have explained above, be verified
+by purely analytic procedures.</p>
+
+<p>If instead of showing that our theorem is true of all numbers,
+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 cascade;
+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.</p>
+
+<p>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 syllogisms
+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.</p>
+
+<p>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.</p>
+
+<p>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 will 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.</p>
+
+<p>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 impracticable.
+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.</p>
+
+<p>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 preponderant
+r&ocirc;le, and without it there would be no science, because there
+would be nothing general.</p>
+<p><span class='pagenum'><a name="Page_39" id="Page_39">[Pg 39]</a></span></p>
+
+<h4>VI</h4>
+
+<p>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.</p>
+
+<p>We can easily pass from one enunciation to the other and thus
+get the illusion of having demonstrated the legitimacy of reasoning
+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.</p>
+
+<p>We can not therefore escape the conclusion that the rule of
+reasoning by recurrence is irreducible to the principle of contradiction.</p>
+
+<p>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 series, more or less long
+but always limited.</p>
+
+<p>Now if it were only a question of that, the principle of contradiction
+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, experience
+becomes powerless. This rule, inaccessible to analytic
+demonstration and to experience, is the veritable type of the
+synthetic <i>a priori</i> judgment. On the other hand, we can not
+think of seeing in it a convention, as in some of the postulates of
+geometry.</p>
+
+<p>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.</p>
+
+<p>But, one will say, if raw experience can not legitimatize
+reasoning by recurrence, is it so of experiment aided by<span class='pagenum'><a name="Page_40" id="Page_40">[Pg 40]</a></span>
+induction? 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.</p>
+
+<p>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, demonstration
+by recurrence, on the contrary, imposes itself necessarily
+because it is only the affirmation of a property of the mind itself.</p>
+
+
+<h4>VII</h4>
+
+<p>Mathematicians, as I have said before, always endeavor to
+<i>generalize</i> the propositions they have obtained, and, to seek no
+other example, we have just proved the equality:</p>
+
+<p class="center">
+<i>a</i> + 1 = 1 + <i>a</i><br />
+</p>
+
+<p class="noidt">and afterwards used it to establish the equality</p>
+
+<p class="center">
+<i>a</i> + <i>b</i> = <i>b</i> + <i>a</i><br />
+</p>
+
+<p class="noidt">which is manifestly more general.</p>
+
+<p>Mathematics can, therefore, like the other sciences, proceed
+from the particular to the general.</p>
+
+<p>This is a fact which would have appeared incomprehensible
+to us at the outset of this study, but which is no longer mysterious
+to us, since we have ascertained the analogies between
+demonstration by recurrence and ordinary induction.</p>
+
+<p>Without doubt recurrent reasoning in mathematics and inductive
+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.</p>
+
+<p>Let us examine the case a little more closely.</p>
+
+<p>To demonstrate the equality</p>
+
+<p class="center">
+<i>a</i> + 2 = 2 + <i>a</i><br />
+</p>
+
+<p class="noidt">it suffices to twice apply the rule</p>
+
+<p class="center">
+<span class="linenum">(1)</span> <i>a</i> + 1 = 1 + <i>a</i><br />
+</p>
+
+<p class="noidt">and write</p>
+
+<p class="center">
+<span class="linenum">(2)</span> <i>a</i> + 2 = <i>a</i> + 1 + 1 = 1 + <i>a</i> + 1 = 1 + 1 + <i>a</i> = 2 + <i>a</i>.</p>
+
+<p><span class='pagenum'><a name="Page_41" id="Page_41">[Pg 41]</a></span></p>
+
+<p>The equality (2) thus deduced in purely analytic way from
+the equality (1) is, however, not simply a particular ease of it;
+it is something quite different.</p>
+
+<p>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.</p>
+
+<p>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.</p>
+
+<p>Mathematicians proceed therefore 'by construction,' they 'construct'
+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 relations
+of these elements and from them deduce the relations of
+the aggregates themselves.</p>
+
+<p>This is a purely analytical proceeding, but it is not, however,
+a proceeding from the general to the particular, because evidently
+the aggregates can not be regarded as more particular
+than their elements.</p>
+
+<p>Great importance, and justly, has been attached to this procedure
+of 'construction,' and some have tried to see in it the
+necessary and sufficient condition for the progress of the exact
+sciences.</p>
+
+<p>Necessary, without doubt; but sufficient, no.</p>
+
+<p>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.</p>
+
+<p>Or, more exactly, there must be some advantage in considering
+the construction rather than its elements themselves.</p>
+
+<p>What can this advantage be?</p>
+
+<p>Why reason on a polygon, for instance, which is always decomposable
+into triangles, and not on the elementary triangles?</p>
+
+<p>It is because there are properties appertaining to polygons
+of any number of sides and that may be immediately applied to
+any particular polygon.</p>
+
+<p>Usually, on the contrary, it is only at the cost of the most<span class='pagenum'><a name="Page_42" id="Page_42">[Pg 42]</a></span>
+prolonged exertions that they could be found by studying
+directly the relations of the elementary triangles. The knowledge
+of the general theorem spares us these efforts.</p>
+
+<p>A construction, therefore, becomes interesting only when it
+can be ranged beside other analogous constructions, forming species
+of the same genus.</p>
+
+<p>If the quadrilateral is something besides the juxtaposition of
+two triangles, this is because it belongs to the genus polygon.</p>
+
+<p>Moreover, one must be able to demonstrate the properties of
+the genus without being forced to establish them successively for
+each of the species.</p>
+
+<p>To attain that, we must necessarily mount from the particular
+to the general, ascending one or more steps.</p>
+
+<p>The analytic procedure 'by construction' does not oblige us
+to descend, but it leaves us at the same level.</p>
+
+<p>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.</p>
+
+<p>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.</p>
+
+
+
+<hr style="width: 65%;" />
+<p><span class='pagenum'><a name="Page_43" id="Page_43">[Pg 43]</a></span></p>
+<h3>CHAPTER II</h3>
+
+<h3><span class="smcap">Mathematical Magnitude and Experience</span></h3>
+
+
+<p>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 importance
+than the whiteness of the chalk.</p>
+
+<p>The pure analyst has not this rock to fear. He has disengaged
+the science of mathematics from all 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 <i>Introduction &agrave; la th&eacute;orie des fonctions d'une variable</i>.</p>
+
+<p>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 <i>exterior</i> to one another. This is not the ordinary conception,
+wherein is supposed between the elements of the continuum
+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
+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<span class='pagenum'><a name="Page_44" id="Page_44">[Pg 44]</a></span>
+enough to apprise us that the veritable mathematical continuum
+is a very different thing from that of the physicists and that of
+the metaphysicians.</p>
+
+<p>It may also be said perhaps that the mathematicians who are
+content with this definition are dupes of words, that it is necessary
+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 property
+of these steps which is used in their reasonings<a name="FNanchor_2_2" id="FNanchor_2_2"></a><a href="#Footnote_2_2" class="fnanchor">[2]</a> is that of
+being before or after such and such steps; therefore also this
+alone should occur in the definition.</p>
+
+<p>So how the intermediary terms should be intercalated need
+not concern us; on the other hand, no one will doubt the possibility
+of this operation, unless from forgetting that possible, in
+the language of geometers, simply means free from contradiction.</p>
+
+<p>Our definition, however, is not yet complete, and I return to
+it after this over-long digression.</p>
+
+
+<p><span class="smcap">Definition of Incommensurables.</span>&mdash;The mathematicians of
+the Berlin school, Kronecker in particular, have devoted themselves
+to constructing this continuous scale of fractional and irrational
+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.</p>
+
+<p>The notion of the rational number seeming to them to present
+no difficulty, they have chiefly striven to define the incommensurable
+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.</p>
+
+<p>Mathematicians study not objects, but relations between objects;
+the replacement of these objects by others is therefore
+indifferent to them, provided the relations do not change. The
+matter is for them unimportant, the form alone interests them.</p>
+
+<p>Without recalling this, it would scarcely be comprehensible
+that Dedekind should designate by the name <i>incommensurable
+number</i> a mere symbol, that is to say, something very different<span class='pagenum'><a name="Page_45" id="Page_45">[Pg 45]</a></span>
+from the ordinary idea of a quantity, which should be measurable
+and almost tangible.</p>
+
+<p>Let us see now what Dedekind's definition is:</p>
+
+<p>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.</p>
+
+<p>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 first class. The number 2 may be
+chosen as symbol of this partition.</p>
+
+<p>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.</p>
+
+<p>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 example,
+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. Evidently
+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.</p>
+
+<p>In Dedekind's view, the incommensurable number</p>
+
+<p class="center">
+&#8730;2 or (2)<sup>&frac12;</sup><br />
+</p>
+
+<p class="noidt">is nothing but the symbol of this particular mode of partition
+of commensurable numbers; and to each mode of partition corresponds
+thus a number, commensurable or not, which serves as
+its symbol.</p>
+
+<p>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,<span class='pagenum'><a name="Page_46" id="Page_46">[Pg 46]</a></span>
+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?</p>
+
+
+<p><span class="smcap">The Physical Continuum.</span>&mdash;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.</p>
+
+<p>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 example,
+that a weight <i>A</i> of 10 grams and a weight <i>B</i> of 11 grams
+produce identical sensations, that the weight <i>B</i> is just as indistinguishable
+from a weight <i>C</i> of 12 grams, but that the weight <i>A</i>
+is easily distinguished from the weight <i>C</i>. Thus the raw results
+of experience may be expressed by the following relations:</p>
+
+<p class="center">
+<i>A</i> = <i>B</i>, &nbsp;&nbsp;&nbsp;&nbsp;&nbsp; <i>B</i> = <i>C</i>, &nbsp;&nbsp;&nbsp;&nbsp;&nbsp; <i>A</i> &lt; <i>C</i>,<br />
+</p>
+
+<p class="noidt">which may be regarded as the formula of the physical continuum.</p>
+
+<p>But here is an intolerable discord with the principle of contradiction,
+and the need of stopping this has compelled us to
+invent the mathematical continuum.</p>
+
+<p>We are, therefore, forced to conclude that this notion has
+been created entirely by the mind, but that experience has given
+the occasion.</p>
+
+<p>We can not believe that two quantities equal to a third are
+not equal to one another, and so we are led to suppose that <i>A</i> is
+different from <i>B</i> and <i>B</i> from <i>C</i>, but that the imperfection of our
+senses has not permitted of our distinguishing them.</p>
+
+
+<p><span class="smcap">Creation of the Mathematical Continuum.</span>&mdash;<i>First Stage.</i>
+So far it would suffice, in accounting for the facts, to intercalate
+between <i>A</i> and <i>B</i> a few terms, which would remain discrete.
+What happens now if we have recourse to some instrument to<span class='pagenum'><a name="Page_47" id="Page_47">[Pg 47]</a></span>
+supplement the feebleness of our senses, if, for example, we
+make use of a microscope? Terms such as <i>A</i> and <i>B</i>, before indistinguishable,
+appear now distinct; but between <i>A</i> and <i>B</i>, now become
+distinct, will be intercalated a new term, <i>D</i>, that we can
+distinguish neither from <i>A</i> nor from <i>B</i>. Despite the employment
+of the most highly perfected methods, the raw results of our
+experience will always present the characteristics of the physical
+continuum with the contradiction which is inherent in it.</p>
+
+<p>We shall escape it only by incessantly intercalating new terms
+between the terms already distinguished, and this operation must
+be continued indefinitely. We might conceive the stopping of
+this operation if we could imagine some instrument sufficiently
+powerful to decompose the physical continuum into discrete elements,
+as the telescope resolves the milky way into stars. But
+this we can not imagine; in fact, it is with the eye we observe the
+image magnified by the microscope, and consequently this image
+must always retain the characteristics of visual sensation and
+consequently those of the physical continuum.</p>
+
+<p>Nothing distinguishes a length observed directly from the
+half of this length doubled by the microscope. The whole is
+homogeneous with the part; this is a new contradiction, or
+rather it would be if the number of terms were supposed finite;
+in fact, it is clear that the part containing fewer terms than the
+whole could not be similar to the whole.</p>
+
+<p>The contradiction ceases when the number of terms is regarded
+as infinite; nothing hinders, for example, considering the aggregate
+of whole numbers as similar to the aggregate of even numbers,
+which, however, is only a part of it; and, in fact, to each
+whole number corresponds an even number, its double.</p>
+
+<p>But it is not only to escape this contradiction contained in the
+empirical data that the mind is led to create the concept of a
+continuum, formed of an indefinite number of terms.</p>
+
+<p>All happens as in the sequence of whole numbers. We have
+the faculty of conceiving that a unit can be added to a collection
+of units; thanks to experience, we have occasion to exercise this
+faculty and we become conscious of it; but from this moment
+we feel that our power has no limit and that we can count indefinitely,
+though we have never had to count more than a finite
+number of objects.<span class='pagenum'><a name="Page_48" id="Page_48">[Pg 48]</a></span></p>
+
+<p>Just so, as soon as we have been led to intercalate means
+between two consecutive terms of a series, we feel that this operation
+can be continued beyond all limit, and that there is, so to
+speak, no intrinsic reason for stopping.</p>
+
+<p>As an abbreviation, let me call a mathematical continuum
+of the first order every aggregate of terms formed according to
+the same law as the scale of commensurable numbers. If we
+afterwards intercalate new steps according to the law of formation
+of incommensurable numbers, we shall obtain what we
+will call a continuum of the second order.</p>
+
+<p><i>Second Stage.</i>&mdash;We have made hitherto only the first stride;
+we have explained the origin of continua of the first order; but it
+is necessary to see why even they are not sufficient and why the
+incommensurable numbers had to be invented.</p>
+
+<p>If we try to imagine a line, it must have the characteristics
+of the physical continuum, that is to say, we shall not be able
+to represent it except with a certain breadth. Two lines then
+will appear to us under the form of two narrow bands, and, if
+we are content with this rough image, it is evident that if the
+two lines cross they will have a common part.</p>
+
+<p>But the pure geometer makes a further effort; without entirely
+renouncing the aid of the senses, he tries to reach the concept of
+the line without breadth, of the point without extension. This
+he can only attain to by regarding the line as the limit toward
+which tends an ever narrowing band, and the point as the limit
+toward which tends an ever lessening area. And then, our two
+bands, however narrow they may be, will always have a common
+area, the smaller as they are the narrower, and whose limit will
+be what the pure geometer calls a point.</p>
+
+<p>This is why it is said two lines which cross have a point in
+common, and this truth seems intuitive.</p>
+
+<p>But it would imply contradiction if lines were conceived as
+continua of the first order, that is to say, if on the lines traced
+by the geometer should be found only points having for coordinates
+rational numbers. The contradiction would be manifest
+as soon as one affirmed, for example, the existence of straights
+and circles.</p>
+
+<p>It is clear, in fact, that if the points whose coordinates are<span class='pagenum'><a name="Page_49" id="Page_49">[Pg 49]</a></span>
+commensurable were alone regarded as real, the circle inscribed
+in a square and the diagonal of this square would not intersect,
+since the coordinates of the point of intersection are incommensurable.</p>
+
+<p>That would not yet be sufficient, because we should get in this
+way only certain incommensurable numbers and not all those
+numbers.</p>
+
+<p>But conceive of a straight line divided into two rays. Each
+of these rays will appear to our imagination as a band of a certain
+breadth; these bands moreover will encroach one on the
+other, since there must be no interval between them. The common
+part will appear to us as a point which will always remain
+when we try to imagine our bands narrower and narrower, so
+that we admit as an intuitive truth that if a straight is cut into
+two rays their common frontier is a point; we recognize here the
+conception of Dedekind, in which an incommensurable number
+was regarded as the common frontier of two classes of rational
+numbers.</p>
+
+<p>Such is the origin of the continuum of the second order, which
+is the mathematical continuum properly so called.</p>
+
+<p><i>R&eacute;sum&eacute;.</i>&mdash;In recapitulation, the mind has the faculty of creating
+symbols, and it is thus that it has constructed the mathematical
+continuum, which is only a particular system of symbols.
+Its power is limited only by the necessity of avoiding all contradiction;
+but the mind only makes use of this faculty if experience
+furnishes it a stimulus thereto.</p>
+
+<p>In the case considered, this stimulus was the notion of the
+physical continuum, drawn from the rough data of the senses.
+But this notion leads to a series of contradictions from which it
+is necessary successively to free ourselves. So we are forced to
+imagine a more and more complicated system of symbols. That
+at which we stop is not only exempt from internal contradiction
+(it was so already at all the stages we have traversed), but
+neither is it in contradiction with various propositions called intuitive,
+which are derived from empirical notions more or less
+elaborated.</p>
+
+<p><span class="smcap">Measurable Magnitude.</span>&mdash;The magnitudes we have studied
+hitherto are not <i>measurable</i>; we can indeed say whether a given<span class='pagenum'><a name="Page_50" id="Page_50">[Pg 50]</a></span>
+one of these magnitudes is greater than another, but not whether
+it is twice or thrice as great.</p>
+
+<p>So far, I have only considered the order in which our terms
+are ranged. But for most applications that does not suffice. We
+must learn to compare the interval which separates any two
+terms. Only on this condition does the continuum become a
+measurable magnitude and the operations of arithmetic applicable.</p>
+
+<p>This can only be done by the aid of a new and special <i>convention</i>.
+We will <i>agree</i> that in such and such a case the interval
+comprised between the terms <i>A</i> and <i>B</i> is equal to the interval
+which separates <i>C</i> and <i>D</i>. For example, at the beginning of our
+work we have set out from the scale of the whole numbers and we
+have supposed intercalated between two consecutive steps <i>n</i>
+intermediary steps; well, these new steps will be by convention
+regarded as equidistant.</p>
+
+<p>This is a way of defining the addition of two magnitudes, because
+if the interval <i>AB</i> is by definition equal to the interval <i>CD</i>,
+the interval <i>AD</i> will be by definition the sum of the intervals
+<i>AB</i> and <i>AC</i>.</p>
+
+<p>This definition is arbitrary in a very large measure. It is not
+completely so, however. It is subjected to certain conditions
+and, for example, to the rules of commutativity and associativity
+of addition. But provided the definition chosen satisfies these
+rules, the choice is indifferent, and it is useless to particularize it.</p>
+
+<p><span class="smcap">Various Remarks.</span>&mdash;We can now discuss several important
+questions:</p>
+
+<p>1&ordm; Is the creative power of the mind exhausted by the creation
+of the mathematical continuum?</p>
+
+<p>No: the works of Du Bois-Reymond demonstrate it in a striking
+way.</p>
+
+<p>We know that mathematicians distinguish between infinitesimals
+of different orders and that those of the second order are
+infinitesimal, not only in an absolute way, but also in relation
+to those of the first order. It is not difficult to imagine infinitesimals
+of fractional or even of irrational order, and thus we find
+again that scale of the mathematical continuum which has been
+dealt with in the preceding pages.<span class='pagenum'><a name="Page_51" id="Page_51">[Pg 51]</a></span></p>
+
+<p>Further, there are infinitesimals which are infinitely small in
+relation to those of the first order, and, on the contrary, infinitely
+great in relation to those of order 1 + &#949;, and that however small
+&#949; may be. Here, then, are new terms intercalated in our series,
+and if I may be permitted to revert to the phraseology lately employed
+which is very convenient though not consecrated by usage,
+I shall say that thus has been created a sort of continuum of the
+third order.</p>
+
+<p>It would be easy to go further, but that would be idle; one
+would only be imagining symbols without possible application,
+and no one will think of doing that. The continuum of the third
+order, to which the consideration of the different orders of infinitesimals
+leads, is itself not useful enough to have won citizenship,
+and geometers regard it only as a mere curiosity. The mind uses
+its creative faculty only when experience requires it.</p>
+
+<p>2&ordm; Once in possession of the concept of the mathematical continuum,
+is one safe from contradictions analogous to those which
+gave birth to it?</p>
+
+<p>No, and I will give an example.</p>
+
+<p>One must be very wise not to regard it as evident that every
+curve has a tangent; and in fact if we picture this curve and a
+straight as two narrow bands we can always so dispose them that
+they have a part in common without crossing. If we imagine
+then the breadth of these two bands to diminish indefinitely, this
+common part will always subsist and, at the limit, so to speak, the
+two lines will have a point in common without crossing, that is to
+say, they will be tangent.</p>
+
+<p>The geometer who reasons in this way, consciously or not, is
+only doing what we have done above to prove two lines which
+cut have a point in common, and his intuition might seem just as
+legitimate.</p>
+
+<p>It would deceive him however. We can demonstrate that
+there are curves which have no tangent, if such a curve is defined
+as an analytic continuum of the second order.</p>
+
+<p>Without doubt some artifice analogous to those we have discussed
+above would have sufficed to remove the contradiction;
+but, as this is met with only in very exceptional cases, it has
+received no further attention.<span class='pagenum'><a name="Page_52" id="Page_52">[Pg 52]</a></span></p>
+
+<p>Instead of seeking to reconcile intuition with analysis, we have
+been content to sacrifice one of the two, and as analysis must
+remain impeccable, we have decided against intuition.</p>
+
+<p><span class="smcap">The Physical Continuum of Several Dimensions.</span>&mdash;We
+have discussed above the physical continuum as derived from the
+immediate data of our senses, or, if you wish, from the rough results
+of Fechner's experiments; I have shown that these results
+are summed up in the contradictory formulas</p>
+
+<p class="center"><i>A</i> = <i>B</i>, &nbsp;&nbsp;&nbsp;&nbsp;&nbsp; <i>B</i> = <i>C</i>, &nbsp;&nbsp;&nbsp;&nbsp;&nbsp; <i>A</i> &lt; <i>C</i>.</p>
+
+<p class="noidt">Let us now see how this notion has been generalized and how
+from it has come the concept of many-dimensional continua.</p>
+
+<p>Consider any two aggregates of sensations. Either we can
+discriminate them one from another, or we can not, just as in
+Fechner's experiments a weight of 10 grams can be distinguished
+from a weight of 12 grams, but not from a weight of 11 grams.
+This is all that is required to construct the continuum of several
+dimensions.</p>
+
+<p>Let us call one of these aggregates of sensations an <i>element</i>.
+That will be something analogous to the <i>point</i> of the mathematicians;
+it will not be altogether the same thing however.
+We can not say our element is without extension, since we can
+not distinguish it from neighboring elements and it is thus
+surrounded by a sort of haze. If the astronomical comparison
+may be allowed, our 'elements' would be like nebulae, whereas
+the mathematical points would be like stars.</p>
+
+<p>That being granted, a system of elements will form a <i>continuum</i>
+if we can pass from any one of them to any other, by a
+series of consecutive elements such that each is indistinguishable
+from the preceding. This <i>linear</i> series is to the <i>line</i> of the
+mathematician what an isolated <i>element</i> was to the point.</p>
+
+<p>Before going farther, I must explain what is meant by a
+<i>cut</i>. Consider a continuum <i>C</i> and remove from it certain of its
+elements which for an instant we shall regard as no longer belonging
+to this continuum. The aggregate of the elements so
+removed will be called a cut. It may happen that, thanks to this
+cut, <i>C</i> may be <i>subdivided</i> into several distinct continua, the aggregate
+of the remaining elements ceasing to form a unique continuum.<span class='pagenum'><a name="Page_53" id="Page_53">[Pg 53]</a></span></p>
+
+<p>There will then be on <i>C</i> two elements, <i>A</i> and <i>B</i>, that must be
+regarded as belonging to two distinct continua, and this will be
+recognized because it will be impossible to find a linear series
+of consecutive elements of <i>C</i>, each of these elements indistinguishable
+from the preceding, the first being <i>A</i> and the last <i>B</i>,
+<i>without one of the elements of this series being indistinguishable
+from one of the elements of the cut</i>.</p>
+
+<p>On the contrary, it may happen that the cut made is insufficient
+to subdivide the continuum <i>C</i>. To classify the physical
+continua, we will examine precisely what are the cuts which must
+be made to subdivide them.</p>
+
+<p>If a physical continuum <i>C</i> can be subdivided by a cut reducing
+to a finite number of elements all distinguishable from one
+another (and consequently forming neither a continuum, nor
+several continua), we shall say <i>C</i> is a <i>one-dimensional</i> continuum.</p>
+
+<p>If, on the contrary, <i>C</i> can be subdivided only by cuts which
+are themselves continua, we shall say <i>C</i> has several dimensions.
+If cuts which are continua of one dimension suffice, we
+shall say <i>C</i> has two dimensions; if cuts of two dimensions suffice,
+we shall say <i>C</i> has three dimensions, and so on.</p>
+
+<p>Thus is defined the notion of the physical continuum of several
+dimensions, thanks to this very simple fact that two aggregates
+of sensations are distinguishable or indistinguishable.</p>
+
+<p><span class="smcap">The Mathematical Continuum of Several Dimensions.</span>&mdash;Thence
+the notion of the mathematical continuum of <i>n</i> dimensions
+has sprung quite naturally by a process very like that we
+discussed at the beginning of this chapter. A point of such a
+continuum, you know, appears to us as defined by a system of
+n distinct magnitudes called its coordinates.</p>
+
+<p>These magnitudes need not always be measurable; there is,
+for instance, a branch of geometry independent of the measurement
+of these magnitudes, in which it is only a question of knowing,
+for example, whether on a curve <i>ABC</i>, the point <i>B</i> is between
+the points <i>A</i> and <i>C</i>, and not of knowing whether the arc
+<i>AB</i> is equal to the arc <i>BC</i> or twice as great. This is what is
+called <i>Analysis Situs</i>.</p>
+
+<p>This is a whole body of doctrine which has attracted the<span class='pagenum'><a name="Page_54" id="Page_54">[Pg 54]</a></span>
+attention of the greatest geometers and where we see flow one
+from another a series of remarkable theorems. What distinguishes
+these theorems from those of ordinary geometry is that
+they are purely qualitative and that they would remain true if
+the figures were copied by a draughtsman so awkward as to
+grossly distort the proportions and replace straights by strokes
+more or less curved.</p>
+
+<p>Through the wish to introduce measure next into the continuum
+just defined this continuum becomes space, and geometry is
+born. But the discussion of this is reserved for Part Second.</p>
+
+
+
+<hr style="width: 65%;" />
+<p><span class='pagenum'><a name="Page_55" id="Page_55">[Pg 55]</a></span></p>
+<h2><b>PART II<br />
+<br />
+<small>SPACE</small></b></h2>
+
+
+
+<hr style="width: 65%;" />
+<h3>CHAPTER III</h3>
+
+<h3><span class="smcap">The Non-euclidean Geometries</span></h3>
+
+
+<p>Every conclusion supposes premises; these premises themselves
+either are self-evident and need no demonstration, or can be
+established only by relying upon other propositions, and since
+we can not go back thus to infinity, every deductive science, and
+in particular geometry, must rest on a certain number of undemonstrable
+axioms. All treatises on geometry begin, therefore,
+by the enunciation of these axioms. But among these there is a
+distinction to be made: Some, for example, 'Things which are
+equal to the same thing are equal to one another,' are not propositions
+of geometry, but propositions of analysis. I regard them
+as analytic judgments <i>a priori</i>, and shall not concern myself with
+them.</p>
+
+<p>But I must lay stress upon other axioms which are peculiar to
+geometry. Most treatises enunciate three of these explicitly:</p>
+
+<p>1&ordm; Through two points can pass only one straight;</p>
+
+<p>2&ordm; The straight line is the shortest path from one point to
+another;</p>
+
+<p>3&ordm; Through a given point there is not more than one parallel
+to a given straight.</p>
+
+<p>Although generally a proof of the second of these axioms is
+omitted, it would be possible to deduce it from the other two and
+from those, much more numerous, which are implicitly admitted
+without enunciating them, as I shall explain further on.</p>
+
+<p>It was long sought in vain to demonstrate likewise the third
+axiom, known as <i>Euclid's Postulate</i>. What vast effort has been
+wasted in this chimeric hope is truly unimaginable. Finally, in<span class='pagenum'><a name="Page_56" id="Page_56">[Pg 56]</a></span>
+the first quarter of the nineteenth century, and almost at the
+same time, a Hungarian and a Russian, Bolyai and Lobachevski,
+established irrefutably that this demonstration is impossible; they
+have almost rid us of inventors of geometries 'sans postulatum';
+since then the Acad&eacute;mie des Sciences receives only about one or
+two new demonstrations a year.</p>
+
+<p>The question was not exhausted; it soon made a great
+stride by the publication of Riemann's celebrated memoir entitled:
+<i>Ueber die Hypothesen welche der Geometrie zu Grunde
+liegen</i>. This paper has inspired most of the recent works of which
+I shall speak further on, and among which it is proper to cite
+those of Beltrami and of Helmholtz.</p>
+
+<p><span class="smcap">The Bolyai-Lobachevski Geometry.</span>&mdash;If it were possible to
+deduce Euclid's postulate from the other axioms, it is evident
+that in denying the postulate and admitting the other axioms, we
+should be led to contradictory consequences; it would therefore
+be impossible to base on such premises a coherent geometry.</p>
+
+<p>Now this is precisely what Lobachevski did.</p>
+
+<p>He assumes at the start that: <i>Through a given point can be
+drawn two parallels to a given straight</i>.</p>
+
+<p>And he retains besides all Euclid's other axioms. From these
+hypotheses he deduces a series of theorems among which it is
+impossible to find any contradiction, and he constructs a
+geometry whose faultless logic is inferior in nothing to that of
+the Euclidean geometry.</p>
+
+<p>The theorems are, of course, very different from those to which
+we are accustomed, and they can not fail to be at first a little
+disconcerting.</p>
+
+<p>Thus the sum of the angles of a triangle is always less than
+two right angles, and the difference between this sum and two
+right angles is proportional to the surface of the triangle.</p>
+
+<p>It is impossible to construct a figure similar to a given figure
+but of different dimensions.</p>
+
+<p>If we divide a circumference into <i>n</i> equal parts, and draw
+tangents at the points of division, these <i>n</i> tangents will form a
+polygon if the radius of the circle is small enough; but if this
+radius is sufficiently great they will not meet.</p>
+
+<p>It is useless to multiply these examples; Lobachevski's<span class='pagenum'><a name="Page_57" id="Page_57">[Pg 57]</a></span>
+propositions have no relation to those of Euclid, but they are not less
+logically bound one to another.</p>
+
+<p><span class="smcap">Riemann's Geometry.</span>&mdash;Imagine a world uniquely peopled
+by beings of no thickness (height); and suppose these 'infinitely
+flat' animals are all in the same plane and can not get out. Admit
+besides that this world is sufficiently far from others to be
+free from their influence. While we are making hypotheses, it
+costs us no more to endow these beings with reason and believe
+them capable of creating a geometry. In that case, they will certainly
+attribute to space only two dimensions.</p>
+
+<p>But suppose now that these imaginary animals, while remaining
+without thickness, have the form of a spherical, and not of a
+plane, figure, and are all on the same sphere without power to get
+off. What geometry will they construct? First it is clear they
+will attribute to space only two dimensions; what will play for
+them the r&ocirc;le of the straight line will be the shortest path from
+one point to another on the sphere, that is to say, an arc of a great
+circle; in a word, their geometry will be the spherical geometry.</p>
+
+<p>What they will call space will be this sphere on which they
+must stay, and on which happen all the phenomena they can
+know. Their space will therefore be <i>unbounded</i> since on a
+sphere one can always go forward without ever being stopped,
+and yet it will be <i>finite</i>; one can never find the end of it, but one
+can make a tour of it.</p>
+
+<p>Well, Riemann's geometry is spherical geometry extended to
+three dimensions. To construct it, the German mathematician
+had to throw overboard, not only Euclid's postulate, but also the
+first axiom: <i>Only one straight can pass through two points</i>.</p>
+
+<p>On a sphere, through two given points we can draw <i>in general</i>
+only one great circle (which, as we have just seen, would play the
+r&ocirc;le of the straight for our imaginary beings); but there is an
+exception: if the two given points are diametrically opposite, an
+infinity of great circles can be drawn through them.</p>
+
+<p>In the same way, in Riemann's geometry (at least in one of
+its forms), through two points will pass in general only a single
+straight; but there are exceptional cases where through two
+points an infinity of straights can pass.<span class='pagenum'><a name="Page_58" id="Page_58">[Pg 58]</a></span></p>
+
+<p>There is a sort of opposition between Riemann's geometry and
+that of Lobachevski.</p>
+
+<p>Thus the sum of the angles of a triangle is:</p>
+
+<p>Equal to two right angles in Euclid's geometry;</p>
+
+<p>Less than two right angles in that of Lobachevski;</p>
+
+<p>Greater than two right angles in that of Riemann.</p>
+
+<p>The number of straights through a given point that can be
+drawn coplanar to a given straight, but nowhere meeting it, is
+equal:</p>
+
+<p>To one in Euclid's geometry;</p>
+
+<p>To zero in that of Riemann;</p>
+
+<p>To infinity in that of Lobachevski.</p>
+
+<p>Add that Riemann's space is finite, although unbounded, in
+the sense given above to these two words.</p>
+
+<p><span class="smcap">The Surfaces of Constant Curvature.</span>&mdash;One objection still
+remained possible. The theorems of Lobachevski and of Riemann
+present no contradiction; but however numerous the consequences
+these two geometers have drawn from their hypotheses,
+they must have stopped before exhausting them, since their number
+would be infinite; who can say then that if they had pushed
+their deductions farther they would not have eventually reached
+some contradiction?</p>
+
+<p>This difficulty does not exist for Riemann's geometry, provided
+it is limited to two dimensions; in fact, as we have seen,
+two-dimensional Riemannian geometry does not differ from spherical
+geometry, which is only a branch of ordinary geometry, and
+consequently is beyond all discussion.</p>
+
+<p>Beltrami, in correlating likewise Lobachevski's two-dimensional
+geometry with a branch of ordinary geometry, has equally
+refuted the objection so far as it is concerned.</p>
+
+<p>Here is how he accomplished it. Consider any figure on a
+surface. Imagine this figure traced on a flexible and inextensible
+canvas applied over this surface in such a way that when the
+canvas is displaced and deformed, the various lines of this figure
+can change their form without changing their length. In general,
+this flexible and inextensible figure can not be displaced
+without leaving the surface; but there are certain particular surfaces<span class='pagenum'><a name="Page_59" id="Page_59">[Pg 59]</a></span>
+for which such a movement would be possible; these are the
+surfaces of constant curvature.</p>
+
+<p>If we resume the comparison made above and imagine beings
+without thickness living on one of these surfaces, they will regard
+as possible the motion of a figure all of whose lines remain constant
+in length. On the contrary, such a movement would appear
+absurd to animals without thickness living on a surface of variable
+curvature.</p>
+
+<p>These surfaces of constant curvature are of two sorts: Some
+are of <i>positive curvature</i>, and can be deformed so as to be applied
+over a sphere. The geometry of these surfaces reduces itself
+therefore to the spherical geometry, which is that of Riemann.</p>
+
+<p>The others are of <i>negative curvature</i>. Beltrami has shown
+that the geometry of these surfaces is none other than that of
+Lobachevski. The two-dimensional geometries of Riemann and
+Lobachevski are thus correlated to the Euclidean geometry.</p>
+
+<p><span class="smcap">Interpretation of Non-Euclidean Geometries.</span>&mdash;So vanishes
+the objection so far as two-dimensional geometries are concerned.</p>
+
+<p>It would be easy to extend Beltrami's reasoning to three-dimensional
+geometries. The minds that space of four dimensions
+does not repel will see no difficulty in it, but they are few.
+I prefer therefore to proceed otherwise.</p>
+
+<p>Consider a certain plane, which I shall call the fundamental
+plane, and construct a sort of dictionary, by making correspond
+each to each a double series of terms written in two columns, just
+as correspond in the ordinary dictionaries the words of two languages
+whose significance is the same:</p>
+
+<p><i>Space</i>: Portion of space situated above the fundamental plane.</p>
+
+<p><i>Plane</i>: Sphere cutting the fundamental plane orthogonally.</p>
+
+<p><i>Straight</i>: Circle cutting the fundamental plane orthogonally.</p>
+
+<p><i>Sphere</i>: Sphere.</p>
+
+<p><i>Circle</i>: Circle.</p>
+
+<p><i>Angle</i>: Angle.</p>
+
+<p><i>Distance between two points</i>: Logarithm of the cross ratio of
+these two points and the intersections of the fundamental plane
+with a circle passing through these two points and cutting it
+orthogonally. Etc., Etc.<span class='pagenum'><a name="Page_60" id="Page_60">[Pg 60]</a></span></p>
+
+<p>Now take Lobachevski's theorems and translate them with
+the aid of this dictionary as we translate a German text with the
+aid of a German-English dictionary. <i>We shall thus obtain theorems
+of the ordinary geometry.</i> For example, that theorem of
+Lobachevski: 'the sum of the angles of a triangle is less than two
+right angles' is translated thus: "If a curvilinear triangle has
+for sides circle-arcs which prolonged would cut orthogonally the
+fundamental plane, the sum of the angles of this curvilinear triangle
+will be less than two right angles." Thus, however far the
+consequences of Lobachevski's hypotheses are pushed, they will
+never lead to a contradiction. In fact, if two of Lobachevski's
+theorems were contradictory, it would be the same with the translations
+of these two theorems, made by the aid of our dictionary,
+but these translations are theorems of ordinary geometry and no
+one doubts that the ordinary geometry is free from contradiction.
+Whence comes this certainty and is it justified? That is a question
+I can not treat here because it would require to be enlarged
+upon, but which is very interesting and I think not insoluble.</p>
+
+<p>Nothing remains then of the objection above formulated.
+This is not all. Lobachevski's geometry, susceptible of a concrete
+interpretation, ceases to be a vain logical exercise and is capable
+of applications; I have not the time to speak here of these applications,
+nor of the aid that Klein and I have gotten from them
+for the integration of linear differential equations.</p>
+
+<p>This interpretation moreover is not unique, and several dictionaries
+analogous to the preceding could be constructed, which
+would enable us by a simple 'translation' to transform Lobachevski's
+theorems into theorems of ordinary geometry.</p>
+
+<p><span class="smcap">The Implicit Axioms.</span>&mdash;Are the axioms explicitly enunciated
+in our treatises the sole foundations of geometry? We may be
+assured of the contrary by noticing that after they are successively
+abandoned there are still left over some propositions common
+to the theories of Euclid, Lobachevski and Riemann. These
+propositions must rest on premises the geometers admit without
+enunciation. It is interesting to try to disentangle them from
+the classic demonstrations.</p>
+
+<p>Stuart Mill has claimed that every definition contains an<span class='pagenum'><a name="Page_61" id="Page_61">[Pg 61]</a></span>
+axiom, because in defining one affirms implicitly the existence
+of the object defined. This is going much too far; it is rare that
+in mathematics a definition is given without its being followed by
+the demonstration of the existence of the object defined, and
+when this is dispensed with it is generally because the reader
+can easily supply it. It must not be forgotten that the word
+existence has not the same sense when it refers to a mathematical
+entity and when it is a question of a material object. A mathematical
+entity exists, provided its definition implies no contradiction,
+either in itself, or with the propositions already admitted.</p>
+
+<p>But if Stuart Mill's observation can not be applied to all
+definitions, it is none the less just for some of them. The plane
+is sometimes defined as follows:</p>
+
+<p>The plane is a surface such that the straight which joins any
+two of its points is wholly on this surface.</p>
+
+<p>This definition manifestly hides a new axiom; it is true we
+might change it, and that would be preferable, but then we
+should have to enunciate the axiom explicitly.</p>
+
+<p>Other definitions would suggest reflections not less important.</p>
+
+<p>Such, for example, is that of the equality of two figures; two
+figures are equal when they can be superposed; to superpose
+them one must be displaced until it coincides with the other; but
+how shall it be displaced? If we should ask this, no doubt we
+should be told that it must be done without altering the shape
+and as a rigid solid. The vicious circle would then be evident.</p>
+
+<p>In fact this definition defines nothing; it would have no meaning
+for a being living in a world where there were only fluids.
+If it seems clear to us, that is because we are used to the properties
+of natural solids which do not differ much from those of the
+ideal solids, all of whose dimensions are invariable.</p>
+
+<p>Yet, imperfect as it may be, this definition implies an axiom.</p>
+
+<p>The possibility of the motion of a rigid figure is not a self-evident
+truth, or at least it is so only in the fashion of Euclid's
+postulate and not as an analytic judgment <i>a priori</i> would be.</p>
+
+<p>Moreover, in studying the definitions and the demonstrations
+of geometry, we see that one is obliged to admit without proof
+not only the possibility of this motion, but some of its properties
+besides.<span class='pagenum'><a name="Page_62" id="Page_62">[Pg 62]</a></span></p>
+
+<p>This is at once seen from the definition of the straight line.
+Many defective definitions have been given, but the true one is
+that which is implied in all the demonstrations where the straight
+line enters:</p>
+
+<p>"It may happen that the motion of a rigid figure is such that
+all the points of a line belonging to this figure remain motionless
+while all the points situated outside of this line move. Such a
+line will be called a straight line." We have designedly, in this
+enunciation, separated the definition from the axiom it implies.</p>
+
+<p>Many demonstrations, such as those of the cases of the equality
+of triangles, of the possibility of dropping a perpendicular from
+a point to a straight, presume propositions which are not enunciated,
+for they require the admission that it is possible to transport
+a figure in a certain way in space.</p>
+
+<p><span class="smcap">The Fourth Geometry.</span>&mdash;Among these implicit axioms, there
+is one which seems to me to merit some attention, because when
+it is abandoned a fourth geometry can be constructed as coherent
+as those of Euclid, Lobachevski and Riemann.</p>
+
+<p>To prove that a perpendicular may always be erected at a
+point <i>A</i> to a straight <i>AB</i>, we consider a straight <i>AC</i> movable
+around the point <i>A</i> and initially coincident with the fixed
+straight <i>AB</i>; and we make it turn about the point <i>A</i> until it
+comes into the prolongation of <i>AB</i>.</p>
+
+<p>Thus two propositions are presupposed: First, that such a rotation
+is possible, and next that it may be continued until the
+two straights come into the prolongation one of the other.</p>
+
+<p>If the first point is admitted and the second rejected, we are
+led to a series of theorems even stranger than those of Lobachevski
+and Riemann, but equally exempt from contradiction.</p>
+
+<p>I shall cite only one of these theorems and that not the most
+singular: <i>A real straight may be perpendicular to itself</i>.</p>
+
+<p><span class="smcap">Lie's Theorem.</span>&mdash;The number of axioms implicitly introduced
+in the classic demonstrations is greater than necessary, and
+it would be interesting to reduce it to a minimum. It may first
+be asked whether this reduction is possible, whether the number
+of necessary axioms and that of imaginable geometries are not
+infinite.<span class='pagenum'><a name="Page_63" id="Page_63">[Pg 63]</a></span></p>
+
+<p>A theorem of Sophus Lie dominates this whole discussion. It
+may be thus enunciated:</p>
+
+<p>Suppose the following premises are admitted:</p>
+
+<p>1&ordm; Space has <i>n</i> dimensions;</p>
+
+<p>2&ordm; The motion of a rigid figure is possible;</p>
+
+<p>3&ordm; It requires <i>p</i> conditions to determine the position of this
+figure in space.</p>
+
+<p><i>The number of geometries compatible with these premises will
+be limited.</i></p>
+
+<p>I may even add that if <i>n</i> is given, a superior limit can be
+assigned to <i>p</i>.</p>
+
+<p>If therefore the possibility of motion is admitted, there can
+be invented only a finite (and even a rather small) number of
+three-dimensional geometries.</p>
+
+<p><span class="smcap">Riemann's Geometries.</span>&mdash;Yet this result seems contradicted
+by Riemann, for this savant constructs an infinity of different
+geometries, and that to which his name is ordinarily given is only
+a particular case.</p>
+
+<p>All depends, he says, on how the length of a curve is defined.
+Now, there is an infinity of ways of defining this length, and each
+of them may be the starting point of a new geometry.</p>
+
+<p>That is perfectly true, but most of these definitions are incompatible
+with the motion of a rigid figure, which in the theorem
+of Lie is supposed possible. These geometries of Riemann, in
+many ways so interesting, could never therefore be other than
+purely analytic and would not lend themselves to demonstrations
+analogous to those of Euclid.</p>
+
+<p><span class="smcap">On the Nature of Axioms.</span>&mdash;Most mathematicians regard
+Lobachevski's geometry only as a mere logical curiosity; some of
+them, however, have gone farther. Since several geometries are
+possible, is it certain ours is the true one? Experience no doubt
+teaches us that the sum of the angles of a triangle is equal to two
+right angles; but this is because the triangles we deal with are
+too little; the difference, according to Lobachevski, is proportional
+to the surface of the triangle; will it not perhaps become
+sensible when we shall operate on larger triangles or when our
+measurements shall become more precise? The Euclidean geometry
+would thus be only a provisional geometry.<span class='pagenum'><a name="Page_64" id="Page_64">[Pg 64]</a></span></p>
+
+<p>To discuss this opinion, we should first ask ourselves what
+is the nature of the geometric axioms.</p>
+
+<p>Are they synthetic <i>a priori</i> judgments, as Kant said?</p>
+
+<p>They would then impose themselves upon us with such force
+that we could not conceive the contrary proposition, nor build
+upon it a theoretic edifice. There would be no non-Euclidean
+geometry.</p>
+
+<p>To be convinced of it take a veritable synthetic <i>a priori</i>
+judgment, the following, for instance, of which we have seen
+the preponderant r&ocirc;le in the first chapter:</p>
+
+<p><i>If a theorem is true for the number 1, and if it has been proved
+that it is true of n + 1 provided it is true of n, it will be true of
+all the positive whole numbers.</i></p>
+
+<p>Then try to escape from that and, denying this proposition,
+try to found a false arithmetic analogous to non-Euclidean
+geometry&mdash;it can not be done; one would even be tempted at first
+blush to regard these judgments as analytic.</p>
+
+<p>Moreover, resuming our fiction of animals without thickness,
+we can hardly admit that these beings, if their minds are like
+ours, would adopt the Euclidean geometry which would be contradicted
+by all their experience.</p>
+
+<p>Should we therefore conclude that the axioms of geometry are
+experimental verities? But we do not experiment on ideal
+straights or circles; it can only be done on material objects. On
+what then could be based experiments which should serve as
+foundation for geometry? The answer is easy.</p>
+
+<p>We have seen above that we constantly reason as if the geometric
+figures behaved like solids. What geometry would borrow
+from experience would therefore be the properties of these
+bodies. The properties of light and its rectilinear propagation
+have also given rise to some of the propositions of geometry,
+and in particular those of projective geometry, so that from this
+point of view one would be tempted to say that metric geometry
+is the study of solids, and projective, that of light.</p>
+
+<p>But a difficulty remains, and it is insurmountable. If geometry
+were an experimental science, it would not be an exact
+science, it would be subject to a continual revision. Nay, it
+would from this very day be convicted of error, since we know
+that there is no rigorously rigid solid.<span class='pagenum'><a name="Page_65" id="Page_65">[Pg 65]</a></span></p>
+
+<p>The <i>axioms of geometry therefore are neither synthetic</i>
+a priori <i>judgments nor experimental facts</i>.</p>
+
+<p>They are <i>conventions</i>; our choice among all possible conventions
+is <i>guided</i> by experimental facts; but it remains <i>free</i> and is
+limited only by the necessity of avoiding all contradiction. Thus
+it is that the postulates can remain <i>rigorously</i> true even though
+the experimental laws which have determined their adoption are
+only approximative.</p>
+
+<p>In other words, <i>the axioms of geometry</i> (I do not speak of
+those of arithmetic) <i>are merely disguised definitions</i>.</p>
+
+<p>Then what are we to think of that question: Is the Euclidean
+geometry true?</p>
+
+<p>It has no meaning.</p>
+
+<p>As well ask whether the metric system is true and the old
+measures false; whether Cartesian coordinates are true and polar
+coordinates false. One geometry can not be more true than another;
+it can only be <i>more convenient</i>.</p>
+
+<p>Now, Euclidean geometry is, and will remain, the most convenient:</p>
+
+<p>1&ordm; Because it is the simplest; and it is so not only in consequence
+of our mental habits, or of I know not what direct intuition
+that we may have of Euclidean space; it is the simplest in
+itself, just as a polynomial of the first degree is simpler than one
+of the second; the formulas of spherical trigonometry are more
+complicated than those of plane trigonometry, and they would
+still appear so to an analyst ignorant of their geometric signification.</p>
+
+<p>2&ordm; Because it accords sufficiently well with the properties of
+natural solids, those bodies which our hands and our eyes compare
+and with which we make our instruments of measure.</p>
+
+
+
+<hr style="width: 65%;" />
+<p><span class='pagenum'><a name="Page_66" id="Page_66">[Pg 66]</a></span></p>
+<h3>CHAPTER IV</h3>
+
+<h3><span class="smcap">Space and Geometry</span></h3>
+
+
+<p>Let us begin by a little paradox.</p>
+
+<p>Beings with minds like ours, and having the same senses as
+we, but without previous education, would receive from a suitably
+chosen external world impressions such that they would be led
+to construct a geometry other than that of Euclid and to localize
+the phenomena of that external world in a non-Euclidean space,
+or even in a space of four dimensions.</p>
+
+<p>As for us, whose education has been accomplished by our
+actual world, if we were suddenly transported into this new
+world, we should have no difficulty in referring its phenomena to
+our Euclidean space. Conversely, if these beings were transported
+into our environment, they would be led to relate our
+phenomena to non-Euclidean space.</p>
+
+<p>Nay more; with a little effort we likewise could do it. A
+person who should devote his existence to it might perhaps attain
+to a realization of the fourth dimension.</p>
+
+<p><span class="smcap">Geometric Space and Perceptual Space.</span>&mdash;It is often said
+the images of external objects are localized in space, even that
+they can not be formed except on this condition. It is also said
+that this space, which serves thus as a ready prepared <i>frame</i> for
+our sensations and our representations, is identical with that of
+the geometers, of which it possesses all the properties.</p>
+
+<p>To all the good minds who think thus, the preceding statement
+must have appeared quite extraordinary. But let us see
+whether they are not subject to an illusion that a more profound
+analysis would dissipate.</p>
+
+<p>What, first of all, are the properties of space, properly so
+called? I mean of that space which is the object of geometry
+and which I shall call <i>geometric space</i>.</p>
+
+<p>The following are some of the most essential:</p>
+
+<p>1&ordm; It is continuous;<span class='pagenum'><a name="Page_67" id="Page_67">[Pg 67]</a></span></p>
+
+<p>2&ordm; It is infinite;</p>
+
+<p>3&ordm; It has three dimensions;</p>
+
+<p>4&ordm; It is homogeneous, that is to say, all its points are identical
+one with another;</p>
+
+<p>5&ordm; It is isotropic, that is to say, all the straights which pass
+through the same point are identical one with another.</p>
+
+<p>Compare it now to the frame of our representations and our
+sensations, which I may call <i>perceptual space</i>.</p>
+
+<p><span class="smcap">Visual Space.</span>&mdash;Consider first a purely visual impression, due
+to an image formed on the bottom of the retina.</p>
+
+<p>A cursory analysis shows us this image as continuous, but as
+possessing only two dimensions; this already distinguishes from
+geometric space what we may call <i>pure visual space</i>.</p>
+
+<p>Besides, this image is enclosed in a limited frame.</p>
+
+<p>Finally, there is another difference not less important: <i>this
+pure visual space is not homogeneous</i>. All the points of the
+retina, aside from the images which may there be formed, do not
+play the same r&ocirc;le. The yellow spot can in no way be regarded
+as identical with a point on the border of the retina. In fact, not
+only does the same object produce there much more vivid impressions,
+but in every <i>limited</i> frame the point occupying the
+center of the frame will never appear as equivalent to a point
+near one of the borders.</p>
+
+<p>No doubt a more profound analysis would show us that this
+continuity of visual space and its two dimensions are only an
+illusion; it would separate it therefore still more from geometric
+space, but we shall not dwell on this remark.</p>
+
+<p>Sight, however, enables us to judge of distances and consequently
+to perceive a third dimension. But every one knows
+that this perception of the third dimension reduces itself to the
+sensation of the effort at accommodation it is necessary to make,
+and to that of the convergence which must be given to the two
+eyes, to perceive an object distinctly.</p>
+
+<p>These are muscular sensations altogether different from the
+visual sensations which have given us the notion of the first two
+dimensions. The third dimension therefore will not appear to
+us as playing the same r&ocirc;le as the other two. What may be
+called <i>complete visual space</i> is therefore not an isotropic space.<span class='pagenum'><a name="Page_68" id="Page_68">[Pg 68]</a></span></p>
+
+<p>It has, it is true, precisely three dimensions, which means that
+the elements of our visual sensations (those at least which combine
+to form the notion of extension) will be completely defined
+when three of them are known; to use the language of
+mathematics, they will be functions of three independent
+variables.</p>
+
+<p>But examine the matter a little more closely. The third
+dimension is revealed to us in two different ways: by the effort
+of accommodation and by the convergence of the eyes.</p>
+
+<p>No doubt these two indications are always concordant, there
+is a constant relation between them, or, in mathematical terms,
+the two variables which measure these two muscular sensations
+do not appear to us as independent; or again, to avoid an appeal
+to mathematical notions already rather refined, we may go back
+to the language of the preceding chapter and enunciate the same
+fact as follows: If two sensations of convergence, <i>A</i> and <i>B</i>, are
+indistinguishable, the two sensations of accommodation, <i>A&acute;</i> and
+<i>B&acute;</i>, which respectively accompany them, will be equally indistinguishable.</p>
+
+<p>But here we have, so to speak, an experimental fact; <i>a priori</i>
+nothing prevents our supposing the contrary, and if the contrary
+takes place, if these two muscular sensations vary independently
+of one another, we shall have to take account of one more independent
+variable, and 'complete visual space' will appear to us
+as a physical continuum of four dimensions.</p>
+
+<p>We have here even, I will add, a fact of <i>external</i> experience.
+Nothing prevents our supposing that a being with a mind like
+ours, having the same sense organs that we have, may be placed
+in a world where light would only reach him after having
+traversed reflecting media of complicated form. The two indications
+which serve us in judging distances would cease to be
+connected by a constant relation. A being who should achieve
+in such a world the education of his senses would no doubt
+attribute four dimensions to complete visual space.</p>
+
+<p><span class="smcap">Tactile Space and Motor Space.</span>&mdash;'Tactile space' is still
+more complicated than visual space and farther removed from
+geometric space. It is superfluous to repeat for touch the discussion
+I have given for sight.<span class='pagenum'><a name="Page_69" id="Page_69">[Pg 69]</a></span></p>
+
+<p>But apart from the data of sight and touch, there are other
+sensations which contribute as much and more than they to the
+genesis of the notion of space. These are known to every one;
+they accompany all our movements, and are usually called muscular
+sensations.</p>
+
+<p>The corresponding frame constitutes what may be called <i>motor
+space</i>.</p>
+
+<p>Each muscle gives rise to a special sensation capable of augmenting
+or of diminishing, so that the totality of our muscular
+sensations will depend upon as many variables as we have
+muscles. From this point of view, <i>motor space would have as
+many dimensions as we have muscles</i>.</p>
+
+<p>I know it will be said that if the muscular sensations contribute
+to form the notion of space, it is because we have the
+sense of the <i>direction</i> of each movement and that it makes an
+integrant part of the sensation. If this were so, if a muscular
+sensation could not arise except accompanied by this geometric
+sense of direction, geometric space would indeed be a form imposed
+upon our sensibility.</p>
+
+<p>But I perceive nothing at all of this when I analyze my sensations.</p>
+
+<p>What I do see is that the sensations which correspond to movements
+in the same direction are connected in my mind by a mere
+<i>association of ideas</i>. It is to this association that what we call
+'the sense of direction' is reducible. This feeling therefore can
+not be found in a single sensation.</p>
+
+<p>This association is extremely complex, for the contraction of
+the same muscle may correspond, according to the position of the
+limbs, to movements of very different direction.</p>
+
+<p>Besides, it is evidently acquired; it is, like all associations of
+ideas, the result of a <i>habit</i>; this habit itself results from very
+numerous <i>experiences</i>; without any doubt, if the education of our
+senses had been accomplished in a different environment, where
+we should have been subjected to different impressions, contrary
+habits would have arisen and our muscular sensations
+would have been associated according to other laws.</p>
+
+<p><span class="smcap">Characteristics of Perceptual Space.</span>&mdash;Thus perceptual
+space, under its triple form, visual, tactile and motor, is essentially
+different from geometric space.<span class='pagenum'><a name="Page_70" id="Page_70">[Pg 70]</a></span></p>
+
+<p>It is neither homogeneous, nor isotropic; one can not even say
+that it has three dimensions.</p>
+
+<p>It is often said that we 'project' into geometric space the
+objects of our external perception; that we 'localize' them.</p>
+
+<p>Has this a meaning, and if so what?</p>
+
+<p>Does it mean that we <i>represent</i> to ourselves external objects in
+geometric space?</p>
+
+<p>Our representations are only the reproduction of our sensations;
+they can therefore be ranged only in the same frame as
+these, that is to say, in perceptual space.</p>
+
+<p>It is as impossible for us to represent to ourselves external
+bodies in geometric space, as it is for a painter to paint on a
+plane canvas objects with their three dimensions.</p>
+
+<p>Perceptual space is only an image of geometric space, an
+image altered in shape by a sort of perspective, and we can represent
+to ourselves objects only by bringing them under the laws of
+this perspective.</p>
+
+<p>Therefore we do not <i>represent</i> to ourselves external bodies in
+geometric space, but we <i>reason</i> on these bodies as if they were
+situated in geometric space.</p>
+
+<p>When it is said then that we 'localize' such and such an object
+at such and such a point of space, what does it mean?</p>
+
+<p><i>It simply means that we represent to ourselves the movements
+it would be necessary to make to reach that object</i>; and one may
+not say that to represent to oneself these movements, it is necessary
+to project the movements themselves in space and that the
+notion of space must, consequently, pre-exist.</p>
+
+<p>When I say that we represent to ourselves these movements,
+I mean only that we represent to ourselves the muscular sensations
+which accompany them and which have no geometric character
+whatever, which consequently do not at all imply the preexistence
+of the notion of space.</p>
+
+<p><span class="smcap">Change of State and Change of Position.</span>&mdash;But, it will
+be said, if the idea of geometric space is not imposed upon our
+mind, and if, on the other hand, none of our sensations can
+furnish it, how could it have come into existence?</p>
+
+<p>This is what we have now to examine, and it will take some
+time, but I can summarize in a few words the attempt at explanation
+that I am about to develop.<span class='pagenum'><a name="Page_71" id="Page_71">[Pg 71]</a></span></p>
+
+<p><i>None of our sensations, isolated, could have conducted us to
+the idea of space; we are led to it only in studying the laws,
+according to which these sensations succeed each other.</i></p>
+
+<p>We see first that our impressions are subject to change; but
+among the changes we ascertain we are soon led to make a distinction.</p>
+
+<p>At one time we say that the objects which cause these impressions
+have changed state, at another time that they have
+changed position, that they have only been displaced.</p>
+
+<p>Whether an object changes its state or merely its position,
+this is always translated for us in the same manner: <i>by a modification
+in an aggregate of impressions</i>.</p>
+
+<p>How then could we have been led to distinguish between the
+two? It is easy to account for. If there has only been a
+change of position, we can restore the primitive aggregate of
+impressions by making movements which replace us opposite the
+mobile object in the same <i>relative</i> situation. We thus <i>correct</i>
+the modification that happened and we reestablish the initial
+state by an inverse modification.</p>
+
+<p>If it is a question of sight, for example, and if an object
+changes its place before our eye, we can 'follow it with the
+eye' and maintain its image on the same point of the retina by
+appropriate movements of the eyeball.</p>
+
+<p>These movements we are conscious of because they are voluntary
+and because they are accompanied by muscular sensations,
+but that does not mean that we represent them to ourselves in
+geometric space.</p>
+
+<p>So what characterizes change of position, what distinguishes
+it from change of state, is that it can always be corrected in this
+way.</p>
+
+<p>It may therefore happen that we pass from the totality of
+impressions <i>A</i> to the totality <i>B</i> in two different ways:</p>
+
+<p>1&ordm; Involuntarily and without experiencing muscular sensations;
+this happens when it is the object which changes place;</p>
+
+<p>2&ordm; Voluntarily and with muscular sensations; this happens
+when the object is motionless, but we move so that the object has
+relative motion with reference to us.</p>
+
+<p>If this be so, the passage from the totality <i>A</i> to the totality <i>B</i>
+is only a change of position.<span class='pagenum'><a name="Page_72" id="Page_72">[Pg 72]</a></span></p>
+
+<p>It follows from this that sight and touch could not have
+given us the notion of space without the aid of the 'muscular
+sense.'</p>
+
+<p>Not only could this notion not be derived from a single sensation
+or even <i>from a series of sensations</i>, but what is more, an
+<i>immobile</i> being could never have acquired it, since, not being
+able to <i>correct</i> by his movements the effects of the changes of
+position of exterior objects, he would have had no reason whatever
+to distinguish them from changes of state. Just as little
+could he have acquired it if his motions had not been voluntary
+or were unaccompanied by any sensations.</p>
+
+<p><span class="smcap">Conditions of Compensation.</span>&mdash;How is a like compensation
+possible, of such sort that two changes, otherwise independent of
+each other, reciprocally correct each other?</p>
+
+<p>A mind already familiar with geometry would reason as follows:
+Evidently, if there is to be compensation, the various
+parts of the external object, on the one hand, and the various
+sense organs, on the other hand, must be in the same <i>relative</i>
+position after the double change. And, for that to be the case,
+the various parts of the external object must likewise have
+retained in reference to each other the same relative position,
+and the same must be true of the various parts of our body in
+regard to each other.</p>
+
+<p>In other words, the external object, in the first change, must
+be displaced as is a rigid solid, and so must it be with the whole
+of our body in the second change which corrects the first.</p>
+
+<p>Under these conditions, compensation may take place.</p>
+
+<p>But we who as yet know nothing of geometry, since for us the
+notion of space is not yet formed, we can not reason thus, we
+can not foresee <i>a priori</i> whether compensation is possible. But
+experience teaches us that it sometimes happens, and it is from
+this experimental fact that we start to distinguish changes of
+state from changes of position.</p>
+
+<p><span class="smcap">Solid Bodies and Geometry.</span>&mdash;Among surrounding objects
+there are some which frequently undergo displacements susceptible
+of being thus corrected by a correlative movement of
+our own body; these are the <i>solid bodies</i>. The other objects,<span class='pagenum'><a name="Page_73" id="Page_73">[Pg 73]</a></span>
+whose form is variable, only exceptionally undergo like displacements
+(change of position without change of form). When a
+body changes its place <i>and its shape</i>, we can no longer, by appropriate
+movements, bring back our sense-organs into the same
+<i>relative</i> situation with regard to this body; consequently we can
+no longer reestablish the primitive totality of impressions.</p>
+
+<p>It is only later, and as a consequence of new experiences, that
+we learn how to decompose the bodies of variable form into
+smaller elements, such that each is displaced almost in accordance
+with the same laws as solid bodies. Thus we distinguish
+'deformations' from other changes of state; in these deformations,
+each element undergoes a mere change of position, which
+can be corrected, but the modification undergone by the aggregate
+is more profound and is no longer susceptible of correction
+by a correlative movement.</p>
+
+<p>Such a notion is already very complex and must have been
+relatively late in appearing; moreover it could not have arisen if
+the observation of solid bodies had not already taught us to distinguish
+changes of position.</p>
+
+<p><i>Therefore, if there were no solid bodies in nature, there would
+be no geometry.</i></p>
+
+<p>Another remark also deserves a moment's attention. Suppose
+a solid body to occupy successively the positions &#945; and &#946;; in its
+first position, it will produce on us the totality of impressions <i>A</i>,
+and in its second position the totality of impressions <i>B</i>. Let
+there be now a second solid body, having qualities entirely different
+from the first, for example, a different color. Suppose it to
+pass from the position &#945;, where it gives us the totality of impressions
+<i>A&acute;</i>, to the position &#946;, where it gives the totality of impressions
+<i>B&acute;</i>.</p>
+
+<p>In general, the totality <i>A</i> will have nothing in common with
+the totality <i>A&acute;</i>, nor the totality <i>B</i> with the totality <i>B&acute;</i>. The transition
+from the totality <i>A</i> to the totality <i>B</i> and that from the
+totality <i>A&acute;</i> to the totality <i>B&acute;</i> are therefore two changes which <i>in
+themselves</i> have in general nothing in common.</p>
+
+<p>And yet we regard these two changes both as displacements
+and, furthermore, we consider them as the <i>same</i> displacement.
+How can that be?<span class='pagenum'><a name="Page_74" id="Page_74">[Pg 74]</a></span></p>
+
+<p>It is simply because they can both be corrected by the <i>same</i>
+correlative movement of our body.</p>
+
+<p>'Correlative movement' therefore constitutes the <i>sole connection</i>
+between two phenomena which otherwise we never should
+have dreamt of likening.</p>
+
+<p>On the other hand, our body, thanks to the number of its
+articulations and muscles, may make a multitude of different
+movements; but all are not capable of 'correcting' a modification
+of external objects; only those will be capable of it in which our
+whole body, or at least all those of our sense-organs which come
+into play, are displaced as a whole, that is, without their relative
+positions varying, or in the fashion of a solid body.</p>
+
+<p>To summarize:</p>
+
+<p>1&ordm; We are led at first to distinguish two categories of phenomena:</p>
+
+<p>Some, involuntary, unaccompanied by muscular sensations, are
+attributed by us to external objects; these are external changes;</p>
+
+<p>Others, opposite in character and attributed by us to the
+movements of our own body, are internal changes;</p>
+
+<p>2&ordm; We notice that certain changes of each of these categories
+may be corrected by a correlative change of the other category;</p>
+
+<p>3&ordm; We distinguish among external changes those which have
+thus a correlative in the other category; these we call displacements;
+and just so among the internal changes, we distinguish
+those which have a correlative in the first category.</p>
+
+<p>Thus are defined, thanks to this reciprocity, a particular class
+of phenomena which we call displacements.</p>
+
+<p><i>The laws of these phenomena constitute the object of geometry.</i></p>
+
+<p><span class="smcap">Law of Homogeneity.</span>&mdash;The first of these laws is the law of
+homogeneity.</p>
+
+<p>Suppose that, by an external change &#945;, we pass from the totality
+of impressions <i>A</i> to the totality <i>B</i>, then that this change
+&#945; is corrected by a correlative voluntary movement &#946;, so that we
+are brought back to the totality <i>A</i>.</p>
+
+<p>Suppose now that another external change &#945;&acute; makes us pass
+anew from the totality <i>A</i> to the totality <i>B</i>.</p>
+
+<p>Experience teaches us that this change &#945;&acute; is, like &#945;, susceptible
+of being corrected by a correlative voluntary movement<span class='pagenum'><a name="Page_75" id="Page_75">[Pg 75]</a></span>
+&#946;&acute; and that this movement &#946;&acute; corresponds to the same muscular
+sensations as the movement &#946; which corrected &#945;.</p>
+
+<p>This fact is usually enunciated by saying that <i>space is homogeneous
+and isotropic</i>.</p>
+
+<p>It may also be said that a movement which has once been produced
+may be repeated a second and a third time, and so on,
+without its properties varying.</p>
+
+<p>In the first chapter, where we discussed the nature of mathematical
+reasoning, we saw the importance which must be
+attributed to the possibility of repeating indefinitely the same
+operation.</p>
+
+<p>It is from this repetition that mathematical reasoning gets its
+power; it is, therefore, thanks to the law of homogeneity, that it
+has a hold on the geometric facts.</p>
+
+<p>For completeness, to the law of homogeneity should be added
+a multitude of other analogous laws, into the details of which I
+do not wish to enter, but which mathematicians sum up in a word
+by saying that displacements form 'a group.'</p>
+
+<p><span class="smcap">The Non-Euclidean World.</span>&mdash;If geometric space were a
+frame imposed on <i>each</i> of our representations, considered individually,
+it would be impossible to represent to ourselves an
+image stripped of this frame, and we could change nothing of
+our geometry.</p>
+
+<p>But this is not the case; geometry is only the r&eacute;sum&eacute; of the
+laws according to which these images succeed each other. Nothing
+then prevents us from imagining a series of representations,
+similar in all points to our ordinary representations, but succeeding
+one another according to laws different from those to
+which we are accustomed.</p>
+
+<p>We can conceive then that beings who received their education
+in an environment where these laws were thus upset might
+have a geometry very different from ours.</p>
+
+<p>Suppose, for example, a world enclosed in a great sphere and
+subject to the following laws:</p>
+
+<p>The temperature is not uniform; it is greatest at the center,
+and diminishes in proportion to the distance from the center, to
+sink to absolute zero when the sphere is reached in which this
+world is enclosed.<span class='pagenum'><a name="Page_76" id="Page_76">[Pg 76]</a></span></p>
+
+<p>To specify still more precisely the law in accordance with
+which this temperature varies: Let <i>R</i> be the radius of the limiting
+sphere; let <i>r</i> be the distance of the point considered from
+the center of this sphere. The absolute temperature shall be
+proportional to <i>R</i><sup>2</sup> &minus; <i>r</i><sup>2</sup>.</p>
+
+<p>I shall further suppose that, in this world, all bodies have
+the same coefficient of dilatation, so that the length of any rule
+is proportional to its absolute temperature.</p>
+
+<p>Finally, I shall suppose that a body transported from one
+point to another of different temperature is put immediately into
+thermal equilibrium with its new environment.</p>
+
+<p>Nothing in these hypotheses is contradictory or unimaginable.</p>
+
+<p>A movable object will then become smaller and smaller in proportion
+as it approaches the limit-sphere.</p>
+
+<p>Note first that, though this world is limited from the point
+of view of our ordinary geometry, it will appear infinite to its
+inhabitants.</p>
+
+<p>In fact, when these try to approach the limit-sphere, they cool
+off and become smaller and smaller. Therefore the steps they
+take are also smaller and smaller, so that they can never reach the
+limiting sphere.</p>
+
+<p>If, for us, geometry is only the study of the laws according
+to which rigid solids move, for these imaginary beings it will be
+the study of the laws of motion of solids <i>distorted by the differences
+of temperature</i> just spoken of.</p>
+
+<p>No doubt, in our world, natural solids likewise undergo variations
+of form and volume due to warming or cooling. But we
+neglect these variations in laying the foundations of geometry,
+because, besides their being very slight, they are irregular and
+consequently seem to us accidental.</p>
+
+<p>In our hypothetical world, this would no longer be the case,
+and these variations would follow regular and very simple laws.</p>
+
+<p>Moreover, the various solid pieces of which the bodies of its
+inhabitants would be composed would undergo the same variations
+of form and volume.</p>
+
+<p>I will make still another hypothesis; I will suppose light
+traverses media diversely refractive and such that the index of
+<span class='pagenum'><a name="Page_77" id="Page_77">[Pg 77]</a></span>refraction is inversely proportional to <i>R</i><sup>2</sup> &minus; <i>r</i><sup>2</sup>. It is easy to
+see that, under these conditions, the rays of light would not be
+rectilinear, but circular.</p>
+
+<p>To justify what precedes, it remains for me to show that
+certain changes in the position of external objects can be <i>corrected</i>
+by correlative movements of the sentient beings inhabiting
+this imaginary world, and that in such a way as to restore the
+primitive aggregate of impressions experienced by these sentient
+beings.</p>
+
+<p>Suppose in fact that an object is displaced, undergoing deformation,
+not as a rigid solid, but as a solid subjected to unequal
+dilatations in exact conformity to the law of temperature above
+supposed. Permit me for brevity to call such a movement a
+<i>non-Euclidean displacement</i>.</p>
+
+<p>If a sentient being happens to be in the neighborhood, his
+impressions will be modified by the displacement of the object,
+but he can reestablish them by moving in a suitable manner. It
+suffices if finally the aggregate of the object and the sentient
+being, considered as forming a single body, has undergone one of
+those particular displacements I have just called non-Euclidean.
+This is possible if it be supposed that the limbs of these beings
+dilate according to the same law as the other bodies of the world
+they inhabit.</p>
+
+<p>Although from the point of view of our ordinary geometry
+there is a deformation of the bodies in this displacement and
+their various parts are no longer in the same relative position,
+nevertheless we shall see that the impressions of the sentient
+being have once more become the same.</p>
+
+<p>In fact, though the mutual distances of the various parts may
+have varied, yet the parts originally in contact are again in
+contact. Therefore the tactile impressions have not changed.</p>
+
+<p>On the other hand, taking into account the hypothesis made
+above in regard to the refraction and the curvature of the rays
+of light, the visual impressions will also have remained the same.</p>
+
+<p>These imaginary beings will therefore like ourselves be led
+to classify the phenomena they witness and to distinguish among
+them the 'changes of position' susceptible of correction by a correlative
+voluntary movement.</p>
+
+<p>If they construct a geometry, it will not be, as ours is, the<span class='pagenum'><a name="Page_78" id="Page_78">[Pg 78]</a></span>
+study of the movements of our rigid solids; it will be the study
+of the changes of position which they will thus have distinguished
+and which are none other than the 'non-Euclidean displacements';
+<i>it will be non-Euclidean geometry</i>.</p>
+
+<p>Thus beings like ourselves, educated in such a world, would
+not have the same geometry as ours.</p>
+
+<p><span class="smcap">The World of Four Dimensions.</span>&mdash;We can represent to ourselves
+a four-dimensional world just as well as a non-Euclidean.</p>
+
+<p>The sense of sight, even with a single eye, together with the
+muscular sensations relative to the movements of the eyeball,
+would suffice to teach us space of three dimensions.</p>
+
+<p>The images of external objects are painted on the retina, which
+is a two-dimensional canvas; they are <i>perspectives</i>.</p>
+
+<p>But, as eye and objects are movable, we see in succession various
+perspectives of the same body, taken from different points
+of view.</p>
+
+<p>At the same time, we find that the transition from one perspective
+to another is often accompanied by muscular sensations.</p>
+
+<p>If the transition from the perspective <i>A</i> to the perspective
+<i>B</i>, and that from the perspective <i>A&acute;</i> to the perspective <i>B&acute;</i> are
+accompanied by the same muscular sensations, we liken them one
+to the other as operations of the same nature.</p>
+
+<p>Studying then the laws according to which these operations
+combine, we recognize that they form a group, which has the
+same structure as that of the movements of rigid solids.</p>
+
+<p>Now, we have seen that it is from the properties of this group
+we have derived the notion of geometric space and that of three
+dimensions.</p>
+
+<p>We understand thus how the idea of a space of three dimensions
+could take birth from the pageant of these perspectives,
+though each of them is of only two dimensions, since <i>they follow
+one another according to certain laws</i>.</p>
+
+<p>Well, just as the perspective of a three-dimensional figure
+can be made on a plane, we can make that of a four-dimensional
+figure on a picture of three (or of two) dimensions. To a
+geometer this is only child's play.</p>
+
+<p>We can even take of the same figure several perspectives from
+several different points of view.<span class='pagenum'><a name="Page_79" id="Page_79">[Pg 79]</a></span></p>
+
+<p>We can easily represent to ourselves these perspectives, since
+they are of only three dimensions.</p>
+
+<p>Imagine that the various perspectives of the same object succeed
+one another, and that the transition from one to the other
+is accompanied by muscular sensations.</p>
+
+<p>We shall of course consider two of these transitions as two
+operations of the same nature when they are associated with the
+same muscular sensations.</p>
+
+<p>Nothing then prevents us from imagining that these operations
+combine according to any law we choose, for example, so as
+to form a group with the same structure as that of the movements
+of a rigid solid of four dimensions.</p>
+
+<p>Here there is nothing unpicturable, and yet these sensations
+are precisely those which would be felt by a being possessed of
+a two-dimensional retina who could move in space of four dimensions.
+In this sense we may say the fourth dimension is
+imaginable.</p>
+
+<p><span class="smcap">Conclusions.</span>&mdash;We see that experience plays an indispensable
+r&ocirc;le in the genesis of geometry; but it would be an error thence
+to conclude that geometry is, even in part, an experimental
+science.</p>
+
+<p>If it were experimental, it would be only approximative and
+provisional. And what rough approximation!</p>
+
+<p>Geometry would be only the study of the movements of solids;
+but in reality it is not occupied with natural solids, it has for
+object certain ideal solids, absolutely rigid, which are only a
+simplified and very remote image of natural solids.</p>
+
+<p>The notion of these ideal solids is drawn from all parts of our
+mind, and experience is only an occasion which induces us to
+bring it forth from them.</p>
+
+<p>The object of geometry is the study of a particular 'group';
+but the general group concept pre-exists, at least potentially, in
+our minds. It is imposed on us, not as form of our sense, but as
+form of our understanding.</p>
+
+<p>Only, from among all the possible groups, that must be chosen
+which will be, so to speak, the <i>standard</i> to which we shall refer
+natural phenomena.</p>
+
+<p>Experience guides us in this choice without forcing it upon<span class='pagenum'><a name="Page_80" id="Page_80">[Pg 80]</a></span>
+us; it tells us not which is the truest geometry, but which is the
+most <i>convenient</i>.</p>
+
+<p>Notice that I have been able to describe the fantastic worlds
+above imagined <i>without ceasing to employ the language of ordinary
+geometry</i>.</p>
+
+<p>And, in fact, we should not have to change it if transported
+thither.</p>
+
+<p>Beings educated there would doubtless find it more convenient
+to create a geometry different from ours, and better adapted to
+their impressions. As for us, in face of the <i>same</i> impressions, it
+is certain we should find it more convenient not to change our
+habits.</p>
+
+
+
+<hr style="width: 65%;" />
+<p><span class='pagenum'><a name="Page_81" id="Page_81">[Pg 81]</a></span></p>
+<h3>CHAPTER V</h3>
+
+<h3><span class="smcap">Experience and Geometry</span></h3>
+
+
+<p>1. Already in the preceding pages I have several times tried
+to show that the principles of geometry are not experimental
+facts and that in particular Euclid's postulate can not be proven
+experimentally.</p>
+
+<p>However decisive appear to me the reasons already given, I
+believe I should emphasize this point because here a false idea
+is profoundly rooted in many minds.</p>
+
+<p>2. If we construct a material circle, measure its radius and
+circumference, and see if the ratio of these two lengths is equal
+to &#960;, what shall we have done? We shall have made an experiment
+on the properties of the matter with which we constructed
+this <i>round thing</i>, and of that of which the measure used was made.</p>
+
+<p>3. <span class="smcap">Geometry and Astronomy.</span>&mdash;The question has also been
+put in another way. If Lobachevski's geometry is true, the parallax
+of a very distant star will be finite; if Riemann's is true, it
+will be negative. These are results which seem within the reach
+of experiment, and there have been hopes that astronomical observations
+might enable us to decide between the three geometries.</p>
+
+<p>But in astronomy 'straight line' means simply 'path of a ray
+of light.'</p>
+
+<p>If therefore negative parallaxes were found, or if it were
+demonstrated that all parallaxes are superior to a certain limit,
+two courses would be open to us; we might either renounce
+Euclidean geometry, or else modify the laws of optics and suppose
+that light does not travel rigorously in a straight line.</p>
+
+<p>It is needless to add that all the world would regard the latter
+solution as the more advantageous.</p>
+
+<p>The Euclidean geometry has, therefore, nothing to fear from
+fresh experiments.</p>
+
+<p>4. Is the position tenable, that certain phenomena, possible
+in Euclidean space, would be impossible in non-Euclidean space,<span class='pagenum'><a name="Page_82" id="Page_82">[Pg 82]</a></span>
+so that experience, in establishing these phenomena, would directly
+contradict the non-Euclidean hypothesis? For my part I
+think no such question can be put. To my mind it is precisely
+equivalent to the following, whose absurdity is patent to all eyes:
+are there lengths expressible in meters and centimeters, but which
+can not be measured in fathoms, feet and inches, so that experience,
+in ascertaining the existence of these lengths, would directly
+contradict the hypothesis that there are fathoms divided into
+six feet?</p>
+
+<p>Examine the question more closely. I suppose that the straight
+line possesses in Euclidean space any two properties which I
+shall call <i>A</i> and <i>B</i>; that in non-Euclidean space it still possesses
+the property <i>A</i>, but no longer has the property <i>B</i>; finally I suppose
+that in both Euclidean and non-Euclidean space the straight
+line is the only line having the property <i>A</i>.</p>
+
+<p>If this were so, experience would be capable of deciding between
+the hypothesis of Euclid and that of Lobachevski. It would be
+ascertained that a definite concrete object, accessible to experiment,
+for example, a pencil of rays of light, possesses the property
+<i>A</i>; we should conclude that it is rectilinear, and then investigate
+whether or not it has the property <i>B</i>.</p>
+
+<p>But <i>this is not so</i>; no property exists which, like this property
+<i>A</i>, can be an absolute criterion enabling us to recognize the
+straight line and to distinguish it from every other line.</p>
+
+<p>Shall we say, for instance: "the following is such a property:
+the straight line is a line such that a figure of which this line
+forms a part can be moved without the mutual distances of its
+points varying and so that all points of this line remain fixed"?</p>
+
+<p>This, in fact, is a property which, in Euclidean or non-Euclidean
+space, belongs to the straight and belongs only to it. But
+how shall we ascertain experimentally whether it belongs to this
+or that concrete object? It will be necessary to measure distances,
+and how shall one know that any concrete magnitude
+which I have measured with my material instrument really represents
+the abstract distance?</p>
+
+<p>We have only pushed back the difficulty.</p>
+
+<p>In reality the property just enunciated is not a property of
+the straight line alone, it is a property of the straight line and<span class='pagenum'><a name="Page_83" id="Page_83">[Pg 83]</a></span>
+distance. For it to serve as absolute criterion, we should have
+to be able to establish not only that it does not also belong to a
+line other than the straight and to distance, but in addition that
+it does not belong to a line other than the straight and to a
+magnitude other than distance. Now this is not true.</p>
+
+<p>It is therefore impossible to imagine a concrete experiment
+which can be interpreted in the Euclidean system and not in the
+Lobachevskian system, so that I may conclude:</p>
+
+<p>No experience will ever be in contradiction to Euclid's postulate;
+nor, on the other hand, will any experience ever contradict
+the postulate of Lobachevski.</p>
+
+<p>5. But it is not enough that the Euclidean (or non-Euclidean)
+geometry can never be directly contradicted by experience. Might
+it not happen that it can accord with experience only by violating
+the principle of sufficient reason or that of the relativity of space?</p>
+
+<p>I will explain myself: consider any material system; we shall
+have to regard, on the one hand, 'the state' of the various bodies
+of this system (for instance, their temperature, their electric
+potential, etc.), and, on the other hand, their position in space;
+and among the data which enable us to define this position we
+shall, moreover, distinguish the mutual distances of these bodies,
+which define their relative positions, from the conditions which
+define the absolute position of the system and its absolute orientation
+in space.</p>
+
+<p>The laws of the phenomena which will happen in this system
+will depend on the state of these bodies and their mutual distances;
+but, because of the relativity and passivity of space, they
+will not depend on the absolute position and orientation of the
+system.</p>
+
+<p>In other words, the state of the bodies and their mutual distances
+at any instant will depend solely on the state of these
+same bodies and on their mutual distances at the initial instant,
+but will not at all depend on the absolute initial position of the
+system or on its absolute initial orientation. This is what for
+brevity I shall call the <i>law of relativity</i>.</p>
+
+<p>Hitherto I have spoken as a Euclidean geometer. As I have
+said, an experience, whatever it be, admits of an interpretation
+on the Euclidean hypothesis; but it admits of one equally on<span class='pagenum'><a name="Page_84" id="Page_84">[Pg 84]</a></span>
+the non-Euclidean hypothesis. Well, we have made a series of
+experiments; we have interpreted them on the Euclidean hypothesis,
+and we have recognized that these experiments thus interpreted
+do not violate this 'law of relativity.'</p>
+
+<p>We now interpret them on the non-Euclidean hypothesis: this
+is always possible; only the non-Euclidean distances of our different
+bodies in this new interpretation will not generally be the
+same as the Euclidean distances in the primitive interpretation.</p>
+
+<p>Will our experiments, interpreted in this new manner, still
+be in accord with our 'law of relativity'? And if there were
+not this accord, should we not have also the right to say experience
+had proven the falsity of the non-Euclidean geometry?</p>
+
+<p>It is easy to see that this is an idle fear; in fact, to apply
+the law of relativity in all rigor, it must be applied to the entire
+universe. For if only a part of this universe were considered,
+and if the absolute position of this part happened to vary, the
+distances to the other bodies of the universe would likewise vary,
+their influence on the part of the universe considered would consequently
+augment or diminish, which might modify the laws
+of the phenomena happening there.</p>
+
+<p>But if our system is the entire universe, experience is powerless
+to give information about its absolute position and orientation
+in space. All that our instruments, however perfected they
+may be, can tell us will be the state of the various parts of the
+universe and their mutual distances.</p>
+
+<p>So our law of relativity may be thus enunciated:</p>
+
+<p>The readings we shall be able to make on our instruments at
+any instant will depend only on the readings we could have made
+on these same instruments at the initial instant.</p>
+
+<p>Now such an enunciation is independent of every interpretation
+of experimental facts. If the law is true in the Euclidean
+interpretation, it will also be true in the non-Euclidean interpretation.</p>
+
+<p>Allow me here a short digression. I have spoken above of
+the data which define the position of the various bodies of the
+system; I should likewise have spoken of those which define their
+velocities; I should then have had to distinguish the velocities
+with which the mutual distances of the different bodies vary;<span class='pagenum'><a name="Page_85" id="Page_85">[Pg 85]</a></span>
+and, on the other hand, the velocities of translation and rotation
+of the system, that is to say, the velocities with which its absolute
+position and orientation vary.</p>
+
+<p>To fully satisfy the mind, the law of relativity should be
+expressible thus:</p>
+
+<p>The state of bodies and their mutual distances at any instant,
+as well as the velocities with which these distances vary at this
+same instant, will depend only on the state of those bodies and
+their mutual distances at the initial instant, and the velocities
+with which these distances vary at this initial instant, but they
+will not depend either upon the absolute initial position of the
+system, or upon its absolute orientation, or upon the velocities
+with which this absolute position and orientation varied at the
+initial instant.</p>
+
+<p>Unhappily the law thus enunciated is not in accord with experiments,
+at least as they are ordinarily interpreted.</p>
+
+<p>Suppose a man be transported to a planet whose heavens were
+always covered with a thick curtain of clouds, so that he could
+never see the other stars; on that planet he would live as if it
+were isolated in space. Yet this man could become aware that it
+turned, either by measuring its oblateness (done ordinarily by
+the aid of astronomic observations, but capable of being done by
+purely geodetic means), or by repeating the experiment of Foucault's
+pendulum. The absolute rotation of this planet could
+therefore be made evident.</p>
+
+<p>That is a fact which shocks the philosopher, but which the
+physicist is compelled to accept.</p>
+
+<p>We know that from this fact Newton inferred the existence
+of absolute space; I myself am quite unable to adopt this view.
+I shall explain why in Part III. For the moment it is not my
+intention to enter upon this difficulty.</p>
+
+<p>Therefore I must resign myself, in the enunciation of the law
+of relativity, to including velocities of every kind among the data
+which define the state of the bodies.</p>
+
+<p>However that may be, this difficulty is the same for Euclid's
+geometry as for Lobachevski's; I therefore need not trouble myself
+with it, and have only mentioned it incidentally.<span class='pagenum'><a name="Page_86" id="Page_86">[Pg 86]</a></span></p>
+
+<p>What is important is the conclusion: experiment can not decide
+between Euclid and Lobachevski.</p>
+
+<p>To sum up, whichever way we look at it, it is impossible to
+discover in geometric empiricism a rational meaning.</p>
+
+<p>6. Experiments only teach us the relations of bodies to one
+another; none of them bears or can bear on the relations of bodies
+with space, or on the mutual relations of different parts of space.</p>
+
+<p>"Yes," you reply, "a single experiment is insufficient, because
+it gives me only a single equation with several unknowns;
+but when I shall have made enough experiments I shall have
+equations enough to calculate all my unknowns."</p>
+
+<p>To know the height of the mainmast does not suffice for calculating
+the age of the captain. When you have measured every
+bit of wood in the ship you will have many equations, but
+you will know his age no better. All your measurements bearing
+only on your bits of wood can reveal to you nothing except
+concerning these bits of wood. Just so your experiments, however
+numerous they may be, bearing only on the relations of
+bodies to one another, will reveal to us nothing about the mutual
+relations of the various parts of space.</p>
+
+<p>7. Will you say that if the experiments bear on the bodies,
+they bear at least upon the geometric properties of the bodies?
+But, first, what do you understand by geometric properties of
+the bodies? I assume that it is a question of the relations of the
+bodies with space; these properties are therefore inaccessible to
+experiments which bear only on the relations of the bodies to one
+another. This alone would suffice to show that there can be no
+question of these properties.</p>
+
+<p>Still let us begin by coming to an understanding about the
+sense of the phrase: geometric properties of bodies. When I
+say a body is composed of several parts, I assume that I do not
+enunciate therein a geometric property, and this would remain
+true even if I agreed to give the improper name of points to the
+smallest parts I consider.</p>
+
+<p>When I say that such a part of such a body is in contact
+with such a part of such another body, I enunciate a proposition
+which concerns the mutual relations of these two bodies and not
+their relations with space.<span class='pagenum'><a name="Page_87" id="Page_87">[Pg 87]</a></span></p>
+
+<p>I suppose you will grant me these are not geometric properties;
+at least I am sure you will grant me these properties are independent
+of all knowledge of metric geometry.</p>
+
+<p>This presupposed, I imagine that we have a solid body formed
+of eight slender iron rods, <i>OA</i>, <i>OB</i>, <i>OC</i>, <i>OD</i>, <i>OE</i>, <i>OF</i>, <i>OG</i>, <i>OH</i>,
+united at one of their extremities <i>O</i>. Let us besides have a second
+solid body, for example a bit of wood, to be marked with three
+little flecks of ink which I shall call &#945;, &#946;, &#947;. I further suppose it
+ascertained that &#945;&#946;&#947; may be brought into contact with <i>AGO</i> (I
+mean &#945; with <i>A</i>, and at the same time &#946; with <i>G</i> and &#947; with <i>O</i>),
+then that we may bring successively into contact &#945;&#946;&#947; with <i>BGO</i>,
+<i>CGO</i>, <i>DGO</i>, <i>EGO</i>, <i>FGO</i>, then with <i>AHO</i>, <i>BHO</i>, <i>CHO</i>, <i>DHO</i>,
+<i>EHO</i>, <i>FHO</i>, then &#945;&#947; successively with <i>AB</i>, <i>BC</i>, <i>CD</i>, <i>DE</i>, <i>EF</i>, <i>FA</i>.</p>
+
+<p>These are determinations we may make without having in
+advance any notion about form or about the metric properties of
+space. They in no wise bear on the 'geometric properties of
+bodies.' And these determinations will not be possible if the
+bodies experimented upon move in accordance with a group
+having the same structure as the Lobachevskian group (I mean
+according to the same laws as solid bodies in Lobachevski's geometry).
+They suffice therefore to prove that these bodies move in
+accordance with the Euclidean group, or at least that they do
+not move according to the Lobachevskian group.</p>
+
+<p>That they are compatible with the Euclidean group is easy
+to see. For they could be made if the body &#945;&#946;&#947; was a rigid
+solid of our ordinary geometry presenting the form of a right-angled
+triangle, and if the points <i>ABCDEFGH</i> were the summits
+of a polyhedron formed of two regular hexagonal pyramids of our
+ordinary geometry, having for common base <i>ABCDEF</i> and for
+apices the one <i>G</i> and the other <i>H</i>.</p>
+
+<p>Suppose now that in place of the preceding determination it
+is observed that as above &#945;&#946;&#947; can be successively applied to <i>AGO</i>,
+<i>BGO</i>, <i>CGO</i>, <i>DGO</i>, <i>EGO</i>, <i>AHO</i>, <i>BHO</i>, <i>CHO</i>, <i>DHO</i>, <i>EHO</i>, <i>FHO</i>,
+then that &#945;&#946; (and no longer &#945;&#947;) can be successively applied to
+<i>AB</i>, <i>BC</i>, <i>CD</i>, <i>DE</i>, <i>EF</i> and <i>FA</i>.</p>
+
+<p>These are determinations which could be made if non-Euclidean
+geometry were true, if the bodies &#945;&#946;&#947; and <i>OABCDEFGH</i>
+were rigid solids, and if the first were a right-angled triangle<span class='pagenum'><a name="Page_88" id="Page_88">[Pg 88]</a></span>
+and the second a double regular hexagonal pyramid of suitable
+dimensions.</p>
+
+<p>Therefore these new determinations are not possible if the
+bodies move according to the Euclidean group; but they become
+so if it be supposed that the bodies move according to the Lobachevskian
+group. They would suffice, therefore (if one made
+them), to prove that the bodies in question do not move according
+to the Euclidean group.</p>
+
+<p>Thus, without making any hypothesis about form, about the
+nature of space, about the relations of bodies to space, and without
+attributing to bodies any geometric property, I have made
+observations which have enabled me to show in one case that
+the bodies experimented upon move according to a group whose
+structure is Euclidean, in the other case that they move according
+to a group whose structure is Lobachevskian.</p>
+
+<p>And one may not say that the first aggregate of determinations
+would constitute an experiment proving that space is Euclidean,
+and the second an experiment proving that space is non-Euclidean.</p>
+
+<p>In fact one could imagine (I say imagine) bodies moving so
+as to render possible the second series of determinations. And
+the proof is that the first mechanician met could construct such
+bodies if he cared to take the pains and make the outlay. You
+will not conclude from that, however, that space is non-Euclidean.</p>
+
+<p>Nay, since the ordinary solid bodies would continue to exist
+when the mechanician had constructed the strange bodies of which
+I have just spoken, it would be necessary to conclude that space is
+at the same time Euclidean and non-Euclidean.</p>
+
+<p>Suppose, for example, that we have a great sphere of radius <i>R</i>
+and that the temperature decreases from the center to the surface
+of this sphere according to the law of which I have spoken in
+describing the non-Euclidean world.</p>
+
+<p>We might have bodies whose expansion would be negligible
+and which would act like ordinary rigid solids; and, on the other
+hand, bodies very dilatable and which would act like non-Euclidean
+solids. We might have two double pyramids <i>OABCDEFGH</i>
+and <i>O&acute;A&acute;B&acute;C&acute;D&acute;E&acute;F&acute;G&acute;H&acute;</i> and two triangles &#945;&#946;&#947; and &#945;&acute;&#946;&acute;&#947;&acute;. The
+first double pyramid might be rectilinear and the second<span class='pagenum'><a name="Page_89" id="Page_89">[Pg 89]</a></span>
+curvilinear; the triangle &#945;&#946;&#947; might be made of inexpansible matter
+and the other of a very dilatable matter.</p>
+
+<p>It would then be possible to make the first observations with
+the double pyramid <i>OAH</i> and the triangle &#945;&#946;&#947;, and the second
+with the double pyramid <i>O&acute;A&acute;H&acute;</i> and the triangle &#945;&acute;&#946;&acute;&#947;&acute;. And
+then experiment would seem to prove first that the Euclidean
+geometry is true and then that it is false.</p>
+
+<p><i>Experiments therefore have a bearing, not on space, but on
+bodies.</i></p>
+
+
+<p><span class="smcap">Supplement</span></p>
+
+<p>8. To complete the matter, I ought to speak of a very delicate
+question, which would require long development; I shall confine
+myself to summarizing here what I have expounded in the <i>Revue
+de M&eacute;taphysique et de Morale</i> and in <i>The Monist</i>. When we
+say space has three dimensions, what do we mean?</p>
+
+<p>We have seen the importance of those 'internal changes'
+revealed to us by our muscular sensations. They may serve to
+characterize the various <i>attitudes</i> of our body. Take arbitrarily
+as origin one of these attitudes <i>A</i>. When we pass from this
+initial attitude to any other attitude <i>B</i>, we feel a series of muscular
+sensations, and this series <i>S</i> will define <i>B</i>. Observe, however,
+that we shall often regard two series <i>S</i> and <i>S&acute;</i> as defining
+the same attitude <i>B</i> (since the initial and final attitudes <i>A</i> and <i>B</i>
+remaining the same, the intermediary attitudes and the corresponding
+sensations may differ). How then shall we recognize
+the equivalence of these two series? Because they may serve to
+compensate the same external change, or more generally because,
+when it is a question of compensating an external change, one
+of the series can be replaced by the other. Among these series,
+we have distinguished those which of themselves alone can compensate
+an external change, and which we have called 'displacements.'
+As we can not discriminate between two displacements
+which are too close together, the totality of these displacements
+presents the characteristics of a physical continuum; experience
+teaches us that they are those of a physical continuum of six
+dimensions; but we do not yet know how many dimensions space
+itself has, we must first solve another question.</p>
+
+<p>What is a point of space? Everybody thinks he knows, but<span class='pagenum'><a name="Page_90" id="Page_90">[Pg 90]</a></span>
+that is an illusion. What we see when we try to represent to ourselves
+a point of space is a black speck on white paper, a speck of
+chalk on a blackboard, always an object. The question should
+therefore be understood as follows:</p>
+
+<p>What do I mean when I say the object <i>B</i> is at the same point
+that the object <i>A</i> occupied just now? Or further, what criterion
+will enable me to apprehend this?</p>
+
+<p>I mean that, <i>although I have not budged</i> (which my muscular
+sense tells me), my first finger which just now touched the object <i>A</i>
+touches at present the object <i>B</i>. I could have used other criteria;
+for instance another finger or the sense of sight. But the first
+criterion is sufficient; I know that if it answers yes, all the other
+criteria will give the same response. I know it <i>by experience</i>, I
+can not know it <i>a priori</i>. For the same reason I say that touch
+can not be exercised at a distance; this is another way of enunciating
+the same experimental fact. And if, on the contrary, I say
+that sight acts at a distance, it means that the criterion furnished
+by sight may respond yes while the others reply no.</p>
+
+<p>And in fact, the object, although moved away, may form its
+image at the same point of the retina. Sight responds yes, the
+object has remained at the same point and touch answers no,
+because my finger which just now touched the object, at present
+touches it no longer. If experience had shown us that one finger
+may respond no when the other says yes, we should likewise
+say that touch acts at a distance.</p>
+
+<p>In short, for each attitude of my body, my first finger determines
+a point, and this it is, and this alone, which defines a point
+of space.</p>
+
+<p>To each attitude corresponds thus a point; but it often happens
+that the same point corresponds to several different attitudes (in
+this case we say our finger has not budged, but the rest of the
+body has moved). We distinguish, therefore, among the changes
+of attitude those where the finger does not budge. How are we
+led thereto? It is because often we notice that in these changes
+the object which is in contact with the finger remains in contact
+with it.</p>
+
+<p>Range, therefore, in the same class all the attitudes obtainable
+from each other by one of the changes we have thus distinguished.<span class='pagenum'><a name="Page_91" id="Page_91">[Pg 91]</a></span>
+To all the attitudes of the class will correspond the same point
+of space. Therefore to each class will correspond a point and to
+each point a class. But one may say that what experience arrives
+at is not the point, it is this class of changes or, better, the corresponding
+class of muscular sensations.</p>
+
+<p>And when we say space has three dimensions, we simply mean
+that the totality of these classes appears to us with the characteristics
+of a physical continuum of three dimensions.</p>
+
+<p>One might be tempted to conclude that it is experience which
+has taught us how many dimensions space has. But in reality
+here also our experiences have bearing, not on space, but on our
+body and its relations with the neighboring objects. Moreover
+they are excessively crude.</p>
+
+<p>In our mind pre-existed the latent idea of a certain number
+of groups&mdash;those whose theory Lie has developed. Which group
+shall we choose, to make of it a sort of standard with which to
+compare natural phenomena? And, this group chosen, which of
+its sub-groups shall we take to characterize a point of space? Experience
+has guided us by showing us which choice best adapts
+itself to the properties of our body. But its r&ocirc;le is limited to that.</p>
+
+
+<h3><span class="smcap">Ancestral Experience</span></h3>
+
+<p>It has often been said that if individual experience could
+not create geometry the same is not true of ancestral experience.
+But what does that mean? Is it meant that we could not experimentally
+demonstrate Euclid's postulate, but that our ancestors
+have been able to do it? Not in the least. It is meant that by
+natural selection our mind has <i>adapted</i> itself to the conditions of
+the external world, that it has adopted the geometry <i>most advantageous</i>
+to the species: or in other words <i>the most convenient</i>.
+This is entirely in conformity with our conclusions; geometry is
+not true, it is advantageous.</p>
+
+
+
+<hr style="width: 65%;" />
+<p><span class='pagenum'><a name="Page_92" id="Page_92">[Pg 92]</a></span></p>
+<h2><b>PART III<br />
+<br />
+<small>FORCE</small></b></h2>
+
+
+
+<hr style="width: 65%;" />
+<h3>CHAPTER VI</h3>
+
+<h3><span class="smcap">The Classic Mechanics</span></h3>
+
+
+<p>The English teach mechanics as an experimental science; on
+the continent it is always expounded as more or less a deductive
+and <i>a priori</i> science. The English are right, that goes without
+saying; but how could the other method have been persisted in
+so long? Why have the continental savants who have sought to
+get out of the ruts of their predecessors been usually unable to
+free themselves completely?</p>
+
+<p>On the other hand, if the principles of mechanics are only of
+experimental origin, are they not therefore only approximate and
+provisional? Might not new experiments some day lead us to
+modify or even to abandon them?</p>
+
+<p>Such are the questions which naturally obtrude themselves,
+and the difficulty of solution comes principally from the fact
+that the treatises on mechanics do not clearly distinguish between
+what is experiment, what is mathematical reasoning, what is convention,
+what is hypothesis.</p>
+
+<p>That is not all:</p>
+
+<p>1&ordm; There is no absolute space and we can conceive only of
+relative motions; yet usually the mechanical facts are enunciated
+as if there were an absolute space to which to refer them.</p>
+
+<p>2&ordm; There is no absolute time; to say two durations are equal
+is an assertion which has by itself no meaning and which can
+acquire one only by convention.</p>
+
+<p>3&ordm; Not only have we no direct intuition of the equality of
+two durations, but we have not even direct intuition of the<span class='pagenum'><a name="Page_93" id="Page_93">[Pg 93]</a></span>
+simultaneity of two events occurring in different places: this I
+have explained in an article entitled <i>La mesure du temps</i>.<a name="FNanchor_3_3" id="FNanchor_3_3"></a><a href="#Footnote_3_3" class="fnanchor">[3]</a></p>
+
+<p>4&ordm; Finally, our Euclidean geometry is itself only a sort of
+convention of language; mechanical facts might be enunciated
+with reference to a non-Euclidean space which would be a guide
+less convenient than, but just as legitimate as, our ordinary
+space; the enunciation would thus become much more complicated,
+but it would remain possible.</p>
+
+<p>Thus absolute space, absolute time, geometry itself, are not
+conditions which impose themselves on mechanics; all these things
+are no more antecedent to mechanics than the French language is
+logically antecedent to the verities one expresses in French.</p>
+
+<p>We might try to enunciate the fundamental laws of mechanics
+in a language independent of all these conventions; we should
+thus without doubt get a better idea of what these laws are in
+themselves; this is what M. Andrade has attempted to do, at least
+in part, in his <i>Le&ccedil;ons de m&eacute;canique physique</i>.</p>
+
+<p>The enunciation of these laws would become of course much
+more complicated, because all these conventions have been devised
+expressly to abridge and simplify this enunciation.</p>
+
+<p>As for me, save in what concerns absolute space, I shall ignore
+all these difficulties; not that I fail to appreciate them, far from
+that; but we have sufficiently examined them in the first two
+parts of the book.</p>
+
+<p>I shall therefore admit, <i>provisionally</i>, absolute time and Euclidean
+geometry.</p>
+
+<p><span class="smcap">The Principle of Inertia.</span>&mdash;A body acted on by no force can
+only move uniformly in a straight line.</p>
+
+<p>Is this a truth imposed <i>a priori</i> upon the mind? If it were
+so, how should the Greeks have failed to recognize it? How could
+they have believed that motion stops when the cause which gave
+birth to it ceases? Or again that every body if nothing prevents,
+will move in a circle, the noblest of motions?</p>
+
+<p>If it is said that the velocity of a body can not change if there
+is no reason for it to change, could it not be maintained just as
+well that the position of this body can not change, or that the<span class='pagenum'><a name="Page_94" id="Page_94">[Pg 94]</a></span>
+curvature of its trajectory can not change, if no external cause
+intervenes to modify them?</p>
+
+<p>Is the principle of inertia, which is not an <i>a priori</i> truth,
+therefore an experimental fact? But has any one ever experimented
+on bodies withdrawn from the action of every force? and,
+if so, how was it known that these bodies were subjected to no
+force? The example ordinarily cited is that of a ball rolling a
+very long time on a marble table; but why do we say it is subjected
+to no force? Is this because it is too remote from all other
+bodies to experience any appreciable action from them? Yet it
+is not farther from the earth than if it were thrown freely into
+the air; and every one knows that in this case it would experience
+the influence of gravity due to the attraction of the earth.</p>
+
+<p>Teachers of mechanics usually pass rapidly over the example
+of the ball; but they add that the principle of inertia is verified
+indirectly by its consequences. They express themselves badly;
+they evidently mean it is possible to verify various consequences
+of a more general principle, of which that of inertia is only a
+particular case.</p>
+
+<p>I shall propose for this general principle the following enunciation:</p>
+
+<p>The acceleration of a body depends only upon the position
+of this body and of the neighboring bodies and upon their
+velocities.</p>
+
+<p>Mathematicians would say the movements of all the material
+molecules of the universe depend on differential equations of the
+second order.</p>
+
+<p>To make it clear that this is really the natural generalization
+of the law of inertia, I shall beg you to permit me a bit of fiction.
+The law of inertia, as I have said above, is not imposed upon us
+<i>a priori</i>; other laws would be quite as compatible with the principle
+of sufficient reason. If a body is subjected to no force, in
+lieu of supposing its velocity not to change, it might be supposed
+that it is its position or else its acceleration which is not to change.</p>
+
+<p>Well, imagine for an instant that one of these two hypothetical
+laws is a law of nature and replaces our law of inertia. What
+would be its natural generalization? A moment's thought will
+show us.<span class='pagenum'><a name="Page_95" id="Page_95">[Pg 95]</a></span></p>
+
+<p>In the first case, we must suppose that the velocity of a body
+depends only upon its position and upon that of the neighboring
+bodies; in the second case that the change of acceleration of a
+body depends only upon the position of this body and of the
+neighboring bodies, upon their velocities and upon their accelerations.</p>
+
+<p>Or to speak the language of mathematics, the differential
+equations of motion would be of the first order in the first case,
+and of the third order in the second case.</p>
+
+<p>Let us slightly modify our fiction. Suppose a world analogous
+to our solar system, but where, by a strange chance, the orbits of
+all the planets are without eccentricity and without inclination.
+Suppose further that the masses of these planets are too slight
+for their mutual perturbations to be sensible. Astronomers inhabiting
+one of these planets could not fail to conclude that the
+orbit of a star can only be circular and parallel to a certain plane;
+the position of a star at a given instant would then suffice to determine
+its velocity and its whole path. The law of inertia which
+they would adopt would be the first of the two hypothetical laws
+I have mentioned.</p>
+
+<p>Imagine now that this system is some day traversed with great
+velocity by a body of vast mass, coming from distant constellations.
+All the orbits would be profoundly disturbed. Still our
+astronomers would not be too greatly astonished; they would very
+well divine that this new star was alone to blame for all the
+mischief. "But," they would say, "when it is gone, order will
+of itself be reestablished; no doubt the distances of the planets
+from the sun will not revert to what they were before the cataclysm,
+but when the perturbing star is gone, the orbits will again
+become circular."</p>
+
+<p>It would only be when the disturbing body was gone and when
+nevertheless the orbits, in lieu of again becoming circular, became
+elliptic, that these astronomers would become conscious of their
+error and the necessity of remaking all their mechanics.</p>
+
+<p>I have dwelt somewhat upon these hypotheses because it seems
+to me one can clearly comprehend what our generalized law of
+inertia really is only in contrasting it with a contrary hypothesis.</p>
+
+<p>Well, now, has this generalized law of inertia been verified by<span class='pagenum'><a name="Page_96" id="Page_96">[Pg 96]</a></span>
+experiment, or can it be? When Newton wrote the <i>Principia</i>
+he quite regarded this truth as experimentally acquired and demonstrated.
+It was so in his eyes, not only through the anthropomorphism
+of which we shall speak further on, but through the
+work of Galileo. It was so even from Kepler's laws themselves;
+in accordance with these laws, in fact, the path of a planet is
+completely determined by its initial position and initial velocity;
+this is just what our generalized law of inertia requires.</p>
+
+<p>For this principle to be only in appearance true, for one to
+have cause to dread having some day to replace it by one of the
+analogous principles I have just now contrasted with it, would be
+necessary our having been misled by some amazing chance, like
+that which, in the fiction above developed, led into error our
+imaginary astronomers.</p>
+
+<p>Such a hypothesis is too unlikely to delay over. No one will
+believe that such coincidences can happen; no doubt the probability
+of two eccentricities being both precisely null, to within
+errors of observation, is not less than the probability of one being
+precisely equal to 0.1, for instance, and the other to 0.2, to within
+errors of observation. The probability of a simple event is not
+less than that of a complicated event; and yet, if the first happens,
+we shall not consent to attribute it to chance; we should not
+believe that nature had acted expressly to deceive us. The hypothesis
+of an error of this sort being discarded, it may therefore
+be admitted that in so far as astronomy is concerned, our law has
+been verified by experiment.</p>
+
+<p>But astronomy is not the whole of physics.</p>
+
+<p>May we not fear lest some day a new experiment should come
+to falsify the law in some domain of physics? An experimental
+law is always subject to revision; one should always expect to see
+it replaced by a more precise law.</p>
+
+<p>Yet no one seriously thinks that the law we are speaking of
+will ever be abandoned or amended. Why? Precisely because
+it can never be subjected to a decisive test.</p>
+
+<p>First of all, in order that this trial should be complete, it
+would be necessary that after a certain time all the bodies in the
+universe should revert to their initial positions with their initial<span class='pagenum'><a name="Page_97" id="Page_97">[Pg 97]</a></span>
+velocities. It might then be seen whether, starting from this
+moment, they would resume their original paths.</p>
+
+<p>But this test is impossible, it can be only partially applied,
+and, however well it is made, there will always be some bodies
+which will not revert to their initial positions; thus every derogation
+of the law will easily find its explanation.</p>
+
+<p>This is not all; in astronomy we <i>see</i> the bodies whose motions
+we study and we usually assume that they are not subjected to the
+action of other invisible bodies. Under these conditions our law
+must indeed be either verified or not verified.</p>
+
+<p>But it is not the same in physics; if the physical phenomena
+are due to motions, it is to the motions of molecules which we do
+not see. If then the acceleration of one of the bodies we see
+appears to us to depend on <i>something else</i> besides the positions
+or velocities of other visible bodies or of invisible molecules whose
+existence we have been previously led to admit, nothing prevents
+our supposing that this <i>something else</i> is the position or the
+velocity of other molecules whose presence we have not before
+suspected. The law will find itself safeguarded.</p>
+
+<p>Permit me to employ mathematical language a moment to
+express the same thought under another form. Suppose we observe
+<i>n</i> molecules and ascertain that their 3<i>n</i> coordinates satisfy
+a system of 3<i>n</i> differential equations of the fourth order (and
+not of the second order as the law of inertia would require). We
+know that by introducing 3<i>n</i> auxiliary variables, a system of 3<i>n</i>
+equations of the fourth order can be reduced to a system of 6<i>n</i>
+equations of the second order. If then we suppose these 3<i>n</i>
+auxiliary variables represent the coordinates of <i>n</i> invisible molecules,
+the result is again in conformity with the law of inertia.</p>
+
+<p>To sum up, this law, verified experimentally in some particular
+cases, may unhesitatingly be extended to the most general cases,
+since we know that in these general cases experiment no longer
+is able either to confirm or to contradict it.</p>
+
+<p><span class="smcap">The Law of Acceleration.</span>&mdash;The acceleration of a body is
+equal to the force acting on it divided by its mass. Can this law
+be verified by experiment? For that it would be necessary to<span class='pagenum'><a name="Page_98" id="Page_98">[Pg 98]</a></span>
+measure the three magnitudes which figure in the enunciation:
+acceleration, force and mass.</p>
+
+<p>I assume that acceleration can be measured, for I pass over
+the difficulty arising from the measurement of time. But how
+measure force, or mass? We do not even know what they are.</p>
+
+<p>What is <i>mass</i>? According to Newton, it is the product of the
+volume by the density. According to Thomson and Tait, it would
+be better to say that density is the quotient of the mass by the
+volume. What is <i>force</i>? It is, replies Lagrange, that which
+moves or tends to move a body. It is, Kirchhoff will say, the
+product of the mass by the <i>acceleration</i>. But then, why not say
+the mass is the quotient of the force by the acceleration?</p>
+
+<p>These difficulties are inextricable.</p>
+
+<p>When we say force is the cause of motion, we talk metaphysics,
+and this definition, if one were content with it, would be absolutely
+sterile. For a definition to be of any use, it must teach us
+to <i>measure</i> force; moreover that suffices; it is not at all necessary
+that it teach us what force is <i>in itself</i>, nor whether it is the cause
+or the effect of motion.</p>
+
+<p>We must therefore first define the equality of two forces.
