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gift of 
Mrs. Clarence I. Lewis 




Edited bt J. McEEEN CATTELL 



SCISNCE AND EDUCATION. A series of volumes for 
the promotion of scientific research and educational 

Volume I. The FonndationB of Science. By H. 
PoincarA. Ck>ntaining the authorised English 
translation by George Bruce Halsted of "Science 
and Hypothesis," "The Value of Science," and 
"Science and Method." 

Volume n. Medical Research and Education. By 
Richard Mills Pearce, William H. Welch, W. H. 
Howell, Franklin P. Mall, Lewellys F. Barker, 
Charles S. Minot, W. B. Cannon, W. T. Council- 
man, Theobald Smith, G. N. Stewart, C. M. Jack- 
son, E. P. Lyon, James B. Herrick, John M. Dod- 
son, C. R. Bardeen, W. Ophtds, S. J. Meltier, James 
Ewing, W. W. Keen, Henry H. Donaldson, Christ- 
ian A. Herter, and Henry P. Bowditch. 

Volume m. UniTenity Control. By J. McKbbn 
Cattbll and other authors. 


SCISNCE. A weekly journal devoted to the advancement 
of science. The official organ of the American Asso- 
ciation for the Advancement of Science. 

magasine devoted to the diffusion of science. 

devoted to the biological sciences, with spedid refer- 
ence to the factors of evolution. 














