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SCIENCE AND HYPOTHESIS
BY
H. POINCARÉ,
MEMBER OF THE INSTITUTE OF FRANCE.
With a Preface by
J. LARMOR, D.Sc., Sec. R.S.,
Lucasian Professor of Mathematics in the University of Cambridge.
London and NewcaĆle-on-Tyne: THE WALTER SCOTT PUBLISHING CO., LTD
NEW YORK: 3 EAST 14TH STREET.
.

CONTENTS.
PAGE
Translator’s Note. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vi Author’s Preface. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xvi
PART I. NUMBER AND MAGNITUDE.
CHAPTER I.
On the Nature of Mathematical Reasoning. . . . . 1
CHAPTER II.
Mathematical Magnitude and Experiment. . . . . . . 22
PART II. SPACE. CHAPTER III.
Non-Euclidean Geometries. . . . . . . . . . . . . . . . . . . . . . . . . 42

contents.

iii

CHAPTER IV.
PAGE
Space and Geometry. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
CHAPTER V.
Experiment and Geometry. . . . . . . . . . . . . . . . . . . . . . . . . 83
PART III. FORCE. CHAPTER VI.
The Classical Mechanics. . . . . . . . . . . . . . . . . . . . . . . . . . . 101
CHAPTER VII.
Relative and Absolute Motion. . . . . . . . . . . . . . . . . . . . 125
CHAPTER VIII.
Energy and Thermo-dynamics. . . . . . . . . . . . . . . . . . . . . 138

contents.

iv

PART IV. NATURE. CHAPTER IX.
PAGE
Hypotheses in Physics. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156
CHAPTER X.
The Theories of Modern Physics.. . . . . . . . . . . . . . . . .178
CHAPTER XI.
The Calculus of Probabilities. . . . . . . . . . . . . . . . . . . . 204
CHAPTER XII.
Optics And Electricity.. . . . . . . . . . . . . . . . . . . . . . . . . . . .235
CHAPTER XIII.
Electro-Dynamics. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251

TRANSLATOR’S NOTE.

The translator wishes to express his indebtedness to Professor Larmor, for kindly consenting to introduce the author of Science and Hypothesis to English readers; to Dr. F. S. Macaulay and Mr. C. S. Jackson, M.A., who have read the whole of the proofs and have greatly helped by suggestions; also to Professor G. H. Bryan, F.R.S., who has read the proofs of Chapter VIII., and whose criticisms have been most valuable.

February 1905.

W. J. G.

INTRODUCTION.
It is to be hoped that, as a consequence of the present active scrutiny of our educational aims and methods, and of the resulting encouragement of the study of modern languages, we shall not remain, as a nation, so much isolated from ideas and tendencies in continental thought and literature as we have been in the past. As things are, however, the translation of this book is doubtless required; at any rate, it brings vividly before us an instructive point of view. Though some of M. Poincaré’s chapters have been collected from well-known treatises written several years ago, and indeed are sometimes in detail not quite up to date, besides occasionally suggesting the suspicion that his views may possibly have been modiﬁed in the interval, yet their publication in a compact form has excited a warm welcome in this country.
It must be confessed that the English language hardly lends itself as a perfect medium for the rendering of the delicate shades of suggestion and allusion characteristic of M. Poincaré’s play around his subject; notwithstanding the excellence of the translation, loss in this respect is inevitable.

introduction.

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There has been of late a growing trend of opinion, prompted in part by general philosophical views, in the direction that the theoretical constructions of physical science are largely factitious, that instead of presenting a valid image of the relations of things on which further progress can be based, they are still little better than a mirage. The best method of abating this scepticism is to become acquainted with the real scope and modes of application of conceptions which, in the popular language of superﬁcial exposition—and even in the unguarded and playful paradox of their authors, intended only for the instructed eye—often look bizarre enough. But much advantage will accrue if men of science become their own epistemologists, and show to the world by critical exposition in non-technical terms of the results and methods of their constructive work, that more than mere instinct is involved in it: the community has indeed a right to expect as much as this.
It would be hard to ﬁnd any one better qualiﬁed for this kind of exposition, either from the profundity of his own mathematical achievements, or from the extent and freshness of his interest in the theories of physical science, than the author of this book. If an appreciation might be ventured on as regards the later chapters, they

introduction.

viii

are, perhaps, intended to present the stern logical analyst quizzing the cultivator of physical ideas as to what he is driving at, and whither he expects to go, rather than any responsible attempt towards a settled confession of faith. Thus, when M. Poincaré allows himself for a moment to indulge in a process of evaporation of the Principle of Energy, he is content to sum up: “Eh bien, quelles que soient les notions nouvelles que les expériences futures nous donneront sur le monde, nous sommes sûrs d’avance qu’il y aura quelque chose qui demeurera constant et que nous pourrons appeler énergie” (p. 185), and to leave the matter there for his readers to think it out. Though hardly necessary in the original French, it may not now be superﬂuous to point out that independent reﬂection and criticism on the part of the reader are tacitly implied here as elsewhere.
An interesting passage is the one devoted to Maxwell’s theory of the functions of the æther, and the comparison of the close-knit theories of the classical French mathematical physicists with the somewhat loosely-connected corpus of ideas by which Maxwell, the interpreter and successor of Faraday, has (posthumously) recast the whole face of physical science. How many times has that theory been re-written since Maxwell’s day? and yet how

introduction.

ix

little has it been altered in essence, except by further developments in the problem of moving bodies, from the form in which he left it! If, as M. Poincaré remarks, the French instinct for precision and lucid demonstration sometimes ﬁnds itself ill at ease with physical theories of the British school, he as readily admits (pp. 248, 250), and indeed fully appreciates, the advantages on the other side. Our own mental philosophers have been shocked at the point of view indicated by the proposition hazarded by Laplace, that a suﬃciently developed intelligence, if it were made acquainted with the positions and motions of the atoms at any instant, could predict all future history: no amount of demur suﬃces sometimes to persuade them that this is not a conception universally entertained in physical science. It was not so even in Laplace’s own day. From the point of view of the study of the evolution of the sciences, there are few episodes more instructive than the collision between Laplace and Young with regard to the theory of capillarity. The precise and intricate mathematical analysis of Laplace, starting from ﬁxed preconceptions regarding atomic forces which were to remain intact throughout the logical development of the argument, came into contrast with the tentative, mobile intuitions of Young; yet the latter was able to

introduction.

x

grasp, by sheer direct mental force, the fruitful though partial analogies of this recondite class of phenomena with more familiar operations of nature, and to form a direct picture of the way things interacted, such as could only have been illustrated, quite possibly damaged or obliterated, by premature eﬀort to translate it into elaborate analytical formulas. The aperçus of Young were apparently devoid of all cogency to Laplace; while Young expressed, doubtless in too extreme a way, his sense of the inanity of the array of mathematical logic of his rival. The subsequent history involved the Nemesis that the fabric of Laplace was taken down and reconstructed in the next generation by Poisson; while the modern cultivator of the subject turns, at any rate in England, to neither of those expositions for illumination, but rather ﬁnds in the partial and succinct indications of Young the best starting-point for further eﬀort.
It seems, however, hard to accept entirely the distinction suggested (p. 237) between the methods of cultivating theoretical physics in the two countries. To mention only two transcendent names which stand at the very front of two of the greatest developments of physical science of the last century, Carnot and Fresnel, their procedure was certainly not on the lines thus described.

introduction.

xi

Possibly it is not devoid of signiﬁcance that each of them attained his ﬁrst eﬀective recognition from the British school.
It may, in fact, be maintained that the part played by mechanical and such-like theories—analogies if you will—is an essential one. The reader of this book will appreciate that the human mind has need of many instruments of comparison and discovery besides the unrelenting logic of the inﬁnitesimal calculus. The dynamical basis which underlies the objects of our most frequent experience has now been systematised into a great calculus of exact thought, and traces of new real relationships may come out more vividly when considered in terms of our familiar acquaintance with dynamical systems than when formulated under the paler shadow of more analytical abstractions. It is even possible for a constructive physicist to conduct his mental operations entirely by dynamical images, though Helmholtz, as well as our author, seems to class a predilection in this direction as a British trait. A time arrives when, as in other subjects, ideas have crystallised out into distinctness; their exact veriﬁcation and development then becomes a problem in mathematical physics. But whether the mechanical analogies still survive, or new terms are

introduction.

xii

now introduced devoid of all naïve mechanical bias, it matters essentially little. The precise determination of the relations of things in the rational scheme of nature in which we ﬁnd ourselves is the fundamental task, and for its fulﬁlment in any direction advantage has to be taken of our knowledge, even when only partial, of new aspects and types of relationship which may have become familiar perhaps in quite diﬀerent ﬁelds. Nor can it be forgotten that the most fruitful and fundamental conceptions of abstract pure mathematics itself have often been suggested from these mechanical ideas of ﬂux and force, where the play of intuition is our most powerful guide. The study of the historical evolution of physical theories is essential to the complete understanding of their import. It is in the mental workshop of a Fresnel, a Kelvin, or a Helmholtz, that profound ideas of the deep things of Nature are struck out and assume form; when pondered over and paraphrased by philosophers we see them react on the conduct of life: it is the business of criticism to polish them gradually to the common measure of human understanding. Oppressed though we are with the necessity of being specialists, if we are to know anything thoroughly in these days of accumulated details, we may at any rate proﬁtably study the historical evolution of

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xiii

knowledge over a ﬁeld wider than our own. The aspect of the subject which has here been dwelt
on is that scientiﬁc progress, considered historically, is not a strictly logical process, and does not proceed by syllogisms. New ideas emerge dimly into intuition, come into consciousness from nobody knows where, and become the material on which the mind operates, forging them gradually into consistent doctrine, which can be welded on to existing domains of knowledge. But this process is never complete: a crude connection can always be pointed to by a logician as an indication of the imperfection of human constructions.
If intuition plays a part which is so important, it is surely necessary that we should possess a ﬁrm grasp of its limitations. In M. Poincaré’s earlier chapters the reader can gain very pleasantly a vivid idea of the various and highly complicated ways of docketing our perceptions of the relations of external things, all equally valid, that were open to the human race to develop. Strange to say, they never tried any of them; and, satisﬁed with the very remarkable practical ﬁtness of the scheme of geometry and dynamics that came naturally to hand, did not consciously trouble themselves about the possible existence of others until recently. Still more recently has it

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xiv

been found that the good Bishop Berkeley’s logical jibes against the Newtonian ideas of ﬂuxions and limiting ratios cannot be adequately appeased in the rigorous mathematical conscience, until our apparent continuities are resolved mentally into discrete aggregates which we only partially apprehend. The irresistible impulse to atomize everything thus proves to be not merely a disease of the physicist; a deeper origin, in the nature of knowledge itself, is suggested.
Everywhere want of absolute, exact adaptation can be detected, if pains are taken, between the various constructions that result from our mental activity and the impressions which give rise to them. The bluntness of our unaided sensual perceptions, which are the source in part of the intuitions of the race, is well brought out in this connection by M. Poincaré. Is there real contradiction? Harmony usually proves to be recovered by shifting our attitude to the phenomena. All experience leads us to interpret the totality of things as a consistent cosmos—undergoing evolution, the naturalists will say—in the large-scale workings of which we are interested spectators and explorers, while of the inner relations and ramiﬁcations we only apprehend dim glimpses. When our formulation of experience is imperfect or even

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xv

paradoxical, we learn to attribute the fault to our point of view, and to expect that future adaptation will put it right. But Truth resides in a deep well, and we shall never get to the bottom. Only, while deriving enjoyment and insight from M. Poincaré’s Socratic exposition of the limitations of the human outlook on the universe, let us beware of counting limitation as imperfection, and drifting into an inadequate conception of the wonderful fabric of human knowledge.
J. LARMOR.