+When shall we say two forces are equal? It is, we are told,
+when, applied to the same mass, they impress upon it the same
+acceleration, or when, opposed directly one to the other, they produce
+equilibrium. This definition is only a sham. A force applied
+to a body can not be uncoupled to hook it up to another body,
+as one uncouples a locomotive to attach it to another train. It
+is therefore impossible to know what acceleration such a force,
+applied to such a body, would impress upon such another body,
+<i>if</i> it were applied to it. It is impossible to know how two forces
+which are not directly opposed would act, <i>if</i> they were directly
+opposed.</p>
+
+<p>It is this definition we try to materialize, so to speak, when
+we measure a force with a dynamometer, or in balancing it with
+a weight. Two forces <i>F</i> and <i>F&acute;</i>, which for simplicity I will suppose
+vertical and directed upward, are applied respectively to two
+bodies <i>C</i> and <i>C&acute;</i>; I suspend the same heavy body <i>P</i> first to the
+body <i>C</i>, then to the body <i>C&acute;</i>; if equilibrium is produced in both
+cases, I shall conclude that the two forces <i>F</i> and <i>F&acute;</i> are equal to<span class='pagenum'><a name="Page_99" id="Page_99">[Pg 99]</a></span>
+one another, since they are each equal to the weight of the body <i>P</i>.</p>
+
+<p>But am I sure the body <i>P</i> has retained the same weight when
+I have transported it from the first body to the second? Far from
+it; <i>I am sure of the contrary</i>; I know the intensity of gravity
+varies from one point to another, and that it is stronger, for
+instance, at the pole than at the equator. No doubt the difference
+is very slight and, in practise, I shall take no account of it; but
+a properly constructed definition should have mathematical
+rigor; this rigor is lacking. What I say of weight would evidently
+apply to the force of the resiliency of a dynamometer,
+which the temperature and a multitude of circumstances may
+cause to vary.</p>
+
+<p>This is not all; we can not say the weight of the body <i>P</i>
+may be applied to the body <i>C</i> and directly balance the force <i>F</i>.
+What is applied to the body <i>C</i> is the action <i>A</i> of the body <i>P</i> on
+the body <i>C</i>; the body <i>P</i> is submitted on its part, on the one hand,
+to its weight; on the other hand, to the reaction <i>R</i> of the body <i>C</i>
+on <i>P</i>. Finally, the force <i>F</i> is equal to the force <i>A</i>, since it balances
+it; the force <i>A</i> is equal to <i>R</i>, in virtue of the principle of
+the equality of action and reaction; lastly, the force <i>R</i> is equal to
+the weight of <i>P</i>, since it balances it. It is from these three equalities
+we deduce as consequence the equality of <i>F</i> and the weight
+of <i>P</i>.</p>
+
+<p>We are therefore obliged in the definition of the equality of
+the two forces to bring in the principle of the equality of action
+and reaction; <i>on this account, this principle must no longer be
+regarded as an experimental law, but as a definition</i>.</p>
+
+<p>For recognizing the equality of two forces here, we are then
+in possession of two rules: equality of two forces which balance;
+equality of action and reaction. But, as we have seen above,
+these two rules are insufficient; we are obliged to have recourse to
+a third rule and to assume that certain forces, as, for instance, the
+weight of a body, are constant in magnitude and direction. But
+this third rule, as I have said, is an experimental law; it is only
+approximately true; <i>it is a bad definition</i>.</p>
+
+<p>We are therefore reduced to Kirchhoff's definition; <i>force is
+equal to the mass multiplied by the acceleration</i>. This 'law of
+Newton' in its turn ceases to be regarded as an experimental law,
+it is now only a definition. But this definition is still insufficient,<span class='pagenum'><a name="Page_100" id="Page_100">[Pg 100]</a></span>
+for we do not know what mass is. It enables us doubtless to calculate
+the relation of two forces applied to the same body at different
+instants; it teaches us nothing about the relation of two
+forces applied to two different bodies.</p>
+
+<p>To complete it, it is necessary to go back anew to Newton's
+third law (equality of action and reaction), regarded again, not
+as an experimental law, but as a definition. Two bodies <i>A</i> and <i>B</i>
+act one upon the other; the acceleration of <i>A</i> multiplied by the
+mass of <i>A</i> is equal to the action of <i>B</i> upon <i>A</i>; in the same way,
+the product of the acceleration of <i>B</i> by its mass is equal to the
+reaction of <i>A</i> upon <i>B</i>. As, by definition, action is equal to reaction,
+the masses of <i>A</i> and <i>B</i> are in the inverse ratio of their
+accelerations. Here we have the ratio of these two masses defined,
+and it is for experiment to verify that this ratio is constant.</p>
+
+<p>That would be all very well if the two bodies <i>A</i> and <i>B</i> alone
+were present and removed from the action of the rest of the
+world. This is not at all the case; the acceleration of <i>A</i> is not due
+merely to the action of <i>B</i>, but to that of a multitude of other
+bodies <i>C</i>, <i>D</i>,... To apply the preceding rule, it is therefore
+necessary to separate the acceleration of <i>A</i> into many components,
+and discern which of these components is due to the action of <i>B</i>.</p>
+
+<p>This separation would still be possible, if we <i>should assume</i>
+that the action of <i>C</i> upon <i>A</i> is simply adjoined to that of <i>B</i>
+upon <i>A</i>, without the presence of the body <i>C</i> modifying the action
+of <i>B</i> upon <i>A</i>; or the presence of <i>B</i> modifying the action of <i>C</i>
+upon <i>A</i>; if we should assume, consequently, that any two bodies
+attract each other, that their mutual action is along their join
+and depends only upon their distance apart; if, in a word, we
+assume <i>the hypothesis of central forces</i>.</p>
+
+<p>You know that to determine the masses of the celestial bodies
+we use a wholly different principle. The law of gravitation
+teaches us that the attraction of two bodies is proportional to
+their masses; if <i>r</i> is their distance apart, <i>m</i> and <i>m&acute;</i> their masses,
+<i>k</i> a constant, their attraction will be <i>kmm&acute;</i>/<i>r</i><sup>2</sup>.</p>
+
+<p>What we are measuring then is not mass, the ratio of force to
+acceleration, but the attracting mass; it is not the inertia of the
+body, but its attracting force.</p>
+
+<p>This is an indirect procedure, whose employment is not<span class='pagenum'><a name="Page_101" id="Page_101">[Pg 101]</a></span>
+theoretically indispensable. It might very well have been that attraction
+was inversely proportional to the square of the distance without
+being proportional to the product of the masses, that it was
+equal to <i>f/r</i><sup>2</sup>, but without our having <i>f = kmm&acute;</i>.</p>
+
+<p>If it were so, we could nevertheless, by observation of the
+<i>relative</i> motions of the heavenly bodies, measure the masses of
+these bodies.</p>
+
+<p>But have we the right to admit the hypothesis of central
+forces? Is this hypothesis rigorously exact? Is it certain it
+will never be contradicted by experiment? Who would dare
+affirm that? And if we must abandon this hypothesis, the whole
+edifice so laboriously erected will crumble.</p>
+
+<p>We have no longer the right to speak of the component of
+the acceleration of <i>A</i> due to the action of <i>B</i>. We have no means
+of distinguishing it from that due to the action of <i>C</i> or of another
+body. The rule for the measurement of masses becomes inapplicable.</p>
+
+<p>What remains then of the principle of the equality of action
+and reaction? If the hypothesis of central forces is rejected,
+this principle should evidently be enunciated thus: the geometric
+resultant of all the forces applied to the various bodies of a
+system isolated from all external action will be null. Or, in
+other words, <i>the motion of the center of gravity of this system
+will be rectilinear and uniform</i>.</p>
+
+<p>There it seems we have a means of defining mass; the position
+of the center of gravity evidently depends on the values attributed
+to the masses; it will be necessary to dispose of these values
+in such a way that the motion of the center of gravity may be
+rectilinear and uniform; this will always be possible if Newton's
+third law is true, and possible in general only in a single way.</p>
+
+<p>But there exists no system isolated from all external action;
+all the parts of the universe are subject more or less to the action
+of all the other parts. <i>The law of the motion of the center of
+gravity is rigorously true only if applied to the entire universe.</i></p>
+
+<p>But then, to get from it the values of the masses, it would be
+necessary to observe the motion of the center of gravity of the
+universe. The absurdity of this consequence is manifest; we
+know only relative motions; the motion of the center of gravity
+of the universe will remain for us eternally unknown.<span class='pagenum'><a name="Page_102" id="Page_102">[Pg 102]</a></span></p>
+
+<p>Therefore nothing remains and our efforts have been fruitless;
+we are driven to the following definition, which is only an
+avowal of powerlessness: <i>masses are coefficients it is convenient
+to introduce into calculations</i>.</p>
+
+<p>We could reconstruct all mechanics by attributing different
+values to all the masses. This new mechanics would not be in
+contradiction either with experience or with the general principles
+of dynamics (principle of inertia, proportionality of
+forces to masses and to accelerations, equality of action and
+reaction, rectilinear and uniform motion of the center of gravity,
+principle of areas).</p>
+
+<p>Only the equations of this new mechanics would be <i>less simple</i>.
+Let us understand clearly: it would only be the first terms which
+would be less simple, that is those experience has already made us
+acquainted with; perhaps one could alter the masses by small
+quantities without the <i>complete</i> equations gaining or losing in
+simplicity.</p>
+
+<p>Hertz has raised the question whether the principles of mechanics
+are rigorously true. "In the opinion of many physicists,"
+he says, "it is inconceivable that the remotest experience
+should ever change anything in the immovable principles of
+mechanics; and yet, what comes from experience may always
+be rectified by experience." After what we have just said, these
+fears will appear groundless.</p>
+
+<p>The principles of dynamics at first appeared to us as experimental
+truths; but we have been obliged to use them as definitions.
+It is <i>by definition</i> that force is equal to the product of
+mass by acceleration; here, then, is a principle which is henceforth
+beyond the reach of any further experiment. It is in the
+same way by definition that action is equal to reaction.</p>
+
+<p>But then, it will be said, these unverifiable principles are absolutely
+devoid of any significance; experiment can not contradict
+them; but they can teach us nothing useful; then what is the
+use of studying dynamics?</p>
+
+<p>This over-hasty condemnation would be unjust. There is not
+in nature any system <i>perfectly</i> isolated, perfectly removed from
+all external action; but there are systems <i>almost</i> isolated.</p>
+
+<p>If such a system be observed, one may study not only the<span class='pagenum'><a name="Page_103" id="Page_103">[Pg 103]</a></span>
+relative motion of its various parts one in reference to another,
+but also the motion of its center of gravity in reference to the
+other parts of the universe. We ascertain then that the motion
+of this center of gravity is <i>almost</i> rectilinear and uniform, in
+conformity with Newton's third law.</p>
+
+<p>That is an experimental truth, but it can not be invalidated
+by experience; in fact, what would a more precise experiment
+teach us? It would teach us that the law was only almost true;
+but that we knew already.</p>
+
+<p><i>We can now understand how experience has been able to serve
+as basis for the principles of mechanics and yet will never be
+able to contradict them.</i></p>
+
+<p><span class="smcap">Anthropomorphic Mechanics.</span>&mdash;"Kirchhoff," it will be said,
+"has only acted in obedience to the general tendency of mathematicians
+toward nominalism; from this his ability as a physicist
+has not saved him. He wanted a definition of force, and he
+took for it the first proposition that presented itself; but we
+need no definition of force: the idea of force is primitive, irreducible,
+indefinable; we all know what it is, we have a direct
+intuition of it. This direct intuition comes from the notion of
+effort, which is familiar to us from infancy."</p>
+
+<p>But first, even though this direct intuition made known to
+us the real nature of force in itself, it would be insufficient as a
+foundation for mechanics; it would besides be wholly useless.
+What is of importance is not to know what force is, but to know
+how to measure it.</p>
+
+<p>Whatever does not teach us to measure it is as useless to
+mechanics as is, for instance, the subjective notion of warmth
+and cold to the physicist who is studying heat. This subjective
+notion can not be translated into numbers, therefore it is of no
+use; a scientist whose skin was an absolutely bad conductor of
+heat and who, consequently, would never have felt either sensations
+of cold or sensations of warmth, could read a thermometer
+just as well as any one else, and that would suffice him for constructing
+the whole theory of heat.</p>
+
+<p>Now this immediate notion of effort is of no use to us for
+measuring force; it is clear, for instance, that I should feel more<span class='pagenum'><a name="Page_104" id="Page_104">[Pg 104]</a></span>
+fatigue in lifting a weight of fifty kilos than a man accustomed
+to carry burdens.</p>
+
+<p>But more than that: this notion of effort does not teach us
+the real nature of force; it reduces itself finally to a remembrance
+of muscular sensations, and it will hardly be maintained
+that the sun feels a muscular sensation when it draws the earth.</p>
+
+<p>All that can there be sought is a symbol, less precise and less
+convenient than the arrows the geometers use, but just as remote
+from the reality.</p>
+
+<p>Anthropomorphism has played a considerable historic r&ocirc;le in
+the genesis of mechanics; perhaps it will still at times furnish
+a symbol which will appear convenient to some minds; but it can
+not serve as foundation for anything of a truly scientific or
+philosophic character.</p>
+
+<p>'<span class="smcap">The School of the Thread.</span>'&mdash;M. Andrade, in his <i>Le&ccedil;ons
+de m&eacute;canique physique</i>, has rejuvenated anthropomorphic mechanics.
+To the school of mechanics to which Kirchhoff belongs,
+he opposes that which he bizarrely calls the school of the thread.</p>
+
+<p>This school tries to reduce everything to "the consideration
+of certain material systems of negligible mass, envisaged in the
+state of tension and capable of transmitting considerable efforts
+to distant bodies, systems of which the ideal type is the <i>thread</i>."</p>
+
+<p>A thread which transmits any force is slightly elongated under
+the action of this force; the direction of the thread tells us the
+direction of the force, whose magnitude is measured by the
+elongation of the thread.</p>
+
+<p>One may then conceive an experiment such as this. A body
+<i>A</i> is attached to a thread; at the other extremity of the thread
+any force acts which varies until the thread takes an elongation
+&#945;; the acceleration of the body <i>A</i> is noted; <i>A</i> is detached and
+the body <i>B</i> attached to the same thread; the same force or
+another force acts anew, and is made to vary until the thread
+takes again the elongation &#945;; the acceleration of the body <i>B</i> is
+noted. The experiment is then renewed with both <i>A</i> and <i>B</i>,
+but so that the thread takes the elongation &szlig;&#946;. The four observed
+accelerations should be proportional. We have thus an experimental
+verification of the law of acceleration above enunciated.</p>
+
+<p>Or still better, a body is submitted to the simultaneous action<span class='pagenum'><a name="Page_105" id="Page_105">[Pg 105]</a></span>
+of several identical threads in equal tension, and by experiment
+it is sought what must be the orientations of all these threads that
+the body may remain in equilibrium. We have then an experimental
+verification of the law of the composition of forces.</p>
+
+<p>But, after all, what have we done? We have defined the
+force to which the thread is subjected by the deformation undergone
+by this thread, which is reasonable enough; we have further
+assumed that if a body is attached to this thread, the effort transmitted
+to it by the thread is equal to the action this body exercises
+on this thread; after all, we have therefore used the principle of
+the equality of action and reaction, in considering it, not as an
+experimental truth, but as the very definition of force.</p>
+
+<p>This definition is just as conventional as Kirchhoff's, but far
+less general.</p>
+
+<p>All forces are not transmitted by threads (besides, to be able
+to compare them, they would all have to be transmitted by identical
+threads). Even if it should be conceded that the earth is
+attached to the sun by some invisible thread, at least it would be
+admitted that we have no means of measuring its elongation.</p>
+
+<p>Nine times out of ten, consequently, our definition would be at
+fault; no sort of sense could be attributed to it, and it would be
+necessary to fall back on Kirchhoff's.</p>
+
+<p>Why then take this d&eacute;tour? You admit a certain definition
+of force which has a meaning only in certain particular cases.
+In these cases you verify by experiment that it leads to the law
+of acceleration. On the strength of this experiment, you then
+take the law of acceleration as a definition of force in all the
+other cases.</p>
+
+<p>Would it not be simpler to consider the law of acceleration as
+a definition in all cases, and to regard the experiments in question,
+not as verifications of this law, but as verifications of the
+principle of reaction, or as demonstrating that the deformations
+of an elastic body depend only on the forces to which this body is
+subjected?</p>
+
+<p>And this is without taking into account that the conditions
+under which your definition could be accepted are never fulfilled
+except imperfectly, that a thread is never without mass, that it
+is never removed from every force except the reaction of the
+bodies attached to its extremities.<span class='pagenum'><a name="Page_106" id="Page_106">[Pg 106]</a></span></p>
+
+<p>Andrade's ideas are nevertheless very interesting; if they
+do not satisfy our logical craving, they make us understand
+better the historic genesis of the fundamental ideas of mechanics.
+The reflections they suggest show us how the human mind has
+raised itself from a na&iuml;ve anthropomorphism to the present conceptions
+of science.</p>
+
+<p>We see at the start a very particular and in sum rather crude
+experiment; at the finish, a law perfectly general, perfectly precise,
+the certainty of which we regard as absolute. This certainty
+we ourselves have bestowed upon it voluntarily, so to
+speak, by looking upon it as a convention.</p>
+
+<p>Are the law of acceleration, the rule of the composition of
+forces then only arbitrary conventions? Conventions, yes; arbitrary,
+no; they would be if we lost sight of the experiments which
+led the creators of the science to adopt them, and which, imperfect
+as they may be, suffice to justify them. It is well that from
+time to time our attention is carried back to the experimental
+origin of these conventions.</p>
+
+
+
+<hr style="width: 65%;" />
+<p><span class='pagenum'><a name="Page_107" id="Page_107">[Pg 107]</a></span></p>
+<h3>CHAPTER VII</h3>
+
+<h3><span class="smcap">Relative Motion and Absolute Motion</span></h3>
+
+
+<p><span class="smcap">The Principle of Relative Motion.</span>&mdash;The attempt has sometimes
+been made to attach the law of acceleration to a more
+general principle. The motion of any system must obey the
+same laws, whether it be referred to fixed axes, or to movable
+axes carried along in a rectilinear and uniform motion. This is
+the principle of relative motion, which forces itself upon us for
+two reasons: first, the commonest experience confirms it, and
+second, the contrary hypothesis is singularly repugnant to the
+mind.</p>
+
+<p>Assume it then, and consider a body subjected to a force;
+the relative motion of this body, in reference to an observer
+moving with a uniform velocity equal to the initial velocity of
+the body, must be identical to what its absolute motion would be
+if it started from rest. We conclude hence that its acceleration
+can not depend upon its absolute velocity; the attempt has even
+been made to derive from this a demonstration of the law of
+acceleration.</p>
+
+<p>There long were traces of this demonstration in the regulations
+for the degree B. &egrave;s Sc. It is evident that this attempt is
+idle. The obstacle which prevented our demonstrating the law
+of acceleration is that we had no definition of force; this obstacle
+subsists in its entirety, since the principle invoked has not furnished
+us the definition we lacked.</p>
+
+<p>The principle of relative motion is none the less highly interesting
+and deserves study for its own sake. Let us first try to
+enunciate it in a precise manner.</p>
+
+<p>We have said above that the accelerations of the different
+bodies forming part of an isolated system depend only on their
+relative velocities and positions, and not on their absolute velocities
+and positions, provided the movable axes to which the relative
+motion is referred move uniformly in a straight line. Or, if<span class='pagenum'><a name="Page_108" id="Page_108">[Pg 108]</a></span>
+we prefer, their accelerations depend only on the differences of
+their velocities and the differences of their coordinates, and not
+on the absolute values of these velocities and coordinates.</p>
+
+<p>If this principle is true for relative accelerations, or rather
+for differences of acceleration, in combining it with the law of
+reaction we shall thence deduce that it is still true of absolute
+accelerations.</p>
+
+<p>It then remains to be seen how we may demonstrate that the
+differences of the accelerations depend only on the differences
+of the velocities and of the coordinates, or, to speak in mathematical
+language, that these differences of coordinates satisfy
+differential equations of the second order.</p>
+
+<p>Can this demonstration be deduced from experiments or from
+<i>a priori</i> considerations?</p>
+
+<p>Recalling what we have said above, the reader can answer for
+himself.</p>
+
+<p>Thus enunciated, in fact, the principle of relative motion
+singularly resembles what I called above the generalized principle
+of inertia; it is not altogether the same thing, since it is a question
+of the differences of coordinates and not of the coordinates
+themselves. The new principle teaches us therefore something
+more than the old, but the same discussion is applicable and
+would lead to the same conclusions; it is unnecessary to return
+to it.</p>
+
+<p><span class="smcap">Newton's Argument.</span>&mdash;Here we encounter a very important
+and even somewhat disconcerting question. I have said the principle
+of relative motion was for us not solely a result of experiment
+and that <i>a priori</i> every contrary hypothesis would be repugnant
+to the mind.</p>
+
+<p>But then, why is the principle true only if the motion of the
+movable axes is rectilinear and uniform? It seems that it ought
+to impose itself upon us with the same force, if this motion is
+varied, or at any rate if it reduces to a uniform rotation. Now,
+in these two cases, the principle is not true. I will not dwell
+long on the case where the motion of the axes is rectilinear without
+being uniform; the paradox does not bear a moment's examination.
+If I am on board, and if the train, striking any<span class='pagenum'><a name="Page_109" id="Page_109">[Pg 109]</a></span>
+obstacle, stops suddenly, I shall be thrown against the seat in front
+of me, although I have not been directly subjected to any force.
+There is nothing mysterious in that; if I have undergone the
+action of no external force, the train itself has experienced an
+external impact. There can be nothing paradoxical in the relative
+motion of two bodies being disturbed when the motion of
+one or the other is modified by an external cause.</p>
+
+<p>I will pause longer on the case of relative motions referred to
+axes which rotate uniformly. If the heavens were always
+covered with clouds, if we had no means of observing the stars,
+we nevertheless might conclude that the earth turns round; we
+could learn this from its flattening or again by the Foucault pendulum
+experiment.</p>
+
+<p>And yet, in this case, would it have any meaning, to say the
+earth turns round? If there is no absolute space, can one turn
+without turning in reference to something else? and, on the other
+hand, how could we admit Newton's conclusion and believe in
+absolute space?</p>
+
+<p>But it does not suffice to ascertain that all possible solutions
+are equally repugnant to us; we must analyze, in each case, the
+reasons for our repugnance, so as to make our choice intelligently.
+The long discussion which follows will therefore be
+excused.</p>
+
+<p>Let us resume our fiction: thick clouds hide the stars from
+men, who can not observe them and are ignorant even of their
+existence; how shall these men know the earth turns round?</p>
+
+<p>Even more than our ancestors, no doubt, they will regard the
+ground which bears them as fixed and immovable; they will
+await much longer the advent of a Copernicus. But in the end
+the Copernicus would come&mdash;how?</p>
+
+<p>The students of mechanics in this world would not at first be
+confronted with an absolute contradiction. In the theory of
+relative motion, besides real forces, two fictitious forces are met
+which are called ordinary and compound centrifugal force. Our
+imaginary scientists could therefore explain everything by regarding
+these two forces as real, and they would not see therein
+any contradiction of the generalized principle of inertia, for
+these forces would depend, the one on the relative positions of<span class='pagenum'><a name="Page_110" id="Page_110">[Pg 110]</a></span>
+the various parts of the system, as real attractions do, the other
+on their relative velocities, as real frictions do.</p>
+
+<p>Many difficulties, however, would soon awaken their attention;
+if they succeeded in realizing an isolated system, the center of
+gravity of this system would not have an almost rectilinear path.
+They would invoke, to explain this fact, the centrifugal forces
+which they would regard as real, and which they would attribute
+no doubt to the mutual actions of the bodies. Only they would
+not see these forces become null at great distances, that is to say
+in proportion as the isolation was better realized; far from it;
+centrifugal force increases indefinitely with the distance.</p>
+
+<p>This difficulty would seem to them already sufficiently great;
+and yet it would not stop them long; they would soon imagine
+some very subtile medium, analogous to our ether, in which all
+bodies would be immersed and which would exert a repellent
+action upon them.</p>
+
+<p>But this is not all. Space is symmetric, and yet the laws of
+motion would not show any symmetry; they would have to distinguish
+between right and left. It would be seen for instance
+that cyclones turn always in the same sense, whereas by reason
+of symmetry these winds should turn indifferently in one sense
+and in the other. If our scientists by their labor had succeeded
+in rendering their universe perfectly symmetric, this symmetry
+would not remain, even though there was no apparent reason
+why it should be disturbed in one sense rather than in the other.</p>
+
+<p>They would get themselves out of the difficulty doubtless, they
+would invent something which would be no more extraordinary
+than the glass spheres of Ptolemy, and so it would go on, complications
+accumulating, until the long-expected Copernicus
+sweeps them all away at a single stroke, saying: It is much
+simpler to assume the earth turns round.</p>
+
+<p>And just as our Copernicus said to us: It is more convenient
+to suppose the earth turns round, since thus the laws of astronomy
+are expressible in a much simpler language; this one would
+say: It is more convenient to suppose the earth turns round,
+since thus the laws of mechanics are expressible in a much
+simpler language.</p>
+
+<p>This does not preclude maintaining that absolute space, that<span class='pagenum'><a name="Page_111" id="Page_111">[Pg 111]</a></span>
+is to say the mark to which it would be necessary to refer the
+earth to know whether it really moves, has no objective existence.
+Hence, this affirmation: 'the earth turns round' has no meaning,
+since it can be verified by no experiment; since such an
+experiment, not only could not be either realized or dreamed by
+the boldest Jules Verne, but can not be conceived of without contradiction;
+or rather these two propositions: 'the earth turns
+round,' and, 'it is more convenient to suppose the earth turns
+round' have the same meaning; there is nothing more in the one
+than in the other.</p>
+
+<p>Perhaps one will not be content even with that, and will find
+it already shocking that among all the hypotheses, or rather
+all the conventions we can make on this subject, there is one more
+convenient than the others.</p>
+
+<p>But if it has been admitted without difficulty when it was a
+question of the laws of astronomy, why should it be shocking in
+that which concerns mechanics?</p>
+
+<p>We have seen that the coordinates of bodies are determined
+by differential equations of the second order, and that so are the
+differences of these coordinates. This is what we have called
+the generalized principle of inertia and the principle of relative
+motion. If the distances of these bodies were determined likewise
+by equations of the second order, it seems that the mind
+ought to be entirely satisfied. In what measure does the mind
+get this satisfaction and why is it not content with it?</p>
+
+<p>To account for this, we had better take a simple example.
+I suppose a system analogous to our solar system, but where one
+can not perceive fixed stars foreign to this system, so that astronomers
+can observe only the mutual distances of the planets and
+the sun, and not the absolute longitudes of the planets. If we
+deduce directly from Newton's law the differential equations
+which define the variation of these distances, these equations will
+not be of the second order. I mean that if, besides Newton's law,
+one knew the initial values of these distances and of their derivatives
+with respect to the time, that would not suffice to determine
+the values of these same distances at a subsequent instant.
+There would still be lacking one datum, and this datum might be
+for instance what astronomers call the area-constant.<span class='pagenum'><a name="Page_112" id="Page_112">[Pg 112]</a></span></p>
+
+<p>But here two different points of view may be taken; we may
+distinguish two sorts of constants. To the eyes of the physicist
+the world reduces to a series of phenomena, depending, on the
+one hand, solely upon the initial phenomena; on the other hand,
+upon the laws which bind the consequents to the antecedents.
+If then observation teaches us that a certain quantity is a constant,
+we shall have the choice between two conceptions.</p>
+
+<p>Either we shall assume that there is a law requiring this
+quantity not to vary, but that by chance, at the beginning of
+the ages, it had, rather than another, this value it has been
+forced to keep ever since. This quantity might then be called
+an <i>accidental</i> constant.</p>
+
+<p>Or else we shall assume, on the contrary, that there is a law
+of nature which imposes upon this quantity such a value and
+not such another.</p>
+
+<p>We shall then have what we may call an <i>essential</i> constant.</p>
+
+<p>For example, in virtue of Newton's laws, the duration of the
+revolution of the earth must be constant. But if it is 366
+sidereal days and something over, and not 300 or 400, this is in
+consequence of I know not what initial chance. This is an
+accidental constant. If, on the contrary, the exponent of the
+distance which figures in the expression of the attractive force is
+equal to &minus;2 and not to &minus;3, this is not by chance, but because
+Newton's law requires it. This is an essential constant.</p>
+
+<p>I know not whether this way of giving chance its part is
+legitimate in itself, and whether this distinction is not somewhat
+artificial; it is certain at least that, so long as nature shall have
+secrets, this distinction will be in application extremely arbitrary
+and always precarious.</p>
+
+<p>As to the area-constant, we are accustomed to regard it as
+accidental. Is it certain our imaginary astronomers would do
+the same? If they could have compared two different solar
+systems, they would have the idea that this constant may have
+several different values; but my very supposition in the beginning
+was that their system should appear as isolated, and that
+they should observe no star foreign to it. Under these conditions,
+they would see only one single constant which would have
+a single value absolutely invariable; they would be led without
+any doubt to regard it as an essential constant.<span class='pagenum'><a name="Page_113" id="Page_113">[Pg 113]</a></span></p>
+
+<p>A word in passing to forestall an objection: the inhabitants
+of this imaginary world could neither observe nor define the
+area-constant as we do, since the absolute longitudes escape them;
+that would not preclude their being quickly led to notice a certain
+constant which would introduce itself naturally into their
+equations and which would be nothing but what we call the area-constant.</p>
+
+<p>But then see what would happen. If the area-constant is
+regarded as essential, as depending upon a law of nature, to calculate
+the distances of the planets at any instant it will suffice
+to know the initial values of these distances and those of their
+first derivatives. From this new point of view, the distances will
+be determined by differential equations of the second order.</p>
+
+<p>Yet would the mind of these astronomers be completely satisfied?
+I do not believe so; first, they would soon perceive that
+in differentiating their equations and thus raising their order,
+these equations became much simpler. And above all they would
+be struck by the difficulty which comes from symmetry. It
+would be necessary to assume different laws, according as the
+aggregate of the planets presented the figure of a certain polyhedron
+or of the symmetric polyhedron, and one would escape from
+this consequence only by regarding the area-constant as accidental.</p>
+
+<p>I have taken a very special example, since I have supposed
+astronomers who did not at all consider terrestrial mechanics,
+and whose view was limited to the solar system. Our universe is
+more extended than theirs, as we have fixed stars, but still it too
+is limited, and so we might reason on the totality of our universe
+as the astronomers on their solar system.</p>
+
+<p>Thus we see that finally we should be led to conclude that the
+equations which define distances are of an order superior to the
+second. Why should we be shocked at that, why do we find it
+perfectly natural for the series of phenomena to depend upon
+the initial values of the first derivatives of these distances, while
+we hesitate to admit that they may depend on the initial values of
+the second derivatives? This can only be because of the habits
+of mind created in us by the constant study of the generalized
+principle of inertia and its consequences.<span class='pagenum'><a name="Page_114" id="Page_114">[Pg 114]</a></span></p>
+
+<p>The values of the distances at any instant depend upon their
+initial values, upon those of their first derivatives and also upon
+something else. What is this <i>something else</i>?</p>
+
+<p>If we will not admit that this may be simply one of the second
+derivatives, we have only the choice of hypotheses. Either it
+may be supposed, as is ordinarily done, that this something else
+is the absolute orientation of the universe in space, or the rapidity
+with which this orientation varies; and this supposition may be
+correct; it is certainly the most convenient solution for geometry;
+it is not the most satisfactory for the philosopher, because
+this orientation does not exist.</p>
+
+<p>Or it may be supposed that this something else is the position
+or the velocity of some invisible body; this has been done by
+certain persons who have even called it the body alpha, although
+we are doomed never to know anything of this body but its
+name. This is an artifice entirely analogous to that of which I
+spoke at the end of the paragraph devoted to my reflections on
+the principle of inertia.</p>
+
+<p>But, after all, the difficulty is artificial. Provided the future
+indications of our instruments can depend only on the indications
+they have given us or would have given us formerly, this
+is all that is necessary. Now as to this we may rest easy.</p>
+
+
+
+<hr style="width: 65%;" />
+<p><span class='pagenum'><a name="Page_115" id="Page_115">[Pg 115]</a></span></p>
+<h3>CHAPTER VIII</h3>
+
+<h3><span class="smcap">Energy and Thermodynamics</span></h3>
+
+
+<p><span class="smcap">Energetics.</span>&mdash;The difficulties inherent in the classic mechanics
+have led certain minds to prefer a new system they call
+<i>energetics</i>.</p>
+
+<p>Energetics took its rise as an outcome of the discovery of the
+principle of the conservation of energy. Helmholtz gave it its
+final form.</p>
+
+<p>It begins by defining two quantities which play the fundamental
+r&ocirc;le in this theory. They are <i>kinetic energy</i>, or <i>vis viva</i>,
+and <i>potential energy</i>.</p>
+
+<p>All the changes which bodies in nature can undergo are regulated
+by two experimental laws:</p>
+
+<p>1&ordm; The sum of kinetic energy and potential energy is constant.
+This is the principle of the conservation of energy.</p>
+
+<p>2&ordm; If a system of bodies is at <i>A</i> at the time <i>t</i><sub>0</sub> and at <i>B</i> at
+the time <i>t</i><sub>1</sub>, it always goes from the first situation to the second
+in such a way that the <i>mean</i> value of the difference between the
+two sorts of energy, in the interval of time which separates the
+two epochs <i>t</i><sub>0</sub> and <i>t</i><sub>1</sub>, may be as small as possible.</p>
+
+<p>This is Hamilton's principle, which is one of the forms of the
+principle of least action.</p>
+
+<p>The energetic theory has the following advantages over the
+classic theory:</p>
+
+<p>1&ordm; It is less incomplete; that is to say, Hamilton's principle
+and that of the conservation of energy teach us more than the
+fundamental principles of the classic theory, and exclude certain
+motions not realized in nature and which would be compatible
+with the classic theory:</p>
+
+<p>2&ordm; It saves us the hypothesis of atoms, which it was almost
+impossible to avoid with the classic theory.</p>
+
+<p>But it raises in its turn new difficulties:</p>
+
+<p>The definitions of the two sorts of energy would raise difficulties
+almost as great as those of force and mass in the first<span class='pagenum'><a name="Page_116" id="Page_116">[Pg 116]</a></span>
+system. Yet they may be gotten over more easily, at least in
+the simplest cases.</p>
+
+<p>Suppose an isolated system formed of a certain number of
+material points; suppose these points subjected to forces depending
+only on their relative position and their mutual distances,
+and independent of their velocities. In virtue of the principle
+of the conservation of energy, a function of forces must exist.</p>
+
+<p>In this simple case the enunciation of the principle of the
+conservation of energy is of extreme simplicity. A certain quantity,
+accessible to experiment, must remain constant. This quantity
+is the sum of two terms; the first depends only on the position
+of the material points and is independent of their velocities;
+the second is proportional to the square of these velocities. This
+resolution can take place only in a single way.</p>
+
+<p>The first of these terms, which I shall call <i>U</i>, will be the
+potential energy; the second, which I shall call <i>T</i>, will be the
+kinetic energy.</p>
+
+<p>It is true that if <i>T</i> + <i>U</i> is a constant, so is any function of
+<i>T</i> + <i>U</i>,</p>
+
+<p class="center">&#934; (<i>T</i> + <i>U</i>).</p>
+
+<p>But this function &#934; (<i>T</i> + <i>U</i>) will not be the sum of two terms the
+one independent of the velocities, the other proportional to the
+square of these velocities. Among the functions which remain
+constant there is only one which enjoys this property, that is
+<i>T</i> + <i>U</i> (or a linear function of <i>T</i> + <i>U</i>, which comes to the same
+thing, since this linear function may always be reduced to <i>T</i> + <i>U</i>
+by change of unit and of origin). This then is what we shall
+call energy; the first term we shall call potential energy and the
+second kinetic energy. The definition of the two sorts of energy
+can therefore be carried through without any ambiguity.</p>
+
+<p>It is the same with the definition of the masses. Kinetic
+energy, or <i>vis viva</i>, is expressed very simply by the aid of the
+masses and the relative velocities of all the material points with
+reference to one of them. These relative velocities are accessible
+to observation, and, when we know the expression of the kinetic
+energy as function of these relative velocities, the coefficients of
+this expression will give us the masses.<span class='pagenum'><a name="Page_117" id="Page_117">[Pg 117]</a></span></p>
+
+<p>Thus, in this simple case, the fundamental ideas may be defined
+without difficulty. But the difficulties reappear in the
+more complicated cases and, for instance, if the forces, in lieu
+of depending only on the distances, depend also on the velocities.
+For example, Weber supposes the mutual action of two electric
+molecules to depend not only on their distance, but on their velocity
+and their acceleration. If material points should attract each
+other according to an analogous law, <i>U</i> would depend on the
+velocity, and might contain a term proportional to the square of
+the velocity.</p>
+
+<p>Among the terms proportional to the squares of the velocities,
+how distinguish those which come from <i>T</i> or from <i>U</i>? Consequently,
+how distinguish the two parts of energy?</p>
+
+<p>But still more; how define energy itself? We no longer have
+any reason to take as definition <i>T</i> + <i>U</i> rather than any other
+function of <i>T</i> + <i>U</i>, when the property which characterized <i>T</i> + <i>U</i>
+has disappeared, that, namely, of being the sum of two terms of
+a particular form.</p>
+
+<p>But this is not all; it is necessary to take account, not only
+of mechanical energy properly so called, but of the other forms
+of energy, heat, chemical energy, electric energy, etc. The principle
+of the conservation of energy should be written:</p>
+
+<p class="center"><i>T</i> + <i>U</i> + <i>Q</i> = const.</p>
+
+<p class="noidt">where <i>T</i> would represent the sensible kinetic energy, <i>U</i> the potential
+energy of position, depending only on the position of the
+bodies, <i>Q</i> the internal molecular energy, under the thermal,
+chemic or electric form.</p>
+
+<p>All would go well if these three terms were absolutely distinct,
+if <i>T</i> were proportional to the square of the velocities, <i>U</i> independent
+of these velocities and of the state of the bodies, <i>Q</i> independent
+of the velocities and of the positions of the bodies and
+dependent only on their internal state.</p>
+
+<p>The expression for the energy could be resolved only in one
+single way into three terms of this form.</p>
+
+<p>But this is not the case; consider electrified bodies; the electrostatic
+energy due to their mutual action will evidently depend
+upon their charge, that is to say, on their state; but it will equally<span class='pagenum'><a name="Page_118" id="Page_118">[Pg 118]</a></span>
+depend upon their position. If these bodies are in motion, they
+will act one upon another electrodynamically and the electrodynamic
+energy will depend not only upon their state and their
+position, but upon their velocities.</p>
+
+<p>We therefore no longer have any means of making the separation
+of the terms which should make part of <i>T</i>, of <i>U</i> and of <i>Q</i>,
+and of separating the three parts of energy.</p>
+
+<p>If (<i>T</i> + <i>U</i> + <i>Q</i>) is constant so is any function &#934; (<i>T</i> + <i>U</i> + <i>Q</i>).</p>
+
+<p>If <i>T</i> + <i>U</i> + <i>Q</i> were of the particular form I have above
+considered, no ambiguity would result; among the functions
+&#934; (<i>T</i> + <i>U</i> + <i>Q</i>) which remain constant, there would only be one
+of this particular form, and that I should convene to call energy.</p>
+
+<p>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.</p>
+
+<p>There only remains for us one enunciation of the principle of
+the conservation of energy: <i>There is something which remains
+constant</i>. 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.</p>
+
+<p>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.</p>
+
+<p>Another objection seems to me still more grave: the principle
+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 principle
+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.<span class='pagenum'><a name="Page_119" id="Page_119">[Pg 119]</a></span></p>
+
+<p>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 statement
+presents the molecule to us, so to speak, as a living and
+free being. Clearly it would be better to replace it by an enunciation
+less objectionable, and where, as the philosophers would
+say, final causes would not seem to be substituted for efficient
+causes.</p>
+
+<p><span class="smcap">Thermodynamics.</span><a name="FNanchor_4_4" id="FNanchor_4_4"></a><a href="#Footnote_4_4" class="fnanchor">[4]</a>&mdash;The r&ocirc;le of the two fundamental principles
+of thermodynamics in all branches of natural philosophy
+becomes daily more important. Abandoning the ambitious theories
+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?</p>
+
+<p>An eminent physicist said to me one day <i>&agrave; propos</i> of the law
+of errors: "All the world believes it firmly, because the mathematicians
+imagine that it is a fact of observation, and the observers
+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.</p>
+
+<p>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 question,
+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 generalization<span class='pagenum'><a name="Page_120" id="Page_120">[Pg 120]</a></span>
+is already necessary to foresee whether this fact will be
+reproduced again after the least of these circumstances shall
+be changed.</p>
+
+<p>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.</p>
+
+<p>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.</p>
+
+<p>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.</p>
+
+<p>But there is something more, and this is why I dwell upon
+the point.</p>
+
+<p>No one doubts that Mayer's principle is destined to survive
+all the particular laws from which it was obtained, just as Newton's
+law has survived Kepler's laws, from which it sprang,
+and which are only approximative if account be taken of
+perturbations.</p>
+
+<p>Why does this principle occupy thus a sort of privileged place
+among all the physical laws? There are many little reasons
+for it.</p>
+
+<p>First of all it is believed that we could not reject it or even
+doubt its absolute rigor without admitting the possibility of perpetual
+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.</p>
+
+<p>That is perhaps not wholly accurate; the impossibility of perpetual
+motion implies the conservation of energy only for reversible
+phenomena.</p>
+
+<p>The imposing simplicity of Mayer's principle likewise contributes
+to strengthen our faith. In a law deduced immediately
+from experiment, like Mariotte's, this simplicity would rather<span class='pagenum'><a name="Page_121" id="Page_121">[Pg 121]</a></span>
+seem to us a reason for distrust; but here this is no longer the
+case; we see elements, at first sight disparate, arrange themselves
+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.</p>
+
+<p>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.</p>
+
+<p>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 impossible
+to find a general definition for it.</p>
+
+<p>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: <i>There is something which remains constant</i>.</p>
+
+<p>But has even this any meaning? In the determinist hypothesis,
+the state of the universe is determined by an extremely great
+number <i>n</i> of parameters which I shall call <i>x</i><sub>1</sub>, <i>x</i><sub>2</sub>, ... <i>x</i><sub><i>n</i></sub>. As
+soon as the values of these <i>n</i> 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 <i>n</i> parameters satisfy <i>n</i> differential equations of the
+first order.</p>
+
+<p>These equations admit of <i>n</i> &minus; 1 integrals and consequently
+there are <i>n</i> &minus; 1 functions of <i>x</i><sub>1</sub>, <i>x</i><sub>2</sub>,... <i>x</i><sub><i>n</i></sub>, which remain
+constant. <i>If then we say there is something which remains
+constant</i>, we only utter a tautology. We should even be puzzled
+to say which among all our integrals should retain the name of
+energy.</p>
+
+<p>Besides, Mayer's principle is not understood in this sense
+when it is applied to a limited system. It is then assumed that
+<i>p</i> of our parameters vary independently, so that we only have
+<i>n</i> &minus; <i>p</i> relations, generally linear, between our <i>n</i> parameters and
+their derivatives.<span class='pagenum'><a name="Page_122" id="Page_122">[Pg 122]</a></span></p>
+
+<p>To simplify the enunciation, suppose that the sum of the
+work of the external forces is null, as well as that of the quantities
+of heat given off to the outside. Then the signification
+of our principle will be:</p>
+
+<p><i>There is a combination of these n &minus; p relations whose first
+member is an exact differential</i>; and then this differential vanishing
+in virtue of our <i>n &minus; p</i> relations, its integral is a constant
+and this integral is called energy.</p>
+
+<p>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 <i>n</i> 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.</p>
+
+<p>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 <i>x</i>i, 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 influence
+of these bodies is too slight for our experimenting to
+detect it?</p>
+
+<p>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 restriction,
+the enunciation becomes still more difficult.</p>
+
+<p>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.</p>
+
+<p>In conclusion, using ordinary language, the law of the<span class='pagenum'><a name="Page_123" id="Page_123">[Pg 123]</a></span>
+conservation of energy can have only one signification, which is
+that there is a property common to all the possibilities; but on
+the determinist hypothesis there is only a single possibility, and
+then the law has no longer any meaning.</p>
+
+<p>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.</p>
+
+<p>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 I 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.</p>
+
+<p>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.</p>
+
+<p>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 approximations,
+and it is hoped to replace them little by little by laws more
+and more precise. If, on the other hand, the principle of Clausius
+reduces to an inequality, this is not caused by the imperfection
+of our means of observation, but by the very nature of
+the question.</p>
+
+
+<h3><span class="smcap">General Conclusions on Part Third</span></h3>
+
+<p>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<span class='pagenum'><a name="Page_124" id="Page_124">[Pg 124]</a></span>
+postulates applicable to the totality of the universe and regarded
+as rigorously true.</p>
+
+<p>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 convention
+which we have the right to make, because we are certain
+beforehand that no experiment can ever contradict it.</p>
+
+<p>This convention, however, is not absolutely arbitrary; it does
+not spring from our caprice; we adopt it because certain experiments
+have shown us that it would be convenient.</p>
+
+<p>Thus is explained how experiment could make the principles
+of mechanics, and yet why it can not overturn them.</p>
+
+<p>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 system
+is true or false.</p>
+
+<p>Only, these conventions are convenient, and it is certain experiments
+which have taught us that.</p>
+
+<p>At first blush, the analogy is complete; the r&ocirc;le of experiment
+seems the same. One will therefore be tempted to say:
+Either mechanics must be regarded as an experimental science,
+and then the same must hold for geometry; or else, on the contrary,
+geometry is a deductive science, and then one may say as
+much of mechanics.</p>
+
+<p>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 fundamental
+experiments are preeminently physiological experiments,
+which bear, not on space which is the object the geometer must<span class='pagenum'><a name="Page_125" id="Page_125">[Pg 125]</a></span>
+study, but on his body, that is to say, on the instrument he must
+use for this study.</p>
+
+<p>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.</p>
+
+<p>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 mechanics
+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 compared
+to that superb body of doctrine called geometry?</p>
+
+<p>One sees now why the teaching of mechanics should remain
+experimental.</p>
+
+<p>Only thus can it make us comprehend the genesis of the science,
+and that is indispensable for the complete understanding of the
+science itself.</p>
+
+<p>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 objectivity.
+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.</p>
+
+<p>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 attributes
+an absolute value.</p>
+
+<p>Some philosophers have generalized too far; they believed the
+principles were the whole science and consequently that the whole
+science was conventional.</p>
+
+<p>This paradoxical doctrine, called nominalism, will not bear
+examination.<span class='pagenum'><a name="Page_126" id="Page_126">[Pg 126]</a></span></p>
+
+<p>How can a law become a principle? It expressed a relation
+between two real terms <i>A</i> and <i>B</i>. But it was not rigorously true,
+it was only approximate. We introduce arbitrarily an intermediary
+term <i>C</i> more or less fictitious, and <i>C</i> is <i>by definition</i> that
+which has with <i>A</i> <i>exactly</i> the relation expressed by the law.</p>
+
+<p>Then our law is separated into an absolute and rigorous principle
+which expresses the relation of <i>A</i> to <i>C</i> and an experimental
+law, approximate and subject to revision, which expresses the
+relation of <i>C</i> to <i>B</i>. It is clear that, however far this partition is
+pushed, some laws will always be left remaining.</p>
+
+<p>We go to enter now the domain of laws properly so called.</p>
+
+
+
+<hr style="width: 65%;" />
+<p><span class='pagenum'><a name="Page_127" id="Page_127">[Pg 127]</a></span></p>
+<h2><b>PART IV<br />
+<br />
+<small>NATURE</small></b></h2>
+
+
+
+<hr style="width: 65%;" />
+<h3>CHAPTER IX</h3>
+
+<h3><span class="smcap">Hypotheses in Physics</span></h3>
+
+
+<p><span class="smcap">The R&ocirc;le of Experiment and Generalization.</span>&mdash;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.</p>
+
+<p>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?</p>
+
+<p>And yet mathematical physics exists, and has done unquestionable
+service. We have here a fact that must be explained.</p>
+
+<p>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.</p>
+
+<p>Every age has ridiculed the one before it, and accused it of
+having generalized too quickly and too na&iuml;vely. Descartes pitied
+the Ionians; Descartes, in his turn, makes us smile. No doubt
+our children will some day laugh at us.</p>
+
+<p>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?</p>
+
+<p>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.<span class='pagenum'><a name="Page_128" id="Page_128">[Pg 128]</a></span></p>
+
+<p>And above all the scientist must foresee. Carlyle has somewhere
+said something like this: "Nothing but facts are of importance.
+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 Lackland
+passed by here; that makes no difference to me, for he
+never will pass this way again."</p>
+
+<p>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 <i>Experimentum crucis</i>. But Carlyle would not have understood
+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? 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.</p>
+
+<p>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 circumstances
+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.</p>
+
+<p>No matter how timid one may be, still it is necessary to interpolate.
+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;<span class='pagenum'><a name="Page_129" id="Page_129">[Pg 129]</a></span>
+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 corrections
+and really be content with the bare experiment, would be
+forced to enunciate some very strange laws.</p>
+
+<p>The bare facts, then, would not be enough for us; and that is
+why we must have science ordered, or rather organized.</p>
+
+<p>It is often said experiments must be made without a preconceived
+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.</p>
+
+<p>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&mdash;I was going to say as antidotes; they will in general accord
+ill with one another&mdash;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.</p>
+
+<p>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 <i>absolutely</i> 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.</p>
+
+<p>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 verifications
+that we can make will never be anything but a negligible
+quantity.<span class='pagenum'><a name="Page_130" id="Page_130">[Pg 130]</a></span></p>
+
+<p>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.</p>
+
+<p>Let us compare science to a library that ought to grow continually.
+The librarian has at his disposal for his purchases only
+insufficient funds. He ought to make an effort not to waste them.</p>
+
+<p>It is experimental physics that is entrusted with the purchases.
+It alone, then, can enrich the library.</p>
+
+<p>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.</p>
+
+<p>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.</p>
+
+<p>Such, then, is the r&ocirc;le 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.</p>
+
+<p><span class="smcap">The Unity of Nature.</span>&mdash;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 difficulty.
+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.</p>
+
+<p>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?</p>
+
+<p>There was a time when the simplicity of Mariotte's law was
+an argument invoked in favor of its accuracy; when Fresnel himself,
+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.<span class='pagenum'><a name="Page_131" id="Page_131">[Pg 131]</a></span></p>
+
+<p>To-day ideas have greatly changed; and yet, those who do not
+believe that natural laws have to be simple, are still often obliged
+to act as if they did. They could not entirely avoid this necessity
+without making impossible all generalization, and consequently
+all science.</p>
+
+<p>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 commonplace
+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.</p>
+
+<p>We may calculate the mass of Jupiter from either the movements
+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.</p>
+
+<p>To sum up, ordinarily every law is held to be simple till the
+contrary is proved.</p>
+
+<p>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.</p>
+
+<p>If we study the history of science, we see happen two inverse
+phenomena, so to speak. Sometimes simplicity hides under complex
+appearances; sometimes it is the simplicity which is apparent,
+and which disguises extremely complicated realities.</p>
+
+<p>What is more complicated than the confused movements of<span class='pagenum'><a name="Page_132" id="Page_132">[Pg 132]</a></span>
+the planets? What simpler than Newton's law? Here nature,
+making sport, as Fresnel said, of analytical difficulties, employs
+only simple means, and by combining them produces I know not
+what inextricable tangle. Here it is the hidden simplicity which
+must be discovered.</p>
+
+<p>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 capricious
+forms and traverse space in every direction. The observable
+result is Mariotte's simple law. Every individual fact was complicated.
+The law of great numbers has reestablished simplicity
+in the average. Here the simplicity is merely apparent, and only
+the coarseness of our senses prevents our perceiving the complexity.</p>
+
+<p>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 analytical
+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 superposition
+of small motions, the use of which is so fruitful, and
+which is the basis of optics.</p>
+
+<p>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 numbers?
+In any case, 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 modifying
+Newton's law, and if the law thus regains its simplicity, it
+would be only because of the immensity of celestial distances.</p>
+
+<p>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.<span class='pagenum'><a name="Page_133" id="Page_133">[Pg 133]</a></span></p>
+
+<p>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, will the ground be firm
+enough? This is what must be investigated.</p>
+
+<p>For that purpose, let us see what part is played in our generalizations
+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.</p>
+
+<p>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.</p>
+
+<p>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 differences,
+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 legitimately
+suppose that it will still be true in analogous cases. To
+refuse to do this would be to attribute to chance an inadmissible
+r&ocirc;le.</p>
+
+<p>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.</p>
+
+<p>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.</p>
+
+<p><span class="smcap">The R&ocirc;le of Hypothesis.</span>&mdash;All generalization is a hypothesis.
+Hypothesis, then, has a necessary r&ocirc;le that no one has ever<span class='pagenum'><a name="Page_134" id="Page_134">[Pg 134]</a></span>
+contested. 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.</p>
+
+<p>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.</p>
+
+<p>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 experiment
+would have been made by chance, so that nothing would
+have been derived from it. One would have seen nothing extraordinary;
+only one fact the more would have been catalogued
+without deducing from it the least consequence.</p>
+
+<p>Now on what condition is the use of hypothesis without danger?</p>
+
+<p>The firm determination to submit to experiment is not enough;
+there are still dangerous hypotheses; first, and above all, those
+which are tacit and unconscious. Since we make them without
+knowing 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
+all the hypotheses that we should make without it, but unconsciously.</p>
+
+<p>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 necessary
+to change? It will be impossible to know. And inversely,
+if the experiment succeeds, shall we believe that we have<span class='pagenum'><a name="Page_135" id="Page_135">[Pg 135]</a></span>
+demonstrated all the hypotheses at once? Shall we believe that with
+one single equation we have determined several unknowns?</p>
+
+<p>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 mathematical
+physics. They are the last that ought to be abandoned.</p>
+
+<p>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?</p>
+
+<p>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 <i>x</i> or <i>y</i>. 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 <i>x</i> and not <i>y</i>.</p>
+
+<p>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 occasion
+to exclude them.<span class='pagenum'><a name="Page_136" id="Page_136">[Pg 136]</a></span></p>
+
+<p>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 will only be fruitful
+if they are not too numerous.</p>
+
+<p><span class="smcap">Origin of Mathematical Physics.</span>&mdash;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 number
+of elementary phenomena.</p>
+
+<p>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 substitute
+Newton's law.</p>
+
+<p>Next we try to analyze the phenomenon in space. What experiment
+gives us is a confused mass of facts presented on a
+stage of considerable extent. We must try to discover the elementary
+phenomenon, which will be, on the contrary, localized in
+a very small region of space.</p>
+
+<p>Some examples will perhaps make my thought better understood.
+If we wished to study in all its complexity the distribution
+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 neighborhood,
+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.<span class='pagenum'><a name="Page_137" id="Page_137">[Pg 137]</a></span></p>
+
+<p>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.</p>
+
+<p>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 verification.
+If it is confirmed, even approximately, it is precious,
+for it will enable us to make mathematical physics, at least by
+successive approximations.</p>
+
+<p>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 elementary
+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 superposition.
+The observed movement will then be decomposed into
+simple movements, for example, sound into its harmonics, white
+light into its monochromatic components.</p>
+
+<p>When we have discovered in what direction it is advisable to
+look for the elementary phenomenon, by what means can we
+reach it?</p>
+
+<p>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 necessary
+to penetrate the mechanism; the law of great numbers will
+suffice.</p>
+
+<p>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<span class='pagenum'><a name="Page_138" id="Page_138">[Pg 138]</a></span>
+symmetry of the medium, all the differences are leveled down, and
+whatever hypothesis may be made, the result is always the same.</p>
+
+<p>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, and that the molecules are very numerous, that
+the medium is symmetrical, and we shall only have to let the law
+of great numbers act.</p>
+
+<p>Here again the simplicity of the elementary phenomenon
+was hidden under the complexity of the resultant observable phenomenon;
+but, in its turn, this simplicity was only apparent, and
+concealed a very complex mechanism.</p>
+
+<p>The best means of arriving at the elementary phenomenon
+would evidently be experiment. We ought by experimental contrivance
+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 decomposed
+into monochromatic lights by the aid of the prism, and
+into polarized light by the aid of the polarizer.</p>
+
+<p>Unfortunately that is neither always possible nor always sufficient,
+and sometimes the mind must outstrip experiment. I shall
+cite only one example, which has always struck me forcibly.</p>
+
+<p>If I decompose white light, I shall be able to isolate a small part
+of the spectrum, but however small it may be, it will retain a
+certain breadth. Likewise the natural lights, called <i>monochromatic</i>,
+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.</p>
+
+<p>That would not be accurate. Suppose that two rays emanate
+from the same source, that we polarize them first in two perpendicular
+planes, then bring them back to the same plane of polarization,
+and try to make them interfere. If the light were <i>strictly</i>
+monochromatic, they would interfere. With our lights, which
+are nearly monochromatic, there will be no interference, and<span class='pagenum'><a name="Page_139" id="Page_139">[Pg 139]</a></span>
+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.</p>
+
+<p>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.</p>
+
+<p>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 <i>integration</i>, and is the business
+of the mathematician.</p>
+
+<p>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 <i>all alike</i>. Thus
+quite naturally are introduced differential equations.</p>
+
+<p>It is not enough that each elementary phenomenon obeys simple
+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 combination
+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 explained
+this above, in the chapter on mathematical reasoning.</p>
+
+<p>But for this, all the operations must be alike. 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.</p>
+
+<p>It is then thanks to the approximate homogeneity of the
+matter studied by physicists that mathematical physics could be
+born.</p>
+
+<p>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.</p>
+
+
+
+<hr style="width: 65%;" />
+<p><span class='pagenum'><a name="Page_140" id="Page_140">[Pg 140]</a></span></p>
+<h3>CHAPTER X</h3>
+
+<h3><span class="smcap">The Theories of Modern Physics</span></h3>
+
+
+<p><span class="smcap">Meaning of Physical Theories.</span>&mdash;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 conclude
+that these are absolutely idle. This is what they call the
+<i>bankruptcy of science</i>.</p>
+
+<p>Their skepticism is superficial; they give no account to themselves
+of the aim and the r&ocirc;le of scientific theories; otherwise
+they would comprehend that the ruins may still be good for
+something.</p>
+
+<p>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.</p>
+
+<p>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.</p>
+
+<p>And let no one say that thus we reduce physical theories to
+the r&ocirc;le of mere practical recipes; these equations express relations,
+and if the equations remain true it is because these relations
+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 <i>motion</i>; we
+now call it <i>electric current</i>. 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<span class='pagenum'><a name="Page_141" id="Page_141">[Pg 141]</a></span>
+the same relations exist between these objects as between the
+images by which we are forced to replace them. If these relations
+are known to us, what matter if we deem it convenient
+to replace one image by another.</p>
+
+<p>That some periodic phenomenon (an electric oscillation, for
+instance) is really due to the vibration of some atom which, acting
+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 reality;
+that this relationship, this similitude, or rather this parallelism
+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 will always remain the
+same under all the costumes in which we may deem it useful to
+deck it out.</p>
+
+<p>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 various
+ways, and its author himself imagined another founded on
+the principles of Maxwell. But, what is remarkable, all the scientists
+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 the
+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.</p>
+
+<p>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.</p>
+
+<p>When a physicist finds a contradiction between two theories<span class='pagenum'><a name="Page_142" id="Page_142">[Pg 142]</a></span>
+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 regarded
+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.</p>
+
+<p>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.</p>
+
+<p>Some philosopher pretends that all 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 billiards. 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 assertion,
+if neither do we wish the restricted sense I have just explained,
+which is good sense, then it has none.</p>
+
+<p>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.</p>
+
+<p>These considerations explain to us why certain theories, supposed
+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.<span class='pagenum'><a name="Page_143" id="Page_143">[Pg 143]</a></span></p>
+
+<p>Scarcely fifteen years ago was there anything more ridiculous,
+more na&iuml;vely antiquated, than Coulomb's fluids? And yet here
+they are reappearing under the name of <i>electrons</i>. 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 reluctance.
+It would be rash to affirm that the belief in electrons
+will not again suffer eclipse; it was none the less curious to note
+this unexpected resurrection.</p>
+
+<p>But the most striking example is Carnot's principle. Carnot
+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, <i>d&eacute;bris</i> 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.</p>
+
+<p>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 objects.
+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.</p>
+
+<p>What I have just said illuminates at the same time the r&ocirc;le
+of general principles such as the principle of least action, or that
+of the conservation of energy.</p>
+
+<p>These principles have a very high value; they were obtained
+in seeking what there was in common in the enunciation of numerous
+physical laws; they represent therefore, as it were, the
+quintessence of innumerable observations.</p>
+
+<p>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<span class='pagenum'><a name="Page_144" id="Page_144">[Pg 144]</a></span>
+signifies simply that there is <i>something</i> 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 <i>energy</i>.</p>
+
+<p>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 <i>energy</i> are connected by a true kinship;
+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 foresee
+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 extension
+of the principle, will yet have condemned it.</p>
+
+<p><span class="smcap">Physics and Mechanism.</span>&mdash;Most theorists have a constant
+predilection for explanations borrowed from mechanics or dynamics.
+Some would be satisfied if they could explain all phenomena
+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 instance,
+to those of our linkages; they try thus to reduce dynamics
+to a sort of kinematics.</p>
+
+<p>In a word, all would bend nature into a certain form outside
+of which their mind could not feel satisfied. Will nature be
+sufficiently flexible for that?</p>
+
+<p>We shall examine this question in Chapter XII, <i>&agrave; propos</i> 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 K&ouml;nig's on<span class='pagenum'><a name="Page_145" id="Page_145">[Pg 145]</a></span>
+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.</p>
+
+<p>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 <i>ether</i> or under other names, have at all
+times played so great a r&ocirc;le in physical theories.</p>
+
+<p>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
+singularities. For instance, what we call <i>matter</i> 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.</p>
+
+<p>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 all its parts. Not
+only did the old physicists, in multiplying fluids, create entities
+unnecessarily, but they broke real ties.</p>
+
+<p>It is not sufficient for a theory to affirm no false relations, it
+must not hide true relations.<span class='pagenum'><a name="Page_146" id="Page_146">[Pg 146]</a></span></p>
+
+<p>And does our ether really exist? We know the origin of our
+belief in the ether. If light reaches us from a distant star, during
+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.</p>
+
+<p>The same idea may be expressed under a more mathematical
+and more abstract form. What we ascertain are the changes undergone
+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 anterior;
+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 immediately
+preceding, but on states much older; the system would
+satisfy equations of finite differences. It is to escape this derogation
+of the general laws of mechanics that we have invented the
+ether.</p>
+
+<p>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 further.
+By the interference of rays which have traversed air or
+water in motion, it seems to show us two different media interpenetrating
+and yet changing place one with regard to the other.</p>
+
+<p>We seem to touch the ether with the finger.</p>
+
+<p>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 <i>alone</i>,
+and 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.</p>
+
+<p>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<span class='pagenum'><a name="Page_147" id="Page_147">[Pg 147]</a></span>
+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 displacements
+with regard to something concrete.</p>
+
+<p>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.</p>
+
+<p>For instance, if the theory of Lorentz, of which I shall speak
+in detail further on in Chapter XIII., were true, Newton's principle
+would not apply to matter <i>alone</i>, and the difference would
+not be very far from being accessible to experiment.</p>
+
+<p>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.</p>
+
+<p>I believe that such a hope is illusory; it was none the less
+interesting to show that a success of this sort would open to us,
+in some sort, a new world.</p>
+
+<p>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 relative
+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.</p>
+
+<p>But this is not enough; who does not feel that this is still to
+leave to chance too great a r&ocirc;le? 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<span class='pagenum'><a name="Page_148" id="Page_148">[Pg 148]</a></span>
+destroy the terms of the first order, and that another circumstance,
+wholly different, but just as opportune, should take upon
+itself to destroy those of the second order? 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.</p>
+
+<p><span class="smcap">Present State of the Science.</span>&mdash;In the history of the development
+of physics we distinguish two inverse tendencies.</p>
+
+<p>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.</p>
+
+<p>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.</p>
+
+<p>Of these two inverse tendencies, which seem to triumph turn
+about, which will win? If it be the first, science is possible;
+but nothing proves this <i>a priori</i>, 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.</p>
+
+<p>To this question we can not reply. All we can do is to observe
+the science of to-day and compare it with that of yesterday.
+From this examination we may doubtless draw some encouragement.</p>
+
+<p>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<span class='pagenum'><a name="Page_149" id="Page_149">[Pg 149]</a></span>
+doubted that it soon would be. For light, the task seemed completely
+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.</p>
+
+<p>But how should electricity in its turn enter into the general
+unity, how should it be reduced to the universal mechanism?</p>
+
+<p>Of that no one had any idea. Yet the possibility of this reduction
+was doubted by none, there was faith. Finally, in what
+concerns the molecular properties of material bodies, the reduction
+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 progress.
+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.</p>
+
+<p>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.</p>
+
+<p>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 unquestionably the
+one which best explains the known facts, the one which illuminates
+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.</p>
+
+<p>Now, from what we know at present, it seems probable that
+things do not happen in this way.</p>
+
+<p>However that may be, thanks to Lorentz, Fizeau's results on<span class='pagenum'><a name="Page_150" id="Page_150">[Pg 150]</a></span>
+the optics of moving bodies, the laws of normal and anomalous dispersion
+and of absorption find themselves linked to one another
+and to the other properties of the ether by bonds which beyond
+any doubt will never more be 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 defied
+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.</p>
+
+<p>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. Larmor
+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.</p>
+
+<p>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 ignorance,
+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.</p>
+
+<p>If Larmor has failed, as it seems to me he has, does that mean
+that a mechanical explanation is impossible? Far 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 mechanical
+explanations; so it is, therefore, with the optical and electrical
+phenomena.</p>
+
+<p>But this is not enough: for a mechanical explanation to be
+good, it must be simple; for choosing it among all which are possible,
+there should be other reasons besides the necessity of making
+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.</p>
+
+<p>We must therefore set bounds to our ambition; let us not try<span class='pagenum'><a name="Page_151" id="Page_151">[Pg 151]</a></span>
+to formulate a mechanical explanation; let us be content with
+showing that we could always find one if we wished to. In this
+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.</p>
+
+<p>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 principle.
+Long did thermodynamics confine itself to the study of
+the dilatation of bodies and their changes of state. For some time
+past 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.</p>
+
+<p>Everywhere the same laws reign; everywhere, under the diversity
+of appearances, is found again Carnot's principle; everywhere
+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.</p>
+
+<p>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 appear to be the type which
+the most various irreversible phenomena copy, and this kinship
+is real and profound.</p>
+
+<p>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<span class='pagenum'><a name="Page_152" id="Page_152">[Pg 152]</a></span>
+go forward in the same sense without hope of return. The apparent
+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.</p>
+
+<p>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.</p>
+
+<p>We signalize, however, M. Gouy's original ideas on the Brownian
+movement. According to this scientist, this singular motion
+should escape Carnot'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.</p>
+
+<p>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.</p>
+
+<p>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?</p>
+
+<p>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.</p>
+
+<p>Nor are they without kinship with the causes which produce
+the electric spark under the action of the ultra-violet light.</p>
+
+<p>Finally, and above all, it is believed that in all these phenomena
+are found true ions, animated, it is true, by velocities incomparably
+greater than in the electrolytes.</p>
+
+<p>That is all very vague, but it will all become more precise.</p>
+
+<p>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<span class='pagenum'><a name="Page_153" id="Page_153">[Pg 153]</a></span>
+constructed which will facilitate their communications with the rest
+of science.</p>
+
+<p>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.</p>
+
+<p>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.</p>
+
+<p>Take, for example, the laws of reflection: Fresnel had established
+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.</p>
+
+<p>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 precise
+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.</p>
+
+<p>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,<span class='pagenum'><a name="Page_154" id="Page_154">[Pg 154]</a></span>
+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.</p>
+
+<p>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.</p>
+
+<p>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.</p>
+
+<p>From this rapid exposition, what shall we conclude?</p>
+
+<p>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.</p>
+
+
+
+<hr style="width: 65%;" />
+<p><span class='pagenum'><a name="Page_155" id="Page_155">[Pg 155]</a></span></p>
+<h3>CHAPTER XI</h3>
+
+<h3><span class="smcap">The Calculus of Probabilities</span></h3>
+
+
+<p>Doubtless it will be astonishing 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 without
+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.'</p>
+
+<p>'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 r&ocirc;le the belief in simplicity plays in our generalizations.
+We have verified a simple law in a great number of particular
+cases; we refuse to admit that this coincidence, so often
+repeated, can be a mere effect of chance...."</p>
+
+<p>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 consciously
+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.</p>
+
+<p>The very name calculus of probabilities is a paradox. Probability
+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.</p>
+
+<p>How can we explain this apparent contradiction?</p>
+
+<p>Has probability been defined? Can it even be defined? And
+if it can not, how dare we reason about it? The definition, it will<span class='pagenum'><a name="Page_156" id="Page_156">[Pg 156]</a></span>
+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.</p>
+
+<p>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? Each die can turn up in six different
+ways; the number of possible cases is 6 &times; 6 = 36; the number of
+favorable cases is 11; the probability is 11/36.</p>
+
+<p>That is the correct solution. But could I not just as well say:
+The points which turn up on the two dice can form 6 &times; 7/2 = 21
+different combinations? Among these combinations 6 are favorable;
+the probability is 6/21.</p>
+
+<p>Now why is the first method of enumerating the possible cases
+more legitimate than the second? In any case it is not our
+definition that tells us.</p>
+
+<p>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.</p>
+
+<p>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.</p>
+
+<p>Will it be said that good sense suffices to show us what convention
+should be adopted? Alas! M. Bertrand has amused
+himself 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.</p>
+
+<p>The conclusion which seems to follow from all this is that the
+calculus of probabilities is a useless science, and that the obscure<span class='pagenum'><a name="Page_157" id="Page_157">[Pg 157]</a></span>
+instinct which we may call good sense, and to which we are wont
+to appeal to legitimatize our conventions, must be distrusted.</p>
+
+<p>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.'</p>
+
+<p>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 enormous
+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.'</p>
+
+<p>From this point of view, all the sciences would be only unconscious
+applications of the calculus of probabilities. To condemn
+this calculus would be to condemn the whole of science.</p>
+
+<p>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.</p>
+
+<p>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.</p>
+
+<p>All 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.</p>
+
+<p>But, as I have said above, it would not be only these partial<span class='pagenum'><a name="Page_158" id="Page_158">[Pg 158]</a></span>
+sacrifices that would be in question; it would be the legitimacy of
+the whole of science that would be challenged.</p>
+
+<p>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 practical
+value.</p>
+
+<p>Unfortunately the difficulty does not thus disappear. A gambler
+wants to try a <i>coup</i>; 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 <i>subjective probability</i>. 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 <i>coups</i>, 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 <i>objective
+probability</i>, and it is this phenomenon which has to be explained.</p>
+
+<p>There are numerous insurance companies which apply the rules
+of the calculus of probabilities, and they distribute to their shareholders
+dividends whose objective reality can not be contested.
+To invoke our ignorance and the necessity to act does not suffice
+to explain them.</p>
+
+<p>Thus absolute skepticism is not admissible. We may distrust,
+but we can not condemn <i>en bloc</i>. Discussion is necessary.</p>
+
+<p><span class="smcap">I. Classification of the Problems of Probability.</span>&mdash;In
+order to classify the problems which present themselves <i>&agrave; propos</i>
+of probabilities, we may look at them from many different points
+of view, and, first, from the <i>point of view of generality</i>. 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<span class='pagenum'><a name="Page_159" id="Page_159">[Pg 159]</a></span>
+possible 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.</p>
+
+<p>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. Generality
+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.</p>
+
+<p>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 according
+to the greater or less depth of this ignorance.</p>
+
+<p>In mathematics even we may set ourselves problems of probability.
+What is the probability that the fifth decimal of a logarithm
+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 logarithm
+without recourse to the table, but we do not wish to give
+ourselves the trouble. This is the first degree of ignorance.</p>
+
+<p>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.</p>
+
+<p>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.<span class='pagenum'><a name="Page_160" id="Page_160">[Pg 160]</a></span></p>
+
+<p>In the kinetic theory of gases, we assume that the gaseous
+molecules follow rectilinear trajectories, and obey the laws of
+impact of elastic bodies. But, as we know nothing of their initial
+velocities, we know nothing of their present velocities.</p>
+
+<p>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.</p>
+
+<p>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.</p>
+
+<p>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 <i>probability of
+causes</i>, the most interesting from the point of view of their scientific
+applications.</p>
+
+<p>I play &eacute;cart&eacute; 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.</p>
+
+<p>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.</p>
+
+<p>It may be said that this is the essential problem of the experimental
+method. I have observed <i>n</i> values of <i>x</i> and the corresponding
+values of <i>y</i>. I have found that the ratio of the latter to
+the former is practically constant. There is the event, what is
+the cause?</p>
+
+<p>Is it probable that there is a general law according to which <i>y</i>
+would be proportional to <i>x</i>, 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.</p>
+
+<p>I am now going to pass in review these different categories of<span class='pagenum'><a name="Page_161" id="Page_161">[Pg 161]</a></span>
+problems, discussing in succession what I have called above subjective
+and objective probability.</p>
+
+<p><span class="smcap">II. Probability in Mathematics.</span>&mdash;The impossibility of squaring
+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 examination the
+alas! too numerous memoirs on this subject, that some unhappy
+madmen sent in every year.</p>
+
+<p>Was the Academy wrong? Evidently not, and it knew well
+that in acting thus it did not run the least risk of stifling a discovery
+of moment. The Academy could not have proved that it
+was right; but it knew quite well that its instinct was not mistaken.
+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.</p>
+
+<p>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 &#960; were a root of an
+algebraic equation, why do you suppose this root to be a period of
+the function sin 2<i>x</i>, and not the same about the other roots of this
+same equation?" To sum up, they would have invoked the principle
+of sufficient reason in its vaguest form.</p>
+
+<p>But what could they deduce from it? At most a rule of conduct
+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.</p>
+
+<p>It is otherwise with the second problem.</p>
+
+<p>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.<span class='pagenum'><a name="Page_162" id="Page_162">[Pg 162]</a></span></p>
+
+<p>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
+&minus;1 if odd. Then take the mean of these 10,000 numbers.</p>
+
+<p>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.</p>
+
+<p>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 <i>Revue g&eacute;n&eacute;rale des Sciences</i>, 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 logarithm
+remain, in the interval considered, between certain limits.</p>
+
+<p>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.</p>
+
+<p>If I was certain beforehand of the result, it is first, because I
+had often observed analogous facts for other continuous functions;
+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.</p>
+
+<p>And besides, since what I call my intuition was only an incomplete
+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.</p>
+
+<p>As a third example I shall choose the following problem: A
+number <i>u</i> is taken at random, and <i>n</i> is a given very large integer.
+What is the probable value of sin <i>nu</i>? This problem has no meaning
+by itself. To give it one a convention is needed. We <i>shall
+agree</i> that the probability for the number <i>u</i> to lie between <i>a</i> and
+<i>a</i>+ is equal to &#981;(<i>a</i>)<i>da</i>; that it is therefore proportional to the
+infinitely small interval <i>da</i>, and equal to this multiplied by <i>a</i>
+function &#981;(<i>a</i>) depending only on <i>a</i>. As for this function, I<span class='pagenum'><a name="Page_163" id="Page_163">[Pg 163]</a></span>
+choose it arbitrarily, but I must assume it to be continuous. The
+value of sin <i>nu</i> remaining the same when <i>u</i> increases by 2&#960;, I may
+without loss of generality assume that <i>u</i> lies between 0 and 2&#960;,
+and I shall thus be led to suppose that &#981;(<i>a</i>) is a periodic function
+whose period is 2&#960;.</p>
+
+<p>The probable value sought is readily expressed by a simple
+integral, and it is easy to show that this integral is less than</p>
+
+<p class="center">2&#960;M<sub><i>k</i></sub> &frasl; <i>n</i><sup><i>k</i></sup>,</p>
+
+<p>M<sub><i>k</i></sub> being the maximum value of the <i>k</i><sup>th</sup> derivative of &#981;(<i>u</i>). We
+see then that if the <i>k</i><sup>th</sup> derivative is finite, our probable value will
+tend toward 0 when <i>n</i> increases indefinitely, and that more rapidly
+than 1/<i>n</i><sup><i>k</i>&minus;1</sup>.</p>
+
+<p>The probable value of sin <i>nu</i> when <i>n</i> is very large is therefore
+naught. To define this value I required a convention; but the
+result remains the same <i>whatever that convention may be</i>. I
+have imposed upon myself only slight restrictions in assuming
+that the function &#981;(<i>a</i>) is continuous and periodic, and these hypotheses
+are so natural that we may ask ourselves how they can
+be escaped.</p>
+
+<p>Examination of the three preceding examples, so different in
+all respects, has already given us a glimpse, on the one hand,
+of the r&ocirc;le 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.</p>
+
+<p><span class="smcap">III. Probability in the Physical Sciences.</span>&mdash;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?</p>
+
+<p>We know they obey the laws of Kepler. We may even, without
+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<span class='pagenum'><a name="Page_164" id="Page_164">[Pg 164]</a></span>
+hesitate to affirm that their distribution is now nearly uniform.
+Why?</p>
+
+<p>Let <i>b</i> be the longitude of a minor planet in the initial epoch,
+that is to say, the epoch zero. Let <i>a</i> be its mean motion. Its
+longitude at the present epoch, that is to say at the epoch <i>t</i>, will
+be <i>at</i> + <i>b</i>. To say that the present distribution is uniform is to
+say that the mean value of the sines and cosines of multiples of
+<i>at</i> + <i>b</i> is zero. Why do we assert this?</p>
+
+<p>Let us represent each minor planet by a point in a plane, to
+wit, by a point whose coordinates are precisely <i>a</i> and <i>b</i>. 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 distribution
+of these points.</p>
+
+<p>What do we do when we wish to apply the calculus of probabilities
+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 arbitrary
+hypothesis. To explain the nature of this hypothesis, allow
+me to use, in lieu of a mathematical formula, a crude but concrete
+image. Let us suppose that over the surface of our plane
+has been spread an imaginary substance, whose density is variable,
+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.</p>
+
+<p>Here then are two distributions, one real, in which the representative
+points are very numerous, very close together, but discrete
+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.</p>
+
+<p>If again we had some idea of the real distribution of the
+representative points, we could arrange it so that in a region<span class='pagenum'><a name="Page_165" id="Page_165">[Pg 165]</a></span>
+of some extent the density of this imaginary continuous matter
+would be nearly proportional to the number of the representative
+points, or, if you wish, to the number of atoms which are contained
+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 sufficient,
+as we shall see, to enable us to reach a conclusion.</p>
+
+<p>What is at the instant <i>t</i> the probable distribution of the minor
+planets? Or rather what is the probable value of the sine of the
+longitude at the instant <i>t</i>, that is to say of sin (<i>at</i> + <i>b</i>)? We
+made at the outset an arbitrary convention, but if we adopt it,
+this probable value is entirely defined. Divide the plane into elements
+of surface. Consider the value of sin (<i>at</i> + <i>b</i>) at the center
+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 &#981; is arbitrary, we can, according to the
+arbitrary choice which we make, obtain any mean value. This
+is not so.</p>
+
+<p>A simple calculation shows that our double integral decreases
+very rapidly when <i>t</i> 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.</p>
+
+<p>Whatever be the function &#981;, the mean value tends toward zero
+as <i>t</i> increases, and as the minor planets have certainly accomplished
+a very great number of revolutions, I may assert that this
+mean value is very small.</p>
+
+<p>I may choose &#981; 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<span class='pagenum'><a name="Page_166" id="Page_166">[Pg 166]</a></span>
+I have for supposing that the initial longitude might be exactly
+0&deg;, but that it could not lie between 0&deg; and 1&deg;?</p>
+
+<p>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.</p>
+
+<p>The mean value of sin (<i>at</i> + <i>b</i>) will be represented quite
+simply by</p>
+
+<p class="center">
+(1/<i>n</i>) &#931; sin (<i>at</i> + <i>b</i>),<br />
+</p>
+
+<p class="noidt"><i>n</i> being the number of minor planets. In lieu of a double integral
+referring to a continuous function, we shall have a sum of
+discrete terms. And yet no one will seriously doubt that this
+mean value is practically very small.</p>
+
+<p>Our representative points being very close together, our discrete
+sum will in general differ very little from an integral.</p>
+
+<p>An integral is the limit toward which a sum of terms tends
+when the number of these terms is indefinitely increased. If the
+terms are very numerous, the sum will differ very little from
+its limit, that is to say from the integral, and what I said of this
+latter will still be true of the sum itself.</p>
+
+<p>Nevertheless, there are exceptions. If, for instance, for all
+the minor planets,</p>
+
+<p class="center">
+<i>b</i> = &#960;/2 &minus; <i>at</i>,<br />
+</p>
+
+<p class="noidt">the longitude for all the planets at the time t would be &#960;/2, and
+the mean value would evidently be equal to unity. For this to
+be the case, it would be necessary that at the epoch 0, the minor
+planets must have all been lying on a spiral of peculiar form, with
+its spires very close together. Every one will admit that such an
+initial distribution is extremely improbable (and, even supposing
+it realized, the distribution would not be uniform at the present
+time, for example, on January 1, 1913, but it would become so
+a few years later).</p>
+
+<p>Why then do we think this initial distribution improbable?
+This must be explained, because if we had no reason for rejecting<span class='pagenum'><a name="Page_167" id="Page_167">[Pg 167]</a></span>
+as improbable this absurd hypothesis everything would break
+down, and we could no longer make any affirmation about the
+probability of this or that present distribution.</p>
+
+<p>Once more we shall invoke the principle of sufficient reason to
+which we must always recur. We might admit that at the beginning
+the planets were distributed almost in a straight line. We
+might admit that they were irregularly distributed. But it seems
+to us that there is no sufficient reason for the unknown cause that
+gave them birth to have acted along a curve so regular and yet so
+complicated, which would appear to have been expressly chosen
+so that the present distribution would not be uniform.</p>
+
+<p><span class="smcap">IV. Rouge et Noir.</span>&mdash;The questions raised by games of chance,
+such as roulette, are, fundamentally, entirely analogous to those
+we have just treated. For example, a wheel is partitioned into
+a great number of equal subdivisions, alternately red and black.
+A needle is whirled with force, and after having made a great
+number of revolutions, it stops before one of these subdivisions.
+The probability that this division is red is evidently 1/2. The
+needle describes an angle &#952;, including several complete revolutions.
+I do not know what is the probability that the needle may
+be whirled with a force such that this angle should lie between &#952;
+and &#952; + <i>d</i>&#952;; but I can make a convention. I can suppose that this
+probability is &#981;(&#952;)<i>d</i>&#952;. As for the function &#981;(&#952;), 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.</p>
+
+<p>Let &#949; be the length (measured on the circumference of radius
+1) of each red and black subdivision. We have to calculate the
+integral of &#981;(&#952;)<i>d</i>&#952;, 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.</p>
+
+<p>Consider an interval 2&#949;, comprising a red division and a black
+division which follows it. Let M and <i>m</i> be the greatest and least
+values of the function &#981;(&#952;) in this interval. The integral extended
+to the red divisions will be smaller than &#931;M&#949;; the integral extended
+to the black divisions will be greater than &#931;<i>m</i>&#949;; the difference
+will therefore be less than &#931;(M &minus; <i>m</i>)&#949;. But, if the function
+&#952; is supposed continuous; if, besides, the interval &#949; is very<span class='pagenum'><a name="Page_168" id="Page_168">[Pg 168]</a></span>
+small with respect to the total angle described by the needle,
+the difference M &minus; <i>m</i> will be very small. The difference of the
+two integrals will therefore be very small, and the probability
+will be very nearly 1/2.</p>
+
+<p>We see that without knowing anything of the function &#952;, 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.</p>
+
+<p>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.</p>
+
+<p>In reality their probability of winning remains 1/2. Observation
+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.</p>
+
+<p>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.</p>
+
+<p><span class="smcap">V. The Probability of Causes.</span>&mdash;We now come to the problems
+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.</p>
+
+<p>I recall first that at the outset of all problems of the probability
+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 <i>a priori</i> discontinuous functions, for example,
+or certain absurd conventions.</p>
+
+<p>We shall find something analogous when we deal with the<span class='pagenum'><a name="Page_169" id="Page_169">[Pg 169]</a></span>
+probability of causes. An effect may be produced by the cause
+<i>A</i> or by the cause <i>B</i>. The effect has just been observed. We
+ask the probability that it is due to the cause <i>A</i>. This is an <i>a
+posteriori</i> probability of cause. But I could not calculate it, if
+a convention more or less justified did not tell me <i>in advance</i>
+what is the <i>a priori</i> probability for the cause <i>A</i> to come into
+play; I mean the probability of this event for some one who had
+not observed the effect.</p>
+
+<p>The better to explain myself I go back to the example of the
+game of &eacute;cart&eacute; 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, <i>before sitting down at the table</i>, I
+had considered that there was one chance in two that my adversary
+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.</p>
+
+<p>The convention about the <i>a priori</i> probability was unjustified,
+and that is why the calculation of the <i>a posteriori</i> probability led
+me to an inadmissible result. We see the importance of this preliminary
+convention. I shall even add that if none were made,
+the problem of the <i>a posteriori</i> probability would have no meaning.
+It must always be made either explicitly or tacitly.</p>
+
+<p>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 inflections
+too accentuated, or brusque variation of the radius of curvature.
+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.<span class='pagenum'><a name="Page_170" id="Page_170">[Pg 170]</a></span></p>
+
+<p>Here is a problem in the probability of causes. The effects
+are the measurements I have recorded; they depend on a combination
+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 distance
+of the corresponding point from this curve.</p>
+
+<p>But the problem would have no meaning if, before any observation,
+I had not fashioned an <i>a priori</i> idea of the probability of
+this or that law, and of the chances of error to which I am exposed.</p>
+
+<p>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.</p>
+
+<p>Why then is it that I seek to trace a curve without sinuosities?
+It is because I consider <i>a priori</i> a law represented by a continuous
+function (or by a function whose derivatives of high order
+are small), as more probable than a law not satisfying these conditions.
+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.</p>
+
+<p>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.</p>
+
+<p><span class="smcap">VI. The Theory of Errors.</span>&mdash;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 <i>effects</i>, to wit,
+a certain number of discordant observations, and we seek to<span class='pagenum'><a name="Page_171" id="Page_171">[Pg 171]</a></span>
+divine the <i>causes</i>, 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
+<i>a posteriori</i> the probable magnitude of each error, and consequently
+the probable value of the quantity to be measured.</p>
+
+<p>But as I have just explained, we should not know how to undertake
+this calculation if we did not admit <i>a priori</i>, that is to
+say, before all observation, a law of probability of errors. Is
+there a law of errors?</p>
+
+<p>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 bell.'</p>
+
+<p>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 will 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.</p>
+
+<p>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 independent
+errors; each of the partial errors is very little and
+besides, obeys any law of probability, provided that the probability
+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.</p>
+
+<p>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, besides
+the systematic errors to which they and the physicists are
+subject alike, have to control with an extremely important source
+of error which is wholly accidental; I mean atmospheric<span class='pagenum'><a name="Page_172" id="Page_172">[Pg 172]</a></span>
+undulations. So it is very curious to hear a physicist discuss with an
+astronomer about a method of observation. The physicist, persuaded
+that one good measurement is worth more than many
+bad ones, is before all concerned with eliminating by dint of
+precautions the least systematic errors, and the astronomer says
+to him: 'But thus you can observe only a small number of stars;
+the accidental errors will not disappear.'</p>
+
+<p>What should we conclude? Must we continue to use the
+method of least squares? We must distinguish. We have eliminated
+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 probability.
+There is nothing more to be said.</p>
+
+<p>But we wish to go farther and affirm that not only is the
+probable value so much, but that the probable error in the result
+is so much. <i>This is absolutely illegitimate</i>; it would be true
+only if we were sure that all the systematic errors were eliminated,
+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 systematic
+error. All we can say is that the first series is <i>probably</i>
+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 than for the other, our ignorance on this
+point being absolute.</p>
+
+<p><span class="smcap">VII. Conclusions.</span>&mdash;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.</p>
+
+<p>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 necessary<span class='pagenum'><a name="Page_173" id="Page_173">[Pg 173]</a></span>
+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
+sufficient 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 under
+which we have met it most often is the belief in continuity, a
+belief which it would be difficult to justify by apodeictic reasoning,
+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.</p>
+
+
+
+<hr style="width: 65%;" />
+<p><span class='pagenum'><a name="Page_174" id="Page_174">[Pg 174]</a></span></p>
+<h3>CHAPTER XII</h3>
+
+<h3><span class="smcap">Optics and Electricity</span></h3>
+
+
+<p><span class="smcap">Fresnel's Theory.</span>&mdash;The best example<a name="FNanchor_5_5" id="FNanchor_5_5"></a><a href="#Footnote_5_5" class="fnanchor">[5]</a> 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-called wave-theory forms a
+whole truly satisfying to the mind. We must not, however, ask
+of it what it can not give us.</p>
+
+<p>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 experiment
+reveals to us, but which, without the help of mathematics,
+we should not be able even to state.</p>
+
+<p>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 will always
+be useful, then, to study a doctrine that unites all these equations.</p>
+
+<p>The undulatory theory rests on a molecular hypothesis. 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 distrust.
+But this distrust seems to me as little justified as the
+illusion of the former.</p>
+
+<p>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.<span class='pagenum'><a name="Page_175" id="Page_175">[Pg 175]</a></span></p>
+
+<p>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.</p>
+
+<p>This explains why most of Fresnel's conclusions remain unchanged
+when we adopt the electromagnetic theory of light.</p>
+
+<p><span class="smcap">Maxwell's Theory.</span>&mdash;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.</p>
+
+<p>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.</p>
+
+<p>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.</p>
+
+<p>The old theories of mathematical physics gave us in this respect
+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 precision
+as celestial mechanics.</p>
+
+<p>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.</p>
+
+<p>This is not all; it will have still other demands, which seem to<span class='pagenum'><a name="Page_176" id="Page_176">[Pg 176]</a></span>
+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 will have
+only purely geometric properties, and whose atoms will be nothing
+but mathematical points, subject to the laws of dynamics
+alone. And yet these atoms, invisible and without color, it will
+seek by an unconscious contradiction to represent to itself and
+consequently to identify as closely as possible with common
+matter.</p>
+
+<p>Then only will it be fully satisfied and imagine that it has
+penetrated the secret of the universe. If this satisfaction is deceitful,
+it is none the less difficult to renounce.</p>
+
+<p>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 himself
+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.</p>
+
+<p>Maxwell does not give a mechanical explanation of electricity
+and magnetism; he confines himself to demonstrating that such
+an explanation is possible.</p>
+
+<p>He shows also that optical phenomena are only a special case
+of electromagnetic phenomena. From every theory of electricity,
+one can therefore deduce immediately a theory of light.</p>
+
+<p>The converse unfortunately is not true; from a complete explanation
+of light, it is not always easy to derive a complete explanation
+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.</p>
+
+<p>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 independent
+constructions, between which communication is difficult
+and sometimes impossible.<span class='pagenum'><a name="Page_177" id="Page_177">[Pg 177]</a></span></p>
+
+<p>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 contains
+a theory complete in itself which one could understand without
+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."</p>
+
+<p>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?</p>
+
+<p>One must not then flatter himself that he can avoid all contradiction;
+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.</p>
+
+<p>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.</p>
+
+<p>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.</p>
+
+<p><span class="smcap">The Mechanical Explanation of Physical Phenomena.</span>&mdash;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 <i>q</i>.</p>
+
+<p>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 <i>q</i> with
+the time.</p>
+
+<p>What is it necessary to do to give a mechanical interpretation
+of such a phenomenon?<span class='pagenum'><a name="Page_178" id="Page_178">[Pg 178]</a></span></p>
+
+<p>One will try to explain it either by the motions of ordinary
+matter, or by those of one or more hypothetical fluids.</p>
+
+<p>These fluids will be considered as formed of a very great number
+of isolated molecules <i>m</i>.</p>
+
+<p>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 coordinates
+of these hypothetical molecules <i>m</i>, equations which, moreover,
+must conform to the principles of dynamics; and, on the
+other hand, when we know the relations that define the coordinates
+of the molecules <i>m</i> as functions of the parameters <i>q</i> accessible
+to experiment.</p>
+
+<p>These equations, as I have said, must conform to the principles
+of dynamics, and, in particular, to the principle of the
+conservation of energy and the principle of least action.</p>
+
+<p>The first of these two principles teaches us that the total energy
+is constant and that this energy is divided into two parts:</p>
+
+<p>1&ordm; The kinetic energy, or <i>vis viva</i>, which depends on the
+masses of the hypothetical molecules <i>m</i>, and their velocities, and
+which I shall call <i>T</i>.</p>
+
+<p>2&ordm; The potential energy, which depends only on the coordinates
+of these molecules and which I shall call <i>U</i>. It is the <i>sum</i>
+of the two energies <i>T</i> and <i>U</i> which is constant.</p>
+
+<p>What now does the principle of least action tell us? It tells
+us that to pass from the initial position occupied at the instant <i>t</i><sub>0</sub>
+to the final position occupied at the instant <i>t</i><sub>1</sub>, the system must
+take such a path that, in the interval of time that elapses between
+the two instants <i>t</i><sub>0</sub> and <i>t</i><sub>1</sub>, the average value of 'the
+action' (that is to say, of the <i>difference</i> between the two energies
+<i>T</i> and <i>U</i>) shall be as small as possible.</p>
+
+<p>If the two functions <i>T</i> and <i>U</i> are known, this principle suffices
+to determine the equations of motion.</p>
+
+<p>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.<span class='pagenum'><a name="Page_179" id="Page_179">[Pg 179]</a></span></p>
+
+<p>Thus we obtain what are called the equations of Lagrange.</p>
+
+<p>In these equations, the independent variables are the coordinates
+of the hypothetical molecules <i>m</i>; but I now suppose that
+one takes as variables the parameters <i>q</i> directly accessible to experiment.</p>
+
+<p>The two parts of the energy must then be expressed as functions
+of the parameters <i>q</i> 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.<a name="FNanchor_6_6" id="FNanchor_6_6"></a><a href="#Footnote_6_6" class="fnanchor">[6]</a></p>
+
+<p>That granted, the system will always go from one position to
+another by a path such that the average action shall be a minimum.</p>
+
+<p>It matters little that <i>T</i> and <i>U</i> are now expressed by the aid
+of the parameters <i>q</i> 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.</p>
+
+<p>Now here again, of all the paths that lead from one position
+to another, there is one for which the average action is a minimum,
+and there is only one. The principle of least action
+suffices, then, to determine the differential equations which define
+the variations of the parameters <i>q</i>.</p>
+
+<p>The equations thus obtained are another form of the equations
+of Lagrange.</p>
+
+<p>To form these equations we need to know neither the relations
+that connect the parameters <i>q</i> with the coordinates of the
+hypothetical molecules, nor the masses of these molecules, nor
+the expression of <i>U</i> as a function of the coordinates of these
+molecules.</p>
+
+<p>All we need to know is the expression of <i>U</i> as a function of
+the parameters, and that of <i>T</i> as a function of the parameters <i>q</i>
+and their derivatives, that is, the expressions of the kinetic and
+of the potential energy as functions of the experimental data.</p>
+
+<p>Then we shall have one of two things: either for a suitable<span class='pagenum'><a name="Page_180" id="Page_180">[Pg 180]</a></span>
+choice of the functions <i>T</i> and <i>U</i>, the equations of Lagrange, constructed
+as we have just said, will be identical with the differential
+equations deduced from experiments; or else there will
+exist no functions <i>T</i> and <i>U</i>, for which this agreement takes place.
+In the latter case it is clear that no mechanical explanation is
+possible.</p>
+
+<p>The <i>necessary</i> condition for a mechanical explanation to be
+possible is therefore that we can choose the functions <i>T</i> and <i>U</i>
+in such a way as to satisfy the principle of least action, which involves
+that of the conservation of energy.</p>
+
+<p>This condition, moreover, is <i>sufficient</i>. Suppose, in fact, that
+we have found a function <i>U</i> of the parameters <i>q</i>, which represents
+one of the parts of the energy; that another part of the
+energy, which we shall represent by <i>T</i>, is a function of the
+parameters <i>q</i> 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, <i>T</i> and <i>U</i>, conform to the data of the
+experiment.</p>
+
+<p>What is necessary in order to deduce from this a mechanical
+explanation? It is necessary that <i>U</i> can be regarded as the potential
+energy of a system and <i>T</i> as the <i>vis viva</i> of the same
+system.</p>
+
+<p>There is no difficulty as to <i>U</i>, but can <i>T</i> be regarded as the
+<i>vis viva</i> of a material system?</p>
+
+<p>It is easy to show that this is always possible, and even in
+an infinity of ways. I will confine myself to referring for more
+details to the preface of my work, '&Eacute;lectricit&eacute; et optique.'</p>
+
+<p>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.</p>
+
+<p>One more observation.</p>
+
+<p>Among the quantities that experiment gives us directly, we
+shall regard some as functions of the coordinates of our hypothetical
+molecules; these are our parameters <i>q</i>. 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<span class='pagenum'><a name="Page_181" id="Page_181">[Pg 181]</a></span>
+of the parameters <i>q</i>, or as combinations of these parameters and
+their derivatives.</p>
+
+<p>And then a question presents itself: among all these quantities
+measured experimentally, which shall we choose to represent the
+parameters <i>q</i>? Which shall we prefer to regard as the derivatives
+of these parameters? This choice remains arbitrary to a
+very large extent; but, for a mechanical explanation to be possible,
+it suffices if we can make the choice in such a way as to
+accord with the principle of least action.</p>
+
+<p>And then Maxwell asked himself whether he could make this
+choice and that of the two energies <i>T</i> and <i>U</i>, in such a way
+that the electrical phenomena would satisfy this principle. Experiment
+shows us that the energy of an electromagnetic field is
+decomposed into two parts, the electrostatic energy and the electrodynamic
+energy. Maxwell observed that if we regard the
+first as representing the potential energy <i>U</i>, the second as representing
+the kinetic energy <i>T</i>; if, moreover, the electrostatic
+charges of the conductors are considered as parameters <i>q</i> and
+the intensities of the currents as the derivatives of other parameters
+<i>q</i>; under these conditions, I say, Maxwell observed that the
+electric phenomena satisfy the principle of least action. Thenceforth
+he was certain of the possibility of a mechanical explanation.</p>
+
+<p>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.</p>
+
+<p>If, then, a phenomenon admits of a complete mechanical explanation,
+it will admit of an infinity of others, that will render
+an account equally well of all the particulars revealed by experiment.</p>
+
+<p>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.</p>
+
+<p>In the same way, without leaving the domain of electricity,
+we may ascertain that the theory of two fluids and that of the<span class='pagenum'><a name="Page_182" id="Page_182">[Pg 182]</a></span>
+single fluid both account in a fashion equally satisfactory for all
+the observed laws of electrostatics.</p>
+
+<p>All these facts are easily explicable, thanks to the properties
+of the equations of Lagrange which I have just recalled.</p>
+
+<p>It is easy now to comprehend what is Maxwell's fundamental
+idea.</p>
+
+<p>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 <i>T</i> and <i>U</i>, 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.</p>
+
+<p>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.</p>
+
+<p>Our choice can therefore be further guided only by considerations
+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 because
+of their simplicity.</p>
+
+<p>In what concerns electricity and magnetism, Maxwell abstains
+from making any choice. It is not that he systematically disdains
+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 <i>Philosophical Magazine</i>. The strangeness and the complexity
+of the hypotheses he had been obliged to make had led
+him afterwards to give this up.</p>
+
+<p>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<span class='pagenum'><a name="Page_183" id="Page_183">[Pg 183]</a></span>
+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.</p>
+
+
+
+<hr style="width: 65%;" />
+<p><span class='pagenum'><a name="Page_184" id="Page_184">[Pg 184]</a></span></p>
+<h3>CHAPTER XIII</h3>
+
+<h3><span class="smcap">Electrodynamics</span></h3>
+
+
+<p>The history of electrodynamics is particularly instructive from
+our point of view.</p>
+
+<p>Amp&egrave;re entitled his immortal work, 'Th&eacute;orie des ph&eacute;nom&egrave;nes
+&eacute;lectrodynamiques, <i>uniquement</i> fond&eacute;e sur l'exp&eacute;rience.' He
+therefore imagined that he had made <i>no</i> hypothesis, but he had
+made them, as we shall soon see; only he made them without
+being conscious of it.</p>
+
+<p>His successors, on the other hand, perceived them, since their
+attention was attracted by the weak points in Amp&egrave;re'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.</p>
+
+<p><span class="smcap">I. Ampere's Theory.</span>&mdash;When Amp&egrave;re studied experimentally
+the mutual actions of currents, he operated and he only could
+operate with closed currents.</p>
+
+<p>It was not that he denied the possibility of open currents.
+If two conductors are charged with positive and negative electricity
+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 Amp&egrave;re's
+time this was an open current; the current was known to go
+from the first conductor to the second, it was not seen to return
+from the second to the first.</p>
+
+<p>So Amp&egrave;re considered as open currents of this nature, for example,
+the currents of discharge of condensers; but he could not
+make them the objects of his experiments because their duration
+is too short.</p>
+
+<p>Another sort of open current may also be imagined. I suppose
+two conductors, <i>A</i> and <i>B</i>, connected by a wire <i>AMB</i>. Small
+conducting masses in motion first come in contact with the<span class='pagenum'><a name="Page_185" id="Page_185">[Pg 185]</a></span>
+conductor <i>B</i>, take from it an electric charge, leave contact with
+<i>B</i> and move along the path <i>BNA</i>, and, transporting with them
+their charge, come into contact with <i>A</i> and give to it their charge,
+which returns then to <i>B</i> along the wire <i>AMB</i>.</p>
+
+<p>Now there we have in a sense a closed circuit, since the electricity
+describes the closed circuit <i>BNAMB</i>; but the two parts
+of this current are very different. In the wire <i>AMB</i>, the electricity
+is displaced through a fixed conductor, like a voltaic current,
+overcoming an ohmic resistance and developing heat; we
+say that it is displaced by conduction. In the part <i>BNA</i>, the
+electricity is carried by a moving conductor; it is said to be displaced
+by convection.</p>
+
+<p>If then the current of convection is considered as altogether
+analogous to the current of conduction, the circuit <i>BNAMB</i> 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 <i>AMB</i>, which is open.</p>
+
+<p>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.</p>
+
+<p>But currents of this sort are very difficult to produce with appreciable
+intensity. With the means at Amp&egrave;re's disposal, we
+may say that this was impossible.</p>
+
+<p>To sum up, Amp&egrave;re 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.</p>
+
+<p>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 displacements
+of the moving part under the action of another closed
+current.</p>
+
+<p>On the other hand, Amp&egrave;re had no means of studying the
+action of an open current, either on a closed current or another
+open current.<span class='pagenum'><a name="Page_186" id="Page_186">[Pg 186]</a></span></p>
+
+<p>1. <i>The Case of Closed Currents.</i>&mdash;In the case of the mutual
+action of two closed currents, experiment revealed to Amp&egrave;re remarkably
+simple laws.</p>
+
+<p>I recall rapidly here those which will be useful to us in the
+sequel:</p>
+
+<p>1&ordm; <i>If the intensity of the currents is kept constant</i>, 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.</p>
+
+<p>In other words, there is an <i>electrodynamic potential</i> 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.</p>
+
+<p>2&ordm; The action of a closed solenoid is null.</p>
+
+<p>3&ordm; The action of a circuit <i>C</i> on another voltaic circuit <i>C&acute;</i> depends
+only on the 'magnetic field' developed by this circuit. At
+each point in space we can in fact define in magnitude and direction
+a certain force called <i>magnetic force</i>, which enjoys the following
+properties:</p>
+
+<p>(<i>a</i>) The force exercised by <i>C</i> on a magnetic pole is applied to
+that pole and is equal to the magnetic force multiplied by the
+magnetic mass of that pole;</p>
+
+<p>(<i>b</i>) 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;</p>
+
+<p>(<i>c</i>) If the circuit <i>C</i> is displaced, the work of the electrodynamic
+action exercised by <i>C</i> on <i>C&acute;</i> will be equal to the increment
+of the 'flow of magnetic force' which passes through the circuit.</p>
+
+<p>2. <i>Action of a Closed Current on a Portion of Current.</i>&mdash;Amp&egrave;re
+not having been able to produce an open current, properly
+so called, had only one way of studying the action of a
+closed current on a portion of current.</p>
+
+<p>This was by operating on a circuit <i>C</i> composed of two parts,
+the one fixed, the other movable. The movable part was, for
+instance, a movable wire &#945;&#946; whose extremities &#945; and &#946; could<span class='pagenum'><a name="Page_187" id="Page_187">[Pg 187]</a></span>
+slide along a fixed wire. In one of the positions of the movable
+wire, the end &#945; rested on the <i>A</i> of the fixed wire and the extremity
+&#946; on the point <i>B</i> of the fixed wire. The current circulated
+from &#945; to &#946;, that is to say, from <i>A</i> to <i>B</i> along the movable wire,
+and then it returned from <i>B</i> to <i>A</i> along the fixed wire. <i>This
+current was therefore closed.</i></p>
+
+<p>In a second position, the movable wire having slipped, the extremity
+&#945; rested on another point <i>A&acute;</i> of the fixed wire, and the
+extremity &#946; on another point <i>B&acute;</i> of the fixed wire. The current
+circulated then from &#945; to &#946;, that is to say from <i>A&acute;</i> to <i>B&acute;</i> along the
+movable wire, and it afterwards returned from <i>B&acute;</i> to <i>B</i>, then
+from <i>B</i> to <i>A</i>, then finally from <i>A</i> to <i>A&acute;</i>, always following the
+fixed wire. The current was therefore also closed.</p>
+
+<p>If a like current is subjected to the action of a closed current
+<i>C</i>, the movable part will be displaced just as if it were acted
+upon by a force. Amp&egrave;re <i>assumes</i> that the apparent force to
+which this movable part <i>AB</i> seems thus subjected, representing
+the action of the <i>C</i> on the portion &#945;&#946; of the current, is the same
+as if &#945;&#946; were traversed by an open current, stopping at &#945; and &#946;,
+in place of being traversed by a closed current which after arriving
+at &#946; returns to &#945; through the fixed part of the circuit.</p>
+
+<p>This hypothesis seems natural enough, and Amp&egrave;re made it
+unconsciously; nevertheless <i>it is not necessary</i>, since we shall see
+further on that Helmholtz rejected it. However that may be, it
+permitted Amp&egrave;re, though he had never been able to produce an
+open current, to enunciate the laws of the action of a closed current
+on an open current, or even on an element of current.</p>
+
+<p>The laws are simple:</p>
+
+<p>1&ordm; 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 magnetic force
+which is normal to the element.</p>
+
+<p>2&ordm; The action of a closed solenoid on an element of current is
+null.</p>
+
+<p>But the electrodynamic potential has disappeared, that is to
+say that, when a closed current and an open current, whose intensities
+have been maintained constant, return to their initial
+positions, the total work is not null.<span class='pagenum'><a name="Page_188" id="Page_188">[Pg 188]</a></span></p>
+
+<p>3. <i>Continuous Rotations.</i>&mdash;Among electrodynamic experiments,
+the most remarkable are those in which continuous rotations
+are produced and which are sometimes called <i>unipolar induction</i>
+experiments. A magnet may turn about its axis; a
+current passes first through a fixed wire, enters the magnet by
+the pole <i>N</i>, for example, passes through half the magnet, emerges
+by a sliding contact and reenters the fixed wire.</p>
+
+<p>The magnet then begins to rotate continuously without being
+able ever to attain equilibrium; this is Faraday's experiment.</p>
+
+<p>How is it possible? If it were a question of two circuits of
+invariable form, the one <i>C</i> fixed, the other <i>C&acute;</i> 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.</p>
+
+<p>Continuous rotations are therefore possible only when the circuit
+<i>C&acute;</i> 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 point of the movable part coming
+successively in contact with diverse points of the fixed part).</p>
+
+<p>It is only in the second case that there can be continuous rotation.
+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 communication
+with a new point of the fixed part; it changes the connections,
+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 indefinitely.</p>
+
+<p>Amp&egrave;re assumes that the action of the circuit on the movable
+part of <i>C&acute;</i> is the same as if the fixed part of <i>C&acute;</i> did not exist, and
+therefore as if the current passing through the movable part
+were open.<span class='pagenum'><a name="Page_189" id="Page_189">[Pg 189]</a></span></p>
+
+<p>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.</p>
+
+<p>But this conclusion depends on the hypothesis I have enunciated
+and which, as I said above, is not admitted by Helmholtz.</p>
+
+<p>4. <i>Mutual Action of Two Open Currents.</i>&mdash;In what concerns
+the mutual actions of two open currents, and in particular that
+of two elements of current, all experiment breaks down. Amp&egrave;re
+has recourse to hypothesis. He supposes:</p>
+
+<p>1&ordm; That the mutual action of two elements reduces to a force
+acting along their join;</p>
+
+<p>2&ordm; 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.</p>
+
+<p>What is remarkable is that here again Amp&egrave;re makes these
+hypotheses unconsciously.</p>
+
+<p>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.</p>
+
+<p>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.</p>
+
+<p>Next there is, properly speaking, no magnetic force.</p>
+
+<p>And, in fact, we have given above three different definitions
+of this force:</p>
+
+<p>1&ordm; By the action on a magnetic pole;</p>
+
+<p>2&ordm; By the director couple which orientates the magnetic
+needle;</p>
+
+<p>3&ordm; By the action on an element of current.</p>
+
+<p>But in the case which now occupies us, not only these three
+definitions are no longer in harmony, but each has lost its meaning,
+and in fact:</p>
+
+<p>1&ordm; 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;<span class='pagenum'><a name="Page_190" id="Page_190">[Pg 190]</a></span></p>
+
+<p>2&ordm; 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;</p>
+
+<p>3&ordm; Finally the force acting on an element of current is not
+normal to this element.</p>
+
+<p>In other words, <i>the unity of the magnetic force has disappeared</i>.</p>
+
+<p>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.</p>
+
+<p>Well, this is true if these two systems contain only closed
+currents; this would no longer be true if these two systems contained
+open currents.</p>
+
+<p>It suffices to remark, for instance, that, if a magnetic pole is
+placed at <i>A</i> and an element at <i>B</i>, the direction of the element
+being along the prolongation of the sect <i>AB</i>, this element which
+will exercise no action on this pole will, on the other hand, exercise
+an action either on a magnetic needle placed at the point <i>A</i>,
+or on an element of current placed at the point <i>A</i>.</p>
+
+<p>5. <i>Induction.</i>&mdash;We know that the discovery of electrodynamic
+induction soon followed the immortal work of Amp&egrave;re.</p>
+
+<p>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 Amp&egrave;re. But
+always on one condition, as Bertrand has well shown; that we
+make besides a certain number of hypotheses.</p>
+
+<p>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.</p>
+
+<p>If we try to apply this mode of analysis to Amp&egrave;re's theory
+of open currents, we reach results calculated to surprise us.</p>
+
+<p>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.<span class='pagenum'><a name="Page_191" id="Page_191">[Pg 191]</a></span></p>
+
+<p>But, further, if a circuit <i>C</i> is subjected to the induction of a
+variable voltaic system <i>S</i>, if this system <i>S</i> be displaced and deformed
+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 situation,
+it seems natural to suppose that the <i>mean</i> electromotive
+force induced in the circuit <i>C</i> is null.</p>
+
+<p>This is true if the circuit <i>C</i> is closed and if the system <i>S</i> contains
+only closed currents. This would no longer be true, if one
+accepts the theory of Amp&egrave;re, 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.</p>
+
+<p><span class="smcap">II. Theory of Helmholtz.</span>&mdash;I have dwelt upon the consequences
+of Amp&egrave;re's theory, and of his method of explaining
+open currents.</p>
+
+<p>It is difficult to overlook the paradoxical and artificial character
+of the propositions to which we are thus led. One can not
+help thinking 'that can not be so.'</p>
+
+<p>We understand therefore why Helmholtz was led to seek something
+else.</p>
+
+<p>Helmholtz rejects Amp&egrave;re'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 Bertrand
+and Helmholtz.</p>
+
+<p>Helmholtz replaces Amp&egrave;re's hypothesis by the following: two
+elements always admit of an electrodynamic potential depending
+solely on their position and orientation; and the work of the
+forces that they exercise, one on the other, is equal to the variation
+of this potential. Thus Helmholtz can no more do without
+hypothesis than Amp&egrave;re; but at least he does not make one without
+explicitly announcing it.</p>
+
+<p>In the case of closed currents, which are alone accessible to
+experiment, the two theories agree.</p>
+
+<p>In all other cases they differ.</p>
+
+<p>In the first place, contrary to what Amp&egrave;re supposed, the force<span class='pagenum'><a name="Page_192" id="Page_192">[Pg 192]</a></span>
+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.</p>
+
+<p>Let us return to the circuit <i>C&acute;</i>, of which we spoke above, and
+which was formed of a movable wire &#945;&#946; sliding on a fixed wire.