Ck)pyright, 1913 
Bt The Sgebngob Pbbsb 







Henri Poincard zi 

Author 'b Preface to the Translation 3 


Introduction hj Bojee 9 

Introduction 27 

Past I. Number and Magnitude 

Chapter I. — On the Nature of Mathematical Beasoning 31 

Sjllogistic Deduction 31 

Verification and Proof 32 

Elements of Arithmetic 33 

Reasoning hj Becurrence 37 

Induction ... * 40 

Mathematical Construction 41 

Chapter II. — ^Mathematical Magnitude and Experience 43 

Definition of Incommensurables 44 

The Physical Continuum 46 

Creation of the Mathematical Continuum 46 

Measurable Magnitude 49 

Various Bemarks (Curves without Tangents) 50 

The Physical Continuum of Several Dimensions 52 

The Mathematical Continuum of Several Dimensions 53 

Part II. Space 

Chapter HE. — The Non-Euclidean Geometries 55 

The Bolyai-Lobachevski Geometry 56 

Riemann 's Geometry 57 

The Surfaces of Constant Curvature 58 

Interpretation of Non-Euclidean Geometries 59 

The Implicit Axioms 60 

The Fourth Geometry 62 

Lie's Theorem 62 

Biemann 's Geometries 63 

On the Nature of Axioms 63 

Chapter IV. — Space and Geometry 66 

Geometric Space and Perceptual Space 66 

Visual Space 67 

Tactile Space and Motor Space 68 

Characteristics of Perceptual Space 69 

Change of State and Change of Position 70 

Conditions of Compensation 72 



Solid Bodies and (Geometry 72 

Law of Homogeneity 74 

The Non-Euclidean World 75 

The World of Pour Dimenaiona 78 

Conclusions 79 

Chaptbb V. — ^Experience and Geometry 81 

Geometry and Aatronomy 81 

The Law of Belativity 83 

Bearing of Experiments 86 

Supplement (What is a Pointf ) 89 

Ancestral Experience 91 

Pabt m. Force 

CHAPm VI. — The Classic Mechanics 92 

The Principle of Inertia 93 

The Law of Acceleration 97 

Anthropomorphic Mechanics 103 

The School of the Thread 104 

Ohaptbr YII. — ^Belatiye Motion and Absolute Motion 107 

The Principle of Belative Motion 107 

Newton 's Argument 108 

Chapter VIII. — ^Energy and Thermodynamics 115 

Energetics 115 

Thermodynamics 119 

General Conclusions on Part HI 123 

Past IV. Natwre 

Chapthi IX.— Hypotheses in Physics 127 

The Bdle of Experiment and (Generalization 127 

The Unity of Nature 130 

The Bdle of Hypothesis 133 

Origin of Mathematical Physics 136 

Chapter X. — The Theories of Modem Physics 140 

Meaning of Physical Theories 140 

Physics and Mechanism 144 

Present State of the Science 148 

Chapter XI. — The Calculus of Probabilities 155 

Classification of the Problems of Probability 158 

Probability in Mathematics 161 

Probability in the Physical Sciences 164 

Bouge et noir 167 

The Probability of Causes 169 

The Theory of Errors 170 

Conclusions 172 

Chapter XII. — Optics and Electricity 174 

Fresnel 's Theory 174 

Maxwell's Theory 175 

The Mechanical Explanation of Physical Phenomena 177 


Xm.— Electrodynamics 184 

Ampere's Theory 184 

Closed Currents 185 

Action of a Closed Current on a Portion of Current 186 

Continuous Botations 187 

Mutual Action of Two Open Currents 189 

Induction 190 

Theory of Helmholtz 191 

Difficulties Baised by these Theories 193 

Maxwell's Theory 193 

Bowland 's Experiment 194 

The Theory of Lorentz 196 


Translator 's Introduction 201 

Does the Scientist Create Sciencef 201 

The Mind Dispelling Optical Illusions 202 

Euclid not Necessary 202 

Without Hypotheses, no Science 203 

What Outcomef 203 

Introduction 205 

Past I. The Mathematical Sciences 

Chaptkb I. — ^Intuition and Logic in Mathematics 210 

Crafteb, II.— -The Measure of Time 223 

Chapter III.— The Notion of Space 235 

Qualitative Geometry 238 

The Physical Continuum of Several Dimensions 240 

The Notion of Point 244 

The Notion of Displacement 247 

Visual Space 252 

Chaptib IV. — Space and its Three Dimensions 256 

The Group of Displacements 256 

Identity of Two Points 259 

Tactile Space 264 

Identity of the Different Spaces 268 

Space and Empiricism 271 

B6le of the Semicircular Canals 276 

Paet II. TTw Physical Sciences 

Chaptee. V.^Analysis and Physics 279 

Chapter VI. — ^Astronomy 289 

Chapter VII. — The History of Mathematical Physics 297 

The Physics of Central Forces 297 

The Physics of the Principles 299 

Chapter Vin. — ^The Present Crisis in Physics 303 

The New Crisis 303 

Camot's Principle 303 


The Principle of Eelativity 305 

Newton's Principle 308 

Lavoisier 's Principle 310 

Majer 's Principle 312 

Chapter IX. — The Future of Mathematical Physics 314 

The Principles and Experiment 314 

The BMe of the Analyst 314 

Aberration and Astronomy 315 

Electrons and Spectra 316 

Conventions preceding Experiment 317 

Futare Mathematical Physics 319 

Part III. The Objective Value of Science 

Chapter X. — Is Science Artificialf 321 

The Philosophy of LeBoy 321 

Science, Bule of Action 323 

The Crude Fact and the Scientific Fact 325 

Nominalism and the Universal Invariant 333 

Chapter XI. — Science and Reality 340 

Contingence and Determinism 340 

Objectivity of Science 347 

The notation of the Earth 353 

Science for Its Own Sake 354 


Introduction 359 

Book I. Science and the Scientist 

Chapter I. — The Choice of Facts 362 

Chapter II. — The Future of Mathematics 369 

Chapter III. — ^Mathematical Creation 383 

Chapter IV. — Chance 395 

Book n. Maihematicdl Seasoning 

Chapter I.— The Belativity of Space 413 

Chapter II. — ^Mathematical Definitions and Teaching 430 

Chapter III. — ^Mathematics and Logic 448 

Chapter IV. — The New Logics 460 

Chapter V. — The Latest Efforts of the Logisticians 472 

Book III. The New Mechanics 

Chapter I. — Mechanics and Badium 486 

Chapter II. — ^Mechanics and Optics 496 

Chapter HJ. — The New Mechanics and Astronomy 515 

Book IV. Astronomic Science 

Chapter I. — The Milky Way and the Theory of Gases 522 

Chapter I. — ^French Geodesy 535 

General Conclusions 544 

Index 547 


Sm George Darwin, worthy son of an immortal father, said, 
referring to what Poincar^ was to him and to his work: **He 
must be regarded as the presiding genius — or, shall I say, my 
patron saint t" 

Henri Poincar6 was born April 29, 1854, at Nancy, where his 
father was a physician highly respected. His schooling was 
broken into by the war of 1870-71, to get news of which he 
learned to read the German newspapers. He outclassed the 
other boys of his age in all subjects and in 1873 passed highest 
into the Ecole Polytechnique, where, like John Bolyai at Maros 
Y&s4rhely, he followed the courses in mathematics without taking 
a note and without the syllabus. He proceeded in 1875 to the 
School of Mines, and was Nomme, March 26, 1879. But he won 
his doctorate in the University of Paris, August 1, 1879, and 
was appointed to teach in the Faculty des Sciences de Caen, 
December 1, 1879, whence he was quickly called to the Uni- 
versity of Paris, teaching there from October 21, 1881, until his 
death, July 17, 1912. So it is an error to say he started as an 
engineer. At the early age of thirty-two he became a member 
of TAcad^mie des Sciences, and, March 5, 1908, was chosen 
Membre de TAcademie Frangaise. July 1, 1909, the number of 
his writings was 436. 

His earliest publication was in 1878, and was not important. 
Afterward came an essay submitted in competition for the 
Grand Prix offered in 1880, but it did not win. Suddenly there 
came a change, a striking fire, a bursting forth, in February, 
1881, and Poincare tells us the very minute it happened. Mount- 
ing an omnibus, **at the moment when I put my foot upon the 
step, the idea came to me, without anything in my previous 
thoughts seeming to foreshadow it, that the transformations I had 
used to define the Fuchsian functions were identical with those 
of non-Euclidean geometry.'' Thereby was opened a perspec- 
tive new and immense. Moreover, the magic wand of his whole 



life-work had been grasped, the Aladdin's lamp had been rubbed, 
non-Euclidean geometry, whose necromancy was to open up a 
new theory of our universe, whose brilliant exposition was com- 
menced in his book Science and Hypothesis, which has been 
translated into six languages and has already had a circulation 
of over 20,000. The non-Euclidean notion is that of the possi- 
bility of alternative laws of nature, which in the Introduction 
to the Electridte et Optique, 1901, is thus put: ''If therefore a 
phenomenon admits of a complete mechanical explanation, it 
will admit of an infinity of others which will account equally 
well for all the peculiarities disclosed by experiment." 