AUTHOR’S PREFACE.
To the superﬁcial observer scientiﬁc truth is unassailable, the logic of science is infallible; and if scientiﬁc men sometimes make mistakes, it is because they have not understood the rules of the game. Mathematical truths are derived from a few self-evident propositions, by a chain of ﬂawless reasonings; they are imposed not only on us, but on Nature itself. By them the Creator is fettered, as it were, and His choice is limited to a relatively small number of solutions. A few experiments, therefore, will be suﬃcient to enable us to determine what choice He has made. From each experiment a number of consequences will follow by a series of mathematical deductions, and in this way each of them will reveal to us a corner of the universe. This, to the minds of most people, and to students who are getting their ﬁrst ideas of physics, is the origin of certainty in science. This is what they take to be the rôle of experiment and mathematics. And thus, too, it was understood a hundred years ago by many men of science who dreamed of constructing the world with the aid of the smallest possible amount of material borrowed from experiment.

author’s preface.

xvii

But upon more mature reﬂection the position held by hypothesis was seen; it was recognised that it is as necessary to the experimenter as it is to the mathematician. And then the doubt arose if all these constructions are built on solid foundations. The conclusion was drawn that a breath would bring them to the ground. This sceptical attitude does not escape the charge of superﬁciality. To doubt everything or to believe everything are two equally convenient solutions; both dispense with the necessity of reﬂection.
Instead of a summary condemnation we should examine with the utmost care the rôle of hypothesis; we shall then recognise not only that it is necessary, but that in most cases it is legitimate. We shall also see that there are several kinds of hypotheses; that some are veriﬁable, and when once conﬁrmed by experiment become truths of great fertility; that others may be useful to us in ﬁxing our ideas; and ﬁnally, that others are hypotheses only in appearance, and reduce to deﬁnitions or to conventions in disguise. The latter are to be met with especially in mathematics and in the sciences to which it is applied. From them, indeed, the sciences derive their rigour; such conventions are the result of the unrestricted activity of the mind, which in this domain recognises no obstacle.

author’s preface.

xviii

For here the mind may aﬃrm because it lays down its own laws; but let us clearly understand that while these laws are imposed on our science, which otherwise could not exist, they are not imposed on Nature. Are they then arbitrary? No; for if they were, they would not be fertile. Experience leaves us our freedom of choice, but it guides us by helping us to discern the most convenient path to follow. Our laws are therefore like those of an absolute monarch, who is wise and consults his council of state. Some people have been struck by this characteristic of free convention which may be recognised in certain fundamental principles of the sciences. Some have set no limits to their generalisations, and at the same time they have forgotten that there is a diﬀerence between liberty and the purely arbitrary. So that they are compelled to end in what is called nominalism; they have asked if the savant is not the dupe of his own deﬁnitions, and if the world he thinks he has discovered is not simply the creation of his own caprice.1 Under these conditions science would retain its certainty, but would not attain its object, and would become powerless. Now, we daily see what science is doing for us. This could not be unless
1Cf. M. le Roy: “Science et Philosophie,” Revue de Métaphysique et de Morale, 1901.

author’s preface.

xix

it taught us something about reality; the aim of science is not things themselves, as the dogmatists in their simplicity imagine, but the relations between things; outside those relations there is no reality knowable.
Such is the conclusion to which we are led; but to reach that conclusion we must pass in review the series of sciences from arithmetic and geometry to mechanics and experimental physics. What is the nature of mathematical reasoning? Is it really deductive, as is commonly supposed? Careful analysis shows us that it is nothing of the kind; that it participates to some extent in the nature of inductive reasoning, and for that reason it is fruitful. But none the less does it retain its character of absolute rigour; and this is what must ﬁrst be shown.
When we know more of this instrument which is placed in the hands of the investigator by mathematics, we have then to analyse another fundamental idea, that of mathematical magnitude. Do we ﬁnd it in nature, or have we ourselves introduced it? And if the latter be the case, are we not running a risk of coming to incorrect conclusions all round? Comparing the rough data of our senses with that extremely complex and subtle conception which mathematicians call magnitude, we are compelled to recognise a divergence. The framework into

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xx

which we wish to make everything ﬁt is one of our own construction; but we did not construct it at random, we constructed it by measurement so to speak; and that is why we can ﬁt the facts into it without altering their essential qualities.
Space is another framework which we impose on the world. Whence are the ﬁrst principles of geometry derived? Are they imposed on us by logic? Lobatschewsky, by inventing non-Euclidean geometries, has shown that this is not the case. Is space revealed to us by our senses? No; for the space revealed to us by our senses is absolutely diﬀerent from the space of geometry. Is geometry derived from experience? Careful discussion will give the answer—no! We therefore conclude that the principles of geometry are only conventions; but these conventions are not arbitrary, and if transported into another world (which I shall call the non-Euclidean world, and which I shall endeavour to describe), we shall ﬁnd ourselves compelled to adopt more of them.
In mechanics we shall be led to analogous conclusions, and we shall see that the principles of this science, although more directly based on experience, still share the conventional character of the geometrical postulates. So far, nominalism triumphs; but we now come to the

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xxi

physical sciences, properly so called, and here the scene changes. We meet with hypotheses of another kind, and we fully grasp how fruitful they are. No doubt at the outset theories seem unsound, and the history of science shows us how ephemeral they are; but they do not entirely perish, and of each of them some traces still remain. It is these traces which we must try to discover, because in them and in them alone is the true reality.
The method of the physical sciences is based upon the induction which leads us to expect the recurrence of a phenomenon when the circumstances which give rise to it are repeated. If all the circumstances could be simultaneously reproduced, this principle could be fearlessly applied; but this never happens; some of the circumstances will always be missing. Are we absolutely certain that they are unimportant? Evidently not! It may be probable, but it cannot be rigorously certain. Hence the importance of the rôle that is played in the physical sciences by the law of probability. The calculus of probabilities is therefore not merely a recreation, or a guide to the baccarat player; and we must thoroughly examine the principles on which it is based. In this connection I have but very incomplete results to lay before the reader, for the vague instinct which enables us to de-

author’s preface.

xxii

termine probability almost deﬁes analysis. After a study of the conditions under which the work of the physicist is carried on, I have thought it best to show him at work. For this purpose I have taken instances from the history of optics and of electricity. We shall thus see how the ideas of Fresnel and Maxwell took their rise, and what unconscious hypotheses were made by Ampère and the other founders of electro-dynamics.

SCIENCE AND HYPOTHESIS
PART I.
NUMBER AND MAGNITUDE.
CHAPTER I.
ON THE NATURE OF MATHEMATICAL REASONING.
I.
The very possibility of mathematical science seems an insoluble contradiction. If this science is only deductive in appearance, from whence is derived that perfect rigour which is challenged by none? If, on the contrary, all the propositions which it enunciates may be derived in order by the rules of formal logic, how is it that mathematics is not reduced to a gigantic tautology? The syllogism can teach us nothing essentially new, and if everything must spring from the principle of identity, then everything should be capable of being reduced to that principle. Are we then to admit that the enunciations of all

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2

the theorems with which so many volumes are ﬁlled, are only indirect ways of saying that A is A?
No doubt we may refer back to axioms which are at the source of all these reasonings. If it is felt that they cannot be reduced to the principle of contradiction, if we decline to see in them any more than experimental facts which have no part or lot in mathematical necessity, there is still one resource left to us: we may class them among à priori synthetic views. But this is no solution of the diﬃculty—it is merely giving it a name; and even if the nature of the synthetic views had no longer for us any mystery, the contradiction would not have disappeared; it would have only been shirked. Syllogistic reasoning remains incapable of adding anything to the data that are given it; the data are reduced to axioms, and that is all we should ﬁnd in the conclusions.
No theorem can be new unless a new axiom intervenes in its demonstration; reasoning can only give us immediately evident truths borrowed from direct intuition; it would only be an intermediary parasite. Should we not therefore have reason for asking if the syllogistic apparatus serves only to disguise what we have borrowed?
The contradiction will strike us the more if we open any book on mathematics; on every page the author an-

nature of mathematical reasoning. 3
nounces his intention of generalising some proposition already known. Does the mathematical method proceed from the particular to the general, and, if so, how can it be called deductive?
Finally, if the science of number were merely analytical, or could be analytically derived from a few synthetic intuitions, it seems that a suﬃciently powerful mind could with a single glance perceive all its truths; nay, one might even hope that some day a language would be invented simple enough for these truths to be made evident to any person of ordinary intelligence.
Even if these consequences are challenged, it must be granted that mathematical reasoning has of itself a kind of creative virtue, and is therefore to be distinguished from the syllogism. The diﬀerence must be profound. We shall not, for instance, ﬁnd the key to the mystery in the frequent use of the rule by which the same uniform operation applied to two equal numbers will give identical results. All these modes of reasoning, whether or not reducible to the syllogism, properly so called, retain the analytical character, and ipso facto, lose their power.

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II.
The argument is an old one. Let us see how Leibnitz tried to show that two and two make four. I assume the number one to be deﬁned, and also the operation x + 1— i.e., the adding of unity to a given number x. These deﬁnitions, whatever they may be, do not enter into the subsequent reasoning. I next deﬁne the numbers 2, 3, 4 by the equalities

(1) 1 + 1 = 2; (2) 2 + 1 = 3; (3) 3 + 1 = 4;

and in the same way I deﬁne the operation x + 2 by the relation
(4) x + 2 = (x + 1) + 1.
Given this, we have

2 + 2 = (2 + 1) + 1, (def. 4);

(2 + 1) + 1 = 3 + 1,

(def. 2);

3 + 1 = 4,

(def. 3);

whence 2 + 2 = 4,

Q.E.D.