+In the only experiment that can be made, the movable portion &#945;&#946;
+is not isolated, but is part of a closed circuit. When it passes
+from <i>AB</i> to <i>A&acute;B&acute;</i>, the total electrodynamic potential varies for
+two reasons:</p>
+
+<p>1&ordm; It undergoes a first increase because the potential of <i>A&acute;B&acute;</i>
+with respect to the circuit <i>C</i> is not the same as that of <i>AB</i>;</p>
+
+<p>2&ordm; It takes a second increment because it must be increased
+by the potentials of the elements <i>AA&acute;</i>, <i>BB&acute;</i> with respect to <i>C</i>.</p>
+
+<p>It is this <i>double</i> increment which represents the work of the
+force to which the portion <i>AB</i> seems subjected.</p>
+
+<p>If, on the contrary, &#945;&#946; 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 <i>AB</i>.</p>
+
+<p>In the second place, there could be no continuous rotation
+without sliding contact, and, in fact, that, as we have seen <i>&agrave;
+propos</i> of closed currents, is an immediate consequence of the
+existence of an electrodynamic potential.</p>
+
+<p>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 <i>open</i> current were to
+run along the wire, the wire would still take a movement of continuous
+rotation.</p>
+
+<p>I have just said in fact that an <i>isolated</i> element is not acted
+upon in the same way as a movable element making part of a
+closed circuit.</p>
+
+<p>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 Amp&egrave;re; it would not be null according to Helmholtz.
+From this follows an important consequence. We have
+given above three definitions of magnetic force. The third has<span class='pagenum'><a name="Page_193" id="Page_193">[Pg 193]</a></span>
+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.</p>
+
+<p>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.</p>
+
+<p>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.</p>
+
+<p><span class="smcap">III. Difficulties Raised by These Theories.</span>&mdash;The theory
+of Helmholtz is in advance of that of Amp&egrave;re; it is necessary,
+however, that all the difficulties should be smoothed away. In
+the one as in the other, the phrase 'magnetic field' has no meaning,
+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.</p>
+
+<p>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 distance,
+electrodynamic phenomena must be explained by a modification
+of the medium. It is precisely this modification that we
+call 'magnetic field.' And then the electrodynamic effects must
+depend only on this field.</p>
+
+<p>All these difficulties arise from the hypothesis of open currents.</p>
+
+<p><span class="smcap">IV. Maxwell's Theory.</span>&mdash;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<span class='pagenum'><a name="Page_194" id="Page_194">[Pg 194]</a></span>
+galvanometer like a current, and which he calls <i>current of displacement</i>.</p>
+
+<p>If then two conductors bearing contrary charges are put in
+communication 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.</p>
+
+<p>We know that Maxwell's theory leads to the explanation of
+optical phenomena, which would be due to extremely rapid electrical
+oscillations.</p>
+
+<p>At that epoch such a conception was only a bold hypothesis,
+which could be supported by no experiment.</p>
+
+<p>At the end of twenty years, Maxwell's ideas received the confirmation
+of experiment. Hertz succeeded in producing systems
+of electric oscillations which reproduce all 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.</p>
+
+<p>It might be said that Hertz has not demonstrated directly
+Maxwell's fundamental idea, the action of the current of displacement
+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.</p>
+
+<p>But to suppose there is no current of displacement, and induction
+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, <i>comes to the same
+thing</i>.</p>
+
+<p>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.</p>
+
+<p><span class="smcap">V. Rowland's Experiment.</span>&mdash;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 whatever.</p>
+
+<p>There are also the cases in which electric discharges describe<span class='pagenum'><a name="Page_195" id="Page_195">[Pg 195]</a></span>
+a closed contour, being displaced by conduction in one part of
+the circuit and by convection in the other part.</p>
+
+<p>For open currents of the first sort, the question might be considered
+as solved; they were closed by the currents of displacement.</p>
+
+<p>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.</p>
+
+<p>But experimental confirmation was lacking. It appeared difficult
+in fact to obtain a sufficient intensity even by augmenting as
+much as possible the charge and the velocity of the conductors.
+It was Rowland, an extremely skillful experimenter, who first triumphed
+over these difficulties. A disc received a strong electrostatic
+charge and a very great speed of rotation. An astatic
+magnetic system placed beside the disc underwent deviations.</p>
+
+<p>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
+making quantitative measurements.</p>
+
+<p>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 electrode 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.</p>
+
+<p>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.</p>
+
+<p>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<span class='pagenum'><a name="Page_196" id="Page_196">[Pg 196]</a></span>
+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, electricity
+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.</p>
+
+<p>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.</p>
+
+<p><span class="smcap">VI. Theory of Lorentz.</span>&mdash;We soon went farther. According
+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 <i>electrons</i>.
+The circulation of these electrons through bodies would
+produce voltaic currents. And what would distinguish conductors
+from insulators would be that the one could be traversed
+by these electrons while the others would arrest their movements.</p>
+
+<p>The theory of Lorentz is very attractive. It gives a very
+simple explanation of certain phenomena which the earlier theories,
+even Maxwell's in its primitive form, could not explain in a
+satisfactory way; for example, the aberration of light, the partial
+carrying away of luminous waves, magnetic polarization and
+the Zeeman effect.</p>
+
+<p>Some objections still remained. The phenomena of an electric
+system seemed to depend on the absolute velocity of translation
+of the center of gravity of this system, which is contrary to
+the idea we have of the relativity of space. Supported by M.
+Cr&eacute;mieu, M. Lippmann 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.<span class='pagenum'><a name="Page_197" id="Page_197">[Pg 197]</a></span></p>
+
+<p>"No!" replied the partisans of Lorentz. "What we could
+measure in that way is not their absolute velocity, but their relative
+velocity <i>with respect to the ether</i>, so that the principle of
+relativity is safe."</p>
+
+<p>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 satisfactory
+aspect. The theories of Amp&egrave;re 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.</p>
+
+<p>This quiescence has been recently disturbed by the experiments
+of M. Cr&eacute;mieu, which for a moment seemed to contradict
+the result previously obtained by Rowland.</p>
+
+<p>But fresh researches have not confirmed them, and the theory
+of Lorentz has victoriously stood the test.</p>
+
+<p>The history of these variations will be none the less instructive;
+it will teach us to what pitfalls the scientist is exposed, and how
+he may hope to escape them.</p>
+<p><span class='pagenum'><a name="Page_198" id="Page_198">[Pg 198]</a></span></p>
+
+
+
+
+<hr style="width: 100%;" />
+<p><span class='pagenum'><a name="Page_199" id="Page_199">[Pg 199]</a></span></p>
+<p>&nbsp;</p>
+<h1><a name="THE_VALUE_OF_SCIENCE" id="THE_VALUE_OF_SCIENCE"></a><b>THE VALUE OF SCIENCE</b></h1>
+<p>&nbsp;</p>
+<p><span class='pagenum'><a name="Page_200" id="Page_200">[Pg 200]</a></span></p>
+
+
+<hr style="width: 100%;" />
+<p><span class='pagenum'><a name="Page_201" id="Page_201">[Pg 201]</a></span></p>
+<h3><b>TRANSLATOR'S INTRODUCTION</b></h3>
+
+
+<p>1. <i>Does the Scientist create Science?</i>&mdash;Professor Rados of
+Budapest in his report to the Hungarian Academy of Science on
+the award to Poincar&eacute; of the Bolyai prize of ten thousand
+crowns, speaking of him as unquestionably the most powerful investigator
+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
+inspiration 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?</p>
+
+<p>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 geometry
+are conventions, disguised definitions or unprovable hypotheses
+precreated by auto-active 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 discovered
+from data." Then does the scientist create science?
+This is a question Poincar&eacute; here dissects with a master hand.</p>
+
+<p>The physiologic-psychologic investigation of the space problem
+<span class='pagenum'><a name="Page_202" id="Page_202">[Pg 202]</a></span>must give the meaning of the words <i>geometric fact</i>, <i>geometric
+reality</i>. Poincar&eacute; here subjects to the most successful analysis
+ever made the tridimensionality of our space.</p>
+
+<p>2. <i>The Mind Dispelling Optical Illusions.</i>&mdash;Actual perception
+of spatial properties is accompanied by movements corresponding
+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 illusions,
+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 perception
+itself successively occur. What is called accuracy of movement
+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 perception
+in the individual helping to make that cerebral organization
+and accurate motor adjustment on which its possibility and
+permanence seem in so far to depend. So-called correct perception
+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.</p>
+
+<p>3. <i>Euclid not Necessary.</i>&mdash;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 confess
+that the Kantian account of space is hopelessly and demonstrably
+antiquated. As Royce says in 'Kant's Doctrine of the
+Basis of Mathematics,' "That very use of intuition which Kant
+regarded as geometrically ideal, the modern geometer regards
+as scientifically defective, because surreptitious. No mathematical
+exactness without explicit proof from assumed principles&mdash;such
+is the motto of the modern geometer. But suppose the
+reasoning of Euclid purified of this comparatively surreptitious
+<span class='pagenum'><a name="Page_203" id="Page_203">[Pg 203]</a></span>appeal to intuition. Suppose that the principles of geometry are
+made quite explicit at the outset of the treatise, as Pieri and
+Hilbert or Professor Halsted or Dr. Veblen makes his principles
+explicit in his recent treatment of geometry. Then, indeed, geometry
+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.</p>
+
+<p>4. <i>Without Hypotheses, no Science.</i>&mdash;Nobody ever observed an
+equidistantial, but also nobody ever observed a straight line.
+Emerson's Uriel</p>
+
+<div class="blockquot">
+<p class="noidt">
+"Gave his sentiment divine<br />
+Against the being of a line.<br />
+Line in Nature is not found."<br />
+</p>
+</div>
+
+<p class="noidt">Clearly not, being an eject from man's mind. What is called 'a
+knowledge of facts' is usually merely a subjective realization that
+the old hypotheses are still sufficiently elastic to serve in some
+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, beating
+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.</p>
+
+<p>5. <i>What Outcome?</i>&mdash;What now is the definite, the permanent
+outcome? What new islets raise their fronded 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 Erd&eacute;ly'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<span class='pagenum'><a name="Page_204" id="Page_204">[Pg 204]</a></span>
+Damascus blade the spikes driven into his door-post, and strewing
+over the sod the thirteen Austrian cavalry officers? This
+book by the greatest mathematician of our time gives weightiest
+and most charming answer.</p>
+
+<p class="ralign"><span class="smcap">George Bruce Halsted.</span></p>
+
+
+
+<hr style="width: 65%;" />
+<p><span class='pagenum'><a name="Page_205" id="Page_205">[Pg 205]</a></span></p>
+<h3><b>INTRODUCTION</b></h3>
+
+
+<p>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.</p>
+
+<p>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.</p>
+
+<p>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 well as to find the other, it is necessary
+to free the soul completely from prejudice and from passion;
+it is necessary to attain absolute sincerity. These two sorts of<span class='pagenum'><a name="Page_206" id="Page_206">[Pg 206]</a></span>
+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.</p>
+
+<p>If we ought not to fear moral truth, still less should we dread
+scientific truth. In the first place it can not conflict 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.</p>
+
+<p>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 in
+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.</p>
+
+<p>But if truth be the sole aim worth pursuing, may we hope to
+attain it? It may well be doubted. Readers of my little book
+'Science and Hypothesis' already know what I think about the
+question. The truth we are permitted to glimpse is not altogether
+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.</p>
+
+<p>In the first place, what instrument have we at our disposal for
+this conquest? Is not human intelligence, more specifically the<span class='pagenum'><a name="Page_207" id="Page_207">[Pg 207]</a></span>
+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 geometer's
+mind is not like the physicist's or the naturalist's, all the
+world would agree; but mathematicians themselves do not resemble
+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.</p>
+
+<p>Next must be examined the frames in which nature seems enclosed
+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 quantitative
+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 'qualitative
+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.</p>
+
+<p>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 without;
+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 intimate
+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.</p>
+
+<p>The best expression of this harmony is law. Law is one of the<span class='pagenum'><a name="Page_208" id="Page_208">[Pg 208]</a></span>
+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 astonished
+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?</p>
+
+<p>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 necessary
+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.</p>
+
+<p>The progress of science has seemed to imperil the best established
+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.</p>
+
+<p>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 needful
+now to set things to rights.</p>
+
+<p>Some people have exaggerated the r&ocirc;le 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<span class='pagenum'><a name="Page_209" id="Page_209">[Pg 209]</a></span>
+artificial creations; we have no reason to regard them as accidental,
+though it be impossible to prove they are not.</p>
+
+<p>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 common
+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.</p>
+
+
+
+<hr style="width: 65%;" />
+<p><span class='pagenum'><a name="Page_210" id="Page_210">[Pg 210]</a></span></p>
+<h2><b>PART I<br />
+
+<br />
+
+<small>THE MATHEMATICAL SCIENCES</small></b></h2>
+
+<hr style="width: 65%;" />
+
+<h3>CHAPTER I</h3>
+
+<h3><span class="smcap">Intuition and Logic in Mathematics</span></h3>
+
+<h4>I</h4>
+
+
+<p>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.</p>
+
+<p>The method is not imposed by the matter treated. Though one
+often says of the first that they are <i>analysts</i> and calls the others
+<i>geometers</i>, 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 intuitionalists, and they can not lay it aside when
+they approach a new subject.</p>
+
+<p>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 born a geometer or an analyst. I should
+like to cite examples and there are surely plenty; but to accentuate
+the contrast I shall begin with an extreme example, taking the
+liberty of seeking it in two living mathematicians.<span class='pagenum'><a name="Page_211" id="Page_211">[Pg 211]</a></span></p>
+
+<p>M. M&eacute;ray wants to prove that a binomial equation always has
+a root, or, in ordinary words, that an angle may always be subdivided.
+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? M. M&eacute;ray 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.</p>
+
+<p>On the other hand, look at Professor Klein: he is studying one
+of the most abstract questions of the theory of functions: to determine
+whether on a given Riemann surface there always exists a
+function admitting of given singularities. What does the celebrated
+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 distribution
+of this current on the surface will define a function whose
+singularities will be precisely those called for by the enunciation.</p>
+
+<p>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 demonstration,
+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.</p>
+
+<p>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 influences;
+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 all their pupils these two
+faces are stamped in deathless lines; for all who have had the
+pleasure of following their teaching, this remembrance is still
+fresh; it is easy for us to evoke it.</p>
+
+<p>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<span class='pagenum'><a name="Page_212" id="Page_212">[Pg 212]</a></span>
+is eager to paint, this is why he calls gesture to his aid. With M.
+Hermite, 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.</p>
+
+<p>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. Weierstrass
+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 conceptions
+is an image that no one can forget, once he has caught its
+meaning.</p>
+
+<p>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.</p>
+
+<p>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.</p>
+
+<p>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 venture
+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 r&ocirc;les. But it is interesting
+to study more closely in the history of science the part
+which belongs to each.</p>
+
+
+<h4>II</h4>
+
+<p>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<span class='pagenum'><a name="Page_213" id="Page_213">[Pg 213]</a></span>
+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.</p>
+
+<p>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.</p>
+
+<p>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."</p>
+
+<p>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 conclude
+that a curve always has a tangent.</p>
+
+<p>I shall take as second example Dirichlet's principle on which
+rest so many theorems of mathematical physics; to-day we establish
+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 <i>function</i> and are familiar with all the singularities
+functions can present when the word is understood in the most
+general sense.<span class='pagenum'><a name="Page_214" id="Page_214">[Pg 214]</a></span></p>
+
+<p>But it would not be the same had we used concrete images,
+had we, for example, considered this function as an electric potential;
+it would have been thought legitimate to affirm that electrostatic
+equilibrium can be attained. Yet perhaps a physical comparison
+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.</p>
+
+<p>Intuition, therefore, does not give us certainty. This is why
+the evolution had to happen; let us now see how it happened.</p>
+
+<p>It was not slow in being noticed that rigor could not be introduced
+in the reasoning unless first made to enter into the definitions.
+For the most part the objects treated of by mathematicians
+were long ill 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.</p>
+
+<p>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.</p>
+
+<p>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.</p>
+
+
+<h4>III</h4>
+
+<p>A first question presents itself. Is this evolution ended? Have
+we finally attained absolute rigor? At each stage of the evolution
+our fathers also thought they had reached it. If they deceived
+themselves, do we not likewise cheat ourselves?</p>
+
+<p>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<span class='pagenum'><a name="Page_215" id="Page_215">[Pg 215]</a></span>
+create nothing new; not from it alone can any science issue. In
+one sense these philosophers 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 <i>intuition</i>. But how many different ideas are
+hidden under this same word?</p>
+
+<p>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 <i>n</i> + 1 if true for <i>n</i>,
+then will it be true of all whole numbers; (3) if on a straight
+the point <i>C</i> is between <i>A</i> and <i>B</i> and the point <i>D</i> between <i>A</i> and
+<i>C</i>, then the point <i>D</i> will be between <i>A</i> and <i>B</i>; (4) through a given
+point there is not more than one parallel to a given straight.</p>
+
+<p>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 <i>a priori</i> judgment, it is the foundation of rigorous
+mathematical induction; the third is an appeal to the imagination;
+the fourth is a disguised definition.</p>
+
+<p>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.</p>
+
+<p>You know what Poncelet understood by the <i>principle of continuity</i>.
+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.</p>
+
+<p>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<span class='pagenum'><a name="Page_216" id="Page_216">[Pg 216]</a></span>
+sciences; 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?</p>
+
+<p>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.</p>
+
+
+<h4>IV</h4>
+
+<p>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."</p>
+
+<p>For example, I seek to show that some property pertains to
+some object whose concept seems to me at first indefinable, because
+it is intuitive. At first I fail or must content myself with
+approximate proofs; finally I decide to give to my object a precise
+definition, and this enables me to establish this property in
+an irreproachable manner.</p>
+
+<p>"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 definition.
+Only then could you affirm that it has the property in
+question. You have only displaced the difficulty."</p>
+
+<p>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 established.
+The second was an experimental verity. Experience
+alone can teach us that some real and concrete object corresponds<span class='pagenum'><a name="Page_217" id="Page_217">[Pg 217]</a></span>
+or 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.</p>
+
+<p>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.</p>
+
+<p>This shows us that logic is not enough; that the science of
+demonstration is not all science and that intuition must retain its
+r&ocirc;le as complement, I was about to say as counterpoise or as
+antidote of logic.</p>
+
+<p>I have already had occasion to insist on the place intuition
+should hold in the teaching of the mathematical sciences. Without
+it young minds could not make a beginning in the understanding
+of mathematics; they could not learn to love it and
+would see in it only a vain logomachy; above all, without intuition
+they would never become capable of applying mathematics.
+But now I wish before all to speak of the r&ocirc;le of intuition in
+science itself. If it is useful to the student it is still more so to
+the creative scientist.</p>
+
+
+<h4>V</h4>
+
+<p>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 interesting
+than that of the isolated elements, and should a naturalist
+who had never studied the elephant except by means of the microscope
+think himself sufficiently acquainted with that animal?</p>
+
+<p>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<span class='pagenum'><a name="Page_218" id="Page_218">[Pg 218]</a></span>
+operations one after the other and ascertained that each is correct,
+are we to think we have grasped the real meaning of the
+demonstration? 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.</p>
+
+<p>Pure analysis puts at our disposal a multitude of procedures
+whose infallibility it guarantees; it opens to us a thousand different
+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 tell
+us which to choose? We need a faculty which makes us see 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.</p>
+
+<p>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 successive
+moves a sort of organized whole. This faculty is still more
+necessary for the player himself, that is, for the inventor.</p>
+
+<p>Let us drop this comparison and return to mathematics. For
+example, see what has happened to the idea of continuous function.
+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 reproduced,
+so to speak, all the lines of the original image; this construction
+finished, the centering of the arch, so to say, was
+removed, that crude representation which had temporarily served<span class='pagenum'><a name="Page_219" id="Page_219">[Pg 219]</a></span>
+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 disappeared
+from our recollection, how could we divine by what
+caprice all these inequalities were erected in this fashion one
+upon another?</p>
+
+<p>Perhaps you think I use too many comparisons; yet pardon still
+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.</p>
+
+<p>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
+<i>analysis</i> and analysis means <i>division</i>, <i>dissection</i>. It can have,
+therefore, no tool other than the scalpel and the microscope.</p>
+
+<p>Thus logic and intuition have each their necessary r&ocirc;le. Each
+is indispensable. Logic, which alone can give certainty, is the
+instrument of demonstration; intuition is the instrument of
+invention.</p>
+
+
+<h4>VI</h4>
+
+<p>But at the moment of formulating this conclusion I am seized
+with scruples. At the outset I distinguished two kinds of mathematical
+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.</p>
+
+<p>Here is a contradiction, at least apparently, which needs explanation.
+And first, do you think these logicians have always proceeded
+from the general to the particular, as the rules of formal<span class='pagenum'><a name="Page_220" id="Page_220">[Pg 220]</a></span>
+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.</p>
+
+<p>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 <i>mathematical induction</i>. 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.</p>
+
+<p>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 employment
+of dominant functions. We know it has already served to
+solve a multitude of problems; in what consists then the r&ocirc;le 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.</p>
+
+<p>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 analogies
+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.</p>
+
+<p>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<span class='pagenum'><a name="Page_221" id="Page_221">[Pg 221]</a></span>
+perceived that they are not an artificial assemblage and that they
+have some principle of internal unity.</p>
+
+<p>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 intuitionalists,
+at least those who are capable of inventing?</p>
+
+<p>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
+mathematical induction, differs from sensible intuition to which
+the imagination, properly so called, is the principal contributor.</p>
+
+<p>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 essential
+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.</p>
+
+<p>It is the intuition of pure number, that of pure logical forms,
+which illumines and directs those we have called <i>analysts</i>. 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 intervene.
+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!</p>
+
+<p>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 intuition
+alone, would soon feel ourselves seized with vertigo. Our
+weakness has need of a staff more solid, and, despite the exceptions
+of which we have just spoken, it is none the less true that
+sensible intuition is in mathematics the most usual instrument of
+invention.</p>
+
+<p>Apropos of these reflections, a question comes up that I have<span class='pagenum'><a name="Page_222" id="Page_222">[Pg 222]</a></span>
+not the time either to solve or even to enunciate with the developments
+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?</p>
+
+<p>M. Hermite, for example, whom I have just cited, can not be
+classed among the geometers who make use of the sensible intuition;
+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.</p>
+
+
+
+<hr style="width: 65%;" />
+<p><span class='pagenum'><a name="Page_223" id="Page_223">[Pg 223]</a></span></p>
+<h3>CHAPTER II</h3>
+
+<h3><span class="smcap">The Measure of Time</span></h3>
+
+<h4>I</h4>
+
+
+<p>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 anterior
+to the other.</p>
+
+<p>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.</p>
+
+<p>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.</p>
+
+<p>I have only a single observation to add. For an aggregate of
+sensations to have become a remembrance capable of classification
+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.</p>
+
+<p>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?</p>
+<p><span class='pagenum'><a name="Page_224" id="Page_224">[Pg 224]</a></span></p>
+
+<h4>II</h4>
+
+<p>But that is not all; into this form we wish to put not only the
+phenomena of our own consciousness, but 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 because
+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.</p>
+
+<p>Think of two consciousnesses, which are like two worlds impenetrable
+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 measuring?
+We know perhaps that some fact is anterior to some other,
+but not <i>by how much</i> it is anterior.</p>
+
+<p>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?</p>
+
+
+<h4>III</h4>
+
+<p>The first difficulty has long been noticed; it has been the subject
+of long discussions and one may say the question is settled.
+<i>We have not a direct intuition of the equality of two intervals
+of time.</i> 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 affirmation?</p>
+
+<p>The least reflection shows that by itself it has none at all. It
+will only have that which I choose to give it, by a definition which
+will certainly possess a certain degree of arbitrariness. Psychologists
+could have done without this definition; physicists and
+astronomers could not; let us see how they have managed.</p>
+
+<p>To measure time they use the pendulum and they suppose by
+definition that all the beats of this pendulum are of equal duration.
+But this is only a first approximation; the temperature,
+the resistance of the air, the barometric pressure, make the pace<span class='pagenum'><a name="Page_225" id="Page_225">[Pg 225]</a></span>
+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, electric,
+magnetic or others, would introduce minute perturbations.</p>
+
+<p>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.</p>
+
+<p>However, the astronomers are still not content with this definition.
+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.</p>
+
+
+<h4>IV</h4>
+
+<p>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.</p>
+
+<p>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? <i>It is that the duration of two identical phenomena
+is the same</i>; or, if you prefer, that the same causes take
+the same time to produce the same effects.</p>
+
+<p>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?</p>
+
+<p>Let me explain myself. I suppose that at a certain place in the
+world the phenomenon &#945; happens, causing as consequence at the
+end of a certain time the effect &#945;&acute;. At another place in the world<span class='pagenum'><a name="Page_226" id="Page_226">[Pg 226]</a></span>
+very far away from the first, happens the phenomenon &#946;, which
+causes as consequence the effect &#946;&acute;. The phenomena &#945; and &#946; are
+simultaneous, as are also the effects &#945;&acute; and &#946;&acute;.</p>
+
+<p>Later, the phenomenon &#945; is reproduced under approximately
+the same conditions as before, and <i>simultaneously</i> the phenomenon
+&#946; is also reproduced at a very distant place in the world
+and almost under the same circumstances. The effects &#945;&acute; and &#946;&acute;
+also take place. Let us suppose that the effect &#945;&acute; happens perceptibly
+before the effect &#946;&acute;.</p>
+
+<p>If experience made us witness such a sight, our postulate
+would be contradicted. For experience would tell us that the
+first duration &#945;&#945;&acute; is equal to the first duration &#946;&#946;&acute; and that the
+second duration &#945;&#945;&acute; is less than the second duration &#946;&#946;&acute;. On the
+other hand, our postulate would require that the two durations
+&#945;&#945;&acute; should be equal to each other, as likewise the two durations
+&#946;&#946;&acute;. The equality and the inequality deduced from experience
+would be incompatible with the two equalities deduced from the
+postulate.</p>
+
+<p>Now can we affirm that the hypotheses I have just made are
+absurd? They are in no wise contrary to the principle of contradiction.
+Doubtless they could not happen without the principle
+of sufficient reason seeming violated. But to justify a
+definition so fundamental I should prefer some other guarantee.</p>
+
+
+<h4>V</h4>
+
+<p>But that is not all. In physical reality one cause does not produce
+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.</p>
+
+<p>Physicists seek to make this distinction; but they make it only
+approximately, and, however they progress, they never will
+make it except approximately. 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 pendulum.</p>
+
+<p>Under these conditions, it is clear that the causes which have
+produced a certain effect will never be reproduced except approximately.
+Then we should modify our postulate and our<span class='pagenum'><a name="Page_227" id="Page_227">[Pg 227]</a></span>
+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.'</p>
+
+<p>Our definition therefore is no longer anything but approximate.
+Besides, as M. Calinon very justly remarks in a recent
+memoir:<a name="FNanchor_7_7" id="FNanchor_7_7"></a><a href="#Footnote_7_7" class="fnanchor">[7]</a></p>
+
+<p><small>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 measure time.</small></p>
+
+<p>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.</p>
+
+<p>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 producing
+heat must destroy <i>vis viva</i>. They invoke therefore the
+principle of <i>vis viva</i>, or of the conservation of energy.</p>
+
+<p>They say next that the secular acceleration of the moon, calculated
+according to Newton's law, would be less than that deduced
+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 <i>vis viva</i> 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.</p>
+
+<p>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 enunciation
+of the law would be different, because it would be translated
+into another language; it would evidently be much less
+simple. So that the definition implicitly adopted by the astronomers
+may be summed up thus: Time should be so defined that<span class='pagenum'><a name="Page_228" id="Page_228">[Pg 228]</a></span>
+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
+<i>convenient</i>. Of two watches, we have no right to say that the
+one goes true, the other wrong; we can only say that it is advantageous
+to conform to the indications of the first.</p>
+
+<p>The difficulty 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.</p>
+
+
+<h4>VI</h4>
+
+<p>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.</p>
+
+<p>Two psychological phenomena happen in two different consciousnesses;
+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 phenomenon,
+what do I mean?</p>
+
+<p>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 discovery
+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 phenomenon
+is anterior to the formation of the visual image of the
+isle of Espa&ntilde;ola in the consciousness of Christopher Columbus,
+what do I mean?</p>
+
+<p>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.</p>
+
+
+<h4>VII</h4>
+
+<p>We should first ask ourselves how one could have had the idea
+of putting into the same frame so many worlds impenetrable to<span class='pagenum'><a name="Page_229" id="Page_229">[Pg 229]</a></span>
+one another. We should like to represent to ourselves the external
+universe, and only by so doing could we feel that we understood
+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 <i>in its time</i>, as we classify, <i>in
+our time</i>, the little we see.</p>
+
+<p>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 all recollections would be equally present to it
+and for it there would be no time. And yet when we speak of
+time, for all which happens outside of us, do we not unconsciously
+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?</p>
+
+<p>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 hypothetical
+intelligence, even if it existed, would be for us impenetrable.
+It is therefore necessary to seek something else.</p>
+
+
+<h4>VIII</h4>
+
+<p>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 without
+mutilating them. Is it the same with two physical facts? Is
+not my present nearer my past of yesterday than the present of
+Sirius?</p>
+
+<p>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.</p>
+<p><span class='pagenum'><a name="Page_230" id="Page_230">[Pg 230]</a></span></p>
+
+<h4>IX</h4>
+
+<p>Let us then seek to give an account of what is understood by
+simultaneity or antecedence, and for this let us analyze some
+examples.</p>
+
+<p>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.</p>
+
+<p>I hear thunder, and I conclude there has been an electric discharge;
+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.</p>
+
+<p>Behold then the rule we follow, 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 antecedent,
+is the cause of the other, of the consequent. It is then by
+time that we define cause. How save ourselves from this <i>petitio
+principii</i>?</p>
+
+<p>We say now <i>post hoc, ergo propter hoc</i>; now <i>propter hoc, ergo
+post hoc</i>; shall we escape from this vicious circle?</p>
+
+
+<h4>X</h4>
+
+<p>Let us see, not how we succeed in escaping, for we do not
+completely succeed, but how we try to escape.</p>
+
+<p>I execute a voluntary act <i>A</i> and I feel afterward a sensation <i>D</i>,
+which I regard as a consequence of the act <i>A</i>; on the other hand,
+for whatever reason, I infer that this consequence is not immediate,
+but that outside my consciousness two facts <i>B</i> and <i>C</i>, which
+I have not witnessed, have happened, and in such a way that
+<i>B</i> is the effect of <i>A</i>, that <i>C</i> is the effect of <i>B</i>, and <i>D</i> of <i>C</i>.</p>
+
+<p>But why? If I think I have reason to regard the four facts
+<i>A</i>, <i>B</i>, <i>C</i>, <i>D</i>, as bound to one another by a causal connection, why<span class='pagenum'><a name="Page_231" id="Page_231">[Pg 231]</a></span>
+range them in the causal order <i>A B C D</i>, and at the same time
+in the chronologic order <i>A B C D</i>, rather than in any other
+order?</p>
+
+<p>I clearly see that in the act <i>A</i> I have the feeling of having
+been active, while in undergoing the sensation <i>D</i> I have that of
+having been passive. This is why I regard <i>A</i> as the initial cause
+and <i>D</i> as the ultimate effect; this is why I put <i>A</i> at the beginning
+of the chain and <i>D</i> at the end; but why put <i>B</i> before <i>C</i> rather
+than <i>C</i> before <i>B</i>?</p>
+
+<p>If this question is put, the reply ordinarily is: we know that it
+is <i>B</i> which is the cause of <i>C</i> because we always see <i>B</i> happen
+before <i>C</i>. These two phenomena, when witnessed, happen in a
+certain order; when analogous phenomena happen without witness,
+there is no reason to invert this order.</p>
+
+<p>Doubtless, but take care; we never know directly the physical
+phenomena <i>B</i> and <i>C</i>. What we know are sensations <i>B&acute;</i> and <i>C&acute;</i>
+produced respectively by <i>B</i> and <i>C</i>. Our consciousness tells us
+immediately that <i>B&acute;</i> precedes <i>C&acute;</i> and we suppose that <i>B</i> and <i>C</i>
+succeed one another in the same order.</p>
+
+<p>This rule appears in fact very natural, and yet we are often
+led to depart from it. We hear the sound of the thunder only
+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?</p>
+
+
+<h4>XI</h4>
+
+<p>Another difficulty; have we really the right to speak of the
+cause of a phenomenon? If all the parts of the universe are interchained
+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 circumstances so complex? And yet it is only thus that these
+rules can be general and rigorous.</p>
+
+<p>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<span class='pagenum'><a name="Page_232" id="Page_232">[Pg 232]</a></span>
+reduced 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
+<i>t</i> + <i>h</i> as well as their positions at the instant <i>t</i> &minus; <i>h</i>.</p>
+
+<p>Even more; the position of Jupiter at the instant <i>t</i>, together
+with that of Saturn at the instant <i>t</i> + <i>a</i>, determines the position
+of Jupiter at any instant and that of Saturn at any instant.</p>
+
+<p>The aggregate of positions occupied by Jupiter at the instant
+<i>t</i> + <i>e</i> and Saturn at the instant <i>t</i> + <i>a</i> + <i>e</i> is bound to the aggregate
+of positions occupied by Jupiter at the instant <i>t</i> and Saturn
+at the instant <i>t</i> + <i>a</i>, by laws as precise as that of Newton, though
+more complicated. Then why not regard one of these aggregates
+as the cause of the other, which would lead to considering
+as simultaneous the instant <i>t</i> of Jupiter and the instant <i>t</i> + <i>a</i> of
+Saturn?</p>
+
+<p>In answer there can only be reasons, very strong, it is true, of
+convenience and simplicity.</p>
+
+
+<h4>XII</h4>
+
+<p>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 simultaneity.</p>
+
+<p>I will take two simple examples, the measurement of the
+velocity of light and the determination of longitude.</p>
+
+<p>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.</p>
+
+<p>He has begun by <i>supposing</i> 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 velocity
+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<span class='pagenum'><a name="Page_233" id="Page_233">[Pg 233]</a></span>
+should think ourselves fortunate that this contradiction has
+not happened and that the slight discordances which may happen
+can be readily explained.</p>
+
+<p>The postulate, at all events, resembling the principle of sufficient
+reason, has been accepted by everybody; what I wish to emphasize
+is that it furnishes us with a new rule for the investigation
+of simultaneity, entirely different from that which we
+have enunciated above.</p>
+
+<p>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.</p>
+
+<p>Could not the observed facts be just as well explained if we attributed
+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 complicated.
+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 discussing;
+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 depend
+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.</p>
+
+<p>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 instantaneous;
+if absolute exactitude were desired, there would be a
+correction to make according to a complicated rule.</p>
+
+<p>Or else finally they use the telegraph. It is clear first that the
+reception of the signal at Berlin, for instance, is after the sending
+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<span class='pagenum'><a name="Page_234" id="Page_234">[Pg 234]</a></span>
+regarded as simultaneous. But, to be rigorous, a little correction
+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 separate
+the qualitative problem of simultaneity from the quantitative
+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 <i>measuring</i> a time.</p>
+
+
+<h4>XIII</h4>
+
+<p>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.</p>
+
+<p>But what is the nature of these rules? No general rule, no
+rigorous rule; a multitude of little rules applicable to each particular
+case.</p>
+
+<p>These rules are not imposed upon us and we might amuse ourselves
+in inventing others; but they could not be cast aside without
+greatly complicating the enunciation of the laws of physics,
+mechanics and astronomy.</p>
+
+<p>We therefore choose these rules, not because they are true,
+but because they are the most convenient, and we may recapitulate
+them as follows: "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."</p>
+
+
+
+<hr style="width: 65%;" />
+<p><span class='pagenum'><a name="Page_235" id="Page_235">[Pg 235]</a></span></p>
+<h3>CHAPTER III</h3>
+
+<h3><span class="smcap">The Notion of Space</span></h3>
+
+<h4>1. <i>Introduction</i></h4>
+
+
+<p>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 difficult
+of approach, such as those which pertain to the number of
+dimensions. All 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.</p>
+
+<p>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 amorphous
+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.</p>
+
+<p>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?</p>
+
+<p>It is evident that experiment can not settle such a question;
+one would not ask, for instance, experiment to decide whether I
+should call <i>AB</i> or <i>CD</i> 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<span class='pagenum'><a name="Page_236" id="Page_236">[Pg 236]</a></span>
+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 <i>true</i> straight and that the non-Euclidean
+straight is not a true straight, we simply mean that
+the first intuitive idea corresponds to a <i>more noteworthy</i> object
+than the second. But how do we decide that this object is more
+noteworthy? This question I have investigated in 'Science and
+Hypothesis.'</p>
+
+<p>It is here that we saw experience come in. If the Euclidean
+straight is more noteworthy than the non-Euclidean straight, it
+is so chiefly because 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 satisfy
+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 conceivable,
+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 motions,
+the Euclidean straights remain Euclidean 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<span class='pagenum'><a name="Page_237" id="Page_237">[Pg 237]</a></span>
+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 triangles;
+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 Euclidean
+triangles if the non-Euclidean motions were called motions
+without deformation.</p>
+
+<p>Now when we say that the Euclidean motions are the <i>true</i>
+motions without deformation, what do we mean? We simply
+mean that they are <i>more noteworthy</i> than the others. And why
+are they more noteworthy? It is because certain noteworthy
+natural bodies, the solid bodies, undergo motions almost similar.</p>
+
+<p>And then when we ask: Can one imagine non-Euclidean space?
+That means: Can we imagine a world where there would be noteworthy
+natural objects affecting almost the form of non-Euclidean
+straights, and noteworthy natural bodies frequently undergoing
+motions almost similar to the non-Euclidean motions? I
+have shown in 'Science and Hypothesis' that to this question we
+must answer yes.</p>
+
+<p>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 themselves
+which they serve to measure. The world, after this dilatation,
+would continue on its course without anything apprising
+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 indistinguishable.
+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<span class='pagenum'><a name="Page_238" id="Page_238">[Pg 238]</a></span>
+coordinates of a point are continuous functions, <i>otherwise altogether
+arbitrary</i>, 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 relativity
+of space is not ordinarily understood in so broad a sense;
+it is thus, however, that it would be proper to understand it.</p>
+
+<p>If one of these universes is our Euclidean world, what its inhabitants
+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, therefore,
+be Euclidean geometry, but their straight will not be our
+Euclidean straight. It will be its transform by the point-transformation
+which carries over from our world to theirs. The
+straights of these men will not be our straights, but they will
+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 themselves,
+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.</p>
+
+
+<h4>2. <i>Qualitative Geometry</i></h4>
+
+<p>All that is relatively easy to understand, and I have already so
+often repeated it that I think it needless to expatiate further on
+the 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
+Lobachevskian space, just as we can, by tracing upon it a proper
+graduation, transform an ungraduated thermometer into a Fahrenheit
+or a R&eacute;aumur thermometer.<span class='pagenum'><a name="Page_239" id="Page_239">[Pg 239]</a></span></p>
+
+<p>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.</p>
+
+<p>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 Riemann and Betti and which
+has received the name of analysis situs. In this science abstraction
+is made of every quantitative idea and, for example, if we
+ascertain that on a line the point <i>B</i> is between the points <i>A</i> and
+<i>C</i>, we shall be content with this ascertainment and shall not
+trouble to know whether the line <i>ABC</i> is straight or curved, nor
+whether the length <i>AB</i> is equal to the length <i>BC</i>, or whether it
+is twice as great.</p>
+
+<p>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 proportions
+and replace the straights by lines more or less sinuous. In
+mathematical terms, they are not altered by any 'point-transformation'
+whatsoever. It has often been said that metric geometry
+was quantitative, while projective geometry was purely qualitative.
+That is not altogether true. The straight is still distinguished
+from other lines by properties which remain quantitative
+in some respects. The real qualitative geometry is, therefore,
+analysis situs.</p>
+
+<p>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 verities?
+Are they the characteristics of a form imposed either
+upon our sensibility or upon our understanding?</p>
+
+<p>I wish simply to observe that the last two solutions exclude
+each other. We can not admit at the same time that it is impossible
+to imagine space of four dimensions and that experience
+proves to us that space has three dimensions. The experimenter
+puts to nature a question: Is it this or that? and he can not put<span class='pagenum'><a name="Page_240" id="Page_240">[Pg 240]</a></span>
+it without imagining the two terms of the alternative. If it were
+impossible to imagine one of these terms, it would be futile and
+besides impossible to consult experience. There is no need of observation
+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.</p>
+
+<p>Note likewise that in analysis situs the empiricists are disembarrassed
+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 experiments
+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.</p>
+
+<p>Of all the theorems of analysis situs, the most important is
+that which is expressed in saying that space has three dimensions.
+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?</p>
+
+
+<h4>3. <i>The Physical Continuum of Several Dimensions</i></h4>
+
+<p>I have explained in 'Science and Hypothesis' whence we
+derive the notion of physical continuity and how that of mathematical
+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 readily
+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 symbols,
+may be written:</p>
+
+<p class="center">
+<i>A = B, &nbsp;&nbsp;&nbsp;&nbsp; B = C, &nbsp;&nbsp;&nbsp;&nbsp; A &lt; C</i>.<br />
+</p>
+
+<p>This would be the formula of the physical continuum, as crude
+experience gives it to us, whence arises an intolerable contradiction<span class='pagenum'><a name="Page_241" id="Page_241">[Pg 241]</a></span>
+that has been obviated by the introduction of the mathematical
+continuum. This is a scale of which the steps (commensurable
+or incommensurable numbers) are infinite in number
+but are exterior to one another, instead of encroaching on one
+another as do the elements of the physical continuum, in conformity
+with the preceding formula.</p>
+
+<p>The physical continuum is, so to speak, a nebula not resolved;
+the most perfect instruments could not attain to its resolution.
+Doubtless if we measured the weights with a good balance instead
+of judging them by the hand, we could distinguish the weight of
+11 grams from those of 10 and 12 grams, and our formula would
+become:</p>
+
+<p class="center">
+<i>A &lt; B, &nbsp;&nbsp;&nbsp;&nbsp; B &lt; C, &nbsp;&nbsp;&nbsp;&nbsp; A &lt; C.</i><br />
+</p>
+
+<p>But we should always find between <i>A</i> and <i>B</i> and between <i>B</i>
+and <i>C</i> new elements <i>D</i> and <i>E</i>, such that</p>
+
+<p class="center">
+<i>A = D, &nbsp;&nbsp;&nbsp;&nbsp; D = B, &nbsp;&nbsp;&nbsp;&nbsp; A &lt; B; &nbsp;&nbsp;&nbsp;&nbsp; B = E, &nbsp;&nbsp;&nbsp;&nbsp; E = C, &nbsp;&nbsp;&nbsp;&nbsp; B &lt; C,</i><br />
+</p>
+
+<p class="noidt">and the difficulty would only have receded and the nebula would
+always remain unresolved; the mind alone can resolve it and the
+mathematical continuum it is which is the nebula resolved into
+stars.</p>
+
+<p>Yet up to this point we have not introduced the notion of the
+number of dimensions. What is meant when we say that a mathematical
+continuum or that a physical continuum has two or
+three dimensions?</p>
+
+<p>First we must introduce the notion of cut, studying first physical
+continua. We have seen what characterizes the physical continuum.
+Each of the elements of this continuum consists of a
+manifold of impressions; and it may happen either that an element
+can not be discriminated from another element of the same
+continuum, if this new element corresponds to a manifold of
+impressions not sufficiently different, or, on the contrary, that
+the discrimination is possible; finally it may happen that two
+elements indistinguishable from a third may, nevertheless, be
+distinguished one from the other.</p>
+
+<p>That postulated, if <i>A</i> and <i>B</i> are two distinguishable elements of
+a continuum <i>C</i>, a series of elements may be found, <i>E</i><sub>1</sub>, <i>E</i><sub>2</sub>, ..., <i>E</i><sub><i>n</i></sub>,
+all belonging to this same continuum <i>C</i> and such that each of<span class='pagenum'><a name="Page_242" id="Page_242">[Pg 242]</a></span>
+them is indistinguishable from the preceding, that <i>E</i><sub>1</sub> is indistinguishable
+from <i>A</i>, and <i>E</i><sub><i>n</i></sub> indistinguishable from <i>B</i>. Therefore
+we can go from <i>A</i> to <i>B</i> by a continuous route and without
+quitting <i>C</i>. If this condition is fulfilled for any two elements
+<i>A</i> and <i>B</i> of the continuum <i>C</i>, we may say that this continuum <i>C</i>
+is all in one piece. Now let us distinguish certain of the elements
+of <i>C</i> which may either be all distinguishable from one another,
+or themselves form one or several continua. The assemblage of
+the elements thus chosen arbitrarily among all those of <i>C</i> will
+form what I shall call the <i>cut</i> or the <i>cuts</i>.</p>
+
+<p>Take on <i>C</i> any two elements <i>A</i> and <i>B</i>. Either we can also find
+a series of elements <i>E</i><sub>1</sub>, <i>E</i><sub>2</sub>, ..., <i>E</i><sub><i>n</i></sub>, such: (1) that they all belong
+to <i>C</i>; (2) that each of them is indistinguishable from the following,
+<i>E</i><sub>1</sub> indistinguishable from <i>A</i> and <i>E</i><sub><i>n</i></sub> from <i>B</i>; (3) <i>and besides
+that none of the elements E is indistinguishable from any element
+of the cut</i>. Or else, on the contrary, in each of the series <i>E</i><sub>1</sub>, <i>E</i><sub>2</sub>,
+..., <i>E</i><sub><i>n</i></sub> satisfying the first two conditions, there will be an element
+<i>E</i> indistinguishable from one of the elements of the cut. In the
+first case we can go from <i>A</i> to <i>B</i> by a continuous route without
+quitting <i>C</i> and <i>without meeting the cuts</i>; in the second case that
+is impossible.</p>
+
+<p>If then for any two elements <i>A</i> and <i>B</i> of the continuum <i>C</i>, it is
+always the first case which presents itself, we shall say that <i>C</i>
+remains all in one piece despite the cuts.</p>
+
+<p>Thus, if we choose the cuts in a certain way, otherwise arbitrary,
+it may happen either that the continuum remains all in one
+piece or that it does not remain all in one piece; in this latter
+hypothesis we shall then say that it is <i>divided</i> by the cuts.</p>
+
+<p>It will be noticed that all these definitions are constructed in
+setting out solely from this very simple fact, that two manifolds
+of impressions sometimes can be discriminated, sometimes can
+not be. That postulated, if, to <i>divide</i> a continuum, it suffices to
+consider as cuts a certain number of elements all distinguishable
+from one another, we say that this continuum <i>is of one dimension</i>;
+if, on the contrary, to divide a continuum, it is necessary to
+consider as cuts a system of elements themselves forming one or
+several continua, we shall say that this continuum is <i>of several
+dimensions</i>.<span class='pagenum'><a name="Page_243" id="Page_243">[Pg 243]</a></span></p>
+
+<p>If to divide a continuum <i>C</i>, cuts forming one or several continua
+of one dimension suffice, we shall say that <i>C</i> is a continuum
+<i>of two dimensions</i>; if cuts suffice which form one or several continua
+of two dimensions at most, we shall say that <i>C</i> is a continuum
+<i>of three dimensions</i>; and so on.</p>
+
+<p>To justify this definition it is proper to see whether it is in this
+way that geometers introduce the notion of three dimensions at
+the beginning of their works. Now, what do we see? Usually
+they begin by defining surfaces as the boundaries of solids or
+pieces of space, lines as the boundaries of surfaces, points as the
+boundaries of lines, and they affirm that the same procedure can
+not be pushed further.</p>
+
+<p>This is just the idea given above: to divide space, cuts that are
+called surfaces are necessary; to divide surfaces, cuts that are
+called lines are necessary; to divide lines, cuts that are called
+points are necessary; we can go no further, the point can not be
+divided, so the point is not a continuum. Then lines which can be
+divided by cuts which are not continua will be continua of one
+dimension; surfaces which can be divided by continuous cuts of
+one dimension will be continua of two dimensions; finally, space
+which can be divided by continuous cuts of two dimensions will
+be a continuum of three dimensions.</p>
+
+<p>Thus the definition I have just given does not differ essentially
+from the usual definitions; I have only endeavored to give it a
+form applicable not to the mathematical continuum, but to the
+physical continuum, which alone is susceptible of representation,
+and yet to retain all its precision. Moreover, we see that this
+definition applies not alone to space; that in all which falls under
+our senses we find the characteristics of the physical continuum,
+which would allow of the same classification; that it would be
+easy to find there examples of continua of four, of five, dimensions,
+in the sense of the preceding definition; such examples
+occur of themselves to the mind.</p>
+
+<p>I should explain finally, if I had the time, that this science,
+of which I spoke above and to which Riemann gave the name of
+analysis situs, teaches us to make distinctions among continua of
+the same number of dimensions and that the classification of these
+continua rests also on the consideration of cuts.<span class='pagenum'><a name="Page_244" id="Page_244">[Pg 244]</a></span></p>
+
+<p>From this notion has arisen that of the mathematical continuum
+of several dimensions in the same way that the physical
+continuum of one dimension engendered the mathematical continuum
+of one dimension. The formula</p>
+
+<p class="center">
+<i>A &gt; C, &nbsp;&nbsp;&nbsp;&nbsp; A = B, &nbsp;&nbsp;&nbsp;&nbsp; B = C,</i><br />
+</p>
+
+<p class="noidt">which summed up the data of crude experience, implied an intolerable
+contradiction. To get free from it, it was necessary to
+introduce a new notion while still respecting the essential characteristics
+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 <i>n</i> dimensions, it will suffice to
+take <i>n</i> like scales whose divisions correspond to different values
+of <i>n</i> independent magnitudes called coordinates. We thus shall
+have an image of the physical continuum of <i>n</i> dimensions, and
+this image will be as faithful as it can be after the determination
+not to allow the contradiction of which I spoke above.</p>
+
+
+<h4>4. <i>The Notion of Point</i></h4>
+
+<p>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 satisfies
+the definition we have just given of the physical continuum
+of three dimensions. To be content with that would be to suppose
+that we know what is the manifold of points of space, or even
+one point of space.</p>
+
+<p>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?</p>
+
+<p>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: <i>See</i> the sheep.<span class='pagenum'><a name="Page_245" id="Page_245">[Pg 245]</a></span></p>
+
+<p>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.</p>
+
+<p>When they try to represent to themselves a point, they represent
+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.</p>
+
+<p>But it is not a question of that; it does not suffice to represent
+<i>one</i> point, it is necessary to represent <i>a certain</i> point and to have
+the means of distinguishing it from an <i>other</i> 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 <i>a specific</i> element and to distinguish it
+from an <i>other</i> element.</p>
+
+<p>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 <i>A</i> at
+the instant &#945; is the same as the point occupied by the object <i>B</i> at
+the instant &#946;, or still better, what this means?</p>
+
+<p>I am seated in my room; an object is placed on my table; during
+a second I do not move, no one touches the object. I am
+tempted to say that the point <i>A</i> which this object occupied at the
+beginning of this second is identical with the point <i>B</i> which it
+occupies at its end. Not at all; from the point <i>A</i> to the point <i>B</i>
+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<span class='pagenum'><a name="Page_246" id="Page_246">[Pg 246]</a></span>
+meaning and in any case can not correspond to any representation.</p>
+
+<p>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 <i>two</i>. 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.</p>
+
+<p>If now we see that two objects have retained their relative position
+with regard to our body, we conclude that the relative position
+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. <i>A fortiori</i> 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.</p>
+
+<p>In a word, the system of coordinate axes to which we naturally
+refer all exterior objects is a system of axes invariably bound to
+our body, and carried around with us.</p>
+
+<p>It is impossible to represent to oneself absolute space; when I
+try to represent to myself simultaneously objects and myself in
+motion in absolute space, in reality I represent to myself my own
+self motionless and seeing move around me different objects and
+a man that is exterior to me, but that I convene to call me.</p>
+
+<p>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 ourselves?
+Many persons will answer yes and will say that they
+'localize' exterior objects.</p>
+
+<p>What does this mean? To localize an object simply means to
+represent to oneself the movements that would be necessary to<span class='pagenum'><a name="Page_247" id="Page_247">[Pg 247]</a></span>
+reach it. I will explain myself. It is not a question of representing
+the movements themselves in space, but solely of representing
+to oneself the muscular sensations which accompany these
+movements and which do not presuppose the preexistence of the
+notion of space.</p>
+
+<p>If we suppose two different objects which successively occupy
+the same relative position with regard to ourselves, the impressions
+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.</p>
+
+<p>But, given an object, we can conceive many different series of
+movements which equally enable us to reach it. If then we represent
+to ourselves a point by representing to ourselves the series
+of muscular sensations which accompany the movements which
+enable us to reach this 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?</p>
+
+<p>Another remark: I have just said that it is to our own body
+that we naturally refer exterior objects; that we carry about
+everywhere with us a system of axes to which we refer all the
+points of space and that this system of axes seems to be invariably
+bound to our body. It should be noticed that rigorously we
+could not speak of axes invariably bound to the body unless the
+different parts of this body were themselves invariably bound to
+one 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.</p>
+
+
+<h4>5. <i>The Notion of Displacement</i></h4>
+
+<p>I have shown in 'Science and Hypothesis' the preponderant
+r&ocirc;le played by the movements of our body in the genesis of the<span class='pagenum'><a name="Page_248" id="Page_248">[Pg 248]</a></span>
+notion of space. For a being completely immovable 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.</p>
+
+<p>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.</p>
+
+<p>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.</p>
+
+<p>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 impossible).
+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.</p>
+
+<p>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<span class='pagenum'><a name="Page_249" id="Page_249">[Pg 249]</a></span>
+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.</p>
+
+<p>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 <i>same</i> displacement? Evidently because,
+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 movements
+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.</p>
+
+<p>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 <i>another</i> nerve fiber starting from the border
+of the retina; these two sensations are qualitatively different;
+otherwise, how could I distinguish them?</p>
+
+<p>Why then am I led to decide that these two sensations, qualitatively
+different, represent the same image, which has been displaced?
+It is because I <i>can follow the object with the eye</i> and by
+a displacement of the eye, voluntary and accompanied by muscular
+sensations, bring back the image to the center of the retina
+and reestablish the primitive sensation.</p>
+
+<p>I suppose that the image of a red object has gone from the
+center <i>A</i> to the border <i>B</i> of the retina, then that the image of a
+blue object goes in its turn from the center <i>A</i> to the border <i>B</i>
+of the retina; I shall decide that these two objects have undergone
+the <i>same</i> displacement. Why? Because in both cases I
+shall have been able to reestablish the primitive sensation, and<span class='pagenum'><a name="Page_250" id="Page_250">[Pg 250]</a></span>
+that to do it I shall have had to execute the <i>same</i> movement of
+the eye, and I shall know that my eye has executed the same
+movement because I shall have felt the <i>same</i> muscular sensations.</p>
+
+<p>If I could not move my eye, should I have any reason to suppose
+that the sensation of red at the center of the retina is to the
+sensation of red at the border of the retina as that of blue at the
+center is to that of blue at the 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.</p>
+
+<p>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.</p>
+
+<p>1. I may suppose that my body has moved from one point to
+another, but that the same <i>attitude</i> is retained; all the parts of
+the body have therefore retained or resumed the same <i>relative</i>
+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.</p>
+
+<p>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.</p>
+
+<p>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 distinction;
+before knowing geometry I ascertain the fact (experimentally,
+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 <i>explain</i> the fact, it suffices me <i>to ascertain</i> it.</p>
+
+<p>However that may be, the explanation is easy. Suppose that<span class='pagenum'><a name="Page_251" id="Page_251">[Pg 251]</a></span>
+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 produced
+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 situation
+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.</p>
+
+<p>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 <i>A</i> to the position <i>B</i> I may
+take several routes. To the first of these routes will correspond
+a series <i>S</i> of muscular sensations; to a second route will correspond
+another series <i>S&acute;&acute;</i>, of muscular sensations which generally
+will be completely different, since other muscles will be used.</p>
+
+<p>How am I led to regard these two series <i>S</i> and <i>S&acute;&acute;</i> as corresponding
+to the same displacement <i>AB</i>? It is because these two
+series are capable of correcting the same external change. Apart
+from that, they have nothing in common.</p>
+
+<p>Let us now consider two external changes: &#945; and &#946;, 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. Consider,
+on the other hand, two series of internal changes <i>S</i> and <i>S&acute;&acute;</i>;
+like the others, they will have nothing in common. And yet I say
+that &#945; and &#946; correspond to the same displacement, and that <i>S</i> and
+<i>S&acute;&acute;</i> correspond also to the same displacement. why? Simply
+because <i>S</i> can correct &#945; as well as &#946; and because &#945; can be corrected
+by <i>S&acute;&acute;</i> as well as by <i>S</i>. And then a question suggests itself:<span class='pagenum'><a name="Page_252" id="Page_252">[Pg 252]</a></span></p>
+
+<p>If I have ascertained that <i>S</i> corrects &#945; and &#946; and that <i>S&acute;&acute;</i> corrects
+&#945;, am I certain that <i>S&acute;&acute;</i> likewise corrects &#946;? Experiment alone
+can teach us 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 classifying
+the internal and external changes as I have just done, and,
+for instance, in distinguishing changes of state from changes of
+position.</p>
+
+<p>It is interesting to see what has been the r&ocirc;le of experience in
+all this. It has shown me that a certain law is approximately
+verified. It has not told me <i>how</i> space is, and that it satisfies
+the condition in question. I knew, in fact, before all experience,
+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.</p>
+
+
+<h4>6. <i>Visual Space</i></h4>
+
+<p>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 r&ocirc;le of visual
+impressions and to investigate how many dimensions 'visual
+space' has, and for that purpose to apply to these impressions
+the definition of &sect; 3.</p>
+
+<p>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 movements
+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 common;
+we should not have been able to disengage from them what<span class='pagenum'><a name="Page_253" id="Page_253">[Pg 253]</a></span>
+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.</p>
+
+<p>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 sensations
+of different color.</p>
+
+<p>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.</p>
+
+<p>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 <i>C</i>, and it is clear 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.</p>
+
+<p>If, therefore, the cut has <i>n</i> dimensions, the total manifold of my
+red sensations, or if you wish, the whole visual space, will have
+<i>n</i> + 1.</p>
+
+<p>Now, I distinguish the red sensations affecting a point of the
+cut <i>C</i>. The assemblage of these sensations will form a new cut
+<i>C&acute;</i>. It is clear that this will divide the cut <i>C</i>, always giving to
+the word divide the same meaning.<span class='pagenum'><a name="Page_254" id="Page_254">[Pg 254]</a></span></p>
+
+<p>If, therefore, the cut <i>C&acute;</i> has <i>n</i> dimensions, the cut <i>C</i> will have
+<i>n</i> + 1 and the whole of visual space <i>n</i> + 2.</p>
+
+<p>If all the red sensations affecting the same point of the retina
+were regarded as identical, the cut <i>C&acute;</i> reducing to a single element
+would have 0 dimensions, and visual space would have 2.</p>
+
+<p>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 recognize
+the distance of objects. When we seek to analyze this feeling,
+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.</p>
+
+<p>Two red sensations affecting the same point of the retina will
+therefore be regarded as identical only if they are accompanied
+by the same sensation of convergence and also by the same sensation
+of effort of accommodation or at least by sensations of
+convergence and accommodation so slightly different as to be
+indistinguishable.</p>
+
+<p>On this account the cut <i>C&acute;</i> is itself a continuum and the cut <i>C</i>
+has more than one dimension.</p>
+
+<p>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 sensation
+of accommodation. If then we form a new cut <i>C&acute;&acute;</i> with all
+those of the sensations of the cut <i>C&acute;</i>, which are accompanied by a
+certain sensation of convergence, in accordance with the preceding
+law they will all be indistinguishable and may be regarded
+as identical. Therefore <i>C&acute;&acute;</i> will not be a continuum and will
+have 0 dimension; and as <i>C&acute;&acute;</i> divides <i>C&acute;</i> it will thence result that
+<i>C&acute;</i> has one, <i>C</i> two and <i>the whole visual space three dimensions</i>.</p>
+
+<p>But would it be the same if experience had taught us the contrary
+and if a certain sensation of convergence were not always
+accompanied by the same sensation of 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 <i>C&acute;&acute;</i>, could
+nevertheless be distinguished since they would be accompanied by
+two different sensations of accommodation. Therefore <i>C&acute;&acute;</i> would
+be in its turn a continuum and would have one dimension (at<span class='pagenum'><a name="Page_255" id="Page_255">[Pg 255]</a></span>
+least); then <i>C&acute;</i> would have two, <i>C</i> three and <i>the whole visual
+space would have four dimensions</i>.</p>
+
+<p>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 dimensions
+and that the optician who constructed them has given one
+more dimension to space? Evidently not; all we can say is that
+experience has taught us that it is convenient to attribute three
+dimensions to space.</p>
+
+<p>But visual space is only one part of space, and in even the
+notion of this space there is something artificial, as I have explained
+at the beginning. The real space is motor space and this
+it is that we shall examine in the following chapter.</p>
+
+
+
+<hr style="width: 65%;" />
+<p><span class='pagenum'><a name="Page_256" id="Page_256">[Pg 256]</a></span></p>
+<h3>CHAPTER IV</h3>
+
+<h3><span class="smcap">Space and its Three Dimensions</span></h3>
+
+
+<h4>1. <i>The Group of Displacements</i></h4>
+
+<p>Let us sum up briefly the results obtained. We proposed to
+investigate what was meant in saying that space has three dimensions
+and we have asked first what is a physical continuum and
+when it may be said to have <i>n</i> dimensions. If we consider different
+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 saying 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 <i>C</i>. And each of these systems is called an
+<i>element</i> of the continuum <i>C</i>.</p>
+
+<p>How many dimensions has this continuum? Take first two
+elements <i>A</i> and <i>B</i> of <i>C</i>, and suppose there exists a series &#931; of
+elements, all belonging to the continuum <i>C</i>, of such a sort that <i>A</i>
+and <i>B</i> are the two extreme terms of this series and that each
+term of the series is indistinguishable from the preceding. If
+such a series &#931; can be found, we say that <i>A</i> and <i>B</i> are joined to
+one another; and if any two elements of <i>C</i> are joined to one
+another, we say that <i>C</i> is all of one piece.</p>
+
+<p>Now take on the continuum <i>C</i> a certain number of elements in
+a way altogether arbitrary. The aggregate of these elements will
+be called a <i>cut</i>. Among the various series &#931; which join <i>A</i> to <i>B</i>,
+we shall distinguish those of which an element is indistinguishable
+from one of the elements of the cut (we shall say that these
+are they which <i>cut</i> the cut) and those of which <i>all</i> the elements
+are distinguishable from all those of the cut. If <i>all</i> the series &#931;
+which join <i>A</i> to <i>B</i> cut the cut, we shall say that <i>A</i> and <i>B</i> are<span class='pagenum'><a name="Page_257" id="Page_257">[Pg 257]</a></span>
+<i>separated</i> by the cut, and that the cut <i>divides</i> <i>C</i>. If we can not
+find on <i>C</i> two elements which are separated by the cut, we shall
+say that the cut <i>does not divide</i> <i>C</i>.</p>
+
+<p>These definitions laid down, if the continuum <i>C</i> can be divided
+by cuts which do not themselves form a continuum, this continuum
+<i>C</i> has only one dimension; in the contrary case it has
+several. If a cut forming a continuum of 1 dimension suffices
+to divide <i>C</i>, <i>C</i> will have 2 dimensions; if a cut forming a continuum
+of 2 dimensions suffices, <i>C</i> 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 element
+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.</p>
+
+<p>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 different
+elements of space, which we call points, are all alike, and,
+to apply our definition, it is necessary that we know how to distinguish
+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).</p>
+
+<p>Then I have sought to form with our visual sensations a physical
+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<span class='pagenum'><a name="Page_258" id="Page_258">[Pg 258]</a></span>
+changes of position and changes of state. Among the changes
+which occur in our impressions, we distinguish, first the <i>internal</i>
+changes, voluntary and accompanied by muscular sensations, and
+the <i>external</i> changes, having opposite characteristics. We ascertain
+that it may happen that an external change may be <i>corrected</i>
+by an internal change which reestablishes the primitive sensations.
+The external changes, capable of being corrected by an
+internal change are called <i>changes of position</i>, those not capable
+of it are called <i>changes of state</i>. The internal changes capable
+of correcting an external change are called <i>displacements of the
+whole body</i>; the others are called <i>changes of attitude</i>.</p>
+
+<p>Now let &#945; and &#946; be two external changes, &#945;&acute; and &#946;&acute; two internal
+changes. Suppose that a may be corrected either by &#945;&acute; or by &#946;',
+and that &#945;&acute; can correct either &#945; or &#946;; experience tells us then that
+&#946;&acute; can likewise correct &#946;. In this case we say that &#945; and &#946; correspond
+to the <i>same</i> displacement and also that &#945;&acute; and &#946;&acute; correspond
+to the <i>same</i> displacement. That postulated, we can
+imagine a physical continuum which we shall call <i>the continuum
+or group of displacements</i> and which we shall define in the following
+manner. The elements of this continuum shall be the internal
+changes capable of correcting an external change. Two of
+these internal changes &#945;&acute; and &#946;&acute; shall be regarded as indistinguishable:
+(1) if they are so naturally, that is, if they are
+too close to one another; (2) if &#945;&acute; is capable of correcting
+the same external change as a third internal change naturally
+indistinguishable from &#946;'. In this second case, they will
+be, so to speak, indistinguishable by convention, I mean by agreeing
+to disregard circumstances which might distinguish them.</p>
+
+<p>Our continuum is now entirely defined, since we know its elements
+and have fixed under what conditions they may be regarded
+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 <i>six</i>. The continuum
+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 displacements
+has six dimensions? We know it <i>by experience</i>.</p>
+
+<p>It would be easy to describe the experiments by which we<span class='pagenum'><a name="Page_259" id="Page_259">[Pg 259]</a></span>
+could arrive at this result. It would be seen that in this continuum
+cuts can be made which divide it and which are continua;
+that these cuts themselves can be divided by other cuts
+of the second order which yet are continua, and that this would
+stop only after cuts of the sixth order which would no longer be
+continua. From our definitions that would mean that the group
+of displacements has six dimensions.</p>
+
+<p>That would be easy, I have said, but that would be rather long;
+and would it not be a little superficial? This group of displacements,
+we have seen, is related to space, and space could be deduced
+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.</p>
+
+
+<h4>2. <i>Identity of Two Points</i></h4>
+
+<p>What is a point? How do we know whether two points of
+space are identical or different? Or, in other words, when I say:
+The object <i>A</i> occupied at the instant &#945; the point which the object
+<i>B</i> occupies at the instant &#946;, what does that mean?</p>
+
+<p>Such is the problem we set ourselves in the preceding chapter,
+&sect;4. As I have explained it, it is not a question of comparing the
+positions of the objects <i>A</i> and <i>B</i> 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 replaced
+in the same attitude.</p>
+
+<p>I suppose that between the instants &#945; and &#946; 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 ascertain
+that at the instant &#945; impressions that I attributed to the
+object <i>A</i> 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 &#946; other impressions
+which I attribute to the object <i>B</i> are transmitted to me, some by<span class='pagenum'><a name="Page_260" id="Page_260">[Pg 260]</a></span>
+this same fiber of the optic nerve, the others by this same tactile
+nerve.</p>
+
+<p>Here I must pause for an explanation; how am I told that this
+impression which I attribute to <i>A</i>, and that which I attribute to
+<i>B</i>, impressions which are qualitatively different, are transmitted
+to me by the same nerve? Must we suppose, to take for example
+the visual sensations, that <i>A</i> produces two simultaneous sensations,
+a sensation purely luminous <i>a</i> and a colored sensation <i>a&acute;</i>,
+that <i>B</i> produces in the same way simultaneously a luminous sensation
+<i>b</i> and a colored sensation <i>b&acute;</i>, that if these different sensations
+are transmitted to me by the same retinal fiber, <i>a</i> is identical
+with <i>b</i>, but that in general the colored sensations <i>a&acute;</i> and <i>b&acute;</i>
+produced by different bodies are different? In that case it would
+be the identity of the sensation <i>a</i> which accompanies <i>a&acute;</i> with the
+sensation <i>b</i> which accompanies <i>b&acute;</i>, which would tell that all these
+sensations are transmitted to me by the same fiber.</p>
+
+<p>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 <i>a</i> + <i>a&acute;</i> and <i>b</i> +<i>b&acute;</i>, without
+which we should have no means of recognizing that the object <i>B</i>
+has taken the place of the object <i>A</i>.</p>
+
+<p>Therefore I do not further insist and I recall the hypothesis I
+have just made: I suppose that I have ascertained that the impressions
+which I attribute to <i>B</i> are transmitted to me at the
+instant &#946; by the same fibers, optic as well as tactile, which, at the
+instant &#945;, had transmitted to me the impressions that I attributed
+to <i>A</i>. If it is so, we shall not hesitate to declare that the point
+occupied by <b>B</b> at the instant &#946; is identical with the point occupied
+by <i>A</i> at the instant &#945;.</p>
+
+<p>I have just enunciated two conditions for these points being
+identical; one is relative to sight, the other to touch. Let us consider
+them separately. The first is necessary, but is not sufficient.
+The second is at once necessary and sufficient. A person
+knowing geometry could easily explain this in the following
+manner: Let <i>O</i> be the point of the retina where is formed at the
+instant &#945; the image of the body <i>A</i>; let <i>M</i> be the point of space
+occupied at the instant &#945; by this body <i>A</i>; let <i>M&acute;</i> be the point of<span class='pagenum'><a name="Page_261" id="Page_261">[Pg 261]</a></span>
+space occupied at the instant &#946; by the body <i>B</i>. For this body <i>B</i>
+to form its image in <i>O</i>, it is not necessary that the points <i>M</i> and
+<i>M&acute;</i> coincide; since vision acts at a distance, it suffices for the
+three points <i>O</i> <i>M</i> <i>M&acute;</i> to be in a straight line. This condition that
+the two objects form their image on <i>O</i> is therefore necessary, but
+not sufficient for the points <i>M</i> and <i>M&acute;</i> to coincide. Let now <i>P</i> 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 <i>A</i> touches my finger at the instant &#945;, it is because <i>M</i> and
+<i>P</i> coincide; if <i>B</i> touches my finger at the instant &#946;, it is because
+<i>M&acute;</i> and <i>P</i> coincide. Therefore <i>M</i> and <i>M&acute;</i> coincide. Thus this
+condition that if <i>A</i> touches my finger at the instant &#945;, <i>B</i> touches
+it at the instant &#946;, is at once necessary and sufficient for <i>M</i> and
+<i>M&acute;</i> to coincide.</p>
+
+<p>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.</p>
+
+<p>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 relative
+to touch may be fulfilled without that of sight being fulfilled
+and that, on the contrary, that of sight can not be fulfilled without
+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 distance,
+and that sight does not operate at a distance.</p>
+
+<p>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.</p>
+
+<p>I suppose that my first finger receives at the instant &#945; a tactile
+impression which I attribute to the object <i>A</i>. I make a series of
+movements, corresponding to a series <i>S</i> of muscular sensations.
+After these movements, at the instant &#945;', my <i>second</i> finger receives
+a tactile impression that I attribute likewise to <i>A</i>. Afterward,
+at the instant &#946;, without my having budged, as my muscular
+sense tells me, this same second finger transmits to me<span class='pagenum'><a name="Page_262" id="Page_262">[Pg 262]</a></span>
+anew a tactile impression which I attribute this time to the
+object <i>B</i>; I then make a series of movements, corresponding to
+a series <i>S&acute;</i> of muscular sensations. I know that this series <i>S&acute;</i> is
+the inverse of the series <i>S</i> and corresponds to contrary movements.
+I know this because many previous experiences have
+shown me that if I made successively the two series of movements
+corresponding to <i>S</i> and to <i>S&acute;</i>, the primitive impressions would be
+reestablished, in other words, that the two series mutually compensate.
+That settled, should I expect that at the instant &#946;',
+when the second series of movements is ended, my <i>first finger</i>
+would feel a tactile impression attributable to the object <i>B</i>?</p>
+
+<p>To answer this question, those already knowing geometry
+would reason as follows: There are chances that the object <i>A</i> has
+not budged, between the instants &#945; and &#945;', nor the object <i>B</i>
+between the instants &#946; and &#946;'; assume this. At the instant &#945;,
+the object <i>A</i> occupied a certain point <i>M</i> of space. Now at this
+instant it touched my first finger, and <i>as touch does not operate
+at a distance</i>, my first finger was likewise at the point <i>M</i>. I
+afterward made the series <i>S</i> of movements and at the end of
+this series, at the instant &#945;', I ascertained that the object <i>A</i>
+touched my second finger. I thence conclude that this second
+finger was then at <i>M</i>, that is, that the movements <i>S</i> had the result
+of bringing the second finger to the place of the first. At the
+instant &#946; the object <i>B</i> has come in contact with my second finger:
+as I have not budged, this second finger has remained at <i>M</i>;
+therefore the object <i>B</i> has come to <i>M</i>; by hypothesis it does not
+budge up to the instant &#946;'. But between the instants &#946; and &#946;'
+I have made the movements <i>S&acute;</i>; as these movements are the inverse
+of the movements <i>S</i>, they must have for effect bringing the
+first finger in the place of the second. At the instant &#946;&acute; this
+first finger will, therefore, be at <i>M</i>; and as the object <i>B</i> is likewise
+at <i>M</i>, this object <i>B</i> will touch my first finger. To the question
+put, the answer should therefore be yes.</p>
+
+<p>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 <i>A</i>
+has moved between the instants &#945; and &#945;', or the object <i>B</i> between
+the instants &#946; and &#946;'.<span class='pagenum'><a name="Page_263" id="Page_263">[Pg 263]</a></span></p>
+
+<p>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 impossible?
+Not the least in the world. We should have contented
+ourselves with concluding <i>that touch can operate at a distance</i>.</p>
+
+<p>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 <i>B</i> occupies at the
+instant &#946; the point occupied by <i>A</i> at the instant &#945;, 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 fingers
+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.</p>
+
+<p>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 sensation
+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 sensations,
+we should obtain 'simple visual space,' of only two dimensions.
+On the other hand, consider tactile space, limiting ourselves
+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.</p>
+
+<p>Now I return to a point over which I passed rapidly in order
+not to interrupt the discussion. How do we know that the impressions
+made on our retina by <i>A</i> at the instant &#945; and <i>B</i> at the<span class='pagenum'><a name="Page_264" id="Page_264">[Pg 264]</a></span>
+instant &#946; are transmitted by the same retinal fiber, although
+these impressions are qualitatively different? I have suggested
+a simple hypothesis, while adding that other hypotheses, decidedly
+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 &#945;, and by the blue object <i>B</i> at the instant &#946;, 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 different
+though contiguous nervous fibers. What means have I
+then of knowing that these fibers are contiguous? It is probable
+that we should have none, if 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 <i>A</i> and the sensation
+of blue at the point <i>B</i> of the retina as between the sensation
+of red at the point <i>A</i> and the sensation of red at the point <i>B</i>.
+They have shown us, in fact, that the same movements, corresponding
+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.</p>
+
+
+<h4>3. <i>Tactile Space</i></h4>
+
+<p>Thus I know how to recognize the identity of two points, the
+point occupied by <i>A</i> at the instant &#945; and the point occupied by
+<i>B</i> at the instant &#946;, but only <i>on one condition</i>, namely, that I have
+not budged between the instants &#945; and &#946;. That does not suffice
+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 <i>A</i> at the instant &#945; is identical
+with the point occupied by <i>B</i> at the instant &#946;? I suppose
+that at the instant &#945;, the object <i>A</i> was in contact with my first
+finger and that in the same way, at the instant &#946;, the object <i>B</i>
+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 <i>S</i> and <i>S&acute;</i>, and<span class='pagenum'><a name="Page_265" id="Page_265">[Pg 265]</a></span>
+I have said it sometimes happens that we are led to consider two
+such series <i>S</i> and <i>S&acute;</i> as inverse one of the other, because we have
+often observed that when these two series succeed one another
+our primitive impressions are reestablished.</p>
+
+<p>If then my muscular sense tells me that I have moved between
+the two instants &#945; and &#946;, but so as to feel successively the two
+series of muscular sensations <i>S</i> and <i>S&acute;</i> that I consider inverses,
+I shall still conclude, just as if I had not budged, that the points
+occupied by <i>A</i> at the instant &#945; and by <i>B</i> at the instant &#946; are
+identical, if I ascertain that my first finger touches <i>A</i> at the
+instant &#945;, and <i>B</i> at the instant &#946;.</p>
+
+<p>This solution is not yet completely satisfactory, as one will see.
+Let us see, in fact, how many dimensions it would make us attribute
+to space. I wish to compare the two points occupied by <i>A</i>
+and <i>B</i> at the instants &#945; and &#946;, or (what amounts to the same
+thing since I suppose that my finger touches <i>A</i> at the instant &#945;
+and <i>B</i> at the instant &#946;) I wish to compare the two points occupied
+by my finger at the two instants &#945; and &#946;. The sole means
+I use for this comparison is the series &#931; of muscular sensations
+which have accompanied the movements of my body between
+these two instants. The different imaginable series &#931; form evidently
+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 &#931; and &#931; + <i>S</i> + <i>S&acute;</i>, when <i>S</i> and <i>S&acute;</i> are inverses
+one of the other in the sense above given to this word;
+in spite of this agreement, the aggregate of distinct series &#931; will
+still form a physical continuum and the number of dimensions
+will be less but still very great.</p>
+
+<p>To each of these series &#931; corresponds a point of space; to two
+series &#931; and &#931;&acute; thus correspond two points <i>M</i> and <i>M&acute;</i>. The means
+we have hitherto used enable us to recognize that <i>M</i> and <i>M&acute;</i> are
+not distinct in two cases: (1) if &#931; is identical with &#931;&acute;; (2) if &#931;&acute; =
+&#931; + <i>S</i> + <i>S&acute;</i>, <i>S</i> and <i>S&acute;</i> being inverses one of the other. If in all
+the other cases we should regard <i>M</i> and <i>M&acute;</i> as distinct, the manifold
+of points would have as many dimensions as the aggregate
+of distinct series &#931;, that is, much more than three.</p>
+
+<p>For those who already know geometry, the following explanation
+would be easily comprehensible. Among the imaginable<span class='pagenum'><a name="Page_266" id="Page_266">[Pg 266]</a></span>
+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 &#931; and &#931; + &#963;,
+where the series &#963; corresponds to movements where the finger
+does not budge, the aggregate of series will constitute a continuum
+of three dimensions, but that if one regards as distinct
+two series &#931; and &#931;&acute; unless &#931;&acute; = &#931; + <i>S</i> + <i>S&acute;</i>, <i>S</i> and <i>S&acute;</i> being inverses,
+the aggregate of series will constitute a continuum of
+more than three dimensions.</p>
+
+<p>In fact, let there be in space a surface <i>A</i>, on this surface a
+line <i>B</i>, on this line a point <i>M</i>. Let <i>C</i><sub>0</sub> be the aggregate of all
+series &#931;. Let <i>C</i><sub>1</sub> be the aggregate of all the series &#931;, such that
+at the end of corresponding movements the finger is found upon
+the surface <i>A</i>, and <i>C</i><sub>2</sub> or <i>C</i><sub>3</sub> the aggregate of series &#931; such that
+at the end the finger is found on <i>B</i>, or at <i>M</i>. It is clear, first that
+<i>C</i><sub>1</sub> will constitute a cut which will divide <i>C</i><sub>0</sub>, that <i>C</i><sub>2</sub> will be a cut
+which will divide <i>C</i><sub>1</sub>, and <i>C</i><sub>3</sub> a cut which will divide <i>C</i><sub>2</sub>. Thence
+it results, in accordance with our definitions, that if <i>C</i><sub>3</sub> is a continuum
+of <i>n</i> dimensions, <i>C</i><sub>0</sub> will be a physical continuum of
+<i>n</i> + 3 dimensions.</p>
+
+<p>Therefore, let &#931; and &#931;&acute; = &#931; + &#963; be two series forming part
+of <i>C</i><sub>3</sub>; for both, at the end of the movements, the finger is found
+at <i>M</i>; thence results that at the beginning and at the end of the
+series &#963; the finger is at the same point <i>M</i>. This series &#963; is therefore
+one of those which correspond to movements where the
+finger does not budge. If &#931; and &#931; + &#963; are not regarded as distinct,
+all the series of <i>C</i><sub>3</sub> blend into one; therefore <i>C</i><sub>3</sub> will have
+0 dimension, and <i>C</i><sub>0</sub> will have 3, as I wished to prove. If, on
+the contrary, I do not regard &#931; and &#931; + &#963; as blending (unless
+&#963; = <i>S</i> + <i>S&acute;</i>, <i>S</i> and <i>S&acute;</i> being inverses), it is clear that <i>C</i><sub>3</sub> will contain
+a great number of series of distinct sensations; because,
+without the finger budging, the body may take a multitude of
+different attitudes. Then <i>C</i><sub>3</sub> will form a continuum and <i>C</i><sub>0</sub> will
+have more than three dimensions, and this also I wished to prove.</p>
+
+<p>We who do not yet know geometry can not reason in this way;
+we can only verify. But then a question arises; how, before
+knowing geometry, have we been led to distinguish from the
+others these series &#963; where the finger does not budge? It is, in<span class='pagenum'><a name="Page_267" id="Page_267">[Pg 267]</a></span>
+fact, only after having made this distinction that we could be led
+to regard &#931; and &#931; + &#963; as identical, and it is on this condition
+alone, as we have just seen, that we can arrive at space of three
+dimensions.</p>
+
+<p>We are led to distinguish the series &#963;, because it often happens
+that when we have executed the movements which correspond to
+these series &#963; of muscular sensations, the tactile sensations which
+are transmitted to us by the nerve of the finger that we have
+called the first finger, persist and are not altered by these movements.
+Experience alone tells us that and it alone could tell us.</p>
+
+<p>If we have distinguished the series of muscular sensations
+<i>S</i> + <i>S&acute;</i> formed by the union of two inverse series, it is because
+they preserve the totality of our impressions; if now we distinguish
+the series &#963;, it is because they preserve <i>certain</i> of our impressions.
+(When I say that a series of muscular sensations <i>S</i>
+'preserves' one of our impressions <i>A</i>, I mean that we ascertain
+that if we feel the impression <i>A</i>, then the muscular sensations <i>S</i>,
+we <i>still</i> feel the impression <i>A</i> <i>after</i> these sensations <i>S</i>.)</p>
+
+<p>I have said above it often happens that the series &#963; do not
+alter the tactile impressions felt by our first finger; I said <i>often</i>,
+I did not say <i>always</i>. This it is that we express in our ordinary
+language by saying that the tactile impressions would not be
+altered if the finger has not moved, <i>on the condition</i> that <i>neither
+has</i> the object <i>A</i>, which was in contact with this finger, moved.
+Before knowing geometry, we could not give this explanation;
+all we could do is to ascertain that the impression often persists,
+but not always.</p>
+
+<p>But that the impression often continues is enough to make the
+series &#963; appear remarkable to us, to lead us to put in the same
+class the series &#931; and &#931; + &#963;, and hence not regard them as distinct.
+Under these conditions we have seen that they will engender
+a physical continuum of three dimensions.</p>
+
+<p>Behold then a space of three dimensions engendered by my
+first finger. Each of my fingers will create one like it. It remains
+to consider how we are led to regard them as identical
+with visual space, as identical with geometric space.</p>
+
+<p>But one reflection before going further; according to the foregoing,
+we know the points of space, or more generally the final<span class='pagenum'><a name="Page_268" id="Page_268">[Pg 268]</a></span>
+situation of our body, only by the series of muscular sensations
+revealing to us the movements which have carried us from a
+certain initial situation to this final situation. But it is clear
+that this final situation will depend, on the one hand, upon
+these movements and, <i>on the other hand, upon the initial situation</i>
+from which we set out. Now these movements are revealed
+to us by our muscular sensations; but nothing tells us the
+initial situation; nothing can distinguish it for us from all the
+other possible situations. This puts well in evidence the essential
+relativity of space.</p>
+
+
+<h4>4. <i>Identity of the Different Spaces</i></h4>
+
+<p>We are therefore led to compare the two continua <i>C</i> and <i>C&acute;</i>
+engendered, for instance, one by my first finger <i>D</i>, the other by
+my second finger <i>D&acute;</i>. These two physical continua both have
+three dimensions. To each element of the continuum <i>C</i>, or, if
+you prefer, to each point of the first tactile space, corresponds a
+series of muscular sensations &#931;, which carry me from a certain
+initial situation to a certain final situation.<a name="FNanchor_8_8" id="FNanchor_8_8"></a><a href="#Footnote_8_8" class="fnanchor">[8]</a> Moreover, the same
+point of this first space will correspond to &#931; and &#931; + &#963;, if &#963;
+is a series of which we know that it does not make the finger <i>D</i>
+move.</p>
+
+<p>Similarly to each element of the continuum <i>C&acute;</i>, or to each point
+of the second tactile space, corresponds a series of sensations &#931;&acute;,
+and the same point will correspond to &#931;&acute; and to &#931;&acute; + &#963;&acute;, if &#963;&acute; is a
+series which does not make the finger <i>D&acute;</i> move.</p>
+
+<p>What makes us distinguish the various series designated &#963; from
+those called &#963;&acute; is that the first do not alter the tactile impressions
+felt by the finger <i>D</i> and the second preserve those the finger <i>D&acute;</i>
+feels.</p>
+
+<p>Now see what we ascertain: in the beginning my finger <i>D&acute;</i> feels
+a sensation <i>A&acute;</i>; I make movements which produce muscular sensations
+<i>S</i>; my finger <i>D</i> feels the impression <i>A</i>; I make movements
+which produce a series of sensations &#963;; my finger <i>D</i> continues
+to feel the impression <i>A</i>, since this is the characteristic<span class='pagenum'><a name="Page_269" id="Page_269">[Pg 269]</a></span>
+property of the series &#963;; I then make movements which produce
+the series <i>S&acute;</i> of muscular sensations, <i>inverse</i> to <i>S</i> in the sense
+above given to this word. I ascertain then that my finger <i>D&acute;</i>
+feels anew the impression <i>A&acute;</i>. (It is of course understood that
+<i>S</i> has been suitably chosen.)</p>
+
+<p>This means that the series <i>S</i> + &#963; + <i>S&acute;</i>, preserving the tactile
+impressions of the finger <i>D&acute;</i>, is one of the series I have called &#963;&acute;.
+Inversely, if one takes any series &#963;&acute;, <i>S&acute;</i> + &#963;&acute; + <i>S</i> will be one of
+the series that we call &#963;&acute;.</p>
+
+<p>Thus if <i>S</i> is suitably chosen, <i>S</i> + &#963; + <i>S&acute;</i> will be a series &#963;&acute;, and
+by making &#963; vary in all possible ways, we shall obtain all the
+possible series &#963;&acute;.</p>
+
+<p>Not yet knowing geometry, we limit ourselves to verifying all
+that, but here is how those who know geometry would explain the
+fact. In the beginning my finger <i>D&acute;</i> is at the point <i>M</i>, in contact
+with the object <i>a</i>, which makes it feel the impression <i>A&acute;</i>. I make
+the movements corresponding to the series <i>S</i>; I have said that
+this series should be suitably chosen, I should so make this
+choice that these movements carry the finger <i>D</i> to the point
+originally occupied by the finger <i>D&acute;</i>, that is, to the point <i>M</i>; this
+finger <i>D</i> will thus be in contact with the object <i>a</i>, which will
+make it feel the impression <i>A</i>.</p>
+
+<p>I then make the movements corresponding to the series &#963;; in
+these movements, by hypothesis, the position of the finger <i>D</i> does
+not change, this finger therefore remains in contact with the object
+a and continues to feel the impression <i>A</i>. Finally I make
+the movements corresponding to the series <i>S&acute;</i>. As <i>S&acute;</i> is inverse
+to <i>S</i>, these movements carry the finger <i>D&acute;</i> to the point previously
+occupied by the finger <i>D</i>, that is, to the point <i>M</i>. If, as may be
+supposed, the object <i>a</i> has not budged, this finger <i>D&acute;</i> will be
+in contact with this object and will feel anew the impression
+<i>A&acute;</i>.... <i>Q.E.D.</i></p>
+
+<p>Let us see the consequences. I consider a series of muscular
+sensations &#931;. To this series will correspond a point <i>M</i> of the
+first tactile space. Now take again the two series <i>S</i> and <i>S&acute;</i>, inverses
+of one another, of which we have just spoken. To the
+series <i>S</i> + &#931; + <i>S&acute;</i> will correspond a point <i>N</i> of the second tactile
+space, since to any series of muscular sensations corresponds,<span class='pagenum'><a name="Page_270" id="Page_270">[Pg 270]</a></span>
+as we have said, a point, whether in the first space or in the
+second.</p>
+
+<p>I am going to consider the two points <i>N</i> and <i>M</i>, thus defined,
+as corresponding. What authorizes me so to do? For this
+correspondence to be admissible, it is necessary that if two points
+<i>M</i> and <i>M&acute;</i>, corresponding in the first space to two series &#931; and &#931;&acute;,
+are identical, so also are the two corresponding points of the
+second space <i>N</i> and <i>N&acute;</i>, that is, the two points which correspond
+to the two series <i>S</i> + &#931; + <i>S&acute;</i> and <i>S</i> + &#931;&acute; + <i>S&acute;</i>. Now we shall see
+that this condition is fulfilled.</p>
+
+<p>First a remark. As <i>S</i> and <i>S&acute;</i> are inverses of one another, we
+shall have <i>S</i> + <i>S&acute;</i> = 0, and consequently <i>S</i> + <i>S&acute;</i> + &#931; = &#931; + <i>S</i> +
+<i>S&acute;</i> = &#931;, or again &#931; + <i>S</i> + <i>S&acute;</i> + &#931;&acute; = &#931; + &#931;&acute;; but it does not follow
+that we have <i>S</i> + &#931; + <i>S&acute;</i> = &#931;; because, though we have used
+the addition sign to represent the succession of our sensations,
+it is clear that the order of this succession is not indifferent:
+we can not, therefore, as in ordinary addition, invert the order
+of the terms; to use abridged language, our operations are associative,
+but not commutative.</p>
+
+<p>That fixed, in order that &#931; and &#931;&acute; should correspond to the
+same point <i>M</i> = <i>M&acute;</i> of the first space, it is necessary and sufficient
+for us to have &#931;&acute; = &#931; + &#963;. We shall then have: <i>S</i> + &#931;&acute; +
+<i>S&acute;</i> = <i>S</i> + &#931; + &#963; + <i>S&acute;</i> = <i>S</i> + &#931; + <i>S&acute;</i> + <i>S</i> + &#963; + <i>S&acute;</i>.</p>
+
+<p>But we have just ascertained that <i>S</i> + &#963; + <i>S&acute;</i> was one of the
+series &#963;&acute;. We shall therefore have: <i>S</i> + &#931;&acute; + <i>S&acute;</i> = <i>S</i> + &#931; + <i>S&acute;</i> + &#963;&acute;,
+which means that the series <i>S</i> + &#931;&acute; + <i>S&acute;</i> and <i>S</i> + &#931; + <i>S&acute;</i> correspond
+to the same point <i>N</i> = <i>N&acute;</i> of the second space. Q.E.D.</p>
+
+<p>Our two spaces therefore correspond point for point; they can
+be 'transformed' one into the other; they are isomorphic. How
+are we led to conclude thence that they are identical?</p>
+
+<p>Consider the two series &#963; and <i>S</i> + &#963; + <i>S&acute;</i> = &#963;&acute;. I have said that
+often, but not always, the series &#963; preserves the tactile impression
+<i>A</i> felt by the finger <i>D</i>; and similarly it often happens, but
+not always, that the series &#963;&acute; preserves the tactile impression <i>A&acute;</i>
+felt by the finger <i>D&acute;</i>. Now I ascertain that it happens <i>very often</i>
+(that is, much more often than what I have just called 'often')
+that when the series &#963; has preserved the impression <i>A</i> of the<span class='pagenum'><a name="Page_271" id="Page_271">[Pg 271]</a></span>
+finger <i>D</i>, the series &#963;&acute; preserves at the same time the impression
+<i>A&acute;</i> of the finger <i>D&acute;</i>; and, inversely, that if the first impression is
+altered, the second is likewise. That happens <i>very often</i>, but not
+always.</p>
+
+<p>We interpret this experimental fact by saying that the unknown
+object <i>a</i> which gives the impression <i>A</i> to the finger <i>D</i> is
+identical with the unknown object <i>a&acute;</i> which gives the impression
+<i>A&acute;</i> to the finger <i>D&acute;</i>. And in fact when the first object moves,
+which the disappearance of the impression <i>A</i> tells us, the second
+likewise moves, since the impression <i>A&acute;</i> disappears likewise.
+When the first object remains motionless, the second remains
+motionless. If these two objects are identical, as the first is at
+the point <i>M</i> of the first space and the second at the point <i>N</i>
+of the second space, these two points are identical. This is how
+we are led to regard these two spaces as identical; or better, this
+is what we mean when we say that they are identical.</p>
+
+<p>What we have just said of the identity of the two tactile
+spaces makes unnecessary our discussing the question of the
+identity of tactile space and visual space, which could be treated
+in the same way.</p>
+
+
+<h4>5. <i>Space and Empiricism</i></h4>
+
+<p>It seems that I am about to be led to conclusions in conformity
+with empiristic ideas. I have, in fact, sought to put in evidence
+the r&ocirc;le of experience and to analyze the experimental facts
+which intervene in the genesis of space of three dimensions. But
+whatever may be the importance of these facts, there is one thing
+we must not forget and to which besides I have more than once
+called attention. These experimental facts are often verified
+but not always. That evidently does not mean that space has
+often three dimensions, but not always.</p>
+
+<p>I know well that it is easy to save oneself and that, if the
+facts do not verify, it will be easily explained by saying that
+the exterior objects have moved. If experience succeeds, we say
+that it teaches us about space; if it does not succeed, we hie to
+exterior objects which we accuse of having moved; in other
+words, if it does not succeed, it is given a fillip.</p>
+
+<p>These fillips are legitimate; I do not refuse to admit them; but<span class='pagenum'><a name="Page_272" id="Page_272">[Pg 272]</a></span>
+they suffice to tell us that the properties of space are not experimental
+truths, properly so called. If we had wished to verify
+other laws, we could have succeeded also, by giving other analogous
+fillips. Should we not always have been able to justify
+these fillips by the same reasons? One could at most have said to
+us: 'Your fillips are doubtless legitimate, but you abuse them;
+why move the exterior objects so often?'</p>
+
+<p>To sum up, experience does not prove to us that space has
+three dimensions; it only proves to us that it is convenient to attribute
+three to it, because thus the number of fillips is reduced
+to a minimum.</p>
+
+<p>I will add that experience brings us into contact only with
+representative space, which is a physical continuum, never with
+geometric space, which is a mathematical continuum. At the
+very most it would appear to tell us that it is convenient to give
+to geometric space three dimensions, so that it may have as
+many as representative space.</p>
+
+<p>The empiric question may be put under another form. Is it
+impossible to conceive physical phenomena, the mechanical phenomena,
+for example, otherwise than in space of three dimensions?
+We should thus have an objective experimental proof,
+so to speak, independent of our physiology, of our modes of
+representation.</p>
+
+<p>But it is not so; I shall not here discuss the question completely,
+I shall confine myself to recalling the striking example
+given us by the mechanics of Hertz. You know that the great
+physicist did not believe in the existence of forces, properly so
+called; he supposed that visible material points are subjected to
+certain invisible bonds which join them to other invisible points
+and that it is the effect of these invisible bonds that we attribute
+to forces.</p>
+
+<p>But that is only a part of his ideas. Suppose a system formed
+of <i>n</i> material points, visible or not; that will give in all 3<i>n</i> coordinates;
+let us regard them as the coordinates of a <i>single</i> point
+in space of 3<i>n</i> dimensions. This single point would be constrained
+to remain upon a surface (of any number of dimensions
+&lt; 3<i>n</i>) in virtue of the bonds of which we have just spoken; to
+go on this surface from one point to another, it would always<span class='pagenum'><a name="Page_273" id="Page_273">[Pg 273]</a></span>
+take the shortest way; this would be the single principle which
+would sum up all mechanics.</p>
+
+<p>Whatever should be thought of this hypothesis, whether we be
+allured by its simplicity, or repelled by its artificial character,
+the simple fact that Hertz was able to conceive it, and to regard
+it as more convenient than our habitual hypotheses, suffices to
+prove that our ordinary ideas, and, in particular, the three dimensions
+of space, are in no wise imposed upon mechanics with
+an invincible force.</p>
+
+
+<h4>6. <i>Mind and Space</i></h4>
+
+<p>Experience, therefore, has played only a single r&ocirc;le, it has
+served as occasion. But this r&ocirc;le was none the less very important;
+and I have thought it necessary to give it prominence.
+This r&ocirc;le would have been useless if there existed an <i>a priori</i>
+form imposing itself upon our sensitivity, and which was space
+of three dimensions.</p>
+
+<p>Does this form exist, or, if you choose, can we represent to ourselves
+space of more than three dimensions? And first what does
+this question mean? In the true sense of the word, it is clear
+that we can not represent to ourselves space of four, nor space
+of three, dimensions; we can not first represent them to ourselves
+empty, and no more can we represent to ourselves an object
+either in space of four, or in space of three, dimensions: (1)
+Because these spaces are both infinite and we can not represent
+to ourselves a figure <i>in</i> space, that is, the part <i>in</i> the whole, without
+representing the whole, and that is impossible, because it is
+infinite; (2) because these spaces are both mathematical continua,
+and we can represent to ourselves only the physical continuum;
+(3) because these spaces are both homogeneous, and
+the frames in which we enclose our sensations, being limited, can
+not be homogeneous.</p>
+
+<p>Thus the question put can only be understood in one way;
+is it possible to imagine that, the results of the experiences
+related above having been different, we might have been led to
+attribute to space more than three dimensions; to imagine, for
+instance, that the sensation of accommodation might not be constantly
+in accord with the sensation of convergence of the eyes;<span class='pagenum'><a name="Page_274" id="Page_274">[Pg 274]</a></span>
+or indeed that the experiences of which we have spoken in &sect; 2,
+and of which we express the result by saying 'that touch does
+not operate at a distance,' might have led us to an inverse conclusion.</p>
+
+<p>And then yes evidently that is possible; from the moment one
+imagines an experience, one imagines just thereby the two contrary
+results it may give. That is possible, but that is difficult,
+because we have to overcome a multitude of associations of
+ideas, which are the fruit of a long personal experience and of
+the still longer experience of the race. Is it these associations
+(or at least those of them that we have inherited from our ancestors),
+which constitute this <i>a priori</i> form of which it is said
+that we have pure intuition? Then I do not see why one should
+declare it refractory to analysis and should deny me the right
+of investigating its origin.</p>
+
+<p>When it is said that our sensations are 'extended' only one
+thing can be meant, that is that they are always associated with
+the idea of certain muscular sensations, corresponding to the
+movements which enable us to reach the object which causes
+them, which enable us, in other words, to defend ourselves against
+it. And it is just because this association is useful for the defense
+of the organism, that it is so old in the history of the species
+and that it seems to us indestructible. Nevertheless, it is only
+an association and we can conceive that it may be broken; so
+that we may not say that sensation can not enter consciousness
+without entering in space, but that in fact it does not enter consciousness
+without entering in space, which means, without being
+entangled in this association.</p>
+
+<p>No more can I understand one's saying that the idea of time
+is logically subsequent to space, since we can represent it to ourselves
+only under the form of a straight line; as well say that
+time is logically subsequent to the cultivation of the prairies,
+since it is usually represented armed with a scythe. That one
+can not represent to himself simultaneously the different parts of
+time, goes without saying, since the essential character of these
+parts is precisely not to be simultaneous. That does not mean
+that we have not the intuition of time. So far as that goes, no
+more should we have that of space, because neither can we<span class='pagenum'><a name="Page_275" id="Page_275">[Pg 275]</a></span>
+represent it, in the proper sense of the word, for the reasons I have
+mentioned. What we represent to ourselves under the name of
+straight is a crude image which as ill resembles the geometric
+straight as it does time itself.</p>
+
+<p>Why has it been said that every attempt to give a fourth dimension
+to space always carries this one back to one of the other
+three? It is easy to understand. Consider our muscular sensations
+and the 'series' they may form. In consequence of numerous
+experiences, the ideas of these series are associated together
+in a very complex woof, our series are <i>classed</i>. Allow
+me, for convenience of language, to express my thought in a
+way altogether crude and even inexact by saying that our series
+of muscular sensations are classed in three classes corresponding
+to the three dimensions of space. Of course this classification
+is much more complicated than that, but that will suffice
+to make my reasoning understood. If I wish to imagine a fourth
+dimension, I shall suppose another series of muscular sensations,
+making part of a fourth class. But as <i>all</i> my muscular sensations
+have already been classed in one of the three pre-existent
+classes, I can only represent to myself a series belonging to one
+of these three classes, so that my fourth dimension is carried
+back to one of the other three.</p>
+
+<p>What does that prove? This: that it would have been necessary
+first to destroy the old classification and replace it by a new
+one in which the series of muscular sensations should have been
+distributed into four classes. The difficulty would have disappeared.</p>
+
+<p>It is presented sometimes under a more striking form. Suppose
+I am enclosed in a chamber between the six impassable
+boundaries formed by the four walls, the floor and the ceiling;
+it will be impossible for me to get out and to imagine my getting
+out. Pardon, can you not imagine that the door opens, or that
+two of these walls separate? But of course, you answer, one
+must suppose that these walls remain immovable. Yes, but it is
+evident that I have the right to move; and then the walls that we
+suppose absolutely at rest will be in motion with regard to me.
+Yes, but such a relative motion can not be arbitrary; when objects
+are at rest, their relative motion with regard to any axes<span class='pagenum'><a name="Page_276" id="Page_276">[Pg 276]</a></span>
+is that of a rigid solid; now, the apparent motions that you
+imagine are not in conformity with the laws of motion of a rigid
+solid. Yes, but it is experience which has taught us the laws
+of motion of a rigid solid; nothing would prevent our <i>imagining</i>
+them different. To sum up, for me to imagine that I get out of
+my prison, I have only to imagine that the walls seem to open,
+when I move.</p>
+
+<p>I believe, therefore, that if by space is understood a mathematical
+continuum of three dimensions, were it otherwise amorphous,
+it is the mind which constructs it, but it does not construct it out
+of nothing; it needs materials and models. These materials,
+like these models, preexist within it. But there is not a single
+model which is imposed upon it; it has <i>choice</i>; it may choose,
+for instance, between space of four and space of three dimensions.
+What then is the r&ocirc;le of experience? It gives the indications
+following which the choice is made.</p>
+
+<p>Another thing: whence does space get its quantitative character?
+It comes from the r&ocirc;le which the series of muscular sensations
+play in its genesis. These are series which may <i>repeat
+themselves</i>, and it is from their repetition that number comes; it
+is because they can repeat themselves indefinitely that space is
+infinite. And finally we have seen, at the end of section 3, that
+it is also because of this that space is relative. So it is repetition
+which has given to space its essential characteristics; now,
+repetition supposes time; this is enough to tell that time is
+logically anterior to space.</p>
+
+<h4>7. <i>R&ocirc;le of the Semicircular Canals</i></h4>
+
+<p>I have not hitherto spoken of the r&ocirc;le of certain organs to
+which the physiologists attribute with reason a capital importance,
+I mean the semicircular canals. Numerous experiments
+have sufficiently shown that these canals are necessary to our
+sense of orientation; but the physiologists are not entirely in
+accord; two opposing theories have been proposed, that of Mach-Delage
+and that of M. de Cyon.</p>
+
+<p>M. de Cyon is a physiologist who has made his name illustrious
+by important discoveries on the innervation of the heart; I can
+not, however, agree with his ideas on the question before us. Not<span class='pagenum'><a name="Page_277" id="Page_277">[Pg 277]</a></span>
+being a physiologist, I hesitate to criticize the experiments he has
+directed against the adverse theory of Mach-Delage; it seems
+to me, however, that they are not convincing, because in many
+of them the <i>total</i> pressure was made to vary in one of the canals,
+while, physiologically, what varies is the <i>difference</i> between the
+pressures on the two extremities of the canal; in others the
+organs were subjected to profound lesions, which must alter their
+functions.</p>
+
+<p>Besides, this is not important; the experiments, if they were
+irreproachable, might be convincing against the old theory. They
+would not be convincing <i>for</i> the new theory. In fact, if I have
+rightly understood the theory, my explaining it will be enough
+for one to understand that it is impossible to conceive of an experiment
+confirming it.</p>
+
+<p>The three pairs of canals would have as sole function to tell us
+that space has three dimensions. Japanese mice have only two
+pairs of canals; they believe, it would seem, that space has only
+two dimensions, and they manifest this opinion in the strangest
+way; they put themselves in a circle, and, so ordered, they spin
+rapidly around. The lampreys, having only one pair of canals,
+believe that space has only one dimension, but their manifestations
+are less turbulent.</p>
+
+<p>It is evident that such a theory is inadmissible. The sense-organs
+are designed to tell us of <i>changes</i> which happen in the
+exterior world. We could not understand why the Creator should
+have given us organs destined to cry without cease: Remember
+that space has three dimensions, since the number of these three
+dimensions is not subject to change.</p>
+
+<p>We must, therefore, come back to the theory of Mach-Delage.
+What the nerves of the canals can tell us is the difference of pressure
+on the two extremities of the same canal, and thereby: (1)
+the direction of the vertical with regard to three axes rigidly
+bound to the head; (2) the three components of the acceleration
+of translation of the center of gravity of the head; (3) the centrifugal
+forces developed by the rotation of the head; (4) the
+acceleration of the motion of rotation of the head.</p>
+
+<p>It follows from the experiments of M. Delage that it is this
+last indication which is much the most important; doubtless<span class='pagenum'><a name="Page_278" id="Page_278">[Pg 278]</a></span>
+because the nerves are less sensible to the difference of pressure
+itself than to the brusque variations of this difference. The first
+three indications may thus be neglected.</p>
+
+<p>Knowing the acceleration of the motion of rotation of the head
+at each instant, we deduce from it, by an unconscious integration,
+the final orientation of the head, referred to a certain initial
+orientation taken as origin. The circular canals contribute, therefore,
+to inform us of the movements that we have executed, and
+that on the same ground as the muscular sensations. When,
+therefore, above we speak of the series <i>S</i> or of the series &#931;, we
+should say, not that these were series of muscular sensations
+alone, but that they were series at the same time of muscular
+sensations and of sensations due to the semicircular canals.
+Apart from this addition, we should have nothing to change in
+what precedes.</p>
+
+<p>In the series <i>S</i> and &#931;, these sensations of the semicircular
+canals evidently hold a very important place. Yet alone they
+would not suffice, because they can tell us only of the movements
+of the head; they tell us nothing of the relative movements of the
+body or of the members in regard to the head. And more, it
+seems that they tell us only of the rotations of the head and not
+of the translations it may undergo.</p>
+
+
+
+<hr style="width: 65%;" />
+<p><span class='pagenum'><a name="Page_279" id="Page_279">[Pg 279]</a></span></p>
+<h2><b>PART II<br />
+
+<br />
+
+<small>THE PHYSICAL SCIENCES</small></b></h2>
+
+
+
+<hr style="width: 65%;" />
+<h3>CHAPTER V</h3>
+
+<h3><span class="smcap">Analysis and Physics</span></h3>
+
+<h4>I</h4>
+
+
+<p>You have doubtless often been asked of what good is mathematics
+and whether these delicate constructions entirely mind-made
+are not artificial and born of our caprice.</p>
+
+<p>Among those who put this question I should make a distinction;
+practical people ask of us only the means of money-making.
+These merit no reply; rather would it be proper to ask of them
+what is the good of accumulating so much wealth and whether,
+to get time to acquire it, we are to neglect art and science, which
+alone give us souls capable of enjoying it, 'and for life's sake to
+sacrifice all reasons for living.'</p>
+
+<p>Besides, a science made solely in view of applications is impossible;
+truths are fecund only if bound together. If we devote
+ourselves solely to those truths whence we expect an immediate
+result, the intermediary links are wanting and there will no
+longer be a chain.</p>
+
+<p>The men most disdainful of theory get from it, without suspecting
+it, their daily bread; deprived of this food, progress
+would quickly cease, and we should soon congeal into the immobility
+of old China.</p>
+
+<p>But enough of uncompromising practicians! Besides these,
+there are those who are only interested in nature and who ask us
+if we can enable them to know it better.</p>
+
+<p>To answer these, we have only to show them the two monuments
+already rough-hewn, Celestial Mechanics and Mathematical
+Physics.<span class='pagenum'><a name="Page_280" id="Page_280">[Pg 280]</a></span></p>
+
+<p>They would doubtless concede that these structures are well
+worth the trouble they have cost us. But this is not enough.
+Mathematics has a triple aim. It must furnish an instrument
+for the study of nature. But that is not all: it has a philosophic
+aim and, I dare maintain, an esthetic aim. It must aid the
+philosopher to fathom the notions of number, of space, of time.
+And above all, its adepts find therein delights analogous to those
+given by painting and music. They admire the delicate harmony
+of numbers and forms; they marvel when a new discovery opens
+to them an unexpected perspective; and has not the joy they thus
+feel the esthetic character, even though the senses take no part
+therein? Only a privileged few are called to enjoy it fully, it is
+true, but is not this the case for all the noblest arts?</p>
+
+<p>This is why I do not hesitate to say that mathematics deserves
+to be cultivated for its own sake, and the theories inapplicable
+to physics as well as the others. Even if the physical aim and
+the esthetic aim were not united, we ought not to sacrifice either.</p>
+
+<p>But more: these two aims are inseparable and the best means
+of attaining one is to aim at the other, or at least never to lose
+sight of it. This is what I am about to try to demonstrate in
+setting forth the nature of the relations between the pure science
+and its applications.</p>
+
+<p>The mathematician should not be for the physicist a mere purveyor
+of formulas; there should be between them a more intimate
+collaboration. Mathematical physics and pure analysis are not
+merely adjacent powers, maintaining good neighborly relations;
+they mutually interpenetrate and their spirit is the same. This
+will be better understood when I have shown what physics gets
+from mathematics and what mathematics, in return, borrows
+from physics.</p>
+
+
+<h4>II</h4>
+
+<p>The physicist can not ask of the analyst to reveal to him a new
+truth; the latter could at most only aid him to foresee it. It is a
+long time since one still dreamt of forestalling experiment, or of
+constructing the entire world on certain premature hypotheses.
+Since all those constructions in which one yet took a na&iuml;ve delight
+it is an age, to-day only their ruins remain.<span class='pagenum'><a name="Page_281" id="Page_281">[Pg 281]</a></span></p>
+
+<p>All laws are therefore deduced from experiment; but to enunciate
+them, a special language is needful; ordinary language is
+too poor, it is besides too vague, to express relations so delicate,
+so rich, and so precise.</p>
+
+<p>This therefore is one reason why the physicist can not do without
+mathematics; it furnishes him the only language he can speak.
+And a well-made language is no indifferent thing; not to go
+beyond physics, the unknown man who invented the word <i>heat</i>
+devoted many generations to error. Heat has been treated as a
+substance, simply because it was designated by a substantive, and
+it has been thought indestructible.</p>
+
+<p>On the other hand, he who invented the word <i>electricity</i> had
+the unmerited good fortune to implicitly endow physics with a
+<i>new</i> law, that of the conservation of electricity, which, by a pure
+chance, has been found exact, at least until now.</p>
+
+<p>Well, to continue the simile, the writers who embellish a language,
+who treat it as an object of art, make of it at the same time
+a more supple instrument, more apt for rendering shades of
+thought.</p>
+
+<p>We understand, then, how the analyst, who pursues a purely
+esthetic aim, helps create, just by that, a language more fit to
+satisfy the physicist.</p>
+
+<p>But this is not all: law springs from experiment, but not immediately.
+Experiment is individual, the law deduced from it is
+general; experiment is only approximate, the law is precise, or at
+least pretends to be. Experiment is made under conditions
+always complex, the enunciation of the law eliminates these complications.