The scheme of laws of nature so largely due to Newton is 
merely one of an infinite number of conceivable rational schemes 
for helping us master and make experience; it is commode, con- 
venient; but perhaps another may be vastly more advantageous. 
The old conception of true has been revised. The first expres- 
sion of the new idea occurs on the title page of John Bolyai's 
marvelous Science Absolute of Space, in the phrase **haud un- 
quam a priori decidenda." 

With bearing on the history of the earth and moon system and 
the origin of double stars, in formulating the geometric criterion 
of stability, Poincar^ proved the existence of a previously un- 
known pear-shaped figure, with the possibility that the progres- 
sive deformation of this figure with increasing angular velocity 
might result in the breaking up of the rotating body into two 
detached masses. Of his treatise Les Methodes nouvelles de la 
Mechanique celeste. Sir George Darwin says: **It is probable that 
for half a century to come it will be the mine from wh^ch humbler 
investigators will excavate their materials." Brilliant was his 
appreciation of Poincar6 in presenting the gold medal of the 
Royal Astronomical Society. The three others most akin in 
genius are linked with him by the Sylvester medal of the Royal 
Society, the Lobachevski medal of the Physico-Mathematical 
Society of Kazan, and the Bolyai prize of the Hungarian Acad- 
emy of Sciences. His work must be reckoned with the greatest 
mathematical achievements of mankind. 

The kernel of Poincar6's power lies in an oracle Sylvester often 
quoted to me as from Hesiod : The whole is less than its part. 


He penetrates at once the divine simplicity of the perfectly 
general case, and thence descends, as from Olympus, to the 
special concrete earthly particulars. 

A combination of seemingly extremely simple analytic and 
geometric concepts gave necessary general conclusions of im- 
mense scope from which sprang a disconcerting wilderness of 
possible deductions. And so he leaves a noble, fruitful heritage. 

Says Love: ''His right is recognized now, and it is not likely 
that future generations will revise the judgment, to rank among 
the greatest mathematicians of all time." 

Geobgb Bruce Halsted. 





I AM exceedingly grateful to Dr. Halsted, who has been so 
good as to present my book to American readers in a translation, 
clear and faithful. 

Every one knows that this savant has already taken the trouble 
to translate many European treatises and thus has powerfully 
contributed to make the new continent understand the thought 
of the old. 

Some people love to repeat that Anglo-Saxons have not the 
same way of thinking as the Latins or as the Germans ; that they 
have quite another way of understanding mathematics or of un- 
derstanding physics ; that this way seems to them superior to all 
others ; that they feel no need of changing it, nor even of know- 
ing the ways of other peoples. 

In that they would beyond question be wrong, but I do not 
believe that is true, or, at least, that is true no longer. For some 
time the English and Americans have been devoting themselves 
much more than formerly to the better understanding of what is 
thought and said on the continent of Europe. 

To be sure, each people will preserve its characteristic genius, 
and it would be a pity if it were otherwise, supposing such a 
thing possible. If the Anglo-Saxons wished to become Latins, 
they would never be more than bad Latins; just as the French, 
in seeking to imitate them, could turn out only pretty poor 

And then the English and Americans have made scientific 
eonquests they alone could have made ; they will make still more 
of which others would be incapable. It would therefore be de- 
plorable if there were no longer Anglo-Saxons. 

But continentals have on their part done things an English- 
man could not have done, so that there is no need either for 
wishing all the world Anglo-Saxon. 

Each has his characteristic aptitudes, and these aptitudes 



should be diverse, else would the scientific concert resemble a 
quartet where every one wanted to play the violin. 

And yet it is not bad for the violin to know what the violon- 
cello is playing, and vice versa. 

This it is that the English and Americans are comprehending 
more and more; and from this point of view the translations 
undertaken by Dr. Halsted are most opportune and timely. 

Consider first what concerns the mathematical sciences. It 
is frequently said the English cultivate them only in view of 
their applications and even that they despise those who have 
other aims; that speculations too abstract repel them as savor- 
ing of metaphysic. 

The English, even in mathematics, are to proceed always 
from the particular to the general, so that they would never have 
an idea of entering mathematics, as do many Germans, by the 
gate of the theory of aggregates. They are always to hold, so to 
speak, one foot in the world of the senses, and never burn the 
bridges keeping them in communication with reality. They thus 
are to be incapable of comprehending or at least of appreciat- 
ing certain theories more interesting than utilitarian, such as the 
non-Euclidean geometries. According to that, the first twK) 
parts of this book, on number and space, should seem to them 
void of all substance and would only baflBe them. 

But that is not true. And first of all, are they such uncom- 
promising realists as has been said? Are they absolutely refrac- 
tory, I do not say to metaphysic, but at least to everything 
metaphysical ? 

Recall the name of Berkeley, bom in Ireland doubtless, but 
immediately adopted by the English, who marked a natural and 
necessary stage in the development of English philosophy. 