It cannot be denied that this reasoning is purely analytical. But if we ask a mathematician, he will reply: “This is not a demonstration properly so called; it is a

nature of mathematical reasoning. 5
veriﬁcation.” We have conﬁned ourselves to bringing together one or other of two purely conventional deﬁnitions, and we have veriﬁed their identity; nothing new has been learned. Veriﬁcation diﬀers from proof precisely because it is analytical, and because it leads to nothing. It leads to nothing because the conclusion is nothing but the premisses translated into another language. A real proof, on the other hand, is fruitful, because the conclusion is in a sense more general than the premisses. The equality 2 + 2 = 4 can be veriﬁed because it is particular. Each individual enunciation in mathematics may be always veriﬁed in the same way. But if mathematics could be reduced to a series of such veriﬁcations it would not be a science. A chess-player, for instance, does not create a science by winning a piece. There is no science but the science of the general. It may even be said that the object of the exact sciences is to dispense with these direct veriﬁcations.
III.
Let us now see the geometer at work, and try to surprise some of his methods. The task is not without diﬃculty; it is not enough to open a book at random and to anal-

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6

yse any proof we may come across. First of all, geometry must be excluded, or the question becomes complicated by diﬃcult problems relating to the rôle of the postulates, the nature and the origin of the idea of space. For analogous reasons we cannot avail ourselves of the inﬁnitesimal calculus. We must seek mathematical thought where it has remained pure—i.e., in Arithmetic. But we still have to choose; in the higher parts of the theory of numbers the primitive mathematical ideas have already undergone so profound an elaboration that it becomes diﬃcult to analyse them.
It is therefore at the beginning of Arithmetic that we must expect to ﬁnd the explanation we seek; but it happens that it is precisely in the proofs of the most elementary theorems that the authors of classic treatises have displayed the least precision and rigour. We may not impute this to them as a crime; they have obeyed a necessity. Beginners are not prepared for real mathematical rigour; they would see in it nothing but empty, tedious subtleties. It would be waste of time to try to make them more exacting; they have to pass rapidly and without stopping over the road which was trodden slowly by the founders of the science.
Why is so long a preparation necessary to habituate

nature of mathematical reasoning. 7
oneself to this perfect rigour, which it would seem should naturally be imposed on all minds? This is a logical and psychological problem which is well worthy of study. But we shall not dwell on it; it is foreign to our subject. All I wish to insist on is, that we shall fail in our purpose unless we reconstruct the proofs of the elementary theorems, and give them, not the rough form in which they are left so as not to weary the beginner, but the form which will satisfy the skilled geometer.
definition of addition.
I assume that the operation x + 1 has been deﬁned; it consists in adding the number 1 to a given number x. Whatever may be said of this deﬁnition, it does not enter into the subsequent reasoning.
We now have to deﬁne the operation x + a, which consists in adding the number a to any given number x. Suppose that 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.

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8

We shall know what x+a is when we know what x+(a−1) is, and as I have assumed that to start with we know 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 distinguishes it even at this stage from the purely logical deﬁnition; the equality (1), in fact, contains an inﬁnite number of distinct deﬁnitions, each having only one meaning when we know the meaning of its predecessor.

properties of addition.
Associative.—I say that
a + (b + c) = (a + b) + c;
in fact, the theorem is true for c = 1. It may then be written
a + (b + 1) = (a + b) + 1; which, remembering the diﬀerence of notation, is nothing but the equality (1) by which I have just deﬁned addition. Assume the theorem true for c = γ, I say that it will be true for c = γ + 1. Let
(a + b) + γ = a + (b + γ);

nature of mathematical reasoning. 9
it follows that
(a + b) + γ + 1 = a + (b + γ) + 1;
or by def. (1),
(a + b) + (γ + 1) = a + (b + γ + 1) = a + b + (γ + 1) ;
which shows by a series of purely analytical deductions that the theorem is true for γ + 1. Being true for c = 1, we see that it is successively true for c = 2, c = 3, etc.
Commutative.—(1) I say that
a + 1 = 1 + a.
The theorem is evidently true for a = 1; we can verify by purely analytical reasoning that if it is true for a = γ it will be true for a = γ + 1.1 Now, it is true for a = 1, and therefore is true for a = 2, a = 3, and so on. This is what is meant by saying that the proof is demonstrated “by recurrence.”
(2) I say that
a + b = b + a.
1For (γ + 1) + 1 = (1 + γ) + 1 = 1 + (γ + 1).—[Tr.]

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10

The theorem has just been shown to hold good for b = 1, and it may be veriﬁed analytically that if it is true for b = β, it will be true for b = β + 1. The proposition is thus established by recurrence.

definition of multiplication.
We shall deﬁne multiplication by the equalities
(1) a × 1 = a; (2) a × b = a × (b − 1) + a.
Both of these include an inﬁnite number of deﬁnitions; having deﬁned a × 1, it enables us to deﬁne in succession a × 2, a × 3, and so on.
properties of multiplication.
Distributive.—I say that
(a + b) × c = (a × c) + (b × c).
We can verify analytically that the theorem is true for c = 1; then if it is true for c = γ, it will be true for c = γ + 1. The proposition is then proved by recurrence.

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Commutative.—(1) I say that
a × 1 = 1 × a.
The theorem is obvious for a = 1. We can verify analytically that if it is true for a = α, it will be true for a = α + 1.
(2) I say that
a × b = b × a.
The theorem has just been proved for b = 1. We can verify analytically that if it be true for b = β it will be true for b = β + 1.
IV.
This monotonous series of reasonings may now be laid aside; but their very monotony brings vividly to light the process, which is uniform, and is met again at every step. The process is proof by recurrence. We ﬁrst show that a theorem is true for n = 1; we then show that if it is true for n−1 it is true for n, and we conclude that it is true for all integers. We have now seen how it may be used for the proof of the rules of addition and multiplication— that is to say, for the rules of the algebraical calculus.

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This calculus is an instrument of transformation which lends itself to many more diﬀerent combinations than the simple syllogism; but it is still a purely analytical instrument, and is incapable of teaching us anything new. If mathematics had no other instrument, it would immediately be arrested in its development; but it has recourse anew to the same process—i.e., to reasoning by recurrence, and it can continue its forward march. Then if we look carefully, we ﬁnd this mode of reasoning at every step, either under the simple form which we have just given to it, or under a more or less modiﬁed form. It is therefore mathematical reasoning par excellence, and we must examine it closer.
V.
The essential characteristic of reasoning by recurrence is that it contains, condensed, so to speak, in a single formula, an inﬁnite number of syllogisms. We shall see this more clearly if we enunciate the syllogisms one after another. They follow one another, if one may use the expression, in a cascade. The following are the 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.

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Now, if it is true of 2, it is true of 3; hence it is true of 3, and so on. We see that the conclusion of each syllogism serves as the minor of its successor. Further, the majors of all our syllogisms may be reduced to a single form. If the theorem is true of n − 1, it is true of n.
We see, then, that in reasoning by recurrence we conﬁne ourselves to the enunciation of the minor of the ﬁrst syllogism, and the general formula which contains as particular cases all the majors. This unending series of syllogisms is thus reduced to a phrase of a few lines.
It is now easy to understand why every particular consequence of a theorem may, as I have above explained, be veriﬁed by purely analytical processes. If, instead of proving that our theorem is true for all numbers, we only wish to show that it is true for the number 6 for instance, it will be enough to establish the ﬁrst ﬁve syllogisms in our cascade. We shall require 9 if we wish to prove it for the number 10; for a greater number we shall require more still; but however great the number may be we shall always reach it, and the analytical veriﬁcation will always be possible. But however far we went we should never reach the general theorem applicable to all numbers, which alone is the object of science. To reach it we should require an inﬁnite number of syllogisms, and we

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should have to cross an abyss which the patience of the analyst, restricted to the resources of formal logic, will never succeed in crossing.
I asked at the outset why we cannot conceive of a mind powerful enough to see at a glance the whole body of mathematical truth. The answer is now easy. A chessplayer can combine for four or ﬁve moves ahead; but, however extraordinary a player he may be, he cannot prepare for more than a ﬁnite number of moves. If he applies his faculties to Arithmetic, he cannot conceive its general truths by direct intuition alone; to prove even the smallest theorem he must use reasoning by recurrence, for that is the only instrument which enables us to pass from the ﬁnite to the inﬁnite. This instrument is always useful, for it enables us to leap over as many stages as we wish; it frees us from the necessity of long, tedious, and monotonous veriﬁcations which would rapidly become impracticable. Then when we take in hand the general theorem it becomes indispensable, for otherwise we should ever be approaching the analytical veriﬁcation without ever actually reaching it. In this domain of Arithmetic we may think ourselves very far from the inﬁnitesimal analysis, but the idea of mathematical inﬁnity is already playing a preponderating part, and without it

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there would be no science at all, because there would be nothing general.
VI.
The views upon which reasoning by recurrence is based may be exhibited in other forms; we may say, for instance, that in any ﬁnite collection of diﬀerent integers there is always one which is smaller than any other. We may readily pass from one enunciation to another, and thus give ourselves the illusion of having proved that reasoning by recurrence is legitimate. But we shall always be brought to a full stop—we shall always come to an indemonstrable axiom, which will at bottom be but the proposition we had to prove translated into another language. We cannot therefore escape the conclusion that the rule of reasoning by recurrence is irreducible to the principle of contradiction. Nor can the rule come to us from experiment. Experiment may teach us that the rule is true for the ﬁrst ten or the ﬁrst hundred numbers, for instance; it will not bring us to the indeﬁnite series of numbers, but only to a more or less long, but always limited, portion of the series.
Now, if that were all that is in question, the principle

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of contradiction would be suﬃcient, it would always enable us to develop as many syllogisms as we wished. It is only when it is a question of a single formula to embrace an inﬁnite number of syllogisms that this principle breaks down, and there, too, experiment is powerless to aid. This rule, inaccessible to analytical proof and to experiment, is the exact type of the à priori synthetic intuition. On the other hand, we cannot see in it a convention as in the case of the postulates of geometry.
Why then is this view imposed upon us with such an irresistible weight of evidence? It is because it is only the aﬃrmation of the power of the mind which knows it can conceive of the indeﬁnite repetition of the same act, when the act is once possible. The mind has a direct intuition of this power, and experiment can only be for it an opportunity of using it, and thereby of becoming conscious of it.
But it will be said, if the legitimacy of reasoning by recurrence cannot be established by experiment alone, is it so with 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 manifest, we say, and it is so on the same ground that every physical law is true which is based on a very large

nature of mathematical reasoning. 17
but limited number of observations. It cannot escape our notice that here is a striking anal-
ogy with the usual processes of induction. But an essential diﬀerence exists. Induction applied to the physical sciences is always uncertain, because it is based on the belief in a general order of the universe, an order which is external to us. Mathematical induction—i.e., proof by recurrence—is, on the contrary, necessarily imposed on us, because it is only the aﬃrmation of a property of the mind itself.
VII.
Mathematicians, as I have said before, always endeavour to generalise the propositions they have obtained. To seek no further example, we have just shown the equality
a + 1 = 1 + a,
and we then used it to establish the equality
a + b = b + a,
which is obviously more general. Mathematics may, therefore, like the other sciences, proceed from the particular to the general. This is a fact which might