+This is what is called 'correcting the systematic errors.'</p>
+
+<p>In a word, to get the law from experiment, it is necessary to
+generalize; this is a necessity imposed upon the most circumspect
+observer. But how generalize? Every particular truth
+may evidently be extended in an infinity of ways. Among these
+thousand routes opening before us, it is necessary to make a
+choice, at least provisional; in this choice, what shall guide us?</p>
+
+<p>It can only be analogy. But how vague is this word! Primitive
+man knew only crude analogies, those which strike the senses,
+those of colors or of sounds. He never would have dreamt of
+likening light to radiant heat.<span class='pagenum'><a name="Page_282" id="Page_282">[Pg 282]</a></span></p>
+
+<p>What has taught us to know the true, profound analogies, those
+the eyes do not see but reason divines?</p>
+
+<p>It is the mathematical spirit, which disdains matter to cling
+only to pure form. This it is which has taught us to give the same
+name to things differing only in material, to call by the same
+name, for instance, the multiplication of quaternions and that of
+whole numbers.</p>
+
+<p>If quaternions, of which I have just spoken, had not been so
+promptly utilized by the English physicists, many persons would
+doubtless see in them only a useless fancy, and yet, in teaching us
+to liken what appearances separate, they would have already
+rendered us more apt to penetrate the secrets of nature.</p>
+
+<p>Such are the services the physicist should expect of analysis;
+but for this science to be able to render them, it must be cultivated
+in the broadest fashion without immediate expectation of
+utility&mdash;the mathematician must have worked as artist.</p>
+
+<p>What we ask of him is to help us to see, to discern our way in
+the labyrinth which opens before us. Now, he sees best who
+stands highest. Examples abound, and I limit myself to the most
+striking.</p>
+
+<p>The first will show us how to change the language suffices to
+reveal generalizations not before suspected.</p>
+
+<p>When Newton's law has been substituted for Kepler's we still
+know only elliptic motion. Now, in so far as concerns this motion,
+the two laws differ only in form; we pass from one to the other
+by a simple differentiation. And yet from Newton's law may be
+deduced by an immediate generalization all the effects of perturbations
+and the whole of celestial mechanics. If, on the other
+hand, Kepler's enunciation had been retained, no one would ever
+have regarded the orbits of the perturbed planets, those complicated
+curves of which no one has ever written the equation, as
+the natural generalizations of the ellipse. The progress of observations
+would only have served to create belief in chaos.</p>
+
+<p>The second example is equally deserving of consideration.</p>
+
+<p>When Maxwell began his work, the laws of electro-dynamics
+admitted up to his time accounted for all the known facts. It was
+not a new experiment which came to invalidate them. But in
+looking at them under a new bias, Maxwell saw that the equations<span class='pagenum'><a name="Page_283" id="Page_283">[Pg 283]</a></span>
+became more symmetrical when a term was added, and
+besides, this term was too small to produce effects appreciable
+with the old methods.</p>
+
+<p>You know that Maxwell's <i>a priori</i> views awaited for twenty
+years an experimental confirmation; or, if you prefer, Maxwell
+was twenty years ahead of experiment. How was this triumph
+obtained?</p>
+
+<p>It was because Maxwell was profoundly steeped in the sense of
+mathematical symmetry; would he have been so, if others before
+him had not studied this symmetry for its own beauty?</p>
+
+<p>It was because Maxwell was accustomed to 'think in vectors,'
+and yet it was through the theory of imaginaries (neomonics)
+that vectors were introduced into analysis. And those who invented
+imaginaries hardly suspected the advantage which would
+be obtained from them for the study of the real world, of this the
+name given them is proof sufficient.</p>
+
+<p>In a word, Maxwell was perhaps not an able analyst, but this
+ability would have been for him only a useless and bothersome
+baggage. On the other hand, he had in the highest degree the
+intimate sense of mathematical analogies. Therefore it is that
+he made good mathematical physics.</p>
+
+<p>Maxwell's example teaches us still another thing.</p>
+
+<p>How should the equations of mathematical physics be treated?
+Should we simply deduce all the consequences and regard them
+as intangible realities? Far from it; what they should teach us
+above all is what can and what should be changed. It is thus
+that we get from them something useful.</p>
+
+<p>The third example goes to show us how we may perceive mathematical
+analogies between phenomena which have physically no
+relation either apparent or real, so that the laws of one of these
+phenomena aid us to divine those of the other.</p>
+
+<p>The very same equation, that of Laplace, is met in the theory
+of Newtonian attraction, in that of the motion of liquids, in that
+of the electric potential, in that of magnetism, in that of the
+propagation of heat and in still many others. What is the result?
+These theories seem images copied one from the other; they are
+mutually illuminating, borrowing their language from each
+other; ask electricians if they do not felicitate themselves on<span class='pagenum'><a name="Page_284" id="Page_284">[Pg 284]</a></span>
+having invented the phrase flow of force, suggested by hydrodynamics
+and the theory of heat.</p>
+
+<p>Thus mathematical analogies not only may make us foresee
+physical analogies, but besides do not cease to be useful when
+these latter fail.</p>
+
+<p>To sum up, the aim of mathematical physics is not only to
+facilitate for the physicist the numerical calculation of certain
+constants or the integration of certain differential equations. It
+is besides, it is above all, to reveal to him the hidden harmony of
+things in making him see them in a new way.</p>
+
+<p>Of all the parts of analysis, the most elevated, the purest, so
+to speak, will be the most fruitful in the hands of those who know
+how to use them.</p>
+
+
+<h4>III</h4>
+
+<p>Let us now see what analysis owes to physics.</p>
+
+<p>It would be necessary to have completely forgotten the history
+of science not to remember that the desire to understand nature
+has had on the development of mathematics the most constant
+and happiest influence.</p>
+
+<p>In the first place the physicist sets us problems whose solution
+he expects of us. But in proposing them to us, he has largely
+paid us in advance for the service we shall render him, if we
+solve them.</p>
+
+<p>If I may be allowed to continue my comparison with the fine
+arts, the pure mathematician who should forget the existence of
+the exterior world would be like a painter who knew how to harmoniously
+combine colors and forms, but who lacked models.
+His creative power would soon be exhausted.</p>
+
+<p>The combinations which numbers and symbols may form are an
+infinite multitude. In this multitude how shall we choose those
+which are worthy to fix our attention? Shall we let ourselves be
+guided solely by our caprice? This caprice, which itself would
+besides soon tire, would doubtless carry us very far apart and we
+should quickly cease to understand each other.</p>
+
+<p>But this is only the smaller side of the question. Physics will
+doubtless prevent our straying, but it will also preserve us from
+a danger much more formidable; it will prevent our ceaselessly
+going around in the same circle.<span class='pagenum'><a name="Page_285" id="Page_285">[Pg 285]</a></span></p>
+
+<p>History proves that physics has not only forced us to choose
+among problems which came in a crowd; it has imposed upon us
+such as we should without it never have dreamed of. However
+varied may be the imagination of man, nature is still a thousand
+times richer. To follow her we must take ways we have
+neglected, and these paths lead us often to summits whence we
+discover new countries. What could be more useful!</p>
+
+<p>It is with mathematical symbols as with physical realities; it is
+in comparing the different aspects of things that we are able to
+comprehend their inner harmony, which alone is beautiful and
+consequently worthy of our efforts.</p>
+
+<p>The first example I shall cite is so old we are tempted to forget
+it; it is nevertheless the most important of all.</p>
+
+<p>The sole natural object of mathematical thought is the whole
+number. It is the external world which has imposed the continuum
+upon us, which we doubtless have invented, but which it
+has forced us to invent. Without it there would be no infinitesimal
+analysis; all mathematical science would reduce itself to
+arithmetic or to the theory of substitutions.</p>
+
+<p>On the contrary, we have devoted to the study of the continuum
+almost all our time and all our strength. Who will regret
+it; who will think that this time and this strength have been
+wasted? Analysis unfolds before us infinite perspectives that
+arithmetic never suspects; it shows us at a glance a majestic
+assemblage whose array is simple and symmetric; on the contrary,
+in the theory of numbers, where reigns the unforeseen, the
+view is, so to speak, arrested at every step.</p>
+
+<p>Doubtless it will be said that outside of the whole number there
+is no rigor, and consequently no mathematical truth; that the
+whole number hides everywhere, and that we must strive to render
+transparent the screens which cloak it, even if to do so we must
+resign ourselves to interminable repetitions. Let us not be such
+purists and let us be grateful to the continuum, which, if <i>all</i>
+springs from the whole number, was alone capable of making
+<i>so much</i> proceed therefrom.</p>
+
+<p>Need I also recall that M. Hermite obtained a surprising advantage
+from the introduction of continuous variables into the
+theory of numbers? Thus the whole number's own domain is<span class='pagenum'><a name="Page_286" id="Page_286">[Pg 286]</a></span>
+itself invaded, and this invasion has established order where disorder
+reigned.</p>
+
+<p>See what we owe to the continuum and consequently to physical
+nature.</p>
+
+<p>Fourier's series is a precious instrument of which analysis
+makes continual use, it is by this means that it has been able to
+represent discontinuous functions; Fourier invented it to solve a
+problem of physics relative to the propagation of heat. If this
+problem had not come up naturally, we should never have dared
+to give discontinuity its rights; we should still long have regarded
+continuous functions as the only true functions.</p>
+
+<p>The notion of function has been thereby considerably extended
+and has received from some logician-analysts an unforeseen development.
+These analysts have thus adventured into regions
+where reigns the purest abstraction and have gone as far away
+as possible from the real world. Yet it is a problem of physics
+which has furnished them the occasion.</p>
+
+<p>After Fourier's series, other analogous series have entered the
+domain of analysis; they have entered by the same door; they
+have been imagined in view of applications.</p>
+
+<p>The theory of partial differential equations of the second
+order has an analogous history. It has been developed chiefly
+by and for physics. But it may take many forms, because such
+an equation does not suffice to determine the unknown function,
+it is necessary to adjoin to it complementary conditions which
+are called conditions at the limits; whence many different
+problems.</p>
+
+<p>If the analysts had abandoned themselves to their natural tendencies,
+they would never have known but one, that which
+Madame Kovalevski has treated in her celebrated memoir. But
+there are a multitude of others which they would have ignored.
+Each of the theories of physics, that of electricity, that of heat,
+presents us these equations under a new aspect. It may, therefore,
+be said that without these theories we should not know
+partial differential equations.</p>
+
+<p>It is needless to multiply examples. I have given enough to
+be able to conclude: when physicists ask of us the solution of a
+problem, it is not a duty-service they impose upon us, it is on
+the contrary we who owe them thanks.</p>
+<p><span class='pagenum'><a name="Page_287" id="Page_287">[Pg 287]</a></span></p>
+
+<h4>IV</h4>
+
+<p>But this is not all; physics not only gives us the occasion to
+solve problems; it aids us to find the means thereto, and that in
+two ways. It makes us foresee the solution; it suggests arguments
+to us.</p>
+
+<p>I have spoken above of Laplace's equation which is met in a
+multitude of diverse physical theories. It is found again in
+geometry, in the theory of conformal representation and in pure
+analysis, in that of imaginaries.</p>
+
+<p>In this way, in the study of functions of complex variables, the
+analyst, alongside of the geometric image, which is his usual instrument,
+finds many physical images which he may make
+use of with the same success. Thanks to these images, he can
+see at a glance what pure deduction would show him only successively.
+He masses thus the separate elements of the solution,
+and by a sort of intuition divines before being able to
+demonstrate.</p>
+
+<p>To divine before demonstrating! Need I recall that thus have
+been made all the important discoveries? How many are the
+truths that physical analogies permit us to present and that we
+are not in condition to establish by rigorous reasoning!</p>
+
+<p>For example, mathematical physics introduces a great number
+of developments in series. No one doubts that these developments
+converge; but the mathematical certitude is lacking. These
+are so many conquests assured for the investigators who shall
+come after us.</p>
+
+<p>On the other hand, physics furnishes us not alone solutions;
+it furnishes us besides, in a certain measure, arguments. It will
+suffice to recall how Felix Klein, in a question relative to Riemann
+surfaces, has had recourse to the properties of electric
+currents.</p>
+
+<p>It is true, the arguments of this species are not rigorous, in
+the sense the analyst attaches to this word. And here a question
+arises: How can a demonstration not sufficiently rigorous for
+the analyst suffice for the physicist? It seems there can not be
+two rigors, that rigor is or is not, and that, where it is not there
+can not be deduction.</p>
+
+<p>This apparent paradox will be better understood by recalling<span class='pagenum'><a name="Page_288" id="Page_288">[Pg 288]</a></span>
+under what conditions number is applied to natural phenomena.
+Whence come in general the difficulties encountered in seeking
+rigor? We strike them almost always in seeking to establish
+that some quantity tends to some limit, or that some function is
+continuous, or that it has a derivative.</p>
+
+<p>Now the numbers the physicist measures by experiment are
+never known except approximately; and besides, any function
+always differs as little as you choose from a discontinuous function,
+and at the same time it differs as little as you choose from
+a continuous function. The physicist may, therefore, at will
+suppose that the function studied is continuous, or that it is discontinuous;
+that it has or has not a derivative; and may do so
+without fear of ever being contradicted, either by present experience
+or by any future experiment. We see that with such
+liberty he makes sport of difficulties which stop the analyst. He
+may always reason as if all the functions which occur in his
+calculations were entire polynomials.</p>
+
+<p>Thus the sketch which suffices for physics is not the deduction
+which analysis requires. It does not follow thence that one
+can not aid in finding the other. So many physical sketches have
+already been transformed into rigorous demonstrations that
+to-day this transformation is easy. There would be plenty of
+examples did I not fear in citing them to tire the reader.</p>
+
+<p>I hope I have said enough to show that pure analysis and
+mathematical physics may serve one another without making any
+sacrifice one to the other, and that each of these two sciences
+should rejoice in all which elevates its associate.</p>
+
+
+
+<hr style="width: 65%;" />
+<p><span class='pagenum'><a name="Page_289" id="Page_289">[Pg 289]</a></span></p>
+<h3>CHAPTER VI</h3>
+
+<h3><span class="smcap">Astronomy</span></h3>
+
+
+<p>Governments and parliaments must find that astronomy is one
+of the sciences which cost most dear: the least instrument costs
+hundreds of thousands of dollars, the least observatory costs
+millions; each eclipse carries with it supplementary appropriations.
+And all that for stars which are so far away, which are
+complete strangers to our electoral contests, and in all probability
+will never take any part in them. It must be that our politicians
+have retained a remnant of idealism, a vague instinct for
+what is grand; truly, I think they have been calumniated; they
+should be encouraged and shown that this instinct does not deceive
+them, that they are not dupes of that idealism.</p>
+
+<p>We might indeed speak to them of navigation, of which no
+one can underestimate the importance, and which has need of
+astronomy. But this would be to take the question by its
+smaller side.</p>
+
+<p>Astronomy is useful because it raises us above ourselves; it is
+useful because it is grand; that is what we should say. It shows
+us how small is man's body, how great his mind, since his intelligence
+can embrace the whole of this dazzling immensity, where
+his body is only an obscure point, and enjoy its silent harmony.
+Thus we attain the consciousness of our power, and this is something
+which can not cost too dear, since this consciousness makes
+us mightier.</p>
+
+<p>But what I should wish before all to show is, to what point
+astronomy has facilitated the work of the other sciences, more
+directly useful, since it has given us a soul capable of comprehending
+nature.</p>
+
+<p>Think how diminished humanity would be if, under heavens
+constantly overclouded, as Jupiter's must be, it had forever
+remained ignorant of the stars. Do you think that in such a
+world we should be what we are? I know well that under this
+somber vault we should have been deprived of the light of the<span class='pagenum'><a name="Page_290" id="Page_290">[Pg 290]</a></span>
+sun, necessary to organisms like those which inhabit the earth.
+But if you please, we shall assume that these clouds are phosphorescent
+and emit a soft and constant light. Since we are
+making hypotheses, another will cost no more. Well! I repeat
+my question: Do you think that in such a world we should be
+what we are?</p>
+
+<p>The stars send us not only that visible and gross light which
+strikes our bodily eyes, but from them also comes to us a light far
+more subtle, which illuminates our minds and whose effects I
+shall try to show you. You know what man was on the earth
+some thousands of years ago, and what he is to-day. Isolated
+amidst a nature where everything was a mystery to him, terrified
+at each unexpected manifestation of incomprehensible forces, he
+was incapable of seeing in the conduct of the universe anything
+but caprice; he attributed all phenomena to the action of a multitude
+of little genii, fantastic and exacting, and to act on the
+world he sought to conciliate them by means analogous to those
+employed to gain the good graces of a minister or a deputy.
+Even his failures did not enlighten him, any more than to-day
+a beggar refused is discouraged to the point of ceasing to beg.</p>
+
+<p>To-day we no longer beg of nature; we command her, because
+we have discovered certain of her secrets and shall discover
+others each day. We command her in the name of laws she can
+not challenge, because they are hers; these laws we do not madly
+ask her to change, we are the first to submit to them. Nature
+can only be governed by obeying her.</p>
+
+<p>What a change must our souls have undergone to pass from the
+one state to the other! Does any one believe that, without the
+lessons of the stars, under the heavens perpetually overclouded
+that I have just supposed, they would have changed so quickly?
+Would the metamorphosis have been possible, or at least would it
+not have been much slower?</p>
+
+<p>And first of all, astronomy it is which taught that there are
+laws. The Chaldeans, who were the first to observe the heavens
+with some attention, saw that this multitude of luminous points
+is not a confused crowd wandering at random, but rather a disciplined
+army. Doubtless the rules of this discipline escaped them,
+but the harmonious spectacle of the starry night sufficed to give<span class='pagenum'><a name="Page_291" id="Page_291">[Pg 291]</a></span>
+them the impression of regularity, and that was in itself already
+a great thing. Besides, these rules were discerned by Hipparchus,
+Ptolemy, Copernicus, Kepler, one after another, and finally,
+it is needless to recall that Newton it was who enunciated the
+oldest, the most precise, the most simple, the most general of all
+natural laws.</p>
+
+<p>And then, taught by this example, we have seen our little terrestrial
+world better and, under the apparent disorder, there also
+we have found again the harmony that the study of the heavens
+had revealed to us. It also is regular, it also obeys immutable
+laws, but they are more complicated, in apparent conflict one with
+another, and an eye untrained by other sights would have seen
+there only chaos and the reign of chance or caprice. If we had
+not known the stars, some bold spirits might perhaps have
+sought to foresee physical phenomena; but their failures would
+have been frequent, and they would have excited only the derision
+of the vulgar; do we not see, that even in our day the
+meteorologists sometimes deceive themselves, and that certain
+persons are inclined to laugh at them.</p>
+
+<p>How often would the physicists, disheartened by so many
+checks, have fallen into discouragement, if they had not had, to
+sustain their confidence, the brilliant example of the success of
+the astronomers! This success showed them that nature obeys
+laws; it only remained to know what laws; for that they only
+needed patience, and they had the right to demand that the
+sceptics should give them credit.</p>
+
+<p>This is not all: astronomy has not only taught us that there are
+laws, but that from these laws there is no escape, that with them
+there is no possible compromise. How much time should we have
+needed to comprehend that fact, if we had known only the terrestrial
+world, where each elemental force would always seem to
+us in conflict with other forces? Astronomy has taught us that
+the laws are infinitely precise, and that if those we enunciate
+are approximative, it is because we do not know them well. Aristotle,
+the most scientific mind of antiquity, still accorded a part
+to accident, to chance, and seemed to think that the laws of nature,
+at least here below, determine only the large features of
+phenomena. How much has the ever-increasing precision of<span class='pagenum'><a name="Page_292" id="Page_292">[Pg 292]</a></span>
+astronomical predictions contributed to correct such an error,
+which would have rendered nature unintelligible!</p>
+
+<p>But are these laws not local, varying in different places, like
+those which men make; does not that which is truth in one corner
+of the universe, on our globe, for instance, or in our little solar
+system, become error a little farther away? And then could it
+not be asked whether laws depending on space do not also depend
+upon time, whether they are not simple habitudes, transitory,
+therefore, and ephemeral? Again it is astronomy that answers
+this question. Consider the double stars; all describe conics;
+thus, as far as the telescope carries, it does not reach the limits
+of the domain which obeys Newton's law.</p>
+
+<p>Even the simplicity of this law is a lesson for us; how many
+complicated phenomena are contained in the two lines of its
+enunciation; persons who do not understand celestial mechanics
+may form some idea of it at least from the size of the treatises
+devoted to this science; and then it may be hoped that the complication
+of physical phenomena likewise hides from us some
+simple cause still unknown.</p>
+
+<p>It is therefore astronomy which has shown us what are the
+general characteristics of natural laws; but among these characteristics
+there is one, the most subtle and the most important of
+all, which I shall ask leave to stress.</p>
+
+<p>How was the order of the universe understood by the
+ancients; for instance, by Pythagoras, Plato or Aristotle? It
+was either an immutable type fixed once for all, or an ideal to
+which the world sought to approach. Kepler himself still
+thought thus when, for instance, he sought whether the distances
+of the planets from the sun had not some relation to the five regular
+polyhedrons. This idea contained nothing absurd, but it
+was sterile, since nature is not so made. Newton has shown us
+that a law is only a necessary relation between the present state
+of the world and its immediately subsequent state. All the
+other laws since discovered are nothing else; they are in sum,
+differential equations; but it is astronomy which furnished the
+first model for them, without which we should doubtless long
+have erred.</p>
+
+<p>Astronomy has also taught us to set at naught appearances.<span class='pagenum'><a name="Page_293" id="Page_293">[Pg 293]</a></span>
+The day Copernicus proved that what was thought the most stable
+was in motion, that what was thought moving was fixed, he
+showed us how deceptive could be the infantile reasonings which
+spring directly from the immediate data of our senses. True,
+his ideas did not easily triumph, but since this triumph there is
+no longer a prejudice so inveterate that we can not shake it off.
+How can we estimate the value of the new weapon thus won?</p>
+
+<p>The ancients thought everything was made for man, and this
+illusion must be very tenacious, since it must ever be combated.
+Yet it is necessary to divest oneself of it; or else one will be only
+an eternal myope, incapable of seeing the truth. To comprehend
+nature one must be able to get out of self, so to speak, and to
+contemplate her from many different points of view; otherwise
+we never shall know more than one side. Now, to get out of
+self is what he who refers everything to himself can not do. Who
+delivered us from this illusion? It was those who showed us that
+the earth is only one of the smallest planets of the solar system,
+and that the solar system itself is only an imperceptible point
+in the infinite spaces of the stellar universe.</p>
+
+<p>At the same time astronomy taught us not to be afraid of big
+numbers. This was needful, not only for knowing the heavens,
+but to know the earth itself; and was not so easy as it seems to
+us to-day. Let us try to go back and picture to ourselves what a
+Greek would have thought if told that red light vibrates four
+hundred millions of millions of times per second. Without any
+doubt, such an assertion would have appeared to him pure madness,
+and he never would have lowered himself to test it. To-day
+a hypothesis will no longer appear absurd to us because it
+obliges us to imagine objects much larger or smaller than those
+our senses are capable of showing us, and we no longer comprehend
+those scruples which arrested our predecessors and prevented
+them from discovering certain truths simply because they
+were afraid of them. But why? It is because we have seen
+the heavens enlarging and enlarging without cease; because we
+know that the sun is 150 millions of kilometers from the earth
+and that the distances of the nearest stars are hundreds of
+thousands of times greater yet. Habituated to the contemplation
+of the infinitely great, we have become apt to comprehend<span class='pagenum'><a name="Page_294" id="Page_294">[Pg 294]</a></span>
+the infinitely small. Thanks to the education it has received,
+our imagination, like the eagle's eye that the sun does not dazzle,
+can look truth in the face.</p>
+
+<p>Was I wrong in saying that it is astronomy which has made
+us a soul capable of comprehending nature; that under heavens
+always overcast and starless, the earth itself would have been for
+us eternally unintelligible; that we should there have seen only
+caprice and disorder; and that, not knowing the world, we should
+never have been able to subdue it? What science could have
+been more useful? And in thus speaking I put myself at the
+point of view of those who only value practical applications.
+Certainly, this point of view is not mine; as for me, on the contrary,
+if I admire the conquests of industry, it is above all because
+if they free us from material cares, they will one day give
+to all the leisure to contemplate nature. I do not say: Science
+is useful, because it teaches us to construct machines. I say:
+Machines are useful, because in working for us, they will some
+day leave us more time to make science. But finally it is worth
+remarking that between the two points of view there is no antagonism,
+and that man having pursued a disinterested aim, all else
+has been added unto him.</p>
+
+<p>Auguste Comte has said somewhere, that it would be idle to
+seek to know the composition of the sun, since this knowledge
+would be of no use to sociology. How could he be so short-sighted?
+Have we not just seen that it is by astronomy that, to
+speak his language, humanity has passed from the theological to
+the positive state? He found an explanation for that because
+it had happened. But how has he not understood that what
+remained to do was not less considerable and would be not less
+profitable? Physical astronomy, which he seems to condemn,
+has already begun to bear fruit, and it will give us much more,
+for it only dates from yesterday.</p>
+
+<p>First was discovered the nature of the sun, what the founder of
+positivism wished to deny us, and there bodies were found which
+exist on the earth, but had here remained undiscovered; for example,
+helium, that gas almost as light as hydrogen. That already
+contradicted Comte. But to the spectroscope we owe a
+lesson precious in a quite different way; in the most distant stars,<span class='pagenum'><a name="Page_295" id="Page_295">[Pg 295]</a></span>
+it shows us the same substances. It might have been asked
+whether the terrestrial elements were not due to some chance
+which had brought together more tenuous atoms to construct of
+them the more complex edifice that the chemists call atom;
+whether, in other regions of the universe, other fortuitous meetings
+had not engendered edifices entirely different. Now we know
+that this is not so, that the laws of our chemistry are the general
+laws of nature, and that they owe nothing to the chance
+which caused us to be born on the earth.</p>
+
+<p>But, it will be said, astronomy has given to the other sciences
+all it can give them, and now that the heavens have procured for
+us the instruments which enable us to study terrestrial nature,
+they could without danger veil themselves forever. After what
+we have just said, is there still need to answer this objection?
+One could have reasoned the same in Ptolemy's time; then also
+men thought they knew everything, and they still had almost
+everything to learn.</p>
+
+<p>The stars are majestic laboratories, gigantic crucibles, such as
+no chemist could dream. There reign temperatures impossible
+for us to realize. Their only defect is being a little far away;
+but the telescope will soon bring them near to us, and then we
+shall see how matter acts there. What good fortune for the
+physicist and the chemist!</p>
+
+<p>Matter will there exhibit itself to us under a thousand different
+states, from those rarefied gases which seem to form the nebul&aelig;
+and which are luminous with I know not what glimmering of
+mysterious origin, even to the incandescent stars and to the
+planets so near and yet so different.</p>
+
+<p>Perchance even, the stars will some day teach us something
+about life; that seems an insensate dream and I do not at all see
+how it can be realized; but, a hundred years ago, would not the
+chemistry of the stars have also appeared a mad dream?</p>
+
+<p>But limiting our views to horizons less distant, there still will
+remain to us promises less contingent and yet sufficiently seductive.
+If the past has given us much, we may rest assured that
+the future will give us still more.</p>
+
+<p>In sum, it is incredible how useful belief in astrology has
+been to humanity. If Kepler and Tycho Brahe made a living,<span class='pagenum'><a name="Page_296" id="Page_296">[Pg 296]</a></span>
+it was because they sold to na&iuml;ve kings predictions founded on
+the conjunctions of the stars. If these princes had not been so
+credulous, we should perhaps still believe that nature obeys
+caprice, and we should still wallow in ignorance.</p>
+
+
+
+<hr style="width: 65%;" />
+<p><span class='pagenum'><a name="Page_297" id="Page_297">[Pg 297]</a></span></p>
+<h3>CHAPTER VII</h3>
+
+<h3><span class="smcap">The History of Mathematical Physics</span></h3>
+
+
+<p><i>The Past and the Future of Physics.</i>&mdash;What is the present
+state of mathematical physics? What are the problems it is led
+to set itself? What is its future? Is its orientation about to be
+modified?</p>
+
+<p>Ten years hence will the aim and the methods of this science
+appear to our immediate successors in the same light as to ourselves;
+or, on the contrary, are we about to witness a profound
+transformation? Such are the questions we are forced to raise
+in entering to-day upon our investigation.</p>
+
+<p>If it is easy to propound them: to answer is difficult. If we
+felt tempted to risk a prediction, we should easily resist this
+temptation, by thinking of all the stupidities the most eminent
+savants of a hundred years ago would have uttered, if some one
+had asked them what the science of the nineteenth century
+would be. They would have thought themselves bold in their
+predictions, and after the event, how very timid we should have
+found them. Do not, therefore, expect of me any prophecy.</p>
+
+<p>But if, like all prudent physicians, I shun giving a prognosis,
+yet I can not dispense with a little diagnostic; well, yes, there are
+indications of a serious crisis, as if we might expect an approaching
+transformation. Still, be not too anxious: we are sure the
+patient will not die of it, and we may even hope that this crisis
+will be salutary, for the history of the past seems to guarantee us
+this. This crisis, in fact, is not the first, and to understand it,
+it is important to recall those which have preceded. Pardon then
+a brief historical sketch.</p>
+
+<p><i>The Physics of Central Forces.</i>&mdash;Mathematical physics, as we
+know, was born of celestial mechanics, which gave birth to it at
+the end of the eighteenth century, at the moment when it itself
+attained its complete development. During its first years especially,
+the infant strikingly resembled its mother.<span class='pagenum'><a name="Page_298" id="Page_298">[Pg 298]</a></span></p>
+
+<p>The astronomic universe is formed of masses, very great, no
+doubt, but separated by intervals so immense that they appear
+to us only as material points. These points attract each other
+inversely as the square of the distance, and this attraction is the
+sole force which influences their movements. But if our senses
+were sufficiently keen to show us all the details of the bodies
+which the physicist studies, the spectacle thus disclosed would
+scarcely differ from the one the astronomer contemplates. There
+also we should see material points, separated from one another
+by intervals, enormous in comparison with their dimensions, and
+describing orbits according to regular laws. These infinitesimal
+stars are the atoms. Like the stars proper, they attract or repel
+each other, and this attraction or this repulsion, following the
+straight line which joins them, depends only on the distance.
+The law according to which this force varies as function of the
+distance is perhaps not the law of Newton, but it is an analogous
+law; in place of the exponent &minus;2, we have probably a different
+exponent, and it is from this change of exponent that arises all
+the diversity of physical phenomena, the variety of qualities and
+of sensations, all the world, colored and sonorous, which surrounds
+us; in a word, all nature.</p>
+
+<p>Such is the primitive conception in all its purity. It only
+remains to seek in the different cases what value should be given
+to this exponent in order to explain all the facts. It is on this
+model that Laplace, for example, constructed his beautiful theory
+of capillarity; he regards it only as a particular case of attraction,
+or, as he says, of universal gravitation, and no one is astonished
+to find it in the middle of one of the five volumes of the
+'M&eacute;canique c&eacute;leste.' More recently Briot believes he penetrated
+the final secret of optics in demonstrating that the atoms of ether
+attract each other in the inverse ratio of the sixth power of the
+distance; and Maxwell himself, does he not say somewhere that
+the atoms of gases repel each other in the inverse ratio of the
+fifth power of the distance? We have the exponent &minus;6, or &minus;5,
+in place of the exponent &minus;2, but it is always an exponent.</p>
+
+<p>Among the theories of this epoch, one alone is an exception,
+that of Fourier; in it are indeed atoms acting at a distance one
+upon the other; they mutually transmit heat, but they do not<span class='pagenum'><a name="Page_299" id="Page_299">[Pg 299]</a></span>
+attract, they never budge. From this point of view, Fourier's
+theory must have appeared to the eyes of his contemporaries, to
+those of Fourier himself, as imperfect and provisional.</p>
+
+<p>This conception was not without grandeur; it was seductive,
+and many among us have not finally renounced it; they know that
+one will attain the ultimate elements of things only by patiently
+disentangling the complicated skein that our senses give us; that
+it is necessary to advance step by step, neglecting no intermediary;
+that our fathers were wrong in wishing to skip stations;
+but they believe that when one shall have arrived at these ultimate
+elements, there again will be found the majestic simplicity
+of celestial mechanics.</p>
+
+<p>Neither has this conception been useless; it has rendered us an
+inestimable service, since it has contributed to make precise the
+fundamental notion of the physical law.</p>
+
+<p>I will explain myself; how did the ancients understand law?
+It was for them an internal harmony, static, so to say, and immutable;
+or else it was like a model that nature tried to imitate.
+For us a law is something quite different; it is a constant relation
+between the phenomenon of to-day and that of to-morrow;
+in a word, it is a differential equation.</p>
+
+<p>Behold the ideal form of physical law; well, it is Newton's law
+which first clothed it forth. If then one has acclimated this form
+in physics, it is precisely by copying as far as possible this law of
+Newton, that is by imitating celestial mechanics. This is, moreover,
+the idea I have tried to bring out in Chapter VI.</p>
+
+<p><i>The Physics of the Principles.</i>&mdash;Nevertheless, a day arrived
+when the conception of central forces no longer appeared sufficient,
+and this is the first of those crises of which I just now
+spoke.</p>
+
+<p>What was done then? The attempt to penetrate into the
+detail of the structure of the universe, to isolate the pieces of this
+vast mechanism, to analyze one by one the forces which put them
+in motion, was abandoned, and we were content to take as guides
+certain general principles the express object of which is to spare
+us this minute study. How so? Suppose we have before us any
+machine; the initial wheel work and the final wheel work alone<span class='pagenum'><a name="Page_300" id="Page_300">[Pg 300]</a></span>
+are visible, but the transmission, the intermediary machinery by
+which the movement is communicated from one to the other, is
+hidden in the interior and escapes our view; we do not know
+whether the communication is made by gearing or by belts, by
+connecting-rods or by other contrivances. Do we say that it
+is impossible for us to understand anything about this machine
+so long as we are not permitted to take it to pieces? You know
+well we do not, and that the principle of the conservation of
+energy suffices to determine for us the most interesting point.
+We easily ascertain that the final wheel turns ten times less
+quickly than the initial wheel, since these two wheels are visible;
+we are able thence to conclude that a couple applied to the one
+will be balanced by a couple ten times greater applied to the
+other. For that there is no need to penetrate the mechanism
+of this equilibrium and to know how the forces compensate each
+other in the interior of the machine; it suffices to be assured
+that this compensation can not fail to occur.</p>
+
+<p>Well, in regard to the universe, the principle of the conservation
+of energy is able to render us the same service. The universe
+is also a machine, much more complicated than all those of
+industry, of which almost all the parts are profoundly hidden
+from us; but in observing the motion of those that we can see,
+we are able, by the aid of this principle, to draw conclusions
+which remain true whatever may be the details of the invisible
+mechanism which animates them.</p>
+
+<p>The principle of the conservation of energy, or Mayer's principle,
+is certainly the most important, but it is not the only one;
+there are others from which we can derive the same advantage.
+These are:</p>
+
+<p>Carnot's principle, or the principle of the degradation of
+energy.</p>
+
+<p>Newton's principle, or the principle of the equality of action
+and reaction.</p>
+
+<p>The principle of relativity, according to which the laws of
+physical phenomena must be the same for a stationary observer
+as for an observer carried along in a uniform motion of translation;
+so that we have not and can not have any means of discerning
+whether or not we are carried along in such a motion.<span class='pagenum'><a name="Page_301" id="Page_301">[Pg 301]</a></span></p>
+
+<p>The principle of the conservation of mass, or Lavoisier's
+principle.</p>
+
+<p>I will add the principle of least action.</p>
+
+<p>The application of these five or six general principles to the
+different physical phenomena is sufficient for our learning of
+them all that we could reasonably hope to know of them. The
+most remarkable example of this new mathematical physics is,
+beyond question, Maxwell's electromagnetic theory of light.</p>
+
+<p>We know nothing as to what the ether is, how its molecules are
+disposed, whether they attract or repel each other; but we know
+that this medium transmits at the same time the optical perturbations
+and the electrical perturbations; we know that this transmission
+must take place in conformity with the general principles
+of mechanics, and that suffices us for the establishment of
+the equations of the electromagnetic field.</p>
+
+<p>These principles are results of experiments boldly generalized;
+but they seem to derive from their very generality a high degree
+of certainty. In fact, the more general they are, the more frequent
+are the opportunities to check them, and the verifications
+multiplying, taking the most varied, the most unexpected forms,
+end by no longer leaving place for doubt.</p>
+
+<p><i>Utility of the Old Physics.</i>&mdash;Such is the second phase of the
+history of mathematical physics and we have not yet emerged
+from it. Shall we say that the first has been useless? that during
+fifty years science went the wrong way, and that there is
+nothing left but to forget so many accumulated efforts that a
+vicious conception condemned in advance to failure? Not the
+least in the world. Do you think the second phase could have
+come into existence without the first? The hypothesis of central
+forces contained all the principles; it involved them as necessary
+consequences; it involved both the conservation of energy and
+that of masses, and the equality of action and reaction, and the
+law of least action, which appeared, it is true, not as experimental
+truths, but as theorems; the enunciation of which had at the
+same time something more precise and less general than under
+their present form.</p>
+
+<p>It is the mathematical physics of our fathers which has familiarized
+us little by little with these various principles; which has<span class='pagenum'><a name="Page_302" id="Page_302">[Pg 302]</a></span>
+habituated us to recognize them under the different vestments in
+which they disguise themselves. They have been compared with
+the data of experience, it has been seen how it was necessary to
+modify their enunciation to adapt them to these data; thereby
+they have been extended and consolidated. Thus they came
+to be regarded as experimental truths; the conception of central
+forces became then a useless support, or rather an embarrassment,
+since it made the principles partake of its hypothetical
+character.</p>
+
+<p>The frames then have not broken, because they are elastic; but
+they have enlarged; our fathers, who established them, did not
+labor in vain, and we recognize in the science of to-day the general
+traits of the sketch which they traced.</p>
+
+
+
+<hr style="width: 65%;" />
+<p><span class='pagenum'><a name="Page_303" id="Page_303">[Pg 303]</a></span></p>
+<h3>CHAPTER VIII</h3>
+
+<h3><span class="smcap">The Present Crisis of Mathematical Physics</span></h3>
+
+
+<p><i>The New Crisis.</i>&mdash;Are we now about to enter upon a third
+period? Are we on the eve of a second crisis? These principles
+on which we have built all, are they about to crumble away in
+their turn? This has been for some time a pertinent question.</p>
+
+<p>When I speak thus, you no doubt think of radium, that grand
+revolutionist of the present time, and in fact I shall come back
+to it presently; but there is something else. It is not alone the
+conservation of energy which is in question; all the other principles
+are equally in danger, as we shall see in passing them successively
+in review.</p>
+
+<p><i>Carnot's Principle.</i>&mdash;Let us commence with the principle of
+Carnot. This is the only one which does not present itself as an
+immediate consequence of the hypothesis of central forces; more
+than that, it seems, if not to directly contradict that hypothesis,
+at least not to be reconciled with it without a certain effort. If
+physical phenomena were due exclusively to the movements of
+atoms whose mutual attraction depended only on the distance, it
+seems that all these phenomena should be reversible; if all the
+initial velocities were reversed, these atoms, always subjected to
+the same forces, ought to go over their trajectories in the contrary
+sense, just as the earth would describe in the retrograde sense
+this same elliptic orbit which it describes in the direct sense, if
+the initial conditions of its motion had been reversed. On this
+account, if a physical phenomenon is possible, the inverse phenomenon
+should be equally so, and one should be able to reascend
+the course of time. Now, it is not so in nature, and this is precisely
+what the principle of Carnot teaches us; heat can pass
+from the warm body to the cold body; it is impossible afterward
+to make it take the inverse route and to reestablish differences
+of temperature which have been effaced. Motion can be wholly
+dissipated and transformed into heat by friction; the contrary
+transformation can never be made except partially.<span class='pagenum'><a name="Page_304" id="Page_304">[Pg 304]</a></span></p>
+
+<p>We have striven to reconcile this apparent contradiction. If
+the world tends toward uniformity, this is not because its ultimate
+parts, at first unlike, tend to become less and less different;
+it is because, shifting at random, they end by blending. For an
+eye which should distinguish all the elements, the variety would
+remain always as great; each grain of this dust preserves its
+originality and does not model itself on its neighbors; but as the
+blend becomes more and more intimate, our gross senses perceive
+only the uniformity. This is why, for example, temperatures
+tend to a level, without the possibility of going backwards.</p>
+
+<p>A drop of wine falls into a glass of water; whatever may be
+the law of the internal motion of the liquid, we shall soon see it
+colored of a uniform rosy tint, and however much from this
+moment one may shake it afterwards, the wine and the water
+do not seem capable of again separating. Here we have the
+type of the irreversible physical phenomenon: to hide a grain of
+barley in a heap of wheat, this is easy; afterwards to find it
+again and get it out, this is practically impossible. All this
+Maxwell and Boltzmann have explained; but the one who has
+seen it most clearly, in a book too little read because it is a little
+difficult to read, is Gibbs, in his `Elementary Principles of Statistical
+Mechanics.'</p>
+
+<p>For those who take this point of view, Carnot's principle is
+only an imperfect principle, a sort of concession to the infirmity
+of our senses; it is because our eyes are too gross that we do not
+distinguish the elements of the blend; it is because our hands are
+too gross that we can not force them to separate; the imaginary
+demon of Maxwell, who is able to sort the molecules one by one,
+could well constrain the world to return backward. Can it return
+of itself? That is not impossible; that is only infinitely
+improbable. The chances are that we should wait a long time
+for the concourse of circumstances which would permit a retrogradation;
+but sooner or later they will occur, after years whose
+number it would take millions of figures to write. These reservations,
+however, all remained theoretic; they were not very disquieting,
+and Carnot's principle retained all its practical value.
+But here the scene changes. The biologist, armed with his microscope,
+long ago noticed in his preparations irregular movements<span class='pagenum'><a name="Page_305" id="Page_305">[Pg 305]</a></span>
+of little particles in suspension; this is the Brownian movement.
+He first thought this was a vital phenomenon, but soon he saw
+that the inanimate bodies danced with no less ardor than the
+others; then he turned the matter over to the physicists. Unhappily,
+the physicists remained long uninterested in this question;
+one concentrates the light to illuminate the microscopic
+preparation, thought they; with light goes heat; thence inequalities
+of temperature and in the liquid interior currents which
+produce the movements referred to. It occurred to M. Gouy to
+look more closely, and he saw, or thought he saw, that this explanation
+is untenable, that the movements become brisker as the
+particles are smaller, but that they are not influenced by the
+mode of illumination. If then these movements never cease, or
+rather are reborn without cease, without borrowing anything
+from an external source of energy, what ought we to believe?
+To be sure, we should not on this account renounce our belief
+in the conservation of energy, but we see under our eyes now
+motion transformed into heat by friction, now inversely heat
+changed into motion, and that without loss since the movement
+lasts forever. This is the contrary of Carnot's principle. If
+this be so, to see the world return backward, we no longer have
+need of the infinitely keen eye of Maxwell's demon; our microscope
+suffices. Bodies too large, those, for example, which are
+a tenth of a millimeter, are hit from all sides by moving atoms,
+but they do not budge, because these shocks are very numerous
+and the law of chance makes them compensate each other; but
+the smaller particles receive too few shocks for this compensation
+to take place with certainty and are incessantly knocked about.
+And behold already one of our principles in peril.</p>
+
+<p><i>The Principle of Relativity.</i>&mdash;Let us pass to the principle of
+relativity; this not only is confirmed by daily experience, not
+only is it a necessary consequence of the hypothesis of central
+forces, but it is irresistibly imposed upon our good sense, and
+yet it also is assailed. Consider two electrified bodies; though
+they seem to us at rest, they are both carried along by the motion
+of the earth; an electric charge in motion, Rowland has
+taught us, is equivalent to a current; these two charged bodies
+are, therefore, equivalent to two parallel currents of the same<span class='pagenum'><a name="Page_306" id="Page_306">[Pg 306]</a></span>
+sense and these two currents should attract each other. In measuring
+this attraction, we shall measure the velocity of the earth;
+not its velocity in relation to the sun or the fixed stars, but its
+absolute velocity.</p>
+
+<p>I well know what will be said: It is not its absolute velocity
+that is measured, it is its velocity in relation to the ether. How
+unsatisfactory that is! Is it not evident that from the principle
+so understood we could no longer infer anything? It could no
+longer tell us anything just because it would no longer fear any
+contradiction. If we succeed in measuring anything, we shall
+always be free to say that this is not the absolute velocity, and if
+it is not the velocity in relation to the ether, it might always be
+the velocity in relation to some new unknown fluid with which
+we might fill space.</p>
+
+<p>Indeed, experiment has taken upon itself to ruin this interpretation
+of the principle of relativity; all attempts to measure the
+velocity of the earth in relation to the ether have led to negative
+results. This time experimental physics has been more
+faithful to the principle than mathematical physics; the theorists,
+to put in accord their other general views, would not have spared
+it; but experiment has been stubborn in confirming it. The
+means have been varied; finally Michelson pushed precision to
+its last limits; nothing came of it. It is precisely to explain
+this obstinacy that the mathematicians are forced to-day to employ
+all their ingenuity.</p>
+
+<p>Their task was not easy, and if Lorentz has got through it, it is
+only by accumulating hypotheses.</p>
+
+<p>The most ingenious idea was that of local time. Imagine two
+observers who wish to adjust their timepieces by optical signals;
+they exchange signals, but as they know that the transmission
+of light is not instantaneous, they are careful to cross them.
+When station B perceives the signal from station A, its clock
+should not mark the same hour as that of station A at the
+moment of sending the signal, but this hour augmented by a
+constant representing the duration of the transmission. Suppose,
+for example, that station A sends its signal when its clock
+marks the hour <i>O</i>, and that station B perceives it when its clock
+marks the hour <i>t</i>. The clocks are adjusted if the slowness equal<span class='pagenum'><a name="Page_307" id="Page_307">[Pg 307]</a></span>
+to <i>t</i> represents the duration of the transmission, and to verify
+it, station B sends in its turn a signal when its clock marks <i>O</i>;
+then station A should perceive it when its clock marks <i>t</i>. The
+timepieces are then adjusted.</p>
+
+<p>And in fact they mark the same hour at the same physical
+instant, but on the one condition, that the two stations are fixed.
+Otherwise the duration of the transmission will not be the same
+in the two senses, since the station A, for example, moves forward
+to meet the optical perturbation emanating from B, whereas
+the station B flees before the perturbation emanating from A.
+The watches adjusted in that way will not mark, therefore, the
+true time; they will mark what may be called the <i>local time</i>, so
+that one of them will be slow of the other. It matters little, since
+we have no means of perceiving it. All the phenomena which
+happen at A, for example, will be late, but all will be equally
+so, and the observer will not perceive it, since his watch is slow;
+so, as the principle of relativity requires, he will have no means
+of knowing whether he is at rest or in absolute motion.</p>
+
+<p>Unhappily, that does not suffice, and complementary hypotheses
+are necessary; it is necessary to admit that bodies in motion
+undergo a uniform contraction in the sense of the motion.
+One of the diameters of the earth, for example, is shrunk by
+one two-hundred-millionth in consequence of our planet's motion,
+while the other diameter retains its normal length. Thus the last
+little differences are compensated. And then, there is still the
+hypothesis about forces. Forces, whatever be their origin, gravity
+as well as elasticity, would be reduced in a certain proportion
+in a world animated by a uniform translation; or, rather,
+this would happen for the components perpendicular to the
+translation; the components parallel would not change. Resume,
+then, our example of two electrified bodies; these bodies
+repel each other, but at the same time if all is carried along in a
+uniform translation, they are equivalent to two parallel currents
+of the same sense which attract each other. This electrodynamic
+attraction diminishes, therefore, the electrostatic repulsion, and
+the total repulsion is feebler than if the two bodies were at rest.
+But since to measure this repulsion we must balance it by another
+force, and all these other forces are reduced in the same<span class='pagenum'><a name="Page_308" id="Page_308">[Pg 308]</a></span>
+proportion, we perceive nothing. Thus all seems arranged, but are
+all the doubts dissipated? What would happen if one could
+communicate by non-luminous signals whose velocity of propagation
+differed from that of light? If, after having adjusted
+the watches by the optical procedure, we wished to verify the
+adjustment by the aid of these new signals, we should observe
+discrepancies which would render evident the common translation
+of the two stations. And are such signals inconceivable, if
+we admit with Laplace that universal gravitation is transmitted
+a million times more rapidly than light?</p>
+
+<p>Thus, the principle of relativity has been valiantly defended
+in these latter times, but the very energy of the defense proves
+how serious was the attack.</p>
+
+<p><i>Newton's Principle.</i>&mdash;Let us speak now of the principle of
+Newton, on the equality of action and reaction. This is intimately
+bound up with the preceding, and it seems indeed that the
+fall of the one would involve that of the other. Thus we must
+not be astonished to find here the same difficulties.</p>
+
+<p>Electrical phenomena, according to the theory of Lorentz, are
+due to the displacements of little charged particles, called electrons,
+immersed in the medium we call ether. The movements
+of these electrons produce perturbations in the neighboring ether;
+these perturbations propagate themselves in every direction with
+the velocity of light, and in turn other electrons, originally at
+rest, are made to vibrate when the perturbation reaches the parts
+of the ether which touch them. The electrons, therefore, act on
+one another, but this action is not direct, it is accomplished
+through the ether as intermediary. Under these conditions can
+there be compensation between action and reaction, at least for
+an observer who should take account only of the movements
+of matter, that is, of the electrons, and who should be ignorant
+of those of the ether that he could not see? Evidently not.
+Even if the compensation should be exact, it could not be simultaneous.
+The perturbation is propagated with a finite velocity;
+it, therefore, reaches the second electron only when the first has
+long ago entered upon its rest. This second electron, therefore,
+will undergo, after a delay, the action of the first, but will certainly
+not at that moment react upon it, since around this first
+electron nothing any longer budges.<span class='pagenum'><a name="Page_309" id="Page_309">[Pg 309]</a></span></p>
+
+<p>The analysis of the facts permits us to be still more precise.
+Imagine, for example, a Hertzian oscillator, like those used in
+wireless telegraphy; it sends out energy in every direction; but
+we can provide it with a parabolic mirror, as Hertz did with his
+smallest oscillators, so as to send all the energy produced in a
+single direction. What happens then according to the theory?
+The apparatus recoils, as if it were a cannon and the projected
+energy a ball; and that is contrary to the principle of Newton,
+since our projectile here has no mass, it is not matter, it is energy.
+The case is still the same, moreover, with a beacon light provided
+with a reflector, since light is nothing but a perturbation of the
+electromagnetic field. This beacon light should recoil as if the
+light it sends out were a projectile. What is the force that
+should produce this recoil? It is what is called the Maxwell-Bartholi
+pressure. It is very minute, and it has been difficult
+to put it in evidence even with the most sensitive radiometers;
+but it suffices that it exists.</p>
+
+<p>If all the energy issuing from our oscillator falls on a receiver,
+this will act as if it had received a mechanical shock, which will
+represent in a sense the compensation of the oscillator's recoil;
+the reaction will be equal to the action, but it will not be simultaneous;
+the receiver will move on, but not at the moment when
+the oscillator recoils. If the energy propagates itself indefinitely
+without encountering a receiver, the compensation will never
+occur.</p>
+
+<p>Shall we say that the space which separates the oscillator from
+the receiver and which the perturbation must pass over in going
+from the one to the other is not void, that it is full not only of
+ether, but of air, or even in the interplanetary spaces of some
+fluid subtile but still ponderable; that this matter undergoes the
+shock like the receiver at the moment when the energy reaches
+it, and recoils in its turn when the perturbation quits it? That
+would save Newton's principle, but that is not true. If energy
+in its diffusion remained always attached to some material substratum,
+then matter in motion would carry along light with it,
+and Fizeau has demonstrated that it does nothing of the sort,
+at least for air. Michelson and Morley have since confirmed
+this. It might be supposed also that the movements of matter<span class='pagenum'><a name="Page_310" id="Page_310">[Pg 310]</a></span>
+proper are exactly compensated by those of the ether; but that
+would lead us to the same reflections as before now. The principle
+so understood will explain everything, since, whatever
+might be the visible movements, we always could imagine hypothetical
+movements which compensate them. But if it is able
+to explain everything, this is because it does not enable us to
+foresee anything; it does not enable us to decide between the
+different possible hypotheses, since it explains everything beforehand.
+It therefore becomes useless.</p>
+
+<p>And then the suppositions that it would be necessary to make
+on the movements of the ether are not very satisfactory. If the
+electric charges double, it would be natural to imagine that the
+velocities of the diverse atoms of ether double also; but, for the
+compensation, it would be necessary that the mean velocity of
+the ether quadruple.</p>
+
+<p>This is why I have long thought that these consequences of
+theory, contrary to Newton's principle, would end some day by
+being abandoned, and yet the recent experiments on the movements
+of the electrons issuing from radium seem rather to confirm
+them.</p>
+
+<p><i>Lavoisier's Principle.</i>&mdash;I arrive at the principle of Lavoisier on
+the conservation of mass. Certainly, this is one not to be
+touched without unsettling all mechanics. And now certain persons
+think that it seems true to us only because in mechanics
+merely moderate velocities are considered, but that it would cease
+to be true for bodies animated by velocities comparable to that
+of light. Now these velocities are believed at present to have
+been realized; the cathode rays and those of radium may be
+formed of very minute particles or of electrons which are displaced
+with velocities smaller no doubt than that of light, but
+which might be its one tenth or one third.</p>
+
+<p>These rays can be deflected, whether by an electric field, or
+by a magnetic field, and we are able, by comparing these deflections,
+to measure at the same time the velocity of the electrons
+and their mass (or rather the relation of their mass to their
+charge). But when it was seen that these velocities approached
+that of light, it was decided that a correction was necessary.
+These molecules, being electrified, can not be displaced without<span class='pagenum'><a name="Page_311" id="Page_311">[Pg 311]</a></span>
+agitating the ether; to put them in motion it is necessary to overcome
+a double inertia, that of the molecule itself and that of the
+ether. The total or apparent mass that one measures is composed,
+therefore, of two parts: the real or mechanical mass of
+the molecule and the electrodynamic mass representing the
+inertia of the ether.</p>
+
+<p>The calculations of Abraham and the experiments of Kaufmann
+have then shown that the mechanical mass, properly so
+called, is null, and that the mass of the electrons, or, at least, of
+the negative electrons, is of exclusively electrodynamic origin.
+This is what forces us to change the definition of mass; we can
+not any longer distinguish mechanical mass and electrodynamic
+mass, since then the first would vanish; there is no mass other
+than electrodynamic inertia. But in this case the mass can no
+longer be constant; it augments with the velocity, and it even
+depends on the direction, and a body animated by a notable
+velocity will not oppose the same inertia to the forces which tend
+to deflect it from its route, as to those which tend to accelerate
+or to retard its progress.</p>
+
+<p>There is still a resource; the ultimate elements of bodies are
+electrons, some charged negatively, the others charged positively.
+The negative electrons have no mass, this is understood; but the
+positive electrons, from the little we know of them, seem much
+greater. Perhaps they have, besides their electrodynamic mass,
+a true mechanical mass. The real mass of a body would, then,
+be the sum of the mechanical masses of its positive electrons, the
+negative electrons not counting; mass so defined might still be
+constant.</p>
+
+<p>Alas! this resource also evades us. Recall what we have said
+of the principle of relativity and of the efforts made to save it.
+And it is not merely a principle which it is a question of saving,
+it is the indubitable results of the experiments of Michelson.</p>
+
+<p>Well, as was above seen, Lorentz, to account for these results,
+was obliged to suppose that all forces, whatever their origin,
+were reduced in the same proportion in a medium animated by a
+uniform translation; this is not sufficient; it is not enough that
+this take place for the real forces, it must also be the same for
+the forces of inertia; it is therefore necessary, he says, that <i>the<span class='pagenum'><a name="Page_312" id="Page_312">[Pg 312]</a></span>
+masses of all the particles be influenced by a translation to the
+same degree as the electromagnetic masses of the electrons</i>.</p>
+
+<p>So the mechanical masses must vary in accordance with the
+same laws as the electrodynamic masses; they can not, therefore,
+be constant.</p>
+
+<p>Need I point out that the fall of Lavoisier's principle involves
+that of Newton's? This latter signifies that the center of gravity
+of an isolated system moves in a straight line; but if there is no
+longer a constant mass, there is no longer a center of gravity,
+we no longer know even what this is. This is why I said above
+that the experiments on the cathode rays appeared to justify
+the doubts of Lorentz concerning Newton's principle.</p>
+
+<p>From all these results, if they were confirmed, would arise an
+entirely new mechanics, which would be, above all, characterized
+by this fact, that no velocity could surpass that of light,<a name="FNanchor_9_9" id="FNanchor_9_9"></a><a href="#Footnote_9_9" class="fnanchor">[9]</a> any
+more than any temperature can fall below absolute zero.</p>
+
+<p>No more for an observer, carried along himself in a translation
+he does not suspect, could any apparent velocity surpass
+that of light; and this would be then a contradiction, if we did
+not recall that this observer would not use the same clocks as a
+fixed observer, but, indeed, clocks marking 'local time.'</p>
+
+<p>Here we are then facing a question I content myself with stating.
+If there is no longer any mass, what becomes of Newton's
+law? Mass has two aspects: it is at the same time a coefficient of
+inertia and an attracting mass entering as factor into Newtonian
+attraction. If the coefficient of inertia is not constant, can the
+attracting mass be? That is the question.</p>
+
+<p><i>Mayer's Principle.</i>&mdash;At least, the principle of the conservation
+of energy yet remained to us, and this seemed more solid. Shall
+I recall to you how it was in its turn thrown into discredit?
+This event has made more noise than the preceding, and it is in
+all the memoirs. From the first words of Becquerel, and, above
+all, when the Curies had discovered radium, it was seen that
+every radioactive body was an inexhaustible source of radiation.
+Its activity seemed to subsist without alteration throughout the
+months and the years. This was in itself a strain on the<span class='pagenum'><a name="Page_313" id="Page_313">[Pg 313]</a></span>
+principles; these radiations were in fact energy, and from the same
+morsel of radium this issued and forever issued. But these
+quantities of energy were too slight to be measured; at least that
+was the belief and we were not much disquieted.</p>
+
+<p>The scene changed when Curie bethought himself to put radium
+in a calorimeter; it was then seen that the quantity of heat
+incessantly created was very notable.</p>
+
+<p>The explanations proposed were numerous; but in such case
+we can not say, the more the better. In so far as no one of them
+has prevailed over the others, we can not be sure there is a good
+one among them. Since some time, however, one of these explanations
+seems to be getting the upper hand and we may reasonably
+hope that we hold the key to the mystery.</p>
+
+<p>Sir W. Ramsay has striven to show that radium is in process
+of transformation, that it contains a store of energy enormous
+but not inexhaustible. The transformation of radium then
+would produce a million times more heat than all known transformations;
+radium would wear itself out in 1,250 years; this is
+quite short, and you see that we are at least certain to have this
+point settled some hundreds of years from now. While waiting,
+our doubts remain.</p>
+
+
+
+<hr style="width: 65%;" />
+<p><span class='pagenum'><a name="Page_314" id="Page_314">[Pg 314]</a></span></p>
+<h3>CHAPTER IX</h3>
+
+<h3><span class="smcap">The Future of Mathematical Physics</span></h3>
+
+
+<p><i>The Principles and Experiment.</i>&mdash;In the midst of so much
+ruin, what remains standing? The principle of least action is
+hitherto intact, and Larmor appears to believe that it will long
+survive the others; in reality, it is still more vague and more
+general.</p>
+
+<p>In presence of this general collapse of the principles, what attitude
+will mathematical physics take? And first, before too
+much excitement, it is proper to ask if all that is really true.
+All these derogations to the principles are encountered only
+among infinitesimals; the microscope is necessary to see the
+Brownian movement; electrons are very light; radium is very
+rare, and one never has more than some milligrams of it at a
+time. And, then, it may be asked whether, besides the infinitesimal
+seen, there was not another infinitesimal unseen counterpoise
+to the first.</p>
+
+<p>So there is an interlocutory question, and, as it seems, only
+experiment can solve it. We shall, therefore, only have to hand
+over the matter to the experimenters, and, while waiting for them
+to finally decide the debate, not to preoccupy ourselves with these
+disquieting problems, and to tranquilly continue our work as if
+the principles were still uncontested. Certes, we have much to
+do without leaving the domain where they may be applied in all
+security; we have enough to employ our activity during this
+period of doubts.</p>
+
+<p><i>The R&ocirc;le of the Analyst.</i>&mdash;And as to these doubts, is it indeed
+true that we can do nothing to disembarrass science of them?
+It must indeed be said, it is not alone experimental physics that
+has given birth to them; mathematical physics has well contributed.
+It is the experimenters who have seen radium throw out
+energy, but it is the theorists who have put in evidence all the
+difficulties raised by the propagation of light across a medium in
+motion; but for these it is probable we should not have become<span class='pagenum'><a name="Page_315" id="Page_315">[Pg 315]</a></span>
+conscious of them. Well, then, if they have done their best to
+put us into this embarrassment, it is proper also that they help us
+to get out of it.</p>
+
+<p>They must subject to critical examination all these new views
+I have just outlined before you, and abandon the principles only
+after having made a loyal effort to save them. What can they
+do in this sense? That is what I will try to explain.</p>
+
+<p>It is a question before all of endeavoring to obtain a more
+satisfactory theory of the electrodynamics of bodies in motion.
+It is there especially, as I have sufficiently shown above, that
+difficulties accumulate. It is useless to heap up hypotheses,
+we can not satisfy all the principles at once; so far, one has
+succeeded in safeguarding some only on condition of sacrificing
+the others; but all hope of obtaining better results is not yet
+lost. Let us take, then, the theory of Lorentz, turn it in all
+senses, modify it little by little, and perhaps everything will
+arrange itself.</p>
+
+<p>Thus in place of supposing that bodies in motion undergo a
+contraction in the sense of the motion, and that this contraction
+is the same whatever be the nature of these bodies and the forces
+to which they are otherwise subjected, could we not make a more
+simple and natural hypothesis? We might imagine, for example,
+that it is the ether which is modified when it is in relative motion
+in reference to the material medium which penetrates it, that,
+when it is thus modified, it no longer transmits perturbations
+with the same velocity in every direction. It might transmit
+more rapidly those which are propagated parallel to the motion
+of the medium, whether in the same sense or in the opposite sense,
+and less rapidly those which are propagated perpendicularly.
+The wave surfaces would no longer be spheres, but ellipsoids,
+and we could dispense with that extraordinary contraction of all
+bodies.</p>
+
+<p>I cite this only as an example, since the modifications that
+might be essayed would be evidently susceptible of infinite variation.</p>
+
+<p><i>Aberration and Astronomy.</i>&mdash;It is possible also that astronomy
+may some day furnish us data on this point; she it was in the
+main who raised the question in making us acquainted with the<span class='pagenum'><a name="Page_316" id="Page_316">[Pg 316]</a></span>
+phenomenon of the aberration of light. If we make crudely the
+theory of aberration, we reach a very curious result. The apparent
+positions of the stars differ from their real positions because
+of the earth's motion, and as this motion is variable, these
+apparent positions vary. The real position we can not ascertain,
+but we can observe the variations of the apparent position. The
+observations of the aberration show us, therefore, not the earth's
+motion, but the variations of this motion; they can not, therefore,
+give us information about the absolute motion of the earth.</p>
+
+<p>At least this is true in first approximation, but the case would
+be no longer the same if we could appreciate the thousandths of
+a second. Then it would be seen that the amplitude of the oscillation
+depends not alone on the variation of the motion, a variation
+which is well known, since it is the motion of our globe on
+its elliptic orbit, but on the mean value of this motion, so that
+the constant of aberration would not be quite the same for all the
+stars, and the differences would tell us the absolute motion of the
+earth in space.</p>
+
+<p>This, then, would be, under another form, the ruin of the principle
+of relativity. We are far, it is true, from appreciating the
+thousandth of a second, but, after all, say some, the earth's total
+absolute velocity is perhaps much greater than its relative velocity
+with respect to the sun. If, for example, it were 300 kilometers
+per second in place of 30, this would suffice to make the
+phenomenon observable.</p>
+
+<p>I believe that in reasoning thus one admits a too simple theory
+of aberration. Michelson has shown us, I have told you, that the
+physical procedures are powerless to put in evidence absolute
+motion; I am persuaded that the same will be true of the astronomic
+procedures, however far precision be carried.</p>
+
+<p>However that may be, the data astronomy will furnish us in
+this regard will some day be precious to the physicist. Meanwhile,
+I believe that the theorists, recalling the experience of
+Michelson, may anticipate a negative result, and that they would
+accomplish a useful work in constructing a theory of aberration
+which would explain this in advance.</p>
+
+<p><i>Electrons and Spectra.</i>&mdash;This dynamics of electrons can be approached
+from many sides, but among the ways leading thither is<span class='pagenum'><a name="Page_317" id="Page_317">[Pg 317]</a></span>
+one which has been somewhat neglected, and yet this is one of
+those which promise us the most surprises. It is movements of
+electrons which produce the lines of the emission spectra; this is
+proved by the Zeeman effect; in an incandescent body what vibrates
+is sensitive to the magnet, therefore electrified. This is a
+very important first point, but no one has gone farther. Why
+are the lines of the spectrum distributed in accordance with a
+regular law? These laws have been studied by the experimenters
+in their least details; they are very precise and comparatively
+simple. A first study of these distributions recalls the harmonics
+encountered in acoustics; but the difference is great. Not
+only are the numbers of vibrations not the successive multiples
+of a single number, but we do not even find anything analogous
+to the roots of those transcendental equations to which we are
+led by so many problems of mathematical physics: that of the
+vibrations of an elastic body of any form, that of the Hertzian
+oscillations in a generator of any form, the problem of Fourier
+for the cooling of a solid body.</p>
+
+<p>The laws are simpler, but they are of wholly other nature, and
+to cite only one of these differences, for the harmonics of high
+order, the number of vibrations tends toward a finite limit,
+instead of increasing indefinitely.</p>
+
+<p>That has not yet been accounted for, and I believe that there
+we have one of the most important secrets of nature. A Japanese
+physicist, M. Nagaoka, has recently proposed an explanation;
+according to him, atoms are composed of a large positive
+electron surrounded by a ring formed of a great number of very
+small negative electrons. Such is the planet Saturn with its
+rings. This is a very interesting attempt, but not yet wholly
+satisfactory; this attempt should be renewed. We will penetrate,
+so to speak, into the inmost recess of matter. And from
+the particular point of view which we to-day occupy, when we
+know why the vibrations of incandescent bodies differ thus from
+ordinary elastic vibrations, why the electrons do not behave like
+the matter which is familiar to us, we shall better comprehend the
+dynamics of electrons and it will be perhaps more easy for us
+to reconcile it with the principles.</p>
+
+<p><i>Conventions Preceding Experiment.</i>&mdash;Suppose, now, that all<span class='pagenum'><a name="Page_318" id="Page_318">[Pg 318]</a></span>
+these efforts fail, and, after all, I do not believe they will, what
+must be done? Will it be necessary to seek to mend the broken
+principles by giving what we French call a <i>coup de pouce</i>? That
+evidently is always possible, and I retract nothing of what I have
+said above.</p>
+
+<p>Have you not written, you might say if you wished to seek a
+quarrel with me&mdash;have you not written that the principles,
+though of experimental origin, are now unassailable by experiment
+because they have become conventions? And now you
+have just told us that the most recent conquests of experiment
+put these principles in danger.</p>
+
+<p>Well, formerly I was right and to-day I am not wrong. Formerly
+I was right, and what is now happening is a new proof of
+it. Take, for example, the calorimetric experiment of Curie on
+radium. Is it possible to reconcile it with the principle of the
+conservation of energy? This has been attempted in many ways.
+But there is among them one I should like you to notice; this is
+not the explanation which tends to-day to prevail, but it is one
+of those which have been proposed. It has been conjectured
+that radium was only an intermediary, that it only stored radiations
+of unknown nature which flashed through space in every
+direction, traversing all bodies, save radium, without being altered
+by this passage and without exercising any action upon
+them. Radium alone took from them a little of their energy and
+afterward gave it out to us in various forms.</p>
+
+<p>What an advantageous explanation, and how convenient!
+First, it is unverifiable and thus irrefutable. Then again it will
+serve to account for any derogation whatever to Mayer's principle;
+it answers in advance not only the objection of Curie, but
+all the objections that future experimenters might accumulate.
+This new and unknown energy would serve for everything.</p>
+
+<p>This is just what I said, and therewith we are shown that our
+principle is unassailable by experiment.</p>
+
+<p>But then, what have we gained by this stroke? The principle
+is intact, but thenceforth of what use is it? It enabled us to foresee
+that in such or such circumstance we could count on such a
+total quantity of energy; it limited us; but now that this indefinite
+provision of new energy is placed at our disposal, we are no<span class='pagenum'><a name="Page_319" id="Page_319">[Pg 319]</a></span>
+longer limited by anything; and, as I have written in 'Science
+and Hypothesis,' if a principle ceases to be fecund, experiment
+without contradicting it directly will nevertheless have condemned
+it.</p>
+
+<p><i>Future Mathematical Physics.</i>&mdash;This, therefore, is not what
+would have to be done; it would be necessary to rebuild anew.
+If we were reduced to this necessity; we could moreover console
+ourselves. It would not be necessary thence to conclude that
+science can weave only a Penelope's web, that it can raise only
+ephemeral structures, which it is soon forced to demolish from
+top to bottom with its own hands.</p>
+
+<p>As I have said, we have already passed through a like crisis.
+I have shown you that in the second mathematical physics, that
+of the principles, we find traces of the first, that of central
+forces; it will be just the same if we must know a third. Just so
+with the animal that exuviates, that breaks its too narrow carapace
+and makes itself a fresh one; under the new envelope one
+will recognize the essential traits of the organism which have
+persisted.</p>
+
+<p>We can not foresee in what way we are about to expand; perhaps
+it is the kinetic theory of gases which is about to undergo
+development and serve as model to the others. Then the facts
+which first appeared to us as simple thereafter would be merely
+resultants of a very great number of elementary facts which only
+the laws of chance would make cooperate for a common end.
+Physical law would then assume an entirely new aspect; it would
+no longer be solely a differential equation, it would take the character
+of a statistical law.</p>
+
+<p>Perhaps, too, we shall have to construct an entirely new mechanics
+that we only succeed in catching a glimpse of, where,
+inertia increasing with the velocity, the velocity of light would
+become an impassable limit. The ordinary mechanics, more
+simple, would remain a first approximation, since it would be
+true for velocities not too great, so that the old dynamics would
+still be found under the new. We should not have to regret having
+believed in the principles, and even, since velocities too great
+for the old formulas would always be only exceptional, the surest
+way in practise would be still to act as if we continued to<span class='pagenum'><a name="Page_320" id="Page_320">[Pg 320]</a></span>
+believe in them. They are so useful, it would be necessary to
+keep a place for them. To determine to exclude them altogether
+would be to deprive oneself of a precious weapon. I hasten to
+say in conclusion that we are not yet there, and as yet nothing
+proves that the principles will not come forth from out the fray
+victorious and intact.<a name="FNanchor_10_10" id="FNanchor_10_10"></a><a href="#Footnote_10_10" class="fnanchor">[10]</a></p>
+
+
+
+<hr style="width: 65%;" />
+<p><span class='pagenum'><a name="Page_321" id="Page_321">[Pg 321]</a></span></p>
+<h2><b>PART III<br />
+
+<br />
+
+<small>THE OBJECTIVE VALUE
+OF SCIENCE</small></b></h2>
+
+
+
+<hr style="width: 65%;" />
+<h3>CHAPTER X</h3>
+
+<h3><span class="smcap">Is Science Artificial?</span></h3>
+
+
+<h4>1. <i>The Philosophy of M. LeRoy</i></h4>
+
+<p>There are many reasons for being sceptics; should we push
+this scepticism to the very end or stop on the way? To go to the
+end is the most tempting solution, the easiest and that which
+many have adopted, despairing of saving anything from the
+shipwreck.</p>
+
+<p>Among the writings inspired by this tendency it is proper to
+place in the first rank those of M. LeRoy. This thinker is not
+only a philosopher and a writer of the greatest merit, but he has
+acquired a deep knowledge of the exact and physical sciences,
+and even has shown rare powers of mathematical invention. Let
+us recapitulate in a few words his doctrine, which has given rise
+to numerous discussions.</p>
+
+<p>Science consists only of conventions, and to this circumstance
+solely does it owe its apparent certitude; the facts of science and,
+<i>a fortiori</i>, its laws are the artificial work of the scientist; science
+therefore can teach us nothing of the truth; it can only serve
+us as rule of action.</p>
+
+<p>Here we recognize the philosophic theory known under the
+name of nominalism; all is not false in this theory; its legitimate
+domain must be left it, but out of this it should not be allowed
+to go.</p>
+
+<p>This is not all; M. LeRoy's doctrine is not only nominalistic;
+it has besides another characteristic which it doubtless owes to M.
+Bergson, it is anti-intellectualistic. According to M. LeRoy, the<span class='pagenum'><a name="Page_322" id="Page_322">[Pg 322]</a></span>
+intellect deforms all it touches, and that is still more true of its
+necessary instrument 'discourse.' There is reality only in our
+fugitive and changing impressions, and even this reality, when
+touched, vanishes.</p>
+
+<p>And yet M. LeRoy is not a sceptic; if he regards the intellect
+as incurably powerless, it is only to give more scope to other
+sources of knowledge, to the heart, for instance, to sentiment, to
+instinct or to faith.</p>
+
+<p>However great my esteem for M. LeRoy's talent, whatever the
+ingenuity of this thesis, I can not wholly accept it. Certes, I
+am in accord on many points with M. LeRoy, and he has even
+cited, in support of his view, various passages of my writings
+which I am by no means disposed to reject. I think myself only
+the more bound to explain why I can not go with him all the way.</p>
+
+<p>M. LeRoy often complains of being accused of scepticism.
+He could not help being, though this accusation is probably unjust.
+Are not appearances against him? Nominalist in doctrine,
+but realist at heart, he seems to escape absolute nominalism
+only by a desperate act of faith.</p>
+
+<p>The fact is that anti-intellectualistic philosophy in rejecting
+analysis and 'discourse,' just by that condemns itself to being
+intransmissible; it is a philosophy essentially internal, or, at the
+very least, only its negations can be transmitted; what wonder
+then that for an external observer it takes the shape of scepticism?</p>
+
+<p>Therein lies the weak point of this philosophy; if it strives to
+remain faithful to itself, its energy is spent in a negation and a
+cry of enthusiasm. Each author may repeat this negation and
+this cry, may vary their form, but without adding anything.</p>
+
+<p>And, yet, would it not be more logical in remaining silent?
+See, you have written long articles; for that, it was necessary
+to use words. And therein have you not been much more 'discursive'
+and consequently much farther from life and truth than
+the animal who simply lives without philosophizing? Would
+not this animal be the true philosopher?</p>
+
+<p>However, because no painter has made a perfect portrait,
+should we conclude that the best painting is not to paint? When
+a zoologist dissects an animal, certainly he 'alters it.' Yes, in
+dissecting it, he condemns himself to never know all of it; but in<span class='pagenum'><a name="Page_323" id="Page_323">[Pg 323]</a></span>
+not dissecting it, he would condemn himself to never know anything
+of it and consequently to never see anything of it.</p>
+
+<p>Certes, in man are other forces besides his intellect; no one
+has ever been mad enough to deny that. The first comer makes
+these blind forces act or lets them act; the philosopher must
+<i>speak</i> of them; to speak of them, he must know of them the little
+that can be known, he should therefore <i>see</i> them act. How?
+With what eyes, if not with his intellect? Heart, instinct, may
+guide it, but not render it useless; they may direct the look, but
+not replace the eye. It may be granted that the heart is the
+workman, and the intellect only the instrument. Yet is it an
+instrument not to be done without, if not for action, at least for
+philosophizing? Therefore a philosopher really anti-intellectualistic
+is impossible. Perhaps we shall have to declare for the
+supremacy of action; always it is our intellect which will thus
+conclude; in allowing precedence to action it will thus retain the
+superiority of the thinking reed. This also is a supremacy not
+to be disdained.</p>
+
+<p>Pardon these brief reflections and pardon also their brevity,
+scarcely skimming the question. The process of intellectualism
+is not the subject I wish to treat: I wish to speak of science, and
+about it there is no doubt; by definition, so to speak, it will be
+intellectualistic or it will not be at all. Precisely the question is,
+whether it will be.</p>
+
+
+<h4>2. <i>Science, Rule of Action</i></h4>
+
+<p>For M. LeRoy, science is only a rule of action. We are powerless
+to know anything and yet we are launched, we must act,
+and at all hazards we have established rules. It is the aggregate
+of these rules that is called science.</p>
+
+<p>It is thus that men, desirous of diversion, have instituted rules
+of play, like those of tric-trac for instance, which, better than
+science itself, could rely upon the proof by universal consent.
+It is thus likewise that, unable to choose, but forced to choose, we
+toss up a coin, head or tail to win.</p>
+
+<p>The rule of tric-trac is indeed a rule of action like science,
+but does any one think the comparison just and not see
+the difference? The rules of the game are arbitrary conventions<span class='pagenum'><a name="Page_324" id="Page_324">[Pg 324]</a></span>
+and the contrary convention might have been adopted,
+<i>which would have been none the less good</i>. On the contrary,
+science is a rule of action which is successful, generally at least,
+and I add, while the contrary rule would not have succeeded.</p>
+
+<p>If I say, to make hydrogen cause an acid to act on zinc, I formulate
+a rule which succeeds; I could have said, make distilled
+water act on gold; that also would have been a rule, only it would
+not have succeeded. If, therefore, scientific 'recipes' have a
+value, as rule of action, it is because we know they succeed, generally
+at least. But to know this is to know something and then
+why tell us we can know nothing?</p>
+
+<p>Science foresees, and it is because it foresees that it can be
+useful and serve as rule of action. I well know that its previsions
+are often contradicted by the event; that shows that
+science is imperfect, and if I add that it will always remain so,
+I am certain that this is a prevision which, at least, will never
+be contradicted. Always the scientist is less often mistaken
+than a prophet who should predict at random. Besides the
+progress though slow is continuous, so that scientists, though
+more and more bold, are less and less misled. This is little, but
+it is enough.</p>
+
+<p>I well know that M. LeRoy has somewhere said that science
+was mistaken oftener than one thought, that comets sometimes
+played tricks on astronomers, that scientists, who apparently are
+men, did not willingly speak of their failures, and that, if they
+should speak of them, they would have to count more defeats
+than victories.</p>
+
+<p>That day, M. LeRoy evidently overreached himself. If science
+did not succeed, it could not serve as rule of action; whence
+would it get its value? Because it is 'lived,' that is, because we
+love it and believe in it? The alchemists had recipes for making
+gold, they loved them and had faith in them, and yet our recipes
+are the good ones, although our faith be less lively, because they
+succeed.</p>
+
+<p>There is no escape from this dilemma; either science does not
+enable us to foresee, and then it is valueless as rule of action; or
+else it enables us to foresee, in a fashion more or less imperfect,
+and then it is not without value as means of knowledge.<span class='pagenum'><a name="Page_325" id="Page_325">[Pg 325]</a></span></p>
+
+<p>It should not even be said that action is the goal of science;
+should we condemn studies of the star Sirius, under pretext that
+we shall probably never exercise any influence on that star? To
+my eyes, on the contrary, it is the knowledge which is the end,
+and the action which is the means. If I felicitate myself on the
+industrial development, it is not alone because it furnishes a
+facile argument to the advocates of science; it is above all because
+it gives to the scientist faith in himself and also because it offers
+him an immense field of experience where he clashes against
+forces too colossal to be tampered with. Without this ballast,
+who knows whether he would not quit solid ground, seduced by
+the mirage of some scholastic novelty, or whether he would not
+despair, believing he had fashioned only a dream?</p>
+
+
+<h4>3. <i>The Crude Fact and the Scientific Fact</i></h4>
+
+<p>What was most paradoxical in M. LeRoy's thesis was that
+affirmation that <i>the scientist creates the fact</i>; this was at the
+same time its essential point and it is one of those which have
+been most discussed.</p>
+
+<p>Perhaps, says he (I well believe that this was a concession),
+it is not the scientist that creates the fact in the rough; it is at
+least he who creates the scientific fact.</p>
+
+<p>This distinction between the fact in the rough and the scientific
+fact does not by itself appear to me illegitimate. But I
+complain first that the boundary has not been traced either
+exactly or precisely; and then that the author has seemed to suppose
+that the crude fact, not being scientific, is outside of science.</p>
+
+<p>Finally, I can not admit that the scientist creates without restraint
+the scientific fact, since it is the crude fact which imposes
+it upon him.</p>
+
+<p>The examples given by M. LeRoy have greatly astonished me.
+The first is taken from the notion of atom. The atom chosen as
+example of fact! I avow that this choice has so disconcerted
+me that I prefer to say nothing about it. I have evidently misunderstood
+the author's thought and I could not fruitfully discuss
+it.</p>
+
+<p>The second case taken as example is that of an eclipse where
+the crude phenomenon is a play of light and shadow, but where<span class='pagenum'><a name="Page_326" id="Page_326">[Pg 326]</a></span>
+the astronomer can not intervene without introducing two foreign
+elements, to wit, a clock and Newton's law.</p>
+
+<p>Finally, M. LeRoy cites the rotation of the earth; it has been
+answered: but this is not a fact, and he has replied: it was one
+for Galileo, who affirmed it, as for the inquisitor, who denied it.
+It always remains that this is not a fact in the same sense as
+those just spoken of and that to give them the same name is to
+expose one's self to many confusions.</p>
+
+<p>Here then are four degrees:</p>
+
+<p>1&ordm;. It grows dark, says the clown.</p>
+
+<p>2&ordm;. The eclipse happened at nine o'clock, says the astronomer.</p>
+
+<p>3&ordm;. The eclipse happened at the time deducible from the tables
+constructed according to Newton's law, says he again.</p>
+
+<p>4&ordm;. That results from the earth's turning around the sun, says
+Galileo finally.</p>
+
+<p>Where then is the boundary between the fact in the rough
+and the scientific fact? To read M. LeRoy one would believe
+that it is between the first and the second stage, but who does not
+see that there is a greater distance from the second to the third,
+and still more from the third to the fourth.</p>
+
+<p>Allow me to cite two examples which perhaps will enlighten us
+a little.</p>
+
+<p>I observe the deviation of a galvanometer by the aid of a movable
+mirror which projects a luminous image or spot on a divided
+scale. The crude fact is this: I see the spot displace itself on the
+scale, and the scientific fact is this: a current passes in the circuit.</p>
+
+<p>Or again: when I make an experiment I should subject the
+result to certain corrections, because I know I must have made
+errors. These errors are of two kinds, some are accidental and
+these I shall correct by taking the mean; the others are systematic
+and I shall be able to correct those only by a thorough study of
+their causes. The first result obtained is then the fact in the
+rough, while the scientific fact is the final result after the
+finished corrections.</p>
+
+<p>Reflecting on this latter example, we are led to subdivide our
+second stage, and in place of saying:</p>
+
+<p>2. The eclipse happened at nine o'clock, we shall say:</p>
+
+<p>2<i>a</i>. The eclipse happened when my clock pointed to nine, and<span class='pagenum'><a name="Page_327" id="Page_327">[Pg 327]</a></span></p>
+
+<p>2<i>b</i>. My clock being ten minutes slow, the eclipse happened at
+ten minutes past nine.</p>
+
+<p>And this is not all: the first stage also should be subdivided,
+and not between these two subdivisions will be the least distance;
+it is necessary to distinguish between the impression of obscurity
+felt by one witnessing an eclipse, and the affirmation: It grows
+dark, which this impression extorts from him. In a sense it is
+the first which is the only true fact in the rough, and the second
+is already a sort of scientific fact.</p>
+
+<p>Now then our scale has six stages, and even though there is no
+reason for halting at this figure, there we shall stop.</p>
+
+<p>What strikes me at the start is this. At the first of our six
+stages, the fact, still completely in the rough, is, so to speak, individual,
+it is completely distinct from all other possible facts.
+From the second stage, already it is no longer the same. The
+enunciation of the fact would suit an infinity of other facts.
+So soon as language intervenes, I have at my command only a
+finite number of terms to express the shades, in number infinite,
+that my impressions might cover. When I say: It grows dark,
+that well expresses the impressions I feel in being present at an
+eclipse; but even in obscurity a multitude of shades could be
+imagined, and if, instead of that actually realized, had happened
+a slightly different shade, yet I should still have enunciated this
+<i>other</i> fact by saying: It grows dark.</p>
+
+<p>Second remark: even at the second stage, the enunciation of
+a fact can only be <i>true or false</i>. This is not so of any proposition;
+if this proposition is the enunciation of a convention, it
+can not be said that this enunciation is <i>true</i>, in the proper sense
+of the word, since it could not be true apart from me and is true
+only because I wish it to be.</p>
+
+<p>When, for instance, I say the unit for length is the meter, this
+is a decree that I promulgate, it is not something ascertained
+which forces itself upon me. It is the same, as I think I have
+elsewhere shown, when it is a question, for example, of Euclid's
+postulate.</p>
+
+<p>When I am asked: Is it growing dark? I always know whether
+I ought to reply yes or no. Although an infinity of possible
+facts may be susceptible of this same enunciation, it grows dark,<span class='pagenum'><a name="Page_328" id="Page_328">[Pg 328]</a></span>
+I shall always know whether the fact realized belongs or does not
+belong among those which answer to this enunciation. Facts are
+classed in categories, and if I am asked whether the fact that I
+ascertain belongs or does not belong in such a category, I shall
+not hesitate.</p>
+
+<p>Doubtless this classification is sufficiently arbitrary to leave a
+large part to man's freedom or caprice. In a word, this classification
+is a convention. <i>This convention being given</i>, if I am
+asked: Is such a fact true? I shall always know what to answer,
+and my reply will be imposed upon me by the witness of my
+senses.</p>
+
+<p>If therefore, during an eclipse, it is asked: Is it growing dark?
+all the world will answer yes. Doubtless those speaking a language
+where bright was called dark, and dark bright, would
+answer no. But of what importance is that?</p>
+
+<p>In the same way, in mathematics, <i>when I have laid down the
+definitions, and the postulates which are conventions</i>, a theorem
+henceforth can only be true or false. But to answer the question:
+Is this theorem true? it is no longer to the witness of my
+senses that I shall have recourse, but to reasoning.</p>
+
+<p>A statement of fact is always verifiable, and for the verification
+we have recourse either to the witness of our senses, or to
+the memory of this witness. This is properly what characterizes
+a fact. If you put the question to me: Is such a fact true? I
+shall begin by asking you, if there is occasion, to state precisely
+the conventions, by asking you, in other words, what language you
+have spoken; then once settled on this point, I shall interrogate
+my senses and shall answer yes or no. But it will be my senses
+that will have made answer, it will not be <i>you</i> when you say to
+me: I have spoken to you in English or in French.</p>
+
+<p>Is there something to change in all that when we pass to the
+following stages? When I observe a galvanometer, as I have
+just said, if I ask an ignorant visitor: Is the current passing?
+he looks at the wire to try to see something pass; but if I put the
+same question to my assistant who understands my language, he
+will know I mean: Does the spot move? and he will look at the
+scale.</p>
+
+<p>What difference is there then between the statement of a fact<span class='pagenum'><a name="Page_329" id="Page_329">[Pg 329]</a></span>
+in the rough and the statement of a scientific fact? The same
+difference as between the statement of the same crude fact in
+French and in German. The scientific statement is the translation
+of the crude statement into a language which is distinguished
+above all from the common German or French, because it is
+spoken by a very much smaller number of people.</p>
+
+<p>Yet let us not go too fast. To measure a current I may use
+a very great number of types of galvanometers or besides an
+electrodynamometer. And then when I shall say there is running
+in this circuit a current of so many amperes, that will mean:
+if I adapt to this circuit such a galvanometer I shall see the
+spot come to the division <i>a</i>; but that will mean equally: if I
+adapt to this circuit such an electrodynamometer, I shall see the
+spot go to the division <i>b</i>. And that will mean still many other
+things, because the current can manifest itself not only by mechanical
+effects, but by effects chemical, thermal, luminous, etc.</p>
+
+<p>Here then is one same statement which suits a very great number
+of facts absolutely different. Why? It is because I assume
+a law according to which, whenever such a mechanical effect shall
+happen, such a chemical effect will happen also. Previous experiments,
+very numerous, have never shown this law to fail, and
+then I have understood that I could express by the same statement
+two facts so invariably bound one to the other.</p>
+
+<p>When I am asked: Is the current passing? I can understand
+that that means: Will such a mechanical effect happen? But I
+can understand also: Will such a chemical effect happen? I
+shall then verify either the existence of the mechanical effect, or
+that of the chemical effect; that will be indifferent, since in both
+cases the answer must be the same.</p>
+
+<p>And if the law should one day be found false? If it was perceived
+that the concordance of the two effects, mechanical and
+chemical, is not constant? That day it would be necessary to
+change the scientific language to free it from a grave ambiguity.</p>
+
+<p>And after that? Is it thought that ordinary language by aid
+of which are expressed the facts of daily life is exempt from
+ambiguity?</p>
+
+<p><i>Shall we thence conclude that the facts of daily life are the
+work of the grammarians?</i><span class='pagenum'><a name="Page_330" id="Page_330">[Pg 330]</a></span></p>
+
+<p>You ask me: Is there a current? I try whether the mechanical
+effect exists, I ascertain it and I answer: Yes, there is a current.
+You understand at once that that means that the mechanical
+effect exists, and that the chemical effect, that I have not investigated,
+exists likewise. Imagine now, supposing an impossibility,
+the law we believe true, not to be, and the chemical effect not to
+exist. Under this hypothesis there will be two distinct facts, the
+one directly observed and which is true, the other inferred and
+which is false. It may strictly be said that we have created the
+second. So that error is the part of man's personal collaboration
+in the creation of the scientific fact.</p>
+
+<p>But if we can say that the fact in question is false, is this not
+just because it is not a free and arbitrary creation of our mind, a
+disguised convention, in which case it would be neither true nor
+false. And in fact it was verifiable; I had not made the verification,
+but I could have made it. If I answered amiss, it was because
+I chose to reply too quickly, without having asked nature,
+who alone knew the secret.</p>
+
+<p>When, after an experiment, I correct the accidental and systematic
+errors to bring out the scientific fact, the case is the same;
+the scientific fact will never be anything but the crude fact translated
+into another language. When I shall say: It is such an
+hour, that will be a short way of saying: There is such a relation
+between the hour indicated by my clock, and the hour it marked
+at the moment of the passing of such a star and such another
+star across the meridian. And this convention of language once
+adopted, when I shall be asked: Is it such an hour? it will not
+depend upon me to answer yes or no.</p>
+
+<p>Let us pass to the stage before the last: the eclipse happened at
+the hour given by the tables deduced from Newton's laws. This
+is still a convention of language which is perfectly clear for those
+who know celestial mechanics or simply for those who have the
+tables calculated by the astronomers. I am asked: Did the
+eclipse happen at the hour predicted? I look in the nautical
+almanac, I see that the eclipse was announced for nine o'clock
+and I understand that the question means: Did the eclipse
+happen at nine o'clock? There still we have nothing to change
+in our conclusions. <i>The scientific fact is only the crude fact
+translated into a convenient language.</i><span class='pagenum'><a name="Page_331" id="Page_331">[Pg 331]</a></span></p>
+
+<p>It is true that at the last stage things change. Does the
+earth rotate? Is this a verifiable fact? Could Galileo and the
+Grand Inquisitor, to settle the matter, appeal to the witness of
+their senses? On the contrary, they were in accord about the
+appearances, and whatever had been the accumulated experiences,
+they would have remained in accord with regard to the
+appearances without ever agreeing on their interpretation. It
+is just on that account that they were obliged to have recourse
+to procedures of discussion so unscientific.</p>
+
+<p>This is why I think they did not disagree about a <i>fact</i>: we
+have not the right to give the same name to the rotation of the
+earth, which was the object of their discussion, and to the facts
+crude or scientific we have hitherto passed in review.</p>
+
+<p>After what precedes, it seems superfluous to investigate
+whether the fact in the rough is outside of science, because there
+can neither be science without scientific fact, nor scientific fact
+without fact in the rough, since the first is only the translation
+of the second.</p>
+
+<p>And then, has one the right to say that the scientist creates the
+scientific fact? First of all, he does not create it from nothing,
+since he makes it with the fact in the rough. Consequently he
+does not make it freely and <i>as he chooses</i>. However able the
+worker may be, his freedom is always limited by the properties of
+the raw material on which he works.</p>
+
+<p>After all, what do you mean when you speak of this free
+creation of the scientific fact and when you take as example the
+astronomer who intervenes actively in the phenomenon of the
+eclipse by bringing his clock? Do you mean: The eclipse happened
+at nine o'clock; but if the astronomer had wished it to
+happen at ten, that depended only on him, he had only to
+advance his clock an hour?</p>
+
+<p>But the astronomer, in perpetrating that bad joke, would
+evidently have been guilty of an equivocation. When he tells
+me: The eclipse happened at nine, I understand that nine is the
+hour deduced from the crude indication of the pendulum by the
+usual series of corrections. If he has given me solely that crude
+indication, or if he has made corrections contrary to the habitual
+rules, he has changed the language agreed upon without forewarning<span class='pagenum'><a name="Page_332" id="Page_332">[Pg 332]</a></span>
+me. If, on the contrary, he took care to forewarn me,
+I have nothing to complain of, but then it is always the same
+fact expressed in another language.</p>
+
+<p>In sum, <i>all the scientist creates in a fact is the language in
+which he enunciates it</i>. If he predicts a fact, he will employ this
+language, and for all those who can speak and understand it, his
+prediction is free from ambiguity. Moreover, this prediction
+once made, it evidently does not depend upon him whether it is
+fulfilled or not.</p>
+
+<p>What then remains of M. LeRoy's thesis? This remains: the
+scientist intervenes actively in choosing the facts worth observing.
+An isolated fact has by itself no interest; it becomes interesting
+if one has reason to think that it may aid in the prediction
+of other facts; or better, if, having been predicted, its verification
+is the confirmation of a law. Who shall choose the facts
+which, corresponding to these conditions, are worthy the freedom
+of the city in science? This is the free activity of the scientist.</p>
+
+<p>And that is not all. I have said that the scientific fact is the
+translation of a crude fact into a certain language; I should add
+that every scientific fact is formed of many crude facts. This is
+sufficiently shown by the examples cited above. For instance,
+for the hour of the eclipse my clock marked the hour &#945; at the
+instant of the eclipse; it marked the hour &#946; at the moment of the
+last transit of the meridian of a certain star that we take as
+origin of right ascensions; it marked the hour &#947; at the moment
+of the preceding transit of this same star. There are three distinct
+facts (still it will be noticed that each of them results itself
+from two simultaneous facts in the rough; but let us pass this
+over). In place of that I say: The eclipse happened at the hour
+24 (&#945;&minus;&#946;) / (&#946;&minus;&#947;), and the three facts are combined in a single
+scientific fact. I have concluded that the three readings, &#945;, &#946;, &#947;
+made on my clock at three different moments lacked interest and
+that the only thing interesting was the combination (&#945;&minus;&#946;) / (&#946;&minus;&#947;)
+of the three. In this conclusion is found the free activity of my
+mind.</p>
+
+<p>But I have thus used up my power; I can not make this combination
+(&#945;&minus;&#946;) / (&#946;&minus;&#947;) have such a value and not such another,
+since I can not influence either the value of &#945;, or that of &#946;, or
+that of &#947;, which are imposed upon me as crude facts.<span class='pagenum'><a name="Page_333" id="Page_333">[Pg 333]</a></span></p>
+
+<p>In sum, facts are facts, and <i>if it happens that they satisfy a
+prediction, this is not an effect of our free activity</i>. There is no
+precise frontier between the fact in the rough and the scientific
+fact; it can only be said that such an enunciation of fact is <i>more
+crude</i> or, on the contrary, <i>more scientific</i> than such another.</p>
+
+
+<h4>4. <i>'Nominalism' and 'the Universal Invariant'</i></h4>
+
+<p>If from facts we pass to laws, it is clear that the part of the
+free activity of the scientist will become much greater. But
+did not M. LeRoy make it still too great? This is what we are
+about to examine.</p>
+
+<p>Recall first the examples he has given. When I say: Phosphorus
+melts at 44&deg;, I think I am enunciating a law; in reality
+it is just the definition of phosphorus; if one should discover a
+body which, possessing otherwise all the properties of phosphorus,
+did not melt at 44&deg;, we should give it another name, that is all,
+and the law would remain true.</p>
+
+<p>Just so when I say: Heavy bodies falling freely pass over
+spaces proportional to the squares of the times, I only give the
+definition of free fall. Whenever the condition shall not be
+fulfilled, I shall say that the fall is not free, so that the law
+will never be wrong. It is clear that if laws were reduced to that,
+they could not serve in prediction; then they would be good for
+nothing, either as means of knowledge or as principle of action.</p>
+
+<p>When I say: Phosphorus melts at 44&deg;, I mean by that: All
+bodies possessing such or such a property (to wit, all the properties
+of phosphorus, save fusing-point) fuse at 44&deg;. So understood,
+my proposition is indeed a law, and this law may be useful
+to me, because if I meet a body possessing these properties
+I shall be able to predict that it will fuse at 44&deg;.</p>
+
+<p>Doubtless the law may be found to be false. Then we shall
+read in the treatises on chemistry: "There are two bodies which
+chemists long confounded under the name of phosphorus; these
+two bodies differ only by their points of fusion." That would
+evidently not be the first time for chemists to attain to the separation
+of two bodies they were at first not able to distinguish; such,
+for example, are neodymium and praseodymium, long confounded
+under the name of didymium.<span class='pagenum'><a name="Page_334" id="Page_334">[Pg 334]</a></span></p>
+
+<p>I do not think the chemists much fear that a like mischance
+will ever happen to phosphorus. And if, to suppose the impossible,
+it should happen, the two bodies would probably not have
+<i>identically</i> the same density, <i>identically</i> the same specific heat,
+etc., so that after having determined with care the density, for
+instance, one could still foresee the fusion point.</p>
+
+<p>It is, moreover, unimportant; it suffices to remark that there
+is a law, and that this law, true or false, does not reduce to a
+tautology.</p>
+
+<p>Will it be said that if we do not know on the earth a body
+which does not fuse at 44&deg; while having all the other properties
+of phosphorus, we can not know whether it does not exist on other
+planets? Doubtless that may be maintained, and it would then
+be inferred that the law in question, which may serve as a rule
+of action to us who inhabit the earth, has yet no general value
+from the point of view of knowledge, and owes its interest only
+to the chance which has placed us on this globe. This is possible,
+but, if it were so, the law would be valueless, not because it reduced
+to a convention, but because it would be false.</p>
+
+<p>The same is true in what concerns the fall of bodies. It would
+do me no good to have given the name of free fall to falls which
+happen in conformity with Galileo's law, if I did not know that
+elsewhere, in such circumstances, the fall will be <i>probably</i> free or
+<i>approximately</i> free. That then is a law which may be true or
+false, but which does not reduce to a convention.</p>
+
+<p>Suppose the astronomers discover that the stars do not exactly
+obey Newton's law. They will have the choice between two
+attitudes; they may say that gravitation does not vary exactly
+as the inverse of the square of the distance, or else they may say
+that gravitation is not the only force which acts on the stars and
+that there is in addition a different sort of force.</p>
+
+<p>In the second case, Newton's law will be considered as the
+definition of gravitation. This will be the nominalist attitude.
+The choice between the two attitudes is free, and is made from
+considerations of convenience, though these considerations are
+most often so strong that there remains practically little of this
+freedom.</p>
+
+<p>We can break up this proposition: (1) The stars obey Newton's<span class='pagenum'><a name="Page_335" id="Page_335">[Pg 335]</a></span>
+law, into two others; (2) gravitation obeys Newton's law; (3)
+gravitation is the only force acting on the stars. In this case
+proposition (2) is no longer anything but a definition and is
+beyond the test of experiment; but then it will be on proposition
+(3) that this check can be exercised. This is indeed necessary,
+since the resulting proposition (1) predicts verifiable facts in the
+rough.</p>
+
+<p>It is thanks to these artifices that by an unconscious nominalism
+the scientists have elevated above the laws what they call
+principles. When a law has received a sufficient confirmation
+from experiment, we may adopt two attitudes: either we may
+leave this law in the fray; it will then remain subjected to an
+incessant revision, which without any doubt will end by demonstrating
+that it is only approximative. Or else we may elevate
+it into a <i>principle</i> by adopting conventions such that the proposition
+may be certainly true. For that the procedure is always
+the same. The primitive law enunciated a relation between two
+facts in the rough, <i>A</i> and <i>B</i>; between these two crude facts is
+introduced an abstract intermediary <i>C</i>, more or less fictitious
+(such was in the preceding example the impalpable entity, gravitation).
+And then we have a relation between <i>A</i> and <i>C</i> that we
+may suppose rigorous and which is the <i>principle</i>; and another
+between <i>C</i> and <i>B</i> which remains a <i>law</i> subject to revision.</p>
+
+<p>The principle, henceforth crystallized, so to speak, is no longer
+subject to the test of experiment. It is not true or false, it is
+convenient.</p>
+
+<p>Great advantages have often been found in proceeding in that
+way, but it is clear that if <i>all</i> the laws had been transformed
+into principles <i>nothing</i> would be left of science. Every law may
+be broken up into a principle and a law, but thereby it is very
+clear that, however far this partition be pushed, there will always
+remain laws.</p>
+
+<p>Nominalism has therefore limits, and this is what one might
+fail to recognize if one took to the very letter M. LeRoy's
+assertions.</p>
+
+<p>A rapid review of the sciences will make us comprehend better
+what are these limits. The nominalist attitude is justified only
+when it is convenient; when is it so?<span class='pagenum'><a name="Page_336" id="Page_336">[Pg 336]</a></span></p>
+
+<p>Experiment teaches us relations between bodies; this is the fact
+in the rough; these relations are extremely complicated. Instead
+of envisaging directly the relation of the body <i>A</i> and the body <i>B</i>,
+we introduce between them an intermediary, which is space, and
+we envisage three distinct relations: that of the body <i>A</i> with the
+figure <i>A&acute;</i> of space, that of the body <i>B</i> with the figure <i>B&acute;</i> of space,
+that of the two figures <i>A&acute;</i> and <i>B&acute;</i> to each other. Why is this
+detour advantageous? Because the relation of <i>A</i> and <i>B</i> was complicated,
+but differed little from that of <i>A&acute;</i> and <i>B&acute;</i>, which is
+simple; so that this complicated relation may be replaced by the
+simple relation between <i>A&acute;</i> and <i>B&acute;</i> and by two other relations
+which tell us that the differences between <i>A</i> and <i>A&acute;</i>, on the one
+hand, between <i>B</i> and <i>B&acute;</i>, on the other hand, are <i>very small</i>. For
+example, if <i>A</i> and <i>B</i> are two natural solid bodies which are displaced
+with slight deformation, we envisage two movable <i>rigid</i>
+figures <i>A&acute;</i> and <i>B&acute;</i>. The laws of the relative displacement of these
+figures <i>A&acute;</i> and <i>B&acute;</i> will be very simple; they will be those of geometry.
+And we shall afterward add that the body <i>A</i>, which always
+differs very little from <i>A&acute;</i>, dilates from the effect of heat and
+bends from the effect of elasticity. These dilatations and flexions,
+just because they are very small, will be for our mind relatively
+easy to study. Just imagine to what complexities of language
+it would have been necessary to be resigned if we had wished to
+comprehend in the same enunciation the displacement of the
+solid, its dilatation and its flexure?</p>
+
+<p>The relation between <i>A</i> and <i>B</i> was a rough law, and was broken
+up; we now have two laws which express the relations of <i>A</i> and <i>A&acute;</i>,
+of <i>B</i> and <i>B&acute;</i>, and a principle which expresses that of <i>A&acute;</i> with <i>B&acute;</i>.
+It is the aggregate of these principles that is called geometry.</p>
+
+<p>Two other remarks. We have a relation between two bodies <i>A</i>
+and <i>B</i>, which we have replaced by a relation between two figures
+<i>A&acute;</i> and <i>B&acute;</i>; but this same relation between the same two figures
+<i>A&acute;</i> and <i>B&acute;</i> could just as well have replaced advantageously a
+relation between two other bodies <i>A&acute;&acute;</i> and <i>B&acute;&acute;</i>, entirely different
+from <i>A</i> and <i>B</i>. And that in many ways. If the principles of
+geometry had not been invented, after having studied the relation
+of <i>A</i> and <i>B</i>, it would be necessary to begin again <i>ab ovo</i> the
+study of the relation of <i>A&acute;&acute;</i> and <i>B&acute;&acute;</i>. That is why geometry is so<span class='pagenum'><a name="Page_337" id="Page_337">[Pg 337]</a></span>
+precious. A geometrical relation can advantageously replace a
+relation which, considered in the rough state, should be regarded
+as mechanical, it can replace another which should be regarded
+as optical, etc.</p>
+
+<p>Yet let no one say: But that proves geometry an experimental
+science; in separating its principles from laws whence they have
+been drawn, you artificially separate it itself from the sciences
+which have given birth to it. The other sciences have likewise
+principles, but that does not preclude our having to call them
+experimental.</p>
+
+<p>It must be recognized that it would have been difficult not to
+make this separation that is pretended to be artificial. We know
+the r&ocirc;le that the kinematics of solid bodies has played in the
+genesis of geometry; should it then be said that geometry is only
+a branch of experimental kinematics? But the laws of the rectilinear
+propagation of light have also contributed to the formation
+of its principles. Must geometry be regarded both as a
+branch of kinematics and as a branch of optics? I recall besides
+that our Euclidean space which is the proper object of geometry
+has been chosen, for reasons of convenience, from among a certain
+number of types which preexist in our mind and which are
+called groups.</p>
+
+<p>If we pass to mechanics, we still see great principles whose
+origin is analogous, and, as their 'radius of action,' so to speak,
+is smaller, there is no longer reason to separate them from
+mechanics proper and to regard this science as deductive.</p>
+
+<p>In physics, finally, the r&ocirc;le of the principles is still more diminished.
+And in fact they are only introduced when it is of advantage.
+Now they are advantageous precisely because they are
+few, since each of them very nearly replaces a great number
+of laws. Therefore it is not of interest to multiply them. Besides
+an outcome is necessary, and for that it is needful to end by leaving
+abstraction to take hold of reality.</p>
+
+<p>Such are the limits of nominalism, and they are narrow.</p>
+
+<p>M. LeRoy has insisted, however, and he has put the question
+under another form.</p>
+
+<p>Since the enunciation of our laws may vary with the conventions
+that we adopt, since these conventions may modify even the<span class='pagenum'><a name="Page_338" id="Page_338">[Pg 338]</a></span>
+natural relations of these laws, is there in the manifold of these
+laws something independent of these conventions and which may,
+so to speak, play the r&ocirc;le of <i>universal invariant</i>? For instance,
+the fiction has been introduced of beings who, having been educated
+in a world different from ours, would have been led to
+create a non-Euclidean geometry. If these beings were afterward
+suddenly transported into our world, they would observe
+the same laws as we, but they would enunciate them in an
+entirely different way. In truth there would still be something
+in common between the two enunciations, but this is because these
+beings do not yet differ enough from us. Beings still more strange
+may be imagined, and the part common to the two systems of
+enunciations will shrink more and more. Will it thus shrink
+in convergence toward zero, or will there remain an irreducible
+residue which will then be the universal invariant sought?</p>
+
+<p>The question calls for precise statement. Is it desired that
+this common part of the enunciations be expressible in words?
+It is clear, then, that there are not words common to all languages,
+and we can not pretend to construct I know not what universal
+invariant which should be understood both by us and by the
+fictitious non-Euclidean geometers of whom I have just spoken;
+no more than we can construct a phrase which can be understood
+both by Germans who do not understand French and by French
+who do not understand German. But we have fixed rules which
+permit us to translate the French enunciations into German,
+and inversely. It is for that that grammars and dictionaries
+have been made. There are also fixed rules for translating the
+Euclidean language into the non-Euclidean language, or, if there
+are not, they could be made.</p>
+
+<p>And even if there were neither interpreter nor dictionary, if
+the Germans and the French, after having lived centuries in
+separate worlds, found themselves all at once in contact, do you
+think there would be nothing in common between the science
+of the German books and that of the French books? The French
+and the Germans would certainly end by understanding each
+other, as the American Indians ended by understanding the
+language of their conquerors after the arrival of the Spanish.</p>
+
+<p>But, it will be said, doubtless the French would be capable of<span class='pagenum'><a name="Page_339" id="Page_339">[Pg 339]</a></span>
+understanding the Germans even without having learned German,
+but this is because there remains between the French and
+the Germans something in common, since both are men. We
+should still attain to an understanding with our hypothetical non-Euclideans,
+though they be not men, because they would still
+retain something human. But in any case a minimum of humanity
+is necessary.</p>
+
+<p>This is possible, but I shall observe first that this little humanness
+which would remain in the non-Euclideans would suffice not
+only to make possible the translation of <i>a little</i> of their language,
+but to make possible the translation of <i>all</i> their language.</p>
+
+<p>Now, that there must be a minimum is what I concede; suppose
+there exists I know not what fluid which penetrates between the
+molecules of our matter, without having any action on it and
+without being subject to any action coming from it. Suppose
+beings sensible to the influence of this fluid and insensible to
+that of our matter. It is clear that the science of these beings
+would differ absolutely from ours and that it would be idle to
+seek an 'invariant' common to these two sciences. Or again, if
+these beings rejected our logic and did not admit, for instance,
+the principle of contradiction.</p>
+
+<p>But truly I think it without interest to examine such
+hypotheses.</p>
+
+<p>And then, if we do not push whimsicality so far, if we introduce
+only fictitious beings having senses analogous to ours and
+sensible to the same impressions, and moreover admitting the
+principles of our logic, we shall then be able to conclude that
+their language, however different from ours it may be, would
+always be capable of translation. Now the possibility of translation
+implies the existence of an invariant. To translate is
+precisely to disengage this invariant. Thus, to decipher a cryptogram
+is to seek what in this document remains invariant, when
+the letters are permuted.</p>
+
+<p>What now is the nature of this invariant it is easy to understand,
+and a word will suffice us. The invariant laws are the
+relations between the crude facts, while the relations between the
+'scientific facts' remain always dependent on certain conventions.</p>
+
+
+
+<hr style="width: 65%;" />
+<p><span class='pagenum'><a name="Page_340" id="Page_340">[Pg 340]</a></span></p>
+<h3>CHAPTER XI</h3>
+
+<h3><span class="smcap">Science and Reality</span></h3>
+
+
+<h4>5. <i>Contingence and Determinism</i></h4>
+
+<p>I do not intend to treat here the question of the contingence of
+the laws of nature, which is evidently insoluble, and on which so
+much has already been written. I only wish to call attention to
+what different meanings have been given to this word, contingence,
+and how advantageous it would be to distinguish them.</p>
+
+<p>If we look at any particular law, we may be certain in advance
+that it can only be approximate. It is, in fact, deduced from
+experimental verifications, and these verifications were and could
+be only approximate. We should always expect that more precise
+measurements will oblige us to add new terms to our formulas;
+this is what has happened, for instance, in the case of Mariotte's
+law.</p>
+
+<p>Moreover the statement of any law is necessarily incomplete.
+This enunciation should comprise the enumeration of <i>all</i> the
+antecedents in virtue of which a given consequent can happen.
+I should first describe <i>all</i> the conditions of the experiment to be
+made and the law would then be stated: If all the conditions are
+fulfilled, the phenomenon will happen.</p>
+
+<p>But we shall be sure of not having forgotten <i>any</i> of these conditions
+only when we shall have described the state of the entire
+universe at the instant <i>t</i>; all the parts of this universe may, in
+fact, exercise an influence more or less great on the phenomenon
+which must happen at the instant <i>t</i> + <i>dt</i>.</p>
+
+<p>Now it is clear that such a description could not be found in
+the enunciation of the law; besides, if it were made, the law
+would become incapable of application; if one required so many
+conditions, there would be very little chance of their ever being
+all realized at any moment.</p>
+
+<p>Then as one can never be certain of not having forgotten some
+essential condition, it can not be said: If such and such conditions<span class='pagenum'><a name="Page_341" id="Page_341">[Pg 341]</a></span>
+are realized, such a phenomenon will occur; it can only be
+said: If such and such conditions are realized, it is probable that
+such a phenomenon will occur, very nearly.</p>
+
+<p>Take the law of gravitation, which is the least imperfect of all
+known laws. It enables us to foresee the motions of the planets.
+When I use it, for instance, to calculate the orbit of Saturn, I
+neglect the action of the stars, and in doing so I am certain of
+not deceiving myself, because I know that these stars are too far
+away for their action to be sensible.</p>
+
+<p>I announce, then, with a quasi-certitude that the coordinates
+of Saturn at such an hour will be comprised between such and
+such limits. Yet is that certitude absolute? Could there not
+exist in the universe some gigantic mass, much greater than that
+of all the known stars and whose action could make itself felt
+at great distances? That mass might be animated by a colossal
+velocity, and after having circulated from all time at such distances
+that its influence had remained hitherto insensible to us,
+it might come all at once to pass near us. Surely it would produce
+in our solar system enormous perturbations that we could
+not have foreseen. All that can be said is that such an event is
+wholly improbable, and then, instead of saying: Saturn will be
+near such a point of the heavens, we must limit ourselves to saying:
+Saturn will probably be near such a point of the heavens.
+Although this probability may be practically equivalent to certainty,
+it is only a probability.</p>
+
+<p>For all these reasons, no particular law will ever be more than
+approximate and probable. Scientists have never failed to recognize
+this truth; only they believe, right or wrong, that every law
+may be replaced by another closer and more probable, that this
+new law will itself be only provisional, but that the same movement
+can continue indefinitely, so that science in progressing will
+possess laws more and more probable, that the approximation
+will end by differing as little as you choose from exactitude and
+the probability from certitude.</p>
+
+<p>If the scientists who think thus are right, still could it be said
+that <i>the</i> laws of nature are contingent, even though <i>each</i> law,
+taken in particular, may be qualified as contingent? Or must one
+require, before concluding the contingence <i>of the</i> natural laws,<span class='pagenum'><a name="Page_342" id="Page_342">[Pg 342]</a></span>
+that this progress have an end, that the scientist finish some day
+by being arrested in his search for a closer and closer approximation,
+and that, beyond a certain limit, he thereafter meet in
+nature only caprice?</p>
+
+<p>In the conception of which I have just spoken (and which I
+shall call the scientific conception), every law is only a statement
+imperfect and provisional, but it must one day be replaced by
+another, a superior law, of which it is only a crude image. No
+place therefore remains for the intervention of a free will.</p>
+
+<p>It seems to me that the kinetic theory of gases will furnish
+us a striking example.</p>
+
+<p>You know that in this theory all the properties of gases are
+explained by a simple hypothesis; it is supposed that all the
+gaseous molecules move in every direction with great velocities
+and that they follow rectilineal paths which are disturbed only
+when one molecule passes very near the sides of the vessel or
+another molecule. The effects our crude senses enable us to
+observe are the mean effects, and in these means, the great deviations
+compensate, or at least it is very improbable that they do
+not compensate; so that the observable phenomena follow simple
+laws such as that of Mariotte or of Gay-Lussac. But this compensation
+of deviations is only probable. The molecules incessantly
+change place and in these continual displacements the
+figures they form pass successively through all possible combinations.