Is this not enough to show they are capable of making ascen- 
sions otherwise than in a captive balloon? 

And to return to America, is not the Monist published at 
Chicago, that review which even to us seems bold and yet which 
finds readers? 

And in mathematics? Do you think American geometers 
are concerned only about applications? Far from it. The part 
of the science they cultivate most devotedly is the theory of 


groups of snbstitations, and under its most abstract form, the 
farthest removed from the practical. 

Moreover, Dr. Halsted gives regularly each year a review of 
all productioDS relative to the non-Euclidean geometry, and he 
has about him a public deeply interested in his work. He has 
initiated this public into the ideaa of Hilbert, and he has even 
written an elementary treatise on 'Rational Geometry,' based 
on the principles of the renowned German savant. 

To introduce this principle into teaching is surely this time 
to bum all bridges of reliance upon sensory intuition, and this is, 
I confess, a boldness which seems to me almost rash&ess. 

The American public is therefore much better prepared than 
has been thought for investigating the origin of the notion of 

Moreover, to analyze this concept is not to sactifiee reality to 
I know not what phantom. The geometric language is after all 
only a language. Space is only a word that we have believed 
a thing. "What is the origin of this word and of other words 
alsol What things do they hidet To ask this is permissible; 
to forbid it would be, on the contrary, to be a dupe of words ; 
it would be to adore a metaphysical idol, like savage peoples who 
prostrate themselves before a statue of wood without daring to 
take a look at what is within. 

Iq the study of nature, the contrast between the Anglo-Saxon 
spirit and the Latin spirit is still greater. 

The Latins seek in general to put their thought in mathe- 
matical form; the English prefer to express it by a material 

Both doubtless rely only on experience for knowing the world; 
when they happen to go beyond this, they consider their fore- 
knowledge as only provisional, and they hasten to ask its defini- 
tive confirmation from nature herself. 

But experience is not all, and the savant is not passive; he 
does not wait for the truth to come and find him, or for a 
chance meeting to bring him face to face with it. He must go 
to meet it, and it is for his thinking to reveal to him the way 
leading thither. For that there is need of an instrument ; well, 
just there begins the difference — the instrument the Latins ordi- 
narily choose is not that preferred by the Anglo-Saxons. 


For a Latin, truth can be expressed only by equations; it 
must obey laws simple, logical, symmetric and fitted to satisfy 
minds in love with mathematical elegance. 

The Anglo-Saxon to depict a phenomenon will first be en- 
grossed in making a model, and he will make it with common 
materials, such as our crude, unaided senses show us them. He 
also makes a hypothesis, he assumes implicitly that nature, in her 
finest elements, is the same as in the complicated aggregates 
which alone are within the reach of our senses. He concludes 
from the body to the atom. 

Both thelrefore make hypotheses, and this indeed is necessary, 
since no scientist has ever been able to get on without them. The 
essential thing is never to make them unconsciously. 

From this point of view again, it would be well for these two 
sorts of physicists to know something of each other; in study- 
ing the work of minds so unlike their own, they will immedi- 
ately recognize that in this work there has been an accumulation 
of hypotheses. 

Doubtless this will not suffice to make them comprehend that 
they on their part have made just as many; each sees the mote 
without seeing the beam ; but by their criticisms they will warn 
their rivals, and it may be supposed these will not fail to render 
them the same service. 

The English procedure often seems to us crude, the analogies 
they think they discover to us seem at times superficial ; they are 
not sufficiently interlocked, not precise enough; they sometimes 
permit incoherences, contradictions in terms, which shock a geo- 
metric spirit and which the employment of the mathematical 
method would immediately have put in evidence. But most often 
it is, on the other hand, very fortunate that they have not per- 
ceived these contradictions; else would they have rejected their 
model and could not have deduced from it the brilliant results 
they have often made to come out of it. 

And then these very contradictions, when they end by per- 
ceiving them, have the advantage of showing them the hypothet- 
ical character of their conceptions, whereas the mathematical 
method, by its apparent rigor and inflexible course, often inspires 
in us a confidence nothing warrants, and prevents our looking 
about us. 


From another point of view, however, the two conceptions are 
very unlike, and if all must be said, they are very unlike because 
of a common fault. 

The English wish to make the world out of what we see. I 
mean what we see with the unaided eye, not the microscope, nor 
that still more subtile microscope, the human head guided by 
scientific induction. 

The Latin wants to make it out of formulas, but these for- 
mulas are still the quintessenced expression of what we see. In 
a word, both would make the unknown out of the known, and 
their excuse is that there is no way of doing otherwise. 

And yet is this legitimate, if the unknown be the simple and 
the known the complex? 

Shall we not get of the simple a false idea, if we think it like 
the complex, or worse yet if we strive to make it out of elements 
which are themselves compounds? 

Is not each great advance accomplished precisely the day some 
one has discovered under the complex aggregate shown by our 
senses something far more simple, not even resembling it — as 
when Newton replaced Kepler's three laws by the single law of 
gravitation, which was something simpler, equivalent, yet unlike ? 