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otherwise have appeared incomprehensible to us at the beginning of this study, but which has no longer anything mysterious about it, since we have ascertained the analogies between proof by recurrence and ordinary induction.
No doubt mathematical recurrent reasoning and physical inductive reasoning are based on diﬀerent foundations, but they move in parallel lines and in the same direction—namely, from the particular to the general.
Let us examine the case a little more closely. To prove the equality
(1) a + 2 = 2 + a,
we need only apply the rule
a+1=1+a
twice, and write
(2) a + 2 = a + 1 + 1 = 1 + a + 1 = 1 + 1 + a = 2 + a.
The equality thus deduced by purely analytical means is not, however, a simple particular case. It is something quite diﬀerent. We may not therefore even say in the really analytical and deductive part of mathematical reasoning that we proceed from the general to the particular

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in the ordinary sense of the words. The two sides of the equality (2) are merely more complicated combinations than the two sides of the equality (1), and analysis only serves to separate the elements which enter into these combinations and to study their relations.
Mathematicians therefore proceed “by construction,” they “construct” more complicated combinations. When they analyse these combinations, these aggregates, so to speak, into their primitive elements, they see the relations of the elements and deduce the relations of the aggregates themselves. The process is purely analytical, but it is not a passing from the general to the particular, for the aggregates obviously cannot be regarded as more particular than their elements.
Great importance has been rightly attached to this process of “construction,” and some claim to see in it the necessary and suﬃcient condition of the progress of the exact sciences. Necessary, no doubt, but not suﬃcient! For a construction to be useful and not mere waste of mental eﬀort, for it to serve as a stepping-stone to higher things, it must ﬁrst of all possess a kind of unity enabling us to see something more than the juxtaposition of its elements. Or more accurately, there must be some advantage in considering the construction rather than the

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elements themselves. What can this advantage be? Why reason on a polygon, for instance, which is always decomposable into triangles, and not on elementary triangles? It is because there are properties of polygons of any number of sides, and they can be immediately applied to any particular kind of polygon. In most cases it is only after long eﬀorts that those properties can be discovered, by directly studying the relations of elementary triangles. If the quadrilateral is anything more than the juxtaposition of two triangles, it is because it is of the polygon type.
A construction only becomes interesting when it can be placed side by side with other analogous constructions for forming species of the same genus. To do this we must necessarily go back from the particular to the general, ascending one or more steps. The analytical process “by construction” does not compel us to descend, but it leaves us at the same level. We can only ascend by mathematical induction, for from it alone can we learn something new. Without the aid of this induction, which in certain respects diﬀers from, but is as fruitful as, physical induction, construction would be powerless to create science.
Let me observe, in conclusion, that this induction is only possible if the same operation can be repeated indefinitely. That is why the theory of chess can never become

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a science, for the diﬀerent moves of the same piece are limited and do not resemble each other.

CHAPTER II.
MATHEMATICAL MAGNITUDE AND EXPERIMENT.
If we want to know what the mathematicians mean by a continuum, it is useless to appeal to geometry. The geometer is always seeking, more or less, to represent to himself the ﬁgures he is studying, but his representations are only instruments to him; he uses space in his geometry just as he uses chalk; and further, too much importance must not be attached to accidents which are often nothing more than the whiteness of the chalk.
The pure analyst has not to dread this pitfall. He has disengaged mathematics from all extraneous elements, and he is in a position to answer our question:—“Tell me exactly what this continuum is, about which mathematicians reason.” Many analysts who reﬂect on their art have already done so—M. Tannery, for instance, in his Introduction à la théorie des Fonctions d’une variable.
Let us start with the integers. Between any two consecutive sets, intercalate one or more intermediary sets, and then between these sets others again, and so on indeﬁnitely. We thus get an unlimited number of terms, and these will be the numbers which we call fractional, rational, or commensurable. But this is not yet all; be-

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tween these terms, which, be it marked, are already inﬁnite in number, other terms are intercalated, and these are called irrational or incommensurable.
Before going any further, let me make a preliminary remark. The continuum thus conceived is no longer a collection of individuals arranged in a certain order, inﬁnite in number, it is true, but external the one to the other. This is not the ordinary conception in which it is supposed that between the elements of the continuum exists an intimate connection making of it one whole, in which the point has no existence previous to the line, but the line does exist previous to the point. Multiplicity alone subsists, unity has disappeared—“the continuum is unity in multiplicity,” according to the celebrated formula. The analysts have even less reason to deﬁne their continuum as they do, since it is always on this that they reason when they are particularly proud of their rigour. It is enough to warn the reader that the real mathematical continuum is quite diﬀerent from that of the physicists and from that of the metaphysicians.
It may also be said, perhaps, that mathematicians who are contented with this deﬁnition are the dupes of words, that the nature of each of these sets should be precisely indicated, that it should be explained how they

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are to be intercalated, and that it should be shown how it is possible to do it. This, however, would be wrong; the only property of the sets which comes into the reasoning is that of preceding or succeeding these or those other sets; this alone should therefore intervene in the deﬁnition. So we need not concern ourselves with the manner in which the sets are intercalated, and no one will doubt the possibility of the operation if he only remembers that “possible” in the language of geometers simply means exempt from contradiction. But our deﬁnition is not yet complete, and we come back to it after this rather long digression.
Deﬁnition of Incommensurables.—The mathematicians of the Berlin school, and Kronecker in particular, have devoted themselves to constructing this continuous scale of irrational and fractional numbers without using any other materials than the integer. The mathematical continuum from this point of view would be a pure creation of the mind in which experiment would have no part.
The idea of rational number not seeming to present to them any diﬃculty, they have conﬁned their attention mainly to deﬁning incommensurable numbers. But before reproducing their deﬁnition here, I must make an

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observation that will allay the astonishment which this will not fail to provoke in readers who are but little familiar with the habits of geometers.
Mathematicians do not study objects, but the relations between objects; to them it is a matter of indifference if these objects are replaced by others, provided that the relations do not change. Matter does not engage their attention, they are interested by form alone.
If we did not remember it, we could hardly understand that Kronecker gives the name of incommensurable number to a simple symbol—that is to say, something very diﬀerent from the idea we think we ought to have of a quantity which should be measurable and almost tangible.
Let us see now what is Kronecker’s deﬁnition. Commensurable numbers may be divided into classes in an inﬁnite number of ways, subject to the condition that any number whatever of the ﬁrst class is greater than any number of the second. It may happen that among the numbers of the ﬁrst class there is one which is smaller than all the rest; if, for instance, we arrange in the ﬁrst class all the numbers greater than 2, and 2 itself, and in the second class all the numbers smaller than 2, it is clear that 2 will be the smallest of all the numbers of the

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ﬁrst class. The number 2 may therefore be chosen as the symbol of this division.
It may happen, on the contrary, that in the second class there is one which is greater than all the rest. This is what takes place, for example, if the ﬁrst class comprises all the numbers greater than 2, and if, in the second, are all the numbers less than 2, and 2 itself. Here again the number 2 might be chosen as the symbol of this division.
But it may equally well happen that we can ﬁnd neither in the ﬁrst class a number smaller than all the rest, nor in the second class a number greater than all the rest. Suppose, for instance, we place in the ﬁrst class all the numbers whose squares are greater than 2, and in the second all the numbers whose squares are smaller than 2. We know that in neither of them is a number whose square is equal to 2. Evidently there will be in the ﬁrst class no number which is smaller than all the rest, for however near the square of a number may be to 2, we can always ﬁnd a commensurable whose square is still nearer to 2. From K√ronecker’s point of view, the incommensurable number 2 is nothing but the symbol of this particular method of division of commensurable numbers; and to each mode of repartition corresponds in this way a number, commensurable or not, which serves

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as a symbol. But to be satisﬁed with this would be to forget the origin of these symbols; it remains to explain how we have been led to attribute to them a kind of concrete existence, and on the other hand, does not the diﬃculty begin with fractions? Should we have the notion of these numbers if we did not previously know a matter which we conceive as inﬁnitely divisible—i.e., as a continuum?
The Physical Continuum.—We are next led to ask if the idea of the mathematical continuum is not simply drawn from experiment. If that be so, the rough data of experiment, which are our sensations, could be measured. We might, indeed, be tempted to believe that this is so, for in recent times there has been an attempt 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 the experiments by which the endeavour has been made to establish this law, we shall be led to a diametrically opposite conclusion. It has, for instance, been observed that a weight A of 10 grammes and a weight B of 11 grammes produced identical sensations, that the weight B could no longer be distinguished from a weight C of 12 grammes, but that the weight A was readily distinguished from the weight C. Thus the rough results of the experiments may

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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 disagreement with the law of contradiction, and the necessity of banishing this disagreement 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 it is experiment that has provided the opportunity. We cannot believe that two quantities which are equal to a third are not equal to one another, and we are thus led to suppose that A is diﬀerent from B, and B from C, and that if we have not been aware of this, it is due to the imperfections of our senses.
The Creation of the Mathematical Continuum: First Stage.—So far it would suﬃce, in order to account for facts, to intercalate between A and B a small number of terms which would remain discrete. What happens now if we have recourse to some instrument to make up for the weakness of our senses? If, for example, we use a microscope? Such terms as A and B, which before were indistinguishable from one another, appear now to be distinct: but between A and B, which are distinct, is inter-

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calated another new term D, which we can distinguish neither from A nor from B. Although we may use the most delicate methods, the rough results of our experiments will always present the characters of the physical continuum with the contradiction which is inherent in it. We only escape from it by incessantly intercalating new terms between the terms already distinguished, and this operation must be pursued indeﬁnitely. We might conceive that it would be possible to stop if we could imagine an instrument powerful enough to decompose the physical continuum into discrete elements, just as the telescope resolves the Milky Way into stars. But this we cannot imagine; it is always with our senses that we use our instruments; it is with the eye that we observe the image magniﬁed by the microscope, and this image must therefore always retain the characters of visual sensation, and therefore those of the physical continuum.
Nothing distinguishes a length directly observed from half that length doubled by the microscope. The whole is homogeneous to the part; and there is a fresh contradiction—or rather there would be one if the number of the terms were supposed to be ﬁnite; it is clear that the part containing less terms than the whole cannot be similar to the whole. The contradiction ceases as soon