+Singly these combinations are very numerous; almost all
+are in conformity with Mariotte's law, only a few deviate from
+it. These also will happen, only it would be necessary to wait
+a long time for them. If a gas were observed during a sufficiently
+long time it would certainly be finally seen to deviate,
+for a very short time, from Mariotte's law. How long would it
+be necessary to wait? If it were desired to calculate the probable
+number of years, it would be found that this number is so
+great that to write only the number of places of figures employed
+would still require half a score places of figures. No matter;
+enough that it may be done.</p>
+
+<p>I do not care to discuss here the value of this theory. It is
+evident that if it be adopted, Mariotte's law will thereafter
+appear only as contingent, since a day will come when it will not<span class='pagenum'><a name="Page_343" id="Page_343">[Pg 343]</a></span>
+be true. And yet, think you the partisans of the kinetic theory
+are adversaries of determinism? Far from it; they are the
+most ultra of mechanists. Their molecules follow rigid paths,
+from which they depart only under the influence of forces which
+vary with the distance, following a perfectly determinate law.
+There remains in their system not the smallest place either for
+freedom, or for an evolutionary factor, properly so-called, or for
+anything whatever that could be called contingence. I add, to
+avoid mistake, that neither is there any evolution of Mariotte's
+law itself; it ceases to be true after I know not how many centuries;
+but at the end of a fraction of a second it again becomes
+true and that for an incalculable number of centuries.</p>
+
+<p>And since I have pronounced the word evolution, let us clear
+away another mistake. It is often said: Who knows whether
+the laws do not evolve and whether we shall not one day discover
+that they were not at the Carboniferous epoch what they are
+to-day? What are we to understand by that? What we think
+we know about the past state of our globe, we deduce from its
+present state. And how is this deduction made? It is by means
+of laws supposed known. The law, being a relation between the
+antecedent and the consequent, enables us equally well to deduce
+the consequent from the antecedent, that is, to foresee the future,
+and to deduce the antecedent from the consequent, that is, to
+conclude from the present to the past. The astronomer who
+knows the present situation of the stars can from it deduce their
+future situation by Newton's law, and this is what he does when
+he constructs ephemerides; and he can equally deduce from it
+their past situation. The calculations he thus can make can not
+teach him that Newton's law will cease to be true in the future,
+since this law is precisely his point of departure; not more can
+they tell him it was not true in the past. Still, in what concerns
+the future, his ephemerides can one day be tested and our descendants
+will perhaps recognize that they were false. But in
+what concerns the past, the geologic past which had no witnesses,
+the results of his calculation, like those of all speculations where
+we seek to deduce the past from the present, escape by their
+very nature every species of test. So that if the laws of nature
+were not the same in the Carboniferous age as at the present<span class='pagenum'><a name="Page_344" id="Page_344">[Pg 344]</a></span>
+epoch, we shall never be able to know it, since we can know
+nothing of this age, only what we deduce from the hypothesis of
+the permanence of these laws.</p>
+
+<p>Perhaps it will be said that this hypothesis might lead to contradictory
+results and that we shall be obliged to abandon it.
+Thus, in what concerns the origin of life, we may conclude that
+there have always been living beings, since the present world
+shows us always life springing from life; and we may also conclude
+that there have not always been, since the application of
+the existent laws of physics to the present state of our globe
+teaches us that there was a time when this globe was so warm that
+life on it was impossible. But contradictions of this sort can
+always be removed in two ways; it may be supposed that the
+actual laws of nature are not exactly what we have assumed;
+or else it may be supposed that the laws of nature actually are
+what we have assumed, but that it has not always been so.</p>
+
+<p>It is evident that the actual laws will never be sufficiently well
+known for us not to be able to adopt the first of these two solutions
+and for us to be constrained to infer the evolution of
+natural laws.</p>
+
+<p>On the other hand, suppose such an evolution; assume, if you
+wish, that humanity lasts sufficiently long for this evolution to
+have witnesses. The <i>same</i> antecedent shall produce, for instance,
+different consequents at the Carboniferous epoch and at the
+Quaternary. That evidently means that the antecedents are
+closely alike; if all the circumstances were identical, the Carboniferous
+epoch would be indistinguishable from the Quaternary.
+Evidently this is not what is supposed. What remains is that
+such antecedent, accompanied by such accessory circumstance,
+produces such consequent; and that the same antecedent, accompanied
+by such other accessory circumstance, produces such
+other consequent. Time does not enter into the affair.</p>
+
+<p>The law, such as ill-informed science would have stated it, and
+which would have affirmed that this antecedent always produces
+this consequent, without taking account of the accessory circumstances,
+this law, which was only approximate and probable,
+must be replaced by another law more approximate and more
+probable, which brings in these accessory circumstances. We<span class='pagenum'><a name="Page_345" id="Page_345">[Pg 345]</a></span>
+always come back, therefore, to that same process which we have
+analyzed above, and if humanity should discover something of
+this sort, it would not say that it is the laws which have evoluted,
+but the circumstances which have changed.</p>
+
+<p>Here, therefore, are several different senses of the word contingence.
+M. LeRoy retains them all and he does not sufficiently
+distinguish them, but he introduces a new one. Experimental
+laws are only approximate, and if some appear to us as exact, it
+is because we have artificially transformed them into what I have
+above called a principle. We have made this transformation
+freely, and as the caprice which has determined us to make it
+is something eminently contingent, we have communicated this
+contingence to the law itself. It is in this sense that we have the
+right to say that determinism supposes freedom, since it is freely
+that we become determinists. Perhaps it will be found that this
+is to give large scope to nominalism and that the introduction
+of this new sense of the word contingence will not help much to
+solve all those questions which naturally arise and of which we
+have just been speaking.</p>
+
+<p>I do not at all wish to investigate here the foundations of the
+principle of induction; I know very well that I should not succeed;
+it is as difficult to justify this principle as to get on without
+it. I only wish to show how scientists apply it and are
+forced to apply it.</p>
+
+<p>When the same antecedent recurs, the same consequent must
+likewise recur; such is the ordinary statement. But reduced
+to these terms this principle could be of no use. For one to be
+able to say that the same antecedent recurred, it would be necessary
+for the circumstances <i>all</i> to be reproduced, since no one
+is absolutely indifferent, and for them to be <i>exactly</i> reproduced.
+And, as that will never happen, the principle can have no
+application.</p>
+
+<p>We should therefore modify the enunciation and say: If an
+antecedent <i>A</i> has once produced a consequent <i>B</i>, an antecedent
+<i>A&acute;</i>, slightly different from <i>A</i>, will produce a consequent <i>B&acute;</i>,
+slightly different from <i>B</i>. But how shall we recognize that the
+antecedents <i>A</i> and <i>A&acute;</i> are 'slightly different'? If some one of the
+circumstances can be expressed by a number, and this number<span class='pagenum'><a name="Page_346" id="Page_346">[Pg 346]</a></span>
+has in the two cases values very near together, the sense of the
+phrase 'slightly different' is relatively clear; the principle then
+signifies that the consequent is a continuous function of the antecedent.
+And as a practical rule, we reach this conclusion that
+we have the right to interpolate. This is in fact what scientists
+do every day, and without interpolation all science would be
+impossible.</p>
+
+<p>Yet observe one thing. The law sought may be represented by
+a curve. Experiment has taught us certain points of this curve.
+In virtue of the principle we have just stated, we believe these
+points may be connected by a continuous graph. We trace this
+graph with the eye. New experiments will furnish us new points
+of the curve. If these points are outside of the graph traced in
+advance, we shall have to modify our curve, but not to abandon
+our principle. Through any points, however numerous they may
+be, a continuous curve may always be passed. Doubtless, if this
+curve is too capricious, we shall be shocked (and we shall even
+suspect errors of experiment), but the principle will not be
+directly put at fault.</p>
+
+<p>Furthermore, among the circumstances of a phenomenon, there
+are some that we regard as negligible, and we shall consider <i>A</i>
+and <i>A&acute;</i> as slightly different if they differ only by these accessory
+circumstances. For instance, I have ascertained that hydrogen
+unites with oxygen under the influence of the electric spark, and
+I am certain that these two gases will unite anew, although the
+longitude of Jupiter may have changed considerably in the
+interval. We assume, for instance, that the state of distant
+bodies can have no sensible influence on terrestrial phenomena,
+and that seems in fact requisite, but there are cases where the
+choice of these practically indifferent circumstances admits of
+more arbitrariness or, if you choose, requires more tact.</p>
+
+<p>One more remark: The principle of induction would be inapplicable
+if there did not exist in nature a great quantity of
+bodies like one another, or almost alike, and if we could not
+infer, for instance, from one bit of phosphorus to another bit of
+phosphorus.</p>
+
+<p>If we reflect on these considerations, the problem of determinism
+and of contingence will appear to us in a new light.<span class='pagenum'><a name="Page_347" id="Page_347">[Pg 347]</a></span></p>
+
+<p>Suppose we were able to embrace the series of all phenomena
+of the universe in the whole sequence of time. We could envisage
+what might be called the <i>sequences</i>; I mean relations between
+antecedent and consequent. I do not wish to speak of constant
+relations or laws, I envisage separately (individually, so to
+speak) the different sequences realized.</p>
+
+<p>We should then recognize that among these sequences there
+are no two altogether alike. But, if the principle of induction,
+as we have just stated it, is true, there will be those almost alike
+and that can be classed alongside one another. In other words,
+it is possible to make a classification of sequences.</p>
+
+<p>It is to the possibility and the legitimacy of such a classification
+that determinism, in the end, reduces. This is all that the
+preceding analysis leaves of it. Perhaps under this modest form
+it will seem less appalling to the moralist.</p>
+
+<p>It will doubtless be said that this is to come back by a detour
+to M. LeRoy's conclusion which a moment ago we seemed to
+reject: we are determinists voluntarily. And in fact all classification
+supposes the active intervention of the classifier. I agree
+that this may be maintained, but it seems to me that this detour
+will not have been useless and will have contributed to enlighten
+us a little.</p>
+
+
+<h4>6. <i>Objectivity of Science</i></h4>
+
+<p>I arrive at the question set by the title of this article: What is
+the objective value of science? And first what should we understand
+by objectivity?</p>
+
+<p>What guarantees the objectivity of the world in which we live
+is that this world is common to us with other thinking beings.
+Through the communications that we have with other men, we
+receive from them ready-made reasonings; we know that these
+reasonings do not come from us and at the same time we recognize
+in them the work of reasonable beings like ourselves. And
+as these reasonings appear to fit the world of our sensations, we
+think we may infer that these reasonable beings have seen the
+same thing as we; thus it is we know we have not been dreaming.</p>
+
+<p>Such, therefore, is the first condition of objectivity; what is
+objective must be common to many minds and consequently transmissible
+from one to the other, and as this transmission can only<span class='pagenum'><a name="Page_348" id="Page_348">[Pg 348]</a></span>
+come about by that 'discourse' which inspires so much distrust
+in M. LeRoy, we are even forced to conclude: no discourse, no
+objectivity.</p>
+
+<p>The sensations of others will be for us a world eternally closed.
+We have no means of verifying that the sensation I call red is
+the same as that which my neighbor calls red.</p>
+
+<p>Suppose that a cherry and a red poppy produce on me the
+sensation <i>A</i> and on him the sensation <i>B</i> and that, on the contrary,
+a leaf produces on me the sensation <i>B</i> and on him the
+sensation <i>A</i>. It is clear we shall never know anything about it;
+since I shall call red the sensation <i>A</i> and green the sensation <i>B</i>,
+while he will call the first green and the second red. In compensation,
+what we shall be able to ascertain is that, for him as
+for me, the cherry and the red poppy produce the <i>same</i> sensation,
+since he gives the same name to the sensations he feels and
+I do the same.</p>
+
+<p>Sensations are therefore intransmissible, or rather all that is
+pure quality in them is intransmissible and forever impenetrable.
+But it is not the same with relations between these sensations.</p>
+
+<p>From this point of view, all that is objective is devoid of all
+quality and is only pure relation. Certes, I shall not go so far
+as to say that objectivity is only pure quantity (this would be
+to particularize too far the nature of the relations in question),
+but we understand how some one could have been carried away
+into saying that the world is only a differential equation.</p>
+
+<p>With due reserve regarding this paradoxical proposition, we
+must nevertheless admit that nothing is objective which is not
+transmissible, and consequently that the relations between the
+sensations can alone have an objective value.</p>
+
+<p>Perhaps it will be said that the esthetic emotion, which is
+common to all mankind, is proof that the qualities of our sensations
+are also the same for all men and hence are objective. But
+if we think about this, we shall see that the proof is not complete;
+what is proved is that this emotion is aroused in John as
+in James by the sensations to which James and John give the
+same name or by the corresponding combinations of these sensations;
+either because this emotion is associated in John with
+the sensation <i>A</i>, which John calls red, while parallelly it is<span class='pagenum'><a name="Page_349" id="Page_349">[Pg 349]</a></span>
+associated in James with the sensation <i>B</i>, which James calls red;
+or better because this emotion is aroused, not by the qualities
+themselves of the sensations, but by the harmonious combination
+of their relations of which we undergo the unconscious
+impression.</p>
+
+<p>Such a sensation is beautiful, not because it possesses such a
+quality, but because it occupies such a place in the woof of our
+associations of ideas, so that it can not be excited without putting
+in motion the 'receiver' which is at the other end of the thread
+and which corresponds to the artistic emotion.</p>
+
+<p>Whether we take the moral, the esthetic or the scientific point
+of view, it is always the same thing. Nothing is objective except
+what is identical for all; now we can only speak of such an
+identity if a comparison is possible, and can be translated into a
+'money of exchange' capable of transmission from one mind to
+another. Nothing, therefore, will have objective value except
+what is transmissible by 'discourse,' that is, intelligible.</p>
+
+<p>But this is only one side of the question. An absolutely disordered
+aggregate could not have objective value since it would
+be unintelligible, but no more can a well-ordered assemblage
+have it, if it does not correspond to sensations really experienced.
+It seems to me superfluous to recall this condition, and I should
+not have dreamed of it, if it had not lately been maintained that
+physics is not an experimental science. Although this opinion
+has no chance of being adopted either by physicists or by philosophers,
+it is well to be warned so as not to let oneself slip over
+the declivity which would lead thither. Two conditions are
+therefore to be fulfilled, and if the first separates reality<a name="FNanchor_11_11" id="FNanchor_11_11"></a><a href="#Footnote_11_11" class="fnanchor">[11]</a> from
+the dream, the second distinguishes it from the romance.</p>
+
+<p>Now what is science? I have explained in the preceding
+article, it is before all a classification, a manner of bringing
+together facts which appearances separate, though they were
+bound together by some natural and hidden kinship. Science,
+in other words, is a system of relations. Now we have just said,
+it is in the relations alone that objectivity must be sought; it<span class='pagenum'><a name="Page_350" id="Page_350">[Pg 350]</a></span>
+would be vain to seek it in beings considered as isolated from one
+another.</p>
+
+<p>To say that science can not have objective value since it teaches
+us only relations, this is to reason backward, since, precisely, it
+is relations alone which can be regarded as objective.</p>
+
+<p>External objects, for instance, for which the word <i>object</i> was
+invented, are really <i>objects</i> and not fleeting and fugitive appearances,
+because they are not only groups of sensations, but groups
+cemented by a constant bond. It is this bond, and this bond
+alone, which is the object in itself, and this bond is a relation.</p>
+
+<p>Therefore, when we ask what is the objective value of science,
+that does not mean: Does science teach us the true nature of
+things? but it means: Does it teach us the true relations of
+things?</p>
+
+<p>To the first question, no one would hesitate to reply, no; but I
+think we may go farther; not only science can not teach us the
+nature of things; but nothing is capable of teaching it to us, and
+if any god knew it, he could not find words to express it. Not
+only can we not divine the response, but if it were given to us
+we could understand nothing of it; I ask myself even whether
+we really understand the question.</p>
+
+<p>When, therefore, a scientific theory pretends to teach us what
+heat is, or what is electricity, or life, it is condemned beforehand;
+all it can give us is only a crude image. It is, therefore, provisional
+and crumbling.</p>
+
+<p>The first question being out of reason, the second remains.
+Can science teach us the true relations of things? What it joins
+together should that be put asunder, what it puts asunder should
+that be joined together?</p>
+
+<p>To understand the meaning of this new question, it is needful
+to refer to what was said above on the conditions of objectivity.
+Have these relations an objective value? That means: Are
+these relations the same for all? Will they still be the same for
+those who shall come after us?</p>
+
+<p>It is clear that they are not the same for the scientist and the
+ignorant person. But that is unimportant, because if the ignorant
+person does not see them all at once, the scientist may succeed
+in making him see them by a series of experiments and<span class='pagenum'><a name="Page_351" id="Page_351">[Pg 351]</a></span>
+reasonings. The thing essential is that there are points on which all
+those acquainted with the experiments made can reach accord.</p>
+
+<p>The question is to know whether this accord will be durable and
+whether it will persist for our successors. It may be asked
+whether the unions that the science of to-day makes will be confirmed
+by the science of to-morrow. To affirm that it will be so
+we can not invoke any <i>a priori</i> reason; but this is a question of
+fact, and science has already lived long enough for us to be able
+to find out by asking its history whether the edifices it builds
+stand the test of time, or whether they are only ephemeral constructions.</p>
+
+<p>Now what do we see? At the first blush, it seems to us that the
+theories last only a day and that ruins upon ruins accumulate.
+To-day the theories are born, to-morrow they are the fashion, the
+day after to-morrow they are classic, the fourth day they are
+superannuated, and the fifth they are forgotten. But if we look
+more closely, we see that what thus succumb are the theories
+properly so called, those which pretend to teach us what things
+are. But there is in them something which usually survives.
+If one of them taught us a true relation, this relation is definitively
+acquired, and it will be found again under a new disguise
+in the other theories which will successively come to reign in
+place of the old.</p>
+
+<p>Take only a single example: The theory of the undulations of
+the ether taught us that light is a motion; to-day fashion favors
+the electromagnetic theory which teaches us that light is a current.
+We do not consider whether we could reconcile them and
+say that light is a current, and that this current is a motion. As
+it is probable in any case that this motion would not be identical
+with that which the partisans of the old theory presume, we might
+think ourselves justified in saying that this old theory is dethroned.
+And yet something of it remains, since between the
+hypothetical currents which Maxwell supposes there are the same
+relations as between the hypothetical motions that Fresnel supposed.
+There is, therefore, something which remains over and
+this something is the essential. This it is which explains how
+we see the present physicists pass without any embarrassment
+from the language of Fresnel to that of Maxwell. Doubtless<span class='pagenum'><a name="Page_352" id="Page_352">[Pg 352]</a></span>
+many connections that were believed well established have been
+abandoned, but the greatest number remain and it would seem
+must remain.</p>
+
+<p>And for these, then, what is the measure of their objectivity?
+Well, it is precisely the same as for our belief in external objects.
+These latter are real in this, that the sensations they make us feel
+appear to us as united to each other by I know not what indestructible
+cement and not by the hazard of a day. In the same
+way science reveals to us between phenomena other bonds finer
+but not less solid; these are threads so slender that they long
+remained unperceived, but once noticed there remains no way of
+not seeing them; they are therefore not less real than those which
+give their reality to external objects; small matter that they are
+more recently known, since neither can perish before the other.</p>
+
+<p>It may be said, for instance, that the ether is no less real than
+any external body; to say this body exists is to say there is
+between the color of this body, its taste, its smell, an intimate
+bond, solid and persistent; to say the ether exists is to say there
+is a natural kinship between all the optical phenomena, and
+neither of the two propositions has less value than the other.</p>
+
+<p>And the scientific syntheses have in a sense even more reality
+than those of the ordinary senses, since they embrace more terms
+and tend to absorb in them the partial syntheses.</p>
+
+<p>It will be said that science is only a classification and that a
+classification can not be true, but convenient. But it is true that
+it is convenient, it is true that it is so not only for me, but for
+all men; it is true that it will remain convenient for our descendants;
+it is true finally that this can not be by chance.</p>
+
+<p>In sum, the sole objective reality consists in the relations of
+things whence results the universal harmony. Doubtless these
+relations, this harmony, could not be conceived outside of a mind
+which conceives them. But they are nevertheless objective because
+they are, will become, or will remain, common to all thinking
+beings.</p>
+
+<p>This will permit us to revert to the question of the rotation of
+the earth which will give us at the same time a chance to make
+clear what precedes by an example.</p>
+<p><span class='pagenum'><a name="Page_353" id="Page_353">[Pg 353]</a></span></p>
+
+<h4>7. <i>The Rotation of the Earth</i></h4>
+
+<p>"... Therefore," have I said in <i>Science and Hypothesis</i>,
+"this affirmation, the earth turns round, has no meaning ... or
+rather these two propositions, the earth turns round, and, it is
+more convenient to suppose that the earth turns round, have one
+and the same meaning."</p>
+
+<p>These words have given rise to the strangest interpretations.
+Some have thought they saw in them the rehabilitation of
+Ptolemy's system, and perhaps the justification of Galileo's
+condemnation.</p>
+
+<p>Those who had read attentively the whole volume could not,
+however, delude themselves. This truth, the earth turns round,
+was put on the same footing as Euclid's postulate, for example.
+Was that to reject it? But better; in the same language it may
+very well be said: These two propositions, the external world
+exists, or, it is more convenient to suppose that it exists, have one
+and the same meaning. So the hypothesis of the rotation of the
+earth would have the same degree of certitude as the very existence
+of external objects.</p>
+
+<p>But after what we have just explained in the fourth part, we
+may go farther. A physical theory, we have said, is by so much
+the more true as it puts in evidence more true relations. In the
+light of this new principle, let us examine the question which
+occupies us.</p>
+
+<p>No, there is no absolute space; these two contradictory propositions:
+'The earth turns round' and 'The earth does not turn
+round' are, therefore, neither of them more true than the other.
+To affirm one while denying the other, <i>in the kinematic sense</i>,
+would be to admit the existence of absolute space.</p>
+
+<p>But if the one reveals true relations that the other hides from
+us, we can nevertheless regard it as physically more true than the
+other, since it has a richer content. Now in this regard no doubt
+is possible.</p>
+
+<p>Behold the apparent diurnal motion of the stars, and the
+diurnal motion of the other heavenly bodies, and besides, the
+flattening of the earth, the rotation of Foucault's pendulum, the
+gyration of cyclones, the trade-winds, what not else? For the<span class='pagenum'><a name="Page_354" id="Page_354">[Pg 354]</a></span>
+Ptolemaist all these phenomena have no bond between them; for
+the Copernican they are produced by the one same cause. In
+saying, the earth turns round, I affirm that all these phenomena
+have an intimate relation, and <i>that is true</i>, and that remains true,
+although there is not and can not be absolute space.</p>
+
+<p>So much for the rotation of the earth upon itself; what shall we
+say of its revolution around the sun? Here again, we have three
+phenomena which for the Ptolemaist are absolutely independent
+and which for the Copernican are referred back to the same
+origin; they are the apparent displacements of the planets on
+the celestial sphere, the aberration of the fixed stars, the parallax
+of these same stars. Is it by chance that all the planets admit an
+inequality whose period is a year, and that this period is precisely
+equal to that of aberration, precisely equal besides to that of
+parallax? To adopt Ptolemy's system is to answer, yes; to adopt
+that of Copernicus is to answer, no; this is to affirm that there is
+a bond between the three phenomena, and that also is true,
+although there is no absolute space.</p>
+
+<p>In Ptolemy's system, the motions of the heavenly bodies can
+not be explained by the action of central forces, celestial
+mechanics is impossible. The intimate relations that celestial
+mechanics reveals to us between all the celestial phenomena are
+true relations; to affirm the immobility of the earth would be to
+deny these relations, that would be to fool ourselves.</p>
+
+<p>The truth for which Galileo suffered remains, therefore, the
+truth, although it has not altogether the same meaning as for
+the vulgar, and its true meaning is much more subtle, more profound
+and more rich.</p>
+
+
+<h4>8. <i>Science for Its Own Sake</i></h4>
+
+<p>Not against M. LeRoy do I wish to defend science for its own
+sake; maybe this is what he condemns, but this is what he cultivates,
+since he loves and seeks truth and could not live without it.
+But I have some thoughts to express.</p>
+
+<p>We can not know all facts and it is necessary to choose those
+which are worthy of being known. According to Tolstoi, scientists
+make this choice at random, instead of making it, which
+would be reasonable, with a view to practical applications. On<span class='pagenum'><a name="Page_355" id="Page_355">[Pg 355]</a></span>
+the contrary, scientists think that certain facts are more interesting
+than others, because they complete an unfinished harmony,
+or because they make one foresee a great number of other facts.
+If they are wrong, if this hierarchy of facts that they implicitly
+postulate is only an idle illusion, there could be no science for its
+own sake, and consequently there could be no science. As for
+me, I believe they are right, and, for example, I have shown above
+what is the high value of astronomical facts, not because they
+are capable of practical applications, but because they are the
+most instructive of all.</p>
+
+<p>It is only through science and art that civilization is of value.
+Some have wondered at the formula: science for its own sake;
+and yet it is as good as life for its own sake, if life is only misery;
+and even as happiness for its own sake, if we do not believe that
+all pleasures are of the same quality, if we do not wish to admit
+that the goal of civilization is to furnish alcohol to people who
+love to drink.</p>
+
+<p>Every act should have an aim. We must suffer, we must work,
+we must pay for our place at the game, but this is for seeing's
+sake; or at the very least that others may one day see.</p>
+
+<p>All that is not thought is pure nothingness; since we can think
+only thoughts and all the words we use to speak of things can
+express only thoughts, to say there is something other than
+thought, is therefore an affirmation which can have no meaning.</p>
+
+<p>And yet&mdash;strange contradiction for those who believe in time&mdash;geologic
+history shows us that life is only a short episode between
+two eternities of death, and that, even in this episode, conscious
+thought has lasted and will last only a moment. Thought is only
+a gleam in the midst of a long night.</p>
+
+<p>But it is this gleam which is everything.</p>
+<p><span class='pagenum'><a name="Page_356" id="Page_356">[Pg 356]</a></span></p>
+
+
+<hr style="width: 100%;" />
+<p><span class='pagenum'><a name="Page_357" id="Page_357">[Pg 357]</a></span></p>
+<p>&nbsp;</p>
+<h1><a name="SCIENCE_AND_METHOD" id="SCIENCE_AND_METHOD"></a><b>SCIENCE AND METHOD</b></h1>
+<p>&nbsp;</p>
+<p><span class='pagenum'><a name="Page_358" id="Page_358">[Pg 358]</a></span></p>
+
+
+<hr style="width: 100%;" />
+<p><span class='pagenum'><a name="Page_359" id="Page_359">[Pg 359]</a></span></p>
+<h3><b>INTRODUCTION</b></h3>
+
+
+<p>I bring together here different studies relating more or less
+directly to questions of scientific methodology. The scientific
+method consists in observing and experimenting; if the scientist
+had at his disposal infinite time, it would only be necessary to
+say to him: 'Look and notice well'; but, as there is not time to
+see everything, and as it is better not to see than to see wrongly,
+it is necessary for him to make choice. The first question, therefore,
+is how he should make this choice. This question presents
+itself as well to the physicist as to the historian; it presents
+itself equally to the mathematician, and the principles which
+should guide each are not without analogy. The scientist conforms
+to them instinctively, and one can, reflecting on these principles,
+foretell the future of mathematics.</p>
+
+<p>We shall understand them better yet if we observe the scientist
+at work, and first of all it is necessary to know the psychologic
+mechanism of invention and, in particular, that of mathematical
+creation. Observation of the processes of the work of
+the mathematician is particularly instructive for the psychologist.</p>
+
+<p>In all the sciences of observation account must be taken of the
+errors due to the imperfections of our senses and our instruments.
+Luckily, we may assume that, under certain conditions,
+these errors are in part self-compensating, so as to disappear in
+the average; this compensation is due to chance. But what is
+chance? This idea is difficult to justify or even to define; and
+yet what I have just said about the errors of observation, shows
+that the scientist can not neglect it. It therefore is necessary to
+give a definition as precise as possible of this concept, so indispensable
+yet so illusive.</p>
+
+<p>These are generalities applicable in sum to all the sciences;
+and for example the mechanism of mathematical invention does
+not differ sensibly from the mechanism of invention in general.
+Later I attack questions relating more particularly to certain
+special sciences and first to pure mathematics.<span class='pagenum'><a name="Page_360" id="Page_360">[Pg 360]</a></span></p>
+
+<p>In the chapters devoted to these, I have to treat subjects
+a little more abstract. I have first to speak of the notion of
+space; every one knows space is relative, or rather every one says
+so, but many think still as if they believed it absolute; it suffices
+to reflect a little however to perceive to what contradictions they
+are exposed.</p>
+
+<p>The questions of teaching have their importance, first in themselves,
+then because reflecting on the best way to make new
+ideas penetrate virgin minds is at the same time reflecting on
+how these notions were acquired by our ancestors, and consequently
+on their true origin, that is to say, in reality on their
+true nature. Why do children usually understand nothing of
+the definitions which satisfy scientists? Why is it necessary to
+give them others? This is the question I set myself in the succeeding
+chapter and whose solution should, I think, suggest useful
+reflections to the philosophers occupied with the logic of
+the sciences.</p>
+
+<p>On the other hand, many geometers believe we can reduce
+mathematics to the rules of formal logic. Unheard-of efforts
+have been made to do this; to accomplish it, some have not
+hesitated, for example, to reverse the historic order of the genesis
+of our conceptions and to try to explain the finite by the infinite.
+I believe I have succeeded in showing, for all those who attack
+the problem unprejudiced, that here there is a fallacious illusion.
+I hope the reader will understand the importance of the question
+and pardon me the aridity of the pages devoted to it.</p>
+
+<p>The concluding chapters relative to mechanics and astronomy
+will be easier to read.</p>
+
+<p>Mechanics seems on the point of undergoing a complete revolution.
+Ideas which appeared best established are assailed by
+bold innovators. Certainly it would be premature to decide in
+their favor at once simply because they are innovators.</p>
+
+<p>But it is of interest to make known their doctrines, and this
+is what I have tried to do. As far as possible I have followed
+the historic order; for the new ideas would seem too astonishing
+unless we saw how they arose.</p>
+
+<p>Astronomy offers us majestic spectacles and raises gigantic
+problems. We can not dream of applying to them directly the<span class='pagenum'><a name="Page_361" id="Page_361">[Pg 361]</a></span>
+experimental method; our laboratories are too small. But analogy
+with phenomena these laboratories permit us to attain may
+nevertheless guide the astronomer. The Milky Way, for example,
+is an assemblage of suns whose movements seem at first
+capricious. But may not this assemblage be compared to that of
+the molecules of a gas, whose properties the kinetic theory of
+gases has made known to us? It is thus by a roundabout way
+that the method of the physicist may come to the aid of the
+astronomer.</p>
+
+<p>Finally I have endeavored to give in a few lines the history
+of the development of French geodesy; I have shown through
+what persevering efforts, and often what dangers, the geodesists
+have procured for us the knowledge we have of the figure of the
+earth. Is this then a question of method? Yes, without doubt,
+this history teaches us in fact by what precautions it is necessary
+to surround a serious scientific operation and how much time and
+pains it costs to conquer one new decimal.</p>
+
+
+
+<hr style="width: 65%;" />
+<p><span class='pagenum'><a name="Page_362" id="Page_362">[Pg 362]</a></span></p>
+<h2><b>BOOK I<br />
+
+<br />
+
+<small>SCIENCE AND THE SCIENTIST</small></b></h2>
+
+
+
+<hr style="width: 65%;" />
+<h3>CHAPTER I</h3>
+
+<h3><span class="smcap">The Choice of Facts</span></h3>
+
+
+<p>Tolstoi somewhere explains why 'science for its own sake' is
+in his eyes an absurd conception. We can not know <i>all</i> facts,
+since their number is practically infinite. It is necessary to
+choose; then we may let this choice depend on the pure caprice
+of our curiosity; would it not be better to let ourselves be guided
+by utility, by our practical and above all by our moral needs;
+have we nothing better to do than to count the number of lady-bugs
+on our planet?</p>
+
+<p>It is clear the word utility has not for him the sense men of
+affairs give it, and following them most of our contemporaries.
+Little cares he for industrial applications, for the marvels of
+electricity or of automobilism, which he regards rather as obstacles
+to moral progress; utility for him is solely what can make
+man better.</p>
+
+<p>For my part, it need scarce be said, I could never be content
+with either the one or the other ideal; I want neither that plutocracy
+grasping and mean, nor that democracy goody and mediocre,
+occupied solely in turning the other cheek, where would dwell
+sages without curiosity, who, shunning excess, would not die of
+disease, but would surely die of ennui. But that is a matter of
+taste and is not what I wish to discuss.</p>
+
+<p>The question nevertheless remains and should fix our attention;
+if our choice can only be determined by caprice or by immediate
+utility, there can be no science for its own sake, and consequently
+no science. But is that true? That a choice must be made is
+incontestable; whatever be our activity, facts go quicker than we,
+and we can not catch them; while the scientist discovers one fact,<span class='pagenum'><a name="Page_363" id="Page_363">[Pg 363]</a></span>
+there happen milliards of milliards in a cubic millimeter of his
+body. To wish to comprise nature in science would be to want
+to put the whole into the part.</p>
+
+<p>But scientists believe there is a hierarchy of facts and that
+among them may be made a judicious choice. They are right,
+since otherwise there would be no science, yet science exists. One
+need only open the eyes to see that the conquests of industry which
+have enriched so many practical men would never have seen the
+light, if these practical men alone had existed and if they had not
+been preceded by unselfish devotees who died poor, who never
+thought of utility, and yet had a guide far other than caprice.</p>
+
+<p>As Mach says, these devotees have spared their successors the
+trouble of thinking. Those who might have worked solely in
+view of an immediate application would have left nothing behind
+them, and, in face of a new need, all must have been begun over
+again. Now most men do not love to think, and this is perhaps
+fortunate when instinct guides them, for most often, when they
+pursue an aim which is immediate and ever the same, instinct
+guides them better than reason would guide a pure intelligence.
+But instinct is routine, and if thought did not fecundate it, it
+would no more progress in man than in the bee or ant. It is
+needful then to think for those who love not thinking, and, as
+they are numerous, it is needful that each of our thoughts be as
+often useful as possible, and this is why a law will be the more
+precious the more general it is.</p>
+
+<p>This shows us how we should choose: the most interesting facts
+are those which may serve many times; these are the facts which
+have a chance of coming up again. We have been so fortunate as
+to be born in a world where there are such. Suppose that instead
+of 60 chemical elements there were 60 milliards of them,
+that they were not some common, the others rare, but that they
+were uniformly distributed. Then, every time we picked up a
+new pebble there would be great probability of its being formed
+of some unknown substance; all that we knew of other pebbles
+would be worthless for it; before each new object we should be
+as the new-born babe; like it we could only obey our caprices or
+our needs. Biologists would be just as much at a loss if there
+were only individuals and no species and if heredity did not
+make sons like their fathers.<span class='pagenum'><a name="Page_364" id="Page_364">[Pg 364]</a></span></p>
+
+<p>In such a world there would be no science; perhaps thought
+and even life would be impossible, since evolution could not there
+develop the preservational instincts. Happily it is not so; like
+all good fortune to which we are accustomed, this is not appreciated
+at its true worth.</p>
+
+<p>Which then are the facts likely to reappear? They are first
+the simple facts. It is clear that in a complex fact a thousand
+circumstances are united by chance, and that only a chance still
+much less probable could reunite them anew. But are there any
+simple facts? And if there are, how recognize them? What
+assurance is there that a thing we think simple does not hide a
+dreadful complexity? All we can say is that we ought to prefer
+the facts which <i>seem</i> simple to those where our crude eye discerns
+unlike elements. And then one of two things: either this simplicity
+is real, or else the elements are so intimately mingled as not
+to be distinguishable. In the first case there is chance of our
+meeting anew this same simple fact, either in all its purity or
+entering itself as element in a complex manifold. In the second
+case this intimate mixture has likewise more chances of recurring
+than a heterogeneous assemblage; chance knows how to mix, it
+knows not how to disentangle, and to make with multiple elements
+a well-ordered edifice in which something is distinguishable, it
+must be made expressly. The facts which appear simple, even
+if they are not so, will therefore be more easily revived by chance.
+This it is which justifies the method instinctively adopted by the
+scientist, and what justifies it still better, perhaps, is that oft-recurring
+facts appear to us simple, precisely because we are
+used to them.</p>
+
+<p>But where is the simple fact? Scientists have been seeking
+it in the two extremes, in the infinitely great and in the infinitely
+small. The astronomer has found it because the distances of
+the stars are immense, so great that each of them appears but
+as a point, so great that the qualitative differences are effaced,
+and because a point is simpler than a body which has form and
+qualities. The physicist on the other hand has sought the elementary
+phenomenon in fictively cutting up bodies into infinitesimal
+cubes, because the conditions of the problem, which undergo
+slow and continuous variation in passing from one point of the<span class='pagenum'><a name="Page_365" id="Page_365">[Pg 365]</a></span>
+body to another, may be regarded as constant in the interior of
+each of these little cubes. In the same way the biologist has
+been instinctively led to regard the cell as more interesting than
+the whole animal, and the outcome has shown his wisdom, since
+cells belonging to organisms the most different are more alike,
+for the one who can recognize their resemblances, than are these
+organisms themselves. The sociologist is more embarrassed; the
+elements, which for him are men, are too unlike, too variable, too
+capricious, in a word, too complex; besides, history never begins
+over again. How then choose the interesting fact, which is that
+which begins again? Method is precisely the choice of facts; it
+is needful then to be occupied first with creating a method, and
+many have been imagined, since none imposes itself, so that sociology
+is the science which has the most methods and the fewest
+results.</p>
+
+<p>Therefore it is by the regular facts that it is proper to begin;
+but after the rule is well established, after it is beyond all doubt,
+the facts in full conformity with it are erelong without interest
+since they no longer teach us anything new. It is then the exception
+which becomes important. We cease to seek resemblances;
+we devote ourselves above all to the differences, and
+among the differences are chosen first the most accentuated, not
+only because they are the most striking, but because they will
+be the most instructive. A simple example will make my thought
+plainer: Suppose one wishes to determine a curve by observing
+some of its points. The practician who concerns himself only
+with immediate utility would observe only the points he might
+need for some special object. These points would be badly distributed
+on the curve; they would be crowded in certain regions,
+rare in others, so that it would be impossible to join them by a
+continuous line, and they would be unavailable for other applications.
+The scientist will proceed differently; as he wishes to
+study the curve for itself, he will distribute regularly the points
+to be observed, and when enough are known he will join them
+by a regular line and then he will have the entire curve. But
+for that how does he proceed? If he has determined an extreme
+point of the curve, he does not stay near this extremity, but goes
+first to the other end; after the two extremities the most instructive
+point will be the mid-point, and so on.<span class='pagenum'><a name="Page_366" id="Page_366">[Pg 366]</a></span></p>
+
+<p>So when a rule is established we should first seek the cases
+where this rule has the greatest chance of failing. Thence,
+among other reasons, come the interest of astronomic facts, and
+the interest of the geologic past; by going very far away in space
+or very far away in time, we may find our usual rules entirely
+overturned, and these grand overturnings aid us the better to see
+or the better to understand the little changes which may happen
+nearer to us, in the little corner of the world where we are called
+to live and act. We shall better know this corner for having
+traveled in distant countries with which we have nothing to do.</p>
+
+<p>But what we ought to aim at is less the ascertainment of resemblances
+and differences than the recognition of likenesses hidden
+under apparent divergences. Particular rules seem at first discordant,
+but looking more closely we see in general that they
+resemble each other; different as to matter, they are alike as to
+form, as to the order of their parts. When we look at them with
+this bias, we shall see them enlarge and tend to embrace everything.
+And this it is which makes the value of certain facts
+which come to complete an assemblage and to show that it is the
+faithful image of other known assemblages.</p>
+
+<p>I will not further insist, but these few words suffice to show
+that the scientist does not choose at random the facts he observes.
+He does not, as Tolstoi says, count the lady-bugs, because, however
+interesting lady-bugs may be, their number is subject to
+capricious variations. He seeks to condense much experience
+and much thought into a slender volume; and that is why a little
+book on physics contains so many past experiences and a thousand
+times as many possible experiences whose result is known
+beforehand.</p>
+
+<p>But we have as yet looked at only one side of the question.
+The scientist does not study nature because it is useful; he studies
+it because he delights in it, and he delights in it because it is
+beautiful. If nature were not beautiful, it would not be worth
+knowing, and if nature were not worth knowing, life would not
+be worth living. Of course I do not here speak of that beauty
+which strikes the senses, the beauty of qualities and of appearances;
+not that I undervalue such beauty, far from it, but it has
+nothing to do with science; I mean that profounder beauty which<span class='pagenum'><a name="Page_367" id="Page_367">[Pg 367]</a></span>
+comes from the harmonious order of the parts and which a pure
+intelligence can grasp. This it is which gives body, a structure
+so to speak, to the iridescent appearances which flatter our senses,
+and without this support the beauty of these fugitive dreams
+would be only imperfect, because it would be vague and always
+fleeting. On the contrary, intellectual beauty is sufficient unto
+itself, and it is for its sake, more perhaps than for the future
+good of humanity, that the scientist devotes himself to long and
+difficult labors.</p>
+
+<p>It is, therefore, the quest of this especial beauty, the sense of
+the harmony of the cosmos, which makes us choose the facts
+most fitting to contribute to this harmony, just as the artist
+chooses from among the features of his model those which perfect
+the picture and give it character and life. And we need not
+fear that this instinctive and unavowed prepossession will turn
+the scientist aside from the search for the true. One may dream
+a harmonious world, but how far the real world will leave it
+behind! The greatest artists that ever lived, the Greeks, made
+their heavens; how shabby it is beside the true heavens, ours!</p>
+
+<p>And it is because simplicity, because grandeur, is beautiful,
+that we preferably seek simple facts, sublime facts, that we delight
+now to follow the majestic course of the stars, now to examine
+with the microscope that prodigious littleness which is
+also a grandeur, now to seek in geologic time the traces of a past
+which attracts because it is far away.</p>
+
+<p>We see too that the longing for the beautiful leads us to the
+same choice as the longing for the useful. And so it is that this
+economy of thought, this economy of effort, which is, according
+to Mach, the constant tendency of science, is at the same time
+a source of beauty and a practical advantage. The edifices that
+we admire are those where the architect has known how to proportion
+the means to the end, where the columns seem to carry
+gaily, without effort, the weight placed upon them, like the
+gracious caryatids of the Erechtheum.</p>
+
+<p>Whence comes this concordance? Is it simply that the things
+which seem to us beautiful are those which best adapt themselves
+to our intelligence, and that consequently they are at the same
+time the implement this intelligence knows best how to use?<span class='pagenum'><a name="Page_368" id="Page_368">[Pg 368]</a></span>
+Or is there here a play of evolution and natural selection? Have
+the peoples whose ideal most conformed to their highest interest
+exterminated the others and taken their place? All pursued
+their ideals without reference to consequences, but while this
+quest led some to destruction, to others it gave empire. One is
+tempted to believe it. If the Greeks triumphed over the barbarians
+and if Europe, heir of Greek thought, dominates the
+world, it is because the savages loved loud colors and the clamorous
+tones of the drum which occupied only their senses, while the
+Greeks loved the intellectual beauty which hides beneath sensuous
+beauty, and this intellectual beauty it is which makes intelligence
+sure and strong.</p>
+
+<p>Doubtless such a triumph would horrify Tolstoi, and he would
+not like to acknowledge that it might be truly useful. But this
+disinterested quest of the true for its own beauty is sane also and
+able to make man better. I well know that there are mistakes,
+that the thinker does not always draw thence the serenity he
+should find therein, and even that there are scientists of bad
+character. Must we, therefore, abandon science and study only
+morals? What! Do you think the moralists themselves are irreproachable
+when they come down from their pedestal?</p>
+
+
+
+<hr style="width: 65%;" />
+<p><span class='pagenum'><a name="Page_369" id="Page_369">[Pg 369]</a></span></p>
+<h3>CHAPTER II</h3>
+
+<h3><span class="smcap">The Future of Mathematics</span></h3>
+
+
+<p>To foresee the future of mathematics, the true method is to
+study its history and its present state.</p>
+
+<p>Is this not for us mathematicians in a way a professional procedure?
+We are accustomed to <i>extrapolate</i>, which is a means
+of deducing the future from the past and present, and as we well
+know what this amounts to, we run no risk of deceiving ourselves
+about the range of the results it gives us.</p>
+
+<p>We have had hitherto prophets of evil. They blithely reiterate
+that all problems capable of solution have already been solved,
+and that nothing is left but gleaning. Happily the case of the
+past reassures us. Often it was thought all problems were solved
+or at least an inventory was made of all admitting solution.
+And then the sense of the word solution enlarged, the insoluble
+problems became the most interesting of all, and others unforeseen
+presented themselves. For the Greeks a good solution was
+one employing only ruler and compasses; then it became one
+obtained by the extraction of roots, then one using only algebraic
+or logarithmic functions. The pessimists thus found themselves
+always outflanked, always forced to retreat, so that at present I
+think there are no more.</p>
+
+<p>My intention, therefore, is not to combat them, as they are
+dead; we well know that mathematics will continue to develop,
+but the question is how, in what direction? You will answer,
+'in every direction,' and that is partly true; but if it were
+wholly true it would be a little appalling. Our riches would
+soon become encumbering and their accumulation would produce
+a medley as impenetrable as the unknown true was for the
+ignorant.</p>
+
+<p>The historian, the physicist, even, must make a choice among
+facts; the head of the scientist, which is only a corner of the
+universe, could never contain the universe entire; so that among
+the innumerable facts nature offers, some will be passed by,
+others retained.<span class='pagenum'><a name="Page_370" id="Page_370">[Pg 370]</a></span></p>
+
+<p>Just so, <i>a fortiori</i>, in mathematics; no more can the geometer
+hold fast pell-mell all the facts presenting themselves to him;
+all the more because he it is, almost I had said his caprice, that
+creates these facts. He constructs a wholly new combination by
+putting together its elements; nature does not in general give it
+to him ready made.</p>
+
+<p>Doubtless it sometimes happens that the mathematician undertakes
+a problem to satisfy a need in physics; that the physicist
+or engineer asks him to calculate a number for a certain application.
+Shall it be said that we geometers should limit ourselves
+to awaiting orders, and, in place of cultivating our science for
+our own delectation, try only to accommodate ourselves to the
+wants of our patrons? If mathematics has no other object besides
+aiding those who study nature, it is from these we should
+await orders. Is this way of looking at it legitimate? Certainly
+not; if we had not cultivated the exact sciences for themselves,
+we should not have created mathematics the instrument, and the
+day the call came from the physicist we should have been
+helpless.</p>
+
+<p>Nor do the physicists wait to study a phenomenon until some
+urgent need of material life has made it a necessity for them;
+and they are right. If the scientists of the eighteenth century
+had neglected electricity as being in their eyes only a curiosity
+without practical interest, we should have had in the twentieth
+century neither telegraphy, nor electro-chemistry, nor electro-technics.
+The physicists, compelled to choose, are therefore not
+guided in their choice solely by utility. How then do they choose
+between the facts of nature? We have explained it in the preceding
+chapter: the facts which interest them are those capable
+of leading to the discovery of a law, and so they are analogous
+to many other facts which do not seem to us isolated, but closely
+grouped with others. The isolated fact attracts all eyes, those of
+the layman as well as of the scientist. But what the genuine
+physicist alone knows how to see, is the bond which unites many
+facts whose analogy is profound but hidden. The story of Newton's
+apple is probably not true, but it is symbolic; let us speak
+of it then as if it were true. Well then, we must believe that
+before Newton plenty of men had seen apples fall; not one knew<span class='pagenum'><a name="Page_371" id="Page_371">[Pg 371]</a></span>
+how to conclude anything therefrom. Facts would be sterile
+were there not minds capable of choosing among them, discerning
+those behind which something was hidden, and of recognizing
+what is hiding, minds which under the crude fact perceive the
+soul of the fact.</p>
+
+<p>We find just the same thing in mathematics. From the varied
+elements at our disposal we can get millions of different combinations;
+but one of these combinations, in so far as it is isolated,
+is absolutely void of value. Often we have taken great pains to
+construct it, but it serves no purpose, if not perhaps to furnish a
+task in secondary education. Quite otherwise will it be when
+this combination shall find place in a class of analogous combinations
+and we shall have noticed this analogy. We are no longer
+in the presence of a fact, but of a law. And upon that day the
+real discoverer will not be the workman who shall have patiently
+built up certain of these combinations; it will be he who brings
+to light their kinship. The first will have seen merely the crude
+fact, only the other will have perceived the soul of the fact.
+Often to fix this kinship it suffices him to make a new word, and
+this word is creative. The history of science furnishes us a
+crowd of examples familiar to all.</p>
+
+<p>The celebrated Vienna philosopher Mach has said that the r&ocirc;le
+of science is to produce economy of thought, just as machines
+produce economy of effort. And that is very true. The savage
+reckons on his fingers or by heaping pebbles. In teaching children
+the multiplication table we spare them later innumerable
+pebble bunchings. Some one has already found out, with pebbles
+or otherwise, that 6 times 7 is 42 and has had the idea of noting
+the result, and so we need not do it over again. He did not
+waste his time even if he reckoned for pleasure: his operation
+took him only two minutes; it would have taken in all two milliards
+if a milliard men had had to do it over after him.</p>
+
+<p>The importance of a fact then is measured by its yield, that is
+to say, by the amount of thought it permits us to spare.</p>
+
+<p>In physics the facts of great yield are those entering into a
+very general law, since from it they enable us to foresee a great
+number of others, and just so it is in mathematics. Suppose I
+have undertaken a complicated calculation and laboriously<span class='pagenum'><a name="Page_372" id="Page_372">[Pg 372]</a></span>
+reached a result: I shall not be compensated for my trouble if
+thereby I have not become capable of foreseeing the results of
+other analogous calculations and guiding them with a certainty
+that avoids the gropings to which one must be resigned in a
+first attempt. On the other hand, I shall not have wasted my
+time if these gropings themselves have ended by revealing to me
+the profound analogy of the problem just treated with a much
+more extended class of other problems; if they have shown me
+at once the resemblances and differences of these, if in a word
+they have made me perceive the possibility of a generalization.
+Then it is not a new result I have won, it is a new power.</p>
+
+<p>The simple example that comes first to mind is that of an algebraic
+formula which gives us the solution of a type of numeric
+problems when finally we replace the letters by numbers. Thanks
+to it, a single algebraic calculation saves us the pains of ceaselessly
+beginning over again new numeric calculations. But this
+is only a crude example; we all know there are analogies inexpressible
+by a formula and all the more precious.</p>
+
+<p>A new result is of value, if at all, when in unifying elements
+long known but hitherto separate and seeming strangers one to
+another it suddenly introduces order where apparently disorder
+reigned. It then permits us to see at a glance each of these
+elements and its place in the assemblage. This new fact is not
+merely precious by itself, but it alone gives value to all the old
+facts it combines. Our mind is weak as are the senses; it would
+lose itself in the world's complexity were this complexity not harmonious;
+like a near-sighted person, it would see only the details
+and would be forced to forget each of these details before examining
+the following, since it would be incapable of embracing all.
+The only facts worthy our attention are those which introduce
+order into this complexity and so make it accessible.</p>
+
+<p>Mathematicians attach great importance to the elegance of
+their methods and their results. This is not pure dilettantism.
+What is it indeed that gives us the feeling of elegance in a solution,
+in a demonstration? It is the harmony of the diverse parts,
+their symmetry, their happy balance; in a word it is all that
+introduces order, all that gives unity, that permits us to see
+clearly and to comprehend at once both the <i>ensemble</i> and the<span class='pagenum'><a name="Page_373" id="Page_373">[Pg 373]</a></span>
+details. But this is exactly what yields great results; in fact the
+more we see this aggregate clearly and at a single glance, the
+better we perceive its analogies with other neighboring objects,
+consequently the more chances we have of divining the possible
+generalizations. Elegance may produce the feeling of the unforeseen
+by the unexpected meeting of objects we are not accustomed
+to bring together; there again it is fruitful, since it thus unveils
+for us kinships before unrecognized. It is fruitful even when it
+results only from the contrast between the simplicity of the
+means and the complexity of the problem set; it makes us then
+think of the reason for this contrast and very often makes us
+see that chance is not the reason; that it is to be found in some
+unexpected law. In a word, the feeling of mathematical elegance
+is only the satisfaction due to any adaptation of the solution
+to the needs of our mind, and it is because of this very
+adaptation that this solution can be for us an instrument. Consequently
+this esthetic satisfaction is bound up with the economy
+of thought. Again the comparison of the Erechtheum
+comes to my mind, but I must not use it too often.</p>
+
+<p>It is for the same reason that, when a rather long calculation
+has led to some simple and striking result, we are not satisfied
+until we have shown that we should have been <i>able to foresee</i>,
+if not this entire result, at least its most characteristic traits.
+Why? What prevents our being content with a calculation
+which has told us, it seems, all we wished to know? It is because,
+in analogous cases, the long calculation might not again
+avail, and that this is not so about the reasoning often half intuitive
+which would have enabled us to foresee. This reasoning
+being short, we see at a single glance all its parts, so that we immediately
+perceive what must be changed to adapt it to all the
+problems of the same nature which can occur. And then it
+enables us to foresee if the solution of these problems will be
+simple, it shows us at least if the calculation is worth undertaking.</p>
+
+<p>What we have just said suffices to show how vain it would be
+to seek to replace by any mechanical procedure the free initiative
+of the mathematician. To obtain a result of real value, it is not
+enough to grind out calculations, or to have a machine to put<span class='pagenum'><a name="Page_374" id="Page_374">[Pg 374]</a></span>
+things in order; it is not order alone, it is unexpected order,
+which is worth while. The machine may gnaw on the crude fact,
+the soul of the fact will always escape it.</p>
+
+<p>Since the middle of the last century, mathematicians are more
+and more desirous of attaining absolute rigor; they are right,
+and this tendency will be more and more accentuated. In mathematics
+rigor is not everything, but without it there is nothing.
+A demonstration which is not rigorous is nothingness. I think
+no one will contest this truth. But if it were taken too literally,
+we should be led to conclude that before 1820, for example, there
+was no mathematics; this would be manifestly excessive; the
+geometers of that time understood voluntarily what we explain
+by prolix discourse. This does not mean that they did not see it
+at all; but they passed over it too rapidly, and to see it well
+would have necessitated taking the pains to say it.</p>
+
+<p>But is it always needful to say it so many times? Those who
+were the first to emphasize exactness before all else have given
+us arguments that we may try to imitate; but if the demonstrations
+of the future are to be built on this model, mathematical
+treatises will be very long; and if I fear the lengthenings, it is
+not solely because I deprecate encumbering libraries, but because
+I fear that in being lengthened out, our demonstrations may lose
+that appearance of harmony whose usefulness I have just
+explained.</p>
+
+<p>The economy of thought is what we should aim at, so it is not
+enough to supply models for imitation. It is needful for those
+after us to be able to dispense with these models and, in place of
+repeating an argument already made, summarize it in a few
+words. And this has already been attained at times. For instance,
+there was a type of reasoning found everywhere, and
+everywhere alike. They were perfectly exact but long. Then
+all at once the phrase 'uniformity of convergence' was hit upon
+and this phrase made those arguments needless; we were no
+longer called upon to repeat them, since they could be understood.
+Those who conquer difficulties then do us a double service:
+first they teach us to do as they at need, but above all they
+enable us as often as possible to avoid doing as they, yet without
+sacrifice of exactness.</p>
+<p><span class='pagenum'><a name="Page_375" id="Page_375">[Pg 375]</a></span></p>
+
+<p>We have just seen by one example the importance of words in
+mathematics, but many others could be cited. It is hard to believe
+how much a well-chosen word can economize thought, as
+Mach says. Perhaps I have already said somewhere that mathematics
+is the art of giving the same name to different things. It
+is proper that these things, differing in matter, be alike in
+form, that they may, so to speak, run in the same mold. When
+the language has been well chosen, we are astonished to see that
+all the proofs made for a certain object apply immediately to
+many new objects; there is nothing to change, not even the words,
+since the names have become the same.</p>
+
+<p>A well-chosen word usually suffices to do away with the exceptions
+from which the rules stated in the old way suffer; this
+is why we have created negative quantities, imaginaries, points
+at infinity, and what not. And exceptions, we must not forget,
+are pernicious because they hide the laws.</p>
+
+<p>Well, this is one of the characteristics by which we recognize
+the facts which yield great results. They are those which allow
+of these happy innovations of language. The crude fact then
+is often of no great interest; we may point it out many times
+without having rendered great service to science. It takes value
+only when a wiser thinker perceives the relation for which it
+stands, and symbolizes it by a word.</p>
+
+<p>Moreover the physicists do just the same. They have invented
+the word 'energy,' and this word has been prodigiously
+fruitful, because it also made the law by eliminating the exceptions,
+since it gave the same name to things differing in matter
+and like in form.</p>
+
+<p>Among words that have had the most fortunate influence I
+would select 'group' and 'invariant.' They have made us see
+the essence of many mathematical reasonings; they have shown
+us in how many cases the old mathematicians considered groups
+without knowing it, and how, believing themselves far from one
+another, they suddenly found themselves near without knowing
+why.</p>
+
+<p>To-day we should say that they had dealt with isomorphic
+groups. We now know that in a group the matter is of little
+interest, the form alone counts, and that when we know a group<span class='pagenum'><a name="Page_376" id="Page_376">[Pg 376]</a></span>
+we thus know all the isomorphic groups; and thanks to these
+words 'group' and 'isomorphism,' which condense in a few syllables
+this subtile rule and quickly make it familiar to all minds,
+the transition is immediate and can be done with every economy
+of thought effort. The idea of group besides attaches to that
+of transformation. Why do we put such a value on the invention
+of a new transformation? Because from a single theorem
+it enables us to get ten or twenty; it has the same value as
+a zero adjoined to the right of a whole number.</p>
+
+<p>This then it is which has hitherto determined the direction of
+mathematical advance, and just as certainly will determine it in
+the future. But to this end the nature of the problems which
+come up contributes equally. We can not forget what must be
+our aim. In my opinion this aim is double. Our science borders
+upon both philosophy and physics, and we work for our two
+neighbors; so we have always seen and shall still see mathematicians
+advancing in two opposite directions.</p>
+
+<p>On the one hand, mathematical science must reflect upon itself,
+and that is useful since reflecting on itself is reflecting on the
+human mind which has created it, all the more because it is the
+very one of its creations for which it has borrowed least from
+without. This is why certain mathematical speculations are
+useful, such as those devoted to the study of the postulates, of
+unusual geometries, of peculiar functions. The more these speculations
+diverge from ordinary conceptions, and consequently
+from nature and applications, the better they show us what the
+human mind can create when it frees itself more and more from
+the tyranny of the external world, the better therefore they let
+us know it in itself.</p>
+
+<p>But it is toward the other side, the side of nature, that we must
+direct the bulk of our army. There we meet the physicist or
+the engineer, who says to us: "Please integrate this differential
+equation for me; I might need it in a week in view of a construction
+which should be finished by that time." "This equation,"
+we answer, "does not come under one of the integrable types;
+you know there are not many." "Yes, I know; but then what
+good are you?" Usually to understand each other is enough;
+the engineer in reality does not need the integral in finite terms;<span class='pagenum'><a name="Page_377" id="Page_377">[Pg 377]</a></span>
+he needs to know the general look of the integral function, or he
+simply wants a certain number which could readily be deduced
+from this integral if it were known. Usually it is not known,
+but the number can be calculated without it if we know exactly
+what number the engineer needs and with what approximation.</p>
+
+<p>Formerly an equation was considered solved only when its
+solution had been expressed by aid of a finite number of known
+functions; but that is possible scarcely once in a hundred times.
+What we always can do, or rather what we should always seek
+to do, is to solve the problem <i>qualitatively</i> so to speak; that is to
+say, seek to know the general form of the curve which represents
+the unknown function.</p>
+
+<p>It remains to find the <i>quantitative</i> solution of the problem;
+but if the unknown can not be determined by a finite calculation,
+it may always be represented by a convergent infinite series
+which enables us to calculate it. Can that be regarded as a true
+solution? We are told that Newton sent Leibnitz an anagram
+almost like this: aaaaabbbeeeeij, etc. Leibnitz naturally understood
+nothing at all of it; but we, who have the key, know that
+this anagram meant, translated into modern terms: "I can integrate
+all differential equations"; and we are tempted to say that
+Newton had either great luck or strange delusions. He merely
+wished to say he could form (by the method of indeterminate
+coefficients) a series of powers formally satisfying the proposed
+equation.</p>
+
+<p>Such a solution would not satisfy us to-day, and for two
+reasons: because the convergence is too slow and because the
+terms follow each other without obeying any law. On the contrary,
+the series &#920; seems to us to leave nothing to be desired, first
+because it converges very quickly (this is for the practical man
+who wishes to get at a number as quickly as possible) and next
+because we see at a glance the law of the terms (this is to satisfy
+the esthetic need of the theorist).</p>
+
+<p>But then there are no longer solved problems and others
+which are not; there are only problems <i>more or less</i> solved,
+according as they are solved by a series converging more or less
+rapidly, or ruled by a law more or less harmonious. It often
+happens however that an imperfect solution guides us toward a<span class='pagenum'><a name="Page_378" id="Page_378">[Pg 378]</a></span>
+better one. Sometimes the series converges so slowly that the
+computation is impracticable and we have only succeeded in
+proving the possibility of the problem.</p>
+
+<p>And then the engineer finds this a mockery, and justly, since
+it will not aid him to complete his construction by the date fixed.
+He little cares to know if it will benefit engineers of the twenty-second
+century. But as for us, we think differently and we are
+sometimes happier to have spared our grandchildren a day's
+work than to have saved our contemporaries an hour.</p>
+
+<p>Sometimes by groping, empirically, so to speak, we reach a
+formula sufficiently convergent. "What more do you want?"
+says the engineer. And yet, in spite of all, we are not satisfied;
+we should have liked <i>to foresee</i> that convergence. Why? Because
+if we had known how to foresee it once, we would know how
+to foresee it another time. We have succeeded; that is a small
+matter in our eyes if we can not validly expect to do so again.</p>
+
+<p>In proportion as science develops, its total comprehension
+becomes more difficult; then we seek to cut it in pieces and to
+be satisfied with one of these pieces: in a word, to specialize.
+If we went on in this way, it would be a grievous obstacle to the
+progress of science. As we have said, it is by unexpected union
+between its diverse parts that it progresses. To specialize too
+much would be to forbid these drawings together. It is to be
+hoped that congresses like those of Heidelberg and Rome, by
+putting us in touch with one another, will open for us vistas over
+neighboring domains and oblige us to compare them with our
+own, to range somewhat abroad from our own little village; thus
+they will be the best remedy for the danger just mentioned.</p>
+
+<p>But I have lingered too long over generalities; it is time to
+enter into detail.</p>
+
+<p>Let us pass in review the various special sciences which combined
+make mathematics; let us see what each has accomplished,
+whither it tends and what we may hope from it. If the preceding
+views are correct, we should see that the greatest advances
+in the past have happened when two of these sciences have united,
+when we have become conscious of the similarity of their form,
+despite the difference of their matter, when they have so modeled
+themselves upon each other that each could profit by the other's<span class='pagenum'><a name="Page_379" id="Page_379">[Pg 379]</a></span>
+conquests. We should at the same time foresee in combinations
+of the same sort the progress of the future.</p>
+
+
+<h3><span class="smcap">Arithmetic</span></h3>
+
+<p>Progress in arithmetic has been much slower than in algebra
+and analysis, and it is easy to see why. The feeling of continuity
+is a precious guide which the arithmetician lacks; each whole
+number is separated from the others&mdash;it has, so to speak, its own
+individuality. Each of them is a sort of exception and this is
+why general theorems are rarer in the theory of numbers; this
+is also why those which exist are more hidden and longer elude
+the searchers.</p>
+
+<p>If arithmetic is behind algebra and analysis, the best thing for
+it to do is to seek to model itself upon these sciences so as to
+profit by their advance. The arithmetician ought therefore to
+take as guide the analogies with algebra. These analogies are
+numerous and if, in many cases, they have not yet been studied
+sufficiently closely to become utilizable, they at least have long
+been foreseen, and even the language of the two sciences shows
+they have been recognized. Thus we speak of transcendent
+numbers and thus we account for the future classification of
+these numbers already having as model the classification of transcendent
+functions, and still we do not as yet very well see how
+to pass from one classification to the other; but had it been seen,
+it would already have been accomplished and would no longer
+be the work of the future.</p>
+
+<p>The first example that comes to my mind is the theory of congruences,
+where is found a perfect parallelism to the theory of
+algebraic equations. Surely we shall succeed in completing this
+parallelism, which must hold for instance between the theory of
+algebraic curves and that of congruences with two variables.
+And when the problems relative to congruences with several
+variables shall be solved, this will be a first step toward the solution
+of many questions of indeterminate analysis.</p>
+
+
+<h3><span class="smcap">Algebra</span></h3>
+
+<p>The theory of algebraic equations will still long hold the attention
+of geometers; numerous and very different are the sides
+whence it may be attacked.<span class='pagenum'><a name="Page_380" id="Page_380">[Pg 380]</a></span></p>
+
+<p>We need not think algebra is ended because it gives us rules
+to form all possible combinations; it remains to find the interesting
+combinations, those which satisfy such and such a condition.
+Thus will be formed a sort of indeterminate analysis where the
+unknowns will no longer be whole numbers, but polynomials.
+This time it is algebra which will model itself upon arithmetic,
+following the analogy of the whole number to the integral polynomial
+with any coefficients or to the integral polynomial with
+integral coefficients.</p>
+
+
+<h3><span class="smcap">Geometry</span></h3>
+
+<p>It looks as if geometry could contain nothing which is not
+already included in algebra or analysis; that geometric facts are
+only algebraic or analytic facts expressed in another language.
+It might then be thought that after our review there would
+remain nothing more for us to say relating specially to geometry.
+This would be to fail to recognize the importance of well-constructed
+language, not to comprehend what is added to the things
+themselves by the method of expressing these things and consequently
+of grouping them.</p>
+
+<p>First the geometric considerations lead us to set ourselves new
+problems; these may be, if you choose, analytic problems, but
+such as we never would have set ourselves in connection with
+analysis. Analysis profits by them however, as it profits by those
+it has to solve to satisfy the needs of physics.</p>
+
+<p>A great advantage of geometry lies in the fact that in it the
+senses can come to the aid of thought, and help find the path to
+follow, and many minds prefer to put the problems of analysis
+into geometric form. Unhappily our senses can not carry us very
+far, and they desert us when we wish to soar beyond the classical
+three dimensions. Does this mean that, beyond the restricted
+domain wherein they seem to wish to imprison us, we should
+rely only on pure analysis and that all geometry of more than
+three dimensions is vain and objectless? The greatest masters
+of a preceding generation would have answered 'yes'; to-day we
+are so familiarized with this notion that we can speak of it, even
+in a university course, without arousing too much astonishment.</p>
+
+<p>But what good is it? That is easy to see: First it gives us a<span class='pagenum'><a name="Page_381" id="Page_381">[Pg 381]</a></span>
+very convenient terminology, which expresses concisely what the
+ordinary analytic language would say in prolix phrases. Moreover,
+this language makes us call like things by the same name
+and emphasize analogies it will never again let us forget. It
+enables us therefore still to find our way in this space which is
+too big for us and which we can not see, always recalling visible
+space, which is only an imperfect image of it doubtless, but which
+is nevertheless an image. Here again, as in all the preceding
+examples, it is analogy with the simple which enables us to comprehend
+the complex.</p>
+
+<p>This geometry of more than three dimensions is not a simple
+analytic geometry; it is not purely quantitative, but qualitative
+also, and it is in this respect above all that it becomes interesting.
+There is a science called <i>analysis situs</i> and which has for its
+object the study of the positional relations of the different elements
+of a figure, apart from their sizes. This geometry is purely
+qualitative; its theorems would remain true if the figures, instead
+of being exact, were roughly imitated by a child. We may also
+make an <i>analysis situs</i> of more than three dimensions. The
+importance of <i>analysis situs</i> is enormous and can not be too much
+emphasized; the advantage obtained from it by Riemann, one of
+its chief creators, would suffice to prove this. We must achieve
+its complete construction in the higher spaces; then we shall have
+an instrument which will enable us really to see in hyperspace
+and supplement our senses.</p>
+
+<p>The problems of <i>analysis situs</i> would perhaps not have suggested
+themselves if the analytic language alone had been spoken;
+or rather, I am mistaken, they would have occurred surely, since
+their solution is essential to a crowd of questions in analysis, but
+they would have come singly, one after another, and without our
+being able to perceive their common bond.</p>
+
+
+<h3><span class="smcap">Cantorism</span></h3>
+
+<p>I have spoken above of our need to go back continually to the
+first principles of our science, and of the advantage of this for
+the study of the human mind. This need has inspired two endeavors
+which have taken a very prominent place in the most
+recent annals of mathematics. The first is Cantorism, which has<span class='pagenum'><a name="Page_382" id="Page_382">[Pg 382]</a></span>
+rendered our science such conspicuous service. Cantor introduced
+into science a new way of considering mathematical infinity.
+One of the characteristic traits of Cantorism is that in
+place of going up to the general by building up constructions
+more and more complicated and defining by construction, it starts
+from the <i>genus supremum</i> and defines only, as the scholastics
+would have said, <i>per genus proximum et differentiam specificam</i>.
+Thence comes the horror it has sometimes inspired in certain
+minds, for instance in Hermite, whose favorite idea was to compare
+the mathematical to the natural sciences. With most of
+us these prejudices have been dissipated, but it has come to
+pass that we have encountered certain paradoxes, certain apparent
+contradictions that would have delighted Zeno, the Eleatic
+and the school of Megara. And then each must seek the remedy.
+For my part, I think, and I am not the only one, that the important
+thing is never to introduce entities not completely definable
+in a finite number of words. Whatever be the cure adopted, we
+may promise ourselves the joy of the doctor called in to follow
+a beautiful pathologic case.</p>
+
+
+<h3><span class="smcap">The Investigation of the Postulates</span></h3>
+
+<p>On the other hand, efforts have been made to enumerate the
+axioms and postulates, more or less hidden, which serve as foundation
+to the different theories of mathematics. Professor Hilbert
+has obtained the most brilliant results. It seems at first that this
+domain would be very restricted and there would be nothing
+more to do when the inventory should be ended, which could not
+take long. But when we shall have enumerated all, there will be
+many ways of classifying all; a good librarian always finds something
+to do, and each new classification will be instructive for
+the philosopher.</p>
+
+<p>Here I end this review which I could not dream of making
+complete. I think these examples will suffice to show by what
+mechanism the mathematical sciences have made their progress
+in the past and in what direction they must advance in the future.</p>
+
+
+
+<hr style="width: 65%;" />
+<p><span class='pagenum'><a name="Page_383" id="Page_383">[Pg 383]</a></span></p>
+<h3>CHAPTER III</h3>
+
+<h3><span class="smcap">Mathematical Creation</span></h3>
+
+
+<p>The genesis of mathematical creation is a problem which
+should intensely interest the psychologist. It is the activity in
+which the human mind seems to take least from the outside
+world, in which it acts or seems to act only of itself and on itself,
+so that in studying the procedure of geometric thought we may
+hope to reach what is most essential in man's mind.</p>
+
+<p>This has long been appreciated, and some time back the journal
+called <i>L'enseignement math&eacute;matique</i>, edited by Laisant and
+Fehr, began an investigation of the mental habits and methods
+of work of different mathematicians. I had finished the main
+outlines of this article when the results of that inquiry were
+published, so I have hardly been able to utilize them and shall
+confine myself to saying that the majority of witnesses confirm
+my conclusions; I do not say all, for when the appeal is to universal
+suffrage unanimity is not to be hoped.</p>
+
+<p>A first fact should surprise us, or rather would surprise us if
+we were not so used to it. How does it happen there are people
+who do not understand mathematics? If mathematics invokes
+only the rules of logic, such as are accepted by all normal minds;
+if its evidence is based on principles common to all men, and that
+none could deny without being mad, how does it come about that
+so many persons are here refractory?</p>
+
+<p>That not every one can invent is nowise mysterious. That
+not every one can retain a demonstration once learned may also
+pass. But that not every one can understand mathematical
+reasoning when explained appears very surprising when we think
+of it. And yet those who can follow this reasoning only with
+difficulty are in the majority: that is undeniable, and will surely
+not be gainsaid by the experience of secondary-school teachers.</p>
+
+<p>And further: how is error possible in mathematics? A sane
+mind should not be guilty of a logical fallacy, and yet there are<span class='pagenum'><a name="Page_384" id="Page_384">[Pg 384]</a></span>
+very fine minds who do not trip in brief reasoning such as occurs
+in the ordinary doings of life, and who are incapable of following
+or repeating without error the mathematical demonstrations
+which are longer, but which after all are only an accumulation
+of brief reasonings wholly analogous to those they make so easily.
+Need we add that mathematicians themselves are not infallible?</p>
+
+<p>The answer seems to me evident. Imagine a long series of
+syllogisms, and that the conclusions of the first serve as premises
+of the following: we shall be able to catch each of these syllogisms,
+and it is not in passing from premises to conclusion that
+we are in danger of deceiving ourselves. But between the
+moment in which we first meet a proposition as conclusion of one
+syllogism, and that in which we reencounter it as premise of
+another syllogism occasionally some time will elapse, several links
+of the chain will have unrolled; so it may happen that we have
+forgotten it, or worse, that we have forgotten its meaning. So
+it may happen that we replace it by a slightly different proposition,
+or that, while retaining the same enunciation, we attribute
+to it a slightly different meaning, and thus it is that we are
+exposed to error.</p>
+
+<p>Often the mathematician uses a rule. Naturally he begins by
+demonstrating this rule; and at the time when this proof is fresh
+in his memory he understands perfectly its meaning and its bearing,
+and he is in no danger of changing it. But subsequently he
+trusts his memory and afterward only applies it in a mechanical
+way; and then if his memory fails him, he may apply it all
+wrong. Thus it is, to take a simple example, that we sometimes
+make slips in calculation because we have forgotten our multiplication
+table.</p>
+
+<p>According to this, the special aptitude for mathematics would
+be due only to a very sure memory or to a prodigious force of
+attention. It would be a power like that of the whist-player who
+remembers the cards played; or, to go up a step, like that of the
+chess-player who can visualize a great number of combinations
+and hold them in his memory. Every good mathematician ought
+to be a good chess-player, and inversely; likewise he should be a
+good computer. Of course that sometimes happens; thus Gauss<span class='pagenum'><a name="Page_385" id="Page_385">[Pg 385]</a></span>
+was at the same time a geometer of genius and a very precocious
+and accurate computer.</p>
+
+<p>But there are exceptions; or rather I err; I can not call them
+exceptions without the exceptions being more than the rule.
+Gauss it is, on the contrary, who was an exception. As for myself,
+I must confess, I am absolutely incapable even of adding
+without mistakes. In the same way I should be but a poor chess-player;
+I would perceive that by a certain play I should expose
+myself to a certain danger; I would pass in review several other
+plays, rejecting them for other reasons, and then finally I should
+make the move first examined, having meantime forgotten the
+danger I had foreseen.</p>
+
+<p>In a word, my memory is not bad, but it would be insufficient
+to make me a good chess-player. Why then does it not fail me in
+a difficult piece of mathematical reasoning where most chess-players
+would lose themselves? Evidently because it is guided
+by the general march of the reasoning. A mathematical demonstration
+is not a simple juxtaposition of syllogisms, it is syllogisms
+<i>placed in a certain order</i>, and the order in which these
+elements are placed is much more important than the elements
+themselves. If I have the feeling, the intuition, so to speak, of
+this order, so as to perceive at a glance the reasoning as a whole,
+I need no longer fear lest I forget one of the elements, for each
+of them will take its allotted place in the array, and that without
+any effort of memory on my part.</p>
+
+<p>It seems to me then, in repeating a reasoning learned, that I
+could have invented it. This is often only an illusion; but even
+then, even if I am not so gifted as to create it by myself, I myself
+re-invent it in so far as I repeat it.</p>
+
+<p>We know that this feeling, this intuition of mathematical
+order, that makes us divine hidden harmonies and relations, can
+not be possessed by every one. Some will not have either this
+delicate feeling so difficult to define, or a strength of memory
+and attention beyond the ordinary, and then they will be absolutely
+incapable of understanding higher mathematics. Such are
+the majority. Others will have this feeling only in a slight
+degree, but they will be gifted with an uncommon memory and
+a great power of attention. They will learn by heart the details<span class='pagenum'><a name="Page_386" id="Page_386">[Pg 386]</a></span>
+one after another; they can understand mathematics and sometimes
+make applications, but they cannot create. Others, finally,
+will possess in a less or greater degree the special intuition
+referred to, and then not only can they understand mathematics
+even if their memory is nothing extraordinary, but they may
+become creators and try to invent with more or less success
+according as this intuition is more or less developed in them.</p>
+
+<p>In fact, what is mathematical creation? It does not consist
+in making new combinations with mathematical entities already
+known. Any one could do that, but the combinations so made
+would be infinite in number and most of them absolutely without
+interest. To create consists precisely in not making useless
+combinations and in making those which are useful and which
+are only a small minority. Invention is discernment, choice.</p>
+
+<p>How to make this choice I have before explained; the mathematical
+facts worthy of being studied are those which, by their
+analogy with other facts, are capable of leading us to the knowledge
+of a mathematical law just as experimental facts lead us to
+the knowledge of a physical law. They are those which reveal
+to us unsuspected kinship between other facts, long known, but
+wrongly believed to be strangers to one another.</p>
+
+<p>Among chosen combinations the most fertile will often be those
+formed of elements drawn from domains which are far apart.
+Not that I mean as sufficing for invention the bringing together
+of objects as disparate as possible; most combinations so formed
+would be entirely sterile. But certain among them, very rare,
+are the most fruitful of all.</p>
+
+<p>To invent, I have said, is to choose; but the word is perhaps
+not wholly exact. It makes one think of a purchaser before whom
+are displayed a large number of samples, and who examines
+them, one after the other, to make a choice. Here the samples
+would be so numerous that a whole lifetime would not suffice to
+examine them. This is not the actual state of things. The sterile
+combinations do not even present themselves to the mind of the
+inventor. Never in the field of his consciousness do combinations
+appear that are not really useful, except some that he rejects
+but which have to some extent the characteristics of useful combinations.
+All goes on as if the inventor were an examiner for<span class='pagenum'><a name="Page_387" id="Page_387">[Pg 387]</a></span>
+the second degree who would only have to question the candidates
+who had passed a previous examination.</p>
+
+<p>But what I have hitherto said is what may be observed or
+inferred in reading the writings of the geometers, reading
+reflectively.</p>
+
+<p>It is time to penetrate deeper and to see what goes on in the
+very soul of the mathematician. For this, I believe, I can do best
+by recalling memories of my own. But I shall limit myself to
+telling how I wrote my first memoir on Fuchsian functions. I
+beg the reader's pardon; I am about to use some technical expressions,
+but they need not frighten him, for he is not obliged to
+understand them. I shall say, for example, that I have found
+the demonstration of such a theorem under such circumstances.
+This theorem will have a barbarous name, unfamiliar to many,
+but that is unimportant; what is of interest for the psychologist
+is not the theorem but the circumstances.</p>
+
+<p>For fifteen days I strove to prove that there could not be any
+functions like those I have since called Fuchsian functions. I
+was then very ignorant; every day I seated myself at my work
+table, stayed an hour or two, tried a great number of combinations
+and reached no results. One evening, contrary to my
+custom, I drank black coffee and could not sleep. Ideas rose in
+crowds; I felt them collide until pairs interlocked, so to speak,
+making a stable combination. By the next morning I had established
+the existence of a class of Fuchsian functions, those which
+come from the hypergeometric series; I had only to write out
+the results, which took but a few hours.</p>
+
+<p>Then I wanted to represent these functions by the quotient of
+two series; this idea was perfectly conscious and deliberate, the
+analogy with elliptic functions guided me. I asked myself what
+properties these series must have if they existed, and I succeeded
+without difficulty in forming the series I have called theta-Fuchsian.</p>
+
+<p>Just at this time I left Caen, where I was then living, to go on
+a geologic excursion under the auspices of the school of mines.
+The changes of travel made me forget my mathematical work.
+Having reached Coutances, we entered an omnibus to go some
+place or other. At the moment when I put my foot on the step<span class='pagenum'><a name="Page_388" id="Page_388">[Pg 388]</a></span>
+the idea came to me, without anything in my former thoughts
+seeming to have paved the way for it, that the transformations
+I had used to define the Fuchsian functions were identical with
+those of non-Euclidean geometry. I did not verify the idea; I
+should not have had time, as, upon taking my seat in the omnibus,
+I went on with a conversation already commenced, but I
+felt a perfect certainty. On my return to Caen, for conscience'
+sake I verified the result at my leisure.</p>
+
+<p>Then I turned my attention to the study of some arithmetical
+questions apparently without much success and without a suspicion
+of any connection with my preceding researches. Disgusted
+with my failure, I went to spend a few days at the seaside,
+and thought of something else. One morning, walking on
+the bluff, the idea came to me, with just the same characteristics
+of brevity, suddenness and immediate certainty, that the arithmetic
+transformations of indeterminate ternary quadratic forms
+were identical with those of non-Euclidean geometry.</p>
+
+<p>Returned to Caen, I meditated on this result and deduced the
+consequences. The example of quadratic forms showed me that
+there were Fuchsian groups other than those corresponding to
+the hypergeometric series; I saw that I could apply to them the
+theory of theta-Fuchsian series and that consequently there
+existed Fuchsian functions other than those from the hypergeometric
+series, the ones I then knew. Naturally I set myself
+to form all these functions. I made a systematic attack upon
+them and carried all the outworks, one after another. There was
+one however that still held out, whose fall would involve that of
+the whole place. But all my efforts only served at first the better
+to show me the difficulty, which indeed was something. All this
+work was perfectly conscious.</p>
+
+<p>Thereupon I left for Mont-Val&eacute;rien, where I was to go through
+my military service; so I was very differently occupied. One
+day, going along the street, the solution of the difficulty which
+had stopped me suddenly appeared to me. I did not try to go
+deep into it immediately, and only after my service did I again
+take up the question. I had all the elements and had only to
+arrange them and put them together. So I wrote out my final
+memoir at a single stroke and without difficulty.<span class='pagenum'><a name="Page_389" id="Page_389">[Pg 389]</a></span></p>
+
+<p>I shall limit myself to this single example; it is useless to
+multiply them. In regard to my other researches I would have
+to say analogous things, and the observations of other mathematicians
+given in <i>L'enseignement math&eacute;matique</i> would only
+confirm them.</p>
+
+<p>Most striking at first is this appearance of sudden illumination,
+a manifest sign of long, unconscious prior work. The r&ocirc;le
+of this unconscious work in mathematical invention appears to
+me incontestable, and traces of it would be found in other cases
+where it is less evident. Often when one works at a hard question,
+nothing good is accomplished at the first attack. Then
+one takes a rest, longer or shorter, and sits down anew to the
+work. During the first half-hour, as before, nothing is found,
+and then all of a sudden the decisive idea presents itself to the
+mind. It might be said that the conscious work has been more
+fruitful because it has been interrupted and the rest has given
+back to the mind its force and freshness. But it is more probable
+that this rest has been filled out with unconscious work and
+that the result of this work has afterward revealed itself to the
+geometer just as in the cases I have cited; only the revelation,
+instead of coming during a walk or a journey, has happened
+during a period of conscious work, but independently of this
+work which plays at most a r&ocirc;le of excitant, as if it were the goad
+stimulating the results already reached during rest, but remaining
+unconscious, to assume the conscious form.</p>
+
+<p>There is another remark to be made about the conditions of
+this unconscious work: it is possible, and of a certainty it is only
+fruitful, if it is on the one hand preceded and on the other hand
+followed by a period of conscious work. These sudden inspirations
+(and the examples already cited sufficiently prove this)
+never happen except after some days of voluntary effort which
+has appeared absolutely fruitless and whence nothing good seems
+to have come, where the way taken seems totally astray. These
+efforts then have not been as sterile as one thinks; they have set
+agoing the unconscious machine and without them it would not
+have moved and would have produced nothing.</p>
+
+<p>The need for the second period of conscious work, after the
+inspiration, is still easier to understand. It is necessary to put<span class='pagenum'><a name="Page_390" id="Page_390">[Pg 390]</a></span>
+in shape the results of this inspiration, to deduce from them the
+immediate consequences, to arrange them, to word the demonstrations,
+but above all is verification necessary. I have spoken of
+the feeling of absolute certitude accompanying the inspiration;
+in the cases cited this feeling was no deceiver, nor is it usually.
+But do not think this a rule without exception; often this feeling
+deceives us without being any the less vivid, and we only find it
+out when we seek to put on foot the demonstration. I have
+especially noticed this fact in regard to ideas coming to me in the
+morning or evening in bed while in a semi-hypnagogic state.</p>
+
+<p>Such are the realities; now for the thoughts they force upon
+us. The unconscious, or, as we say, the subliminal self plays an
+important r&ocirc;le in mathematical creation; this follows from what
+we have said. But usually the subliminal self is considered as
+purely automatic. Now we have seen that mathematical work is
+not simply mechanical, that it could not be done by a machine,
+however perfect. It is not merely a question of applying rules,
+of making the most combinations possible according to certain
+fixed laws. The combinations so obtained would be exceedingly
+numerous, useless and cumbersome. The true work of the inventor
+consists in choosing among these combinations so as to
+eliminate the useless ones or rather to avoid the trouble of making
+them, and the rules which must guide this choice are extremely
+fine and delicate. It is almost impossible to state them precisely;
+they are felt rather than formulated. Under these conditions,
+how imagine a sieve capable of applying them mechanically?</p>
+
+<p>A first hypothesis now presents itself: the subliminal self is in
+no way inferior to the conscious self; it is not purely automatic;
+it is capable of discernment; it has tact, delicacy; it knows how
+to choose, to divine. What do I say? It knows better how to
+divine than the conscious self, since it succeeds where that has
+failed. In a word, is not the subliminal self superior to the
+conscious self? You recognize the full importance of this question.
+Boutroux in a recent lecture has shown how it came up
+on a very different occasion, and what consequences would follow
+an affirmative answer. (See also, by the same author, <i>Science
+et Religion</i>, pp. 313 ff.)</p>
+
+<p>Is this affirmative answer forced upon us by the facts I have<span class='pagenum'><a name="Page_391" id="Page_391">[Pg 391]</a></span>
+just given? I confess that, for my part, I should hate to accept
+it. Reexamine the facts then and see if they are not compatible
+with another explanation.</p>
+
+<p>It is certain that the combinations which present themselves to
+the mind in a sort of sudden illumination, after an unconscious
+working somewhat prolonged, are generally useful and fertile
+combinations, which seem the result of a first impression. Does
+it follow that the subliminal self, having divined by a delicate
+intuition that these combinations would be useful, has formed
+only these, or has it rather formed many others which were
+lacking in interest and have remained unconscious?</p>
+
+<p>In this second way of looking at it, all the combinations would
+be formed in consequence of the automatism of the subliminal
+self, but only the interesting ones would break into the domain
+of consciousness. And this is still very mysterious. What is the
+cause that, among the thousand products of our unconscious
+activity, some are called to pass the threshold, while others remain
+below? Is it a simple chance which confers this privilege? Evidently
+not; among all the stimuli of our senses, for example, only
+the most intense fix our attention, unless it has been drawn to
+them by other causes. More generally the privileged unconscious
+phenomena, those susceptible of becoming conscious, are
+those which, directly or indirectly, affect most profoundly our
+emotional sensibility.</p>
+
+<p>It may be surprising to see emotional sensibility invoked
+<i>&agrave; propos</i> of mathematical demonstrations which, it would seem,
+can interest only the intellect. This would be to forget the feeling
+of mathematical beauty, of the harmony of numbers and
+forms, of geometric elegance. This is a true esthetic feeling that
+all real mathematicians know, and surely it belongs to emotional
+sensibility.</p>
+
+<p>Now, what are the mathematic entities to which we attribute
+this character of beauty and elegance, and which are capable of
+developing in us a sort of esthetic emotion? They are those
+whose elements are harmoniously disposed so that the mind without
+effort can embrace their totality while realizing the details.
+This harmony is at once a satisfaction of our esthetic needs and
+an aid to the mind, sustaining and guiding; And at the same<span class='pagenum'><a name="Page_392" id="Page_392">[Pg 392]</a></span>
+time, in putting under our eyes a well-ordered whole, it makes
+us foresee a mathematical law. Now, as we have said above, the
+only mathematical facts worthy of fixing our attention and
+capable of being useful are those which can teach us a mathematical
+law. So that we reach the following conclusion: The
+useful combinations are precisely the most beautiful, I mean
+those best able to charm this special sensibility that all mathematicians
+know, but of which the profane are so ignorant as
+often to be tempted to smile at it.</p>
+
+<p>What happens then? Among the great numbers of combinations
+blindly formed by the subliminal self, almost all are without
+interest and without utility; but just for that reason they are
+also without effect upon the esthetic sensibility. Consciousness
+will never know them; only certain ones are harmonious, and,
+consequently, at once useful and beautiful. They will be capable
+of touching this special sensibility of the geometer of which I
+have just spoken, and which, once aroused, will call our attention
+to them, and thus give them occasion to become conscious.</p>
+
+<p>This is only a hypothesis, and yet here is an observation which
+may confirm it: when a sudden illumination seizes upon the
+mind of the mathematician, it usually happens that it does not
+deceive him, but it also sometimes happens, as I have said, that
+it does not stand the test of verification; well, we almost always
+notice that this false idea, had it been true, would have gratified
+our natural feeling for mathematical elegance.</p>
+
+<p>Thus it is this special esthetic sensibility which plays the r&ocirc;le
+of the delicate sieve of which I spoke, and that sufficiently explains
+why the one lacking it will never be a real creator.</p>
+
+<p>Yet all the difficulties have not disappeared. The conscious
+self is narrowly limited, and as for the subliminal self we know
+not its limitations, and this is why we are not too reluctant in
+supposing that it has been able in a short time to make more
+different combinations than the whole life of a conscious being
+could encompass. Yet these limitations exist. Is it likely that
+it is able to form all the possible combinations, whose number
+would frighten the imagination? Nevertheless that would seem
+necessary, because if it produces only a small part of these combinations,
+and if it makes them at random, there would be small<span class='pagenum'><a name="Page_393" id="Page_393">[Pg 393]</a></span>
+chance that the <i>good</i>, the one we should choose, would be found
+among them.</p>
+
+<p>Perhaps we ought to seek the explanation in that preliminary
+period of conscious work which always precedes all fruitful
+unconscious labor. Permit me a rough comparison. Figure
+the future elements of our combinations as something like the
+hooked atoms of Epicurus. During the complete repose of the
+mind, these atoms are motionless, they are, so to speak, hooked
+to the wall; so this complete rest may be indefinitely prolonged
+without the atoms meeting, and consequently without any combination
+between them.</p>
+
+<p>On the other hand, during a period of apparent rest and
+unconscious work, certain of them are detached from the wall and
+put in motion. They flash in every direction through the space
+(I was about to say the room) where they are enclosed, as would,
+for example, a swarm of gnats or, if you prefer a more learned
+comparison, like the molecules of gas in the kinematic theory of
+gases. Then their mutual impacts may produce new combinations.</p>
+
+<p>What is the r&ocirc;le of the preliminary conscious work? It is
+evidently to mobilize certain of these atoms, to unhook them from
+the wall and put them in swing. We think we have done no
+good, because we have moved these elements a thousand different
+ways in seeking to assemble them, and have found no satisfactory
+aggregate. But, after this shaking up imposed upon them by our
+will, these atoms do not return to their primitive rest. They
+freely continue their dance.</p>
+
+<p>Now, our will did not choose them at random; it pursued a
+perfectly determined aim. The mobilized atoms are therefore
+not any atoms whatsoever; they are those from which we might
+reasonably expect the desired solution. Then the mobilized atoms
+undergo impacts which make them enter into combinations among
+themselves or with other atoms at rest which they struck against
+in their course. Again I beg pardon, my comparison is very
+rough, but I scarcely know how otherwise to make my thought
+understood.</p>
+
+<p>However it may be, the only combinations that have a chance
+of forming are those where at least one of the elements is one
+of those atoms freely chosen by our will. Now, it is evidently<span class='pagenum'><a name="Page_394" id="Page_394">[Pg 394]</a></span>
+among these that is found what I called the <i>good combination</i>.
+Perhaps this is a way of lessening the paradoxical in the original
+hypothesis.</p>
+
+<p>Another observation. It never happens that the unconscious
+work gives us the result of a somewhat long calculation <i>all made</i>,
+where we have only to apply fixed rules. We might think the
+wholly automatic subliminal self particularly apt for this sort of
+work, which is in a way exclusively mechanical. It seems that
+thinking in the evening upon the factors of a multiplication we
+might hope to find the product ready made upon our awakening,
+or again that an algebraic calculation, for example a verification,
+would be made unconsciously. Nothing of the sort, as
+observation proves. All one may hope from these inspirations,
+fruits of unconscious work, is a point of departure for such calculations.
+As for the calculations themselves, they must be made
+in the second period of conscious work, that which follows the
+inspiration, that in which one verifies the results of this inspiration
+and deduces their consequences. The rules of these calculations
+are strict and complicated. They require discipline, attention,
+will, and therefore consciousness. In the subliminal self,
+on the contrary, reigns what I should call liberty, if we might
+give this name to the simple absence of discipline and to the
+disorder born of chance. Only, this disorder itself permits unexpected
+combinations.</p>
+
+<p>I shall make a last remark: when above I made certain personal
+observations, I spoke of a night of excitement when I worked in
+spite of myself. Such cases are frequent, and it is not necessary
+that the abnormal cerebral activity be caused by a physical excitant
+as in that I mentioned. It seems, in such cases, that one is
+present at his own unconscious work, made partially perceptible
+to the over-excited consciousness, yet without having changed its
+nature. Then we vaguely comprehend what distinguishes the
+two mechanisms or, if you wish, the working methods of the two
+egos. And the psychologic observations I have been able thus
+to make seem to me to confirm in their general outlines the views
+I have given.</p>
+
+<p>Surely they have need of it, for they are and remain in spite
+of all very hypothetical: the interest of the questions is so great
+that I do not repent of having submitted them to the reader.</p>
+
+
+
+<hr style="width: 65%;" />
+<p><span class='pagenum'><a name="Page_395" id="Page_395">[Pg 395]</a></span></p>
+<h3>CHAPTER IV</h3>
+
+<h3><span class="smcap">Chance</span></h3>
+
+
+<h4>I</h4>
+
+<p>"How dare we speak of the laws of chance? Is not chance
+the antithesis of all law?" So says Bertrand at the beginning of
+his <i>Calcul des probabiltit&eacute;s</i>. Probability is opposed to certitude;
+so it is what we do not know and consequently it seems what we
+could not calculate. Here is at least apparently a contradiction,
+and about it much has already been written.</p>
+
+<p>And first, what is chance? The ancients distinguished between
+phenomena seemingly obeying harmonious laws, established once
+for all, and those which they attributed to chance; these were
+the ones unpredictable because rebellious to all law. In each
+domain the precise laws did not decide everything, they only
+drew limits between which chance might act. In this conception
+the word chance had a precise and objective meaning; what was
+chance for one was also chance for another and even for the gods.</p>
+
+<p>But this conception is not ours to-day. We have become absolute
+determinists, and even those who want to reserve the rights
+of human free will let determinism reign undividedly in the inorganic
+world at least. Every phenomenon, however minute, has
+a cause; and a mind infinitely powerful, infinitely well-informed
+about the laws of nature, could have foreseen it from the beginning
+of the centuries. If such a mind existed, we could not play
+with it at any game of chance; we should always lose.</p>
+
+<p>In fact for it the word chance would not have any meaning,
+or rather there would be no chance. It is because of our weakness
+and our ignorance that the word has a meaning for us. And,
+even without going beyond our feeble humanity, what is chance
+for the ignorant is not chance for the scientist. Chance is only
+the measure of our ignorance. Fortuitous phenomena are, by
+definition, those whose laws we do not know.</p>
+
+<p>But is this definition altogether satisfactory? When the first<span class='pagenum'><a name="Page_396" id="Page_396">[Pg 396]</a></span>
+Chaldean shepherds followed with their eyes the movements of
+the stars, they knew not as yet the laws of astronomy; would they
+have dreamed of saying that the stars move at random? If a
+modern physicist studies a new phenomenon, and if he discovers
+its law Tuesday, would he have said Monday that this phenomenon
+was fortuitous? Moreover, do we not often invoke what
+Bertrand calls the laws of chance, to predict a phenomenon?
+For example, in the kinetic theory of gases we obtain the known
+laws of Mariotte and of Gay-Lussac by means of the hypothesis
+that the velocities of the molecules of gas vary irregularly, that
+is to say at random. All physicists will agree that the observable
+laws would be much less simple if the velocities were ruled by
+any simple elementary law whatsoever, if the molecules were,
+as we say, <i>organized</i>, if they were subject to some discipline. It
+is due to chance, that is to say, to our ignorance, that we can draw
+our conclusions; and then if the word chance is simply synonymous
+with ignorance what does that mean? Must we therefore
+translate as follows?</p>
+
+<p>"You ask me to predict for you the phenomena about to
+happen. If, unluckily, I knew the laws of these phenomena I
+could make the prediction only by inextricable calculations and
+would have to renounce attempting to answer you; but as I have
+the good fortune not to know them, I will answer you at once.
+And what is most surprising, my answer will be right."</p>
+
+<p>So it must well be that chance is something other than the
+name we give our ignorance, that among phenomena whose
+causes are unknown to us we must distinguish fortuitous phenomena
+about which the calculus of probabilities will provisionally
+give information, from those which are not fortuitous and of
+which we can say nothing so long as we shall not have determined
+the laws governing them. For the fortuitous phenomena themselves,
+it is clear that the information given us by the calculus
+of probabilities will not cease to be true upon the day when these
+phenomena shall be better known.</p>
+
+<p>The director of a life insurance company does not know when
+each of the insured will die, but he relies upon the calculus of
+probabilities and on the law of great numbers, and he is not
+deceived, since he distributes dividends to his stockholders. These<span class='pagenum'><a name="Page_397" id="Page_397">[Pg 397]</a></span>
+dividends would not vanish if a very penetrating and very indiscreet
+physician should, after the policies were signed, reveal to
+the director the life chances of the insured. This doctor would
+dissipate the ignorance of the director, but he would have no
+influence on the dividends, which evidently are not an outcome
+of this ignorance.</p>
+
+
+<h4>II</h4>
+
+<p>To find a better definition of chance we must examine some of
+the facts which we agree to regard as fortuitous, and to which
+the calculus of probabilities seems to apply; we then shall investigate
+what are their common characteristics.</p>
+
+<p>The first example we select is that of unstable equilibrium; if
+a cone rests upon its apex, we know well that it will fall, but we
+do not know toward what side; it seems to us chance alone will
+decide. If the cone were perfectly symmetric, if its axis were
+perfectly vertical, if it were acted upon by no force other than
+gravity, it would not fall at all. But the least defect in symmetry
+will make it lean slightly toward one side or the other, and if it
+leans, however little, it will fall altogether toward that side.
+Even if the symmetry were perfect, a very slight tremor, a breath
+of air could make it incline some seconds of arc; this will be
+enough to determine its fall and even the sense of its fall which
+will be that of the initial inclination.</p>
+
+<p>A very slight cause, which escapes us, determines a considerable
+effect which we can not help seeing, and then we say this
+effect is due to chance. If we could know exactly the laws of
+nature and the situation of the universe at the initial instant,
+we should be able to predict exactly the situation of this same
+universe at a subsequent instant. But even when the natural
+laws should have no further secret for us, we could know the
+initial situation only <i>approximately</i>. If that permits us to foresee
+the subsequent situation <i>with the same degree of approximation</i>,
+this is all we require, we say the phenomenon has been
+predicted, that it is ruled by laws. But this is not always the
+case; it may happen that slight differences in the initial conditions
+produce very great differences in the final phenomena; a
+slight error in the former would make an enormous error in the<span class='pagenum'><a name="Page_398" id="Page_398">[Pg 398]</a></span>
+latter. Prediction becomes impossible and we have the fortuitous
+phenomenon.</p>
+
+<p>Our second example will be very analogous to the first and we
+shall take it from meteorology. Why have the meteorologists
+such difficulty in predicting the weather with any certainty?
+Why do the rains, the tempests themselves seem to us to come by
+chance, so that many persons find it quite natural to pray for
+rain or shine, when they would think it ridiculous to pray for
+an eclipse? We see that great perturbations generally happen in
+regions where the atmosphere is in unstable equilibrium. The
+meteorologists are aware that this equilibrium is unstable, that a
+cyclone is arising somewhere; but where they can not tell; one-tenth
+of a degree more or less at any point, and the cyclone
+bursts here and not there, and spreads its ravages over countries
+it would have spared. This we could have foreseen if we had
+known that tenth of a degree, but the observations were neither
+sufficiently close nor sufficiently precise, and for this reason all
+seems due to the agency of chance. Here again we find the same
+contrast between a very slight cause, unappreciable to the observer,
+and important effects, which are sometimes tremendous
+disasters.</p>
+
+<p>Let us pass to another example, the distribution of the minor
+planets on the zodiac. Their initial longitudes may have been
+any longitudes whatever; but their mean motions were different
+and they have revolved for so long a time that we may say they
+are now distributed <i>at random</i> along the zodiac. Very slight
+initial differences between their distances from the sun, or, what
+comes to the same thing, between their mean motions, have
+ended by giving enormous differences between their present
+longitudes. An excess of the thousandth of a second in the daily
+mean motion will give in fact a second in three years, a degree
+in ten thousand years, an entire circumference in three or four
+million years, and what is that to the time which has passed since
+the minor planets detached themselves from the nebula of
+Laplace? Again therefore we see a slight cause and a great
+effect; or better, slight differences in the cause and great differences
+in the effect.</p>
+
+<p>The game of roulette does not take us as far as might seem<span class='pagenum'><a name="Page_399" id="Page_399">[Pg 399]</a></span>
+from the preceding example. Assume a needle to be turned on a
+pivot over a dial divided into a hundred sectors alternately red
+and black. If it stops on a red sector I win; if not, I lose. Evidently
+all depends upon the initial impulse I give the needle.
+The needle will make, suppose, ten or twenty turns, but it will
+stop sooner or not so soon, according as I shall have pushed it
+more or less strongly. It suffices that the impulse vary only by
+a thousandth or a two thousandth to make the needle stop over a
+black sector or over the following red one. These are differences
+the muscular sense can not distinguish and which elude even the
+most delicate instruments. So it is impossible for me to foresee
+what the needle I have started will do, and this is why my heart
+throbs and I hope everything from luck. The difference in the
+cause is imperceptible, and the difference in the effect is for me
+of the highest importance, since it means my whole stake.</p>
+
+
+<h4>III</h4>
+
+<p>Permit me, in this connection, a thought somewhat foreign to
+my subject. Some years ago a philosopher said that the future
+is determined by the past, but not the past by the future; or, in
+other words, from knowledge of the present we could deduce the
+future, but not the past; because, said he, a cause can have only
+one effect, while the same effect might be produced by several
+different causes. It is clear no scientist can subscribe to this
+conclusion. The laws of nature bind the antecedent to the consequent
+in such a way that the antecedent is as well determined by
+the consequent as the consequent by the antecedent. But whence
+came the error of this philosopher? We know that in virtue of
+Carnot's principle physical phenomena are irreversible and the
+world tends toward uniformity. When two bodies of different
+temperature come in contact, the warmer gives up heat to the
+colder; so we may foresee that the temperature will equalize.
+But once equal, if asked about the anterior state, what can we
+answer? We might say that one was warm and the other cold,
+but not be able to divine which formerly was the warmer.</p>
+
+<p>And yet in reality the temperatures will never reach perfect
+equality. The difference of the temperatures only tends asymptotically
+toward zero. There comes a moment when our<span class='pagenum'><a name="Page_400" id="Page_400">[Pg 400]</a></span>
+thermometers are powerless to make it known. But if we had thermometers
+a thousand times, a hundred thousand times as sensitive,
+we should recognize that there still is a slight difference, and
+that one of the bodies remains a little warmer than the other, and
+so we could say this it is which formerly was much the warmer.</p>
+
+<p>So then there are, contrary to what we found in the former
+examples, great differences in cause and slight differences in
+effect. Flammarion once imagined an observer going away from
+the earth with a velocity greater than that of light; for him time
+would have changed sign. History would be turned about, and
+Waterloo would precede Austerlitz. Well, for this observer,
+effects and causes would be inverted; unstable equilibrium would
+no longer be the exception. Because of the universal irreversibility,
+all would seem to him to come out of a sort of chaos in
+unstable equilibrium. All nature would appear to him delivered
+over to chance.</p>
+
+
+<h4>IV</h4>
+
+<p>Now for other examples where we shall see somewhat different
+characteristics. Take first the kinetic theory of gases. How
+should we picture a receptacle filled with gas? Innumerable
+molecules, moving at high speeds, flash through this receptacle
+in every direction. At every instant they strike against its walls
+or each other, and these collisions happen under the most diverse
+conditions. What above all impresses us here is not the littleness
+of the causes, but their complexity, and yet the former element
+is still found here and plays an important r&ocirc;le. If a molecule
+deviated right or left from its trajectory, by a very small
+quantity, comparable to the radius of action of the gaseous molecules,
+it would avoid a collision or sustain it under different conditions,
+and that would vary the direction of its velocity after
+the impact, perhaps by ninety degrees or by a hundred and
+eighty degrees.</p>
+
+<p>And this is not all; we have just seen that it is necessary to
+deflect the molecule before the clash by only an infinitesimal, to
+produce its deviation after the collision by a finite quantity. If
+then the molecule undergoes two successive shocks, it will suffice
+to deflect it before the first by an infinitesimal of the second
+order, for it to deviate after the first encounter by an infinitesimal<span class='pagenum'><a name="Page_401" id="Page_401">[Pg 401]</a></span>
+of the first order, and after the second hit, by a finite quantity.
+And the molecule will not undergo merely two shocks; it
+will undergo a very great number per second. So that if the
+first shock has multiplied the deviation by a very large number
+<i>A</i>, after <i>n</i> shocks it will be multiplied by <i>A<sup>n</sup></i>. It will therefore
+become very great not merely because <i>A</i> is large, that is to say
+because little causes produce big effects, but because the exponent
+<i>n</i> is large, that is to say because the shocks are very numerous
+and the causes very complex.</p>
+
+<p>Take a second example. Why do the drops of rain in a
+shower seem to be distributed at random? This is again because
+of the complexity of the causes which determine their formation.
+Ions are distributed in the atmosphere. For a long while they
+have been subjected to air-currents constantly changing, they
+have been caught in very small whirlwinds, so that their final
+distribution has no longer any relation to their initial distribution.
+Suddenly the temperature falls, vapor condenses, and each
+of these ions becomes the center of a drop of rain. To know
+what will be the distribution of these drops and how many will
+fall on each paving-stone, it would not be sufficient to know the
+initial situation of the ions, it would be necessary to compute
+the effect of a thousand little capricious air-currents.</p>
+
+<p>And again it is the same if we put grains of powder in suspension
+in water. The vase is ploughed by currents whose law
+we know not, we only know it is very complicated. At the
+end of a certain time the grains will be distributed at random,
+that is to say uniformly, in the vase; and this is due precisely to
+the complexity of these currents. If they obeyed some simple
+law, if for example the vase revolved and the currents circulated
+around the axis of the vase, describing circles, it would no
+longer be the same, since each grain would retain its initial altitude
+and its initial distance from the axis.</p>
+
+<p>We should reach the same result in considering the mixing of
+two liquids or of two fine-grained powders. And to take a
+grosser example, this is also what happens when we shuffle playing-cards.
+At each stroke the cards undergo a permutation
+(analogous to that studied in the theory of substitutions). What
+will happen? The probability of a particular permutation (for<span class='pagenum'><a name="Page_402" id="Page_402">[Pg 402]</a></span>
+example, that bringing to the <i>n</i>th place the card occupying the
+&#981;(<i>n</i>)th place before the permutation) depends upon the player's
+habits. But if this player shuffles the cards long enough, there
+will be a great number of successive permutations, and the resulting
+final order will no longer be governed by aught but
+chance; I mean to say that all possible orders will be equally
+probable. It is to the great number of successive permutations,
+that is to say to the complexity of the phenomenon, that this
+result is due.</p>
+
+<p>A final word about the theory of errors. Here it is that the
+causes are complex and multiple. To how many snares is not
+the observer exposed, even with the best instrument! He should
+apply himself to finding out the largest and avoiding them.
+These are the ones giving birth to systematic errors. But when
+he has eliminated those, admitting that he succeeds, there remain
+many small ones which, their effects accumulating, may become
+dangerous. Thence come the accidental errors; and we attribute
+them to chance because their causes are too complicated
+and too numerous. Here again we have only little causes, but
+each of them would produce only a slight effect; it is by their
+union and their number that their effects become formidable.</p>
+
+
+<h4>V</h4>
+
+<p>We may take still a third point of view, less important than
+the first two and upon which I shall lay less stress. When we
+seek to foresee an event and examine its antecedents, we strive
+to search into the anterior situation. This could not be done for
+all parts of the universe and we are content to know what is
+passing in the neighborhood of the point where the event should
+occur, or what would appear to have some relation to it. An
+examination can not be complete and we must know how to
+choose. But it may happen that we have passed by circumstances
+which at first sight seemed completely foreign to the
+foreseen happening, to which one would never have dreamed of
+attributing any influence and which nevertheless, contrary to all
+anticipation, come to play an important r&ocirc;le.</p>
+
+<p>A man passes in the street going to his business; some one
+knowing the business could have told why he started at such a<span class='pagenum'><a name="Page_403" id="Page_403">[Pg 403]</a></span>
+time and went by such a street. On the roof works a tiler.
+The contractor employing him could in a certain measure foresee
+what he would do. But the passer-by scarcely thinks of the
+tiler, nor the tiler of him; they seem to belong to two worlds
+completely foreign to one another. And yet the tiler drops a
+tile which kills the man, and we do not hesitate to say this is
+chance.</p>
+
+<p>Our weakness forbids our considering the entire universe
+and makes us cut it up into slices. We try to do this as little
+artificially as possible. And yet it happens from time to time
+that two of these slices react upon each other. The effects
+of this mutual action then seem to us to be due to chance.</p>
+
+<p>Is this a third way of conceiving chance? Not always; in
+fact most often we are carried back to the first or the second.
+Whenever two worlds usually foreign to one another come thus
+to react upon each other, the laws of this reaction must be very
+complex. On the other hand, a very slight change in the initial
+conditions of these two worlds would have been sufficient for the
+reaction not to have happened. How little was needed for the
+man to pass a second later or the tiler to drop his tile a second
+sooner.</p>
+
+
+<h4>VI</h4>
+
+<p>All we have said still does not explain why chance obeys laws.
+Does the fact that the causes are slight or complex suffice for
+our foreseeing, if not their effects <i>in each case</i>, at least what their
+effects will be, <i>on the average</i>? To answer this question we had
+better take up again some of the examples already cited.</p>
+
+<p>I shall begin with that of the roulette. I have said that the
+point where the needle will stop depends upon the initial push
+given it. What is the probability of this push having this or
+that value? I know nothing about it, but it is difficult for me
+not to suppose that this probability is represented by a continuous
+analytic function. The probability that the push is comprised
+between &#945; and &#945; + &#949; will then be sensibly equal to the probability
+of its being comprised between &#945; + &#949; and &#945; + 2&#949;, <i>provided</i> &#949; <i>be
+very small</i>. This is a property common to all analytic functions.
+Minute variations of the function are proportional to minute
+variations of the variable.<span class='pagenum'><a name="Page_404" id="Page_404">[Pg 404]</a></span></p>
+
+<p>But we have assumed that an exceedingly slight variation of
+the push suffices to change the color of the sector over which the
+needle finally stops. From &#945; to &#945; + &#949; it is red, from &#945; + &#949; to
+&#945; + 2&#949; it is black; the probability of each red sector is therefore
+the same as of the following black, and consequently the total
+probability of red equals the total probability of black.</p>
+
+<p>The datum of the question is the analytic function representing
+the probability of a particular initial push. But the theorem
+remains true whatever be this datum, since it depends upon a
+property common to all analytic functions. From this it follows
+finally that we no longer need the datum.</p>
+
+<p>What we have just said for the case of the roulette applies
+also to the example of the minor planets. The zodiac may be
+regarded as an immense roulette on which have been tossed many
+little balls with different initial impulses varying according to
+some law. Their present distribution is uniform and independent
+of this law, for the same reason as in the preceding case.