One is justified in asking if we are not on the eve of just such 
a revolution or one even more important. Matter seems on 
the point of losing its mass, its solidest attribute, and resolving 
itself into electrons. Mechanics must then give place to a 
broader conception which will explain it, but which it will not 

So it was in vain the attempt was made in England to con- 
struct the ether by material models, or in Prance to apply to 
it the laws of dynamic. 

The ether it is, the unknown, which explains matter, the 
known; matter is incapable of explaining the ether. 



Habvasd University 

The treatise of a master needs no commendation through the 
words of a mere learner. But, since my friend and former fellow 
student, the translator of this volume, has joined with another 
of my colleagues. Professor Cattell, in asking me to undertake 
the task of calling the attention of my fellow students to the 
importance and to the scope of M. Poincare's volume, I accept 
the office, not as one competent to pass judgment upon the book, 
but simply as a learner, desirous to increase the number of those 
amongst us who are already interested in the type of researches 
to which M. Poincare has so notably contributed. 

The branches of inquiry collectively known as the Philosophy 
of Science have undergone great changes since the appearance of 
Herbert Spencer's First Principles, that volume which a large 
part of the general public in this country used to regard as the 
representative compend of all modern wisdom relating to the 
foundations of scientific knowledge. The summary which M. 
Poincare gives, at the outset of his own introduction to the 
present work, where he states the view which the 'superficial 
observer' takes of scientific truth, suggests, not indeed Spencer's 
own most characteristic theories, but something of the spirit in 
which many disciples of Spencer interpreting their master's 
formulas used to conceive the position which science occupies in 
dealing with experience. It was well known to them, indeed, 
that experience is a constant guide, and an inexhaustible source 
both of novel scientific results and of unsolved problems; but 
the fundamental Spencerian principles of science, such as *the 
persistence of force,' the 'rhythm of motion' and the rest, were 
treated by Spencer himself as demonstrably objective, although 



indeed 'relative' truths, capable of being tested once for all by 
the 'inconceivability of the opposite,' and certain to hold true for 
the whole 'knowable' universe. Thus, whether one dwelt upon 
the results of such a mathematical procedure as that to which M. 
Poincar6 refers in his opening paragraphs, or whether, like Spen- 
cer himself, one applied the 'first principles' to regions of less 
exact science, this confidence that a certain orthodoxy regarding 
the principles of science was established forever was characteristic 
of the followers of the movement in question. Experience, 
lighted up by reason, seemed to them to have predetermined for 
all future time certain great theoretical results regarding the real 
constitution of the 'knowable' cosmos. Whoever doubted this 
doubted 'the verdict of science.' 

Some of us well remember how, when Stallo's 'Principles and 
Theories of Modem Physics' first appeared, this sense of scien- 
tific orthodoxy was shocked amongst many of our American read- 
ers and teachers of science. I myself can recall to mind some 
highly authoritative reviews of that work in which the author 
was more or less sharply taken to task for his ignorant presump- 
tion in speaking with the freedom that he there used regarding 
such sacred possessions of humanity as the fundamental concepts 
of physics. That very book, however, has quite lately been 
translated into German as a valuable contribution to some of the 
most recent efforts to reconstitute a modem 'philosophy of 
nature.' And whatever may be otherwise thought of Stallo's 
critical methods, or of his results, there can be no doubt that, at 
the present moment, if his book were to appear for the first 
time, nobody would attempt to discredit the work merely on 
account of its disposition to be agnostic regarding the objective 
reality of the concepts of the kinetic theory of gases, or on 
account of its call for a logical rearrangement of the fundamental 
concepts of the theory of energy. We are no longer able so easily 
to know heretics at first sight. 

For we now appear to stand in this position: The control 
of natural phenomena, which through the sciences men have 
attained, grows daily vaster and more detailed, and in its de- 
tails more assured. Phenomena men know and predict better 
than ever. But regarding the most general theories, and the 


most fundamental, of science, there ia no longer an? notablt 
Kieatific orthodoxy. Thus, as knowledge grows firmer and wider, 
conceptual construction becomes less rigid. The field of the 
theoretical philosophy of nature — ^yes, the field of the logic of 
science — this whole region is to-day an open one. "Whoever will 
work there must indeed accept the verdict of experience regard- 
ing what happens in the natural world. So far he is indeed 
bound. But he may undertake without hindrance from mere 
tradition the task of trying afresh to reduce what happens 
to conceptual unity. The cdrele-squares and the inventors of 
devices for perpetual motion are indeed still as unwelcome in 
scientific company as they were in the days when scientific 
orthodoxy was more rigidly defined ; but that is not because the 
foundations of geometry are now viewed as completely settled, 
beyond controversy, nor yet because the 'persistence of force' 
has been finally so defined>as to make the 'opposite ineonceiT- 
able ' and the doctrine of energy beyond the reach of novel formu- 
lations. No, the circle-squarers and the inventors of devices for 
perpetual motion are to-day discredited, not because of any 
unorthodoxy of their general philosophy of nature, but because 
their views regarding special facts and processes stand in 
conflict with certain equally special results of science which 
themselves admit of very various general theoretical interpre- 
tations. Certain properties of the irrational number ir are 
known, in suificient multitude to justify the mathematician in 
declining to listen to the arguments of the circle-squarer ; but, 
despite great advances, and despite the assured results of Dede- 
kind, of Cantor, of "Weierstrass and of various others, the gen- 
eral theory of the logic of the numbers, rational and irrational, 
still presents several important features of great obscurity ; and 
the philosophy of the concepts of geometry yet remains, in sev- 
eral very notable respects, unconquered territory, despite the 
work of Hilbert and of Fieri, and of our author himself. The 
ordinary inventors of the perpetual motion machines still stand 
in conflict with accepted generalizations; but nobody knows as 
yet what the final form of the theory of energy will be, nor can 
any one say precisely what place the phenomena of the radioac- 
tive bodies will occupy in that theory. The alchemists would not 