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as the number of terms is regarded as inﬁnite. There is nothing, for example, to prevent us from regarding the aggregate of integers as similar to the aggregate of even numbers, which is however only a part of it; in fact, to each integer corresponds another even number which is its double. But it is not only to escape this contradiction contained in the empiric data that the mind is led to create the concept of a continuum formed of an indeﬁnite number of terms.
Here everything takes place just as in the series of the integers. We have the faculty of conceiving that a unit may be added to a collection of units. Thanks to experiment, we have had the opportunity of exercising this faculty and are conscious of it; but from this fact we feel that our power is unlimited, and that we can count indefinitely, although we have never had to count more than a ﬁnite number of objects. In the same way, as soon as we have intercalated terms between two consecutive terms of a series, we feel that this operation may be continued without limit, and that, so to speak, there is no intrinsic reason for stopping. As an abbreviation, I may give the name of a mathematical continuum of the ﬁrst order to every aggregate of terms formed after the same law as the scale of commensurable numbers. If, then, we inter-

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calate new sets according to the laws of incommensurable numbers, we obtain what may be called a continuum of the second order.
Second Stage.—We have only taken our ﬁrst step. We have explained the origin of continuums of the ﬁrst order; we must now see why this is not suﬃcient, and why the incommensurable numbers had to be invented.
If we try to imagine a line, it must have the characters of the physical continuum—that is to say, our representation must have a certain breadth. Two lines will therefore appear to us under the form of two narrow bands, and if we are content with this rough image, it is clear that where two lines cross they must have some common part. But the pure geometer makes one further eﬀort; without entirely renouncing the aid of his senses, he tries to imagine a line without breadth and a point without size. This he can do only by imagining a line as the limit towards which tends a band that is growing thinner and thinner, and the point as the limit towards which is tending an area that is growing smaller and smaller. Our two bands, however narrow they may be, will always have a common area; the smaller they are the smaller it will be, and its limit is what the geometer calls a point. This is why it is said that the two lines which cross must have a common

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point, and this truth seems intuitive. But a contradiction would be implied if we conceived
of lines as continuums of the ﬁrst order—i.e., the lines traced by the geometer should only give us points, the co-ordinates of which are rational numbers. The contradiction would be manifest if we were, for instance, to assert the existence of lines and circles. It is clear, in fact, that if the points whose co-ordinates are commensurable were alone regarded as real, the in-circle of a square and the diagonal of the square would not intersect, since the co-ordinates of the point of intersection are incommensurable.
Even then we should have only certain incommensurable numbers, and not all these numbers.
But let us imagine a line divided into two half-rays (demi-droites). Each of these half-rays will appear to our minds as a band of a certain breadth; these bands will ﬁt close together, because there must be no interval between them. The common part will appear to us to be a point which will still remain as we imagine the bands to become thinner and thinner, so that we admit as an intuitive truth that if a line be divided into two half-rays the common frontier of these half-rays is a point. Here we recognise the conception of Kronecker, in which an

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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.
Summary.—To sum up, 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. The only limit to its power is the necessity of avoiding all contradiction; but the mind only makes use of it when experiment gives a reason for it.
In the case with which we are concerned, the reason is given by the idea of the physical continuum, drawn from the rough data of the senses. But this idea leads to a series of contradictions from each of which in turn we must be freed. In this way we are forced to imagine a more and more complicated system of symbols. That on which we shall dwell is not merely exempt from internal contradiction,—it was so already at all the steps we have taken,—but it is no longer in contradiction with the various propositions which are called intuitive, and which are derived from more or less elaborate empirical notions.
Measurable Magnitude.—So far we have not spoken of the measure of magnitudes; we can tell if any one of

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them is greater than any other, but we cannot say that it is two or three times as large.
So far, I have only considered the order in which the terms are arranged; but that is not suﬃcient for most applications. We must learn how to compare the interval which separates any two terms. On this condition alone will the continuum become measurable, and the operations of arithmetic be applicable. This can only be done by the aid of a new and special convention; and this convention is, that in such a case the interval between the terms A and B is equal to the interval which separates C and D. For instance, we started with the integers, and between two consecutive sets we intercalated n intermediary sets; by convention we now assume these new sets to be equidistant. This is one of the ways of deﬁning the addition of two magnitudes; for if the interval AB is by deﬁnition equal to the interval CD, the interval AD will by deﬁnition be the sum of the intervals AB and AC. This deﬁnition is very largely, but not altogether, arbitrary. It must satisfy certain conditions—the commutative and associative laws of addition, for instance; but, provided the deﬁnition we choose satisﬁes these laws, the choice is indiﬀerent, and we need not state it precisely.
Remarks.—We are now in a position to discuss several

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important questions. (1) Is the creative power of the mind exhausted by the
creation of the mathematical continuum? The answer is in the negative, and this is shown in a very striking manner by the work of Du Bois Reymond.
We know that mathematicians distinguish between inﬁnitesimals of diﬀerent orders, and that inﬁnitesimals of the second order are inﬁnitely small, not only absolutely so, 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 here once more we ﬁnd the mathematical continuum which has been dealt with in the preceding pages. Further, there are inﬁnitesimals which are inﬁnitely small with reference to those of the ﬁrst order, and inﬁnitely large with respect to the order 1 + , however small may be. Here, then, are new terms intercalated in our series; and if I may be permitted to revert to the terminology used in the preceding pages, a terminology which is very convenient, although it has not been consecrated by usage, I shall say that we have created a kind of continuum of the third order.
It is an easy matter to go further, but it is idle to do so, for we would only be imagining symbols without any possible application, and no one will dream of doing that.

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This continuum of the third order, to which we are led by the consideration of the diﬀerent orders of inﬁnitesimals, is in itself of but little use and hardly worth quoting. Geometers look on it as a mere curiosity. The mind only uses its creative faculty when experiment requires it.
(2) When we are once in possession of the conception of the mathematical continuum, are we protected from contradictions analogous to those which gave it birth? No, and the following is an instance:—
He is a savant indeed who will not take it as evident that every curve has a tangent; and, in fact, if we think of a curve and a straight line as two narrow bands, we can always arrange them in such a way that they have a common part without intersecting. Suppose now that the breadth of the bands diminishes indeﬁnitely: the common part will still remain, and in the limit, so to speak, the two lines will have a common point, although they do not intersect—i.e., they will touch. The geometer who reasons in this way is only doing what we have done when we proved that two lines which intersect have a common point, and his intuition might also seem to be quite legitimate. But this is not the case. We can show that there are curves which have no tangent, if we deﬁne such a curve as an analytical continuum of the sec-

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ond order. No doubt some artiﬁce analogous to those we have discussed above would enable us to get rid of this contradiction, but as the latter is only met with in very exceptional cases, we need not trouble to do so. Instead of endeavouring to reconcile intuition and analysis, we are content to sacriﬁce one of them, and as analysis must be ﬂawless, intuition must go to the wall.
The Physical Continuum of several Dimensions.—We have discussed above the physical continuum as it is derived from the immediate evidence of our senses—or, if the reader prefers, from the rough results of Fechner’s experiments; I have shown that these results are summed up in the contradictory formulæ
A = B, B = C, A < C.
Let us now see how this notion is generalised, and how from it may be derived the concept of continuums of several dimensions. Consider any two aggregates of sensations. We can either distinguish between them, or we cannot; just as in Fechner’s experiments the weight of 10 grammes could be distinguished from the weight of 12 grammes, but not from the weight of 11 grammes. This is all that is required to construct the continuum of several dimensions.

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Let us call one of these aggregates of sensations an element. It will be in a measure analogous to the point of the mathematicians, but will not be, however, the same thing. We cannot say that our element has no size, for we cannot distinguish it from its immediate neighbours, and it is thus surrounded by a kind of fog. If the astronomical comparison may be allowed, our “elements” would be like nebulæ, whereas the mathematical points would be like stars.
If this be 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 cannot be distinguished from its predecessor. This linear series is to the line of the mathematician what the isolated element was to the point.
Before going further, I must explain what is meant by a cut. Let us consider a continuum C, and remove from it certain of its elements, which for a moment we shall regard as no longer belonging to the continuum. We shall call the aggregate of elements thus removed a cut. By means of this cut, the continuum C will be subdivided into several distinct continuums; the aggregate of elements which remain will cease to form a single continuum. There will then be on C two elements, A and B,

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which we must look upon as belonging to two distinct continuums; and we see that this must be so, because it will be impossible to ﬁnd a linear series of consecutive elements of C (each of the elements indistinguishable from the preceding, the ﬁrst being A and the last B), unless one of the elements of this series is indistinguishable from one of the elements of the cut.
It may happen, on the contrary, that the cut may not be suﬃcient to subdivide the continuum C. To classify the physical continuums, we must ﬁrst of all ascertain the nature of the cuts which must be made in order to subdivide them. If a physical continuum, C, may be subdivided by a cut reducing to a ﬁnite number of elements, all distinguishable the one from the other (and therefore forming neither one continuum nor several continuums), we shall call C a continuum of one dimension. If, on the contrary, C can only be subdivided by cuts which are themselves continuums, we shall say that C is of several dimensions; if the cuts are continuums of one dimension, then we shall say that C has two dimensions; if cuts of two dimensions are suﬃcient, we shall say that C is of three dimensions, and so on. Thus the notion of the physical continuum of several dimensions is deﬁned, thanks to the very simple fact, that two aggregates of sensations may

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be distinguishable or indistinguishable. The Mathematical Continuum of Several Dimen-
sions.—The conception of the mathematical continuum of n dimensions may be led up to quite naturally by a process similar to that which we discussed at the beginning of this chapter. A point of such a continuum is deﬁned by a system of n distinct magnitudes which we call its co-ordinates.
The magnitudes need not always be measurable; there is, for instance, one branch of geometry independent of the measure of magnitudes, in which we are only concerned with knowing, for example, if, on a curve ABC, the point B is between the points A and C, and in which it is immaterial whether the arc AB is equal to or twice the arc BC. This branch is called Analysis Situs. It contains quite a large body of doctrine which has attracted the attention of the greatest geometers, and from which are derived, one from another, a whole series of remarkable theorems. What distinguishes these theorems from those of ordinary geometry is that they are purely qualitative. They are still true if the ﬁgures are copied by an unskilful draughtsman, with the result that the proportions are distorted and the straight lines replaced by lines which are more or less curved.

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As soon as measurement is introduced into the continuum we have just deﬁned, the continuum becomes space, and geometry is born. But the discussion of this is reserved for Part II.