+Thus we see why phenomena obey the laws of chance when
+slight differences in the causes suffice to bring on great differences
+in the effects. The probabilities of these slight differences may
+then be regarded as proportional to these differences themselves,
+just because these differences are minute, and the infinitesimal
+increments of a continuous function are proportional to those of
+the variable.</p>
+
+<p>Take an entirely different example, where intervenes especially
+the complexity of the causes. Suppose a player shuffles a pack
+of cards. At each shuffle he changes the order of the cards, and
+he may change them in many ways. To simplify the exposition,
+consider only three cards. The cards which before the shuffle
+occupied respectively the places 123, may after the shuffle occupy
+the places</p>
+
+<p class="center">
+123, 231, 312, 321, 132, 213.<br />
+</p>
+
+<p>Each of these six hypotheses is possible and they have respectively
+for probabilities:</p>
+
+<p class="center">
+<i>p</i><sub>1</sub>, <i>p</i><sub>2</sub>, <i>p</i><sub>3</sub>, <i>p</i><sub>4</sub>, <i>p</i><sub>5</sub>, <i>p</i><sub>6</sub>.<br />
+</p>
+
+<p>The sum of these six numbers equals 1; but this is all we know
+of them; these six probabilities depend naturally upon the habits
+of the player which we do not know.<span class='pagenum'><a name="Page_405" id="Page_405">[Pg 405]</a></span></p>
+
+<p>At the second shuffle and the following, this will recommence,
+and under the same conditions; I mean that <i>p</i><sub>4</sub> for example represents
+always the probability that the three cards which occupied
+after the <i>n</i>th shuffle and before the <i>n</i> + 1th the places 123,
+occupy the places 321 after the <i>n</i> + 1th shuffle. And this remains
+true whatever be the number <i>n</i>, since the habits of the
+player and his way of shuffling remain the same.</p>
+
+<p>But if the number of shuffles is very great, the cards which
+before the first shuffle occupied the places 123 may, after the
+last shuffle, occupy the places</p>
+
+<p class="center">
+123, 231, 312, 321, 132, 213<br />
+</p>
+
+<p class="noidt">and the probability of these six hypotheses will be sensibly the
+same and equal to 1/6; and this will be true whatever be the
+numbers <i>p</i><sub>1</sub> ... <i>p</i><sub>6</sub> which we do not know. The great number
+of shuffles, that is to say the complexity of the causes, has
+produced uniformity.</p>
+
+<p>This would apply without change if there were more than
+three cards, but even with three cards the demonstration would
+be complicated; let it suffice to give it for only two cards. Then
+we have only two possibilities 12, 21 with the probabilities <i>p</i><sub>1</sub> and
+<i>p</i><sub>2</sub> = 1 &minus; <i>p</i><sub>1</sub>.</p>
+
+<p>Suppose <i>n</i> shuffles and suppose I win one franc if the cards
+are finally in the initial order and lose one if they are finally
+inverted. Then, my mathematical expectation will be (<i>p</i><sub>1</sub> &minus; <i>p</i><sub>2</sub>)<sup><i>n</i></sup>.</p>
+
+<p>The difference <i>p</i><sub>1</sub> &minus; <i>p</i><sub>2</sub> is certainly less than 1; so that if <i>n</i>
+is very great my expectation will be zero; we need not learn <i>p</i><sub>1</sub>
+and <i>p</i><sub>2</sub> to be aware that the game is equitable.</p>
+
+<p>There would always be an exception if one of the numbers
+<i>p</i><sub>1</sub> and <i>p</i><sub>2</sub> was equal to 1 and the other naught. <i>Then it would
+not apply because our initial hypotheses would be too simple.</i></p>
+
+<p>What we have just seen applies not only to the mixing of
+cards, but to all mixings, to those of powders and of liquids;
+and even to those of the molecules of gases in the kinetic theory
+of gases.</p>
+
+<p>To return to this theory, suppose for a moment a gas whose
+molecules can not mutually clash, but may be deviated by hitting
+the insides of the vase wherein the gas is confined. If the form<span class='pagenum'><a name="Page_406" id="Page_406">[Pg 406]</a></span>
+of the vase is sufficiently complex the distribution of the molecules
+and that of the velocities will not be long in becoming uniform.
+But this will not be so if the vase is spherical or if it
+has the shape of a cuboid. Why? Because in the first case the
+distance from the center to any trajectory will remain constant;
+in the second case this will be the absolute value of the angle of
+each trajectory with the faces of the cuboid.</p>
+
+<p>So we see what should be understood by conditions <i>too simple</i>;
+they are those which conserve something, which leave an invariant
+remaining. Are the differential equations of the problem too
+simple for us to apply the laws of chance? This question would
+seem at first view to lack precise meaning; now we know what it
+means. They are too simple if they conserve something, if they
+admit a uniform integral. If something in the initial conditions
+remains unchanged, it is clear the final situation can no longer
+be independent of the initial situation.</p>
+
+<p>We come finally to the theory of errors. We know not to
+what are due the accidental errors, and precisely because we do
+not know, we are aware they obey the law of Gauss. Such is the
+paradox. The explanation is nearly the same as in the preceding
+cases. We need know only one thing: that the errors are very
+numerous, that they are very slight, that each may be as well
+negative as positive. What is the curve of probability of each
+of them? We do not know; we only suppose it is symmetric.
+We prove then that the resultant error will follow Gauss's law,
+and this resulting law is independent of the particular laws
+which we do not know. Here again the simplicity of the result
+is born of the very complexity of the data.</p>
+
+
+<h4>VII</h4>
+
+<p>But we are not through with paradoxes. I have just recalled
+the figment of Flammarion, that of the man going quicker than
+light, for whom time changes sign. I said that for him all phenomena
+would seem due to chance. That is true from a certain
+point of view, and yet all these phenomena at a given moment
+would not be distributed in conformity with the laws of chance,
+since the distribution would be the same as for us, who, seeing
+them unfold harmoniously and without coming out of a primal
+chaos, do not regard them as ruled by chance.<span class='pagenum'><a name="Page_407" id="Page_407">[Pg 407]</a></span></p>
+
+<p>What does that mean? For Lumen, Flammarion's man, slight
+causes seem to produce great effects; why do not things go on as
+for us when we think we see grand effects due to little causes?
+Would not the same reasoning be applicable in his case?</p>
+
+<p>Let us return to the argument. When slight differences in the
+causes produce vast differences in the effects, why are these effects
+distributed according to the laws of chance? Suppose a difference
+of a millimeter in the cause produces a difference of a kilometer
+in the effect. If I win in case the effect corresponds to a
+kilometer bearing an even number, my probability of winning
+will be 1/2. Why? Because to make that, the cause must correspond
+to a millimeter with an even number. Now, according to
+all appearance, the probability of the cause varying between
+certain limits will be proportional to the distance apart of these
+limits, provided this distance be very small. If this hypothesis
+were not admitted there would no longer be any way of representing
+the probability by a continuous function.</p>
+
+<p>What now will happen when great causes produce small
+effects? This is the case where we should not attribute the phenomenon
+to chance and where on the contrary Lumen would
+attribute it to chance. To a difference of a kilometer in the
+cause would correspond a difference of a millimeter in the effect.
+Would the probability of the cause being comprised between two
+limits <i>n</i> kilometers apart still be proportional to <i>n</i>? We have
+no reason to suppose so, since this distance, <i>n</i> kilometers, is
+great. But the probability that the effect lies between two
+limits <i>n</i> millimeters apart will be precisely the same, so it will not
+be proportional to <i>n</i>, even though this distance, <i>n</i> millimeters,
+be small. There is no way therefore of representing the law of
+probability of effects by a continuous curve. This curve, understand,
+may remain continuous in the <i>analytic</i> sense of the
+word; to <i>infinitesimal</i> variations of the abscissa will correspond
+infinitesimal variations of the ordinate. But <i>practically</i> it will
+not be continuous, since <i>very small</i> variations of the ordinate
+would not correspond to very small variations of the abscissa. It
+would become impossible to trace the curve with an ordinary
+pencil; that is what I mean.</p>
+
+<p>So what must we conclude? Lumen has no right to say that<span class='pagenum'><a name="Page_408" id="Page_408">[Pg 408]</a></span>
+the probability of the cause (<i>his</i> cause, our effect) should be
+represented necessarily by a continuous function. But then why
+have we this right? It is because this state of unstable equilibrium
+which we have been calling initial is itself only the final
+outcome of a long previous history. In the course of this history
+complex causes have worked a great while: they have contributed
+to produce the mixture of elements and they have tended to make
+everything uniform at least within a small region; they have
+rounded off the corners, smoothed down the hills and filled up
+the valleys. However capricious and irregular may have been the
+primitive curve given over to them, they have worked so much
+toward making it regular that finally they deliver over to us a
+continuous curve. And this is why we may in all confidence
+assume its continuity.</p>
+
+<p>Lumen would not have the same reasons for such a conclusion.
+For him complex causes would not seem agents of equalization
+and regularity, but on the contrary would create only inequality
+and differentiation. He would see a world more and more varied
+come forth from a sort of primitive chaos. The changes he
+could observe would be for him unforeseen and impossible to
+foresee. They would seem to him due to some caprice or another;
+but this caprice would be quite different from our chance, since
+it would be opposed to all law, while our chance still has its laws.
+All these points call for lengthy explications, which perhaps
+would aid in the better comprehension of the irreversibility of
+the universe.</p>
+
+
+<h4>VIII</h4>
+
+<p>We have sought to define chance, and now it is proper to put a
+question. Has chance thus defined, in so far as this is possible,
+objectivity?</p>
+
+<p>It may be questioned. I have spoken of very slight or very
+complex causes. But what is very little for one may be very
+big for another, and what seems very complex to one may seem
+simple to another. In part I have already answered by saying
+precisely in what cases differential equations become too simple
+for the laws of chance to remain applicable. But it is fitting to
+examine the matter a little more closely, because we may take
+still other points of view.<span class='pagenum'><a name="Page_409" id="Page_409">[Pg 409]</a></span></p>
+
+
+<p>What means the phrase 'very slight'? To understand it we
+need only go back to what has already been said. A difference
+is very slight, an interval is very small, when within the limits
+of this interval the probability remains sensibly constant. And
+why may this probability be regarded as constant within a
+small interval? It is because we assume that the law of probability
+is represented by a continuous curve, continuous not only
+in the analytic sense, but <i>practically</i> continuous, as already explained.
+This means that it not only presents no absolute hiatus,
+but that it has neither salients nor reentrants too acute or too
+accentuated.</p>
+
+<p>And what gives us the right to make this hypothesis? We
+have already said it is because, since the beginning of the ages,
+there have always been complex causes ceaselessly acting in the
+same way and making the world tend toward uniformity without
+ever being able to turn back. These are the causes which little
+by little have flattened the salients and filled up the reentrants,
+and this is why our probability curves now show only gentle undulations.
+In milliards of milliards of ages another step will
+have been made toward uniformity, and these undulations will be
+ten times as gentle; the radius of mean curvature of our curve
+will have become ten times as great. And then such a length as
+seems to us to-day not very small, since on our curve an arc of
+this length can not be regarded as rectilineal, should on the contrary
+at that epoch be called very little, since the curvature will
+have become ten times less and an arc of this length may be
+sensibly identified with a sect.</p>
+
+<p>Thus the phrase 'very slight' remains relative; but it is not
+relative to such or such a man, it is relative to the actual state of
+the world. It will change its meaning when the world shall have
+become more uniform, when all things shall have blended still
+more. But then doubtless men can no longer live and must give
+place to other beings&mdash;should I say far smaller or far larger?
+So that our criterion, remaining true for all men, retains an
+objective sense.</p>
+
+<p>And on the other hand what means the phrase 'very complex'?
+I have already given one solution, but there are others. Complex
+causes we have said produce a blend more and more intimate,<span class='pagenum'><a name="Page_410" id="Page_410">[Pg 410]</a></span>
+but after how long a time will this blend satisfy us? When
+will it have accumulated sufficient complexity? When shall we
+have sufficiently shuffled the cards? If we mix two powders, one
+blue, the other white, there comes a moment when the tint of the
+mixture seems to us uniform because of the feebleness of our
+senses; it will be uniform for the presbyte, forced to gaze from
+afar, before it will be so for the myope. And when it has become
+uniform for all eyes, we still could push back the limit by the use
+of instruments. There is no chance for any man ever to discern
+the infinite variety which, if the kinetic theory is true, hides
+under the uniform appearance of a gas. And yet if we accept
+Gouy's ideas on the Brownian movement, does not the microscope
+seem on the point of showing us something analogous?</p>
+
+<p>This new criterion is therefore relative like the first; and if it
+retains an objective character, it is because all men have approximately
+the same senses, the power of their instruments is
+limited, and besides they use them only exceptionally.</p>
+
+
+<h4>IX</h4>
+
+<p>It is just the same in the moral sciences and particularly in
+history. The historian is obliged to make a choice among the
+events of the epoch he studies; he recounts only those which
+seem to him the most important. He therefore contents himself
+with relating the most momentous events of the sixteenth century,
+for example, as likewise the most remarkable facts of the
+seventeenth century. If the first suffice to explain the second,
+we say these conform to the laws of history. But if a great event
+of the seventeenth century should have for cause a small fact of
+the sixteenth century which no history reports, which all the
+world has neglected, then we say this event is due to chance.
+This word has therefore the same sense as in the physical sciences;
+it means that slight causes have produced great effects.</p>
+
+<p>The greatest bit of chance is the birth of a great man. It is
+only by chance that meeting of two germinal cells, of different
+sex, containing precisely, each on its side, the mysterious elements
+whose mutual reaction must produce the genius. One will
+agree that these elements must be rare and that their meeting is
+still more rare. How slight a thing it would have required to
+deflect from its route the carrying spermatozoon. It would have<span class='pagenum'><a name="Page_411" id="Page_411">[Pg 411]</a></span>
+sufficed to deflect it a tenth of a millimeter and Napoleon would
+not have been born and the destinies of a continent would have
+been changed. No example can better make us understand the
+veritable characteristics of chance.</p>
+
+<p>One more word about the paradoxes brought out by the application
+of the calculus of probabilities to the moral sciences. It
+has been proven that no Chamber of Deputies will ever fail to
+contain a member of the opposition, or at least such an event
+would be so improbable that we might without fear wager the
+contrary, and bet a million against a sou.</p>
+
+<p>Condorcet has striven to calculate how many jurors it would
+require to make a judicial error practically impossible. If we
+had used the results of this calculation, we should certainly have
+been exposed to the same disappointments as in betting, on the
+faith of the calculus, that the opposition would never be without
+a representative.</p>
+
+<p>The laws of chance do not apply to these questions. If justice
+be not always meted out to accord with the best reasons, it uses
+less than we think the method of Bridoye. This is perhaps to
+be regretted, for then the system of Condorcet would shield us
+from judicial errors.</p>
+
+<p>What is the meaning of this? We are tempted to attribute
+facts of this nature to chance because their causes are obscure;
+but this is not true chance. The causes are unknown to us, it is
+true, and they are even complex; but they are not sufficiently so,
+since they conserve something. We have seen that this it is which
+distinguishes causes 'too simple.' When men are brought together
+they no longer decide at random and independently one
+of another; they influence one another. Multiplex causes come
+into action. They worry men, dragging them to right or left,
+but one thing there is they can not destroy, this is their Panurge
+flock-of-sheep habits. And this is an invariant.</p>
+
+
+<h4>X</h4>
+
+<p>Difficulties are indeed involved in the application of the
+calculus of probabilities to the exact sciences. Why are the
+decimals of a table of logarithms, why are those of the number
+&#960; distributed in accordance with the laws of chance? Elsewhere
+I have already studied the question in so far as it concerns<span class='pagenum'><a name="Page_412" id="Page_412">[Pg 412]</a></span>
+logarithms, and there it is easy. It is clear that a slight difference
+of argument will give a slight difference of logarithm, but a great
+difference in the sixth decimal of the logarithm. Always we find
+again the same criterion.</p>
+
+<p>But as for the number &#960;, that presents more difficulties, and I
+have at the moment nothing worth while to say.</p>
+
+<p>There would be many other questions to resolve, had I wished
+to attack them before solving that which I more specially set
+myself. When we reach a simple result, when we find for example
+a round number, we say that such a result can not be due
+to chance, and we seek, for its explanation, a non-fortuitous
+cause. And in fact there is only a very slight probability that
+among 10,000 numbers chance will give a round number; for
+example, the number 10,000. This has only one chance in 10,000.
+But there is only one chance in 10,000 for the occurrence of any
+other one number; and yet this result will not astonish us, nor
+will it be hard for us to attribute it to chance; and that simply
+because it will be less striking.</p>
+
+<p>Is this a simple illusion of ours, or are there cases where this
+way of thinking is legitimate? We must hope so, else were all
+science impossible. When we wish to check a hypothesis, what
+do we do? We can not verify all its consequences, since they
+would be infinite in number; we content ourselves with verifying
+certain ones and if we succeed we declare the hypothesis confirmed,
+because so much success could not be due to chance.
+And this is always at bottom the same reasoning.</p>
+
+<p>I can not completely justify it here, since it would take too
+much time; but I may at least say that we find ourselves confronted
+by two hypotheses, either a simple cause or that aggregate
+of complex causes we call chance. We find it natural to
+suppose that the first should produce a simple result, and then,
+if we find that simple result, the round number for example, it
+seems more likely to us to be attributable to the simple cause
+which must give it almost certainly, than to chance which could
+only give it once in 10,000 times. It will not be the same if we
+find a result which is not simple; chance, it is true, will not give
+this more than once in 10,000 times; but neither has the simple
+cause any more chance of producing it.</p>
+
+
+
+<hr style="width: 65%;" />
+<p><span class='pagenum'><a name="Page_413" id="Page_413">[Pg 413]</a></span></p>
+<h2><b>BOOK II<br />
+
+<br />
+<small>MATHEMATICAL REASONING</small></b></h2>
+
+
+
+<hr style="width: 65%;" />
+<h3>CHAPTER I</h3>
+
+<h3><span class="smcap">The Relativity of Space</span></h3>
+
+
+<h4>I</h4>
+
+<p>It is impossible to represent to oneself empty space; all our
+efforts to imagine a pure space, whence should be excluded the
+changing images of material objects, can result only in a representation
+where vividly colored surfaces, for example, are replaced
+by lines of faint coloration, and we can not go to the very
+end in this way without all vanishing and terminating in nothingness.
+Thence comes the irreducible relativity of space.</p>
+
+<p>Whoever speaks of absolute space uses a meaningless phrase.
+This is a truth long proclaimed by all who have reflected upon
+the matter, but which we are too often led to forget.</p>
+
+<p>I am at a determinate point in Paris, place du Panth&eacute;on for instance,
+and I say: I shall come back <i>here</i> to-morrow. If I be
+asked: Do you mean you will return to the same point of space,
+I shall be tempted to answer: yes; and yet I shall be wrong,
+since by to-morrow the earth will have journeyed hence, carrying
+with it the place du Panth&eacute;on, which will have traveled over
+more than two million kilometers. And if I tried to speak more
+precisely, I should gain nothing, since our globe has run over
+these two million kilometers in its motion with relation to the sun,
+while the sun in its turn is displaced with reference to the Milky
+Way, while the Milky Way itself is doubtless in motion without
+our being able to perceive its velocity. So that we are completely
+ignorant, and always shall be, of how much the place du Panth&eacute;on
+is displaced in a day.</p>
+
+<p>In sum, I meant to say: To-morrow I shall see again the dome<span class='pagenum'><a name="Page_414" id="Page_414">[Pg 414]</a></span>
+and the pediment of the Panth&eacute;on, and if there were no Panth&eacute;on
+my phrase would be meaningless and space would vanish.</p>
+
+<p>This is one of the most commonplace forms of the principle
+of the relativity of space; but there is another, upon which
+Delbeuf has particularly insisted. Suppose that in the night
+all the dimensions of the universe become a thousand times
+greater: the world will have remained <i>similar</i> to itself, giving to
+the word <i>similitude</i> the same meaning as in Euclid, Book VI.
+Only what was a meter long will measure thenceforth a kilometer,
+what was a millimeter long will become a meter. The bed whereon
+I lie and my body itself will be enlarged in the same proportion.</p>
+
+<p>When I awake to-morrow morning, what sensation shall I feel
+in presence of such an astounding transformation? Well, I shall
+perceive nothing at all. The most precise measurements will be
+incapable of revealing to me anything of this immense convulsion,
+since the measures I use will have varied precisely in the
+same proportion as the objects I seek to measure. In reality,
+this convulsion exists only for those who reason as if space were
+absolute. If I for a moment have reasoned as they do, it is the
+better to bring out that their way of seeing implies contradiction.
+In fact it would be better to say that, space being relative,
+nothing at all has happened, which is why we have perceived
+nothing.</p>
+
+<p>Has one the right, therefore, to say he knows the distance between
+two points? No, since this distance could undergo enormous
+variations without our being able to perceive them, provided
+the other distances have varied in the same proportion.
+We have just seen that when I say: I shall be here to-morrow,
+this does not mean: To-morrow I shall be at the same point of
+space where I am to-day, but rather: To-morrow I shall be at the
+same distance from the Panth&eacute;on as to-day. And we see that
+this statement is no longer sufficient and that I should say: To-morrow
+and to-day my distance from the Panth&eacute;on will be equal
+to the same number of times the height of my body.</p>
+
+<p>But this is not all; I have supposed the dimensions of the world
+to vary, but that at least the world remained always similar to
+itself. We might go much further, and one of the most astonishing
+theories of modern physics furnishes us the occasion.<span class='pagenum'><a name="Page_415" id="Page_415">[Pg 415]</a></span></p>
+
+<p>According to Lorentz and Fitzgerald, all the bodies borne along
+in the motion of the earth undergo a deformation.</p>
+
+<p>This deformation is, in reality, very slight, since all dimensions
+parallel to the movement of the earth diminish by a hundred
+millionth, while the dimensions perpendicular to this movement
+are unchanged. But it matters little that it is slight, that it
+exists suffices for the conclusion I am about to draw. And besides,
+I have said it was slight, but in reality I know nothing
+about it; I have myself been victim of the tenacious illusion
+which makes us believe we conceive an absolute space; I have
+thought of the motion of the earth in its elliptic orbit around
+the sun, and I have allowed thirty kilometers as its velocity.
+But its real velocity (I mean, this time, not its absolute velocity,
+which is meaningless, but its velocity with relation to the ether),
+I do not know that, and have no means of knowing it: it is perhaps,
+10, 100 times greater, and then the deformation will be 100,
+10,000 times more.</p>
+
+<p>Can we show this deformation? Evidently not; here is a cube
+with edge one meter; in consequence of the earth's displacement
+it is deformed, one of its edges, that parallel to the motion,
+becomes smaller, the others do not change. If I wish to assure
+myself of it by aid of a meter measure, I shall measure first
+one of the edges perpendicular to the motion and shall find that
+my standard meter fits this edge exactly; and in fact neither of
+these two lengths is changed, since both are perpendicular to
+the motion. Then I wish to measure the other edge, that parallel
+to the motion; to do this I displace my meter and turn it so as to
+apply it to the edge. But the meter, having changed orientation
+and become parallel to the motion, has undergone, in its
+turn, the deformation, so that though the edge be not a meter
+long, it will fit exactly, I shall find out nothing.</p>
+
+<p>You ask then of what use is the hypothesis of Lorentz and
+of Fitzgerald if no experiment can permit of its verification?
+It is my exposition that has been incomplete; I have spoken only
+of measurements that can be made with a meter; but we can
+also measure a length by the time it takes light to traverse it, on
+condition we suppose the velocity of light constant and independent
+of direction. Lorentz could have accounted for the<span class='pagenum'><a name="Page_416" id="Page_416">[Pg 416]</a></span>
+facts by supposing the velocity of light greater in the direction
+of the earth's motion than in the perpendicular direction.
+He preferred to suppose that the velocity is the same in these
+different directions but that the bodies are smaller in the one
+than in the other. If the wave surfaces of light had undergone
+the same deformations as the material bodies we should never
+have perceived the Lorentz-Fitzgerald deformation.</p>
+
+<p>In either case, it is not a question of absolute magnitude, but
+of the measure of this magnitude by means of some instrument;
+this instrument may be a meter, or the path traversed by light;
+it is only the relation of the magnitude to the instrument that
+we measure; and if this relation is altered, we have no way of
+knowing whether it is the magnitude or the instrument which
+has changed.</p>
+
+<p>But what I wish to bring out is, that in this deformation the
+world has not remained similar to itself; squares have become
+rectangles, circles ellipses, spheres ellipsoids. And yet we have
+no way of knowing whether this deformation be real.</p>
+
+<p>Evidently one could go much further: in place of the Lorentz-Fitzgerald
+deformation, whose laws are particularly simple, we
+could imagine any deformation whatsoever. Bodies could be
+deformed according to any laws, as complicated as we might wish,
+we never should notice it provided all bodies without exception
+were deformed according to the same laws. In saying, all bodies
+without exception, I include of course our own body and the
+light rays emanating from different objects.</p>
+
+<p>If we look at the world in one of those mirrors of complicated
+shape which deform objects in a bizarre way, the mutual relations
+of the different parts of this world would not be altered; if,
+in fact two real objects touch, their images likewise seem to touch.
+Of course when we look in such a mirror we see indeed the
+deformation, but this is because the real world subsists alongside
+of its deformed image; and then even were this real world
+hidden from us, something there is could not be hidden, ourself;
+we could not cease to see, or at least to feel, our body and our
+limbs which have not been deformed and which continue to serve
+us as instruments of measure.</p>
+
+<p>But if we imagine our body itself deformed in the same way<span class='pagenum'><a name="Page_417" id="Page_417">[Pg 417]</a></span>
+as if seen in the mirror, these instruments of measure in their
+turn will fail us and the deformation will no longer be ascertainable.</p>
+
+<p>Consider in the same way two worlds images of one another;
+to each object <i>P</i> of the world <i>A</i> corresponds in the world <i>B</i> an
+object <i>P&acute;</i>, its image; the coordinates of this image <i>P&acute;</i> are determinate
+functions of those of the object <i>P</i>; moreover these functions
+may be any whatsoever; I only suppose them chosen once
+for all. Between the position of <i>P</i> and that of <i>P&acute;</i> there is a
+constant relation; what this relation is, matters not; enough that
+it be constant.</p>
+
+<p>Well, these two worlds will be indistinguishable one from the
+other. I mean the first will be for its inhabitants what the
+second is for its. And so it will be as long as the two worlds
+remain strangers to each other. Suppose we lived in world <i>A</i>, we
+shall have constructed our science and in particular our geometry;
+during this time the inhabitants of world <i>B</i> will have constructed
+a science, and as their world is the image of ours, their
+geometry will also be the image of ours or, better, it will be the
+same. But if for us some day a window is opened upon world
+<i>B</i>, how we shall pity them: "Poor things," we shall say, "they
+think they have made a geometry, but what they call so is only
+a grotesque image of ours; their straights are all twisted, their
+circles are humped, their spheres have capricious inequalities."
+And we shall never suspect they say the same of us, and one
+never will know who is right.</p>
+
+<p>We see in how broad a sense should be understood the relativity
+of space; space is in reality amorphous and the things
+which are therein alone give it a form. What then should be
+thought of that direct intuition we should have of the straight
+or of distance? So little have we intuition of distance in itself
+that in the night, as we have said, a distance might become a
+thousand times greater without our being able to perceive it, if
+all other distances had undergone the same alteration. And even
+in a night the world <i>B</i> might be substituted for the world <i>A</i>
+without our having any way of knowing it, and then the straight
+lines of yesterday would have ceased to be straight and we
+should never notice.<span class='pagenum'><a name="Page_418" id="Page_418">[Pg 418]</a></span></p>
+
+<p>One part of space is not by itself and in the absolute sense of
+the word equal to another part of space; because if so it is for
+us, it would not be for the dwellers in world <i>B</i>; and these have
+just as much right to reject our opinion as we to condemn theirs.</p>
+
+<p>I have elsewhere shown what are the consequences of these
+facts from the viewpoint of the idea we should form of non-Euclidean
+geometry and other analogous geometries; to that I
+do not care to return; and to-day I shall take a somewhat different
+point of view.</p>
+
+
+<h4>II</h4>
+
+<p>If this intuition of distance, of direction, of the straight line,
+if this direct intuition of space in a word does not exist, whence
+comes our belief that we have it? If this is only an illusion,
+why is this illusion so tenacious? It is proper to examine into
+this. We have said there is no direct intuition of size and we
+can only arrive at the relation of this magnitude to our instruments
+of measure. We should therefore not have been able to
+construct space if we had not had an instrument to measure it;
+well, this instrument to which we relate everything, which we
+use instinctively, it is our own body. It is in relation to our
+body that we place exterior objects, and the only spatial relations
+of these objects that we can represent are their relations
+to our body. It is our body which serves us, so to speak, as
+system of axes of coordinates.</p>
+
+<p>For example, at an instant &#945;, the presence of the object <i>A</i> is
+revealed to me by the sense of sight; at another instant, &#946;, the
+presence of another object, <i>B</i>, is revealed to me by another sense,
+that of hearing or of touch, for instance. I judge that this
+object <i>B</i> occupies the same place as the object <i>A</i>. What does
+that mean? First that does not signify that these two objects
+occupy, at two different moments, the same point of an absolute
+space, which even if it existed would escape our cognition, since,
+between the instants &#945; and &#946;, the solar system has moved and
+we can not know its displacement. That means these two objects
+occupy the same relative position with reference to our body.</p>
+
+<p>But even this, what does it mean? The impressions that have
+come to us from these objects have followed paths absolutely<span class='pagenum'><a name="Page_419" id="Page_419">[Pg 419]</a></span>
+different, the optic nerve for the object <i>A</i>, the acoustic nerve for
+the object <i>B</i>. They have nothing in common from the qualitative
+point of view. The representations we are able to make of
+these two objects are absolutely heterogeneous, irreducible one to
+the other. Only I know that to reach the object <i>A</i> I have just
+to extend the right arm in a certain way; even when I abstain
+from doing it, I represent to myself the muscular sensations and
+other analogous sensations which would accompany this extension,
+and this representation is associated with that of the
+object <i>A</i>.</p>
+
+<p>Now, I likewise know I can reach the object <i>B</i> by extending my
+right arm in the same manner, an extension accompanied by the
+same train of muscular sensations. And when I say these two
+objects occupy the same place, I mean nothing more.</p>
+
+<p>I also know I could have reached the object <i>A</i> by another
+appropriate motion of the left arm and I represent to myself the
+muscular sensations which would have accompanied this movement;
+and by this same motion of the left arm, accompanied by
+the same sensations, I likewise could have reached the object <i>B</i>.</p>
+
+<p>And that is very important, since thus I can defend myself
+against dangers menacing me from the object <i>A</i> or the object <i>B</i>.
+With each of the blows we can be hit, nature has associated
+one or more parries which permit of our guarding ourselves.
+The same parry may respond to several strokes; and so it is, for
+instance, that the same motion of the right arm would have
+allowed us to guard at the instant &#945; against the object <i>A</i> and at
+the instant &#946; against the object <i>B</i>. Just so, the same stroke can
+be parried in several ways, and we have said, for instance, the
+object <i>A</i> could be reached indifferently either by a certain movement
+of the right arm or by a certain movement of the left arm.</p>
+
+<p>All these parries have nothing in common except warding off
+the same blow, and this it is, and nothing else, which is meant
+when we say they are movements terminating at the same point
+of space. Just so, these objects, of which we say they occupy
+the same point of space, have nothing in common, except that the
+same parry guards against them.</p>
+
+<p>Or, if you choose, imagine innumerable telegraph wires, some
+centripetal, others centrifugal. The centripetal wires warn us of<span class='pagenum'><a name="Page_420" id="Page_420">[Pg 420]</a></span>
+accidents happening without; the centrifugal wires carry the
+reparation. Connections are so established that when a centripetal
+wire is traversed by a current this acts on a relay and so
+starts a current in one of the centrifugal wires, and things are
+so arranged that several centripetal wires may act on the same
+centrifugal wire if the same remedy suits several ills, and that a
+centripetal wire may agitate different centrifugal wires, either
+simultaneously or in lieu one of the other when the same ill may
+be cured by several remedies.</p>
+
+<p>It is this complex system of associations, it is this table of distribution,
+so to speak, which is all our geometry or, if you wish,
+all in our geometry that is instinctive. What we call our intuition
+of the straight line or of distance is the consciousness we
+have of these associations and of their imperious character.</p>
+
+<p>And it is easy to understand whence comes this imperious
+character itself. An association will seem to us by so much the
+more indestructible as it is more ancient. But these associations
+are not, for the most part, conquests of the individual, since their
+trace is seen in the new-born babe: they are conquests of the race.
+Natural selection had to bring about these conquests by so much
+the more quickly as they were the more necessary.</p>
+
+<p>On this account, those of which we speak must have been of
+the earliest in date, since without them the defense of the organism
+would have been impossible. From the time when the cellules
+were no longer merely juxtaposed, but were called upon to
+give mutual aid, it was needful that a mechanism organize analogous
+to what we have described, so that this aid miss not its
+way, but forestall the peril.</p>
+
+<p>When a frog is decapitated, and a drop of acid is placed on a
+point of its skin, it seeks to wipe off the acid with the nearest foot,
+and, if this foot be amputated, it sweeps it off with the foot of
+the opposite side. There we have the double parry of which I
+have just spoken, allowing the combating of an ill by a second
+remedy, if the first fails. And it is this multiplicity of parries,
+and the resulting coordination, which is space.</p>
+
+<p>We see to what depths of the unconscious we must descend
+to find the first traces of these spatial associations, since only
+the inferior parts of the nervous system are involved. Why be<span class='pagenum'><a name="Page_421" id="Page_421">[Pg 421]</a></span>
+astonished then at the resistance we oppose to every attempt
+made to dissociate what so long has been associated? Now, it is
+just this resistance that we call the evidence for the geometric
+truths; this evidence is nothing but the repugnance we feel toward
+breaking with very old habits which have always proved good.</p>
+
+
+<h4>III</h4>
+
+<p>The space so created is only a little space extending no farther
+than my arm can reach; the intervention of the memory is necessary
+to push back its limits. There are points which will remain
+out of my reach, whatever effort I make to stretch forth my hand;
+if I were fastened to the ground like a hydra polyp, for instance,
+which can only extend its tentacles, all these points would be
+outside of space, since the sensations we could experience from
+the action of bodies there situated, would be associated with the
+idea of no movement allowing us to reach them, of no appropriate
+parry. These sensations would not seem to us to have
+any spatial character and we should not seek to localize them.</p>
+
+<p>But we are not fixed to the ground like the lower animals; we
+can, if the enemy be too far away, advance toward him first and
+extend the hand when we are sufficiently near. This is still a
+parry, but a parry at long range. On the other hand, it is a
+complex parry, and into the representation we make of it enter
+the representation of the muscular sensations caused by the
+movements of the legs, that of the muscular sensations caused
+by the final movement of the arm, that of the sensations of the
+semicircular canals, etc. We must, besides, represent to ourselves,
+not a complex of simultaneous sensations, but a complex
+of successive sensations, following each other in a determinate
+order, and this is why I have just said the intervention of memory
+was necessary. Notice moreover that, to reach the same point,
+I may approach nearer the mark to be attained, so as to have to
+stretch my arm less. What more? It is not one, it is a thousand
+parries I can oppose to the same danger. All these parries are
+made of sensations which may have nothing in common and yet
+we regard them as defining the same point of space, since they
+may respond to the same danger and are all associated with the
+notion of this danger. It is the potentiality of warding off the<span class='pagenum'><a name="Page_422" id="Page_422">[Pg 422]</a></span>
+same stroke which makes the unity of these different parries, as
+it is the possibility of being parried in the same way which makes
+the unity of the strokes so different in kind, which may menace
+us from the same point of space. It is this double unity which
+makes the individuality of each point of space, and, in the
+notion of point, there is nothing else.</p>
+
+<p>The space before considered, which might be called <i>restricted
+space</i>, was referred to coordinate axes bound to my body; these
+axes were fixed, since my body did not move and only my members
+were displaced. What are the axes to which we naturally
+refer the <i>extended space</i>? that is to say the new space just
+defined. We define a point by the sequence of movements to be
+made to reach it, starting from a certain initial position of the
+body. The axes are therefore fixed to this initial position of the
+body.</p>
+
+<p>But the position I call initial may be arbitrarily chosen among
+all the positions my body has successively occupied; if the memory
+more or less unconscious of these successive positions is necessary
+for the genesis of the notion of space, this memory may go back
+more or less far into the past. Thence results in the definition
+itself of space a certain indetermination, and it is precisely this
+indetermination which constitutes its relativity.</p>
+
+<p>There is no absolute space, there is only space relative to a
+certain initial position of the body. For a conscious being fixed
+to the ground like the lower animals, and consequently knowing
+only restricted space, space would still be relative (since it would
+have reference to his body), but this being would not be conscious
+of this relativity, because the axes of reference for this restricted
+space would be unchanging! Doubtless the rock to which this
+being would be fettered would not be motionless, since it would
+be carried along in the movement of our planet; for us consequently
+these axes would change at each instant; but for him they
+would be changeless. We have the faculty of referring our
+extended space now to the position <i>A</i> of our body, considered as
+initial, again to the position <i>B</i>, which it had some moments
+afterward, and which we are free to regard in its turn as initial;
+we make therefore at each instant unconscious transformations
+of coordinates. This faculty would be lacking in our imaginary<span class='pagenum'><a name="Page_423" id="Page_423">[Pg 423]</a></span>
+being, and from not having traveled, he would think space absolute.
+At every instant, his system of axes would be imposed
+upon him; this system would have to change greatly in reality,
+but for him it would be always the same, since it would be
+always the <i>only</i> system. Quite otherwise is it with us, who at
+each instant have many systems between which we may choose at
+will, on condition of going back by memory more or less far into
+the past.</p>
+
+<p>This is not all; restricted space would not be homogeneous;
+the different points of this space could not be regarded as equivalent,
+since some could be reached only at the cost of the greatest
+efforts, while others could be easily attained. On the contrary,
+our extended space seems to us homogeneous, and we say all its
+points are equivalent. What does that mean?</p>
+
+<p>If we start from a certain place <i>A</i>, we can, from this position,
+make certain movements, <i>M</i>, characterized by a certain complex
+of muscular sensations. But, starting from another position, <i>B</i>,
+we make movements <i>M&acute;</i> characterized by the same muscular sensations.
+Let <i>a</i>, then, be the situation of a certain point of the
+body, the end of the index finger of the right hand for example,
+in the initial position <i>A</i>, and <i>b</i> the situation of this same index
+when, starting from this position <i>A</i>, we have made the motions <i>M</i>.
+Afterwards, let <i>a&acute;</i> be the situation of this index in the position <i>B</i>,
+and <i>b&acute;</i> its situation when, starting from the position <i>B</i>, we have
+made the motions <i>M&acute;</i>.</p>
+
+<p>Well, I am accustomed to say that the points of space <i>a</i> and <i>b</i>
+are related to each other just as the points <i>a&acute;</i> and <i>b&acute;</i>, and this
+simply means that the two series of movements <i>M</i> and <i>M&acute;</i> are
+accompanied by the same muscular sensations. And as I am
+conscious that, in passing from the position <i>A</i> to the position <i>B</i>,
+my body has remained capable of the same movements, I know
+there is a point of space related to the point <i>a&acute;</i> just as any point
+<i>b</i> is to the point <i>a</i>, so that the two points <i>a</i> and <i>a&acute;</i> are equivalent.
+This is what is called the homogeneity of space. And, at the same
+time, this is why space is relative, since its properties remain the
+same whether it be referred to the axes <i>A</i> or to the axes <i>B</i>. So
+that the relativity of space and its homogeneity are one sole and
+same thing.<span class='pagenum'><a name="Page_424" id="Page_424">[Pg 424]</a></span></p>
+
+<p>Now, if I wish to pass to the great space, which no longer
+serves only for me, but where I may lodge the universe, I get
+there by an act of imagination. I imagine how a giant would
+feel who could reach the planets in a few steps; or, if you choose,
+what I myself should feel in presence of a miniature world where
+these planets were replaced by little balls, while on one of these
+little balls moved a liliputian I should call myself. But this act
+of imagination would be impossible for me had I not previously
+constructed my restricted space and my extended space for my
+own use.</p>
+
+
+<h4>IV</h4>
+
+<p>Why now have all these spaces three dimensions? Go back
+to the "table of distribution" of which we have spoken. We
+have on the one side the list of the different possible dangers;
+designate them by <i>A1</i>, <i>A2</i>, etc.; and, on the other side, the list
+of the different remedies which I shall call in the same way
+<i>B1</i>, <i>B2</i>, etc. We have then connections between the contact studs
+or push buttons of the first list and those of the second, so that
+when, for instance, the announcer of danger <i>A3</i> functions, it
+will put or may put in action the relay corresponding to the
+parry <i>B4</i>.</p>
+
+<p>As I have spoken above of centripetal or centrifugal wires, I
+fear lest one see in all this, not a simple comparison, but a description
+of the nervous system. Such is not my thought, and that
+for several reasons: first I should not permit myself to put forth
+an opinion on the structure of the nervous system which I do
+not know, while those who have studied it speak only circumspectly;
+again because, despite my incompetence, I well know
+this scheme would be too simplistic; and finally because on my
+list of parries, some would figure very complex, which might even,
+in the case of extended space, as we have seen above, consist of
+many steps followed by a movement of the arm. It is not a question
+then of physical connection between two real conductors
+but of psychologic association between two series of sensations.</p>
+
+<p>If <i>A1</i> and <i>A2</i> for instance are both associated with the parry
+<i>B1</i>, and if <i>A1</i> is likewise associated with the parry <i>B2</i>, it will
+generally happen that <i>A2</i> and <i>B2</i> will also themselves be associated.
+If this fundamental law were not generally true, there<span class='pagenum'><a name="Page_425" id="Page_425">[Pg 425]</a></span>
+would exist only an immense confusion and there would be
+nothing resembling a conception of space or a geometry. How
+in fact have we defined a point of space. We have done it in two
+ways: it is on the one hand the aggregate of announcers <i>A</i> in
+connection with the same parry <i>B</i>; it is on the other hand the
+aggregate of parries <i>B</i> in connection with the same announcer <i>A</i>.
+If our law was not true, we should say <i>A1</i> and <i>A2</i> correspond
+to the same point since they are both in connection with <i>B1</i>; but
+we should likewise say they do not correspond to the same point,
+since <i>A1</i> would be in connection with <i>B2</i> and the same would
+not be true of <i>A2</i>. This would be a contradiction.</p>
+
+<p>But, from another side, if the law were rigorously and always
+true, space would be very different from what it is. We should
+have categories strongly contrasted between which would be
+portioned out on the one hand the announcers <i>A</i>, on the other
+hand the parries <i>B</i>; these categories would be excessively numerous,
+but they would be entirely separated one from another.
+Space would be composed of points very numerous, but discrete;
+it would be <i>discontinuous</i>. There would be no reason for ranging
+these points in one order rather than another, nor consequently
+for attributing to space three dimensions.</p>
+
+<p>But it is not so; permit me to resume for a moment the language
+of those who already know geometry; this is quite proper
+since this is the language best understood by those I wish to make
+understand me.</p>
+
+<p>When I desire to parry the stroke, I seek to attain the point
+whence comes this blow, but it suffices that I approach quite near.
+Then the parry <i>B1</i> may answer for <i>A1</i> and for <i>A2</i>, if the point
+which corresponds to <i>B1</i> is sufficiently near both to that corresponding
+to <i>A1</i> and to that corresponding to <i>A2</i>. But it may
+happen that the point corresponding to another parry <i>B2</i> may be
+sufficiently near to the point corresponding to A1 and not sufficiently
+near the point corresponding to <i>A2</i>; so that the parry <i>B2</i>
+may answer for <i>A1</i> without answering for <i>A2</i>. For one who
+does not yet know geometry, this translates itself simply by a
+derogation of the law stated above. And then things will happen
+thus:</p>
+
+<p>Two parries <i>B1</i> and <i>B2</i> will be associated with the same warning<span class='pagenum'><a name="Page_426" id="Page_426">[Pg 426]</a></span>
+<i>A1</i> and with a large number of warnings which we shall
+range in the same category as <i>A1</i> and which we shall make correspond
+to the same point of space. But we may find warnings
+<i>A2</i> which will be associated with <i>B2</i> without being associated
+with <i>B1</i>, and which in compensation will be associated with <i>B3</i>,
+which <i>B3</i> was not associated with <i>A1</i>, and so forth, so that we
+may write the series</p>
+
+<p class="center">
+<i>B1</i>, <i>A1</i>, <i>B2</i>, <i>A2</i>, <i>B3</i>, <i>A3</i>, <i>B4</i>, <i>A4</i>,<br />
+</p>
+
+<p class="noidt">where each term is associated with the following and the preceding,
+but not with the terms several places away.</p>
+
+<p>Needless to add that each of the terms of these series is not
+isolated, but forms part of a very numerous category of other
+warnings or of other parries which have the same connections as
+it, and which may be regarded as belonging to the same point of
+space.</p>
+
+<p>The fundamental law, though admitting of exceptions, remains
+therefore almost always true. Only, in consequence of these
+exceptions, these categories, in place of being entirely separated,
+encroach partially one upon another and mutually penetrate in
+a certain measure, so that space becomes continuous.</p>
+
+<p>On the other hand, the order in which these categories are to
+be ranged is no longer arbitrary, and if we refer to the preceding
+series, we see it is necessary to put <i>B2</i> between <i>A1</i> and <i>A2</i> and
+consequently between <i>B1</i> and <i>B3</i> and that we could not for
+instance put it between <i>B3</i> and <i>B4</i>.</p>
+
+<p>There is therefore an order in which are naturally arranged
+our categories which correspond to the points of space, and experience
+teaches us that this order presents itself under the
+form of a table of triple entry, and this is why space has three
+dimensions.</p>
+
+
+<h4>V</h4>
+
+<p>So the characteristic property of space, that of having three
+dimensions, is only a property of our table of distribution, an
+internal property of the human intelligence, so to speak. It
+would suffice to destroy certain of these connections, that is to
+say of the associations of ideas to give a different table of distribution,
+and that might be enough for space to acquire a fourth
+dimension.<span class='pagenum'><a name="Page_427" id="Page_427">[Pg 427]</a></span></p>
+
+<p>Some persons will be astonished at such a result. The external
+world, they will think, should count for something. If the number
+of dimensions comes from the way we are made, there might
+be thinking beings living in our world, but who might be made
+differently from us and who would believe space has more or less
+than three dimensions. Has not M. de Cyon said that the Japanese
+mice, having only two pair of semicircular canals, believe
+that space is two-dimensional? And then this thinking being, if
+he is capable of constructing a physics, would he not make a physics
+of two or of four dimensions, and which in a sense would
+still be the same as ours, since it would be the description of the
+same world in another language?</p>
+
+<p>It seems in fact that it would be possible to translate our physics
+into the language of geometry of four dimensions; to attempt
+this translation would be to take great pains for little profit, and
+I shall confine myself to citing the mechanics of Hertz where we
+have something analogous. However, it seems that the translation
+would always be less simple than the text, and that it would
+always have the air of a translation, that the language of three
+dimensions seems the better fitted to the description of our world,
+although this description can be rigorously made in another
+idiom. Besides, our table of distribution was not made at random.
+There is connection between the warning <i>A1</i> and the
+parry <i>B1</i>, this is an internal property of our intelligence; but
+why this connection? It is because the parry <i>B1</i> affords means
+effectively to guard against the danger <i>A1</i>; and this is a fact
+exterior to us, this is a property of the exterior world. Our
+table of distribution is therefore only the translation of an aggregate
+of exterior facts; if it has three dimensions, this is because
+it has adapted itself to a world having certain properties;
+and the chief of these properties is that there exist natural solids
+whose displacements follow sensibly the laws we call laws of
+motion of rigid solids. If therefore the language of three dimensions
+is that which permits us most easily to describe our
+world, we should not be astonished; this language is copied from
+our table of distribution; and it is in order to be able to live in
+this world that this table has been established.</p>
+
+<p>I have said we could conceive, living in our world, thinking<span class='pagenum'><a name="Page_428" id="Page_428">[Pg 428]</a></span>
+beings whose table of distribution would be four-dimensional
+and who consequently would think in hyperspace. It is not
+certain however that such beings, admitting they were born there,
+could live there and defend themselves against the thousand
+dangers by which they would there be assailed.</p>
+
+
+<h4>VI</h4>
+
+<p>A few remarks to end with. There is a striking contrast between
+the roughness of this primitive geometry, reducible to
+what I call a table of distribution, and the infinite precision of
+the geometers' geometry. And yet this is born of that; but not
+of that alone; it must be made fecund by the faculty we have of
+constructing mathematical concepts, such as that of group, for
+instance; it was needful to seek among the pure concepts that
+which best adapts itself to this rough space whose genesis I have
+sought to explain and which is common to us and the higher
+animals.</p>
+
+<p>The evidence for certain geometric postulates, we have said, is
+only our repugnance to renouncing very old habits. But these
+postulates are infinitely precise, while these habits have something
+about them essentially pliant. When we wish to think, we
+need postulates infinitely precise, since this is the only way to
+avoid contradiction; but among all the possible systems of postulates,
+there are some we dislike to choose because they are not
+sufficiently in accord with our habits; however pliant, however
+elastic they may be, these have a limit of elasticity.</p>
+
+<p>We see that if geometry is not an experimental science, it is a
+science born apropos of experience; that we have created the
+space it studies, but adapting it to the world wherein we live.
+We have selected the most convenient space, but experience has
+guided our choice; as this choice has been unconscious, we think
+it has been imposed upon us; some say experience imposes it,
+others that we are born with our space ready made; we see from
+the preceding considerations, what in these two opinions is the
+part of truth, what of error.</p>
+
+<p>In this progressive education whose outcome has been the construction
+of space, it is very difficult to determine what is the<span class='pagenum'><a name="Page_429" id="Page_429">[Pg 429]</a></span>
+part of the individual, what the part of the race. How far could
+one of us, transported from birth to an entirely different world,
+where were dominant, for instance, bodies moving in conformity
+to the laws of motion of non-Euclidean solids, renounce the ancestral
+space to build a space completely new?</p>
+
+<p>The part of the race seems indeed preponderant; yet if to it we
+owe rough space, the soft space I have spoken of, the space of
+the higher animals, is it not to the unconscious experience of the
+individual we owe the infinitely precise space of the geometer?
+This is a question not easy to solve. Yet we cite a fact showing
+that the space our ancestors have bequeathed us still retains a
+certain plasticity. Some hunters learn to shoot fish under water,
+though the image of these fish be turned up by refraction. Besides
+they do it instinctively: they therefore have learned to
+modify their old instinct of direction; or, if you choose, to substitute
+for the association <i>A1</i>, <i>B1</i>, another association <i>A1</i>, <i>B2</i>,
+because experience showed them the first would not work.</p>
+
+
+
+<hr style="width: 65%;" />
+<p><span class='pagenum'><a name="Page_430" id="Page_430">[Pg 430]</a></span></p>
+<h3>CHAPTER II</h3>
+
+<h3><span class="smcap">Mathematical Definitions and Teaching</span></h3>
+
+
+<p>1. I should speak here of general definitions in mathematics;
+at least that is the title, but it will be impossible to confine myself
+to the subject as strictly as the rule of unity of action would
+require; I shall not be able to treat it without touching upon a
+few other related questions, and if thus I am forced from time
+to time to walk on the bordering flower-beds on the right or left,
+I pray you bear with me.</p>
+
+<p>What is a good definition? For the philosopher or the scientist
+it is a definition which applies to all the objects defined, and only
+those; it is the one satisfying the rules of logic. But in teaching
+it is not that; a good definition is one understood by the
+scholars.</p>
+
+<p>How does it happen that so many refuse to understand mathematics?
+Is that not something of a paradox? Lo and behold!
+a science appealing only to the fundamental principles of logic,
+to the principle of contradiction, for instance, to that which is
+the skeleton, so to speak, of our intelligence, to that of which we
+can not divest ourselves without ceasing to think, and there are
+people who find it obscure! and they are even in the majority!
+That they are incapable of inventing may pass, but that they do
+not understand the demonstrations shown them, that they remain
+blind when we show them a light which seems to us flashing
+pure flame, this it is which is altogether prodigious.</p>
+
+<p>And yet there is no need of a wide experience with examinations
+to know that these blind men are in no wise exceptional
+beings. This is a problem not easy to solve, but which should
+engage the attention of all those wishing to devote themselves to
+teaching.</p>
+
+<p>What is it, to understand? Has this word the same meaning
+for all the world? To understand the demonstration of a theorem,
+is that to examine successively each of the syllogisms composing
+it and to ascertain its correctness, its conformity to the rules of<span class='pagenum'><a name="Page_431" id="Page_431">[Pg 431]</a></span>
+the game? Likewise, to understand a definition, is this merely
+to recognize that one already knows the meaning of all the terms
+employed and to ascertain that it implies no contradiction?</p>
+
+<p>For some, yes; when they have done this, they will say: I understand.</p>
+
+<p>For the majority, no. Almost all are much more exacting;
+they wish to know not merely whether all the syllogisms of a
+demonstration are correct, but why they link together in this
+order rather than another. In so far as to them they seem engendered
+by caprice and not by an intelligence always conscious
+of the end to be attained, they do not believe they understand.</p>
+
+<p>Doubtless they are not themselves just conscious of what they
+crave and they could not formulate their desire, but if they do
+not get satisfaction, they vaguely feel that something is lacking.
+Then what happens? In the beginning they still perceive the
+proofs one puts under their eyes; but as these are connected
+only by too slender a thread to those which precede and those
+which follow, they pass without leaving any trace in their head;
+they are soon forgotten; a moment bright, they quickly vanish in
+night eternal. When they are farther on, they will no longer see
+even this ephemeral light, since the theorems lean one upon
+another and those they would need are forgotten; thus it is they
+become incapable of understanding mathematics.</p>
+
+<p>This is not always the fault of their teacher; often their mind,
+which needs to perceive the guiding thread, is too lazy to seek
+and find it. But to come to their aid, we first must know just
+what hinders them.</p>
+
+<p>Others will always ask of what use is it; they will not have
+understood if they do not find about them, in practise or in
+nature, the justification of such and such a mathematical concept.
+Under each word they wish to put a sensible image; the definition
+must evoke this image, so that at each stage of the demonstration
+they may see it transform and evolve. Only upon this condition
+do they comprehend and retain. Often these deceive themselves;
+they do not listen to the reasoning, they look at the figures; they
+think they have understood and they have only seen.</p>
+
+<p>2. How many different tendencies! Must we combat them?
+Must we use them? And if we wish to combat them, which should<span class='pagenum'><a name="Page_432" id="Page_432">[Pg 432]</a></span>
+be favored? Must we show those content with the pure logic that
+they have seen only one side of the matter? Or need we say to
+those not so cheaply satisfied that what they demand is not
+necessary?</p>
+
+<p>In other words, should we constrain the young people to change
+the nature of their minds? Such an attempt would be vain; we
+do not possess the philosopher's stone which would enable us to
+transmute one into another the metals confided to us; all we
+can do is to work with them, adapting ourselves to their
+properties.</p>
+
+<p>Many children are incapable of becoming mathematicians, to
+whom however it is necessary to teach mathematics; and the
+mathematicians themselves are not all cast in the same mold.
+To read their works suffices to distinguish among them two
+sorts of minds, the logicians like Weierstrass for example, the
+intuitives like Riemann. There is the same difference among
+our students. The one sort prefer to treat their problems 'by
+analysis' as they say, the others 'by geometry.'</p>
+
+<p>It is useless to seek to change anything of that, and besides
+would it be desirable? It is well that there are logicians and
+that there are intuitives; who would dare say whether he preferred
+that Weierstrass had never written or that there never
+had been a Riemann? We must therefore resign ourselves to the
+diversity of minds, or better we must rejoice in it.</p>
+
+<p>3. Since the word understand has many meanings, the definitions
+which will be best understood by some will not be best
+suited to others. We have those which seek to produce an image,
+and those where we confine ourselves to combining empty forms,
+perfectly intelligible, but purely intelligible, which abstraction
+has deprived of all matter.</p>
+
+<p>I know not whether it be necessary to cite examples. Let us
+cite them, anyhow, and first the definition of fractions will furnish
+us an extreme case. In the primary schools, to define a fraction,
+one cuts up an apple or a pie; it is cut up mentally of
+course and not in reality, because I do not suppose the budget
+of the primary instruction allows of such prodigality. At the
+Normal School, on the other hand, or at the college, it is said:
+a fraction is the combination of two whole numbers separated by<span class='pagenum'><a name="Page_433" id="Page_433">[Pg 433]</a></span>
+a horizontal bar; we define by conventions the operations to
+which these symbols may be submitted; it is proved that the rules
+of these operations are the same as in calculating with whole
+numbers, and we ascertain finally that multiplying the fraction,
+according to these rules, by the denominator gives the numerator.
+This is all very well because we are addressing young people
+long familiarized with the notion of fractions through having cut
+up apples or other objects, and whose mind, matured by a hard
+mathematical education, has come little by little to desire a purely
+logical definition. But the d&eacute;butant to whom one should try to
+give it, how dumfounded!</p>
+
+<p>Such also are the definitions found in a book justly admired
+and greatly honored, the <i>Foundations of Geometry</i> by Hilbert.
+See in fact how he begins: <i>We think three systems of</i> <span class="smcap">things</span>
+<i>which we shall call points, straights and planes</i>. What are these
+'things'?</p>
+
+<p>We know not, nor need we know; it would even be a pity to
+seek to know; all we have the right to know of them is what the
+assumptions tell us; this for example: <i>Two distinct points always
+determine a straight</i>, which is followed by this remark: <i>in place
+of determine, we may say the two points are on the straight, or
+the straight goes through these two points or joins the two points</i>.</p>
+
+<p>Thus 'to be on a straight' is simply defined as synonymous
+with 'determine a straight.' Behold a book of which I think
+much good, but which I should not recommend to a school boy.
+Yet I could do so without fear, he would not read much of it.
+I have taken extreme examples and no teacher would dream of
+going that far. But even stopping short of such models, does
+he not already expose himself to the same danger?</p>
+
+<p>Suppose we are in a class; the professor dictates: the circle is
+the locus of points of the plane equidistant from an interior point
+called the center. The good scholar writes this phrase in his
+note-book; the bad scholar draws faces; but neither understands;
+then the professor takes the chalk and draws a circle on the board.
+"Ah!" think the scholars, "why did he not say at once: a circle
+is a ring, we should have understood." Doubtless the professor
+is right. The scholars' definition would have been of no avail,
+since it could serve for no demonstration, since besides it would<span class='pagenum'><a name="Page_434" id="Page_434">[Pg 434]</a></span>
+not give them the salutary habit of analyzing their conceptions.
+But one should show them that they do not comprehend what
+they think they know, lead them to be conscious of the roughness
+of their primitive conception, and of themselves to wish it purified
+and made precise.</p>
+
+<p>4. I shall return to these examples; I only wished to show you
+the two opposed conceptions; they are in violent contrast. This
+contrast the history of science explains. If we read a book
+written fifty years ago, most of the reasoning we find there seems
+lacking in rigor. Then it was assumed a continuous function
+can change sign only by vanishing; to-day we prove it. It was
+assumed the ordinary rules of calculation are applicable to
+incommensurable numbers; to-day we prove it. Many other
+things were assumed which sometimes were false.</p>
+
+<p>We trusted to intuition; but intuition can not give rigor, nor
+even certainty; we see this more and more. It tells us for instance
+that every curve has a tangent, that is to say that every
+continuous function has a derivative, and that is false. And as
+we sought certainty, we had to make less and less the part of
+intuition.</p>
+
+<p>What has made necessary this evolution? We have not been
+slow to perceive that rigor could not be established in the
+reasonings, if it were not first put into the definitions.</p>
+
+<p>The objects occupying mathematicians were long ill defined;
+we thought we knew them because we represented them with the
+senses or the imagination; but we had of them only a rough
+image and not a precise concept upon which reasoning could take
+hold. It is there that the logicians would have done well to direct
+their efforts.</p>
+
+<p>So for the incommensurable number, the vague idea of continuity,
+which we owe to intuition, has resolved itself into a complicated
+system of inequalities bearing on whole numbers. Thus
+have finally vanished all those difficulties which frightened our
+fathers when they reflected upon the foundations of the infinitesimal
+calculus. To-day only whole numbers are left in analysis,
+or systems finite or infinite of whole numbers, bound by a
+plexus of equalities and inequalities. Mathematics we say is
+arithmetized.<span class='pagenum'><a name="Page_435" id="Page_435">[Pg 435]</a></span></p>
+
+<p>5. But do you think mathematics has attained absolute rigor
+without making any sacrifice? Not at all; what it has gained in
+rigor it has lost in objectivity. It is by separating itself from
+reality that it has acquired this perfect purity. We may freely
+run over its whole domain, formerly bristling with obstacles, but
+these obstacles have not disappeared. They have only been
+moved to the frontier, and it would be necessary to vanquish
+them anew if we wished to break over this frontier to enter the
+realm of the practical.</p>
+
+<p>We had a vague notion, formed of incongruous elements, some
+<i>a priori</i>, others coming from experiences more or less digested;
+we thought we knew, by intuition, its principal properties. To-day
+we reject the empiric elements, retaining only the <i>a priori</i>;
+one of the properties serves as definition and all the others are
+deduced from it by rigorous reasoning. This is all very well,
+but it remains to be proved that this property, which has become
+a definition, pertains to the real objects which experience had
+made known to us and whence we drew our vague intuitive
+notion. To prove that, it would be necessary to appeal to experience,
+or to make an effort of intuition, and if we could not prove
+it, our theorems would be perfectly rigorous, but perfectly
+useless.</p>
+
+<p>Logic sometimes makes monsters. Since half a century we
+have seen arise a crowd of bizarre functions which seem to try
+to resemble as little as possible the honest functions which serve
+some purpose. No longer continuity, or perhaps continuity, but
+no derivatives, etc. Nay more, from the logical point of view,
+it is these strange functions which are the most general, those
+one meets without seeking no longer appear except as particular
+case. There remains for them only a very small corner.</p>
+
+<p>Heretofore when a new function was invented, it was for some
+practical end; to-day they are invented expressly to put at fault
+the reasonings of our fathers, and one never will get from them
+anything more than that.</p>
+
+<p>If logic were the sole guide of the teacher, it would be necessary
+to begin with the most general functions, that is to say with
+the most bizarre. It is the beginner that would have to be set<span class='pagenum'><a name="Page_436" id="Page_436">[Pg 436]</a></span>
+grappling with this teratologic museum. If you do not do it,
+the logicians might say, you will achieve rigor only by stages.</p>
+
+<p>6. Yes, perhaps, but we can not make so cheap of reality, and
+I mean not only the reality of the sensible world, which however
+has its worth, since it is to combat against it that nine tenths of
+your students ask of you weapons. There is a reality more
+subtile, which makes the very life of the mathematical beings,
+and which is quite other than logic.</p>
+
+<p>Our body is formed of cells, and the cells of atoms; are these
+cells and these atoms then all the reality of the human body?
+The way these cells are arranged, whence results the unity of the
+individual, is it not also a reality and much more interesting?</p>
+
+<p>A naturalist who never had studied the elephant except in
+the microscope, would he think he knew the animal adequately?
+It is the same in mathematics. When the logician shall have
+broken up each demonstration into a multitude of elementary
+operations, all correct, he still will not possess the whole reality;
+this I know not what which makes the unity of the demonstration
+will completely escape him.</p>
+
+<p>In the edifices built up by our masters, of what use to admire
+the work of the mason if we can not comprehend the plan of
+the architect? Now pure logic can not give us this appreciation
+of the total effect; this we must ask of intuition.</p>
+
+<p>Take for instance the idea of continuous function. This is at
+first only a sensible image, a mark traced by the chalk on the
+blackboard. Little by little it is refined; we use it to construct
+a complicated system of inequalities, which reproduces all the
+features of the primitive image; when all is done, we have
+<i>removed the centering</i>, as after the construction of an arch;
+this rough representation, support thenceforth useless, has disappeared
+and there remains only the edifice itself, irreproachable
+in the eyes of the logician. And yet, if the professor did not
+recall the primitive image, if he did not restore momentarily the
+<i>centering</i>, how could the student divine by what caprice all these
+inequalities have been scaffolded in this fashion one upon another?
+The definition would be logically correct, but it would
+not show him the veritable reality.</p>
+
+<p>7. So back we must return; doubtless it is hard for a master<span class='pagenum'><a name="Page_437" id="Page_437">[Pg 437]</a></span>
+to teach what does not entirely satisfy him; but the satisfaction
+of the master is not the unique object of teaching; we should first
+give attention to what the mind of the pupil is and to what we
+wish it to become.</p>
+
+<p>Zoologists maintain that the embryonic development of an
+animal recapitulates in brief the whole history of its ancestors
+throughout geologic time. It seems it is the same in the development
+of minds. The teacher should make the child go over the
+path his fathers trod; more rapidly, but without skipping stations.
+For this reason, the history of science should be our first
+guide.</p>
+
+<p>Our fathers thought they knew what a fraction was, or continuity,
+or the area of a curved surface; we have found they did
+not know it. Just so our scholars think they know it when they
+begin the serious study of mathematics. If without warning I
+tell them: "No, you do not know it; what you think you understand,
+you do not understand; I must prove to you what seems
+to you evident," and if in the demonstration I support myself
+upon premises which to them seem less evident than the conclusion,
+what shall the unfortunates think? They will think that
+the science of mathematics is only an arbitrary mass of useless
+subtilities; either they will be disgusted with it, or they will play
+it as a game and will reach a state of mind like that of the Greek
+sophists.</p>
+
+<p>Later, on the contrary, when the mind of the scholar, familiarized
+with mathematical reasoning, has been matured by this long
+frequentation, the doubts will arise of themselves and then your
+demonstration will be welcome. It will awaken new doubts, and
+the questions will arise successively to the child, as they arose successively
+to our fathers, until perfect rigor alone can satisfy him.
+To doubt everything does not suffice, one must know why he
+doubts.</p>
+
+<p>8. The principal aim of mathematical teaching is to develop
+certain faculties of the mind, and among them intuition is not the
+least precious. It is through it that the mathematical world
+remains in contact with the real world, and if pure mathematics
+could do without it, it would always be necessary to have recourse
+to it to fill up the chasm which separates the symbol from reality.<span class='pagenum'><a name="Page_438" id="Page_438">[Pg 438]</a></span>
+The practician will always have need of it, and for one pure
+geometer there should be a hundred practicians.</p>
+
+<p>The engineer should receive a complete mathematical education,
+but for what should it serve him?</p>
+
+<p>To see the different aspects of things and see them quickly;
+he has no time to hunt mice. It is necessary that, in the complex
+physical objects presented to him, he should promptly recognize
+the point where the mathematical tools we have put in his
+hands can take hold. How could he do it if we should leave
+between instruments and objects the deep chasm hollowed out
+by the logicians?</p>
+
+<p>9. Besides the engineers, other scholars, less numerous, are in
+their turn to become teachers; they therefore must go to the
+very bottom; a knowledge deep and rigorous of the first principles
+is for them before all indispensable. But this is no reason
+not to cultivate in them intuition; for they would get a false idea
+of the science if they never looked at it except from a single side,
+and besides they could not develop in their students a quality
+they did not themselves possess.</p>
+
+<p>For the pure geometer himself, this faculty is necessary; it
+is by logic one demonstrates, by intuition one invents. To know
+how to criticize is good, to know how to create is better. You
+know how to recognize if a combination is correct; what a predicament
+if you have not the art of choosing among all the possible
+combinations. Logic tells us that on such and such a way
+we are sure not to meet any obstacle; it does not say which way
+leads to the end. For that it is necessary to see the end from
+afar, and the faculty which teaches us to see is intuition. Without
+it the geometer would be like a writer who should be versed
+in grammar but had no ideas. Now how could this faculty
+develop if, as soon as it showed itself, we chase it away and proscribe
+it, if we learn to set it at naught before knowing the
+good of it.</p>
+
+<p>And here permit a parenthesis to insist upon the importance of
+written exercises. Written compositions are perhaps not sufficiently
+emphasized in certain examinations, at the polytechnic
+school, for instance. I am told they would close the door<span class='pagenum'><a name="Page_439" id="Page_439">[Pg 439]</a></span>
+against very good scholars who have mastered the course, thoroughly
+understanding it, and who nevertheless are incapable of
+making the slightest application. I have just said the word
+understand has several meanings: such students only understand
+in the first way, and we have seen that suffices neither to make an
+engineer nor a geometer. Well, since choice must be made, I prefer
+those who understand completely.</p>
+
+<p>10. But is the art of sound reasoning not also a precious
+thing, which the professor of mathematics ought before all to
+cultivate? I take good care not to forget that. It should occupy
+our attention and from the very beginning. I should be
+distressed to see geometry degenerate into I know not what tachymetry
+of low grade and I by no means subscribe to the extreme
+doctrines of certain German Oberlehrer. But there are occasions
+enough to exercise the scholars in correct reasoning in the
+parts of mathematics where the inconveniences I have pointed
+out do not present themselves. There are long chains of theorems
+where absolute logic has reigned from the very first and,
+so to speak, quite naturally, where the first geometers have given
+us models we should constantly imitate and admire.</p>
+
+<p>It is in the exposition of first principles that it is necessary
+to avoid too much subtility; there it would be most discouraging
+and moreover useless. We can not prove everything and we can
+not define everything; and it will always be necessary to borrow
+from intuition; what does it matter whether it be done a little
+sooner or a little later, provided that in using correctly premises
+it has furnished us, we learn to reason soundly.</p>
+
+<p>11. Is it possible to fulfill so many opposing conditions? Is
+this possible in particular when it is a question of giving a definition?
+How find a concise statement satisfying at once the uncompromising
+rules of logic, our desire to grasp the place of the
+new notion in the totality of the science, our need of thinking
+with images? Usually it will not be found, and this is why it is
+not enough to state a definition; it must be prepared for and
+justified.</p>
+
+<p>What does that mean? You know it has often been said:
+every definition implies an assumption, since it affirms the existence
+of the object defined. The definition then will not be<span class='pagenum'><a name="Page_440" id="Page_440">[Pg 440]</a></span>
+justified, from the purely logical point of view, until one shall have
+<i>proved</i> that it involves no contradiction, neither in the terms,
+nor with the verities previously admitted.</p>
+
+<p>But this is not enough; the definition is stated to us as a convention;
+but most minds will revolt if we wish to impose it upon
+them as an <i>arbitrary</i> convention. They will be satisfied only
+when you have answered numerous questions.</p>
+
+<p>Usually mathematical definitions, as M. Liard has shown, are
+veritable constructions built up wholly of more simple notions.
+But why assemble these elements in this way when a thousand
+other combinations were possible?</p>
+
+<p>Is it by caprice? If not, why had this combination more right
+to exist than all the others? To what need does it respond?
+How was it foreseen that it would play an important r&ocirc;le in the
+development of the science, that it would abridge our reasonings
+and our calculations? Is there in nature some familiar
+object which is so to speak the rough and vague image of it?</p>
+
+<p>This is not all; if you answer all these questions in a satisfactory
+manner, we shall see indeed that the new-born had the
+right to be baptized; but neither is the choice of a name arbitrary;
+it is needful to explain by what analogies one has been
+guided and that if analogous names have been given to different
+things, these things at least differ only in material and are allied
+in form; that their properties are analogous and so to say
+parallel.</p>
+
+<p>At this cost we may satisfy all inclinations. If the statement
+is correct enough to please the logician, the justification will
+satisfy the intuitive. But there is still a better procedure;
+wherever possible, the justification should precede the statement
+and prepare for it; one should be led on to the general statement
+by the study of some particular examples.</p>
+
+<p>Still another thing: each of the parts of the statement of a
+definition has as aim to distinguish the thing to be defined from
+a class of other neighboring objects. The definition will be understood
+only when you have shown, not merely the object defined,
+but the neighboring objects from which it is proper to distinguish
+it, when you have given a grasp of the difference and
+when you have added explicitly: this is why in stating the definition
+I have said this or that.<span class='pagenum'><a name="Page_441" id="Page_441">[Pg 441]</a></span></p>
+
+<p>But it is time to leave generalities and examine how the somewhat
+abstract principles I have expounded may be applied in
+arithmetic, geometry, analysis and mechanics.</p>
+
+
+<h3><span class="smcap">Arithmetic</span></h3>
+
+<p>12. The whole number is not to be defined; in return, one ordinarily
+defines the operations upon whole numbers; I believe
+the scholars learn these definitions by heart and attach no meaning
+to them. For that there are two reasons: first they are made
+to learn them too soon, when their mind as yet feels no need of
+them; then these definitions are not satisfactory from the logical
+point of view. A good definition for addition is not to be found
+just simply because we must stop and can not define everything.
+It is not defining addition to say it consists in adding. All that
+can be done is to start from a certain number of concrete examples
+and say: the operation we have performed is called addition.</p>
+
+<p>For subtraction it is quite otherwise; it may be logically defined
+as the operation inverse to addition; but should we begin
+in that way? Here also start with examples, show on these examples
+the reciprocity of the two operations; thus the definition
+will be prepared for and justified.</p>
+
+<p>Just so again for multiplication; take a particular problem;
+show that it may be solved by adding several equal numbers;
+then show that we reach the result more quickly by a multiplication,
+an operation the scholars already know how to do by routine
+and out of that the logical definition will issue naturally.</p>
+
+<p>Division is defined as the operation inverse to multiplication;
+but begin by an example taken from the familiar notion of partition
+and show on this example that multiplication reproduces
+the dividend.</p>
+
+<p>There still remain the operations on fractions. The only
+difficulty is for multiplication. It is best to expound first the
+theory of proportion; from it alone can come a logical definition;
+but to make acceptable the definitions met at the beginning of
+this theory, it is necessary to prepare for them by numerous examples
+taken from classic problems of the rule of three, taking
+pains to introduce fractional data.</p>
+
+<p>Neither should we fear to familiarize the scholars with the<span class='pagenum'><a name="Page_442" id="Page_442">[Pg 442]</a></span>
+notion of proportion by geometric images, either by appealing to
+what they remember if they have already studied geometry, or
+in having recourse to direct intuition, if they have not studied
+it, which besides will prepare them to study it. Finally I shall
+add that after defining multiplication of fractions, it is needful
+to justify this definition by showing that it is commutative, associative
+and distributive, and calling to the attention of the
+auditors that this is established to justify the definition.</p>
+
+<p>One sees what a r&ocirc;le geometric images play in all this; and
+this r&ocirc;le is justified by the philosophy and the history of the
+science. If arithmetic had remained free from all admixture
+of geometry, it would have known only the whole number; it is
+to adapt itself to the needs of geometry that it invented anything
+else.</p>
+
+
+<h3><span class="smcap">Geometry</span></h3>
+
+<p>In geometry we meet forthwith the notion of the straight line.
+Can the straight line be defined? The well-known definition,
+the shortest path from one point to another, scarcely satisfies
+me. I should start simply with the <i>ruler</i> and show at first to
+the scholar how one may verify a ruler by turning; this verification
+is the true definition of the straight line; the straight
+line is an axis of rotation. Next he should be shown how to
+verify the ruler by sliding and he would have one of the most
+important properties of the straight line.</p>
+
+<p>As to this other property of being the shortest path from one
+point to another, it is a theorem which can be demonstrated
+apodictically, but the demonstration is too delicate to find a place
+in secondary teaching. It will be worth more to show that a
+ruler previously verified fits on a stretched thread. In presence
+of difficulties like these one need not dread to multiply assumptions,
+justifying them by rough experiments.</p>
+
+<p>It is needful to grant these assumptions, and if one admits a
+few more of them than is strictly necessary, the evil is not very
+great; the essential thing is to learn to reason soundly on the
+assumptions admitted. Uncle Sarcey, who loved to repeat, often
+said that at the theater the spectator accepts willingly all the
+postulates imposed upon him at the beginning, but the curtain<span class='pagenum'><a name="Page_443" id="Page_443">[Pg 443]</a></span>
+once raised, he becomes uncompromising on the logic. Well, it
+is just the same in mathematics.</p>
+
+<p>For the circle, we may start with the compasses; the scholars
+will recognize at the first glance the curve traced; then make
+them observe that the distance of the two points of the instrument
+remains constant, that one of these points is fixed and the
+other movable, and so we shall be led naturally to the logical
+definition.</p>
+
+<p>The definition of the plane implies an axiom and this need not
+be hidden. Take a drawing board and show that a moving ruler
+may be kept constantly in complete contact with this plane and
+yet retain three degrees of freedom. Compare with the cylinder
+and the cone, surfaces on which an applied straight retains
+only two degrees of freedom; next take three drawing boards;
+show first that they will glide while remaining applied to one another
+and this with three degrees of freedom; and finally to distinguish
+the plane from the sphere, show that two of these boards
+which fit a third will fit each other.</p>
+
+<p>Perhaps you are surprised at this incessant employment of
+moving things; this is not a rough artifice; it is much more
+philosophic than one would at first think. What is geometry
+for the philosopher? It is the study of a group. And what
+group? That of the motions of solid bodies. How define this
+group then without moving some solids?</p>
+
+<p>Should we retain the classic definition of parallels and say
+parallels are two coplanar straights which do not meet, however
+far they be prolonged? No, since this definition is negative,
+since it is unverifiable by experiment, and consequently can not
+be regarded as an immediate datum of intuition. No, above all
+because it is wholly strange to the notion of group, to the consideration
+of the motion of solid bodies which is, as I have said, the
+true source of geometry. Would it not be better to define first
+the rectilinear translation of an invariable figure, as a motion
+wherein all the points of this figure have rectilinear trajectories;
+to show that such a translation is possible by making a square
+glide on a ruler?</p>
+
+<p>From this experimental ascertainment, set up as an assumption,
+it would be easy to derive the notion of parallel and
+Euclid's postulate itself.</p>
+<p><span class='pagenum'><a name="Page_444" id="Page_444">[Pg 444]</a></span></p>
+
+<h3><span class="smcap">Mechanics</span></h3>
+
+<p>I need not return to the definition of velocity, or acceleration,
+or other kinematic notions; they may be advantageously connected
+with that of the derivative.</p>
+
+<p>I shall insist, on the other hand, upon the dynamic notions of
+force and mass.</p>
+
+<p>I am struck by one thing: how very far the young people who
+have received a high-school education are from applying to the
+real world the mechanical laws they have been taught. It is not
+only that they are incapable of it; they do not even think of it.
+For them the world of science and the world of reality are separated
+by an impervious partition wall.</p>
+
+<p>If we try to analyze the state of mind of our scholars, this will
+astonish us less. What is for them the real definition of force?
+Not that which they recite, but that which, crouching in a nook
+of their mind, from there directs it wholly. Here is the definition:
+forces are arrows with which one makes parallelograms. These
+arrows are imaginary things which have nothing to do with anything
+existing in nature. This would not happen if they had
+been shown forces in reality before representing them by arrows.</p>
+
+<p>How shall we define force?</p>
+
+<p>I think I have elsewhere sufficiently shown there is no good
+logical definition. There is the anthropomorphic definition, the
+sensation of muscular effort; this is really too rough and nothing
+useful can be drawn from it.</p>
+
+<p>Here is how we should go: first, to make known the genus
+force, we must show one after the other all the species of this
+genus; they are very numerous and very different; there is the
+pressure of fluids on the insides of the vases wherein they are
+contained; the tension of threads; the elasticity of a spring; the
+gravity working on all the molecules of a body; friction; the
+normal mutual action and reaction of two solids in contact.</p>
+
+<p>This is only a qualitative definition; it is necessary to learn
+to measure force. For that begin by showing that one force may
+be replaced by another without destroying equilibrium; we may
+find the first example of this substitution in the balance and
+Borda's double weighing.</p>
+
+<p>Then show that a weight may be replaced, not only by another<span class='pagenum'><a name="Page_445" id="Page_445">[Pg 445]</a></span>
+weight, but by force of a different nature; for instance, Prony's
+brake permits replacing weight by friction.</p>
+
+<p>From all this arises the notion of the equivalence of two forces.</p>
+
+<p>The direction of a force must be defined. If a force <i>F</i> is equivalent
+to another force <i>F&acute;</i> applied to the body considered by means
+of a stretched string, so that <i>F</i> may be replaced by <i>F&acute;</i> without
+affecting the equilibrium, then the point of attachment of the
+string will be by definition the point of application of the force
+<i>F&acute;</i>, and that of the equivalent force <i>F</i>; the direction of the string
+will be the direction of the force <i>F&acute;</i> and that of the equivalent
+force <i>F</i>.</p>
+
+<p>From that, pass to the comparison of the magnitude of forces.
+If a force can replace two others with the same direction, it
+equals their sum; show for example that a weight of 20 grams
+may replace two 10-gram weights.</p>
+
+<p>Is this enough? Not yet. We now know how to compare the
+intensity of two forces which have the same direction and same
+point of application; we must learn to do it when the directions
+are different. For that, imagine a string stretched by a weight
+and passing over a pulley; we shall say that the tensor of the
+two legs of the string is the same and equal to the tension weight.</p>
+
+<p>This definition of ours enables us to compare the tensions of
+the two pieces of our string, and, using the preceding definitions,
+to compare any two forces having the same direction as
+these two pieces. It should be justified by showing that the
+tension of the last piece of the string remains the same for the
+same tensor weight, whatever be the number and the disposition
+of the reflecting pulleys. It has still to be completed by showing
+this is only true if the pulleys are frictionless.</p>
+
+<p>Once master of these definitions, it is to be shown that the
+point of application, the direction and the intensity suffice to
+determine a force; that two forces for which these three elements
+are the same are <i>always</i> equivalent and may <i>always</i> be replaced
+by one another, whether in equilibrium or in movement, and this
+whatever be the other forces acting.</p>
+
+<p>It must be shown that two concurrent forces may always be
+replaced by a unique resultant; and that <i>this resultant remains<span class='pagenum'><a name="Page_446" id="Page_446">[Pg 446]</a></span>
+the same</i>, whether the body be at rest or in motion and whatever
+be the other forces applied to it.</p>
+
+<p>Finally it must be shown that forces thus defined satisfy the
+principle of the equality of action and reaction.</p>
+
+<p>Experiment it is, and experiment alone, which can teach us
+all that. It will suffice to cite certain common experiments,
+which the scholars make daily without suspecting it, and to perform
+before them a few experiments, simple and well chosen.</p>
+
+<p>It is after having passed through all these meanders that one
+may represent forces by arrows, and I should even wish that in
+the development of the reasonings return were made from time
+to time from the symbol to the reality. For instance it would
+not be difficult to illustrate the parallelogram of forces by aid
+of an apparatus formed of three strings, passing over pulleys,
+stretched by weights and in equilibrium while pulling on the
+same point.</p>
+
+<p>Knowing force, it is easy to define mass; this time the definition
+should be borrowed from dynamics; there is no way of doing
+otherwise, since the end to be attained is to give understanding
+of the distinction between mass and weight. Here again, the
+definition should be led up to by experiments; there is in fact a
+machine which seems made expressly to show what mass is,
+Atwood's machine; recall also the laws of the fall of bodies, that
+the acceleration of gravity is the same for heavy as for light
+bodies, and that it varies with the latitude, etc.</p>
+
+<p>Now, if you tell me that all the methods I extol have long been
+applied in the schools, I shall rejoice over it more than be surprised
+at it. I know that on the whole our mathematical teaching
+is good. I do not wish it overturned; that would even distress
+me. I only desire betterments slowly progressive. This
+teaching should not be subjected to brusque oscillations under
+the capricious blast of ephemeral fads. In such tempests its
+high educative value would soon founder. A good and sound
+logic should continue to be its basis. The definition by example
+is always necessary, but it should prepare the way for the logical
+definition, it should not replace it; it should at least make this
+wished for, in the cases where the true logical definition can be
+advantageously given only in advanced teaching.<span class='pagenum'><a name="Page_447" id="Page_447">[Pg 447]</a></span></p>
+
+<p>Understand that what I have here said does not imply giving
+up what I have written elsewhere. I have often had occasion to
+criticize certain definitions I extol to-day. These criticisms hold
+good completely. These definitions can only be provisory. But
+it is by way of them that we must pass.</p>
+
+
+
+<hr style="width: 65%;" />
+<p><span class='pagenum'><a name="Page_448" id="Page_448">[Pg 448]</a></span></p>
+<h3>CHAPTER III</h3>
+
+<h3><span class="smcap">Mathematics and Logic</span></h3>
+
+
+<h4><span class="smcap">Introduction</span></h4>
+
+<p>Can mathematics be reduced to logic without having to appeal
+to principles peculiar to mathematics? There is a whole school,
+abounding in ardor and full of faith, striving to prove it. They
+have their own special language, which is without words, using
+only signs. This language is understood only by the initiates,
+so that commoners are disposed to bow to the trenchant affirmations
+of the adepts. It is perhaps not unprofitable to examine
+these affirmations somewhat closely, to see if they justify the
+peremptory tone with which they are presented.</p>
+
+<p>But to make clear the nature of the question it is necessary to
+enter upon certain historical details and in particular to recall
+the character of the works of Cantor.</p>
+
+<p>Since long ago the notion of infinity had been introduced
+into mathematics; but this infinite was what philosophers call
+a <i>becoming</i>. The mathematical infinite was only a quantity
+capable of increasing beyond all limit: it was a variable quantity
+of which it could not be said that it <i>had passed</i> all limits, but
+only that it <i>could pass</i> them.</p>
+
+<p>Cantor has undertaken to introduce into mathematics an
+<i>actual infinite</i>, that is to say a quantity which not only is capable
+of passing all limits, but which is regarded as having already
+passed them. He has set himself questions like these: Are there
+more points in space than whole numbers? Are there more
+points in space than points in a plane? etc.</p>
+
+<p>And then the number of whole numbers, that of the points of
+space, etc., constitutes what he calls a <i>transfinite cardinal number</i>,
+that is to say a cardinal number greater than all the ordinary
+cardinal numbers. And he has occupied himself in comparing
+these transfinite cardinal numbers. In arranging in a proper
+order the elements of an aggregate containing an infinity of<span class='pagenum'><a name="Page_449" id="Page_449">[Pg 449]</a></span>
+them, he has also imagined what he calls transfinite ordinal numbers
+upon which I shall not dwell.</p>
+
+<p>Many mathematicians followed his lead and set a series of
+questions of the sort. They so familiarized themselves with
+transfinite numbers that they have come to make the theory of
+finite numbers depend upon that of Cantor's cardinal numbers.
+In their eyes, to teach arithmetic in a way truly logical, one
+should begin by establishing the general properties of transfinite
+cardinal numbers, then distinguish among them a very
+small class, that of the ordinary whole numbers. Thanks to this
+d&eacute;tour, one might succeed in proving all the propositions relative
+to this little class (that is to say all our arithmetic and our
+algebra) without using any principle foreign to logic. This
+method is evidently contrary to all sane psychology; it is certainly
+not in this way that the human mind proceeded in constructing
+mathematics; so its authors do not dream, I think, of
+introducing it into secondary teaching. But is it at least logic,
+or, better, is it correct? It may be doubted.</p>
+
+<p>The geometers who have employed it are however very numerous.
+They have accumulated formulas and they have thought
+to free themselves from what was not pure logic by writing
+memoirs where the formulas no longer alternate with explanatory
+discourse as in the books of ordinary mathematics, but where
+this discourse has completely disappeared.</p>
+
+<p>Unfortunately they have reached contradictory results, what
+are called the <i>cantorian antinomies</i>, to which we shall have
+occasion to return. These contradictions have not discouraged
+them and they have tried to modify their rules so as to make
+those disappear which had already shown themselves, without
+being sure, for all that, that new ones would not manifest
+themselves.</p>
+
+<p>It is time to administer justice on these exaggerations. I do
+not hope to convince them; for they have lived too long in this
+atmosphere. Besides, when one of their demonstrations has
+been refuted, we are sure to see it resurrected with insignificant
+alterations, and some of them have already risen several times
+from their ashes. Such long ago was the Lern&aelig;an hydra with its
+famous heads which always grew again. Hercules got through,<span class='pagenum'><a name="Page_450" id="Page_450">[Pg 450]</a></span>
+since his hydra had only nine heads, or eleven; but here there are
+too many, some in England, some in Germany, in Italy, in
+France, and he would have to give up the struggle. So I appeal
+only to men of good judgment unprejudiced.</p>
+
+
+<h4>I</h4>
+
+<p>In these latter years numerous works have been published on
+pure mathematics and the philosophy of mathematics, trying to
+separate and isolate the logical elements of mathematical reasoning.
+These works have been analyzed and expounded very
+clearly by M. Couturat in a book entitled: <i>The Principles of
+Mathematics</i>.</p>
+
+<p>For M. Couturat, the new works, and in particular those of
+Russell and Peano, have finally settled the controversy, so long
+pending between Leibnitz and Kant. They have shown that
+there are no synthetic judgments a priori (Kant's phrase to
+designate judgments which can neither be demonstrated analytically,
+nor reduced to identities, nor established experimentally),
+they have shown that mathematics is entirely reducible to logic
+and that intuition here plays no r&ocirc;le.</p>
+
+<p>This is what M. Couturat has set forth in the work just cited;
+this he says still more explicitly in his Kant jubilee discourse,
+so that I heard my neighbor whisper: "I well see this is the
+centenary of Kant's <i>death</i>."</p>
+
+<p>Can we subscribe to this conclusive condemnation? I think
+not, and I shall try to show why.</p>
+
+
+<h4>II</h4>
+
+<p>What strikes us first in the new mathematics is its purely
+formal character: "We think," says Hilbert, "three sorts of
+<i>things</i>, which we shall call points, straights and planes. We
+convene that a straight shall be determined by two points, and
+that in place of saying this straight is determined by these two
+points, we may say it passes through these two points, or that
+these two points are situated on this straight." What these
+<i>things</i> are, not only we do not know, but we should not seek to
+know. We have no need to, and one who never had seen either
+point or straight or plane could geometrize as well as we. That<span class='pagenum'><a name="Page_451" id="Page_451">[Pg 451]</a></span>
+the phrase <i>to pass through</i>, or the phrase <i>to be situated upon</i>
+may arouse in us no image, the first is simply a synonym of to
+<i>be determined</i> and the second of <i>to determine</i>.</p>
+
+<p>Thus, be it understood, to demonstrate a theorem, it is neither
+necessary nor even advantageous to know what it means. The
+geometer might be replaced by the <i>logic piano</i> imagined by
+Stanley Jevons; or, if you choose, a machine might be imagined
+where the assumptions were put in at one end, while the theorems
+came out at the other, like the legendary Chicago machine where
+the pigs go in alive and come out transformed into hams and
+sausages. No more than these machines need the mathematician
+know what he does.</p>
+
+<p>I do not make this formal character of his geometry a reproach
+to Hilbert. This is the way he should go, given the problem he
+set himself. He wished to reduce to a minimum the number of
+the fundamental assumptions of geometry and completely enumerate
+them; now, in reasonings where our mind remains active,
+in those where intuition still plays a part, in living reasonings,
+so to speak, it is difficult not to introduce an assumption or a
+postulate which passes unperceived. It is therefore only after
+having carried back all the geometric reasonings to a form purely
+mechanical that he could be sure of having accomplished his
+design and finished his work.</p>
+
+<p>What Hilbert did for geometry, others have tried to do for
+arithmetic and analysis. Even if they had entirely succeeded,
+would the Kantians be finally condemned to silence? Perhaps
+not, for in reducing mathematical thought to an empty form,
+it is certainly mutilated.</p>
+
+<p>Even admitting it were established that all the theorems could
+be deduced by procedures purely analytic, by simple logical
+combinations of a finite number of assumptions, and that these
+assumptions are only conventions; the philosopher would still
+have the right to investigate the origins of these conventions,
+to see why they have been judged preferable to the contrary
+conventions.</p>
+
+<p>And then the logical correctness of the reasonings leading
+from the assumptions to the theorems is not the only thing
+which should occupy us. The rules of perfect logic, are they<span class='pagenum'><a name="Page_452" id="Page_452">[Pg 452]</a></span>
+the whole of mathematics? As well say the whole art of playing
+chess reduces to the rules of the moves of the pieces. Among
+all the constructs which can be built up of the materials furnished
+by logic, choice must be made; the true geometer makes
+this choice judiciously because he is guided by a sure instinct,
+or by some vague consciousness of I know not what more profound
+and more hidden geometry, which alone gives value to the
+edifice constructed.</p>
+
+<p>To seek the origin of this instinct, to study the laws of this
+deep geometry, felt, not stated, would also be a fine employment
+for the philosophers who do not want logic to be all. But it is
+not at this point of view I wish to put myself, it is not thus I
+wish to consider the question. The instinct mentioned is necessary
+for the inventor, but it would seem at first we might do
+without it in studying the science once created. Well, what I
+wish to investigate is if it be true that, the principles of logic
+once admitted, one can, I do not say discover, but demonstrate,
+all the mathematical verities without making a new appeal to
+intuition.</p>
+
+
+<h4>III</h4>
+
+<p>I once said no to this question:<a name="FNanchor_12_12" id="FNanchor_12_12"></a><a href="#Footnote_12_12" class="fnanchor">[12]</a> should our reply be modified
+by the recent works? My saying no was because "the principle
+of complete induction" seemed to me at once necessary to the
+mathematician and irreducible to logic. The statement of this
+principle is: "If a property be true of the number 1, and if we
+establish that it is true of <i>n</i> + 1 provided it be of <i>n</i>, it will be
+true of all the whole numbers." Therein I see the mathematical
+reasoning par excellence. I did not mean to say, as has been
+supposed, that all mathematical reasonings can be reduced to
+an application of this principle. Examining these reasonings
+closely, we there should see applied many other analogous principles,
+presenting the same essential characteristics. In this category
+of principles, that of complete induction is only the simplest
+of all and this is why I have chosen it as type.</p>
+
+<p>The current name, principle of complete induction, is not
+justified. This mode of reasoning is none the less a true<span class='pagenum'><a name="Page_453" id="Page_453">[Pg 453]</a></span>
+mathematical induction which differs from ordinary induction only by
+its certitude.</p>
+
+
+<h4>IV</h4>
+
+<p><span class="smcap">Definitions and Assumptions</span></p>
+
+<p>The existence of such principles is a difficulty for the uncompromising
+logicians; how do they pretend to get out of it? The
+principle of complete induction, they say, is not an assumption
+properly so called or a synthetic judgment <i>a priori</i>; it is just
+simply the definition of whole number. It is therefore a simple
+convention. To discuss this way of looking at it, we must examine
+a little closely the relations between definitions and
+assumptions.</p>
+
+<p>Let us go back first to an article by M. Couturat on mathematical
+definitions which appeared in <i>l'Enseignement math&eacute;matique</i>,
+a magazine published by Gauthier-Villars and by Georg
+at Geneva. We shall see there a distinction between the <i>direct
+definition and the definition by postulates</i>.</p>
+
+<p>"The definition by postulates," says M. Couturat, "applies
+not to a single notion, but to a system of notions; it consists in
+enumerating the fundamental relations which unite them and
+which enable us to demonstrate all their other properties; these
+relations are postulates."</p>
+
+<p>If previously have been defined all these notions but one, then
+this last will be by definition the thing which verifies these postulates.
+Thus certain indemonstrable assumptions of mathematics
+would be only disguised definitions. This point of view
+is often legitimate; and I have myself admitted it in regard for
+instance to Euclid's postulate.</p>
+
+<p>The other assumptions of geometry do not suffice to completely
+define distance; the distance then will be, by definition, among all
+the magnitudes which satisfy these other assumptions, that which
+is such as to make Euclid's postulate true.</p>
+
+<p>Well the logicians suppose true for the principle of complete
+induction what I admit for Euclid's postulate; they want to see
+in it only a disguised definition.</p>
+
+<p>But to give them this right, two conditions must be fulfilled.
+Stuart Mill says every definition implies an assumption, that by
+which the existence of the defined object is affirmed. According<span class='pagenum'><a name="Page_454" id="Page_454">[Pg 454]</a></span>
+to that, it would no longer be the assumption which might be a
+disguised definition, it would on the contrary be the definition
+which would be a disguised assumption. Stuart Mill meant the
+word existence in a material and empirical sense; he meant to
+say that in defining the circle we affirm there are round things in
+nature.</p>
+
+<p>Under this form, his opinion is inadmissible. Mathematics is
+independent of the existence of material objects; in mathematics
+the word exist can have only one meaning, it means free from
+contradiction. Thus rectified, Stuart Mill's thought becomes
+exact; in defining a thing, we affirm that the definition implies no
+contradiction.</p>
+
+<p>If therefore we have a system of postulates, and if we can
+demonstrate that these postulates imply no contradiction, we
+shall have the right to consider them as representing the definition
+of one of the notions entering therein. If we can not demonstrate
+that, it must be admitted without proof, and that then
+will be an assumption; so that, seeking the definition under the
+postulate, we should find the assumption under the definition.</p>
+
+<p>Usually, to show that a definition implies no contradiction, we
+proceed by <i>example</i>, we try to make an example of a thing satisfying
+the definition. Take the case of a definition by postulates;
+we wish to define a notion <i>A</i>, and we say that, by definition, an
+<i>A</i> is anything for which certain postulates are true. If we can
+prove directly that all these postulates are true of a certain object
+<i>B</i>, the definition will be justified; the object <i>B</i> will be an <i>example</i>
+of an <i>A</i>. We shall be certain that the postulates are not contradictory,
+since there are cases where they are all true at the same
+time.</p>
+
+<p>But such a direct demonstration by example is not always
+possible.</p>
+
+<p>To establish that the postulates imply no contradiction, it is
+then necessary to consider all the propositions deducible from
+these postulates considered as premises, and to show that, among
+these propositions, no two are contradictory. If these propositions
+are finite in number, a direct verification is possible. This
+case is infrequent and uninteresting. If these propositions are
+infinite in number, this direct verification can no longer be made;<span class='pagenum'><a name="Page_455" id="Page_455">[Pg 455]</a></span>
+recourse must be had to procedures where in general it is necessary
+to invoke just this principle of complete induction which is
+precisely the thing to be proved.</p>
+
+<p>This is an explanation of one of the conditions the logicians
+should satisfy, <i>and further on we shall see they have not done it</i>.</p>
+
+
+<h4>V</h4>
+
+<p>There is a second. When we give a definition, it is to use it.</p>
+
+<p>We therefore shall find in the sequel of the exposition the
+word defined; have we the right to affirm, of the thing represented
+by this word, the postulate which has served for definition?
+Yes, evidently, if the word has retained its meaning, if we do
+not attribute to it implicitly a different meaning. Now this is
+what sometimes happens and it is usually difficult to perceive it;
+it is needful to see how this word comes into our discourse, and
+if the gate by which it has entered does not imply in reality a
+definition other than that stated.</p>
+
+<p>This difficulty presents itself in all the applications of mathematics.
+The mathematical notion has been given a definition
+very refined and very rigorous; and for the pure mathematician
+all doubt has disappeared; but if one wishes to apply it to the
+physical sciences for instance, it is no longer a question of this
+pure notion, but of a concrete object which is often only a rough
+image of it. To say that this object satisfies, at least approximately,
+the definition, is to state a new truth, which experience
+alone can put beyond doubt, and which no longer has the character
+of a conventional postulate.</p>
+
+<p>But without going beyond pure mathematics, we also meet the
+same difficulty.</p>
+
+<p>You give a subtile definition of numbers; then, once this definition
+given, you think no more of it; because, in reality, it is not
+it which has taught you what number is; you long ago knew
+that, and when the word number further on is found under your
+pen, you give it the same sense as the first comer. To know what
+is this meaning and whether it is the same in this phrase or that,
+it is needful to see how you have been led to speak of number and
+to introduce this word into these two phrases. I shall not for
+the moment dilate upon this point, because we shall have occasion
+to return to it.<span class='pagenum'><a name="Page_456" id="Page_456">[Pg 456]</a></span></p>
+
+<p>Thus consider a word of which we have given explicitly a definition
+<i>A</i>; afterwards in the discourse we make a use of it which
+implicitly supposes another definition <i>B</i>. It is possible that
+these two definitions designate the same thing. But that this is
+so is a new truth which must either be demonstrated or admitted
+as an independent assumption.</p>
+
+<p><i>We shall see farther on that the logicians have not fulfilled the
+second condition any better than the first.</i></p>
+
+
+<h4>VI</h4>
+
+<p>The definitions of number are very numerous and very different;
+I forego the enumeration even of the names of their authors.
+We should not be astonished that there are so many. If one
+among them was satisfactory, no new one would be given. If
+each new philosopher occupying himself with this question has
+thought he must invent another one, this was because he was not
+satisfied with those of his predecessors, and he was not satisfied
+with them because he thought he saw a petitio principii.</p>
+
+<p>I have always felt, in reading the writings devoted to this problem,
+a profound feeling of discomfort; I was always expecting to
+run against a petitio principii, and when I did not immediately
+perceive it, I feared I had overlooked it.</p>
+
+<p>This is because it is impossible to give a definition without
+using a sentence, and difficult to make a sentence without using
+a number word, or at least the word several, or at least a word
+in the plural. And then the declivity is slippery and at each
+instant there is risk of a fall into petitio principii.</p>
+
+<p>I shall devote my attention in what follows only to those of
+these definitions where the petitio principii is most ably concealed.</p>
+
+
+<h4>VII</h4>
+
+<h4><span class="smcap">Pasigraphy</span></h4>
+
+<p>The symbolic language created by Peano plays a very grand
+r&ocirc;le in these new researches. It is capable of rendering some
+service, but I think M. Couturat attaches to it an exaggerated
+importance which must astonish Peano himself.</p>
+
+<p>The essential element of this language is certain algebraic<span class='pagenum'><a name="Page_457" id="Page_457">[Pg 457]</a></span>
+signs which represent the different conjunctions: if, and, or,
+therefore. That these signs may be convenient is possible; but
+that they are destined to revolutionize all philosophy is a different
+matter. It is difficult to admit that the word <i>if</i> acquires,
+when written C, a virtue it had not when written if. This invention
+of Peano was first called <i>pasigraphy</i>, that is to say the
+art of writing a treatise on mathematics without using a single
+word of ordinary language. This name defined its range very
+exactly. Later, it was raised to a more eminent dignity by conferring
+on it the title of <i>logistic</i>. This word is, it appears, employed
+at the Military Academy, to designate the art of the
+quartermaster of cavalry, the art of marching and cantoning
+troops; but here no confusion need be feared, and it is at once
+seen that this new name implies the design of revolutionizing
+logic.</p>
+
+<p>We may see the new method at work in a mathematical memoir
+by Burali-Forti, entitled: <i>Una Questione sui numeri transfiniti</i>,
+inserted in Volume XI of the <i>Rendiconti del circolo matematico
+di Palermo</i>.</p>
+
+<p>I begin by saying this memoir is very interesting, and my taking
+it here as example is precisely because it is the most important
+of all those written in the new language. Besides, the uninitiated
+may read it, thanks to an Italian interlinear translation.</p>
+
+<p>What constitutes the importance of this memoir is that it has
+given the first example of those antinomies met in the study of
+transfinite numbers and making since some years the despair of
+mathematicians. The aim, says Burali-Forti, of this note is to
+show there may be two transfinite numbers (ordinals), <i>a</i> and <i>b</i>,
+such that <i>a</i> is neither equal to, greater than, nor less than <i>b</i>.</p>
+
+<p>To reassure the reader, to comprehend the considerations which
+follow, he has no need of knowing what a transfinite ordinal
+number is.</p>
+
+<p>Now, Cantor had precisely proved that between two transfinite
+numbers as between two finite, there can be no other relation
+than equality or inequality in one sense or the other. But it is
+not of the substance of this memoir that I wish to speak here;
+that would carry me much too far from my subject; I only wish
+to consider the form, and just to ask if this form makes it gain<span class='pagenum'><a name="Page_458" id="Page_458">[Pg 458]</a></span>
+much in rigor and whether it thus compensates for the efforts it
+imposes upon the writer and the reader.</p>
+
+<p>First we see Burali-Forti define the number 1 as follows:</p>
+
+<div class="figcenter" style="width: 300px;">
+<img src="images/img470def1.png" width="300" height="40" alt="" title="" />
+</div>
+
+
+<p class="noidt">a definition eminently fitted to give an idea of the number 1 to
+persons who had never heard speak of it.</p>
+
+<p>I understand Peanian too ill to dare risk a critique, but still I
+fear this definition contains a petitio principii, considering that
+I see the figure 1 in the first member and Un in letters in the
+second.</p>
+
+<p>However that may be, Burali-Forti starts from this definition
+and, after a short calculation, reaches the equation:</p>
+
+<div class="figcenter" style="width: 600px;">
+<img src="images/img470eq27.png" width="600" height="40" alt="" title="" />
+</div>
+
+<p class="noidt">which tells us that One is a number.</p>
+
+<p>And since we are on these definitions of the first numbers, we
+recall that M. Couturat has also defined 0 and 1.</p>
+
+<p>What is zero? It is the number of elements of the null class.
+And what is the null class? It is that containing no element.</p>
+
+<p>To define zero by null, and null by no, is really to abuse the
+wealth of language; so M. Couturat has introduced an improvement
+in his definition, by writing:</p>
+
+<div class="figcenter" style="width: 300px;">
+<img src="images/img470def0.png" width="300" height="40" alt="" title="" />
+</div>
+
+
+<p class="noidt">which means: zero is the number of things satisfying a condition
+never satisfied.</p>
+
+<p>But as never means <i>in no case</i> I do not see that the progress is
+great.</p>
+
+<p>I hasten to add that the definition M. Couturat gives of the
+number 1 is more satisfactory.</p>
+
+<p>One, says he in substance, is the number of elements in a class
+in which any two elements are identical.</p>
+
+<p>It is more satisfactory, I have said, in this sense that to define
+1, he does not use the word one; in compensation, he uses the
+word two. But I fear, if asked what is two, M. Couturat would
+have to use the word one.</p>
+<p><span class='pagenum'><a name="Page_459" id="Page_459">[Pg 459]</a></span></p>
+
+<h4>VIII</h4>
+
+<p>But to return to the memoir of Burali-Forti; I have said his
+conclusions are in direct opposition to those of Cantor. Now, one
+day M. Hadamard came to see me and the talk fell upon this
+antinomy.</p>
+
+<p>"Burali-Forti's reasoning," I said, "does it not seem to you
+irreproachable?" "No, and on the contrary I find nothing to
+object to in that of Cantor. Besides, Burali-Forti had no right
+to speak of the aggregate of <i>all</i> the ordinal numbers."</p>
+
+<p>"Pardon, he had the right, since he could always put</p>
+
+<div class="figcenter" style="width: 300px;">
+<img src="images/img471.png" width="300" height="40" alt="" title="" />
+</div>
+
+<p class="noidt">I should like to know who was to prevent him, and can it be
+said a thing does not exist, when we have called it &#937;?"</p>
+
+<p>It was in vain, I could not convince him (which besides would
+have been sad, since he was right). Was it merely because I do
+not speak the Peanian with enough eloquence? Perhaps; but
+between ourselves I do not think so.</p>
+
+<p>Thus, despite all this pasigraphic apparatus, the question was
+not solved. What does that prove? In so far as it is a question
+only of proving one a number, pasigraphy suffices, but if a difficulty
+presents itself, if there is an antinomy to solve, pasigraphy
+becomes impotent.</p>
+
+
+
+<hr style="width: 65%;" />
+<p><span class='pagenum'><a name="Page_460" id="Page_460">[Pg 460]</a></span></p>
+<h3>CHAPTER IV</h3>
+
+<h3><span class="smcap">The New Logics</span></h3>
+
+
+<h4>I</h4>
+
+<h4><i>The Russell Logic</i></h4>
+
+<p>To justify its pretensions, logic had to change. We have seen
+new logics arise of which the most interesting is that of Russell.
+It seems he has nothing new to write about formal logic, as if
+Aristotle there had touched bottom. But the domain Russell
+attributes to logic is infinitely more extended than that of the
+classic logic, and he has put forth on the subject views which are
+original and at times well warranted.</p>
+
+<p>First, Russell subordinates the logic of classes to that of propositions,
+while the logic of Aristotle was above all the logic of
+classes and took as its point of departure the relation of subject
+to predicate. The classic syllogism, "Socrates is a man," etc.,
+gives place to the hypothetical syllogism: "If <i>A</i> is true, <i>B</i> is
+true; now if <i>B</i> is true, <i>C</i> is true," etc. And this is, I think, a
+most happy idea, because the classic syllogism is easy to carry
+back to the hypothetical syllogism, while the inverse transformation
+is not without difficulty.</p>
+
+<p>And then this is not all. Russell's logic of propositions is the
+study of the laws of combination of the conjunctions <i>if</i>, <i>and</i>, <i>or</i>,
+and the negation <i>not</i>.</p>
+
+<p>In adding here two other conjunctions, <i>and</i> and <i>or</i>, Russell
+opens to logic a new field. The symbols <i>and</i>, <i>or</i> follow the same
+laws as the two signs &times; and +, that is to say the commutative
+associative and distributive laws. Thus <i>and</i> represents logical
+multiplication, while <i>or</i> represents logical addition. This also is
+very interesting.</p>
+
+<p>Russell reaches the conclusion that any false proposition implies
+all other propositions true or false. M. Couturat says this
+conclusion will at first seem paradoxical. It is sufficient however
+to have corrected a bad thesis in mathematics to recognize<span class='pagenum'><a name="Page_461" id="Page_461">[Pg 461]</a></span>
+how right Russell is. The candidate often is at great pains to
+get the first false equation; but that once obtained, it is only
+sport then for him to accumulate the most surprising results,
+some of which even may be true.</p>
+
+
+<h4>II</h4>
+
+<p>We see how much richer the new logic is than the classic logic;
+the symbols are multiplied and allow of varied combinations
+<i>which are no longer limited in number</i>. Has one the right to
+give this extension to the meaning of the word <i>logic</i>? It would
+be useless to examine this question and to seek with Russell a
+mere quarrel about words. Grant him what he demands; but be
+not astonished if certain verities declared irreducible to logic in
+the old sense of the word find themselves now reducible to logic
+in the new sense&mdash;something very different.</p>
+
+<p>A great number of new notions have been introduced, and
+these are not simply combinations of the old. Russell knows
+this, and not only at the beginning of the first chapter, 'The
+Logic of Propositions,' but at the beginning of the second and
+third, 'The Logic of Classes' and 'The Logic of Relations,' he
+introduces new words that he declares indefinable.</p>
+
+<p>And this is not all; he likewise introduces principles he declares
+indemonstrable. But these indemonstrable principles are
+appeals to intuition, synthetic judgments <i>a priori</i>. We regard
+them as intuitive when we meet them more or less explicitly
+enunciated in mathematical treatises; have they changed character
+because the meaning of the word logic has been enlarged
+and we now find them in a book entitled <i>Treatise on Logic</i>?
+<i>They have not changed nature; they have only changed place.</i></p>
+
+
+<h4>III</h4>
+
+<p>Could these principles be considered as disguised definitions?
+It would then be necessary to have some way of proving that
+they imply no contradiction. It would be necessary to establish
+that, however far one followed the series of deductions, he would
+never be exposed to contradicting himself.</p>
+
+<p>We might attempt to reason as follows: We can verify that<span class='pagenum'><a name="Page_462" id="Page_462">[Pg 462]</a></span>
+the operations of the new logic applied to premises exempt from
+contradiction can only give consequences equally exempt from
+contradiction. If therefore after <i>n</i> operations we have not met
+contradiction, we shall not encounter it after <i>n</i> + 1. Thus it is
+impossible that there should be a moment when contradiction
+<i>begins</i>, which shows we shall never meet it. Have we the right to
+reason in this way? No, for this would be to make use of complete
+induction; and <i>remember, we do not yet know the principle
+of complete induction</i>.</p>
+
+<p>We therefore have not the right to regard these assumptions
+as disguised definitions and only one resource remains for us, to
+admit a new act of intuition for each of them. Moreover I believe
+this is indeed the thought of Russell and M. Couturat.</p>
+
+<p>Thus each of the nine indefinable notions and of the twenty
+indemonstrable propositions (I believe if it were I that did the
+counting, I should have found some more) which are the foundation
+of the new logic, logic in the broad sense, presupposes a new
+and independent act of our intuition and (why not say it?) a
+veritable synthetic judgment <i>a priori</i>. On this point all seem
+agreed, but what Russell claims, and <i>what seems to me doubtful,
+is that after these appeals to intuition, that will be the end of it;
+we need make no others and can build all mathematics without
+the intervention of any new element</i>.</p>
+
+
+<h4>IV</h4>
+
+<p>M. Couturat often repeats that this new logic is altogether independent
+of the idea of number. I shall not amuse myself by
+counting how many numeral adjectives his exposition contains,
+both cardinal and ordinal, or indefinite adjectives such as several.