be welcome workers in modem laboratories; yet some sorts of 
transformation and of evolution of the elements are to-day 
matters which theory can find it convenient, upon occasion, to 
treat as more or less exactly definable possibilities; while some 
newly observed phenomena tend to indicate, not indeed that the 
ancient hopes of the alchemists were well founded, but that the 
ultimate constitution of matter is something more fluent, less in- 
variant, than the theoretical orthodoxy of a recent period sap- 
posed. Again, regarding the foundations of biology, a theoret- 
ical orthodoxy grows less possible, less definable, less conceiv- 
able (even as a hope) the more knowledge advances. Once 
'mechanism' and 'vitalism' were mutually contradictory theories 
regarding the ultimate constitution of living bodies. Now they 
are obviously becoming more and more 'points of view,' diverse 
but not necessarily conflicting. So far as you find it convenient 
to limit your study of vital processes to those phenomena which 
distinguish living matter from all other natural obects, you may 
assume, in the modern 'pragmatic' sense, the attitude of a 'neo- 
vitalist. ' So far, however, as you are able to lay stress, with good 
results, upon the many ways in which the life processes can be 
assimilated to those studied in physics and in chemistry, yon 
work as if you were a partisan of 'mechanics.' In any case, 
your special science prospers by reason of the empirical discov- 
eries that jou make. And your theories, whatever they are, 
must not run counter to any positive empirical results. But 
otherwise, scientific orthodoxy no longer predetermines what 
alone it is respectable for you to think about the nature of living 

This gain in the freedom of theory, coming, as it does, side by 
side with a constant increase of a positive knowledge of nature, 
lends itself to various interpretations, and raises various obvious 


One of the most natural of these interpretations, one of the 
most obvious of these questions, may be readily stated. Is not 
the lesson of all these recent discussions simply this, that general 
theories are simply vain, that a philosophy of nature is an idle 


dream, and that the results of science are coextensive with the 
range of actual empirical observation and of successful predic- 
tion? If this is indeed the lesson, then the decline of theoretical 
orthodoxy in science is — ^like the eclipse of dogma in religion — 
merely a further lesson in pure positivism, another proof that 
nttn does best when he limits himself to thinking about what can 
be found in human experience, and in trying to plan what can 
be done to make human life more controllable and more reason- 
able. What we are free to do as we please — ^is it any longer a 
serious business? What we are free to think as we please — ^is it 
of any further interest to one who is in search of truth? If 
certain general theories are mere conceptual constructions, which 
to-day are, and to-morrow are cast into the oven, why dignify 
them by the name of philosophy? Has science any place for 
such theories? Why be a *neo-vitalist,' or an 'evolutionist,' or 
an * atomist, ' or an ' Energetiker ' ? Why not say, plainly : * * Such 
and such phenomena, thus and thus described, have been ob- 
served; such and such experiences are to be expected, since the 
hypotheses by the terms of which we are required to expect 
them have been verified too often to let us regard the agreement 
with experience as due merely to chance; so much then with 
reasonable assurance we know; all else is silence— or else is 
some matter to be tested by another experiment?" Why not 
limit our philosophy of science strictly to such a counsel of resig- 
nation? Why not substitute, for the old scientific orthodoxy, 
simply a confession of ignorance, and a resolution to devote our- 
selves to the business of enlarging the bounds of actual em- 
pirical knowledge? 

Such comments upon the situation just characterized are fre- 
quently made. Unfortunately, they seem not to content the 
very age whose revolt from the orthodoxy of traditional theory, 
whose uncertainty about all theoretical formulations, and whose 
vast wealth of empirical discoveries and of rapidly advancing 
special researches, would seem most to justify tliese very com- 
ments. Never has there been better reason than there is to-day 
to be content, if rational man could be content, with a pure pos- 
itivism. The splendid triumphs of special research in the most 
various fields, the constant increase in our practical control over 


nature — ^these, our positive and growing possessions, stand in 
glaring contrast to the failure of the scientific orthodoxy of a 
former period to fix the outlines of an ultimate creed about the 
nature of the knowable universe. Why not 'take the cash and 
let the credit go'f Why pursue the elusive theoretical 'unifica- 
tion' any further, when what we daily get from our sciences is 
an increasing wealth of detailed information and of practical 
guidance T 