PART II.
SPACE.
CHAPTER III.
NON-EUCLIDEAN GEOMETRIES.
Every conclusion presumes premisses. These premisses are either self-evident and need no demonstration, or can be established only if based on other propositions; and, as we cannot go back in this way to inﬁnity, every deductive science, and geometry in particular, must rest upon a certain number of indemonstrable axioms. All treatises of geometry begin therefore with the enunciation of these axioms. But there is a distinction to be drawn between them. Some of these, for example, “Things which are equal to the same thing are equal to one another,” are not propositions in geometry but propositions in analysis. I look upon them as analytical à priori intuitions, and they concern me no further. But I must insist on other axioms which are special to geometry. Of these most treatises explicitly enunciate three:—(1) Only one line can pass through two points; (2) a straight line is the shortest distance between two points; (3) through one point only one parallel can be drawn to a given straight

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line. Although we generally dispense with proving the second of these axioms, it would be possible to deduce it from the other two, and from those much more numerous axioms which are implicitly admitted without enunciation, as I shall explain further on. For a long time a proof of the third axiom known as Euclid’s postulate was sought in vain. It is impossible to imagine the eﬀorts that have been spent in pursuit of this chimera. Finally, at the beginning of the nineteenth century, and almost simultaneously, two scientists, a Russian and a Hungarian, Lobatschewsky and Bolyai, showed irrefutably that this proof is impossible. They have nearly rid us of inventors of geometries without a postulate, and ever since the Académic des Sciences receives only about one or two new demonstrations a year. But the question was not exhausted, and it was not long before a great step was taken by the celebrated memoir of Riemann, entitled: Ueber die Hypothesen welche der Geometrie zum Grunde liegen. This little work has inspired most of the recent treatises to which I shall later on refer, and among which I may mention those of Beltrami and Helmholtz.
The Geometry of Lobatschewsky.—If it were possible to deduce Euclid’s postulate from the several axioms, it is evident that by rejecting the postulate and retaining

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the other axioms we should be led to contradictory consequences. It would be, therefore, impossible to found on those premisses a coherent geometry. Now, this is precisely what Lobatschewsky has done. He assumes at the outset that several parallels may be drawn through a point to a given straight line, and he retains all the other axioms of Euclid. From these hypotheses he deduces a series of theorems between which it is impossible to ﬁnd any contradiction, and he constructs a geometry as impeccable in its logic as Euclidean geometry. The theorems are very diﬀerent, however, from those to which we are accustomed, and at ﬁrst will be found a little disconcerting. For instance, the sum of the angles of a triangle is always less than two right angles, and the diﬀerence between that sum and two right angles is proportional to the area of the triangle. It is impossible to construct a ﬁgure similar to a given ﬁgure but of diﬀerent dimensions. If the circumference of a circle be divided into n equal parts, and tangents be drawn at the points of intersection, the n tangents will form a polygon if the radius of the circle is small enough, but if the radius is large enough they will never meet. We need not multiply these examples. Lobatschewsky’s propositions have no relation to those of Euclid, but they are none the less

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logically interconnected. Riemann’s Geometry.—Let us imagine to ourselves
a world only peopled with beings of no thickness, and suppose these “inﬁnitely ﬂat” animals are all in one and the same plane, from which they cannot emerge. Let us further admit that this world is suﬃciently distant from other worlds to be withdrawn from their inﬂuence, and while we are making these hypotheses it will not cost us much to endow these beings with reasoning power, and to believe them capable of making a geometry. In that case they will certainly attribute to space only two dimensions. But now suppose 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, from which they cannot escape. What kind of a geometry will they construct? In the ﬁrst place, it is clear that they will attribute to space only two dimensions. The straight line to them will be the shortest distance from one point on the sphere to another—that is to say, an arc of a great circle. In a word, their geometry will be spherical geometry. What they will call space will be the sphere on which they are conﬁned, and on which take place all the phenomena with which they are acquainted. Their space will therefore be unbounded,

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since on a sphere one may always walk forward without ever being brought to a stop, and yet it will be ﬁnite; the end will never be found, but the complete tour can be made. Well, Riemann’s geometry is spherical geometry extended to three dimensions. To construct it, the German mathematician had ﬁrst of all to throw overboard, not only Euclid’s postulate but also the ﬁrst axiom that only one line can pass through two points. On a sphere, through two given points, we can in general draw only one great circle which, as we have just seen, would be to our imaginary beings a straight line. But there was one exception. If the two given points are at the ends of a diameter, an inﬁnite number 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 only one straight line can in general be drawn, but there are exceptional cases in which through two points an inﬁnite number of straight lines can be drawn. So there is a kind of opposition between the geometries of Riemann and Lobatschewsky. For instance, 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 Lobatschewsky, and greater than two right angles in that of Riemann. The number of parallel lines that can

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be drawn through a given point to a given line is one in Euclid’s geometry, none in Riemann’s, and an inﬁnite number in the geometry of Lobatschewsky. Let us add that Riemann’s space is ﬁnite, although unbounded in the sense which we have above attached to these words.
Surfaces with Constant Curvature.—One objection, however, remains possible. There is no contradiction between the theorems of Lobatschewsky and Riemann; but however numerous are the other consequences that these geometers have deduced from their hypotheses, they had to arrest their course before they exhausted them all, for the number would be inﬁnite; and who can say that if they had carried their deductions further they would not have eventually reached some contradiction? This difﬁculty does not exist for Riemann’s geometry, provided it is limited to two dimensions. As we have seen, the two-dimensional geometry of Riemann, in fact, does not diﬀer from spherical geometry, which is only a branch of ordinary geometry, and is therefore outside all contradiction. Beltrami, by showing that Lobatschewsky’s two-dimensional geometry was only a branch of ordinary geometry, has equally refuted the objection as far as it is concerned. This is the course of his argument: Let us consider any ﬁgure whatever on a surface. Imagine this

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ﬁgure to be traced on a ﬂexible and inextensible canvas applied to the surface, in such a way that when the canvas is displaced and deformed the diﬀerent lines of the ﬁgure change their form without changing their length. As a rule, this ﬂexible and inextensible ﬁgure cannot be displaced without leaving the surface. But there are certain surfaces for which such a movement would be possible. They are surfaces of constant curvature. If we resume the comparison that we made just now, and imagine beings without thickness living on one of these surfaces, they will regard as possible the motion of a ﬁgure all the lines of which remain of a constant length. Such a movement would appear absurd, on the other hand, to animals without thickness living on a surface of variable curvature. These surfaces of constant curvature are of two kinds. The curvature of some is positive, and they may be deformed so as to be applied to a sphere. The geometry of these surfaces is therefore reduced to spherical geometry—namely, Riemann’s. The curvature of others is negative. Beltrami has shown that the geometry of these surfaces is identical with that of Lobatschewsky. Thus the two-dimensional geometries of Riemann and Lobatschewsky are connected with Euclidean geometry.
Interpretation of Non-Euclidean Geometries.—Thus

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vanishes the objection so far as two-dimensional geometries are concerned. It would be easy to extend Beltrami’s reasoning to three-dimensional geometries, and minds which do not recoil before space of four dimensions will see no diﬃculty in it; but such minds are few in number. I prefer, then, to proceed otherwise. Let us consider a certain plane, which I shall call the fundamental plane, and let us construct a kind of dictionary by making a double series of terms written in two columns, and corresponding each to each, just as in ordinary dictionaries the words in two languages which have the same signiﬁcation correspond to one another:—
Space . . . . . . . . . . . . . . The portion of space situated above the fundamental plane.
Plane . . . . . . . . . . . . . . Sphere cutting orthogonally the fundamental plane.
Line . . . . . . . . . . . . . . . Circle cutting orthogonally the fundamental plane.
Sphere . . . . . . . . . . . . . Sphere.
Circle . . . . . . . . . . . . . . Circle.
Angle . . . . . . . . . . . . . . Angle.

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Distance between two points . . . . . . . . . .

Logarithm of the anharmonic ratio of these two points and of the intersection of the fundamental plane with the circle passing through these two points and cutting it orthogonally.

Etc. . . . . . . . . . . . . . . . . Etc. Let us now take Lobatschewsky’s theorems and trans-
late them by the aid of this dictionary, as we would translate a German text with the aid of a German-French dictionary. We shall then obtain the theorems of ordinary geometry. For instance, Lobatschewsky’s theorem: “The sum of the angles of a triangle is less than two right angles,” may be translated thus: “If a curvilinear triangle has for its sides arcs of circles which if produced 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 Lobatschewsky’s hypotheses are carried, they will never lead to a contradiction; in fact, if two of Lobatschewsky’s theorems were contradictory, the translations of these two theorems made by the aid of our dictionary would be contradictory also. But these translations are theorems

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of ordinary geometry, and no one doubts that ordinary geometry is exempt from contradiction. Whence is the certainty derived, and how far is it justiﬁed? That is a question upon which I cannot enter here, but it is a very interesting question, and I think not insoluble. Nothing, therefore, is left of the objection I formulated above. But this is not all. Lobatschewsky’s geometry being susceptible of a concrete interpretation, ceases to be a useless logical exercise, and may be applied. I have no time here to deal with these applications, nor with what Herr Klein and myself have done by using them in the integration of linear equations. Further, this interpretation is not unique, and several dictionaries may be constructed analogous to that above, which will enable us by a simple translation to convert Lobatschewsky’s theorems into the theorems of ordinary geometry.
Implicit Axioms.—Are the axioms implicitly enunciated in our text-books the only foundation of geometry? We may be assured of the contrary when we see that, when they are abandoned one after another, there are still left standing some propositions which are common to the geometries of Euclid, Lobatschewsky, and Riemann. These propositions must be based on premisses that geometers admit without enunciation. It is interesting to

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try and extract them from the classical proofs. John Stuart Mill asserted1 that every deﬁnition con-
tains an axiom, because by deﬁning we implicitly aﬃrm the existence of the object deﬁned. That is going rather too far. It is but rarely in mathematics that a deﬁnition is given without following it up by the proof of the existence of the object deﬁned, and when this is not done it is generally because the reader can easily supply it; and it must not be forgotten that the word “existence” has not the same meaning when it refers to a mathematical entity as when it refers to a material object.
A mathematical entity exists provided there is no contradiction implied in its deﬁnition, either in itself, or with the propositions previously admitted. But if the observation of John Stuart Mill cannot be applied to all deﬁnitions, it is none the less true for some of them. A plane is sometimes deﬁned in the following manner:—The plane is a surface such that the line which joins any two points upon it lies wholly on that surface. Now, there is obviously a new axiom concealed in this deﬁnition. It is true we might change it, and that would be preferable, but then we should have to enunciate the axiom explicitly. Other deﬁnitions may give rise to no less important re-
1Logic, c. viii., cf. Deﬁnitions, §5–6.—[Tr.]