+We may cite, however, some examples:</p>
+
+<p>"The logical product of <i>two</i> or <i>more</i> propositions is....";</p>
+
+<p>"All propositions are capable only of <i>two</i> values, true and
+false";</p>
+
+<p>"The relative product of <i>two</i> relations is a relation";</p>
+
+<p>"A relation exists between two terms," etc., etc.</p>
+
+<p>Sometimes this inconvenience would not be unavoidable, but
+sometimes also it is essential. A relation is incomprehensible<span class='pagenum'><a name="Page_463" id="Page_463">[Pg 463]</a></span>
+without two terms; it is impossible to have the intuition of the
+relation, without having at the same time that of its two terms,
+and without noticing they are two, because, if the relation is to
+be conceivable, it is necessary that there be two and only two.</p>
+
+
+<h4>V</h4>
+
+<h4><i>Arithmetic</i></h4>
+
+<p>I reach what M. Couturat calls the <i>ordinal theory</i> which is
+the foundation of arithmetic properly so called. M. Couturat
+begins by stating Peano's five assumptions, which are independent,
+as has been proved by Peano and Padoa.</p>
+
+<p>1. Zero is an integer.</p>
+
+<p>2. Zero is not the successor of any integer.</p>
+
+<p>3. The successor of an integer is an integer.</p>
+
+
+<p class="noidt">To this it would be proper to add,</p>
+
+<p>Every integer has a successor.</p>
+
+<p>4. Two integers are equal if their successors are.</p>
+
+<p>The fifth assumption is the principle of complete induction.</p>
+
+<p>M. Couturat considers these assumptions as disguised definitions;
+they constitute the definition by postulates of zero, of
+successor, and of integer.</p>
+
+<p>But we have seen that for a definition by postulates to be
+acceptable we must be able to prove that it implies no contradiction.</p>
+
+<p>Is this the case here? Not at all.</p>
+
+<p>The demonstration can not be made <i>by example</i>. We can not
+take a part of the integers, for instance the first three, and
+prove they satisfy the definition.</p>
+
+<p>If I take the series 0, 1, 2, I see it fulfils the assumptions 1,
+2, 4 and 5; but to satisfy assumption 3 it still is necessary that
+3 be an integer, and consequently that the series 0, 1, 2, 3, fulfil
+the assumptions; we might prove that it satisfies assumptions
+1, 2, 4, 5, but assumption 3 requires besides that 4 be an integer
+and that the series 0, 1, 2, 3, 4 fulfil the assumptions, and so on.</p>
+
+<p>It is therefore impossible to demonstrate the assumptions for
+certain integers without proving them for all; we must give up
+proof by example.<span class='pagenum'><a name="Page_464" id="Page_464">[Pg 464]</a></span></p>
+
+<p>It is necessary then to take all the consequences of our assumptions
+and see if they contain no contradiction.</p>
+
+<p>If these consequences were finite in number, this would be
+easy; but they are infinite in number; they are the whole of
+mathematics, or at least all arithmetic.</p>
+
+<p>What then is to be done? Perhaps strictly we could repeat
+the reasoning of number III.</p>
+
+<p>But as we have said, this reasoning is complete induction, and
+it is precisely the principle of complete induction whose justification
+would be the point in question.</p>
+
+
+<h4>VI</h4>
+
+<h4><i>The Logic of Hilbert</i></h4>
+
+<p>I come now to the capital work of Hilbert which he communicated
+to the Congress of Mathematicians at Heidelberg, and
+of which a French translation by M. Pierre Boutroux appeared
+in <i>l'Enseignement math&eacute;matique</i>, while an English translation
+due to Halsted appeared in <i>The Monist</i>.<a name="FNanchor_13_13" id="FNanchor_13_13"></a><a href="#Footnote_13_13" class="fnanchor">[13]</a> In this work, which
+contains profound thoughts, the author's aim is analogous to
+that of Russell, but on many points he diverges from his
+predecessor.</p>
+
+<p>"But," he says (<i>Monist</i>, p. 340), "on attentive consideration
+we become aware that in the usual exposition of the laws of logic
+certain fundamental concepts of arithmetic are already employed;
+for example, the concept of the aggregate, in part also the concept
+of number.</p>
+
+<p>"We fall thus into a vicious circle and therefore to avoid paradoxes
+a partly simultaneous development of the laws of logic and
+arithmetic is requisite."</p>
+
+<p>We have seen above that what Hilbert says of the principles
+of logic <i>in the usual exposition</i> applies likewise to the logic of
+Russell. So for Russell logic is prior to arithmetic; for Hilbert
+they are 'simultaneous.' We shall find further on other differences
+still greater, but we shall point them out as we come
+to them. I prefer to follow step by step the development
+of Hilbert's thought, quoting textually the most important
+passages.<span class='pagenum'><a name="Page_465" id="Page_465">[Pg 465]</a></span></p>
+
+<p>"Let us take as the basis of our consideration first of all a
+thought-thing 1 (one)" (p. 341). Notice that in so doing we in
+no wise imply the notion of number, because it is understood that
+1 is here only a symbol and that we do not at all seek to know
+its meaning. "The taking of this thing together with itself
+respectively two, three or more times...." Ah! this time it is
+no longer the same; if we introduce the words 'two,' 'three,' and
+above all 'more,' 'several,' we introduce the notion of number;
+and then the definition of finite whole number which we shall
+presently find, will come too late. Our author was too circumspect
+not to perceive this begging of the question. So at the end
+of his work he tries to proceed to a truly patching-up process.</p>
+
+<p>Hilbert then introduces two simple objects 1 and =, and considers
+all the combinations of these two objects, all the combinations
+of their combinations, etc. It goes without saying that we
+must forget the ordinary meaning of these two signs and not
+attribute any to them.</p>
+
+<p>Afterwards he separates these combinations into two classes,
+the class of the existent and the class of the non-existent, and
+till further orders this separation is entirely arbitrary. Every
+affirmative statement tells us that a certain combination belongs
+to the class of the existent; every negative statement tells us that
+a certain combination belongs to the class of the non-existent.</p>
+
+
+<h4>VII</h4>
+
+<p>Note now a difference of the highest importance. For Russell
+any object whatsoever, which he designates by <i>x</i>, is an object
+absolutely undetermined and about which he supposes nothing;
+for Hilbert it is one of the combinations formed with the symbols
+1 and =; he could not conceive of the introduction of anything
+other than combinations of objects already defined. Moreover
+Hilbert formulates his thought in the neatest way, and I think
+I must reproduce <i>in extenso</i> his statement (p. 348):</p>
+
+<p>"In the assumptions the arbitraries (as equivalent for the
+concept 'every' and 'all' in the customary logic) represent only
+those thought-things and their combinations with one another,
+which at this stage are laid down as fundamental or are to be<span class='pagenum'><a name="Page_466" id="Page_466">[Pg 466]</a></span>
+newly defined. Therefore in the deduction of inferences from
+the assumptions, the arbitraries, which occur in the assumptions,
+can be replaced only by such thought-things and their
+combinations.</p>
+
+<p>"Also we must duly remember, that through the super-addition
+and making fundamental of a new thought-thing the preceding
+assumptions undergo an enlargement of their validity,
+and where necessary, are to be subjected to a change in conformity
+with the sense."</p>
+
+<p>The contrast with Russell's view-point is complete. For this
+philosopher we may substitute for <i>x</i> not only objects already
+known, but anything.</p>
+
+<p>Russell is faithful to his point of view, which is that of comprehension.
+He starts from the general idea of being, and
+enriches it more and more while restricting it, by adding new
+qualities. Hilbert on the contrary recognizes as possible beings
+only combinations of objects already known; so that (looking at
+only one side of his thought) we might say he takes the view-point
+of extension.</p>
+
+
+<h4>VIII</h4>
+
+<p>Let us continue with the exposition of Hilbert's ideas. He
+introduces two assumptions which he states in his symbolic
+language but which signify, in the language of the uninitiated,
+that every quality is equal to itself and that every operation performed
+upon two identical quantities gives identical results.</p>
+
+<p>So stated, they are evident, but thus to present them would
+be to misrepresent Hilbert's thought. For him mathematics
+has to combine only pure symbols, and a true mathematician
+should reason upon them without preconceptions as to their
+meaning. So his assumptions are not for him what they are for
+the common people.</p>
+
+<p>He considers them as representing the definition by postulates
+of the symbol (=) heretofore void of all signification. But to
+justify this definition we must show that these two assumptions
+lead to no contradiction. For this Hilbert used the reasoning of
+our number III, without appearing to perceive that he is using
+complete induction.</p>
+<p><span class='pagenum'><a name="Page_467" id="Page_467">[Pg 467]</a></span></p>
+
+<h4>IX</h4>
+
+<p>The end of Hilbert's memoir is altogether enigmatic and I
+shall not lay stress upon it. Contradictions accumulate; we feel
+that the author is dimly conscious of the <i>petitio principii</i> he has
+committed, and that he seeks vainly to patch up the holes in his
+argument.</p>
+
+<p>What does this mean? At the point of proving that the definition
+of the whole number by the assumption of complete induction
+implies no contradiction, Hilbert withdraws as Russell and
+Couturat withdrew, because the difficulty is too great.</p>
+
+
+<h4>X</h4>
+
+<h4><i>Geometry</i></h4>
+
+<p>Geometry, says M. Couturat, is a vast body of doctrine wherein
+the principle of complete induction does not enter. That is true
+in a certain measure; we can not say it is entirely absent, but it
+enters very slightly. If we refer to the <i>Rational Geometry</i> of
+Dr. Halsted (New York, John Wiley and Sons, 1904) built up
+in accordance with the principles of Hilbert, we see the principle
+of induction enter for the first time on page 114 (unless I have
+made an oversight, which is quite possible).<a name="FNanchor_14_14" id="FNanchor_14_14"></a><a href="#Footnote_14_14" class="fnanchor">[14]</a></p>
+
+<p>So geometry, which only a few years ago seemed the domain
+where the reign of intuition was uncontested, is to-day the realm
+where the logicians seem to triumph. Nothing could better
+measure the importance of the geometric works of Hilbert and
+the profound impress they have left on our conceptions.</p>
+
+<p>But be not deceived. What is after all the fundamental
+theorem of geometry? It is that the assumptions of geometry
+imply no contradiction, and this we can not prove without the
+principle of induction.</p>
+
+<p>How does Hilbert demonstrate this essential point? By leaning
+upon analysis and through it upon arithmetic and through
+it upon the principle of induction.</p>
+
+<p>And if ever one invents another demonstration, it will still
+be necessary to lean upon this principle, since the possible consequences
+of the assumptions, of which it is necessary to show
+that they are not contradictory, are infinite in number.</p>
+<p><span class='pagenum'><a name="Page_468" id="Page_468">[Pg 468]</a></span></p>
+
+<h4>XI</h4>
+
+<h4><i>Conclusion</i></h4>
+
+<p>Our conclusion straightway is that the principle of induction
+can not be regarded as the disguised definition of the entire
+world.</p>
+
+<p>Here are three truths: (1) The principle of complete induction;
+(2) Euclid's postulate; (3) the physical law according
+to which phosphorus melts at 44&deg; (cited by M. Le Roy).</p>
+
+<p>These are said to be three disguised definitions: the first, that
+of the whole number; the second, that of the straight line; the
+third, that of phosphorus.</p>
+
+<p>I grant it for the second; I do not admit it for the other two.
+I must explain the reason for this apparent inconsistency.</p>
+
+<p>First, we have seen that a definition is acceptable only on condition
+that it implies no contradiction. We have shown likewise
+that for the first definition this demonstration is impossible;
+on the other hand, we have just recalled that for the second
+Hilbert has given a complete proof.</p>
+
+<p>As to the third, evidently it implies no contradiction. Does
+this mean that the definition guarantees, as it should, the existence
+of the object defined? We are here no longer in the mathematical
+sciences, but in the physical, and the word existence has
+no longer the same meaning. It no longer signifies absence of
+contradiction; it means objective existence.</p>
+
+<p>You already see a first reason for the distinction I made between
+the three cases; there is a second. In the applications we
+have to make of these three concepts, do they present themselves
+to us as defined by these three postulates?</p>
+
+<p>The possible applications of the principle of induction are
+innumerable; take, for example, one of those we have expounded
+above, and where it is sought to prove that an aggregate of
+assumptions can lead to no contradiction. For this we consider
+one of the series of syllogisms we may go on with in starting
+from these assumptions as premises. When we have finished
+the <i>n</i>th syllogism, we see we can make still another and this is
+the <i>n</i> + 1th. Thus the number <i>n</i> serves to count a series of successive
+operations; it is a number obtainable by successive additions.
+<span class='pagenum'><a name="Page_469" id="Page_469">[Pg 469]</a></span>This therefore is a number from which we may go back
+to unity by <i>successive subtractions</i>. Evidently we could not do
+this if we had <i>n</i> = <i>n</i> &minus; 1, since then by subtraction we should
+always obtain again the same number. So the way we have been
+led to consider this number <i>n</i> implies a definition of the finite
+whole number and this definition is the following: A finite whole
+number is that which can be obtained by successive additions;
+it is such that <i>n</i> is not equal to <i>n</i> &minus; 1.</p>
+
+<p>That granted, what do we do? We show that if there has
+been no contradiction up to the <i>n</i>th syllogism, no more will there
+be up to the <i>n</i> + 1th, and we conclude there never will be. You
+say: I have the right to draw this conclusion, since the whole
+numbers are by definition those for which a like reasoning is
+legitimate. But that implies another definition of the whole
+number, which is as follows: A whole number is that on which we
+may reason by recurrence. In the particular case it is that of
+which we may say that, if the absence of contradiction up to the
+time of a syllogism of which the number is an integer carries
+with it the absence of contradiction up to the time of the syllogism
+whose number is the following integer, we need fear no
+contradiction for any of the syllogisms whose number is an
+integer.</p>
+
+<p>The two definitions are not identical; they are doubtless equivalent,
+but only in virtue of a synthetic judgment <i>a priori</i>; we can
+not pass from one to the other by a purely logical procedure.
+Consequently we have no right to adopt the second, after having
+introduced the whole number by a way that presupposes the first.</p>
+
+<p>On the other hand, what happens with regard to the straight
+line? I have already explained this so often that I hesitate to
+repeat it again, and shall confine myself to a brief recapitulation
+of my thought. We have not, as in the preceding case, two
+equivalent definitions logically irreducible one to the other. We
+have only one expressible in words. Will it be said there is
+another which we feel without being able to word it, since we
+have the intuition of the straight line or since we represent to
+ourselves the straight line? First of all, we can not represent it
+to ourselves in geometric space, but only in representative space,
+and then we can represent to ourselves just as well the objects<span class='pagenum'><a name="Page_470" id="Page_470">[Pg 470]</a></span>
+which possess the other properties of the straight line, save that
+of satisfying Euclid's postulate. These objects are 'the non-Euclidean
+straights,' which from a certain point of view are not
+entities void of sense, but circles (true circles of true space)
+orthogonal to a certain sphere. If, among these objects equally
+capable of representation, it is the first (the Euclidean straights)
+which we call straights, and not the latter (the non-Euclidean
+straights), this is properly by definition.</p>
+
+<p>And arriving finally at the third example, the definition of
+phosphorus, we see the true definition would be: Phosphorus is
+the bit of matter I see in yonder flask.</p>
+
+
+<h4>XII</h4>
+
+<p>And since I am on this subject, still another word. Of the
+phosphorus example I said: "This proposition is a real verifiable
+physical law, because it means that all bodies having all the other
+properties of phosphorus, save its point of fusion, melt like it at
+44&deg;." And it was answered: "No, this law is not verifiable,
+because if it were shown that two bodies resembling phosphorus
+melt one at 44&deg; and the other at 50&deg;, it might always be said
+that doubtless, besides the point of fusion, there is some other
+unknown property by which they differ."</p>
+
+<p>That was not quite what I meant to say. I should have written,
+"All bodies possessing such and such properties finite in number
+(to wit, the properties of phosphorus stated in the books on
+chemistry, the fusion-point excepted) melt at 44&deg;."</p>
+
+<p>And the better to make evident the difference between the case
+of the straight and that of phosphorus, one more remark. The
+straight has in nature many images more or less imperfect, of
+which the chief are the light rays and the rotation axis of the
+solid. Suppose we find the ray of light does not satisfy Euclid's
+postulate (for example by showing that a star has a negative
+parallax), what shall we do? Shall we conclude that the straight
+being by definition the trajectory of light does not satisfy the
+postulate, or, on the other hand, that the straight by definition
+satisfying the postulate, the ray of light is not straight?</p>
+
+<p>Assuredly we are free to adopt the one or the other definition
+and consequently the one or the other conclusion; but to adopt<span class='pagenum'><a name="Page_471" id="Page_471">[Pg 471]</a></span>
+the first would be stupid, because the ray of light probably
+satisfies only imperfectly not merely Euclid's postulate, but the
+other properties of the straight line, so that if it deviates from
+the Euclidean straight, it deviates no less from the rotation axis
+of solids which is another imperfect image of the straight line;
+while finally it is doubtless subject to change, so that such a line
+which yesterday was straight will cease to be straight to-morrow
+if some physical circumstance has changed.</p>
+
+<p>Suppose now we find that phosphorus does not melt at 44&deg;,
+but at 43.9&deg;. Shall we conclude that phosphorus being by definition
+that which melts at 44&deg;, this body that we did call phosphorus
+is not true phosphorus, or, on the other hand, that phosphorous
+melts at 43.9&deg;? Here again we are free to adopt the one
+or the other definition and consequently the one or the other
+conclusion; but to adopt the first would be stupid because we
+can not be changing the name of a substance every time we
+determine a new decimal of its fusion-point.</p>
+
+
+<h4>XIII</h4>
+
+<p>To sum up, Russell and Hilbert have each made a vigorous
+effort; they have each written a work full of original views,
+profound and often well warranted. These two works give us
+much to think about and we have much to learn from them.
+Among their results, some, many even, are solid and destined to
+live.</p>
+
+<p>But to say that they have finally settled the debate between
+Kant and Leibnitz and ruined the Kantian theory of mathematics
+is evidently incorrect. I do not know whether they really
+believed they had done it, but if they believed so, they deceived
+themselves.</p>
+
+
+
+<hr style="width: 65%;" />
+<p><span class='pagenum'><a name="Page_472" id="Page_472">[Pg 472]</a></span></p>
+<h3>CHAPTER V</h3>
+
+<h3><span class="smcap">The Latest Efforts of the Logisticians</span></h3>
+
+
+<h4>I</h4>
+
+<p>The logicians have attempted to answer the preceding considerations.
+For that, a transformation of logistic was necessary,
+and Russell in particular has modified on certain points his
+original views. Without entering into the details of the debate,
+I should like to return to the two questions to my mind most important:
+Have the rules of logistic demonstrated their fruitfulness
+and infallibility? Is it true they afford means of proving
+the principle of complete induction without any appeal to
+intuition?</p>
+
+
+<h4>II</h4>
+
+<h4><i>The Infallibility of Logistic</i></h4>
+
+<p>On the question of fertility, it seems M. Couturat has na&iuml;ve
+illusions. Logistic, according to him, lends invention 'stilts and
+wings,' and on the next page: "<i>Ten years ago</i>, Peano published
+the first edition of his <i>Formulaire</i>." How is that, ten years of
+wings and not to have flown!</p>
+
+<p>I have the highest esteem for Peano, who has done very pretty
+things (for instance his 'space-filling curve,' a phrase now discarded);
+but after all he has not gone further nor higher nor
+quicker than the majority of wingless mathematicians, and would
+have done just as well with his legs.</p>
+
+<p>On the contrary I see in logistic only shackles for the inventor.
+It is no aid to conciseness&mdash;far from it, and if twenty-seven
+equations were necessary to establish that 1 is a number, how
+many would be needed to prove a real theorem? If we distinguish,
+with Whitehead, the individual <i>x</i>, the class of which the
+only member is <i>x</i> and which shall be called &#953;<i>x</i>, then the class of
+which the only member is the class of which the only member is <i>x</i>
+and which shall be called &#956;<i>x</i>, do you think these distinctions,
+useful as they may be, go far to quicken our pace?<span class='pagenum'><a name="Page_473" id="Page_473">[Pg 473]</a></span></p>
+
+<p>Logistic forces us to say all that is ordinarily left to be understood;
+it makes us advance step by step; this is perhaps surer
+but not quicker.</p>
+
+<p>It is not wings you logisticians give us, but leading-strings.
+And then we have the right to require that these leading-strings
+prevent our falling. This will be their only excuse. When a
+bond does not bear much interest, it should at least be an investment
+for a father of a family.</p>
+
+<p>Should your rules be followed blindly? Yes, else only intuition
+could enable us to distinguish among them; but then they
+must be infallible; for only in an infallible authority can one
+have a blind confidence. This, therefore, is for you a necessity.
+Infallible you shall be, or not at all.</p>
+
+<p>You have no right to say to us: "It is true we make mistakes,
+but so do you." For us to blunder is a misfortune, a very great
+misfortune; for you it is death.</p>
+
+<p>Nor may you ask: Does the infallibility of arithmetic prevent
+errors in addition? The rules of calculation are infallible, and
+yet we see those blunder <i>who do not apply these rules</i>; but in
+checking their calculation it is at once seen where they went
+wrong. Here it is not at all the case; the logicians <i>have applied</i>
+their rules, and they have fallen into contradiction; and so true
+is this, that they are preparing to change these rules and to
+"sacrifice the notion of class." Why change them if they were
+infallible?</p>
+
+<p>"We are not obliged," you say, "to solve <i>hic et nunc</i> all possible
+problems." Oh, we do not ask so much of you. If, in face
+of a problem, you would give <i>no</i> solution, we should have nothing
+to say; but on the contrary you give us <i>two</i> of them and those
+contradictory, and consequently at least one false; this it is which
+is failure.</p>
+
+<p>Russell seeks to reconcile these contradictions, which can only
+be done, according to him, "by restricting or even sacrificing the
+notion of class." And M. Couturat, discovering the success of
+his attempt, adds: "If the logicians succeed where others have
+failed, M. Poincar&eacute; will remember this phrase, and give the honor
+of the solution to logistic."</p>
+
+<p>But no! Logistic exists, it has its code which has already had<span class='pagenum'><a name="Page_474" id="Page_474">[Pg 474]</a></span>
+four editions; or rather this code is logistic itself. Is Mr. Russell
+preparing to show that one at least of the two contradictory
+reasonings has transgressed the code? Not at all; he is preparing
+to change these laws and to abrogate a certain number of
+them. If he succeeds, I shall give the honor of it to Russell's
+intuition and not to the Peanian logistic which he will have
+destroyed.</p>
+
+
+<h4>III</h4>
+
+<h4><i>The Liberty of Contradiction</i></h4>
+
+<p>I made two principal objections to the definition of whole
+number adopted in logistic. What says M. Couturat to the first
+of these objections?</p>
+
+<p>What does the word <i>exist</i> mean in mathematics? It means,
+I said, to be free from contradiction. This M. Couturat contests.
+"Logical existence," says he, "is quite another thing
+from the absence of contradiction. It consists in the fact that
+a class is not empty." To say: <i>a</i>'s exist, is, by definition, to
+affirm that the class <i>a</i> is not null.</p>
+
+<p>And doubtless to affirm that the class <i>a</i> is not null, is, by definition,
+to affirm that <i>a</i>'s exist. But one of the two affirmations
+is as denuded of meaning as the other, if they do not both signify,
+either that one may see or touch <i>a</i>'s which is the meaning physicists
+or naturalists give them, or that one may conceive an <i>a</i>
+without being drawn into contradictions, which is the meaning
+given them by logicians and mathematicians.</p>
+
+<p>For M. Couturat, "it is not non-contradiction that proves existence,
+but it is existence that proves non-contradiction." To establish
+the existence of a class, it is necessary therefore to establish,
+by an <i>example</i>, that there is an individual belonging to this class:
+"But, it will be said, how is the existence of this individual
+proved? Must not this existence be established, in order that
+the existence of the class of which it is a part may be deduced?
+Well, no; however paradoxical may appear the assertion, we
+never demonstrate the existence of an individual. Individuals,
+just because they are individuals, are always considered as existent....
+We never have to express that an individual exists,
+absolutely speaking, but only that it exists in a class." M.<span class='pagenum'><a name="Page_475" id="Page_475">[Pg 475]</a></span>
+Couturat finds his own assertion paradoxical, and he will certainly
+not be the only one. Yet it must have a meaning. It
+doubtless means that the existence of an individual, alone in the
+world, and of which nothing is affirmed, can not involve contradiction;
+in so far as it is all alone it evidently will not embarrass
+any one. Well, so let it be; we shall admit the existence of the
+individual, 'absolutely speaking,' but nothing more. It remains
+to prove the existence of the individual 'in a class,' and for that
+it will always be necessary to prove that the affirmation, "Such
+an individual belongs to such a class," is neither contradictory
+in itself, nor to the other postulates adopted.</p>
+
+<p>"It is then," continues M. Couturat, "arbitrary and misleading
+to maintain that a definition is valid only if we first
+prove it is not contradictory." One could not claim in prouder
+and more energetic terms the liberty of contradiction. "In any
+case, the <i>onus probandi</i> rests upon those who believe that these
+principles are contradictory." Postulates are presumed to be
+compatible until the contrary is proved, just as the accused
+person is presumed innocent. Needless to add that I do not
+assent to this claim. But, you say, the demonstration you require
+of us is impossible, and you can not ask us to jump over the
+moon. Pardon me; that is impossible for you, but not for us, who
+admit the principle of induction as a synthetic judgment <i>a priori</i>.
+And that would be necessary for you, as for us.</p>
+
+<p>To demonstrate that a system of postulates implies no contradiction,
+it is necessary to apply the principle of complete induction;
+this mode of reasoning not only has nothing 'bizarre' about
+it, but it is the only correct one. It is not 'unlikely' that it has
+ever been employed; and it is not hard to find 'examples and
+precedents' of it. I have cited two such instances borrowed from
+Hilbert's article. He is not the only one to have used it, and
+those who have not done so have been wrong. What I have
+blamed Hilbert for is not his having recourse to it (a born
+mathematician such as he could not fail to see a demonstration
+was necessary and this the only one possible), but his having
+recourse without recognizing the reasoning by recurrence.</p>
+<p><span class='pagenum'><a name="Page_476" id="Page_476">[Pg 476]</a></span></p>
+
+<h4>IV</h4>
+
+<h4><i>The Second Objection</i></h4>
+
+<p>I pointed out a second error of logistic in Hilbert's article.
+To-day Hilbert is excommunicated and M. Couturat no longer
+regards him as of the logistic cult; so he asks if I have found
+the same fault among the orthodox. No, I have not seen it in the
+pages I have read; I know not whether I should find it in the
+three hundred pages they have written which I have no desire to
+read.</p>
+
+<p>Only, they must commit it the day they wish to make any
+application of mathematics. This science has not as sole object
+the eternal contemplation of its own navel; it has to do with
+nature and some day it will touch it. Then it will be necessary
+to shake off purely verbal definitions and to stop paying oneself
+with words.</p>
+
+<p>To go back to the example of Hilbert: always the point at
+issue is reasoning by recurrence and the question of knowing
+whether a system of postulates is not contradictory. M. Couturat
+will doubtless say that then this does not touch him, but it perhaps
+will interest those who do not claim, as he does, the liberty
+of contradiction.</p>
+
+<p>We wish to establish, as above, that we shall never encounter
+contradiction after any number of deductions whatever, provided
+this number be finite. For that, it is necessary to apply the
+principle of induction. Should we here understand by finite
+number every number to which by definition the principle of
+induction applies? Evidently not, else we should be led to most
+embarrassing consequences. To have the right to lay down a
+system of postulates, we must be sure they are not contradictory.
+This is a truth admitted by <i>most</i> scientists; I should have written
+<i>by all</i> before reading M. Couturat's last article. But what does
+this signify? Does it mean that we must be sure of not meeting
+contradiction after a <i>finite</i> number of propositions, the <i>finite</i>
+number being by definition that which has all properties of
+recurrent nature, so that if one of these properties fails&mdash;if, for
+instance, we come upon a contradiction&mdash;we shall agree to say
+that the number in question is not finite? In other words, do<span class='pagenum'><a name="Page_477" id="Page_477">[Pg 477]</a></span>
+we mean that we must be sure not to meet contradictions, on
+condition of agreeing to stop just when we are about to encounter
+one? To state such a proposition is enough to condemn it.</p>
+
+<p>So, Hilbert's reasoning not only assumes the principle of induction,
+but it supposes that this principle is given us not as
+a simple definition, but as a synthetic judgment <i>a priori</i>.</p>
+
+<p>To sum up:</p>
+
+<p>A demonstration is necessary.</p>
+
+<p>The only demonstration possible is the proof by recurrence.</p>
+
+<p>This is legitimate only if we admit the principle of induction
+and if we regard it not as a definition but as a synthetic judgment.</p>
+
+
+<h4>V</h4>
+
+<h4><i>The Cantor Antinomies</i></h4>
+
+<p>Now to examine Russell's new memoir. This memoir was
+written with the view to conquer the difficulties raised by those
+Cantor antinomies to which frequent allusion has already been
+made. Cantor thought he could construct a science of the
+infinite; others went on in the way he opened, but they soon ran
+foul of strange contradictions. These antinomies are already
+numerous, but the most celebrated are:</p>
+
+<p>1. The Burali-Forti antinomy;</p>
+
+<p>2. The Zermelo-K&ouml;nig antinomy;</p>
+
+<p>3. The Richard antinomy.</p>
+
+<p>Cantor proved that the ordinal numbers (the question is of
+transfinite ordinal numbers, a new notion introduced by him)
+can be ranged in a linear series; that is to say that of two unequal
+ordinals one is always less than the other. Burali-Forti
+proves the contrary; and in fact he says in substance that if one
+could range <i>all</i> the ordinals in a linear series, this series would
+define an ordinal greater than <i>all</i> the others; we could afterwards
+adjoin 1 and would obtain again an ordinal which would
+be <i>still greater</i>, and this is contradictory.</p>
+
+<p>We shall return later to the Zermelo-K&ouml;nig antinomy which is
+of a slightly different nature. The Richard antinomy<a name="FNanchor_15_15" id="FNanchor_15_15"></a><a href="#Footnote_15_15" class="fnanchor">[15]</a> is as follows:
+Consider all the decimal numbers definable by a finite<span class='pagenum'><a name="Page_478" id="Page_478">[Pg 478]</a></span>
+number of words; these decimal numbers form an aggregate <i>E</i>,
+and it is easy to see that this aggregate is countable, that is to
+say we can <i>number</i> the different decimal numbers of this assemblage
+from 1 to infinity. Suppose the numbering effected, and
+define a number <i>N</i> as follows: If the <i>n</i>th decimal of the <i>n</i>th
+number of the assemblage <i>E</i> is</p>
+
+<p class="center">
+0, 1, 2, 3, 4, 5, 6, 7, 8, 9<br />
+</p>
+
+<p class="noidt">the <i>n</i>th decimal of <i>N</i> shall be:</p>
+
+<p class="center">
+1, 2, 3, 4, 5, 6, 7, 8, 1, 1<br />
+</p>
+
+<p class="noidt">As we see, <i>N</i> is not equal to the <i>n</i>th number of <i>E</i>, and as <i>n</i> is
+arbitrary, <i>N</i> does not appertain to <i>E</i> and yet <i>N</i> should belong
+to this assemblage since we have defined it with a finite number
+of words.</p>
+
+<p>We shall later see that M. Richard has himself given with
+much sagacity the explanation of his paradox and that this extends,
+<i>mutatis mutandis</i>, to the other like paradoxes. Again,
+Russell cites another quite amusing paradox: <i>What is the least
+whole number which can not be defined by a phrase composed of
+less than a hundred English words</i>?</p>
+
+<p>This number exists; and in fact the numbers capable of being
+defined by a like phrase are evidently finite in number since the
+words of the English language are not infinite in number. Therefore
+among them will be one less than all the others. And, on the
+other hand, this number does not exist, because its definition
+implies contradiction. This number, in fact, is defined by the
+phrase in italics which is composed of less than a hundred English
+words; and by definition this number should not be capable
+of definition by a like phrase.</p>
+
+
+<h4>VI</h4>
+
+<h4><i>Zigzag Theory and No-class Theory</i></h4>
+
+<p>What is Mr. Russell's attitude in presence of these contradictions?
+After having analyzed those of which we have just
+spoken, and cited still others, after having given them a form recalling
+Epimenides, he does not hesitate to conclude: "A<span class='pagenum'><a name="Page_479" id="Page_479">[Pg 479]</a></span>
+propositional function of one variable does not always determine a
+class." A propositional function (that is to say a definition)
+does not always determine a class. A 'propositional function'
+or 'norm' may be 'non-predicative.' And this does not mean
+that these non-predicative propositions determine an empty class,
+a null class; this does not mean that there is no value of x satisfying
+the definition and capable of being one of the elements
+of the class. The elements exist, but they have no right to unite
+in a syndicate to form a class.</p>
+
+<p>But this is only the beginning and it is needful to know how
+to recognize whether a definition is or is not predicative. To
+solve this problem Russell hesitates between three theories which
+he calls</p>
+
+<p>A. The zigzag theory;</p>
+
+<p>B. The theory of limitation of size;</p>
+
+<p>C. The no-class theory.</p>
+
+<p>According to the zigzag theory "definitions (propositional
+functions) determine a class when they are very simple and cease
+to do so only when they are complicated and obscure." Who,
+now, is to decide whether a definition may be regarded as simple
+enough to be acceptable? To this question there is no answer, if
+it be not the loyal avowal of a complete inability: "The rules
+which enable us to recognize whether these definitions are predicative
+would be extremely complicated and can not commend themselves
+by any plausible reason. This is a fault which might be
+remedied by greater ingenuity or by using distinctions not yet
+pointed out. But hitherto in seeking these rules, I have not
+been able to find any other directing principle than the absence
+of contradiction."</p>
+
+<p>This theory therefore remains very obscure; in this night a
+single light&mdash;the word zigzag. What Russell calls the 'zigzaginess'
+is doubtless the particular characteristic which distinguishes
+the argument of Epimenides.</p>
+
+<p>According to the theory of limitation of size, a class would
+cease to have the right to exist if it were too extended. Perhaps
+it might be infinite, but it should not be too much so. But we
+always meet again the same difficulty; at what precise moment<span class='pagenum'><a name="Page_480" id="Page_480">[Pg 480]</a></span>
+does it begin to be too much so? Of course this difficulty is not
+solved and Russell passes on to the third theory.</p>
+
+<p>In the no-classes theory it is forbidden to speak the word
+'class' and this word must be replaced by various periphrases.
+What a change for logistic which talks only of classes and
+classes of classes! It becomes necessary to remake the whole
+of logistic. Imagine how a page of logistic would look upon suppressing
+all the propositions where it is a question of class.
+There would only be some scattered survivors in the midst of a
+blank page. <i>Apparent rari nantes in gurgite vasto.</i></p>
+
+<p>Be that as it may, we see how Russell hesitates and the modifications
+to which he submits the fundamental principles he has
+hitherto adopted. Criteria are needed to decide whether a definition
+is too complex or too extended, and these criteria can only
+be justified by an appeal to intuition.</p>
+
+<p>It is toward the no-classes theory that Russell finally inclines.
+Be that as it may, logistic is to be remade and it is not clear
+how much of it can be saved. Needless to add that Cantorism
+and logistic are alone under consideration; real mathematics,
+that which is good for something, may continue to develop in
+accordance with its own principles without bothering about the
+storms which rage outside it, and go on step by step with its usual
+conquests which are final and which it never has to abandon.</p>
+
+
+<h4>VII</h4>
+
+<h4><i>The True Solution</i></h4>
+
+<p>What choice ought we to make among these different theories?
+It seems to me that the solution is contained in a letter of M.
+Richard of which I have spoken above, to be found in the <i>Revue
+g&eacute;n&eacute;rale des sciences</i> of June 30, 1905. After having set forth
+the antinomy we have called Richard's antinomy, he gives its
+explanation. Recall what has already been said of this antinomy.
+<i>E</i> is the aggregate of <i>all</i> the numbers definable by a finite number
+of words, <i>without introducing the notion of the aggregate E itself</i>.
+Else the definition of <i>E</i> would contain a vicious circle; we must
+not define <i>E</i> by the aggregate <i>E</i> itself.</p>
+
+<p>Now we have defined <i>N</i> with a finite number of words, it is<span class='pagenum'><a name="Page_481" id="Page_481">[Pg 481]</a></span>
+true, but with the aid of the notion of the aggregate <i>E</i>. And
+this is why <i>N</i> is not part of <i>E</i>. In the example selected by M.
+Richard, the conclusion presents itself with complete evidence
+and the evidence will appear still stronger on consulting the
+text of the letter itself. But the same explanation holds good
+for the other antinomies, as is easily verified. Thus <i>the definitions
+which should be regarded as not predicative are those
+which contain a vicious circle</i>. And the preceding examples sufficiently
+show what I mean by that. Is it this which Russell calls
+the 'zigzaginess'? I put the question without answering it.</p>
+
+
+<h4>VIII</h4>
+
+<h4><i>The Demonstrations of the Principle of Induction</i></h4>
+
+<p>Let us now examine the pretended demonstrations of the
+principle of induction and in particular those of Whitehead and
+of Burali-Forti.</p>
+
+<p>We shall speak of Whitehead's first, and take advantage of
+certain new terms happily introduced by Russell in his recent
+memoir. Call <i>recurrent class</i> every class containing zero, and
+containing <i>n</i> + 1 if it contains <i>n</i>. Call <i>inductive number</i> every
+number which is a part of <i>all</i> the recurrent classes. Upon what
+condition will this latter definition, which plays an essential
+r&ocirc;le in Whitehead's proof, be 'predicative' and consequently
+acceptable?</p>
+
+<p>In accordance with what has been said, it is necessary to
+understand by <i>all</i> the recurrent classes, all those in whose definition
+the notion of inductive number does not enter. Else we fall
+again upon the vicious circle which has engendered the antinomies.</p>
+
+<p>Now <i>Whitehead has not taken this precaution</i>. Whitehead's
+reasoning is therefore fallacious; it is the same which led to the
+antinomies. It was illegitimate when it gave false results; it
+remains illegitimate when by chance it leads to a true result.</p>
+
+<p>A definition containing a vicious circle defines nothing. It is
+of no use to say, we are sure, whatever meaning we may give to
+our definition, zero at least belongs to the class of inductive
+numbers; it is not a question of knowing whether this class is
+void, but whether it can be rigorously deliminated. A<span class='pagenum'><a name="Page_482" id="Page_482">[Pg 482]</a></span>
+'non-predicative' class is not an empty class, it is a class whose
+boundary is undetermined. Needless to add that this particular
+objection leaves in force the general objections applicable to all
+the demonstrations.</p>
+
+
+<h4>IX</h4>
+
+<p>Burali-Forti has given another demonstration.<a name="FNanchor_16_16" id="FNanchor_16_16"></a><a href="#Footnote_16_16" class="fnanchor">[16]</a> But he is
+obliged to assume two postulates: First, there always exists at
+least one infinite class. The second is thus expressed:</p>
+
+<div class="figcenter" style="width: 300px;">
+<img src="images/img494.png" width="300" height="35" alt="" title="" />
+</div>
+
+<p>The first postulate is not more evident than the principle to be
+proved. The second not only is not evident, but it is false, as
+Whitehead has shown; as moreover any recruit would see at the
+first glance, if the axiom had been stated in intelligible language,
+since it means that the number of combinations which can be
+formed with several objects is less than the number of these
+objects.</p>
+
+
+<h4>X</h4>
+
+<h4><i>Zermelo's Assumption</i></h4>
+
+<p>A famous demonstration by Zermelo rests upon the following
+assumption: In any aggregate (or the same in each aggregate
+of an assemblage of aggregates) we can always choose <i>at random</i>
+an element (even if this assemblage of aggregates should contain
+an infinity of aggregates). This assumption had been
+applied a thousand times without being stated, but, once stated,
+it aroused doubts. Some mathematicians, for instance M. Borel,
+resolutely reject it; others admire it. Let us see what, according
+to his last article, Russell thinks of it. He does not speak
+out, but his reflections are very suggestive.</p>
+
+<p>And first a picturesque example: Suppose we have as many
+pairs of shoes as there are whole numbers, and so that we can
+number <i>the pairs</i> from one to infinity, how many shoes shall we
+have? Will the number of shoes be equal to the number of
+pairs? Yes, if in each pair the right shoe is distinguishable
+from the left; it will in fact suffice to give the number 2<i>n</i> &minus; 1 to
+the right shoe of the <i>n</i>th pair, and the number 2<i>n</i> to the left<span class='pagenum'><a name="Page_483" id="Page_483">[Pg 483]</a></span>
+shoe of the <i>n</i>th pair. No, if the right shoe is just like the left,
+because a similar operation would become impossible&mdash;unless
+we admit Zermelo's assumption, since then we could choose <i>at
+random</i> in each pair the shoe to be regarded as the right.</p>
+
+
+<h4>XI</h4>
+
+<h4><i>Conclusions</i></h4>
+
+<p>A demonstration truly founded upon the principles of analytic
+logic will be composed of a series of propositions. Some, serving
+as premises, will be identities or definitions; the others will be
+deduced from the premises step by step. But though the bond
+between each proposition and the following is immediately evident,
+it will not at first sight appear how we get from the first
+to the last, which we may be tempted to regard as a new truth.
+But if we replace successively the different expressions therein by
+their definition and if this operation be carried as far as possible,
+there will finally remain only identities, so that all will
+reduce to an immense tautology. Logic therefore remains sterile
+unless made fruitful by intuition.</p>
+
+<p>This I wrote long ago; logistic professes the contrary and
+thinks it has proved it by actually proving new truths. By
+what mechanism? Why in applying to their reasonings the procedure
+just described&mdash;namely, replacing the terms defined by
+their definitions&mdash;do we not see them dissolve into identities like
+ordinary reasonings? It is because this procedure is not applicable
+to them. And why? Because their definitions are not
+predicative and present this sort of hidden vicious circle which
+I have pointed out above; non-predicative definitions can not be
+substituted for the terms defined. Under these conditions
+<i>logistic is not sterile, it engenders antinomies</i>.</p>
+
+<p>It is the belief in the existence of the actual infinite which has
+given birth to those non-predicative definitions. Let me explain.
+In these definitions the word 'all' figures, as is seen in the
+examples cited above. The word 'all' has a very precise meaning
+when it is a question of a finite number of objects; to
+have another one, when the objects are infinite in number, would
+require there being an actual (given complete) infinity. Otherwise<span class='pagenum'><a name="Page_484" id="Page_484">[Pg 484]</a></span>
+<i>all</i> these objects could not be conceived as postulated anteriorly
+to their definition, and then if the definition of a notion
+<i>N</i> depends upon <i>all</i> the objects <i>A</i>, it may be infected with a
+vicious circle, if among the objects <i>A</i> are some indefinable without
+the intervention of the notion <i>N</i> itself.</p>
+
+<p>The rules of formal logic express simply the properties of all
+possible classifications. But for them to be applicable it is necessary
+that these classifications be immutable and that we have no
+need to modify them in the course of the reasoning. If we have
+to classify only a finite number of objects, it is easy to keep our
+classifications without change. If the objects are <i>indefinite</i> in
+number, that is to say if one is constantly exposed to seeing new
+and unforeseen objects arise, it may happen that the appearance
+of a new object may require the classification to be modified, and
+thus it is we are exposed to antinomies. <i>There is no actual
+(given complete) infinity.</i> The Cantorians have forgotten this,
+and they have fallen into contradiction. It is true that Cantorism
+has been of service, but this was when applied to a real
+problem whose terms were precisely defined, and then we could
+advance without fear.</p>
+
+<p>Logistic also forgot it, like the Cantorians, and encountered
+the same difficulties. But the question is to know whether they
+went this way by accident or whether it was a necessity for them.
+For me, the question is not doubtful; belief in an actual infinity
+is essential in the Russell logic. It is just this which distinguishes
+it from the Hilbert logic. Hilbert takes the view-point
+of extension, precisely in order to avoid the Cantorian antinomies.
+Russell takes the view-point of comprehension. Consequently
+for him the genus is anterior to the species, and the
+<i>summum genus</i> is anterior to all. That would not be inconvenient
+if the <i>summum genus</i> was finite; but if it is infinite, it is
+necessary to postulate the infinite, that is to say to regard the
+infinite as actual (given complete). And we have not only infinite
+classes; when we pass from the genus to the species in
+restricting the concept by new conditions, these conditions are
+still infinite in number. Because they express generally that the
+envisaged object presents such or such a relation with all the
+objects of an infinite class.<span class='pagenum'><a name="Page_485" id="Page_485">[Pg 485]</a></span></p>
+
+<p>But that is ancient history. Russell has perceived the peril
+and takes counsel. He is about to change everything, and, what
+is easily understood, he is preparing not only to introduce new
+principles which shall allow of operations formerly forbidden,
+but he is preparing to forbid operations he formerly thought
+legitimate. Not content to adore what he burned, he is about
+to burn what he adored, which is more serious. He does not add
+a new wing to the building, he saps its foundation.</p>
+
+<p>The old logistic is dead, so much so that already the zigzag
+theory and the no-classes theory are disputing over the succession.
+To judge of the new, we shall await its coming.</p>
+
+
+
+<hr style="width: 65%;" />
+<p><span class='pagenum'><a name="Page_486" id="Page_486">[Pg 486]</a></span></p>
+<h2><b>BOOK III<br />
+
+<br />
+
+<small>THE NEW MECHANICS</small></b></h2>
+
+
+
+<hr style="width: 65%;" />
+<h3>CHAPTER I</h3>
+
+<h3><span class="smcap">Mechanics and Radium</span></h3>
+
+
+<h4>I</h4>
+
+<h4><i>Introduction</i></h4>
+
+<p>The general principles of Dynamics, which have, since Newton,
+served as foundation for physical science, and which appeared
+immovable, are they on the point of being abandoned or
+at least profoundly modified? This is what many people have
+been asking themselves for some years. According to them, the
+discovery of radium has overturned the scientific dogmas we believed
+the most solid: on the one hand, the impossibility of the
+transmutation of metals; on the other hand, the fundamental
+postulates of mechanics.</p>
+
+<p>Perhaps one is too hasty in considering these novelties as
+finally established, and breaking our idols of yesterday; perhaps
+it would be proper, before taking sides, to await experiments
+more numerous and more convincing. None the less is it necessary,
+from to-day, to know the new doctrines and the arguments,
+already very weighty, upon which they rest.</p>
+
+<p>In few words let us first recall in what those principles consist:</p>
+
+<p><i>A.</i> The motion of a material point isolated and apart from all
+exterior force is straight and uniform; this is the principle of
+inertia: without force no acceleration;</p>
+
+<p><i>B.</i> The acceleration of a moving point has the same direction
+as the resultant of all the forces to which it is subjected; it is
+equal to the quotient of this resultant by a coefficient called
+<i>mass</i> of the moving point.</p>
+
+<p>The mass of a moving point, so defined, is a constant; it does<span class='pagenum'><a name="Page_487" id="Page_487">[Pg 487]</a></span>
+not depend upon the velocity acquired by this point; it is the
+same whether the force, being parallel to this velocity, tends only
+to accelerate or to retard the motion of the point, or whether,
+on the contrary, being perpendicular to this velocity, it tends to
+make this motion deviate toward the right, or the left, that is to
+say to <i>curve</i> the trajectory;</p>
+
+<p><i>C.</i> All the forces affecting a material point come from the
+action of other material points; they depend only upon the
+<i>relative</i> positions and velocities of these different material points.</p>
+
+<p>Combining the two principles <i>B</i> and <i>C</i>, we reach the <i>principle
+of relative motion</i>, in virtue of which the laws of the motion
+of a system are the same whether we refer this system to
+fixed axes, or to moving axes animated by a straight and uniform
+motion of translation, so that it is impossible to distinguish absolute
+motion from a relative motion with reference to such moving
+axes;</p>
+
+<p><i>D.</i> If a material point <i>A</i> acts upon another material point <i>B</i>,
+the body <i>B</i> reacts upon <i>A</i>, and these two actions are two equal
+and directly opposite forces. This is <i>the principle of the equality
+of action and reaction</i>, or, more briefly, the <i>principle of reaction</i>.</p>
+
+<p>Astronomic observations and the most ordinary physical phenomena
+seem to have given of these principles a confirmation complete,
+constant and very precise. This is true, it is now said,
+but it is because we have never operated with any but very
+small velocities; Mercury, for example, the fastest of the planets,
+goes scarcely 100 kilometers a second. Would this planet act
+the same if it went a thousand times faster? We see there is yet
+no need to worry; whatever may be the progress of automobilism,
+it will be long before we must give up applying to our machines
+the classic principles of dynamics.</p>
+
+<p>How then have we come to make actual speeds a thousand times
+greater than that of Mercury, equal, for instance, to a tenth or
+a third of the velocity of light, or approaching still more closely
+to that velocity? It is by aid of the cathode rays and the rays
+from radium.</p>
+
+<p>We know that radium emits three kinds of rays, designated
+by the three Greek letters &#945;, &#946;, &#947;; in what follows, unless the
+contrary be expressly stated, it will always be a question of the
+&#946; rays, which are analogous to the cathode rays.<span class='pagenum'><a name="Page_488" id="Page_488">[Pg 488]</a></span></p>
+
+<p>After the discovery of the cathode rays two theories appeared.
+Crookes attributed the phenomena to a veritable molecular bombardment;
+Hertz, to special undulations of the ether. This was
+a renewal of the debate which divided physicists a century ago
+about light; Crookes took up the emission theory, abandoned
+for light; Hertz held to the undulatory theory. The facts seem
+to decide in favor of Crookes.</p>
+
+<p>It has been recognized, in the first place, that the cathode
+rays carry with them a negative electric charge; they are deviated
+by a magnetic field and by an electric field; and these deviations
+are precisely such as these same fields would produce upon projectiles
+animated by a very high velocity and strongly charged
+with electricity. These two deviations depend upon two quantities:
+one the velocity, the other the relation of the electric charge
+of the projectile to its mass; we cannot know the absolute value
+of this mass, nor that of the charge, but only their relation; in
+fact, it is clear that if we double at the same time the charge and
+the mass, without changing the velocity, we shall double the
+force which tends to deviate the projectile, but, as its mass is also
+doubled, the acceleration and deviation observable will not be
+changed. The observation of the two deviations will give us
+therefore two equations to determine these two unknowns. We
+find a velocity of from 10,000 to 30,000 kilometers a second; as
+to the ratio of the charge to the mass, it is very great. We may
+compare it to the corresponding ratio in regard to the hydrogen
+ion in electrolysis; we then find that a cathodic projectile carries
+about a thousand times more electricity than an equal mass of
+hydrogen would carry in an electrolyte.</p>
+
+<p>To confirm these views, we need a direct measurement of this
+velocity to compare with the velocity so calculated. Old experiments
+of J. J. Thomson had given results more than a hundred
+times too small; but they were exposed to certain causes of error.
+The question was taken up again by Wiechert in an arrangement
+where the Hertzian oscillations were utilized; results were found
+agreeing with the theory, at least as to order of magnitude; it
+would be of great interest to repeat these experiments. However
+that may be, the theory of undulations appears powerless
+to account for this complex of facts.<span class='pagenum'><a name="Page_489" id="Page_489">[Pg 489]</a></span></p>
+
+<p>The same calculations made with reference to the &#946; rays of
+radium have given velocities still greater: 100,000 or 200,000
+kilometers or more yet. These velocities greatly surpass all those
+we know. It is true that light has long been known to go 300,000
+kilometers a second; but it is not a carrying of matter, while, if
+we adopt the emission theory for the cathode rays, there would
+be material molecules really impelled at the velocities in question,
+and it is proper to investigate whether the ordinary laws of mechanics
+are still applicable to them.</p>
+
+
+<h4>II</h4>
+
+<h4><i>Mass Longitudinal and Mass Transversal</i></h4>
+
+<p>We know that electric currents produce the phenomena of induction,
+in particular <i>self-induction</i>. When a current increases,
+there develops an electromotive force of self-induction which
+tends to oppose the current; on the contrary, when the current
+decreases, the electromotive force of self-induction tends to maintain
+the current. The self-induction therefore opposes every
+variation of the intensity of the current, just as in mechanics the
+inertia of a body opposes every variation of its velocity.</p>
+
+<p><i>Self-induction is a veritable inertia.</i> Everything happens as if
+the current could not establish itself without putting in motion
+the surrounding ether and as if the inertia of this ether tended,
+in consequence, to keep constant the intensity of this current.
+It would be requisite to overcome this inertia to establish the
+current, it would be necessary to overcome it again to make the
+current cease.</p>
+
+<p>A cathode ray, which is a rain of projectiles charged with negative
+electricity, may be likened to a current; doubtless this current
+differs, at first sight at least, from the currents of ordinary
+conduction, where the matter does not move and where the electricity
+circulates through the matter. This is a <i>current of convection</i>,
+where the electricity, attached to a material vehicle, is
+carried along by the motion of this vehicle. But Rowland has
+proved that currents of convection produce the same magnetic
+effects as currents of conduction; they should produce also the
+same effects of induction. First, if this were not so, the principle
+of the conservation of energy would be violated; besides,<span class='pagenum'><a name="Page_490" id="Page_490">[Pg 490]</a></span>
+Cr&eacute;mieu and Pender have employed a method putting in evidence
+<i>directly</i> these effects of induction.</p>
+
+<p>If the velocity of a cathode corpuscle varies, the intensity of the
+corresponding current will likewise vary; and there will develop
+effects of self-induction which will tend to oppose this variation.
+These corpuscles should therefore possess a double inertia: first
+their own proper inertia, and then the apparent inertia, due to
+self-induction, which produces the same effects. They will therefore
+have a total apparent mass, composed of their real mass and
+of a fictitious mass of electromagnetic origin. Calculation shows
+that this fictitious mass varies with the velocity, and that the
+force of inertia of self-induction is not the same when the velocity
+of the projectile accelerates or slackens, or when it is deviated;
+therefore so it is with the force of the total apparent inertia.</p>
+
+<p>The total apparent mass is therefore not the same when the real
+force applied to the corpuscle is parallel to its velocity and tends
+to accelerate the motion as when it is perpendicular to this velocity
+and tends to make the direction vary. It is necessary therefore
+to distinguish the <i>total longitudinal mass</i> from the <i>total
+transversal mass</i>. These two total masses depend, moreover,
+upon the velocity. This follows from the theoretical work of
+Abraham.</p>
+
+<p>In the measurements of which we speak in the preceding section,
+what is it we determine in measuring the two deviations?
+It is the velocity on the one hand, and on the other hand the
+ratio of the charge to the <i>total transversal mass</i>. How, under
+these conditions, can we make out in this total mass the part
+of the real mass and that of the fictitious electromagnetic mass?
+If we had only the cathode rays properly so called, it could not
+be dreamed of; but happily we have the rays of radium which,
+as we have seen, are notably swifter. These rays are not all identical
+and do not behave in the same way under the action of an
+electric field and a magnetic field. It is found that the electric
+deviation is a function of the magnetic deviation, and we are able,
+by receiving on a sensitive plate radium rays which have been
+subjected to the action of the two fields, to photograph the curve
+which represents the relation between these two deviations. This
+is what Kaufmann has done, deducing from it the relation<span class='pagenum'><a name="Page_491" id="Page_491">[Pg 491]</a></span>
+between the velocity and the ratio of the charge to the total apparent
+mass, a ratio we shall call &#949;.</p>
+
+<p>One might suppose there are several species of rays, each characterized
+by a fixed velocity, by a fixed charge and by a fixed
+mass. But this hypothesis is improbable; why, in fact, would all
+the corpuscles of the same mass take always the same velocity?
+It is more natural to suppose that the charge as well as the <i>real</i>
+mass are the same for all the projectiles, and that these differ
+only by their velocity. If the ratio &#949; is a function of the velocity,
+this is not because the real mass varies with this velocity; but,
+since the fictitious electromagnetic mass depends upon this velocity,
+the total apparent mass, alone observable, must depend upon
+it, though the real mass does not depend upon it and may be
+constant.</p>
+
+<p>The calculations of Abraham let us know the law according to
+which the <i>fictitious</i> mass varies as a function of the velocity;
+Kaufmann's experiment lets us know the law of variation of the
+<i>total</i> mass.</p>
+
+<p>The comparison of these two laws will enable us therefore to
+determine the ratio of the real mass to the total mass.</p>
+
+<p>Such is the method Kaufmann used to determine this ratio.
+The result is highly surprising: <i>the real mass is naught</i>.</p>
+
+<p>This has led to conceptions wholly unexpected. What had
+only been proved for cathode corpuscles was extended to all
+bodies. What we call mass would be only semblance; all inertia
+would be of electromagnetic origin. But then mass would no
+longer be constant, it would augment with the velocity; sensibly
+constant for velocities up to 1,000 kilometers a second, it
+then would increase and would become infinite for the velocity
+of light. The transversal mass would no longer be equal to the
+longitudinal: they would only be nearly equal if the velocity is
+not too great. The principle <i>B</i> of mechanics would no longer
+be true.</p>
+
+
+<h4>III</h4>
+
+<h4><i>The Canal Rays</i></h4>
+
+<p>At the point where we now are, this conclusion might seem
+premature. Can one apply to all matter what has been proved<span class='pagenum'><a name="Page_492" id="Page_492">[Pg 492]</a></span>
+only for such light corpuscles, which are a mere emanation of
+matter and perhaps not true matter? But before entering upon
+this question, a word must be said of another sort of rays. I
+refer to the <i>canal rays</i>, the <i>Kanalstrahlen</i> of Goldstein.</p>
+
+<p>The cathode, together with the cathode rays charged with negative
+electricity, emits canal rays charged with positive electricity.
+In general, these canal rays not being repelled by the cathode, are
+confined to the immediate neighborhood of this cathode, where
+they constitute the `chamois cushion,' not very easy to perceive;
+but, if the cathode is pierced with holes and if it almost completely
+blocks up the tube, the canal rays spread <i>back</i> of the
+cathode, in the direction opposite to that of the cathode rays, and
+it becomes possible to study them. It is thus that it has been
+possible to show their positive charge and to show that the magnetic
+and electric deviations still exist, as for the cathode rays,
+but are much feebler.</p>
+
+<p>Radium likewise emits rays analogous to the canal rays, and
+relatively very absorbable, called &#945; rays.</p>
+
+<p>We can, as for the cathode rays, measure the two deviations
+and thence deduce the velocity and the ratio &#949;. The results are
+less constant than for the cathode rays, but the velocity is less,
+as well as the ratio &#949;; the positive corpuscles are less charged
+than the negative; or if, which is more natural, we suppose the
+charges equal and of opposite sign, the positive corpuscles are
+much the larger. These corpuscles, charged the ones positively,
+the others negatively, have been called <i>electrons</i>.</p>
+
+
+<h4>IV</h4>
+
+<h4><i>The Theory of Lorentz</i></h4>
+
+<p>But the electrons do not merely show us their existence in
+these rays where they are endowed with enormous velocities.
+We shall see them in very different r&ocirc;les, and it is they that
+account for the principal phenomena of optics and electricity.
+The brilliant synthesis about to be noticed is due to Lorentz.</p>
+
+<p>Matter is formed solely of electrons carrying enormous charges,
+and, if it seems to us neutral, this is because the charges of
+opposite sign of these electrons compensate each other. We<span class='pagenum'><a name="Page_493" id="Page_493">[Pg 493]</a></span>
+may imagine, for example, a sort of solar system formed of a
+great positive electron, around which gravitate numerous little
+planets, the negative electrons, attracted by the electricity of
+opposite name which charges the central electron. The negative
+charges of these planets would balance the positive charge
+of this sun, so that the algebraic sum of all these charges would
+be naught.</p>
+
+<p>All these electrons swim in the ether. The ether is everywhere
+identically the same, and perturbations in it are propagated
+according to the same laws as light or the Hertzian oscillations
+<i>in vacuo</i>. There is nothing but electrons and ether.
+When a luminous wave enters a part of the ether where electrons
+are numerous, these electrons are put in motion under the influence
+of the perturbation of the ether, and they then react
+upon the ether. So would be explained refraction, dispersion,
+double refraction and absorption. Just so, if for any cause an
+electron be put in motion, it would trouble the ether around it
+and would give rise to luminous waves, and this would explain
+the emission of light by incandescent bodies.</p>
+
+<p>In certain bodies, the metals for example, we should have
+fixed electrons, between which would circulate moving electrons
+enjoying perfect liberty, save that of going out from the metallic
+body and breaking the surface which separates it from the exterior
+void or from the air, or from any other non-metallic body.</p>
+
+<p>These movable electrons behave then, within the metallic body,
+as do, according to the kinetic theory of gases, the molecules of
+a gas within the vase where this gas is confined. But, under the
+influence of a difference of potential, the negative movable electrons
+would tend to go all to one side, and the positive movable
+electrons to the other. This is what would produce electric currents,
+and <i>this is why these bodies would be conductors</i>. On the
+other hand, the velocities of our electrons would be the greater
+the higher the temperature, if we accept the assimilation with
+the kinetic theory of gases. When one of these movable electrons
+encounters the surface of the metallic body, whose boundary it
+can not pass, it is reflected like a billiard ball which has hit the
+cushion, and its velocity undergoes a sudden change of direction.
+But when an electron changes direction, as we shall see further<span class='pagenum'><a name="Page_494" id="Page_494">[Pg 494]</a></span>
+on, it becomes the source of a luminous wave, and this is why hot
+metals are incandescent.</p>
+
+<p>In other bodies, the dielectrics and the transparent bodies, the
+movable electrons enjoy much less freedom. They remain as if
+attached to fixed electrons which attract them. The farther they
+go away from them the greater becomes this attraction and
+tends to pull them back. They therefore can make only small
+excursions; they can no longer circulate, but only oscillate about
+their mean position. This is why these bodies would not be conductors;
+moreover they would most often be transparent, and
+they would be refractive, since the luminous vibrations would be
+communicated to the movable electrons, susceptible of oscillation,
+and thence a perturbation would result.</p>
+
+<p>I can not here give the details of the calculations; I confine
+myself to saying that this theory accounts for all the known
+facts, and has predicted new ones, such as the Zeeman effect.</p>
+
+
+<h4>V</h4>
+
+<h4><i>Mechanical Consequences</i></h4>
+
+<p>We now may face two hypotheses:</p>
+
+<p>1&ordm; The positive electrons have a real mass, much greater than
+their fictitious electromagnetic mass; the negative electrons alone
+lack real mass. We might even suppose that apart from electrons
+of the two signs, there are neutral atoms which have only their
+real mass. In this case, mechanics is not affected; there is no
+need of touching its laws; the real mass is constant; simply, motions
+are deranged by the effects of self-induction, as has always
+been known; moreover, these perturbations are almost negligible,
+except for the negative electrons which, not having real mass, are
+not true matter.</p>
+
+<p>2&ordm; But there is another point of view; we may suppose there
+are no neutral atoms, and the positive electrons lack real mass
+just as the negative electrons. But then, real mass vanishing,
+either the word <i>mass</i> will no longer have any meaning, or else
+it must designate the fictitious electromagnetic mass; in this
+case, mass will no longer be constant, the transversal <i>mass</i> will
+no longer be equal to the longitudinal, the principles of mechanics
+will be overthrown.<span class='pagenum'><a name="Page_495" id="Page_495">[Pg 495]</a></span></p>
+
+<p>First a word of explanation. We have said that, for the
+same charge, the <i>total</i> mass of a positive electron is much greater
+than that of a negative. And then it is natural to think that this
+difference is explained by the positive electron having, besides
+its fictitious mass, a considerable real mass; which takes us back
+to the first hypothesis. But we may just as well suppose that the
+real mass is null for these as for the others, but that the fictitious
+mass of the positive electron is much the greater since this electron
+is much the smaller. I say advisedly: much the smaller.
+And, in fact, in this hypothesis inertia is exclusively electromagnetic
+in origin; it reduces itself to the inertia of the ether; the
+electrons are no longer anything by themselves; they are solely
+holes in the ether and around which the ether moves; the smaller
+these holes are, the more will there be of ether, the greater, consequently,
+will be the inertia of the ether.</p>
+
+<p>How shall we decide between these two hypotheses? By operating
+upon the canal rays as Kaufmann did upon the &#946; rays?
+This is impossible; the velocity of these rays is much too slight.
+Should each therefore decide according to his temperament, the
+conservatives going to one side and the lovers of the new to the
+other? Perhaps, but, to fully understand the arguments of the
+innovators, other considerations must come in.</p>
+
+
+
+<hr style="width: 65%;" />
+<p><span class='pagenum'><a name="Page_496" id="Page_496">[Pg 496]</a></span></p>
+<h3>CHAPTER II</h3>
+
+<h3><span class="smcap">Mechanics and Optics</span></h3>
+
+
+<h4>I</h4>
+
+<h4><i>Aberration</i></h4>
+
+<p>You know in what the phenomenon of aberration, discovered
+by Bradley, consists. The light issuing from a star takes a certain
+time to go through a telescope; during this time, the telescope,
+carried along by the motion of the earth, is displaced. If
+therefore the telescope were pointed in the <i>true</i> direction of the
+star, the image would be formed at the point occupied by the
+crossing of the threads of the network when the light has reached
+the objective; and this crossing would no longer be at this same
+point when the light reached the plane of the network. We
+would therefore be led to mis-point the telescope to bring the
+image upon the crossing of the threads. Thence results that the
+astronomer will not point the telescope in the direction of the
+absolute velocity of the light, that is to say toward the true
+position of the star, but just in the direction of the relative velocity
+of the light with reference to the earth, that is to say toward
+what is called the apparent position of the star.</p>
+
+<p>The velocity of light is known; we might therefore suppose
+that we have the means of calculating the <i>absolute</i> velocity of the
+earth. (I shall soon explain my use here of the word absolute.)
+Nothing of the sort; we indeed know the apparent position of the
+star we observe; but we do not know its true position; we know
+the velocity of the light only in magnitude and not in direction.</p>
+
+<p>If therefore the absolute velocity of the earth were straight
+and uniform, we should never have suspected the phenomenon of
+aberration; but it is variable; it is composed of two parts: the
+velocity of the solar system, which is straight and uniform; the
+velocity of the earth with reference to the sun, which is variable.
+If the velocity of the solar system, that is to say if the constant
+part existed alone, the observed direction would be invariable.<span class='pagenum'><a name="Page_497" id="Page_497">[Pg 497]</a></span>
+This position that one would thus observe is called the <i>mean</i>
+apparent position of the star.</p>
+
+<p>Taking account now at the same time of the two parts of the
+velocity of the earth, we shall have the actual apparent position,
+which describes a little ellipse around the mean apparent position,
+and it is this ellipse that we observe.</p>
+
+<p>Neglecting very small quantities, we shall see that the dimensions
+of this ellipse depend only upon the ratio of the velocity of
+the earth with reference to the sun to the velocity of light, so
+that the relative velocity of the earth with regard to the sun has
+alone come in.</p>
+
+<p>But wait! This result is not exact, it is only approximate; let
+us push the approximation a little farther. The dimensions of
+the ellipse will depend then upon the absolute velocity of the
+earth. Let us compare the major axes of the ellipse for the
+different stars: we shall have, theoretically at least, the means of
+determining this absolute velocity.</p>
+
+<p>That would be perhaps less shocking than it at first seems; it
+is a question, in fact, not of the velocity with reference to an
+absolute void, but of the velocity with regard to the ether, which
+is taken <i>by definition</i> as being absolutely at rest.</p>
+
+<p>Besides, this method is purely theoretical. In fact, the aberration
+is very small; the possible variations of the ellipse of aberration
+are much smaller yet, and, if we consider the aberration
+as of the first order, they should therefore be regarded as of the
+second order: about a millionth of a second; they are absolutely
+inappreciable for our instruments. We shall finally see, further
+on, why the preceding theory should be rejected, and why we
+could not determine this absolute velocity even if our instruments
+were ten thousand times more precise!</p>
+
+<p>One might imagine some other means, and in fact, so one has.
+The velocity of light is not the same in water as in air; could
+we not compare the two apparent positions of a star seen through
+a telescope first full of air, then full of water? The results have
+been negative; the apparent laws of reflection and refraction
+are not altered by the motion of the earth. This phenomenon
+is capable of two explanations:</p>
+
+<p>1&ordm; It might be supposed that the ether is not at rest, but that
+<span class='pagenum'><a name="Page_498" id="Page_498">[Pg 498]</a></span>
+it is carried along by the body in motion. It would then not be
+astonishing that the phenomena of refraction are not altered
+by the motion of the earth, since all, prisms, telescopes and
+ether, are carried along together in the same translation. As to
+the aberration itself, it would be explained by a sort of refraction
+happening at the surface of separation of the ether at rest
+in the interstellar spaces and the ether carried along by the
+motion of the earth. It is upon this hypothesis (bodily carrying
+along of the ether) that is founded the <i>theory of Hertz</i> on the
+electrodynamics of moving bodies.</p>
+
+<p>2&ordm; Fresnel, on the contrary, supposes that the ether is at
+absolute rest in the void, at rest almost absolute in the air, whatever
+be the velocity of this air, and that it is partially carried
+along by refractive media. Lorentz has given to this theory a
+more satisfactory form. For him, the ether is at rest, only the
+electrons are in motion; in the void, where it is only a question
+of the ether, in the air, where this is almost the case, the carrying
+along is null or almost null; in refractive media, where perturbation
+is produced at the same time by vibrations of the ether and
+those of electrons put in swing by the agitation of the ether,
+the undulations are <i>partially</i> carried along.</p>
+
+<p>To decide between the two hypotheses, we have Fizeau's experiment,
+comparing by measurements of the fringes of interference,
+the velocity of light in air at rest or in motion. These experiments
+have confirmed Fresnel's hypothesis of partial carrying
+along. They have been repeated with the same result by Michelson.
+<i>The theory of Hertz must therefore be rejected.</i></p>
+
+
+<h4>II</h4>
+
+<h4><i>The Principle of Relativity</i></h4>
+
+<p>But if the ether is not carried along by the motion of the
+earth, is it possible to show, by means of optical phenomena, the
+absolute velocity of the earth, or rather its velocity with respect
+to the unmoving ether? Experiment has answered negatively,
+and yet the experimental procedures have been varied in all
+possible ways. Whatever be the means employed there will
+never be disclosed anything but relative velocities; I mean the
+<span class='pagenum'><a name="Page_499" id="Page_499">[Pg 499]</a></span>
+velocities of certain material bodies with reference to other
+material bodies. In fact, if the source of light and the apparatus
+of observation are on the earth and participate in its
+motion, the experimental results have always been the same,
+whatever be the orientation of the apparatus with reference to
+the orbital motion of the earth. If astronomic aberration
+happens, it is because the source, a star, is in motion with
+reference to the observer.</p>
+
+<p>The hypotheses so far made perfectly account for this general
+result, <i>if we neglect very small quantities of the order of the
+square of the aberration</i>. The explanation rests upon the notion
+of <i>local time</i>, introduced by Lorentz, which I shall try to make
+clear. Suppose two observers, placed one at <i>A</i>, the other at <i>B</i>,
+and wishing to set their watches by means of optical signals.
+They agree that <i>B</i> shall send a signal to <i>A</i> when his watch marks
+an hour determined upon, and <i>A</i> is to put his watch to that
+hour the moment he sees the signal. If this alone were done,
+there would be a systematic error, because as the light takes a
+certain time <i>t</i> to go from <i>B</i> to <i>A</i>, <i>A</i>'s watch would be behind
+<i>B</i>'s the time <i>t</i>. This error is easily corrected. It suffices to cross
+the signals. <i>A</i> in turn must signal <i>B</i>, and, after this new adjustment,
+<i>B</i>'s watch will be behind <i>A</i>'s the time <i>t</i>. Then it will be
+sufficient to take the arithmetic mean of the two adjustments.</p>
+
+<p>But this way of doing supposes that light takes the same time
+to go from <i>A</i> to <i>B</i> as to return from <i>B</i> to <i>A</i>. That is true if
+the observers are motionless; it is no longer so if they are carried
+along in a common translation, since then <i>A</i>, for example, will
+go to meet the light coming from <i>B</i>, while <i>B</i> will flee before the
+light coming from <i>A</i>. If therefore the observers are borne along
+in a common translation and if they do not suspect it, their
+adjustment will be defective; their watches will not indicate
+the same time; each will show the <i>local time</i> belonging to the
+point where it is.</p>
+
+<p>The two observers will have no way of perceiving this, if the
+unmoving ether can transmit to them only luminous signals all
+of the same velocity, and if the other signals they might send
+are transmitted by media carried along with them in their translation.
+The phenomenon each observes will be too soon or too<span class='pagenum'><a name="Page_500" id="Page_500">[Pg 500]</a></span>
+late; it would be seen at the same instant only if the translation
+did not exist; but as it will be observed with a watch that is
+wrong, this will not be perceived and the appearances will not
+be altered.</p>
+
+<p>It results from this that the compensation is easy to explain
+so long as we neglect the square of the aberration, and for a
+long time the experiments were not sufficiently precise to warrant
+taking account of it. But the day came when Michelson imagined
+a much more delicate procedure: he made rays interfere which
+had traversed different courses, after being reflected by mirrors;
+each of the paths approximating a meter and the fringes of
+interference permitting the recognition of a fraction of a thousandth
+of a millimeter, the square of the aberration could no
+longer be neglected, and <i>yet the results were still negative</i>.
+Therefore the theory required to be completed, and it has been
+by the <i>Lorentz-Fitzgerald hypothesis</i>.</p>
+
+<p>These two physicists suppose that all bodies carried along in a
+translation undergo a contraction in the sense of this translation,
+while their dimensions perpendicular to this translation remain
+unchanged. <i>This contraction is the same for all bodies</i>; moreover,
+it is very slight, about one two-hundred-millionth for a
+velocity such as that of the earth. Furthermore our measuring
+instruments could not disclose it, even if they were much more
+precise; our measuring rods in fact undergo the same contraction
+as the objects to be measured. If the meter exactly fits when
+applied to a body, if we point the body and consequently the
+meter in the sense of the motion of the earth, it will not cease
+to exactly fit in another orientation, and that although the
+body and the meter have changed in length as well as orientation,
+and precisely because the change is the same for one as
+for the other. But it is quite different if we measure a length,
+not now with a meter, but by the time taken by light to pass along
+it, and this is just what Michelson has done.</p>
+
+<p>A body, spherical when at rest, will take thus the form of a
+flattened ellipsoid of revolution when in motion; but the observer
+will always think it spherical, since he himself has undergone
+an analogous deformation, as also all the objects serving as points
+of reference. On the contrary, the surfaces of the waves of<span class='pagenum'><a name="Page_501" id="Page_501">[Pg 501]</a></span>
+light, remaining rigorously spherical, will seem to him elongated
+ellipsoids.</p>
+
+<p>What happens then? Suppose an observer and a source of
+light carried along together in the translation: the wave surfaces
+emanating from the source will be spheres having as centers the
+successive positions of the source; the distance from this center
+to the actual position of the source will be proportional to the
+time elapsed after the emission, that is to say to the radius of the
+sphere. All these spheres are therefore homothetic one to the
+other, with relation to the actual position <i>S</i> of the source. But,
+for our observer, because of the contraction, all these spheres
+will seem elongated ellipsoids, and all these ellipsoids will moreover
+be homothetic, with reference to the point <i>S</i>; the excentricity
+of all these ellipsoids is the same and depends solely upon
+the velocity of the earth. <i>We shall so select the law of contraction
+that the point S may be at the focus of the meridian section
+of the ellipsoid.</i></p>
+
+<p>This time the compensation is <i>rigorous</i>, and this it is which
+explains Michelson's experiment.</p>
+
+<p>I have said above that, according to the ordinary theories,
+observations of the astronomic aberration would give us the
+absolute velocity of the earth, if our instruments were a thousand
+times more precise. I must modify this statement. Yes, the
+observed angles would be modified by the effect of this absolute
+velocity, but the graduated circles we use to measure the angles
+would be deformed by the translation: they would become
+ellipses; thence would result an error in regard to the angle
+measured, and <i>this second error would exactly compensate the
+first</i>.</p>
+
+<p>This Lorentz-Fitzgerald hypothesis seems at first very extraordinary;
+all we can say for the moment, in its favor, is that
+it is only the immediate translation of Michelson's experimental
+result, if we <i>define</i> lengths by the time taken by light to run
+along them.</p>
+
+<p>However that may be, it is impossible to escape the impression
+that the principle of relativity is a general law of nature,
+that one will never be able by any imaginable means to show
+any but relative velocities, and I mean by that not only the<span class='pagenum'><a name="Page_502" id="Page_502">[Pg 502]</a></span>
+velocities of bodies with reference to the ether, but the velocities
+of bodies with regard to one another. Too many different experiments
+have given concordant results for us not to feel tempted
+to attribute to this principle of relativity a value comparable to
+that, for example, of the principle of equivalence. In any case,
+it is proper to see to what consequences this way of looking at
+things would lead us and then to submit these consequences to
+the control of experiment.</p>
+
+
+<h4>III</h4>
+
+<h4><i>The Principle of Reaction</i></h4>
+
+<p>Let us see what the principle of the equality of action and
+reaction becomes in the theory of Lorentz. Consider an electron
+<i>A</i> which for any cause begins to move; it produces a perturbation
+in the ether; at the end of a certain time, this perturbation
+reaches another electron <i>B</i>, which will be disturbed from its position
+of equilibrium. In these conditions there can not be equality
+between action and reaction, at least if we do not consider the
+ether, but only the electrons, <i>which alone are observable</i>, since
+our matter is made of electrons.</p>
+
+<p>In fact it is the electron <i>A</i> which has disturbed the electron
+<i>B</i>; even in case the electron <i>B</i> should react upon <i>A</i>, this reaction
+could be equal to the action, but in no case simultaneous, since
+the electron <i>B</i> can begin to move only after a certain time,
+necessary for the propagation. Submitting the problem to a
+more exact calculation, we reach the following result: Suppose
+a Hertz discharger placed at the focus of a parabolic mirror to
+which it is mechanically attached; this discharger emits electromagnetic
+waves, and the mirror reflects all these waves in the
+same direction; the discharger therefore will radiate energy in a
+determinate direction. Well, the calculation shows that <i>the discharger
+recoils</i> like a cannon which has shot out a projectile.
+In the case of the cannon, the recoil is the natural result of the
+equality of action and reaction. The cannon recoils because the
+projectile upon which it has acted reacts upon it. But here it
+is no longer the same. What has been sent out is no longer a
+material projectile: it is energy, and energy has no mass: it has<span class='pagenum'><a name="Page_503" id="Page_503">[Pg 503]</a></span>
+no counterpart. And, in place of a discharger, we could have
+considered just simply a lamp with a reflector concentrating its
+rays in a single direction.</p>
+
+<p>It is true that, if the energy sent out from the discharger or
+from the lamp meets a material object, this object receives a
+mechanical push as if it had been hit by a real projectile, and
+this push will be equal to the recoil of the discharger and of
+the lamp, if no energy has been lost on the way and if the object
+absorbs the whole of the energy. Therefore one is tempted to
+say that there still is compensation between the action and the
+reaction. But this compensation, even should it be complete,
+is always belated. It never happens if the light, after leaving
+its source, wanders through interstellar spaces without ever meeting
+a material body; it is incomplete, if the body it strikes is not
+perfectly absorbent.</p>
+
+<p>Are these mechanical actions too small to be measured, or are
+they accessible to experiment? These actions are nothing other
+than those due to the <i>Maxwell-Bartholi</i> pressures; Maxwell had
+predicted these pressures from calculations relative to electrostatics
+and magnetism; Bartholi reached the same result by
+thermodynamic considerations.</p>
+
+<p>This is how the <i>tails of comets</i> are explained. Little particles
+detach themselves from the nucleus of the comet; they are struck
+by the light of the sun, which pushes them back as would a rain
+of projectiles coming from the sun. The mass of these particles
+is so little that this repulsion sweeps it away against the Newtonian
+attraction; so in moving away from the sun they form
+the tails.</p>
+
+<p>The direct experimental verification was not easy to obtain.
+The first endeavor led to the construction of the <i>radiometer</i>. But
+this instrument <i>turns backward</i>, in the sense opposite to the theoretic
+sense, and the explanation of its rotation, since discovered,
+is wholly different. At last success came, by making the vacuum
+more complete, on the one hand, and on the other by not blackening
+one of the faces of the paddles and directing a pencil of
+luminous rays upon one of the faces. The radiometric effects and
+the other disturbing causes are eliminated by a series of pains-taking
+precautions, and one obtains a deviation which is very<span class='pagenum'><a name="Page_504" id="Page_504">[Pg 504]</a></span>
+minute, but which is, it would seem, in conformity with the
+theory.</p>
+
+<p>The same effects of the Maxwell-Bartholi pressure are forecast
+likewise by the theory of Hertz of which we have before
+spoken, and by that of Lorentz. But there is a difference. Suppose
+that the energy, under the form of light, for example, proceeds
+from a luminous source to any body through a transparent
+medium. The Maxwell-Bartholi pressure will act, not alone
+upon the source at the departure, and on the body lit up at the
+arrival, but upon the matter of the transparent medium which it
+traverses. At the moment when the luminous wave reaches a
+new region of this medium, this pressure will push forward the
+matter there distributed and will put it back when the wave
+leaves this region. So that the recoil of the source has for
+counterpart the forward movement of the transparent matter
+which is in contact with this source; a little later, the recoil of
+this same matter has for counterpart the forward movement of
+the transparent matter which lies a little further on, and so on.</p>
+
+<p>Only, is the compensation perfect? Is the action of the Maxwell-Bartholi
+pressure upon the matter of the transparent
+medium equal to its reaction upon the source, and that whatever
+be this matter? Or is this action by so much the less as the
+medium is less refractive and more rarefied, becoming null in
+the void?</p>
+
+<p>If we admit the theory of Hertz, who regards matter as
+mechanically bound to the ether, so that the ether may be entirely
+carried along by matter, it would be necessary to answer yes to
+the first question and no to the second.</p>
+
+<p>There would then be perfect compensation, as required by the
+principle of the equality of action and reaction, even in the least
+refractive media, even in the air, even in the interplanetary
+void, where it would suffice to suppose a residue of matter, however
+subtile. If on the contrary we admit the theory of Lorentz,
+the compensation, always imperfect, is insensible in the air and
+becomes null in the void.</p>
+
+<p>But we have seen above that Fizeau's experiment does not
+permit of our retaining the theory of Hertz; it is necessary<span class='pagenum'><a name="Page_505" id="Page_505">[Pg 505]</a></span>
+therefore to adopt the theory of Lorentz, and consequently <i>to renounce
+the principle of reaction</i>.</p>
+
+
+<h4>IV</h4>
+
+<h4><i>Consequences of the Principle of Relativity</i></h4>
+
+<p>We have seen above the reasons which impel us to regard the
+principle of relativity as a general law of nature. Let us see
+to what consequences this principle would lead, should it be
+regarded as finally demonstrated.</p>
+
+<p>First, it obliges us to generalize the hypothesis of Lorentz and
+Fitzgerald on the contraction of all bodies in the sense of the
+translation. In particular, we must extend this hypothesis to
+the electrons themselves. Abraham considered these electrons as
+spherical and indeformable; it will be necessary for us to admit
+that these electrons, spherical when in repose, undergo the
+Lorentz contraction when in motion and take then the form of
+flattened ellipsoids.</p>
+
+<p>This deformation of the electrons will influence their mechanical
+properties. In fact I have said that the displacement of
+these charged electrons is a veritable current of convection and
+that their apparent inertia is due to the self-induction of this
+current: exclusively as concerns the negative electrons; exclusively
+or not, we do not yet know, for the positive electrons.
+Well, the deformation of the electrons, a deformation which
+depends upon their velocity, will modify the distribution of the
+electricity upon their surface, consequently the intensity of the
+convection current they produce, consequently the laws according
+to which the self-induction of this current will vary as a
+function of the velocity.</p>
+
+<p>At this price, the compensation will be perfect and will conform
+to the requirements of the principle of relativity, but only
+upon two conditions:</p>
+
+<p>1&ordm; That the positive electrons have no real mass, but only a
+fictitious electromagnetic mass; or at least that their real mass,
+if it exists, is not constant and varies with the velocity according
+to the same laws as their fictitious mass;</p>
+
+<p>2&ordm; That all forces are of electromagnetic origin, or at least<span class='pagenum'><a name="Page_506" id="Page_506">[Pg 506]</a></span>
+that they vary with the velocity according to the same laws as
+the forces of electromagnetic origin.</p>
+
+<p>It still is Lorentz who has made this remarkable synthesis;
+stop a moment and see what follows therefrom. First, there is
+no more matter, since the positive electrons no longer have real
+mass, or at least no constant real mass. The present principles
+of our mechanics, founded upon the constancy of mass, must
+therefore be modified. Again, an electromagnetic explanation
+must be sought of all the known forces, in particular of gravitation,
+or at least the law of gravitation must be so modified that
+this force is altered by velocity in the same way as the electromagnetic
+forces. We shall return to this point.</p>
+
+<p>All that appears, at first sight, a little artificial. In particular,
+this deformation of electrons seems quite hypothetical. But
+the thing may be presented otherwise, so as to avoid putting this
+hypothesis of deformation at the foundation of the reasoning.
+Consider the electrons as material points and ask how their mass
+should vary as function of the velocity not to contravene the
+principle of relativity. Or, still better, ask what should be their
+acceleration under the influence of an electric or magnetic field,
+that this principle be not violated and that we come back to the
+ordinary laws when we suppose the velocity very slight. We
+shall find that the variations of this mass, or of these accelerations,
+must be <i>as if</i> the electron underwent the Lorentz
+deformation.</p>
+
+
+<h4>V</h4>
+
+<h4><i>Kaufmann's Experiment</i></h4>
+
+<p>We have before us, then, two theories: one where the electrons
+are indeformable, this is that of Abraham; the other where they
+undergo the Lorentz deformation. In both cases, their mass
+increases with the velocity, becoming infinite when this velocity
+becomes equal to that of light; but the law of the variation is
+not the same. The method employed by Kaufmann to bring to
+light the law of variation of the mass seems therefore to give us
+an experimental means of deciding between the two theories.</p>
+
+<p>Unhappily, his first experiments were not sufficiently precise
+for that; so he decided to repeat them with more precautions, and<span class='pagenum'><a name="Page_507" id="Page_507">[Pg 507]</a></span>
+measuring with great care the intensity of the fields. Under
+their new form <i>they are in favor of the theory of Abraham</i>.
+Then the principle of relativity would not have the rigorous
+value we were tempted to attribute to it; there would no longer
+be reason for believing the positive electrons denuded of real
+mass like the negative electrons. However, before definitely
+adopting this conclusion, a little reflection is necessary. The
+question is of such importance that it is to be wished Kaufmann's
+experiment were repeated by another experimenter.<a name="FNanchor_17_17" id="FNanchor_17_17"></a><a href="#Footnote_17_17" class="fnanchor">[17]</a> Unhappily,
+this experiment is very delicate and could be carried out successfully
+only by a physicist of the same ability as Kaufmann.
+All precautions have been properly taken and we hardly see
+what objection could be made.</p>
+
+<p>There is one point however to which I wish to draw attention:
+that is to the measurement of the electrostatic field, a measurement
+upon which all depends. This field was produced between
+the two armatures of a condenser; and, between these armatures,
+there was to be made an extremely perfect vacuum, in order to
+obtain a complete isolation. Then the difference of potential of
+the two armatures was measured, and the field obtained by dividing
+this difference by the distance apart of the armatures. That
+supposes the field uniform; is this certain? Might there not be
+an abrupt fall of potential in the neighborhood of one of the
+armatures, of the negative armature, for example? There may
+be a difference of potential at the meeting of the metal and the
+vacuum, and it may be that this difference is not the same on the
+positive side and on the negative side; what would lead me to
+think so is the electric valve effects between mercury and vacuum.
+However slight the probability that it is so, it seems that it
+should be considered.</p>
+
+
+<h4>VI</h4>
+
+<h4><i>The Principle of Inertia</i></h4>
+
+<p>In the new dynamics, the principle of inertia is still true, that
+is to say that an <i>isolated</i> electron will have a straight and uniform
+motion. At least this is generally assumed; however,<span class='pagenum'><a name="Page_508" id="Page_508">[Pg 508]</a></span>
+Lindemann has made objections to this view; I do not wish to
+take part in this discussion, which I can not here expound
+because of its too difficult character. In any case, slight modifications
+to the theory would suffice to shelter it from Lindemann's
+objections.</p>
+
+<p>We know that a body submerged in a fluid experiences, when
+in motion, considerable resistance, but this is because our fluids
+are viscous; in an ideal fluid, perfectly free from viscosity, the
+body would stir up behind it a liquid hill, a sort of wake; upon
+departure, a great effort would be necessary to put it in motion,
+since it would be necessary to move not only the body itself, but
+the liquid of its wake. But, the motion once acquired, it would
+perpetuate itself without resistance, since the body, in advancing,
+would simply carry with it the perturbation of the liquid,
+without the total vis viva of the liquid augmenting. Everything
+would happen therefore as if its inertia was augmented. An
+electron advancing in the ether would behave in the same way:
+around it, the ether would be stirred up, but this perturbation
+would accompany the body in its motion; so that, for an observer
+carried along with the electron, the electric and magnetic fields
+accompanying this electron would appear invariable, and would
+change only if the velocity of the electron varied. An effort
+would therefore be necessary to put the electron in motion, since
+it would be necessary to create the energy of these fields; on the
+contrary, once the movement acquired, no effort would be necessary
+to maintain it, since the created energy would only have to
+go along behind the electron as a wake. This energy, therefore,
+could only augment the inertia of the electron, as the agitation of
+the liquid augments that of the body submerged in a perfect
+fluid. And anyhow, the negative electrons at least have no other
+inertia except that.</p>
+
+<p>In the hypothesis of Lorentz, the vis viva, which is only the
+energy of the ether, is not proportional to <i>v</i><sup>2</sup>. Doubtless if <i>v</i> is
+very slight, the vis viva is sensibly proportional to <i>v</i><sup>2</sup>, the quantity
+of motion sensibly proportional to <i>v</i>, the two masses sensibly
+constant and equal to each other. But <i>when the velocity tends
+toward the velocity of light, the vis viva, the quantity of motion
+and the two masses increase beyond all limit</i>.<span class='pagenum'><a name="Page_509" id="Page_509">[Pg 509]</a></span></p>
+
+<p>In the hypothesis of Abraham, the expressions are a little
+more complicated; but what we have just said remains true in
+essentials.</p>
+
+<p>So the mass, the quantity of motion, the vis viva become
+infinite when the velocity is equal to that of light.</p>
+
+<p>Thence results that <i>no body can attain in any way a velocity
+beyond that of light</i>. And in fact, in proportion as its velocity
+increases, its mass increases, so that its inertia opposes to any
+new increase of velocity a greater and greater obstacle.</p>
+
+<p>A question then suggests itself: let us admit the principle of
+relativity; an observer in motion would not have any means of
+perceiving his own motion. If therefore no body in its absolute
+motion can exceed the velocity of light, but may approach it as
+nearly as you choose, it should be the same concerning its relative
+motion with reference to our observer. And then we might be
+tempted to reason as follows: The observer may attain a velocity
+of 200,000 kilometers; the body in its relative motion with reference
+to the observer may attain the same velocity; its absolute
+velocity will then be 400,000 kilometers, which is impossible,
+since this is beyond the velocity of light. This is only a seeming,
+which vanishes when account is taken of how Lorentz evaluates
+local time.</p>
+
+
+<h4>VII</h4>
+
+<h4><i>The Wave of Acceleration</i></h4>
+
+<p>When an electron is in motion, it produces a perturbation in
+the ether surrounding it; if its motion is straight and uniform,
+this perturbation reduces to the wake of which we have spoken
+in the preceding section. But it is no longer the same, if the
+motion be curvilinear or varied. The perturbation may then be
+regarded as the superposition of two others, to which Langevin
+has given the names <i>wave of velocity</i> and <i>wave of acceleration</i>.
+The wave of velocity is only the wave which happens in uniform
+motion.</p>
+
+<p>As to the wave of acceleration, this is a perturbation altogether
+analogous to light waves, which starts from the electron at the
+instant when it undergoes an acceleration, and which is then<span class='pagenum'><a name="Page_510" id="Page_510">[Pg 510]</a></span>
+propagated by successive spherical waves with the velocity of
+light. Whence follows: in a straight and uniform motion, the
+energy is wholly conserved; but, when there is an acceleration,
+there is loss of energy, which is dissipated under the form of
+luminous waves and goes out to infinity across the ether.</p>
+
+<p>However, the effects of this wave of acceleration, in particular
+the corresponding loss of energy, are in most cases negligible,
+that is to say not only in ordinary mechanics and in the motions
+of the heavenly bodies, but even in the radium rays, where the
+velocity is very great without the acceleration being so. We may
+then confine ourselves to applying the laws of mechanics, putting
+the force equal to the product of acceleration by mass, this mass,
+however, varying with the velocity according to the laws explained
+above. We then say the motion is <i>quasi-stationary</i>.</p>
+
+<p>It would not be the same in all cases where the acceleration
+is great, of which the chief are the following:</p>
+
+<p>1&ordm; In incandescent gases certain electrons take an oscillatory
+motion of very high frequency; the displacements are very small,
+the velocities are finite, and the accelerations very great; energy
+is then communicated to the ether, and this is why these gases
+radiate light of the same period as the oscillations of the electron;</p>
+
+<p>2&ordm; Inversely, when a gas receives light, these same electrons
+are put in swing with strong accelerations and they absorb
+light;</p>
+
+<p>3&ordm; In the Hertz discharger, the electrons which circulate in
+the metallic mass undergo, at the instant of the discharge, an
+abrupt acceleration and take then an oscillatory motion of high
+frequency. Thence results that a part of the energy radiates
+under the form of Hertzian waves;</p>
+
+<p>4&ordm; In an incandescent metal, the electrons enclosed in this
+metal are impelled with great velocity; upon reaching the surface
+of the metal, which they can not get through, they are reflected
+and thus undergo a considerable acceleration. This is why the
+metal emits light. The details of the laws of the emission of
+light by dark bodies are perfectly explained by this hypothesis;</p>
+
+<p>5&ordm; Finally when the cathode rays strike the anticathode, the
+negative electrons, constituting these rays, which are impelled
+with very great velocity, are abruptly arrested. Because of the<span class='pagenum'><a name="Page_511" id="Page_511">[Pg 511]</a></span>
+acceleration they thus undergo, they produce undulations in the
+ether. This, according to certain physicists, is the origin of the
+R&ouml;ntgen rays, which would only be light rays of very short
+wave-length.</p>
+
+
+
+<hr style="width: 65%;" />
+<p><span class='pagenum'><a name="Page_512" id="Page_512">[Pg 512]</a></span></p>
+<h3>CHAPTER III</h3>
+
+<h3><span class="smcap">The New Mechanics and Astronomy</span></h3>
+
+
+<h4>I</h4>
+
+<h4><i>Gravitation</i></h4>
+
+<p>Mass may be defined in two ways:</p>
+
+<p>1&ordm; By the quotient of the force by the acceleration; this is the
+true definition of the mass, which measures the inertia of the
+body.</p>
+
+<p>2&ordm; By the attraction the body exercises upon an exterior body,
+in virtue of Newton's law. We should therefore distinguish the
+mass coefficient of inertia and the mass coefficient of attraction.
+According to Newton's law, there is rigorous proportionality
+between these two coefficients. But that is demonstrated only
+for velocities to which the general principles of dynamics are
+applicable. Now, we have seen that the mass coefficient of inertia
+increases with the velocity; should we conclude that the mass
+coefficient of attraction increases likewise with the velocity and
+remains proportional to the coefficient of inertia, or, on the contrary,
+that this coefficient of attraction remains constant? This
+is a question we have no means of deciding.</p>
+
+<p>On the other hand, if the coefficient of attraction depends
+upon the velocity, since the velocities of two bodies which mutually
+attract are not in general the same, how will this coefficient
+depend upon these two velocities?</p>
+
+<p>Upon this subject we can only make hypotheses, but we are
+naturally led to investigate which of these hypotheses would be
+compatible with the principle of relativity. There are a great
+number of them; the only one of which I shall here speak is that
+of Lorentz, which I shall briefly expound.</p>
+
+<p>Consider first electrons at rest. Two electrons of the same sign
+repel each other and two electrons of contrary sign attract each
+other; in the ordinary theory, their mutual actions are proportional
+to their electric charges; if therefore we have four<span class='pagenum'><a name="Page_513" id="Page_513">[Pg 513]</a></span>
+electrons, two positive <i>A</i> and <i>A&acute;</i>, and two negative <i>B</i> and <i>B&acute;</i>, the
+charges of these four being the same in absolute value, the repulsion
+of <i>A</i> for <i>A&acute;</i> will be, at the same distance, equal to the repulsion
+of <i>B</i> for <i>B&acute;</i> and equal also to the attraction of <i>A</i> for <i>B&acute;</i>, or
+of <i>A&acute;</i> for <i>B</i>. If therefore <i>A</i> and <i>B</i> are very near each other, as
+also <i>A&acute;</i> and <i>B&acute;</i>, and we examine the action of the system <i>A</i> + <i>B</i>
+upon the system <i>A&acute;</i> + <i>B&acute;</i>, we shall have two repulsions and two
+attractions which will exactly compensate each other and the
+resulting action will be null.</p>
+
+<p>Now, material molecules should just be regarded as species of
+solar systems where circulate the electrons, some positive, some
+negative, and <i>in such a way that the algebraic sum of all the
+charges is null</i>. A material molecule is therefore wholly analogous
+to the system <i>A</i> + <i>B</i> of which we have spoken, so that the
+total electric action of two molecules one upon the other should
+be null.</p>
+
+<p>But experiment shows us that these molecules attract each
+other in consequence of Newtonian gravitation; and then we may
+make two hypotheses: we may suppose gravitation has no relation
+to the electrostatic attractions, that it is due to a cause
+entirely different, and is simply something additional; or else
+we may suppose the attractions are not proportional to the
+charges and that the attraction exercised by a charge +1 upon
+a charge &minus;1 is greater than the mutual repulsion of two +1
+charges, or two &minus;1 charges.</p>
+
+<p>In other words, the electric field produced by the positive
+electrons and that which the negative electrons produce might
+be superposed and yet remain distinct. The positive electrons
+would be more sensitive to the field produced by the negative
+electrons than to the field produced by the positive electrons;
+the contrary would be the case for the negative electrons. It is
+clear that this hypothesis somewhat complicates electrostatics,
+but that it brings back into it gravitation. This was, in sum,
+Franklin's hypothesis.</p>
+
+<p>What happens now if the electrons are in motion? The
+positive electrons will cause a perturbation in the ether and
+produce there an electric and magnetic field. The same will
+be the case for the negative electrons. The electrons, positive as<span class='pagenum'><a name="Page_514" id="Page_514">[Pg 514]</a></span>
+well as negative, undergo then a mechanical impulsion by the
+action of these different fields. In the ordinary theory, the
+electromagnetic field, due to the motion of the positive electrons,
+exercises, upon two electrons of contrary sign and of the same
+absolute charge, equal actions with contrary sign. We may then
+without inconvenience not distinguish the field due to the motion
+of the positive electrons and the field due to the motion of the
+negative electrons and consider only the algebraic sum of these
+two fields, that is to say the resulting field.</p>
+
+<p>In the new theory, on the contrary, the action upon the positive
+electrons of the electromagnetic field due to the positive
+electrons follows the ordinary laws; it is the same with the action
+upon the negative electrons of the field due to the negative electrons.
+Let us now consider the action of the field due to the
+positive electrons upon the negative electrons (or inversely); it
+will still follow the same laws, but <i>with a different coefficient</i>.
+Each electron is more sensitive to the field created by the electrons
+of contrary name than to the field created by the electrons
+of the same name.</p>
+
+<p>Such is the hypothesis of Lorentz, which reduces to Franklin's
+hypothesis for slight velocities; it will therefore explain, for
+these small velocities, Newton's law. Moreover, as gravitation
+goes back to forces of electrodynamic origin, the general theory
+of Lorentz will apply, and consequently the principle of relativity
+will not be violated.</p>
+
+<p>We see that Newton's law is no longer applicable to great
+velocities and that it must be modified, for bodies in motion,
+precisely in the same way as the laws of electrostatics for electricity
+in motion.</p>
+
+<p>We know that electromagnetic perturbations spread with the
+velocity of light. We may therefore be tempted to reject the
+preceding theory upon remembering that gravitation spreads,
+according to the calculations of Laplace, at least ten million
+times more quickly than light, and that consequently it can not
+be of electromagnetic origin. The result of Laplace is well
+known, but one is generally ignorant of its signification. Laplace
+supposed that, if the propagation of gravitation is not instantaneous,
+its velocity of spread combines with that of the body<span class='pagenum'><a name="Page_515" id="Page_515">[Pg 515]</a></span>
+attracted, as happens for light in the phenomenon of astronomic
+aberration, so that the effective force is not directed along
+the straight joining the two bodies, but makes with this straight
+a small angle. This is a very special hypothesis, not well justified,
+and, in any case, entirely different from that of Lorentz.
+Laplace's result proves nothing against the theory of Lorentz.</p>
+
+
+<h4>II</h4>
+
+<h4><i>Comparison with Astronomic Observations</i></h4>
+
+<p>Can the preceding theories be reconciled with astronomic
+observations?</p>
+
+<p>First of all, if we adopt them, the energy of the planetary
+motions will be constantly dissipated by the effect of the <i>wave
+of acceleration</i>. From this would result that the mean motions
+of the stars would constantly accelerate, as if these stars were
+moving in a resistant medium. But this effect is exceedingly
+slight, far too much so to be discerned by the most precise observations.
+The acceleration of the heavenly bodies is relatively
+slight, so that the effects of the wave of acceleration are negligible
+and the motion may be regarded as <i>quasi stationary</i>. It is
+true that the effects of the wave of acceleration constantly accumulate,
+but this accumulation itself is so slow that thousands
+of years of observation would be necessary for it to become
+sensible. Let us therefore make the calculation considering the
+motion as quasi-stationary, and that under the three following
+hypotheses:</p>
+
+<p>A. Admit the hypothesis of Abraham (electrons indeformable)
+and retain Newton's law in its usual form;</p>
+
+<p>B. Admit the hypothesis of Lorentz about the deformation of
+electrons and retain the usual Newton's law;</p>
+
+<p>C. Admit the hypothesis of Lorentz about electrons and modify
+Newton's law as we have done in the preceding paragraph, so as
+to render it compatible with the principle of relativity.</p>
+
+<p>It is in the motion of Mercury that the effect will be most
+sensible, since this planet has the greatest velocity. Tisserand
+formerly made an analogous calculation, admitting Weber's law;
+I recall that Weber had sought to explain at the same time the<span class='pagenum'><a name="Page_516" id="Page_516">[Pg 516]</a></span>
+electrostatic and electrodynamic phenomena in supposing that
+electrons (whose name was not yet invented) exercise, one upon
+another, attractions and repulsions directed along the straight
+joining them, and depending not only upon their distances, but
+upon the first and second derivatives of these distances, consequently
+upon their velocities and their accelerations. This law
+of Weber, different enough from those which to-day tend to prevail,
+none the less presents a certain analogy with them.</p>
+
+<p>Tisserand found that, if the Newtonian attraction conformed to
+Weber's law there resulted, for Mercury's perihelion, secular
+variation of 14&acute;&acute;, <i>of the same sense as that which has been
+observed and could not be explained</i>, but smaller, since this
+is 38&acute;&acute;.</p>
+
+<p>Let us recur to the hypotheses A, B and C, and study first
+the motion of a planet attracted by a fixed center. The hypotheses
+B and C are no longer distinguished, since, if the attracting point
+is fixed, the field it produces is a purely electrostatic field, where
+the attraction varies inversely as the square of the distance, in
+conformity with Coulomb's electrostatic law, identical with that
+of Newton.</p>
+
+<p>The vis viva equation holds good, taking for vis viva the new
+definition; in the same way, the equation of areas is replaced by
+another equivalent to it; the moment of the quantity of motion
+is a constant, but the quantity of motion must be defined as in
+the new dynamics.</p>
+
+<p>The only sensible effect will be a secular motion of the perihelion.
+With the theory of Lorentz, we shall find, for this motion,
+half of what Weber's law would give; with the theory of Abraham,
+two fifths.</p>
+
+<p>If now we suppose two moving bodies gravitating around their
+common center of gravity, the effects are very little different,
+though the calculations may be a little more complicated. The
+motion of Mercury's perihelion would therefore be 7&acute;&acute; in the
+theory of Lorentz and 5&acute;&acute;.6 in that of Abraham.</p>
+
+<p>The effect moreover is proportional to <i>n</i><sup>3</sup><i>a</i><sup>2</sup>, where <i>n</i> is the star's
+mean motion and a the radius of its orbit. For the planets, in
+virtue of Kepler's law, the effect varies then inversely as &#8730;<i>a</i><sup>5</sup>;
+it is therefore insensible, save for Mercury.<span class='pagenum'><a name="Page_517" id="Page_517">[Pg 517]</a></span></p>
+
+<p>It is likewise insensible for the moon though <i>n</i> is great, because
+<i>a</i> is extremely small; in sum, it is five times less for Venus, and
+six hundred times less for the moon than for Mercury. We may
+add that as to Venus and the earth, the motion of the perihelion
+(for the same angular velocity of this motion) would be much
+more difficult to discern by astronomic observations, because the
+excentricity of their orbits is much less than for Mercury.</p>
+
+<p>To sum up, <i>the only sensible effect upon astronomic observations
+would be a motion of Mercury's perihelion, in the same
+sense as that which has been observed without being explained,
+but notably slighter</i>.</p>
+
+<p>That can not be regarded as an argument in favor of the new
+dynamics, since it will always be necessary to seek another explanation
+for the greater part of Mercury's anomaly; but still less
+can it be regarded as an argument against it.</p>
+
+
+<h4>III</h4>
+
+<h4><i>The Theory of Lesage</i></h4>
+
+<p>It is interesting to compare these considerations with a theory
+long since proposed to explain universal gravitation.</p>
+
+<p>Suppose that, in the interplanetary spaces, circulate in every
+direction, with high velocities, very tenuous corpuscles. A body
+isolated in space will not be affected, apparently, by the impacts
+of these corpuscles, since these impacts are equally distributed
+in all directions. But if two bodies <i>A</i> and <i>B</i> are present, the
+body <i>B</i> will play the r&ocirc;le of screen and will intercept part of the
+corpuscles which, without it, would have struck <i>A</i>. Then, the
+impacts received by <i>A</i> in the direction opposite that from <i>B</i> will
+no longer have a counterpart, or will now be only partially compensated,
+and this will push <i>A</i> toward <i>B</i>.</p>
+
+<p>Such is the theory of Lesage; and we shall discuss it, taking
+first the view-point of ordinary mechanics.</p>
+
+<p>First, how should the impacts postulated by this theory take
+place; is it according to the laws of perfectly elastic bodies, or
+according to those of bodies devoid of elasticity, or according
+to an intermediate law? The corpuscles of Lesage can not act
+as perfectly elastic bodies; otherwise the effect would be null,<span class='pagenum'><a name="Page_518" id="Page_518">[Pg 518]</a></span>
+since the corpuscles intercepted by the body <i>B</i> would be replaced
+by others which would have rebounded from <i>B</i>, and calculation
+proves that the compensation would be perfect. It is necessary
+then that the impact make the corpuscles lose energy, and this
+energy should appear under the form of heat. But how much
+heat would thus be produced? Note that attraction passes
+through bodies; it is necessary therefore to represent to ourselves
+the earth, for example, not as a solid screen, but as formed of
+a very great number of very small spherical molecules, which
+play individually the r&ocirc;le of little screens, but between which the
+corpuscles of Lesage may freely circulate. So, not only the earth
+is not a solid screen, but it is not even a cullender, since the voids
+occupy much more space than the plenums. To realize this,
+recall that Laplace has demonstrated that attraction, in traversing
+the earth, is weakened at most by one ten-millionth part, and
+his proof is perfectly satisfactory: in fact, if attraction were
+absorbed by the body it traverses, it would no longer be proportional
+to the masses; it would be <i>relatively</i> weaker for great
+bodies than for small, since it would have a greater thickness to
+traverse. The attraction of the sun for the earth would therefore
+be <i>relatively</i> weaker than that of the sun for the moon, and
+thence would result, in the motion of the moon, a very sensible
+inequality. We should therefore conclude, if we adopt the theory
+of Lesage, that the total surface of the spherical molecules which
+compose the earth is at most the ten-millionth part of the total
+surface of the earth.</p>
+
+<p>Darwin has proved that the theory of Lesage only leads exactly
+to Newton's law when we postulate particles entirely devoid of
+elasticity. The attraction exerted by the earth on a mass 1 at a
+distance 1 will then be proportional, at the same time, to the
+total surface <i>S</i> of the spherical molecules composing it, to the
+velocity <i>v</i> of the corpuscles, to the square root of the density &#961; of
+the medium formed by the corpuscles. The heat produced will
+be proportional to <i>S</i>, to the density &#961;, and to the cube of the
+velocity <i>v</i>.</p>
+
+<p>But it is necessary to take account of the resistance experienced
+by a body moving in such a medium; it can not move, in fact,
+without going against certain impacts, in fleeing, on the contrary,<span class='pagenum'><a name="Page_519" id="Page_519">[Pg 519]</a></span>
+before those coming in the opposite direction, so that the compensation
+realized in the state of rest can no longer subsist. The
+calculated resistance is proportional to <i>S</i>, to &#961; and to <i>v</i>; now, we
+know that the heavenly bodies move as if they experienced no
+resistance, and the precision of observations permits us to fix a
+limit to the resistance of the medium.</p>
+
+<p>This resistance varying as <i>S</i>&#961;<i>v</i>, while the attraction varies as
+<i>S</i>&#8730;(&#961;<i>v</i>), we see that the ratio of the resistance to the square of the
+attraction is inversely as the product <i>Sv</i>.</p>
+
+<p>We have therefore a lower limit of the product <i>Sv</i>. We have
+already an upper limit of <i>S</i> (by the absorption of attraction by
+the body it traverses); we have therefore a lower limit of the
+velocity <i>v</i>, which must be at least 24&middot;10<sup>17</sup> times that of light.</p>
+
+<p>From this we are able to deduce &#961; and the quantity of heat
+produced; this quantity would suffice to raise the temperature
+10<sup>26</sup> degrees a second; the earth would receive in a given time
+10<sup>20</sup> times more heat than the sun emits in the same time; I am
+not speaking of the heat the sun sends to the earth, but of that
+it radiates in all directions.</p>
+
+<p>It is evident the earth could not long stand such a r&eacute;gime.</p>
+
+<p>We should not be led to results less fantastic if, contrary to
+Darwin's views, we endowed the corpuscles of Lesage with an
+elasticity imperfect without being null. In truth, the vis viva of
+these corpuscles would not be entirely converted into heat, but
+the attraction produced would likewise be less, so that it would be
+only the part of this vis viva converted into heat, which would
+contribute to produce the attraction and that would come to the
+same thing; a judicious employment of the theorem of the viriel
+would enable us to account for this.</p>
+
+<p>The theory of Lesage may be transformed; suppress the corpuscles
+and imagine the ether overrun in all senses by luminous
+waves coming from all points of space. When a material object
+receives a luminous wave, this wave exercises upon it a mechanical
+action due to the Maxwell-Bartholi pressure, just as if it
+had received the impact of a material projectile. The waves in
+question could therefore play the r&ocirc;le of the corpuscles of Lesage.
+This is what is supposed, for example, by M. Tommasina.</p>
+
+<p>The difficulties are not removed for all that; the velocity of<span class='pagenum'><a name="Page_520" id="Page_520">[Pg 520]</a></span>
+propagation can be only that of light, and we are thus led, for
+the resistance of the medium, to an inadmissible figure. Besides,
+if the light is all reflected, the effect is null, just as in the
+hypothesis of the perfectly elastic corpuscles.</p>
+
+<p>That there should be attraction, it is necessary that the light
+be partially absorbed; but then there is production of heat. The
+calculations do not differ essentially from those made in the ordinary
+theory of Lesage, and the result retains the same fantastic
+character.</p>
+
+<p>On the other hand, attraction is not absorbed by the body it
+traverses, or hardly at all; it is not so with the light we know.
+Light which would produce the Newtonian attraction would have
+to be considerably different from ordinary light and be, for
+example, of very short wave length. This does not count that,
+if our eyes were sensible of this light, the whole heavens should
+appear to us much more brilliant than the sun, so that the sun
+would seem to us to stand out in black, otherwise the sun would
+repel us instead of attracting us. For all these reasons, light
+which would permit of the explanation of attraction would be
+much more like R&ouml;ntgen rays than like ordinary light.</p>
+
+<p>And besides, the X-rays would not suffice; however penetrating
+they may seem to us, they could not pass through the whole
+earth; it would be necessary therefore to imagine X&acute;-rays much
+more penetrating than the ordinary X-rays. Moreover a part of
+the energy of these X&acute;-rays would have to be destroyed, otherwise
+there would be no attraction. If you do not wish it transformed
+into heat, which would lead to an enormous heat production,
+you must suppose it radiated in every direction under the
+form of secondary rays, which might be called X&acute;&acute; and which
+would have to be much more penetrating still than the X&acute;-rays,
+otherwise they would in their turn derange the phenomena of
+attraction.</p>
+
+<p>Such are the complicated hypotheses to which we are led when
+we try to give life to the theory of Lesage.</p>
+
+<p>But all we have said presupposes the ordinary laws of
+mechanics.</p>
+
+<p>Will things go better if we admit the new dynamics? And
+first, can we conserve the principles of relativity? Let us give at<span class='pagenum'><a name="Page_521" id="Page_521">[Pg 521]</a></span>
+first to the theory of Lesage its primitive form, and suppose space
+ploughed by material corpuscles; if these corpuscles were perfectly
+elastic, the laws of their impact would conform to this
+principle of relativity, but we know that then their effect would
+be null. We must therefore suppose these corpuscles are not
+elastic, and then it is difficult to imagine a law of impact compatible
+with the principle of relativity. Besides, we should still
+find a production of considerable heat, and yet a very sensible
+resistance of the medium.</p>
+
+<p>If we suppress these corpuscles and revert to the hypothesis of
+the Maxwell-Bartholi pressure, the difficulties will not be less.