As a fact, however, the known answer of our own age to these 
very obvious comments is a constant multiplication of new 
efforts towards large and unifying theories. If theoretical ortho- 
doxy is no longer clearly definable, theoretical construction was 
never more rife. The history of the doctrine of evolution, even 
in its most recent phases, when the theoretical uncertainties re- 
garding the 'factors of evolution' are most insisted upon, is full 
of illustrations of this remarkable union of scepticism in critical 
work with courage regarding the use of the scientific imagination. 
The history of those controversies regarding theoretical physics, 
some of whose principal phases M. Poincare, in his book, sketches 
with the hand of the master, is another illustration of the con- 
sciousness of the time. Men have their freedom of thought in 
these regions; and they feel the need of making constant and 
constructive use of this freedom. And the men who most feel 
this need are by no means in the majority of cases professional 
metaphysicians — or students who, like myself, have to view all 
these controversies amongst the scientific theoreticians from 
without as learners. These large theoretical constructions are 
due, on the contrary, in a great many cases to special workers, 
who have been driven to the freedom of philosophy by the oppres- 
sion of experience, and who have learned in the conflict with 
special problems the lesson that they now teach in the form of 
general ideas regarding the philosophical aspects of science. 

Why, then, does science actually need general theories, despite 
the fact that these theories inevitably alter and pass awayf 
What is the service of a philosophy of science, when it is certain 
that the philosophy of science which is best suited to the needs 
of one generation must be superseded by the advancing insight 
of the next generation? Why must that which endlessly grows^ 


namdy, man's knowledge of the phenomenal order of natnre^ 
be constantly united in men's minds with that which is certain 
to decay, namely, the theoretical formulation of special knowl- 
edge in more or less completely unified systems of doctrine T 

I understand our author's volume to be in the main an 
answer to this question. To be sure, the compact and manifold 
teachings which this text contains relate to a great many dif- 
ferent special issues. A student interested in the problems of 
the philosophy of mathematics, or in the theory of probabilities, 
or in the nature and office of mathematical physics, or in still 
other problems belonging to the wide field here discussed, may 
find what he wants here and there in the text, even in case the 
general issues which give the volume its unity mean little to 
him, or even if he differs from the author's views regarding the 
principal issues of the book. But in the main, this volume must 
be regarded as what its title indicates — a critique of the nature 
and place of hypothesis in the work of science and a study of the 
logical relations of theory and fact. The result of the book is a 
substantial justification of the scientific utility of theoretical con- 
struction — an abandonment of dogma, but a vindication of the 
rights of the constructive reason. 


The most notable of the results of our author's investigation 
of the logic of scientific theories relates, as I understand his work, 
to a topic which the present state of logical investigation, just 
summarized, makes especially important, but which has thus far 
been very inadequately treated in the text-books of inductive 
logic. The useful hypotheses of science are of two kinds : 

1. The hypotheses which are valuable precisely because they 
are either verifiable or else refutable through a definite appeal 
to the tests furnished by experience ; and 

2. The hypotheses which, despite the fact that experience sug- 
gests them, are valuable despite, or even because, of the fact that 
experience can neither confirm nor refute them. The contrast 
between these two kinds of hypotheses is a prominent topic of 
our author's discussion. 

Hypotheses of the general type which I have here placed first 


in order are the ones which the text-books of inductive logic and 
those summaries of scientific method which are customary in the 
course of the elementary treatises upon physical science are 
already accustomed to recognize and to characterize. The value 
of such hypotheses is indeed undoubted. But hypotheses of the 
type which I have here named in the second place are far less 
frequentiy recognized in a perfectly explicit way as useful aids 
in the work of special science. One usually either fails to admit 
their presence in scientific work, or else remains silent as to the 
reasons of their usefulness. Our author's treatment of the work 
of science is therefore especially marked by the fact that he ex- 
plicitiy makes prominent both the existence and the scientific 
importance of hypotheses of this second type. They occupy in 
his discussion a place somewhat analogous to each of the two dis- 
tinct positions occupied by the 'categories' and the 'forms of 
sensibility/ on the one hand, and by the 'regulative principles of 
the reason,' on the other hand, in the Kantian theory of our 
knowledge of nature. That is, these hypotheses which can 
neither be confirmed nor refuted by experience appear, in M. 
Poincar6's account, partly (like the conception of * continuous 
quantity') as devices of the understanding whereby we give 
conceptual unity and an invisible connectedness to certain types 
of phenomenal facts which come to us in a discrete form and in 
a confused variety; and partly (like the larger organizing con- 
cepts of science) as principles regarding the structure of the 
world in its wholeness ; i. e., as principles in the light of which we 
try to interpret our experience, so as to give to it a totality and 
an inclusive unity such as Euclidean space, or such as the world 
of the theory of energy is conceived to possess. Thus viewed, M. 
Poincare's logical theory of this second class of hypotheses under- 
takes to accomplish, with modem means and in the light of 
to-day's issues, a part of what Kant endeavored to accomplish 
in his theory of scientific knowledge with the limited means 
which were at his disposal. Those aspects of science which are 
determined by the use of the hypotheses of this second kind 
appear in our author's account as constituting an essential 
human way of viewing nature, an interpretation rather than 
a portrayal or a prediction of the objective facts of nature, an 


adjustment of our conceptions of things to the internal needs 
of our intelligence, rather than a grasping of things as they are 
in themselves. 