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ﬂections, such as, for example, that of the equality of two ﬁgures. Two ﬁgures are equal when they can be superposed. To superpose them, one of them must be displaced until it coincides with the other. But how must it be displaced? If we asked that question, no doubt we should be told that it ought to be done without deforming it, and as an invariable solid is displaced. The vicious circle would then be evident. As a matter of fact, this deﬁnition deﬁnes nothing. It has no meaning to a being living in a world in which there are only ﬂuids. If it seems clear to us, it is because we are accustomed to the properties of natural solids which do not much diﬀer from those of the ideal solids, all of whose dimensions are invariable. However, imperfect as it may be, this deﬁnition implies an axiom. The possibility of the motion of an invariable ﬁgure is not a self-evident truth. At least it is only so in the application to Euclid’s postulate, and not as an analytical à priori intuition would be. Moreover, when we study the deﬁnitions and the proofs of geometry, we see that we are compelled to admit without proof not only the possibility of this motion, but also some of its properties. This ﬁrst arises in the deﬁnition of the straight line. Many defective deﬁnitions have been given, but the true one is that which is understood in all the proofs in

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which the straight line intervenes. “It may happen that the motion of an invariable ﬁgure may be such that all the points of a line belonging to the ﬁgure are motionless, while all the points situate outside that line are in motion. Such a line would be called a straight line.” We have deliberately in this enunciation separated the definition from the axiom which it implies. Many proofs such as those of the cases of the equality of triangles, of the possibility of drawing a perpendicular from a point to a straight line, assume propositions the enunciations of which are dispensed with, for they necessarily imply that it is possible to move a ﬁgure in space in a certain way.
The Fourth Geometry.—Among these explicit axioms there is one which seems to me to deserve some attention, because when we abandon it we can construct a fourth geometry as coherent as those of Euclid, Lobatschewsky, and Riemann. To prove that we can always draw a perpendicular at a point A to a straight line AB, we consider a straight line AC movable about the point A, and initially identical with the ﬁxed straight line AB. We then can make it turn about the point A until it lies in AB produced. Thus we assume two propositions—ﬁrst, that such a rotation is possible, and then that it may continue

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until the two lines lie the one in the other produced. If the ﬁrst point is conceded and the second rejected, we are led to a series of theorems even stranger than those of Lobatschewsky and Riemann, but equally free from contradiction. I shall give only one of these theorems, and I shall not choose the least remarkable of them. A real straight line may be perpendicular to itself.
Lie’s Theorem.—The number of axioms implicitly introduced into classical proofs is greater than necessary, and it would be interesting to reduce them to a minimum. It may be asked, in the ﬁrst place, if this reduction is possible—if the number of necessary axioms and that of imaginable geometries is not inﬁnite? A theorem due to Sophus Lie is of weighty importance in this discussion. It may be enunciated in the following manner:—Suppose the following premisses are admitted: (1) space has n dimensions; (2) the movement of an invariable ﬁgure is possible; (3) p conditions are necessary to determine the position of this ﬁgure in space.
The number of geometries compatible with these premisses 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 the movement is granted, we can only invent a ﬁnite and even a rather restricted number of three-

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dimensional geometries. Riemann’s Geometries.—However, this result seems
contradicted by Riemann, for that scientist constructs an inﬁnite number of geometries, and that to which his name is usually attached is only a particular case of them. All depends, he says, on the manner in which the length of a curve is deﬁned. Now, there is an inﬁnite number 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 movement of a variable ﬁgure such as we assume to be possible in Lie’s theorem. These geometries of Riemann, so interesting on various grounds, can never be, therefore, purely analytical, and would not lend themselves to proofs analogous to those of Euclid.
On the Nature of Axioms.—Most mathematicians regard Lobatschewsky’s geometry as a mere logical curiosity. Some of them have, however, gone further. If several geometries are possible, they say, is it certain that our geometry is the one that is true? Experiment 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 small. According to Lobatschewsky, the diﬀerence is proportional to the area of

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the triangle, and will not this become sensible when we operate on much larger triangles, and when our measurements become more accurate? Euclid’s geometry would thus be a provisory geometry. Now, to discuss this view we must ﬁrst of all ask ourselves, what is the nature of geometrical axioms? Are they synthetic à priori intuitions, as Kant aﬃrmed? They would then be imposed upon us with such a force that we could not conceive of the contrary proposition, nor could we build upon it a theoretical ediﬁce. There would be no non-Euclidean geometry. To convince ourselves of this, let us take a true synthetic à priori intuition—the following, for instance, which played an important part 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 for all positive integers. Let us next try to get rid of this, and while rejecting this proposition let us construct a false arithmetic analogous to non-Euclidean geometry. We shall not be able to do it. We shall be even tempted at the outset to look upon these intuitions as analytical. Besides, to take up again our ﬁction of animals without thickness, we can scarcely admit that these beings, if their minds are like ours, would adopt the Euclidean geometry, which would be contradicted by

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all their experience. Ought we, then, to conclude that the axioms of geometry are experimental truths? But we do not make experiments on ideal lines or ideal circles; we can only make them on material objects. On what, therefore, would experiments serving as a foundation for geometry be based? The answer is easy. We have seen above that we constantly reason as if the geometrical ﬁgures behaved like solids. What geometry would borrow from experiment would be therefore the properties of these bodies. The properties of light and its propagation in a straight line have also given rise to some of the propositions of geometry, and in particular to those of projective geometry, so that from that point of view one would be tempted to say that metrical geometry is the study of solids, and projective geometry that of light. But a diﬃculty remains, and is unsurmountable. If geometry were an experimental science, it would not be an exact science. It would be subjected to continual revision. Nay, it would from that day forth be proved to be erroneous, for we know that no rigorously invariable solid exists. The geometrical axioms are therefore neither synthetic à priori intuitions nor experimental facts. They are conventions. Our choice among all possible conventions is guided by experimental facts; but it remains free,

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and is only limited by the necessity of avoiding every contradiction, and thus it is that postulates may remain rigorously true even when the experimental laws which have determined their adoption are only approximate. In other words, the axioms of geometry (I do not speak of those of arithmetic) are only deﬁnitions in disguise. What, then, are we to think of the question: Is Euclidean geometry true? It has no meaning. We might as well ask if the metric system is true, and if the old weights and measures are false; if Cartesian co-ordinates are true and polar co-ordinates false. One geometry cannot be more true than another; it can only be more convenient. Now, Euclidean geometry is, and will remain, the most convenient: 1st, because it is the simplest, and it is not so only because of our mental habits or because of the kind of direct intuition that we have of Euclidean space; it is the simplest in itself, just as a polynomial of the ﬁrst degree is simpler than a polynomial of the second degree; 2nd, because it suﬃciently agrees with the properties of natural solids, those bodies which we can compare and measure by means of our senses.

CHAPTER IV.
SPACE AND GEOMETRY.
Let us begin with a little paradox. Beings whose minds were made as ours, and with senses like ours, but without any preliminary education, might receive from a suitablychosen external world impressions which would lead them to construct a geometry other than that of Euclid, and to localise the phenomena of this external world in a nonEuclidean space, or even in space of four dimensions. As for us, whose education has been made by our actual world, if we were suddenly transported into this new world, we should have no diﬃculty in referring phenomena to our Euclidean space. Perhaps somebody may appear on the scene some day who will devote his life to it, and be able to represent to himself the fourth dimension.
Geometrical Space and Representative Space.—It is often said that the images we form of external objects are localised in space, and even that they can only be formed on this condition. It is also said that this space, which thus serves as a kind of framework ready prepared for our sensations and representations, is identical with the space of the geometers, having all the properties of that space. To all clear-headed men who think in this

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way, the preceding statement might well appear extraordinary; but it is as well to see if they are not the victims of some illusion which closer analysis may be able to dissipate. In the ﬁrst place, what are the properties of space properly so called? I mean of that space which is the object of geometry, and which I shall call geometrical space. The following are some of the more essential:—
1st, it is continuous; 2nd, it is inﬁnite; 3rd, it is of three dimensions; 4th, it is homogeneous—that is to say, all its points are identical one with another; 5th, it is isotropic. Compare this now with the framework of our representations and sensations, which I may call representative space.
Visual Space.—First of all let us consider a purely visual impression, due to an image formed on the back of the retina. A cursory analysis shows us this image as continuous, but as possessing only two dimensions, which already distinguishes purely visual from what may be called geometrical space. On the other hand, the image is enclosed within a limited framework; and there is a no less important diﬀerence: this pure visual space is not homogeneous. All the points on the retina, apart from the images which may be formed, do not play the same rôle. The yellow spot can in no way be regarded as identical

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with a point on the edge of the retina. Not only does the same object produce on it much brighter impressions, but in the whole of the limited framework the point which occupies the centre will not appear identical with a point near one of the edges. Closer analysis no doubt would show us that this continuity of visual space and its two dimensions are but an illusion. It would make visual space even more diﬀerent than before from geometrical space, but we may treat this remark as incidental.
However, sight enables us to appreciate distance, and therefore to perceive a third dimension. But every one knows that this perception of the third dimension reduces to a sense of the eﬀort of accommodation which must be made, and to a sense of the convergence of the two eyes, that must take place in order to perceive an object distinctly. These are muscular sensations quite diﬀerent from the visual sensations which have given us the concept of the two ﬁrst dimensions. The third dimension will therefore not appear to us as playing the same rôle as the two others. What may be called complete visual space is not therefore an isotropic space. It has, it is true, exactly three dimensions; which means that the elements of our visual sensations (those at least which concur in forming the concept of extension) will be completely deﬁned if we

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know three of them; or, in mathematical language, they will be functions of three independent variables. But let us look at the matter a little closer. 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 in harmony; there is between them a constant relation; or, in mathematical language, the two variables which measure these two muscular sensations do not appear to us as independent. Or, again, to avoid an appeal to mathematical ideas which are already rather too 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 accompany them respectively will also be indistinguishable. But that is, so to speak, an experimental fact. Nothing prevents us à priori from assuming the contrary, and if the contrary takes place, if these two muscular sensations both vary independently, we must take into account one more independent variable, and complete visual space will appear to us as a physical continuum of four dimensions. And so in this there is also a fact of external experiment. Nothing prevents us from assuming that a being with a mind like

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ours, with the same sense-organs as ourselves, may be placed in a world in which light would only reach him after being passed through refracting media of complicated form. The two indications which enable us to appreciate distances would cease to be connected by a constant relation. A being educating his senses in such a world would no doubt attribute four dimensions to complete visual space.
Tactile and Motor Space.—“Tactile space” is more complicated still than visual space, and diﬀers even more widely from geometrical space. It is useless to repeat for the sense of touch my remarks on the sense of sight. But outside the data of sight and touch there are other sensations which contribute as much and more than they do to the genesis of the concept of space. They are those which everybody knows, which accompany all our movements, and which we usually call muscular sensations. The corresponding framework constitutes what may be called motor space. Each muscle gives rise to a special sensation which may be increased or diminished so that the aggregate 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 that it is said that if