+This is what Lorentz himself has attempted in his Memoir to the
+Amsterdam Academy of Sciences of April 25, 1900.</p>
+
+<p>Consider a system of electrons immersed in an ether permeated
+in every sense by luminous waves; one of these electrons,
+struck by one of these waves, begins to vibrate; its vibration will
+be synchronous with that of light; but it may have a difference of
+phase, if the electron absorbs a part of the incident energy. In
+fact, if it absorbs energy, this is because the vibration of the
+ether <i>impels</i> the electron; the electron must therefore be slower
+than the ether. An electron in motion is analogous to a convection
+current; therefore every magnetic field, in particular that
+due to the luminous perturbation itself, must exert a mechanical
+action upon this electron. This action is very slight; moreover,
+it changes sign in the current of the period; nevertheless, the
+mean action is not null if there is a difference of phase between
+the vibrations of the electron and those of the ether. The mean
+action is proportional to this difference, consequently to the
+energy absorbed by the electron. I can not here enter into the
+detail of the calculations; suffice it to say only that the final
+result is an attraction of any two electrons, varying inversely as
+the square of the distance and proportional to the energy
+absorbed by the two electrons.</p>
+
+<p>Therefore there can not be attraction without absorption of
+light and, consequently, without production of heat, and this it
+is which determined Lorentz to abandon this theory, which, at
+bottom, does not differ from that of Lesage-Maxwell-Bartholi.
+He would have been much more dismayed still if he had pushed<span class='pagenum'><a name="Page_522" id="Page_522">[Pg 522]</a></span>
+the calculation to the end. He would have found that the temperature
+of the earth would have to increase 10<sup>12</sup> degrees a second.</p>
+
+
+<h4>IV</h4>
+
+<h4><i>Conclusions</i></h4>
+
+<p>I have striven to give in few words an idea as complete as
+possible of these new doctrines; I have sought to explain how
+they took birth; otherwise the reader would have had ground
+to be frightened by their boldness. The new theories are not
+yet demonstrated; far from it; only they rest upon an aggregate
+of probabilities sufficiently weighty for us not to have the right
+to treat them with disregard.</p>
+
+<p>New experiments will doubtless teach us what we should
+finally think of them. The knotty point of the question lies in
+Kaufmann's experiment and those that may be undertaken to
+verify it.</p>
+
+<p>In conclusion, permit me a word of warning. Suppose that,
+after some years, these theories undergo new tests and triumph;
+then our secondary education will incur a great danger; certain
+professors will doubtless wish to make a place for the new
+theories.</p>
+
+<p>Novelties are so attractive, and it is so hard not to seem
+highly advanced! At least there will be the wish to open vistas
+to the pupils and, before teaching them the ordinary mechanics,
+to let them know it has had its day and was at best good enough
+for that old dolt Laplace. And then they will not form the habit
+of the ordinary mechanics.</p>
+
+<p>Is it well to let them know this is only approximative? Yes;
+but later, when it has penetrated to their very marrow, when
+they shall have taken the bent of thinking only through it, when
+there shall no longer be risk of their unlearning it, then one may,
+without inconvenience, show them its limits.</p>
+
+<p>It is with the ordinary mechanics that they must live; this
+alone will they ever have to apply. Whatever be the progress of
+automobilism, our vehicles will never attain speeds where it is
+not true. The other is only a luxury, and we should think of
+the luxury only when there is no longer any risk of harming
+the necessary.</p>
+
+
+
+<hr style="width: 65%;" />
+<p><span class='pagenum'><a name="Page_523" id="Page_523">[Pg 523]</a></span></p>
+<h2><b>BOOK IV<br />
+
+<br />
+<small>ASTRONOMIC SCIENCE</small></b></h2>
+
+
+
+<hr style="width: 65%;" />
+<h3>CHAPTER I</h3>
+
+<h3><span class="smcap">The Milky Way and the Theory of Gases</span></h3>
+
+
+<p>The considerations to be here developed have scarcely as yet
+drawn the attention of astronomers; there is hardly anything to
+cite except an ingenious idea of Lord Kelvin's, which has opened
+a new field of research, but still waits to be followed out. Nor
+have I original results to impart, and all I can do is to give an
+idea of the problems presented, but which no one hitherto has
+undertaken to solve. Every one knows how a large number of
+modern physicists represent the constitution of gases; gases are
+formed of an innumerable multitude of molecules which, at high
+speeds, cross and crisscross in every direction. These molecules
+probably act at a distance one upon another, but this action
+decreases very rapidly with distance, so that their trajectories
+remain sensibly straight; they cease to be so only when two
+molecules happen to pass very near to each other; in this case,
+their mutual attraction or repulsion makes them deviate to
+right or left. This is what is sometimes called an impact; but
+the word <i>impact</i> is not to be understood in its usual sense; it is
+not necessary that the two molecules come into contact, it suffices
+that they approach sufficiently near each other for their mutual
+attractions to become sensible. The laws of the deviation they
+undergo are the same as for a veritable impact.</p>
+
+<p>It seems at first that the disorderly impacts of this innumerable
+dust can engender only an inextricable chaos before which
+analysis must recoil. But the law of great numbers, that supreme
+law of chance, comes to our aid; in presence of a semi-disorder,
+we must despair, but in extreme disorder, this statistical law<span class='pagenum'><a name="Page_524" id="Page_524">[Pg 524]</a></span>
+reestablishes a sort of mean order where the mind can recover.
+It is the study of this mean order which constitutes the kinetic
+theory of gases; it shows us that the velocities of the molecules
+are equally distributed among all the directions, that the rapidity
+of these velocities varies from one molecule to another, but that
+even this variation is subject to a law called Maxwell's law.
+This law tells us how many of the molecules move with such and
+such a velocity. As soon as the gas departs from this law, the
+mutual impacts of the molecules, in modifying the rapidity and
+direction of their velocities, tend to bring it promptly back.
+Physicists have striven, not without success, to explain in this way
+the experimental properties of gases; for example Mariotte's
+law.</p>
+
+<p>Consider now the milky way; there also we see an innumerable
+dust; only the grains of this dust are not atoms, they are stars;
+these grains move also with high velocities; they act at a distance
+one upon another, but this action is so slight at great distance
+that their trajectories are straight; and yet, from time to time,
+two of them may approach near enough to be deviated from their
+path, like a comet which has passed too near Jupiter. In a word,
+to the eyes of a giant for whom our suns would be as for us our
+atoms, the milky way would seem only a bubble of gas.</p>
+
+<p>Such was Lord Kelvin's leading idea. What may be drawn
+from this comparison? In how far is it exact? This is what we
+are to investigate together; but before reaching a definite conclusion,
+and without wishing to prejudge it, we foresee that the
+kinetic theory of gases will be for the astronomer a model he
+should not follow blindly, but from which he may advantageously
+draw inspiration. Up to the present, celestial mechanics has
+attacked only the solar system or certain systems of double stars.
+Before the assemblage presented by the milky way, or the agglomeration
+of stars, or the resolvable nebulae it recoils, because it
+sees therein only chaos. But the milky way is not more complicated
+than a gas; the statistical methods founded upon the calculus
+of probabilities applicable to a gas are also applicable to it.
+Before all, it is important to grasp the resemblance of the two
+cases, and their difference.</p>
+
+<p>Lord Kelvin has striven to determine in this manner the<span class='pagenum'><a name="Page_525" id="Page_525">[Pg 525]</a></span>
+dimensions of the milky way; for that we are reduced to counting the
+stars visible in our telescopes; but we are not sure that behind
+the stars we see, there are not others we do not see; so that what
+we should measure in this way would not be the size of the milky
+way, it would be the range of our instruments.</p>
+
+<p>The new theory comes to offer us other resources. In fact, we
+know the motions of the stars nearest us, and we can form an
+idea of the rapidity and direction of their velocities. If the ideas
+above set forth are exact, these velocities should follow Maxwell's
+law, and their mean value will tell us, so to speak, that
+which corresponds to the temperature of our fictitious gas. But
+this temperature depends itself upon the dimensions of our gas
+bubble. In fact, how will a gaseous mass let loose in the void
+act, if its elements attract one another according to Newton's
+law? It will take a spherical form; moreover, because of gravitation,
+the density will be greater at the center, the pressure also
+will increase from the surface to the center because of the weight
+of the outer parts drawn toward the center; finally, the temperature
+will increase toward the center: the temperature and the
+pressure being connected by the law called adiabatic, as happens
+in the successive layers of our atmosphere. At the surface itself,
+the pressure will be null, and it will be the same with the absolute
+temperature, that is to say with the velocity of the molecules.</p>
+
+<p>A question comes here: I have spoken of the adiabatic law,
+but this law is not the same for all gases, since it depends upon
+the ratio of their two specific heats; for the air and like gases,
+this ratio is 1.42; but is it to air that it is proper to liken the
+milky way? Evidently not, it should be regarded as a mono-atomic
+gas, like mercury vapor, like argon, like helium, that is
+to say that the ratio of the specific heats should be taken equal
+to 1.66. And, in fact, one of our molecules would be for example
+the solar system; but the planets are very small personages, the
+sun alone counts, so that our molecule is indeed mono-atomic.
+And even if we take a double star, it is probable that the action
+of a strange star which might approach it would become sufficiently
+sensible to deviate the motion of general translation of
+the system much before being able to trouble the relative orbits<span class='pagenum'><a name="Page_526" id="Page_526">[Pg 526]</a></span>
+of the two components; the double star, in a word, would act like
+an indivisible atom.</p>
+
+<p>However that may be, the pressure, and consequently the
+temperature, at the center of the gaseous sphere would be by so
+much the greater as the sphere was larger since the pressure
+increases by the weight of all the superposed layers. We may
+suppose that we are nearly at the center of the milky way, and
+by observing the mean proper velocity of the stars, we shall
+know that which corresponds to the central temperature of our
+gaseous sphere and we shall determine its radius.</p>
+
+<p>We may get an idea of the result by the following considerations:
+make a simpler hypothesis: the milky way is spherical, and
+in it the masses are distributed in a homogeneous manner; thence
+results that the stars in it describe ellipses having the same center.
+If we suppose the velocity becomes nothing at the surface,
+we may calculate this velocity at the center by the equation of
+vis viva. Thus we find that this velocity is proportional to the
+radius of the sphere and to the square root of its density. If
+the mass of this sphere was that of the sun and its radius that
+of the terrestrial orbit, this velocity would be (it is easy to see)
+that of the earth in its orbit. But in the case we have supposed,
+the mass of the sun should be distributed in a sphere of radius
+1,000,000 times greater, this radius being the distance of the
+nearest stars; the density is therefore 10<sup>18</sup> times less; now, the
+velocities are of the same order, therefore it is necessary that the
+radius be 10<sup>9</sup> times greater, be 1,000 times the distance of the
+nearest stars, which would give about a thousand millions of
+stars in the milky way.</p>
+
+<p>But you will say these hypothesis differ greatly from the
+reality; first, the milky way is not spherical and we shall soon
+return to this point, and again the kinetic theory of gases is not
+compatible with the hypothesis of a homogeneous sphere. But
+in making the exact calculation according to this theory, we
+should find a different result, doubtless, but of the same order
+of magnitude; now in such a problem the data are so uncertain
+that the order of magnitude is the sole end to be aimed at.</p>
+
+<p>And here a first remark presents itself; Lord Kelvin's result,
+which I have obtained again by an approximative calculation,<span class='pagenum'><a name="Page_527" id="Page_527">[Pg 527]</a></span>
+agrees sensibly with the evaluations the observers have made with
+their telescopes; so that we must conclude we are very near to
+piercing through the milky way. But that enables us to answer
+another question. There are the stars we see because they
+shine; but may there not be dark stars circulating in the interstellar
+spaces whose existence might long remain unknown?
+But then, what Lord Kelvin's method would give us would be
+the total number of stars, including the dark stars; as his figure
+is comparable to that the telescope gives, this means there is no
+dark matter, or at least not so much as of shining matter.</p>
+
+<p>Before going further, we must look at the problem from another
+angle. Is the milky way thus constituted truly the image
+of a gas properly so called? You know Crookes has introduced
+the notion of a fourth state of matter, where gases having become
+too rarefied are no longer true gases and become what he calls
+radiant matter. Considering the slight density of the milky
+way, is it the image of gaseous matter or of radiant matter?
+The consideration of what is called the <i>free path</i> will furnish us
+the answer.</p>
+
+<p>The trajectory of a gaseous molecule may be regarded as
+formed of straight segments united by very small arcs corresponding
+to the successive impacts. The length of each of these
+segments is what is called the free path; of course this length is
+not the same for all the segments and for all the molecules; but
+we may take a mean; this is what is called the <i>mean path</i>. This
+is the greater the less the density of the gas. The matter will be
+radiant if the mean path is greater than the dimensions of the
+receptacle wherein the gas is enclosed, so that a molecule has a
+chance to go across the whole receptacle without undergoing an
+impact; if the contrary be the case, it is gaseous. From this it
+follows that the same fluid may be radiant in a little receptacle
+and gaseous in a big one; this perhaps is why, in a Crookes tube,
+it is necessary to make the vacuum by so much the more complete
+as the tube is larger.</p>
+
+<p>How is it then for the milky way? This is a mass of gas of
+which the density is very slight, but whose dimensions are very
+great; has a star chances of traversing it without undergoing an
+impact, that is to say without passing sufficiently near another<span class='pagenum'><a name="Page_528" id="Page_528">[Pg 528]</a></span>
+star to be sensibly deviated from its route! What do we mean
+by <i>sufficiently near</i>? That is perforce a little arbitrary; take
+it as the distance from the sun to Neptune, which would represent
+a deviation of a dozen degrees; suppose therefore each of
+our stars surrounded by a protective sphere of this radius;
+could a straight pass between these spheres? At the mean distance
+of the stars of the milky way, the radius of these spheres
+will be seen under an angle of about a tenth of a second; and we
+have a thousand millions of stars. Put upon the celestial sphere
+a thousand million little circles of a tenth of a second radius.
+Are the chances that these circles will cover a great number of
+times the celestial sphere? Far from it; they will cover only its
+sixteen thousandth part. So the milky way is not the image of
+gaseous matter, but of Crookes' radiant matter. Nevertheless, as
+our foregoing conclusions are happily not at all precise, we need
+not sensibly modify them.</p>
+
+<p>But there is another difficulty: the milky way is not spherical,
+and we have reasoned hitherto as if it were, since this is the form
+of equilibrium a gas isolated in space would take. To make
+amends, agglomerations of stars exist whose form is globular and
+to which would better apply what we have hitherto said. Herschel
+has already endeavored to explain their remarkable appearances.
+He supposed the stars of the aggregates uniformly
+distributed, so that an assemblage is a homogeneous sphere; each
+star would then describe an ellipse and all these orbits would be
+passed over in the same time, so that at the end of a period the
+aggregate would take again its primitive configuration and this
+configuration would be stable. Unluckily, the aggregates do not
+appear to be homogeneous; we see a condensation at the center,
+we should observe it even were the sphere homogeneous, since
+it is thicker at the center; but it would not be so accentuated.
+We may therefore rather compare an aggregate to a gas in adiabatic
+equilibrium, which takes the spherical form because this is
+the figure of equilibrium of a gaseous mass.</p>
+
+<p>But, you will say, these aggregates are much smaller than the
+milky way, of which they even in probability make part, and even
+though they be more dense, they will rather present something
+analogous to radiant matter; now, gases attain their adiabatic<span class='pagenum'><a name="Page_529" id="Page_529">[Pg 529]</a></span>
+equilibrium only through innumerable impacts of the molecules.
+That might perhaps be adjusted. Suppose the stars of the aggregate
+have just enough energy for their velocity to become null
+when they reach the surface; then they may traverse the aggregate
+without impact, but arrived at the surface they will go back
+and will traverse it anew; after a great number of crossings, they
+will at last be deviated by an impact; under these conditions, we
+should still have a matter which might be regarded as gaseous;
+if perchance there had been in the aggregate stars whose velocity
+was greater, they have long gone away out of it, they have left
+it never to return. For all these reasons, it would be interesting
+to examine the known aggregates, to seek to account for the law
+of the densities, and to see if it is the adiabatic law of gases.</p>
+
+<p>But to return to the milky way; it is not spherical and would
+rather be represented as a flattened disc. It is clear then that a
+mass starting without velocity from the surface will reach the
+center with different velocities, according as it starts from the
+surface in the neighborhood of the middle of the disc or just on
+the border of the disc; the velocity would be notably greater in
+the latter case. Now, up to the present, we have supposed that
+the proper velocities of the stars, those we observe, must be comparable
+to those which like masses would attain; this involves a
+certain difficulty. We have given above a value for the dimensions
+of the milky way, and we have deduced it from the observed
+proper velocities which are of the same order of magnitude as
+that of the earth in its orbit; but which is the dimension we have
+thus measured? Is it the thickness? Is it the radius of the disc?
+It is doubtless something intermediate; but what can we say then
+of the thickness itself, or of the radius of the disc? Data are
+lacking to make the calculation; I shall confine myself to giving
+a glimpse of the possibility of basing an evaluation at least approximate
+upon a deeper discussion of the proper motions.</p>
+
+<p>And then we find ourselves facing two hypotheses: either the
+stars of the milky way are impelled by velocities for the most
+part parallel to the galactic plane, but otherwise distributed
+uniformly in all directions parallel to this plane. If this be so,
+observation of the proper motions should show a preponderance
+of components parallel to the milky way; this is to be determined,<span class='pagenum'><a name="Page_530" id="Page_530">[Pg 530]</a></span>
+because I do not know whether a systematic discussion has ever
+been made from this view-point. On the other hand, such an
+equilibrium could only be provisory, since because of impacts the
+molecules, I mean the stars, would in the long run acquire notable
+velocities in the sense perpendicular to the milky way and would
+end by swerving from its plane, so that the system would tend
+toward the spherical form, the only figure of equilibrium of an
+isolated gaseous mass.</p>
+
+<p>Or else the whole system is impelled by a common rotation, and
+for that reason is flattened like the earth, like Jupiter, like all
+bodies that twirl. Only, as the flattening is considerable, the
+rotation must be rapid; rapid doubtless, but it must be understood
+in what sense this word is used. The density of the milky
+way is 10<sup>23</sup> times less than that of the sun; a velocity of rotation
+&#8730;10<sup>25</sup> times less than that of the sun, for it would, therefore, be
+the equivalent so far as concerns flattening; a velocity 10<sup>12</sup> times
+slower than that of the earth, say a thirtieth of a second of arc
+in a century, would be a very rapid rotation, almost too rapid for
+stable equilibrium to be possible.</p>
+
+<p>In this hypothesis, the observable proper motions would appear
+to us uniformly distributed, and there would no longer be a preponderance
+of components parallel to the galactic plane.</p>
+
+<p>They will tell us nothing about the rotation itself, since we belong
+to the turning system. If the spiral nebul&aelig; are other
+milky ways, foreign to ours, they are not borne along in this
+rotation, and we might study their proper motions. It is true
+they are very far away; if a nebula has the dimensions of the
+milky way and if its apparent radius is for example 20&acute;&acute;, its
+distance is 10,000 times the radius of the milky way.</p>
+
+<p>But that makes no difference, since it is not about the translation
+of our system that we ask information from them, but
+about its rotation. The fixed stars, by their apparent motion,
+reveal to us the diurnal rotation of the earth, though their distance
+is immense. Unluckily, the possible rotation of the milky
+way, however rapid it may be relatively, is very slow viewed
+absolutely, and besides the pointings on nebul&aelig; can not be very
+precise; therefore thousands of years of observations would be
+necessary to learn anything.<span class='pagenum'><a name="Page_531" id="Page_531">[Pg 531]</a></span></p>
+
+<p>However that may be, in this second hypothesis, the figure of
+the milky way would be a figure of final equilibrium.</p>
+
+<p>I shall not further discuss the relative value of these two hypotheses
+since there is a third which is perhaps more probable.
+We know that among the irresolvable nebul&aelig;, several kinds may
+be distinguished: the irregular nebul&aelig; like that of Orion, the
+planetary and annular nebul&aelig;, the spiral nebul&aelig;. The spectra
+of the first two families have been determined, they are discontinuous;
+these nebul&aelig; are therefore not formed of stars; besides,
+their distribution on the heavens seems to depend upon the milky
+way; whether they have a tendency to go away from it, or on
+the contrary to approach it, they make therefore a part of the
+system. On the other hand, the spiral nebul&aelig; are generally
+considered as independent of the milky way; it is supposed that
+they, like it, are formed of a multitude of stars, that they are,
+in a word, other milky ways very far away from ours. The
+recent investigations of Stratonoff tend to make us regard the
+milky way itself as a spiral nebula, and this is the third hypothesis
+of which I wish to speak.</p>
+
+<p>How can we explain the very singular appearances presented
+by the spiral nebul&aelig;, which are too regular and too constant to
+be due to chance? First of all, to take a look at one of these
+representations is enough to see that the mass is in rotation; we
+may even see what the sense of the rotation is; all the spiral radii
+are curved in the same sense; it is evident that the <i>moving wing</i>
+lags behind the pivot and that fixes the sense of the rotation.
+But this is not all; it is evident that these nebul&aelig; can not be
+likened to a gas at rest, nor even to a gas in relative equilibrium
+under the sway of a uniform rotation; they are to be compared
+to a gas in permanent motion in which internal currents prevail.</p>
+
+<p>Suppose, for example, that the rotation of the central nucleus
+is rapid (you know what I mean by this word), too rapid for
+stable equilibrium; then at the equator the centrifugal force will
+drive it away over the attraction, and the stars will tend to
+break away at the equator and will form divergent currents; but
+in going away, as their moment of rotation remains constant,
+while the radius vector augments, their angular velocity will
+diminish, and this is why the moving wing seems to lag back.<span class='pagenum'><a name="Page_532" id="Page_532">[Pg 532]</a></span></p>
+
+<p>From this point of view, there would not be a real permanent
+motion, the central nucleus would constantly lose matter which
+would go out of it never to return, and would drain away progressively.
+But we may modify the hypothesis. In proportion
+as it goes away, the star loses its velocity and ends by stopping;
+at this moment attraction regains possession of it and leads it
+back toward the nucleus; so there will be centripetal currents.
+We must suppose the centripetal currents are the first rank and
+the centrifugal currents the second rank, if we adopt the comparison
+with a troop in battle executing a change of front; and,
+in fact, it is necessary that the composite centrifugal force be
+compensated by the attraction exercised by the central layers of
+the swarm upon the extreme layers.</p>
+
+<p>Besides, at the end of a certain time a permanent r&eacute;gime establishes
+itself; the swarm being curved, the attraction exercised
+upon the pivot by the moving wing tends to slow up the pivot
+and that of the pivot upon the moving wing tends to accelerate
+the advance of this wing which no longer augments its lag, so that
+finally all the radii end by turning with a uniform velocity. We
+may still suppose that the rotation of the nucleus is quicker than
+that of the radii.</p>
+
+<p>A question remains; why do these centripetal and centrifugal
+swarms tend to concentrate themselves in radii instead of disseminating
+themselves a little everywhere? Why do these rays distribute
+themselves regularly? If the swarms concentrate themselves,
+it is because of the attraction exercised by the already
+existing swarms upon the stars which go out from the nucleus
+in their neighborhood. After an inequality is produced, it tends
+to accentuate itself in this way.</p>
+
+<p>Why do the rays distribute themselves regularly? That is less
+obvious. Suppose there is no rotation, that all the stars are in
+two planes at right angles, in such a way that their distribution
+is symmetric with regard to these two planes.</p>
+
+<p>By symmetry, there would be no reason for their going out of
+these planes, nor for the symmetry changing. This configuration
+would give us therefore equilibrium, but <i>this would be an
+unstable equilibrium</i>.</p>
+
+<p>If on the contrary, there is rotation, we shall find an analogous<span class='pagenum'><a name="Page_533" id="Page_533">[Pg 533]</a></span>
+configuration of equilibrium with four curved rays, equal to
+each other and intersecting at 90&deg;, and if the rotation is sufficiently
+rapid, this equilibrium is stable.</p>
+
+<p>I am not in position to make this more precise: enough if you
+see that these spiral forms may perhaps some day be explained
+by only the law of gravitation and statistical consideration recalling
+those of the theory of gases.</p>
+
+<p>What has been said of internal currents shows it is of interest
+to discuss systematically the aggregate of proper motions; this
+may be done in a hundred years, when the second edition is issued
+of the chart of the heavens and compared with the first, that we
+now are making.</p>
+
+<p>But, in conclusion, I wish to call your attention to a question,
+that of the age of the milky way or the nebul&aelig;. If what we
+think we see is confirmed, we can get an idea of it. That sort of
+statistical equilibrium of which gases give us the model is established
+only in consequence of a great number of impacts. If
+these impacts are rare, it can come about only after a very long
+time; if really the milky way (or at least the agglomerations
+which are contained in it), if the nebul&aelig; have attained this equilibrium,
+this means they are very old, and we shall have an inferior
+limit of their age. Likewise we should have of it a superior
+limit; this equilibrium is not final and can not last always.
+Our spiral nebul&aelig; would be comparable to gases impelled by
+permanent motions; but gases in motion are viscous and their
+velocities end by wearing out. What here corresponds to the
+viscosity (and which depends upon the chances of impact of the
+molecules) is excessively slight, so that the present r&eacute;gime may
+persist during an extremely long time, yet not forever, so that our
+milky ways can not live eternally nor become infinitely old.</p>
+
+<p>And this is not all. Consider our atmosphere: at the surface
+must reign a temperature infinitely small and the velocity of the
+molecules there is near zero. But this is a question only of the
+mean velocity; as a consequence of impacts, one of these molecules
+may acquire (rarely, it is true) an enormous velocity, and
+then it will rush out of the atmosphere, and once out, it will
+never return; therefore our atmosphere drains off thus with extreme
+slowness. The milky way also from time to time loses a<span class='pagenum'><a name="Page_534" id="Page_534">[Pg 534]</a></span>
+star by the same mechanism, and that likewise limits its duration.</p>
+
+<p>Well, it is certain that if we compute in this manner the age
+of the milky way, we shall get enormous figures. But here a
+difficulty presents itself. Certain physicists, relying upon other
+considerations, reckon that suns can have only an ephemeral existence,
+about fifty million years; our minimum would be much
+greater than that. Must we believe that the evolution of the
+milky way began when the matter was still dark? But how have
+the stars composing it reached all at the same time adult age,
+an age so briefly to endure? Or must they reach there all successively,
+and are those we see only a feeble minority compared with
+those extinguished or which shall one day light up? But how
+reconcile that with what we have said above on the absence of a
+noteworthy proportion of dark matter? Should we abandon one
+of the two hypotheses, and which? I confine myself to pointing
+out the difficulty without pretending to solve it; I shall end therefore
+with a big interrogation point.</p>
+
+<p>However, it is interesting to set problems, even when their solution
+seems very far away.</p>
+
+
+
+<hr style="width: 65%;" />
+<p><span class='pagenum'><a name="Page_535" id="Page_535">[Pg 535]</a></span></p>
+<h3>CHAPTER II</h3>
+
+<h3><span class="smcap">French Geodesy</span></h3>
+
+
+<p>Every one understands our interest in knowing the form and
+dimensions of our earth; but some persons will perhaps be surprised
+at the exactitude sought after. Is this a useless luxury?
+What good are the efforts so expended by the geodesist?</p>
+
+<p>Should this question be put to a congressman, I suppose he
+would say: "I am led to believe that geodesy is one of the most
+useful of the sciences; because it is one of those costing us most
+dear." I shall try to give you an answer a little more precise.</p>
+
+<p>The great works of art, those of peace as well as those of war,
+are not to be undertaken without long studies which save much
+groping, miscalculation and useless expense. These studies can
+only be based upon a good map. But a map will be only a valueless
+phantasy if constructed without basing it upon a solid framework.
+As well make stand a human body minus the skeleton.</p>
+
+<p>Now, this framework is given us by geodesic measurements;
+so, without geodesy, no good map; without a good map, no great
+public works.</p>
+
+<p>These reasons will doubtless suffice to justify much expense;
+but these are arguments for practical men. It is not upon these
+that it is proper to insist here; there are others higher and,
+everything considered, more important.</p>
+
+<p>So we shall put the question otherwise; can geodesy aid us the
+better to know nature? Does it make us understand its unity
+and harmony? In reality an isolated fact is of slight value,
+and the conquests of science are precious only if they prepare for
+new conquests.</p>
+
+<p>If therefore a little hump were discovered on the terrestrial
+ellipsoid, this discovery would be by itself of no great interest.
+On the other hand, it would become precious if, in seeking the
+cause of this hump, we hoped to penetrate new secrets.</p>
+
+<p>Well, when, in the eighteenth century, Maupertuis and La
+Condamine braved such opposite climates, it was not solely to<span class='pagenum'><a name="Page_536" id="Page_536">[Pg 536]</a></span>
+learn the shape of our planet, it was a question of the whole
+world-system.</p>
+
+<p>If the earth was flattened, Newton triumphed and with him the
+doctrine of gravitation and the whole modern celestial mechanics.</p>
+
+<p>And to-day, a century and a half after the victory of the Newtonians,
+think you geodesy has nothing more to teach us?</p>
+
+<p>We know not what is within our globe. The shafts of mines
+and borings have let us know a layer of 1 or 2 kilometers thickness,
+that is to say, the millionth part of the total mass; but what
+is beneath?</p>
+
+<p>Of all the extraordinary journeys dreamed by Jules Verne,
+perhaps that to the center of the earth took us to regions least
+explored.</p>
+
+<p>But these deep-lying rocks we can not reach, exercise from
+afar their attraction which operates upon the pendulum and deforms
+the terrestrial spheroid. Geodesy can therefore weigh
+them from afar, so to speak, and tell us of their distribution.
+Thus will it make us really see those mysterious regions which
+Jules Verne only showed us in imagination.</p>
+
+<p>This is not an empty illusion. M. Faye, comparing all the
+measurements, has reached a result well calculated to surprise us.
+Under the oceans, in the depths, are rocks of very great density;
+under the continents, on the contrary, are empty spaces.</p>
+
+<p>New observations will modify perhaps the details of these conclusions.</p>
+
+<p>In any case, our venerated dean has shown us where to search
+and what the geodesist may teach the geologist, desirous of knowing
+the interior constitution of the earth, and even the thinker
+wishing to speculate upon the past and the origin of this planet.</p>
+
+<p>And now, why have I entitled this chapter <i>French Geodesy</i>?
+It is because, in each country, this science has taken, more than
+all others, perhaps, a national character. It is easy to see why.</p>
+
+<p>There must be rivalry. The scientific rivalries are always
+courteous, or at least almost always; in any case, they are necessary,
+because they are always fruitful. Well, in those enterprises
+which require such long efforts and so many collaborators,
+the individual is effaced, in spite of himself, of course; no one
+has the right to say: this is my work. Therefore it is not between
+men, but between nations that rivalries go on.<span class='pagenum'><a name="Page_537" id="Page_537">[Pg 537]</a></span></p>
+
+<p>So we are led to seek what has been the part of France. Her
+part I believe we are right to be proud of.</p>
+
+<p>At the beginning of the eighteenth century, long discussions
+arose between the Newtonians who believed the earth flattened,
+as the theory of gravitation requires, and Cassini, who, deceived
+by inexact measurements, believed our globe elongated. Only
+direct observation could settle the question. It was our Academy
+of Sciences that undertook this task, gigantic for the epoch.</p>
+
+<p>While Maupertuis and Clairaut measured a degree of meridian
+under the polar circle, Bouguer and La Condamine went toward
+the Andes Mountains, in regions then under Spain which to-day
+are the Republic of Ecuador.</p>
+
+<p>Our envoys were exposed to great hardships. Traveling was
+not as easy as at present.</p>
+
+<p>Truly, the country where Maupertuis operated was not a desert
+and he even enjoyed, it is said, among the Laplanders those sweet
+satisfactions of the heart that real arctic voyagers never know.
+It was almost the region where, in our days, comfortable steamers
+carry, each summer, hosts of tourists and young English people.
+But in those days Cook's agency did not exist and Maupertuis
+really believed he had made a polar expedition.</p>
+
+<p>Perhaps he was not altogether wrong. The Russians and the
+Swedes carry out to-day analogous measurements at Spitzbergen,
+in a country where there is real ice-cap. But they have quite
+other resources, and the difference of time makes up for that
+of latitude.</p>
+
+<p>The name of Maupertuis has reached us much scratched by the
+claws of Doctor Akakia; the scientist had the misfortune to displease
+Voltaire, who was then the king of mind. He was first
+praised beyond measure; but the flatteries of kings are as much
+to be dreaded as their displeasure, because the days after are
+terrible. Voltaire himself knew something of this.</p>
+
+<p>Voltaire called Maupertuis, my amiable master in thinking,
+marquis of the polar circle, dear flattener out of the world and
+Cassini, and even, flattery supreme, Sir Isaac Maupertuis; he
+wrote him: "Only the king of Prussia do I put on a level with
+you; he only lacks being a geometer." But soon the scene
+changes, he no longer speaks of deifying him, as in days of yore<span class='pagenum'><a name="Page_538" id="Page_538">[Pg 538]</a></span>
+the Argonauts, or of calling down from Olympus the council of
+the gods to contemplate his works, but of chaining him up in a
+madhouse. He speaks no longer of his sublime mind, but of his
+despotic pride, plated with very little science and much
+absurdity.</p>
+
+<p>I care not to relate these comico-heroic combats; but permit me
+some reflections on two of Voltaire's verses. In his 'Discourse
+on Moderation' (no question of moderation in praise and criticism),
+the poet has written:</p>
+
+<div class="blockquot">
+<p class="noidt">
+You have confirmed in regions drear<br />
+What Newton discerned without going abroad.<br />
+</p>
+</div>
+
+<p class="noidt">These two verses (which replace the hyperbolic praises of the first
+period) are very unjust, and doubtless Voltaire was too enlightened
+not to know it.</p>
+
+<p>Then, only those discoveries were esteemed which could be
+made without leaving one's house.</p>
+
+<p>To-day, it would rather be theory that one would make light of.</p>
+
+<p>This is to misunderstand the aim of science.</p>
+
+<p>Is nature governed by caprice, or does harmony rule there?
+That is the question. It is when it discloses to us this harmony
+that science is beautiful and so worthy to be cultivated. But
+whence can come to us this revelation, if not from the accord of
+a theory with experiment? To seek whether this accord exists
+or if it fails, this therefore is our aim. Consequently these two
+terms, which we must compare, are as indispensable the one as
+the other. To neglect one for the other would be nonsense. Isolated,
+theory would be empty, experiment would be blind; each
+would be useless and without interest.</p>
+
+<p>Maupertuis therefore deserves his share of glory. Truly, it
+will not equal that of Newton, who had received the spark divine;
+nor even that of his collaborator Clairaut. Yet it is not to be
+despised, because his work was necessary, and if France, outstripped
+by England in the seventeenth century, has so well
+taken her revenge in the century following, it is not alone to the
+genius of Clairauts, d'Alemberts, Laplaces that she owes it;
+it is also to the long patience of the Maupertuis and the La
+Condamines.<span class='pagenum'><a name="Page_539" id="Page_539">[Pg 539]</a></span></p>
+
+<p>We reach what may be called the second heroic period of
+geodesy. France is torn within. All Europe is armed against
+her; it would seem that these gigantic combats might absorb all
+her forces. Far from it; she still has them for the service of
+science. The men of that time recoiled before no enterprise,
+they were men of faith.</p>
+
+<p>Delambre and M&eacute;chain were commissioned to measure an arc
+going from Dunkerque to Barcelona. This time there was no
+going to Lapland or to Peru; the hostile squadrons had closed to
+us the ways thither. But, though the expeditions are less distant,
+the epoch is so troubled that the obstacles, the perils even,
+are just as great.</p>
+
+<p>In France, Delambre had to fight against the ill-will of suspicious
+municipalities. One knows that the steeples, which are
+visible from so far, and can be aimed at with precision, often
+serve as signal points to geodesists. But in the region Delambre
+traversed there were no longer any steeples. A certain proconsul
+had passed there, and boasted of knocking down all the
+steeples rising proudly above the humble abode of the sans-culottes.
+Pyramids then were built of planks and covered with
+white cloth to make them more visible. That was quite another
+thing: with white cloth! What was this rash person who, upon
+our heights so recently set free, dared to raise the hateful standard
+of the counter-revolution? It was necessary to border the
+white cloth with blue and red bands.</p>
+
+<p>M&eacute;chain operated in Spain; the difficulties were other; but
+they were not less. The Spanish peasants were hostile. There
+steeples were not lacking: but to install oneself in them with
+mysterious and perhaps diabolic instruments, was it not sacrilege?
+The revolutionists were allies of Spain, but allies smelling
+a little of the stake.</p>
+
+<p>"Without cease," writes M&eacute;chain, "they threaten to butcher
+us." Fortunately, thanks to the exhortations of the priests, to
+the pastoral letters of the bishops, these ferocious Spaniards contented
+themselves with threatening.</p>
+
+<p>Some years after M&eacute;chain made a second expedition into Spain:
+he proposed to prolong the meridian from Barcelona to the
+Balearics. This was the first time it had been attempted to make<span class='pagenum'><a name="Page_540" id="Page_540">[Pg 540]</a></span>
+the triangulations overpass a large arm of the sea by observing
+signals installed upon some high mountain of a far-away isle.
+The enterprise was well conceived and well prepared; it failed
+however.</p>
+
+<p>The French scientist encountered all sorts of difficulties of
+which he complains bitterly in his correspondence. "Hell," he
+writes, perhaps with some exaggeration&mdash;"hell and all the
+scourges it vomits upon the earth, tempests, war, the plague and
+black intrigues are therefore unchained against me!"</p>
+
+<p>The fact is that he encountered among his collaborators more
+of proud obstinacy than of good will and that a thousand accidents
+retarded his work. The plague was nothing, the fear of
+the plague was much more redoubtable; all these isles were on
+their guard against the neighboring isles and feared lest they
+should receive the scourge from them. M&eacute;chain obtained permission
+to disembark only after long weeks upon the condition
+of covering all his papers with vinegar; this was the antisepsis
+of that time.</p>
+
+<p>Disgusted and sick, he had just asked to be recalled, when he
+died.</p>
+
+<p>Arago and Biot it was who had the honor of taking up the
+unfinished work and carrying it on to completion.</p>
+
+<p>Thanks to the support of the Spanish government, to the protection
+of several bishops and, above all, to that of a famous
+brigand chief, the operations went rapidly forward. They were
+successfully completed, and Biot had returned to France when
+the storm burst.</p>
+
+<p>It was the moment when all Spain took up arms to defend her
+independence against France. Why did this stranger climb the
+mountains to make signals? It was evidently to call the French
+army. Arago was able to escape the populace only by becoming
+a prisoner. In his prison, his only distraction was reading in
+the Spanish papers the account of his own execution. The papers
+of that time sometimes gave out news prematurely. He had at
+least the consolation of learning that he died with courage and
+like a Christian.</p>
+
+<p>Even the prison was no longer safe; he had to escape and reach
+Algiers. There, he embarked for Marseilles on an Algerian<span class='pagenum'><a name="Page_541" id="Page_541">[Pg 541]</a></span>
+vessel. This ship was captured by a Spanish corsair, and behold
+Arago carried back to Spain and dragged from dungeon to
+dungeon, in the midst of vermin and in the most shocking
+wretchedness.</p>
+
+<p>If it had only been a question of his subjects and his guests,
+the dey would have said nothing. But there were on board two
+lions, a present from the African sovereign to Napoleon. The
+dey threatened war.</p>
+
+<p>The vessel and the prisoners were released. The port should
+have been properly reached, since they had on board an astronomer;
+but the astronomer was seasick, and the Algerian seamen,
+who wished to make Marseilles, came out at Bougie. Thence
+Arago went to Algiers, traversing Kabylia on foot in the midst
+of a thousand perils. He was long detained in Africa and
+threatened with the convict prison. Finally he was able to get
+back to France; his observations, which he had preserved and
+safeguarded under his shirt, and, what is still more remarkable,
+his instruments had traversed unhurt these terrible adventures.
+Up to this point, not only did France hold the foremost place,
+but she occupied the stage almost alone.</p>
+
+<p>In the years which follow, she has not been inactive and our
+staff-office map is a model. However, the new methods of observation
+and calculation have come to us above all from Germany
+and England. It is only in the last forty years that France has
+regained her rank. She owes it to a scientific officer, General
+Perrier, who has successfully executed an enterprise truly audacious,
+the junction of Spain and Africa. Stations were installed
+on four peaks upon the two sides of the Mediterranean.
+For long months they awaited a calm and limpid atmosphere.
+At last was seen the little thread of light which had traversed
+300 kilometers over the sea. The undertaking had succeeded.</p>
+
+<p>To-day have been conceived projects still more bold. From a
+mountain near Nice will be sent signals to Corsica, not now for
+geodesic determinations, but to measure the velocity of light.
+The distance is only 200 kilometers; but the ray of light is to
+make the journey there and return, after reflection by a mirror
+installed in Corsica. And it should not wander on the way, for
+it must return exactly to the point of departure.<span class='pagenum'><a name="Page_542" id="Page_542">[Pg 542]</a></span></p>
+
+<p>Ever since, the activity of French geodesy has never slackened.
+We have no more such astonishing adventures to tell; but the
+scientific work accomplished is immense. The territory of
+France beyond the sea, like that of the mother country, is covered
+by triangles measured with precision.</p>
+
+<p>We have become more and more exacting and what our fathers
+admired does not satisfy us to-day. But in proportion as we seek
+more exactitude, the difficulties greatly increase; we are surrounded
+by snares and must be on our guard against a thousand
+unsuspected causes of error. It is needful, therefore, to create
+instruments more and more faultless.</p>
+
+<p>Here again France has not let herself be distanced. Our
+appliances for the measurement of bases and angles leave nothing
+to desire, and, I may also mention the pendulum of Colonel
+Defforges, which enables us to determine gravity with a precision
+hitherto unknown.</p>
+
+<p>The future of French geodesy is at present in the hands of the
+Geographic Service of the army, successively directed by General
+Bassot and General Berthaut. We can not sufficiently congratulate
+ourselves upon it. For success in geodesy, scientific aptitudes
+are not enough; it is necessary to be capable of standing
+long fatigues in all sorts of climates; the chief must be able to
+win obedience from his collaborators and to make obedient his
+native auxiliaries. These are military qualities. Besides, one
+knows that, in our army, science has always marched shoulder to
+shoulder with courage.</p>
+
+<p>I add that a military organization assures the indispensable
+unity of action. It would be more difficult to reconcile the rival
+pretensions of scientists jealous of their independence, solicitous
+of what they call their fame, and who yet must work in concert,
+though separated by great distances. Among the geodesists of
+former times there were often discussions, of which some aroused
+long echoes. The Academy long resounded with the quarrel of
+Bouguer and La Condamine. I do not mean to say that soldiers
+are exempt from passion, but discipline imposes silence upon a
+too sensitive self-esteem.</p>
+
+<p>Several foreign governments have called upon our officers to<span class='pagenum'><a name="Page_543" id="Page_543">[Pg 543]</a></span>
+organize their geodesic service: this is proof that the scientific
+influence of France abroad has not declined.</p>
+
+<p>Our hydrographic engineers contribute also to the common
+achievement a glorious contingent. The survey of our coasts, of
+our colonies, the study of the tides, offer them a vast domain of
+research. Finally I may mention the general leveling of France
+which is carried out by the ingenious and precise methods of
+M. Lallemand.</p>
+
+<p>With such men we are sure of the future. Moreover, work for
+them will not be lacking; our colonial empire opens for them immense
+expanses illy explored. That is not all: the International
+Geodetic Association has recognized the necessity of a new measurement
+of the arc of Quito, determined in days of yore by La
+Condamine. It is France that has been charged with this operation;
+she had every right to it, since our ancestors had made, so
+to speak, the scientific conquest of the Cordilleras. Besides,
+these rights have not been contested and our government has
+undertaken to exercise them.</p>
+
+<p>Captains Maurain and Lacombe completed a first reconnaissance,
+and the rapidity with which they accomplished their
+mission, crossing the roughest regions and climbing the most
+precipitous summits, is worthy of all praise. It won the admiration
+of General Alfaro, President of the Republic of Ecuador,
+who called them 'los hombres de hierro,' the men of iron.</p>
+
+<p>The final commission then set out under the command of Lieutenant-Colonel
+(then Major) Bourgeois. The results obtained
+have justified the hopes entertained. But our officers have encountered
+unforeseen difficulties due to the climate. More than
+once, one of them has been forced to remain several months at
+an altitude of 4,000 meters, in the clouds and the snow, without
+seeing anything of the signals he had to aim at and which refused
+to unmask themselves. But thanks to their perseverance and
+courage, there resulted from this only a delay and an increase of
+expense, without the exactitude of the measurements suffering
+therefrom.</p>
+
+
+
+<hr style="width: 65%;" />
+<p><span class='pagenum'><a name="Page_544" id="Page_544">[Pg 544]</a></span></p>
+<h3>GENERAL CONCLUSIONS</h3>
+
+
+<p>What I have sought to explain in the preceding pages is how
+the scientist should guide himself in choosing among the innumerable
+facts offered to his curiosity, since indeed the natural
+limitations of his mind compel him to make a choice, even though
+a choice be always a sacrifice. I have expounded it first by
+general considerations, recalling on the one hand the nature
+of the problem to be solved and on the other hand seeking to
+better comprehend that of the human mind, which is the principal
+instrument of the solution. I then have explained it by
+examples; I have not multiplied them indefinitely; I also have
+had to make a choice, and I have chosen naturally the questions
+I had studied most. Others would doubtless have made a different
+choice; but what difference, because I believe they would
+have reached the same conclusions.</p>
+
+<p>There is a hierarchy of facts; some have no reach; they teach
+us nothing but themselves. The scientist who has ascertained
+them has learned nothing but a fact, and has not become more
+capable of foreseeing new facts. Such facts, it seems, come once,
+but are not destined to reappear.</p>
+
+<p>There are, on the other hand, facts of great yield; each of them
+teaches us a new law. And since a choice must be made, it is to
+these that the scientist should devote himself.</p>
+
+<p>Doubtless this classification is relative and depends upon the
+weakness of our mind. The facts of slight outcome are the complex
+facts, upon which various circumstances may exercise a
+sensible influence, circumstances too numerous and too diverse
+for us to discern them all. But I should rather say that these
+are the facts we think complex, since the intricacy of these circumstances
+surpasses the range of our mind. Doubtless a mind vaster
+and finer than ours would think differently of them. But what
+matter; we can not use that superior mind, but only our own.</p>
+
+<p>The facts of great outcome are those we think simple; may be
+they really are so, because they are influenced only by a small<span class='pagenum'><a name="Page_545" id="Page_545">[Pg 545]</a></span>
+number of well-defined circumstances, may be they take on an
+appearance of simplicity because the various circumstances upon
+which they depend obey the laws of chance and so come to mutually
+compensate. And this is what happens most often. And so
+we have been obliged to examine somewhat more closely what
+chance is.</p>
+
+<p>Facts where the laws of chance apply become easy of access to
+the scientist who would be discouraged before the extraordinary
+complication of the problems where these laws are not applicable.
+We have seen that these considerations apply not only to the
+physical sciences, but to the mathematical sciences. The method
+of demonstration is not the same for the physicist and the mathematician.
+But the methods of invention are very much alike.
+In both cases they consist in passing up from the fact to the
+law, and in finding the facts capable of leading to a law.</p>
+
+<p>To bring out this point, I have shown the mind of the mathematician
+at work, and under three forms: the mind of the mathematical
+inventor and creator; that of the unconscious geometer
+who among our far distant ancestors, or in the misty years of
+our infancy, has constructed for us our instinctive notion of
+space; that of the adolescent to whom the teachers of secondary
+education unveil the first principles of the science, seeking to
+give understanding of the fundamental definitions. Everywhere
+we have seen the r&ocirc;le of intuition and of the spirit of generalization
+without which these three stages of mathematicians, if I
+may so express myself, would be reduced to an equal impotence.</p>
+
+<p>And in the demonstration itself, the logic is not all; the true
+mathematical reasoning is a veritable induction, different in
+many regards from the induction of physics, but proceeding like
+it from the particular to the general. All the efforts that have
+been made to reverse this order and to carry back mathematical
+induction to the rules of logic have eventuated only in failures,
+illy concealed by the employment of a language inaccessible to
+the uninitiated. The examples I have taken from the physical
+sciences have shown us very different cases of facts of great
+outcome. An experiment of Kaufmann on radium rays revolutionizes
+at the same time mechanics, optics and astronomy.
+Why? Because in proportion as these sciences have developed,<span class='pagenum'><a name="Page_546" id="Page_546">[Pg 546]</a></span>
+we have the better recognized the bonds uniting them, and then
+we have perceived a species of general design of the chart of universal
+science. There are facts common to several sciences, which
+seem the common source of streams diverging in all directions
+and which are comparable to that knoll of Saint Gothard whence
+spring waters which fertilize four different valleys.</p>
+
+<p>And then we can make choice of facts with more discernment
+than our predecessors who regarded these valleys as distinct and
+separated by impassable barriers.</p>
+
+<p>It is always simple facts which must be chosen, but among
+these simple facts we must prefer those which are situated upon
+these sorts of knolls of Saint Gothard of which I have just
+spoken.</p>
+
+<p>And when sciences have no direct bond, they still mutually
+throw light upon one another by analogy. When we studied
+the laws obeyed by gases we knew we had attacked a fact of great
+outcome; and yet this outcome was still estimated beneath its
+value, since gases are, from a certain point of view, the image
+of the milky way, and those facts which seemed of interest only
+for the physicist, ere long opened new vistas to astronomy quite
+unexpected.</p>
+
+<p>And finally when the geodesist sees it is necessary to move his
+telescope some seconds to see a signal he has set up with great
+pains, this is a very small fact; but this is a fact of great outcome,
+not only because this reveals to him the existence of a
+small protuberance upon the terrestrial globe, that little hump
+would be by itself of no great interest, but because this protuberance
+gives him information about the distribution of matter
+in the interior of the globe, and through that about the past of
+our planet, about its future, about the laws of its development.</p>
+
+
+
+<hr style="width: 100%;" />
+<p><span class='pagenum'><a name="Page_547" id="Page_547">[Pg 547]</a></span></p>
+<h2><a name="INDEX" id="INDEX"></a>INDEX</h2>
+
+
+<p class="indx">
+aberration of light, <a href="#Page_315">315</a>, <a href="#Page_496">496</a><br />
+<br />
+Abraham, <a href="#Page_311">311</a>, <a href="#Page_490">490-1</a>, <a href="#Page_505">505-7</a>, <a href="#Page_509">509</a>, <a href="#Page_515">515-6</a><br />
+<br />
+absolute motion, <a href="#Page_107">107</a><br />
+<span style="margin-left: 1em;">orientation, <a href="#Page_83">83</a></span><br />
+<span style="margin-left: 1em;">space, <a href="#Page_85">85</a>, <a href="#Page_93">93</a>, <a href="#Page_246">246</a>, <a href="#Page_257">257</a>, <a href="#Page_353">353</a></span><br />
+<br />
+acceleration, <a href="#Page_94">94</a>, <a href="#Page_98">98</a>, <a href="#Page_486">486</a>, <a href="#Page_509">509</a><br />
+<br />
+accidental constant, <a href="#Page_112">112</a><br />
+<span style="margin-left: 1em;">errors, <a href="#Page_171">171</a>, <a href="#Page_402">402</a></span><br />
+<br />
+accommodation of the eyes, <a href="#Page_67">67-8</a><br />
+<br />
+action at a distance, <a href="#Page_137">137</a><br />
+<br />
+addition, <a href="#Page_34">34</a><br />
+<br />
+aim of mathematics, <a href="#Page_280">280</a><br />
+<br />
+alchemists, <a href="#Page_11">11</a><br />
+<br />
+Alfaro, <a href="#Page_543">543</a><br />
+<br />
+algebra, <a href="#Page_379">379</a><br />
+<br />
+analogy, <a href="#Page_220">220</a><br />
+<br />
+analysis, <a href="#Page_218">218-9</a>, <a href="#Page_279">279</a><br />
+<br />
+analysis situs, <a href="#Page_53">53</a>, <a href="#Page_239">239</a>, <a href="#Page_381">381</a><br />
+<br />
+analyst, <a href="#Page_210">210</a>, <a href="#Page_221">221</a><br />
+<br />
+ancestral experience, <a href="#Page_91">91</a><br />
+<br />
+Andrade, <a href="#Page_93">93</a>, <a href="#Page_104">104</a>, <a href="#Page_228">228</a><br />
+<br />
+Andrews, <a href="#Page_153">153</a><br />
+<br />
+angle sum of triangle, <a href="#Page_58">58</a><br />
+<br />
+Anglo-Saxons, <a href="#Page_3">3</a><br />
+<br />
+antinomies, <a href="#Page_449">449</a>, <a href="#Page_457">457</a>, <a href="#Page_477">477</a><br />
+<br />
+Arago, <a href="#Page_540">540-1</a><br />
+<br />
+Aristotle, <a href="#Page_205">205</a>, <a href="#Page_292">292</a>, <a href="#Page_460">460</a><br />
+<br />
+arithmetic, <a href="#Page_34">34</a>, <a href="#Page_379">379</a>, <a href="#Page_441">441</a>, <a href="#Page_463">463</a><br />
+<br />
+associativity, <a href="#Page_35">35</a><br />
+<br />
+assumptions, <a href="#Page_451">451</a>, <a href="#Page_453">453</a><br />
+<br />
+astronomy, <a href="#Page_81">81</a>, <a href="#Page_289">289</a>, <a href="#Page_315">315</a>, <a href="#Page_512">512</a><br />
+<br />
+Atwood, <a href="#Page_446">446</a><br />
+<br />
+axiom, <a href="#Page_60">60</a>, <a href="#Page_63">63</a>, <a href="#Page_65">65</a>, <a href="#Page_215">215</a><br />
+<br />
+<br />
+Bacon, <a href="#Page_128">128</a><br />
+<br />
+Bartholi, <a href="#Page_503">503</a><br />
+<br />
+Bassot, <a href="#Page_542">542</a><br />
+<br />
+beauty, <a href="#Page_349">349</a>, <a href="#Page_368">368</a><br />
+<br />
+Becquerel, <a href="#Page_312">312</a><br />
+<br />
+Beltrami, <a href="#Page_56">56</a>, <a href="#Page_58">58</a><br />
+<br />
+Bergson, <a href="#Page_321">321</a><br />
+<br />
+Berkeley, <a href="#Page_4">4</a><br />
+<br />
+Berthaut, <a href="#Page_542">542</a><br />
+<br />
+Bertrand, <a href="#Page_156">156</a>, <a href="#Page_190">190</a>, <a href="#Page_211">211</a>, <a href="#Page_395">395</a><br />
+<br />
+Betti, <a href="#Page_239">239</a><br />
+<br />
+Biot, <a href="#Page_540">540</a><br />
+<br />
+bodies, solid, <a href="#Page_72">72</a><br />
+<br />
+Boltzmann, <a href="#Page_304">304</a><br />
+<br />
+Bolyai, <a href="#Page_56">56</a>, <a href="#Page_201">201</a>, <a href="#Page_203">203</a><br />
+<br />
+Borel, <a href="#Page_482">482</a><br />
+<br />
+Bouguer, <a href="#Page_537">537</a>, <a href="#Page_542">542</a><br />
+<br />
+Bourgeois, <a href="#Page_543">543</a><br />
+<br />
+Boutroux, <a href="#Page_390">390</a>, <a href="#Page_464">464</a><br />
+<br />
+Bradley, <a href="#Page_496">496</a><br />
+<br />
+Briot, <a href="#Page_298">298</a><br />
+<br />
+Brownian movement, <a href="#Page_152">152</a>, <a href="#Page_410">410</a><br />
+<br />
+Bucherer, <a href="#Page_507">507</a><br />
+<br />
+Burali-Forti, <a href="#Page_457">457-9</a>, <a href="#Page_477">477</a>, <a href="#Page_481">481-2</a><br />
+<br />
+<br />
+Caen, <a href="#Page_387">387-8</a><br />
+<br />
+Calinon, <a href="#Page_228">228</a><br />
+<br />
+canal rays, <a href="#Page_491">491-2</a><br />
+<br />
+canals, semicircular, <a href="#Page_276">276</a><br />
+<br />
+Cantor, <a href="#Page_11">11</a>, <a href="#Page_448">448-9</a>, <a href="#Page_457">457</a>, <a href="#Page_459">459</a>, <a href="#Page_477">477</a><br />
+<br />
+Cantorism, <a href="#Page_381">381</a>, <a href="#Page_382">382</a>, <a href="#Page_480">480</a>, <a href="#Page_484">484</a><br />
+<br />
+capillarity, <a href="#Page_298">298</a><br />
+<br />
+Carlyle, <a href="#Page_128">128</a><br />
+<br />
+Carnot's principle, <a href="#Page_143">143</a>, <a href="#Page_151">151</a>, <a href="#Page_300">300</a>, <a href="#Page_303">303-5</a>, <a href="#Page_399">399</a><br />
+<br />
+Cassini, <a href="#Page_537">537</a><br />
+<br />
+cathode rays, <a href="#Page_487">487-92</a><br />
+<br />
+cells, <a href="#Page_217">217</a><br />
+<br />
+center of gravity, <a href="#Page_103">103</a><br />
+<br />
+central forces, <a href="#Page_297">297</a><br />
+<br />
+Chaldeans, <a href="#Page_290">290</a><br />
+<br />
+chance, <a href="#Page_395">395</a>, <a href="#Page_408">408</a><br />
+<br />
+change of position, <a href="#Page_70">70</a><br />
+<span style="margin-left: 1em;">state, <a href="#Page_70">70</a></span><br />
+<br />
+chemistry of the stars, <a href="#Page_295">295</a><br />
+<br />
+circle-squarers, <a href="#Page_11">11</a><br />
+<br />
+Clairaut, <a href="#Page_537">537-8</a><br />
+<br />
+Clausius, <a href="#Page_119">119</a>, <a href="#Page_123">123</a>, <a href="#Page_143">143</a><br />
+<br />
+color sensation, <a href="#Page_252">252</a><br />
+<br />
+Columbus, <a href="#Page_228">228</a><br />
+<br />
+commutativity, <a href="#Page_35">35-6</a><br />
+<span class='pagenum'><a name="Page_548" id="Page_548">[Pg 548]</a></span><br />
+compensation, <a href="#Page_72">72</a><br />
+<br />
+complete induction, <a href="#Page_40">40</a><br />
+<br />
+Comte, <a href="#Page_294">294</a><br />
+<br />
+Condorcet, <a href="#Page_411">411</a><br />
+<br />
+contingence, <a href="#Page_340">340</a><br />
+<br />
+continuity, <a href="#Page_173">173</a><br />
+<br />
+continuum, <a href="#Page_43">43</a><br />
+<span style="margin-left: 1em;">amorphous, <a href="#Page_238">238</a></span><br />
+<span style="margin-left: 1em;">mathematical, <a href="#Page_46">46</a></span><br />
+<span style="margin-left: 1em;">physical, <a href="#Page_46">46</a>, <a href="#Page_240">240</a></span><br />
+<span style="margin-left: 1em;">tridimensional, <a href="#Page_240">240</a></span><br />
+<br />
+convention, <a href="#Page_50">50</a>, <a href="#Page_93">93</a>, <a href="#Page_106">106</a>, <a href="#Page_125">125</a>, <a href="#Page_173">173</a>, <a href="#Page_208">208</a>, <a href="#Page_317">317</a>, <a href="#Page_440">440</a>, <a href="#Page_451">451</a><br />
+<br />
+convergence, <a href="#Page_67">67-8</a><br />
+<br />
+coordinates, <a href="#Page_244">244</a><br />
+<br />
+Copernicus, <a href="#Page_109">109</a>, <a href="#Page_291">291</a>, <a href="#Page_354">354</a><br />
+<br />
+Coulomb, <a href="#Page_143">143</a>, <a href="#Page_516">516</a><br />
+<br />
+Couturat, <a href="#Page_450">450</a>, <a href="#Page_453">453</a>, <a href="#Page_456">456</a>, <a href="#Page_460">460</a>, <a href="#Page_462">462-3</a>, <a href="#Page_467">467</a>, <a href="#Page_472">472-6</a><br />
+<br />
+creation, mathematical, <a href="#Page_383">383</a><br />
+<br />
+creed, <a href="#Page_1">1</a><br />
+<br />
+Cr&eacute;mieu, <a href="#Page_168">168-9</a>, <a href="#Page_490">490</a><br />
+<br />
+crisis, <a href="#Page_303">303</a><br />
+<br />
+Crookes, <a href="#Page_195">195</a>, <a href="#Page_488">488</a>, <a href="#Page_527">527-8</a><br />
+<br />
+crude fact, <a href="#Page_326">326</a>, <a href="#Page_330">330</a><br />
+<br />
+Curie, <a href="#Page_312">312-3</a>, <a href="#Page_318">318</a><br />
+<br />
+current, <a href="#Page_186">186</a><br />
+<br />
+curvature, <a href="#Page_58">58-9</a><br />
+<br />
+curve, <a href="#Page_213">213</a>, <a href="#Page_346">346</a><br />
+<br />
+curves without tangents, <a href="#Page_51">51</a><br />
+<br />
+cut, <a href="#Page_52">52</a>, <a href="#Page_256">256</a><br />
+<br />
+cyclones, <a href="#Page_353">353</a><br />
+<br />
+<br />
+d'Alembert, <a href="#Page_538">538</a><br />
+<br />
+Darwin, <a href="#Page_518">518-9</a><br />
+<br />
+De Cyon, <a href="#Page_276">276</a>, <a href="#Page_427">427</a><br />
+<br />
+Dedekind, <a href="#Page_44">44-5</a><br />
+<br />
+Defforges, <a href="#Page_542">542</a><br />
+<br />
+definitions, <a href="#Page_430">430</a>, <a href="#Page_453">453</a><br />
+<br />
+deformation, <a href="#Page_73">73</a>, <a href="#Page_415">415</a><br />
+<br />
+Delage, <a href="#Page_277">277</a><br />
+<br />
+Delambre, <a href="#Page_539">539</a><br />
+<br />
+Delbeuf, <a href="#Page_414">414</a><br />
+<br />
+Descartes, <a href="#Page_127">127</a><br />
+<br />
+determinism, <a href="#Page_123">123</a>, <a href="#Page_340">340</a><br />
+<br />
+dictionary, <a href="#Page_59">59</a><br />
+<br />
+didymium, <a href="#Page_333">333</a><br />
+<br />
+dilatation, <a href="#Page_76">76</a><br />
+<br />
+dimensions, <a href="#Page_53">53</a>, <a href="#Page_68">68</a>, <a href="#Page_78">78</a>, <a href="#Page_241">241</a>, <a href="#Page_256">256</a>, <a href="#Page_426">426</a><br />
+<br />
+direction, <a href="#Page_69">69</a><br />
+<br />
+Dirichlet, <a href="#Page_213">213</a><br />
+<br />
+dispersion, <a href="#Page_141">141</a><br />
+<br />
+displacement, <a href="#Page_73">73</a>, <a href="#Page_77">77</a>, <a href="#Page_247">247</a>, <a href="#Page_256">256</a><br />
+<br />
+distance, <a href="#Page_59">59</a>, <a href="#Page_292">292</a><br />
+<br />
+distributivity, <a href="#Page_36">36</a><br />
+<br />
+Du Bois-Reymond, <a href="#Page_50">50</a><br />
+<br />
+<br />
+earth, rotation of, <a href="#Page_326">326</a>, <a href="#Page_353">353</a><br />
+<br />
+eclipse, <a href="#Page_326">326</a><br />
+<br />
+electricity, <a href="#Page_174">174</a><br />
+<br />
+electrified bodies, <a href="#Page_117">117</a><br />
+<br />
+electrodynamic attraction, <a href="#Page_308">308</a><br />
+<span style="margin-left: 1em;">induction, <a href="#Page_188">188</a></span><br />
+<span style="margin-left: 1em;">mass, <a href="#Page_311">311</a></span><br />
+<br />
+electrodynamics, <a href="#Page_184">184</a>, <a href="#Page_282">282</a><br />
+<br />
+electromagnetic theory of light, <a href="#Page_301">301</a><br />
+<br />
+electrons, <a href="#Page_316">316</a>, <a href="#Page_492">492-4</a>, <a href="#Page_505">505-8</a>, <a href="#Page_510">510</a>, <a href="#Page_512">512-4</a><br />
+<br />
+elephant, <a href="#Page_217">217</a>, <a href="#Page_436">436</a><br />
+<br />
+ellipse, <a href="#Page_215">215</a><br />
+<br />
+Emerson, <a href="#Page_203">203</a><br />
+<br />
+empiricism, <a href="#Page_86">86</a>, <a href="#Page_271">271</a><br />
+<br />
+Epimenides, <a href="#Page_478">478-9</a><br />
+<br />
+equation of Laplace, <a href="#Page_283">283</a><br />
+<br />
+Erd&eacute;ly, <a href="#Page_203">203</a><br />
+<br />
+errors, accidental, <a href="#Page_171">171</a>, <a href="#Page_402">402</a><br />
+<span style="margin-left: 1em;">law of, <a href="#Page_119">119</a></span><br />
+<span style="margin-left: 1em;">systematic, <a href="#Page_171">171</a>, <a href="#Page_402">402</a></span><br />
+<span style="margin-left: 1em;">theory of, <a href="#Page_402">402</a>, <a href="#Page_406">406</a></span><br />
+<br />
+ether, <a href="#Page_145">145</a>, <a href="#Page_351">351</a><br />
+<br />
+ethics, <a href="#Page_205">205</a><br />
+<br />
+Euclid, <a href="#Page_62">62</a>, <a href="#Page_86">86</a>, <a href="#Page_202">202-3</a>, <a href="#Page_213">213</a><br />
+<br />
+Euclidean geometry, <a href="#Page_65">65</a>, <a href="#Page_235">235-6</a>, <a href="#Page_337">337</a><br />
+<br />
+Euclid's postulate, <a href="#Page_83">83</a>, <a href="#Page_91">91</a>, <a href="#Page_124">124</a>, <a href="#Page_353">353</a>, <a href="#Page_443">443</a>, <a href="#Page_453">453</a>, <a href="#Page_468">468</a>, <a href="#Page_470">470-1</a><br />
+<br />
+experience, <a href="#Page_90">90-1</a><br />
+<br />
+experiment, <a href="#Page_127">127</a>, <a href="#Page_317">317</a>, <a href="#Page_336">336</a>, <a href="#Page_446">446</a><br />
+<br />
+<br />
+fact, crude, <a href="#Page_326">326</a>, <a href="#Page_330">330</a><br />
+<span style="margin-left: 1em;">in the rough, <a href="#Page_327">327</a></span><br />
+<span style="margin-left: 1em;">scientific, <a href="#Page_326">326</a></span><br />
+<br />
+facts, <a href="#Page_362">362</a>, <a href="#Page_371">371</a><br />
+<br />
+Fahrenheit, <a href="#Page_238">238</a><br />
+<br />
+Faraday, <a href="#Page_150">150</a>, <a href="#Page_192">192</a><br />
+<br />
+Faye, <a href="#Page_536">536</a><br />
+<br />
+Fechner, <a href="#Page_46">46</a>, <a href="#Page_52">52</a><br />
+<br />
+Fehr, <a href="#Page_383">383</a><br />
+<br />
+finite, <a href="#Page_57">57</a><br />
+<span class='pagenum'><a name="Page_549" id="Page_549">[Pg 549]</a></span><br />
+Fitzgerald, <a href="#Page_415">415-6</a>, <a href="#Page_500">500-1</a>, <a href="#Page_505">505</a><br />
+<br />
+Fizeau, <a href="#Page_146">146</a>, <a href="#Page_149">149</a>, <a href="#Page_309">309</a>, <a href="#Page_498">498</a>, <a href="#Page_504">504</a><br />
+<br />
+Flammarion, <a href="#Page_400">400</a>, <a href="#Page_406">406-7</a><br />
+<br />
+flattening of the earth, <a href="#Page_353">353</a><br />
+<br />
+force, <a href="#Page_72">72</a>, <a href="#Page_98">98</a>, <a href="#Page_444">444</a><br />
+<span style="margin-left: 1em;">direction of, <a href="#Page_445">445</a></span><br />
+<span style="margin-left: 1em;">-flow, <a href="#Page_284">284</a></span><br />
+<br />
+forces, central, <a href="#Page_297">297</a><br />
+<span style="margin-left: 1em;">equivalence of, <a href="#Page_445">445</a></span><br />
+<span style="margin-left: 1em;">magnitude of, <a href="#Page_445">445</a></span><br />
+<br />
+Foucault's pendulum, <a href="#Page_85">85</a>, <a href="#Page_109">109</a>, <a href="#Page_353">353</a><br />
+<br />
+four dimensions, <a href="#Page_78">78</a><br />
+<br />
+Fourier, <a href="#Page_298">298-9</a><br />
+<br />
+Fourier's problem, <a href="#Page_317">317</a><br />
+<span style="margin-left: 1em;">series, <a href="#Page_286">286</a></span><br />
+<br />
+Franklin, <a href="#Page_513">513-4</a><br />
+<br />
+Fresnel, <a href="#Page_132">132</a>, <a href="#Page_140">140</a>, <a href="#Page_153">153</a>, <a href="#Page_174">174</a>, <a href="#Page_176">176</a>, <a href="#Page_181">181</a>, <a href="#Page_351">351</a>, <a href="#Page_498">498</a><br />
+<br />
+Fuchsian, <a href="#Page_387">387-8</a><br />
+<br />
+function, <a href="#Page_213">213</a><br />
+<span style="margin-left: 1em;">continuous, <a href="#Page_218">218</a>, <a href="#Page_288">288</a></span><br />
+<br />
+<br />
+Galileo, <a href="#Page_97">97</a>, <a href="#Page_331">331</a>, <a href="#Page_353">353-4</a><br />
+<br />
+gaseous pressure, <a href="#Page_141">141</a><br />
+<br />
+gases, theory of, <a href="#Page_400">400</a>, <a href="#Page_405">405</a>, <a href="#Page_523">523</a><br />
+<br />
+Gauss, <a href="#Page_384">384-5</a>, <a href="#Page_406">406</a><br />
+<br />
+Gay-Lussac, <a href="#Page_157">157</a><br />
+<br />
+generalize, <a href="#Page_342">342</a><br />
+<br />
+geodesy, <a href="#Page_535">535</a><br />
+<br />
+geometer, <a href="#Page_83">83</a>, <a href="#Page_210">210</a>, <a href="#Page_438">438</a><br />
+<br />
+geometric space, <a href="#Page_66">66</a><br />
+<br />
+geometry, <a href="#Page_72">72</a>, <a href="#Page_81">81</a>, <a href="#Page_125">125</a>, <a href="#Page_207">207</a>, <a href="#Page_380">380</a>, <a href="#Page_428">428</a>, <a href="#Page_442">442</a>, <a href="#Page_467">467</a><br />
+<span style="margin-left: 1em;">Euclidean, <a href="#Page_65">65</a>, <a href="#Page_93">93</a></span><br />
+<span style="margin-left: 1em;">fourth, <a href="#Page_62">62</a></span><br />
+<span style="margin-left: 1em;">non-Euclidean, <a href="#Page_55">55</a></span><br />
+<span style="margin-left: 1em;">projective, <a href="#Page_201">201</a></span><br />
+<span style="margin-left: 1em;">qualitative, <a href="#Page_238">238</a></span><br />
+<span style="margin-left: 1em;">rational, <a href="#Page_5">5</a>, <a href="#Page_467">467</a></span><br />
+<span style="margin-left: 1em;">Riemann's, <a href="#Page_57">57</a></span><br />
+<span style="margin-left: 1em;">spheric, <a href="#Page_59">59</a></span><br />
+<br />
+Gibbs, <a href="#Page_304">304</a><br />
+<br />
+Goldstein, <a href="#Page_492">492</a><br />
+<br />
+Gouy, <a href="#Page_152">152</a>, <a href="#Page_305">305</a>, <a href="#Page_410">410</a><br />
+<br />
+gravitation, <a href="#Page_512">512</a><br />
+<br />
+Greeks, <a href="#Page_93">93</a>, <a href="#Page_368">368</a><br />
+<br />
+<br />
+Hadamard, <a href="#Page_459">459</a><br />
+<br />
+Halsted, <a href="#Page_3">3</a>, <a href="#Page_203">203</a>, <a href="#Page_464">464</a>, <a href="#Page_467">467</a><br />
+<br />
+Hamilton, <a href="#Page_115">115</a><br />
+<br />
+helium, <a href="#Page_294">294</a><br />
+<br />
+Helmholtz, <a href="#Page_56">56</a>, <a href="#Page_115">115</a>, <a href="#Page_118">118</a>, <a href="#Page_141">141</a>, <a href="#Page_190">190</a>, <a href="#Page_196">196</a><br />
+<br />
+Hercules, <a href="#Page_449">449</a><br />
+<br />
+Hermite, <a href="#Page_211">211</a>, <a href="#Page_220">220</a>, <a href="#Page_222">222</a>, <a href="#Page_285">285</a><br />
+<br />
+Herschel, <a href="#Page_528">528</a><br />
+<br />
+Hertz, <a href="#Page_102">102</a>, <a href="#Page_145">145</a>, <a href="#Page_194">194-5</a>, <a href="#Page_427">427</a>, <a href="#Page_488">488</a>, <a href="#Page_498">498</a>, <a href="#Page_502">502</a>, <a href="#Page_504">504</a>, <a href="#Page_510">510</a><br />
+<br />
+Hertzian oscillator, <a href="#Page_309">309</a>, <a href="#Page_317">317</a><br />
+<br />
+Hilbert, <a href="#Page_5">5</a>, <a href="#Page_11">11</a>, <a href="#Page_203">203</a>, <a href="#Page_433">433</a>, <a href="#Page_450">450-1</a>, <a href="#Page_464">464-8</a>, <a href="#Page_471">471</a>, <a href="#Page_475">475-7</a>, <a href="#Page_484">484</a><br />
+<br />
+Himstedt, <a href="#Page_195">195</a><br />
+<br />
+Hipparchus, <a href="#Page_291">291</a><br />
+<br />
+homogeneity, <a href="#Page_74">74</a>, <a href="#Page_423">423</a><br />
+<br />
+homogeneous, <a href="#Page_67">67</a><br />
+<br />
+hydrodynamics, <a href="#Page_284">284</a><br />
+<br />
+hyperbola, <a href="#Page_215">215</a><br />
+<br />
+hypotheses, <a href="#Page_6">6</a>, <a href="#Page_15">15</a>, <a href="#Page_127">127</a>, <a href="#Page_133">133</a><br />
+<br />
+hysteresis, <a href="#Page_151">151</a><br />
+<br />
+<br />
+identity of spaces, <a href="#Page_268">268</a><br />
+<span style="margin-left: 1em;">of two points, <a href="#Page_259">259</a></span><br />
+<br />
+illusions, optical, <a href="#Page_202">202</a><br />
+<br />
+incommensurable numbers, <a href="#Page_44">44</a><br />
+<br />
+induction, complete, <a href="#Page_40">40</a>, <a href="#Page_452">452-3</a>, <a href="#Page_467">467-8</a><br />
+<span style="margin-left: 1em;">electromagnetic, <a href="#Page_188">188</a></span><br />
+<span style="margin-left: 1em;">mathematical, <a href="#Page_40">40</a>, <a href="#Page_220">220</a></span><br />
+<span style="margin-left: 1em;">principle of, <a href="#Page_481">481</a></span><br />
+<br />
+inertia, <a href="#Page_93">93</a>, <a href="#Page_486">486</a>, <a href="#Page_489">489</a>, <a href="#Page_507">507</a><br />
+<br />
+infinite, <a href="#Page_448">448</a><br />
+<br />
+infinitesimals, <a href="#Page_50">50</a><br />
+<br />
+inquisitor, <a href="#Page_331">331</a><br />
+<br />
+integration, <a href="#Page_139">139</a><br />
+<br />
+interpolation, <a href="#Page_131">131</a><br />
+<br />
+intuition, <a href="#Page_210">210</a>, <a href="#Page_213">213</a>, <a href="#Page_215">215</a><br />
+<br />
+invariant, <a href="#Page_333">333</a><br />
+<br />
+Ionians, <a href="#Page_127">127</a><br />
+<br />
+ions, <a href="#Page_152">152</a><br />
+<br />
+irrational number, <a href="#Page_44">44</a><br />
+<br />
+irreversible phenomena, <a href="#Page_151">151</a><br />
+<br />
+isotropic, <a href="#Page_67">67</a><br />
+<br />
+<br />
+Japanese mice, <a href="#Page_277">277</a>, <a href="#Page_427">427</a><br />
+<br />
+Jevons, <a href="#Page_451">451</a><br />
+<br />
+John Lackland, <a href="#Page_128">128</a><br />
+<br />
+Jules Verne, <a href="#Page_111">111</a>, <a href="#Page_536">536</a><br />
+<br />
+Jupiter, <a href="#Page_131">131</a>, <a href="#Page_157">157</a>, <a href="#Page_231">231</a>, <a href="#Page_289">289</a><br />
+<br />
+<br />
+Kant, <a href="#Page_16">16</a>, <a href="#Page_64">64</a>, <a href="#Page_202">202-3</a>, <a href="#Page_450">450-1</a>, <a href="#Page_471">471</a><br />
+<span class='pagenum'><a name="Page_550" id="Page_550">[Pg 550]</a></span><br />
+Kauffman, <a href="#Page_311">311</a>, <a href="#Page_490">490-1</a>, <a href="#Page_495">495</a>, <a href="#Page_506">506-7</a>, <a href="#Page_522">522</a>, <a href="#Page_545">545</a><br />
+<br />
+Kazan, <a href="#Page_203">203</a><br />
+<br />
+Kelvin, <a href="#Page_145">145</a>, <a href="#Page_523">523-4</a>, <a href="#Page_526">526-7</a><br />
+<br />
+Kepler, <a href="#Page_120">120</a>, <a href="#Page_133">133</a>, <a href="#Page_153">153</a>, <a href="#Page_282">282</a>, <a href="#Page_291">291-2</a><br />
+<br />
+Kepler's laws, <a href="#Page_136">136</a>, <a href="#Page_516">516</a><br />
+<br />
+kinematics, <a href="#Page_337">337</a><br />
+<br />
+kinetic energy, <a href="#Page_116">116</a><br />
+<span style="margin-left: 1em;">theory of gases, <a href="#Page_141">141</a></span><br />
+<br />
+Kirchhoff, <a href="#Page_98">98-9</a>, <a href="#Page_103">103-5</a><br />
+<br />
+Klein, <a href="#Page_60">60</a>, <a href="#Page_211">211</a>, <a href="#Page_287">287</a><br />
+<br />
+knowledge, <a href="#Page_201">201</a><br />
+<br />
+K&ouml;nig, <a href="#Page_144">144</a>, <a href="#Page_477">477</a><br />
+<br />
+Kovalevski, <a href="#Page_212">212</a>, <a href="#Page_286">286</a><br />
+<br />
+Kronecker, <a href="#Page_44">44</a><br />
+<br />
+<br />
+Lacombe, <a href="#Page_543">543</a><br />
+<br />
+La Condamine, <a href="#Page_535">535</a>, <a href="#Page_537">537-8</a>, <a href="#Page_542">542-3</a><br />
+<br />
+Lagrange, <a href="#Page_98">98</a>, <a href="#Page_151">151</a>, <a href="#Page_179">179</a><br />
+<br />
+Laisant, <a href="#Page_383">383</a><br />
+<br />
+Lallamand, <a href="#Page_543">543</a><br />
+<br />
+Langevin, <a href="#Page_509">509</a><br />
+<br />
+Laplace, <a href="#Page_298">298</a>, <a href="#Page_398">398</a>, <a href="#Page_514">514-5</a>, <a href="#Page_518">518</a>, <a href="#Page_522">522</a>, <a href="#Page_538">538</a><br />
+<br />
+Laplace's equation, <a href="#Page_283">283</a>, <a href="#Page_287">287</a><br />
+<br />
+Larmor, <a href="#Page_145">145</a>, <a href="#Page_150">150</a><br />
+<br />
+Lavoisier's principle, <a href="#Page_301">301</a>, <a href="#Page_310">310</a>, <a href="#Page_312">312</a><br />
+<br />
+law, <a href="#Page_207">207</a>, <a href="#Page_291">291</a>, <a href="#Page_395">395</a><br />
+<br />
+Leibnitz, <a href="#Page_32">32</a>, <a href="#Page_450">450</a>, <a href="#Page_471">471</a><br />
+<br />
+Le Roy, <a href="#Page_28">28</a>, <a href="#Page_321">321-6</a>, <a href="#Page_332">332</a>, <a href="#Page_335">335</a>, <a href="#Page_337">337</a>, <a href="#Page_347">347-8</a>, <a href="#Page_354">354</a>, <a href="#Page_468">468</a><br />
+<br />
+Lesage, <a href="#Page_517">517-21</a><br />
+<br />
+Liard, <a href="#Page_440">440</a><br />
+<br />
+Lie, <a href="#Page_62">62-3</a>, <a href="#Page_212">212</a><br />
+<br />
+light sensations, <a href="#Page_252">252</a><br />
+<span style="margin-left: 1em;">theory of, <a href="#Page_351">351</a></span><br />
+<span style="margin-left: 1em;">velocity of, <a href="#Page_232">232</a>, <a href="#Page_312">312</a></span><br />
+<br />
+Lindemann, <a href="#Page_508">508</a><br />
+<br />
+line, <a href="#Page_203">203</a>, <a href="#Page_243">243</a><br />
+<br />
+linkages, <a href="#Page_144">144</a><br />
+<br />
+Lippmann, <a href="#Page_196">196</a><br />
+<br />
+Lobachevski, <a href="#Page_29">29</a>, <a href="#Page_56">56</a>, <a href="#Page_60">60</a>, <a href="#Page_62">62</a>, <a href="#Page_83">83</a>, <a href="#Page_86">86</a>, <a href="#Page_203">203</a><br />
+<br />
+Lobachevski's space, <a href="#Page_239">239</a><br />
+<br />
+local time, <a href="#Page_306">306-7</a>, <a href="#Page_499">499</a><br />
+<br />
+logic, <a href="#Page_214">214</a>, <a href="#Page_435">435</a>, <a href="#Page_448">448</a>, <a href="#Page_460">460-2</a>, <a href="#Page_464">464</a><br />
+<br />
+logistic, <a href="#Page_457">457</a>, <a href="#Page_472">472-4</a><br />
+<br />
+logisticians, <a href="#Page_472">472</a><br />
+<br />
+Lorentz, <a href="#Page_147">147</a>, <a href="#Page_149">149</a>, <a href="#Page_196">196-7</a>, <a href="#Page_306">306</a>, <a href="#Page_308">308</a>, <a href="#Page_311">311</a>, <a href="#Page_315">315</a>, <a href="#Page_415">415-6</a>, <a href="#Page_492">492</a>, <a href="#Page_498">498-502</a>, <a href="#Page_504">504-9</a>, <a href="#Page_512">512</a>, <a href="#Page_514">514-6</a>, <a href="#Page_521">521</a><br />
+<br />
+Lotze, <a href="#Page_264">264</a><br />
+<br />
+luck, <a href="#Page_399">399</a><br />
+<br />
+Lumen, <a href="#Page_407">407-8</a><br />
+<br />
+<br />
+MacCullagh, <a href="#Page_150">150</a><br />
+<br />
+Mach, <a href="#Page_375">375</a><br />
+<br />
+Mach-Delage, <a href="#Page_276">276</a><br />
+<br />
+magnetism, <a href="#Page_149">149</a><br />
+<br />
+magnitude, <a href="#Page_49">49</a><br />
+<br />
+Mariotte's law, <a href="#Page_120">120</a>, <a href="#Page_132">132</a>, <a href="#Page_157">157</a>, <a href="#Page_342">342</a>, <a href="#Page_524">524</a><br />
+<br />
+Maros, <a href="#Page_203">203</a><br />
+<br />
+mass, <a href="#Page_98">98</a>, <a href="#Page_312">312</a>, <a href="#Page_446">446</a>, <a href="#Page_486">486</a>, <a href="#Page_489">489</a>, <a href="#Page_494">494</a>, <a href="#Page_515">515</a><br />
+<br />
+mathematical analysis, <a href="#Page_218">218</a><br />
+<span style="margin-left: 1em;">continuum, <a href="#Page_46">46</a></span><br />
+<span style="margin-left: 1em;">creation, <a href="#Page_383">383</a></span><br />
+<span style="margin-left: 1em;">induction, <a href="#Page_40">40</a>, <a href="#Page_220">220</a></span><br />
+<span style="margin-left: 1em;">physics, <a href="#Page_136">136</a>, <a href="#Page_297">297</a>, <a href="#Page_319">319</a></span><br />
+<br />
+mathematics, <a href="#Page_369">369</a>, <a href="#Page_448">448</a><br />
+<br />
+matter, <a href="#Page_492">492</a><br />
+<br />
+Maupertuis, <a href="#Page_535">535</a>, <a href="#Page_537">537-8</a><br />
+<br />
+Maurain, <a href="#Page_543">543</a><br />
+<br />
+Maxwell, <a href="#Page_140">140</a>, <a href="#Page_152">152</a>, <a href="#Page_175">175</a>, <a href="#Page_177">177</a>, <a href="#Page_181">181</a>, <a href="#Page_193">193</a>, <a href="#Page_282">282-3</a>, <a href="#Page_298">298</a>, <a href="#Page_301">301</a>, <a href="#Page_304">304-5</a>, <a href="#Page_351">351</a>, <a href="#Page_503">503</a>, <a href="#Page_524">524-5</a><br />
+<br />
+Maxwell-Bartholi, <a href="#Page_309">309</a>, <a href="#Page_503">503-4</a>, <a href="#Page_519">519</a>, <a href="#Page_521">521</a><br />
+<br />
+Mayer, <a href="#Page_119">119</a>, <a href="#Page_123">123</a>, <a href="#Page_300">300</a>, <a href="#Page_312">312</a>, <a href="#Page_318">318</a><br />
+<br />
+measurement, <a href="#Page_49">49</a><br />
+<br />
+M&eacute;chain, <a href="#Page_539">539-40</a><br />
+<br />
+mechanical explanation, <a href="#Page_177">177</a><br />
+<span style="margin-left: 1em;">mass, <a href="#Page_312">312</a></span><br />
+<br />
+mechanics, <a href="#Page_92">92</a>, <a href="#Page_444">444</a>, <a href="#Page_486">486</a>, <a href="#Page_496">496</a>, <a href="#Page_512">512</a><br />
+<span style="margin-left: 1em;">anthropomorphic, <a href="#Page_103">103</a></span><br />
+<span style="margin-left: 1em;">celestial, <a href="#Page_279">279</a></span><br />
+<span style="margin-left: 1em;">statistical, <a href="#Page_304">304</a></span><br />
+<br />
+M&eacute;ray, <a href="#Page_211">211</a><br />
+<br />
+metaphysician, <a href="#Page_221">221</a><br />
+<br />
+meteorology, <a href="#Page_398">398</a><br />
+<br />
+mice, <a href="#Page_277">277</a><br />
+<br />
+Michelson, <a href="#Page_306">306</a>, <a href="#Page_309">309</a>, <a href="#Page_311">311</a>, <a href="#Page_316">316</a>, <a href="#Page_498">498</a>, <a href="#Page_500">500-1</a><br />
+<br />
+milky way, <a href="#Page_523">523-30</a><br />
+<br />
+Mill, Stuart, <a href="#Page_60">60-1</a>, <a href="#Page_453">453-4</a><br />
+<br />
+Monist, <a href="#Page_4">4</a>, <a href="#Page_89">89</a>, <a href="#Page_464">464</a><br />
+<br />
+moons of Jupiter, <a href="#Page_233">233</a><br />
+<br />
+Morley, <a href="#Page_309">309</a><br />
+<br />
+motion of liquids, <a href="#Page_283">283</a><br />
+<span class='pagenum'><a name="Page_551" id="Page_551">[Pg 551]</a></span><span style="margin-left: 1em;">of moon, <a href="#Page_28">28</a></span><br />
+<span style="margin-left: 1em;">of planets, <a href="#Page_341">341</a></span><br />
+<span style="margin-left: 1em;">relative, <a href="#Page_107">107</a>, <a href="#Page_487">487</a></span><br />
+<span style="margin-left: 1em;">without deformation, <a href="#Page_236">236</a></span><br />
+<br />
+multiplication, <a href="#Page_36">36</a><br />
+<br />
+muscular sensations, <a href="#Page_69">69</a><br />
+<br />
+<br />
+Nagaoka, <a href="#Page_317">317</a><br />
+<br />
+nature, <a href="#Page_127">127</a><br />
+<br />
+navigation, <a href="#Page_289">289</a><br />
+<br />
+neodymium, <a href="#Page_333">333</a><br />
+<br />
+neomonics, <a href="#Page_283">283</a><br />
+<br />
+Neumann, <a href="#Page_181">181</a><br />
+<br />
+Newton, <a href="#Page_85">85</a>, <a href="#Page_96">96</a>, <a href="#Page_98">98</a>, <a href="#Page_109">109</a>, <a href="#Page_153">153</a>, <a href="#Page_291">291</a>, <a href="#Page_370">370</a>, <a href="#Page_486">486</a>, <a href="#Page_516">516</a>, <a href="#Page_536">536</a>, <a href="#Page_538">538</a><br />
+<br />
+Newton's argument, <a href="#Page_108">108</a>, <a href="#Page_334">334</a>, <a href="#Page_343">343</a><br />
+<span style="margin-left: 1em;">law, <a href="#Page_111">111</a>, <a href="#Page_118">118</a>, <a href="#Page_132">132</a>, <a href="#Page_136">136</a>, <a href="#Page_149">149</a>, <a href="#Page_157">157</a>, <a href="#Page_233">233</a>, <a href="#Page_282">282</a>, <a href="#Page_292">292</a>, <a href="#Page_512">512</a>, <a href="#Page_514">514-5</a>, <a href="#Page_518">518</a>, <a href="#Page_525">525</a></span><br />
+<span style="margin-left: 1em;">principle, <a href="#Page_146">146</a>, <a href="#Page_300">300</a>, <a href="#Page_308">308-9</a>, <a href="#Page_312">312</a></span><br />
+<br />
+no-class theory, <a href="#Page_478">478</a><br />
+<br />
+nominalism, <a href="#Page_28">28</a>, <a href="#Page_125">125</a>, <a href="#Page_321">321</a>, <a href="#Page_333">333</a>, <a href="#Page_335">335</a><br />
+<br />
+non-Euclidean geometry, <a href="#Page_55">55</a>, <a href="#Page_59">59</a>, <a href="#Page_388">388</a><br />
+<span style="margin-left: 1em;">language, <a href="#Page_127">127</a></span><br />
+<span style="margin-left: 1em;">space, <a href="#Page_55">55</a>, <a href="#Page_235">235</a>, <a href="#Page_237">237</a></span><br />
+<span style="margin-left: 1em;">straight, <a href="#Page_236">236</a>, <a href="#Page_470">470</a></span><br />
+<span style="margin-left: 1em;">world, <a href="#Page_75">75</a></span><br />
+<br />
+number, <a href="#Page_31">31</a><br />
+<span style="margin-left: 1em;">big, <a href="#Page_88">88</a></span><br />
+<span style="margin-left: 1em;">imaginary, <a href="#Page_283">283</a></span><br />
+<span style="margin-left: 1em;">incommensurable, <a href="#Page_44">44</a></span><br />
+<span style="margin-left: 1em;">transfinite, <a href="#Page_448">448-9</a></span><br />
+<span style="margin-left: 1em;">whole, <a href="#Page_44">44</a>, <a href="#Page_469">469</a></span><br />
+<br />
+<br />
+objectivity, <a href="#Page_209">209</a>, <a href="#Page_347">347</a>, <a href="#Page_349">349</a>, <a href="#Page_408">408</a><br />
+<br />
+optical illusions, <a href="#Page_202">202</a><br />
+<br />
+optics, <a href="#Page_174">174</a>, <a href="#Page_496">496</a><br />
+<br />
+orbit of Saturn, <a href="#Page_341">341</a><br />
+<br />
+order, <a href="#Page_385">385</a><br />
+<br />
+orientation, <a href="#Page_83">83</a><br />
+<br />
+osmotic, <a href="#Page_141">141</a><br />
+<br />
+<br />
+Padoa, <a href="#Page_463">463</a><br />
+<br />
+Panth&eacute;on, <a href="#Page_414">414</a><br />
+<br />
+parallax, <a href="#Page_470">470</a><br />
+<br />
+parallels, <a href="#Page_56">56</a>, <a href="#Page_443">443</a><br />
+<br />
+Paris time, <a href="#Page_233">233</a><br />
+<br />
+parry, <a href="#Page_419">419-22</a>, <a href="#Page_427">427</a><br />
+<br />
+partition, <a href="#Page_45">45</a><br />
+<br />
+pasigraphy, <a href="#Page_456">456-7</a><br />
+<br />
+Pasteur, <a href="#Page_128">128</a><br />
+<br />
+Peano, <a href="#Page_450">450</a>, <a href="#Page_456">456-9</a>, <a href="#Page_463">463</a>, <a href="#Page_472">472</a><br />
+<br />
+Pender, <a href="#Page_490">490</a><br />
+<br />
+pendulum, <a href="#Page_224">224</a><br />
+<br />
+Perrier, <a href="#Page_541">541</a><br />
+<br />
+Perrin, <a href="#Page_195">195</a><br />
+<br />
+phosphorus, <a href="#Page_333">333</a>, <a href="#Page_468">468</a>, <a href="#Page_470">470-1</a><br />
+<br />
+physical continuum, <a href="#Page_46">46</a><br />
+<br />
+physics, <a href="#Page_127">127</a>, <a href="#Page_140">140</a>, <a href="#Page_144">144</a>, <a href="#Page_279">279</a>, <a href="#Page_297">297</a><br />
+<br />
+physics of central forces, <a href="#Page_297">297</a><br />
+<span style="margin-left: 1em;">of the principles, <a href="#Page_299">299</a></span><br />
+<br />
+Pieri, <a href="#Page_11">11</a>, <a href="#Page_203">203</a><br />
+<br />
+Plato, <a href="#Page_292">292</a><br />
+<br />
+Poincar&eacute;, <a href="#Page_473">473</a><br />
+<br />
+point, <a href="#Page_89">89</a>, <a href="#Page_244">244</a><br />
+<br />
+Poncelet, <a href="#Page_215">215</a><br />
+<br />
+postulates, <a href="#Page_382">382</a><br />
+<br />
+potential energy, <a href="#Page_116">116</a><br />
+<br />
+praseodymium, <a href="#Page_333">333</a><br />
+<br />
+principle, <a href="#Page_125">125</a>, <a href="#Page_299">299</a><br />
+<span style="margin-left: 1em;">Carnot's, <a href="#Page_143">143</a>, <a href="#Page_151">151</a>, <a href="#Page_300">300</a>, <a href="#Page_303">303-5</a>, <a href="#Page_399">399</a></span><br />
+<span style="margin-left: 1em;">Clausius', <a href="#Page_119">119</a>, <a href="#Page_123">123</a>, <a href="#Page_143">143</a></span><br />
+<span style="margin-left: 1em;">Hamilton's, <a href="#Page_115">115</a></span><br />
+<span style="margin-left: 1em;">Lavoisier's, <a href="#Page_300">300</a>, <a href="#Page_310">310</a></span><br />
+<span style="margin-left: 1em;">Mayer's, <a href="#Page_119">119</a>, <a href="#Page_121">121</a>, <a href="#Page_123">123</a>, <a href="#Page_300">300</a>, <a href="#Page_312">312</a>, <a href="#Page_318">318</a></span><br />
+<span style="margin-left: 1em;">Newton's, <a href="#Page_146">146</a>, <a href="#Page_300">300</a>, <a href="#Page_308">308-9</a>, <a href="#Page_312">312</a></span><br />
+<span style="margin-left: 1em;">of action and reaction, <a href="#Page_300">300</a>, <a href="#Page_487">487</a>, <a href="#Page_502">502</a></span><br />
+<span style="margin-left: 1em;">of conservation of energy, <a href="#Page_300">300</a></span><br />
+<span style="margin-left: 1em;">of degradation of energy, <a href="#Page_300">300</a></span><br />
+<span style="margin-left: 1em;">of inertia, <a href="#Page_93">93</a>, <a href="#Page_486">486</a>, <a href="#Page_507">507</a></span><br />
+<span style="margin-left: 1em;">of least action, <a href="#Page_118">118</a>, <a href="#Page_300">300</a></span><br />
+<span style="margin-left: 1em;">of relativity, <a href="#Page_300">300</a>, <a href="#Page_305">305</a>, <a href="#Page_498">498</a>, <a href="#Page_505">505</a></span><br />
+<br />
+Prony, <a href="#Page_445">445</a><br />
+<br />
+psychologist, <a href="#Page_383">383</a><br />
+<br />
+Ptolemy, <a href="#Page_110">110</a>, <a href="#Page_291">291</a>, <a href="#Page_353">353-4</a><br />
+<br />
+Pythagoras, <a href="#Page_292">292</a><br />
+<br />
+<br />
+quadrature of the circle, <a href="#Page_161">161</a><br />
+<br />
+qualitative geometry, <a href="#Page_238">238</a><br />
+<span style="margin-left: 1em;">space, <a href="#Page_207">207</a></span><br />
+<span style="margin-left: 1em;">time, <a href="#Page_224">224</a></span><br />
+<br />
+quaternions, <a href="#Page_282">282</a><br />
+<span class='pagenum'><a name="Page_552" id="Page_552">[Pg 552]</a></span><br />
+<br />
+radiometer, <a href="#Page_503">503</a><br />
+<br />
+radium, <a href="#Page_312">312</a>, <a href="#Page_318">318</a>, <a href="#Page_486">486-7</a><br />
+<br />
+Rados, <a href="#Page_201">201</a><br />
+<br />
+Ramsay, <a href="#Page_313">313</a><br />
+<br />
+rational geometry, <a href="#Page_5">5</a>, <a href="#Page_467">467</a><br />
+<br />
+reaction, <a href="#Page_502">502</a><br />
+<br />
+reality, <a href="#Page_217">217</a>, <a href="#Page_340">340</a>, <a href="#Page_349">349</a><br />
+<br />
+R&eacute;aumur, <a href="#Page_238">238</a><br />
+<br />
+recurrence, <a href="#Page_37">37</a><br />
+<br />
+Regnault, <a href="#Page_170">170</a><br />
+<br />
+relativity, <a href="#Page_83">83</a>, <a href="#Page_305">305</a>, <a href="#Page_417">417</a>, <a href="#Page_423">423</a>, <a href="#Page_498">498</a>, <a href="#Page_505">505</a><br />
+<br />
+Richard, <a href="#Page_477">477-8</a>, <a href="#Page_480">480-1</a><br />
+<br />
+Riemann, <a href="#Page_56">56</a>, <a href="#Page_62">62</a>, <a href="#Page_145">145</a>, <a href="#Page_212">212</a>, <a href="#Page_239">239</a>, <a href="#Page_243">243</a>, <a href="#Page_381">381</a>, <a href="#Page_432">432</a><br />
+<span style="margin-left: 1em;">surface, <a href="#Page_211">211</a>, <a href="#Page_287">287</a></span><br />
+<br />
+Roemer, <a href="#Page_233">233</a><br />
+<br />
+R&ouml;ntgen, <a href="#Page_511">511</a>, <a href="#Page_520">520</a><br />
+<br />
+rotation of earth, <a href="#Page_225">225</a>, <a href="#Page_331">331</a>, <a href="#Page_353">353</a><br />
+<br />
+roulette, <a href="#Page_403">403</a><br />
+<br />
+Rowland, <a href="#Page_194">194-7</a>, <a href="#Page_305">305</a>, <a href="#Page_489">489</a><br />
+<br />
+Royce, <a href="#Page_202">202</a><br />
+<br />
+Russell, <a href="#Page_201">201</a>, <a href="#Page_450">450</a>, <a href="#Page_460">460-2</a>, <a href="#Page_464">464-7</a>, <a href="#Page_471">471-4</a>, <a href="#Page_477">477-82</a>, <a href="#Page_484">484-5</a><br />
+<br />
+<br />
+St. Louis exposition, <a href="#Page_208">208</a>, <a href="#Page_320">320</a><br />
+<br />
+Sarcey, <a href="#Page_442">442</a><br />
+<br />
+Saturn, <a href="#Page_231">231</a>, <a href="#Page_317">317</a><br />
+<br />
+Schiller, <a href="#Page_202">202</a><br />
+<br />
+Schliemann, <a href="#Page_19">19</a><br />
+<br />
+science, <a href="#Page_205">205</a>, <a href="#Page_321">321</a>, <a href="#Page_323">323</a>, <a href="#Page_340">340</a>, <a href="#Page_354">354</a><br />
+<br />
+Science and Hypothesis, <a href="#Page_205">205-7</a>, <a href="#Page_220">220</a>, <a href="#Page_240">240</a>, <a href="#Page_246">246-7</a>, <a href="#Page_319">319</a>, <a href="#Page_353">353</a>, <a href="#Page_452">452</a><br />
+<br />
+semicircular canals, <a href="#Page_276">276</a><br />
+<br />
+series, development in, <a href="#Page_287">287</a><br />
+<span style="margin-left: 1em;">Fourier's, <a href="#Page_286">286</a></span><br />
+<br />
+Sirius, <a href="#Page_226">226</a>, <a href="#Page_229">229</a><br />
+<br />
+solid bodies, <a href="#Page_72">72</a><br />
+<br />
+space, <a href="#Page_55">55</a>, <a href="#Page_66">66</a>, <a href="#Page_89">89</a>, <a href="#Page_235">235</a>, <a href="#Page_256">256</a><br />
+<span style="margin-left: 1em;">absolute, <a href="#Page_85">85</a>, <a href="#Page_93">93</a></span><br />
+<span style="margin-left: 1em;">amorphous, <a href="#Page_417">417</a></span><br />
+<span style="margin-left: 1em;">Bolyai, <a href="#Page_56">56</a></span><br />
+<span style="margin-left: 1em;">Euclidean, <a href="#Page_65">65</a></span><br />
+<span style="margin-left: 1em;">geometric, <a href="#Page_66">66</a></span><br />
+<span style="margin-left: 1em;">Lobachevski's, <a href="#Page_239">239</a></span><br />
+<span style="margin-left: 1em;">motor, <a href="#Page_69">69</a></span><br />
+<span style="margin-left: 1em;">non-Euclidean, <a href="#Page_55">55</a>, <a href="#Page_235">235</a>, <a href="#Page_237">237</a></span><br />
+<span style="margin-left: 1em;">of four dimensions, <a href="#Page_78">78</a></span><br />
+<span style="margin-left: 1em;">perceptual, <a href="#Page_66">66</a>, <a href="#Page_69">69</a></span><br />
+<span style="margin-left: 1em;">tactile, <a href="#Page_68">68</a>, <a href="#Page_264">264</a></span><br />
+<span style="margin-left: 1em;">visual, <a href="#Page_67">67</a>, <a href="#Page_252">252</a></span><br />
+<br />
+spectra, <a href="#Page_316">316</a><br />
+<br />
+spectroscope, <a href="#Page_294">294</a><br />
+<br />
+Spencer, <a href="#Page_9">9</a><br />
+<br />
+sponge, <a href="#Page_219">219</a><br />
+<br />
+Stallo, <a href="#Page_10">10</a><br />
+<br />
+stars, <a href="#Page_292">292</a><br />
+<br />
+statistical mechanics, <a href="#Page_304">304</a><br />
+<br />
+straight, <a href="#Page_62">62</a>, <a href="#Page_82">82</a>, <a href="#Page_236">236</a>, <a href="#Page_433">433</a>, <a href="#Page_450">450</a>, <a href="#Page_470">470</a><br />
+<br />
+Stratonoff, <a href="#Page_531">531</a><br />
+<br />
+surfaces, <a href="#Page_58">58</a><br />
+<br />
+systematic errors, <a href="#Page_171">171</a><br />
+<br />
+<br />
+tactile space, <a href="#Page_68">68</a>, <a href="#Page_264">264</a><br />
+<br />
+Tait, <a href="#Page_98">98</a><br />
+<br />
+tangent, <a href="#Page_51">51</a><br />
+<br />
+Tannery, <a href="#Page_43">43</a><br />
+<br />
+teaching, <a href="#Page_430">430</a>, <a href="#Page_437">437</a><br />
+<br />
+thermodynamics, <a href="#Page_115">115</a>, <a href="#Page_119">119</a><br />
+<br />
+Thomson, <a href="#Page_98">98</a>, <a href="#Page_488">488</a><br />
+<br />
+thread, <a href="#Page_104">104</a><br />
+<br />
+time, <a href="#Page_223">223</a><br />
+<span style="margin-left: 1em;">equality, <a href="#Page_225">225</a></span><br />
+<span style="margin-left: 1em;">local, <a href="#Page_306">306</a>, <a href="#Page_307">307</a></span><br />
+<span style="margin-left: 1em;">measure of, <a href="#Page_223">223-4</a></span><br />
+<br />
+Tisserand, <a href="#Page_515">515-6</a><br />
+<br />
+Tolstoi, <a href="#Page_354">354</a>, <a href="#Page_362">362</a>, <a href="#Page_368">368</a><br />
+<br />
+Tommasina, <a href="#Page_519">519</a><br />
+<br />
+Transylvania, <a href="#Page_203">203</a><br />
+<br />
+triangle, <a href="#Page_58">58</a><br />
+<span style="margin-left: 1em;">angle sum of, <a href="#Page_58">58</a></span><br />
+<br />
+truth, <a href="#Page_205">205</a><br />
+<br />
+Tycho Brahe, <a href="#Page_133">133</a>, <a href="#Page_153">153</a>, <a href="#Page_228">228</a><br />
+<br />
+<br />
+unity of nature, <a href="#Page_130">130</a><br />
+<br />
+universal invariant, <a href="#Page_333">333</a><br />
+<br />
+Uriel, <a href="#Page_203">203</a><br />
+<br />
+<br />
+van der Waals, <a href="#Page_153">153</a><br />
+<br />
+Vauban, <a href="#Page_210">210</a><br />
+<br />
+Veblen, <a href="#Page_203">203</a><br />
+<br />
+velocity of light, <a href="#Page_232">232</a>, <a href="#Page_312">312</a><br />
+<br />
+Venus of Milo, <a href="#Page_201">201</a><br />
+<br />
+verification, <a href="#Page_33">33</a><br />
+<br />
+Virchow, <a href="#Page_21">21</a><br />
+<span class='pagenum'><a name="Page_553" id="Page_553">[Pg 553]</a></span><br />
+visual impressions, <a href="#Page_252">252</a><br />
+<span style="margin-left: 1em;">space, <a href="#Page_67">67</a>, <a href="#Page_252">252</a></span><br />
+<br />
+Volga, <a href="#Page_203">203</a><br />
+<br />
+Voltaire, <a href="#Page_537">537-8</a><br />
+<br />
+<br />
+Weber, <a href="#Page_117">117</a>, <a href="#Page_515">515-6</a><br />
+<br />
+Weierstrass, <a href="#Page_11">11</a>, <a href="#Page_212">212</a>, <a href="#Page_432">432</a><br />
+<br />
+Whitehead, <a href="#Page_472">472</a>, <a href="#Page_481">481-2</a><br />
+<br />
+whole numbers, <a href="#Page_44">44</a><br />
+<br />
+Wiechert, <a href="#Page_145">145</a>, <a href="#Page_488">488</a><br />
+<br />
+<br />
+x-rays, <a href="#Page_152">152</a>, <a href="#Page_511">511</a>, <a href="#Page_520">520</a><br />
+<br />
+<br />
+Zeeman effect, <a href="#Page_152">152</a>, <a href="#Page_196">196</a>, <a href="#Page_317">317</a>, <a href="#Page_494">494</a><br />
+<br />
+Zeno, <a href="#Page_382">382</a><br />
+<br />
+Zermelo, <a href="#Page_477">477</a>, <a href="#Page_482">482-3</a><br />
+<br />
+zigzag theory, <a href="#Page_478">478</a><br />
+<br />
+zodiac, <a href="#Page_398">398</a>, <a href="#Page_404">404</a><br />
+</p>
+
+
+
+
+
+
+<hr style="width: 100%;" />
+<h2>FOOTNOTES</h2>
+
+<div class="footnote"><p><a name="Footnote_1_1" id="Footnote_1_1"></a><a href="#FNanchor_1_1"><span class="label">[1]</span></a> See Le Roy, 'Science et Philosophie,' <i>Revue de M&eacute;taphysique et de
+Morale</i>, 1901.</p></div>
+
+<div class="footnote"><p><a name="Footnote_2_2" id="Footnote_2_2"></a><a href="#FNanchor_2_2"><span class="label">[2]</span></a> With those contained in the special conventions which serve to define
+addition and of which we shall speak later.</p></div>
+
+<div class="footnote"><p><a name="Footnote_3_3" id="Footnote_3_3"></a><a href="#FNanchor_3_3"><span class="label">[3]</span></a> <i>Revue de M&eacute;taphysique et de Morale</i>, t. VI., pp. 1-13 (January, 1898).</p></div>
+
+<div class="footnote"><p><a name="Footnote_4_4" id="Footnote_4_4"></a><a href="#FNanchor_4_4"><span class="label">[4]</span></a> The following lines are a partial reproduction of the preface of my
+book <i>Thermodynamique</i>.</p></div>
+
+<div class="footnote"><p><a name="Footnote_5_5" id="Footnote_5_5"></a><a href="#FNanchor_5_5"><span class="label">[5]</span></a> This chapter is a partial reproduction of the prefaces of two of my
+works: <i>Th&eacute;orie math&eacute;matique de la lumi&egrave;re</i> (Paris, Naud, 1889), and <i>&Eacute;lectricit&eacute;
+et optique</i> (Paris, Naud, 1901).</p></div>
+
+<div class="footnote"><p><a name="Footnote_6_6" id="Footnote_6_6"></a><a href="#FNanchor_6_6"><span class="label">[6]</span></a> We add that <i>U</i> will depend only on the parameters <i>q</i>, that <i>T</i> will depend
+on the parameters <i>q</i> and their derivatives with respect to the time and will
+be a homogeneous polynomial of the second degree with respect to these
+derivatives.</p></div>
+
+<div class="footnote"><p><a name="Footnote_7_7" id="Footnote_7_7"></a><a href="#FNanchor_7_7"><span class="label">[7]</span></a> <i>Etude sur les diverses grandeurs</i>, Paris, Gauthier-Villars, 1897.</p></div>
+
+<div class="footnote"><p><a name="Footnote_8_8" id="Footnote_8_8"></a><a href="#FNanchor_8_8"><span class="label">[8]</span></a> In place of saying that we refer space to axes rigidly bound to our
+body, perhaps it would be better to say, in conformity to what precedes,
+that we refer it to axes rigidly bound to the initial situation of our body.</p></div>
+
+<div class="footnote"><p><a name="Footnote_9_9" id="Footnote_9_9"></a><a href="#FNanchor_9_9"><span class="label">[9]</span></a> Because bodies would oppose an increasing inertia to the causes which
+would tend to accelerate their motion; and this inertia would become infinite
+when one approached the velocity of light.</p></div>
+
+<div class="footnote"><p><a name="Footnote_10_10" id="Footnote_10_10"></a><a href="#FNanchor_10_10"><span class="label">[10]</span></a> These considerations on mathematical physics are borrowed from my
+St. Louis address.</p></div>
+
+<div class="footnote"><p><a name="Footnote_11_11" id="Footnote_11_11"></a><a href="#FNanchor_11_11"><span class="label">[11]</span></a> I here use the word real as a synonym of objective; I thus conform to
+common usage; perhaps I am wrong, our dreams are real, but they are not
+objective.</p></div>
+
+<div class="footnote"><p><a name="Footnote_12_12" id="Footnote_12_12"></a><a href="#FNanchor_12_12"><span class="label">[12]</span></a> See <i>Science and Hypothesis</i>, chapter I.</p></div>
+
+<div class="footnote"><p><a name="Footnote_13_13" id="Footnote_13_13"></a><a href="#FNanchor_13_13"><span class="label">[13]</span></a> 'The Foundations of Logic and Arithmetic,' <i>Monist</i>, XV., 338-352.</p></div>
+
+<div class="footnote"><p><a name="Footnote_14_14" id="Footnote_14_14"></a><a href="#FNanchor_14_14"><span class="label">[14]</span></a> Second ed., 1907, p. 86; French ed., 1911, p. 97. G. B. H.</p></div>
+
+<div class="footnote"><p><a name="Footnote_15_15" id="Footnote_15_15"></a><a href="#FNanchor_15_15"><span class="label">[15]</span></a> <i>Revue g&eacute;n&eacute;rale des sciences</i>, June 30, 1905.</p></div>
+
+<div class="footnote"><p><a name="Footnote_16_16" id="Footnote_16_16"></a><a href="#FNanchor_16_16"><span class="label">[16]</span></a> In his article 'Le classi finite,' <i>Atti di Torino</i>, Vol. XXXII.</p></div>
+
+<div class="footnote"><p><a name="Footnote_17_17" id="Footnote_17_17"></a><a href="#FNanchor_17_17"><span class="label">[17]</span></a> At the moment of going to press we learn that M. Bucherer has repeated
+the experiment, taking new precautions, and that he has obtained, contrary
+to Kaufmann, results confirming the views of Lorentz.</p></div>
+
+
+
+
+
+
+
+
+
+<pre>
+
+
+
+
+
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+and Hypothesis, The Value of Science, Science and Method, by Henri Poincaré
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