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the muscular sensations contribute to form the concept of space, it is because we have the sense of the direction of each movement, and that this is an integral part of the sensation. If this were so, and if a muscular sense could not be aroused unless it were accompanied by this geometrical sense of direction, geometrical space would certainly be a form imposed upon our sensitiveness. But I do not see this at all when I analyse 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 simple association of ideas. It is to this association that what we call the sense of direction is reduced. We cannot therefore discover this sense 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 very diﬀerent movements of direction. Moreover, it is evidently acquired; it is like all associations of ideas, the result of a habit. This habit itself is the result of a very large number of experiments, and no doubt if the education of our senses had taken place in a diﬀerent medium, where we would have been subjected to diﬀerent impressions, then contrary habits would have been acquired, and our muscular sensations would have been associated

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according to other laws. Characteristics of Representative Space.—Thus rep-
resentative space in its triple form—visual, tactile, and motor—diﬀers essentially from geometrical space. It is neither homogeneous nor isotropic; we cannot even say that it is of three dimensions. It is often said that we “project” into geometrical space the objects of our external perception; that we “localise” them. Now, has that any meaning, and if so what is that meaning? Does it mean that we represent to ourselves external objects in geometrical space? Our representations are only the reproduction of our sensations; they cannot therefore be arranged in the same framework—that is to say, in representative space. It is also just as impossible for us to represent to ourselves external objects in geometrical space, as it is impossible for a painter to paint on a ﬂat surface objects with their three dimensions. Representative space is only an image of geometrical space, an image deformed by a kind of perspective, and we can only represent to ourselves objects by making them obey the laws of this perspective. Thus we do not represent to ourselves external bodies in geometrical space, but we reason about these bodies as if they were situated in geometrical space. When it is said, on the other hand, that we “lo-

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calise” such an object in such a point of space, what does it mean? It simply means that we represent to ourselves the movements that must take place to reach that object. And it does not mean that to represent to ourselves these movements they must be projected into space, and that the concept of space must therefore pre-exist. When I say that we represent to ourselves these movements, I only mean that we represent to ourselves the muscular sensations which accompany them, and which have no geometrical character, and which therefore in no way imply the pre-existence of the concept of space.
Changes of State and Changes of Position.—But, it may be said, if the concept of geometrical space is not imposed upon our minds, and if, on the other hand, none of our sensations can furnish us with that concept, how then did it ever come into existence? This is what we have now to examine, and it will take some time; but I can sum up in a few words the attempt at explanation which I am going to develop. None of our sensations, if isolated, could have brought us to the concept of space; we are brought to it solely by studying the laws by which those sensations succeed one another. We see at ﬁrst that our impressions are subject to change; but among the changes that we ascertain, we are very soon

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led to make a distinction. Sometimes we say that the objects, the causes of these impressions, have changed their state, sometimes that they have changed their position, that they have only been displaced. Whether an object changes its state or only its position, this is always translated for us in the same manner, by a modiﬁcation in an aggregate of impressions. How then have we been enabled to distinguish them? If there were only change of position, we could restore the primitive aggregate of impressions by making movements which would confront us with the movable object in the same relative situation. We thus correct the modiﬁcation which was produced, and we re-establish the initial state by an inverse modiﬁcation. If, for example, it were a question of the sight, and if an object be displaced before our eyes, we can “follow it with the eye,” and retain 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 geometrical space. So what characterises change of position, what distinguishes it from change of state, is that it can always be corrected by this means. It may therefore happen that we pass

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from the aggregate of impressions A to the aggregate B in two diﬀerent ways. First, involuntarily and without experiencing muscular sensations—which happens when it is the object that is displaced; secondly, voluntarily, and with muscular sensation—which happens when the object is motionless, but when we displace ourselves in such a way that the object has relative motion with respect to us. If this be so, the translation of the aggregate A to the aggregate B is only a change of position. It follows that sight and touch could not have given us the idea of space without the help of the “muscular sense.” Not only could this concept not be derived from a single sensation, or even from a series of sensations; but a motionless being could never have acquired it, because, not being able to correct by his movements the eﬀects of the change of position of external objects, he would have had no reason to distinguish them from changes of state. Nor would he have been able to acquire it if his movements had not been voluntary, or if they were unaccompanied by any sensations whatever.
Conditions of Compensation.—How is such a compensation possible in such a way that two changes, otherwise mutually independent, may be reciprocally corrected? A mind already familiar with geometry would

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reason as follows:—If there is to be compensation, the diﬀerent parts of the external object on the one hand, and the diﬀerent organs of our senses on the other, must be in the same relative position after the double change. And for that to be the case, the diﬀerent parts of the external body on the one hand, and the diﬀerent organs of our senses on the other, must have the same relative position to each other after the double change; and so with the diﬀerent parts of our body with respect to each other. In other words, the external object in the ﬁrst change must be displaced as an invariable solid would be displaced, and it must also be so with the whole of our body in the second change, which is to correct the ﬁrst. Under these conditions compensation may be produced. But we who as yet know nothing of geometry, whose ideas of space are not yet formed, we cannot reason in this way—we cannot predict à priori if compensation is possible. But experiment shows us that it sometimes does take place, and we start from this experimental fact in order to distinguish changes of state from changes of position.
Solid Bodies and Geometry.—Among surrounding objects there are some which frequently experience displacements that may be thus corrected by a correlative movement of our own body—namely, solid bodies. The other

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objects, whose form is variable, only in exceptional circumstances undergo similar displacement (change of position without change of form). When the displacement of a body takes place with deformation, we can no longer by appropriate movements place the organs of our body in the same relative situation with respect to this body; we can no longer, therefore, reconstruct the primitive aggregate of impressions.
It is only later, and after a series of new experiments, that we learn how to decompose a body of variable form into smaller elements such that each is displaced approximately according to the same laws as solid bodies. We thus distinguish “deformations” from other changes of state. In these deformations each element undergoes a simple change of position which may be corrected; but the modiﬁcation of the aggregate is more profound, and can no longer be corrected by a correlative movement. Such a concept is very complex even at this stage, and has been relatively slow in its appearance. It would not have been conceived at all had not the observation of solid bodies shown us beforehand how to distinguish changes of position.
If, then, there were no solid bodies in nature there would be no geometry.

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Another remark deserves a moment’s attention. Suppose a solid body to occupy successively the positions α and β; in the ﬁrst position it will give us an aggregate of impressions A, and in the second position the aggregate of impressions B. Now let there be a second solid body, of qualities entirely diﬀerent from the ﬁrst—of diﬀerent colour, for instance. Assume it to pass from the position α, where it gives us the aggregate of impressions A to the position β, where it gives the aggregate of impressions B . In general, the aggregate A will have nothing in common with the aggregate A , nor will the aggregate B have anything in common with the aggregate B . The transition from the aggregate A to the aggregate B, and that of the aggregate A to the aggregate B , are therefore two changes which in themselves have in general nothing in common. Yet we consider both these changes as displacements; and, further, we consider them the same displacement. How can this be? It is simply because they may be both corrected by the same correlative movement of our body. “Correlative movement,” therefore, constitutes the sole connection between two phenomena which otherwise we should never have dreamed of connecting.
On the other hand, our body, thanks to the number of its articulations and muscles, may have a multitude

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of diﬀerent movements, but all are not capable of “correcting” a modiﬁcation of external objects; those alone are capable of it in which our whole body, or at least all those in which the organs of our senses enter into play are displaced en bloc—i.e., without any variation of their relative positions, as in the case of a solid body.
To sum up:— 1. In the ﬁrst place, we distinguish two categories of phenomena:—The ﬁrst involuntary, unaccompanied by muscular sensations, and attributed to external objects— they are external changes; the second, of opposite character and attributed to the movements of our own body, are internal changes. 2. We notice that certain changes of each in these categories may be corrected by a correlative change of the other category. 3. We distinguish among external changes those that have a correlative in the other category—which we call displacements; and in the same way we distinguish among the internal changes those which have a correlative in the ﬁrst category. Thus by means of this reciprocity is deﬁned a particular class of phenomena called displacements. The laws of these phenomena are the object of geometry.

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Law of Homogeneity.—The ﬁrst of these laws is the law of homogeneity. Suppose that by an external change we pass from the aggregate of impressions A to the aggregate B, and that then this change α is corrected by a correlative voluntary movement β, so that we are brought back to the aggregate A. Suppose now that another external change α brings us again from the aggregate A to the aggregate B. Experiment then shows us that this change α , like the change α, may be corrected by a voluntary correlative movement β , and that this movement β corresponds to the same muscular sensations as the movement β which corrected α.
This fact is usually enunciated as follows:—Space is homogeneous and isotropic. We may also say that a movement which is once produced may be repeated a second and a third time, and so on, without any variation of its properties. In the ﬁrst chapter, in which we discussed the nature of mathematical reasoning, we saw the importance that should be attached to the possibility of repeating the same operation indeﬁnitely. The virtue of mathematical reasoning is due to this repetition; by means of the law of homogeneity geometrical facts are apprehended. To be complete, to the law of homogeneity must be added a multitude of other laws, into the details

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of which I do not propose to enter, but which mathematicians sum up by saying that these displacements form a “group.”
The Non-Euclidean World.—If geometrical space were a framework imposed on each of our representations considered individually, it would be impossible to represent to ourselves an image without this framework, and we should be quite unable to change our geometry. But this is not the case; geometry is only the summary of the laws by which these images succeed each other. There is nothing, therefore, to prevent us from imagining a series of representations, similar in every way to our ordinary representations, but succeeding one another according to laws which diﬀer from those to which we are accustomed. We may thus conceive that beings whose education has taken place in a medium in which those laws would be so diﬀerent, might have a very diﬀerent geometry from ours.
Suppose, for example, a world enclosed in a large sphere and subject to the following laws:—The temperature is not uniform; it is greatest at the centre, and gradually decreases as we move towards the circumference of the sphere, where it is absolute zero. The law of this temperature is as follows:—If R be the radius of the

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sphere, and r the distance of the point considered from the centre, the absolute temperature will be proportional to R2 − r2. Further, I shall suppose that in this world all bodies have the same co-eﬃcient of dilatation, so that the linear dilatation of any body is proportional to its absolute temperature. Finally, I shall assume that a body transported from one point to another of diﬀerent temperature is instantaneously in thermal equilibrium with its new environment. There is nothing in these hypotheses either contradictory or unimaginable. A moving object will become smaller and smaller as it approaches the circumference of the sphere. Let us observe, in the ﬁrst place, that although from the point of view of our ordinary geometry this world is ﬁnite, to its inhabitants it will appear inﬁnite. As they approach the surface of the sphere they become colder, and at the same time smaller and smaller. The steps they take are therefore also smaller and smaller, so that they can never reach the boundary of the sphere. If to us geometry is only the study of the laws according to which invariable solids move, to these imaginary beings it will be the study of the laws of motion of solids deformed by the diﬀerences of temperature alluded to.
No doubt, in our world, natural solids also experience