To be sure, M. Poincare's view, in this portion of his work, 
obviously differs, meanwhile, from that of Kant, as well as this 
agrees, in a measure, with the spirit of the Kantian epistemology. 
I do not mean therefore to class our author as a Kantian. For 
Kant, the interpretations imposed by the * forms of sensibility,' 
and by the 'categories of the understanding,' upon our doctrine 
of nature are rigidly predetermined by the unalterable 'form' 
of our intellectual powers. We 'must' thus view facts, whatever 
tiie data of sense must be. This, of course, is not M. Poincar^'s 
view. A similarly rigid predetermination also limits the Kantian 
'ideas of the reason' to a certain set of principles whose guidance 
of the course of our theoretical investigations is indeed only 
'regulative,' but is 'a priori,' and so unchangeable. For M. 
Poincar^, on the contrary, all this adjustment of our interpre- 
tations of experience to the needs of our intellect is something 
far less rigid and unalterable, and is constantly subject to the 
suggestions of experience. We must indeed interpret in our own 
way; but our way is itself only relatively determinate; it is 
essentially more or less plastic ; other interpretations of experience 
are conceivable. Those that we use are merely the ones found to 
be most convenient. But this convenience is not absolute neces- 
sity. Unverifiable and irrefutable hypotheses in science are in- 
deed, in general, indispensable aids to the organization and to the 
guidance of our interpretation of experience. But it is expe- 
rience itself which points out to us what lines of interpretation 
will prove most convenient. Instead of Kant's rigid list of 
a priori 'forms,' we consequently have in M. Poincare's account 
a set of conventions, neither wholly subjective and arbitrary, nor 
yet imposed upon us unambiguously by the external compulsion 
of experience. The organization of science, so far as this organ- 
ization is due to hypotheses of the kind here in question, thus 
resembles that of a constitutional government — neither abso- 
lutely necessary, nor yet determined apart from the will of the 
subjects, nor yet accidental — a free, yet not a capricious estab- 
lishment of good order, in conformity with empirical needs. 



Characteristic remains, however, for our author, as, in his 
decidedly contrasting way, for Kant, the thought that without 
principles which at every stage transcend precise confirmation 
through such experience as is then accessible the organization of 
experience is impossible. Whether one views these principles as 
conventions or as a priori 'forms,' they may therefore be de- 
scribed as hypotheses, but as hypotheses that, while lying at the 
basis of our actual physical sciences, at once refer to experience 
and help us in dealing with experience, and are yet neither con- 
firmed nor refuted by the experiences which we possess or which 
we can hope to attain. 

Three special instances or classes of instances, according to 
our author's account, may be used as illustrations of this general 
type of hypotheses. They are: (1) The hypothesis of the exist- 
ence of continuous extensive quanta in nature; (2) The prin- 
ciples of geometry; (3) The principles of mechanics and of the 
general theory of energy. In case of each of these special types 
of hypotheses we are at first disposed, apart from reflection, to 
say that we find the world to be thus or thus, so that, for instance, 
we can confirm the thesis according to which nature contains 
continuous magnitudes; or can prove or disprove the physical 
truth of the postulates of Euclidean geometry ; or can confirm by 
definite experience the objective validity of the principles of 
mechanics. A closer examination reveals, according to our 
author, the incorrectness of all such opinions. H3rpotheses of 
these various special types are needed ; and their usefulness can 
be empirically shown. They are in touch with experience; and 
that they are not merely arbitrary conventions is also verifiable. 
They are not a priori necessities ; and we can easily conceive in- 
telligent beings whose experience could be best interpreted with- 
out using these hypotheses. Yet these hypotheses are not sub- 
ject to direct confirmation or refutation by experience. They 
stand then in sharp contrast to the scientific hypotheses of the 
other, and more frequently recognized, type, i. e., to the hy- 
potheses which can be tested by a definite appeal to experience. 
To these other hypotheses our author attaches, of course, great 
importance. His treatment of them is full of a living apprecia- 
tion of the significance of empirical investigation. But the cen- 


tral problem of the logic of science thus becomes the problem of 
the relation between the two fundamentally distinct types of 
hypotheses, ♦. e., between those which can not be verified or re- 
futed through experience, and those which can be empirically 


The detailed treatment which M. Poincar6 gives to the problem 
thus defined must be learned from his text. It is no part of my 
purpose to expound, to defend or to traverse any of his special 
conclusions regarding this matter. Yet I can not avoid observ- 
ing that, while M. Poincar^ strictly confines his illustrations and 
his expressions of opinion to those regions of science wherein, as 
special investigator, he is himself most at home, the issues which 
he thus raises regarding the logic of science are of even more 
critical importance and of more impressive interest when one 
applies M. Poincare's methods to the study of the concepts and 
presuppositions of the organic and of the historical and social 
sciences, than when one confines one's attention, as our author 
here does, to the physical sciences. It belongs to the province of 
an introduction like the present to point out, however briefiy and 
inadequately, that the significance of our author's ideas extends 
far beyond the scope to which he chooses to confine their discussion. 

The historical sciences, and in fact all those sciences such as 
geology, and such as the evolutionary sciences in general, un- 
dertake theoretical constructions which relate to past time. Hy- 
potheses relating to the more or less remote past stand, however, 
in a position which is very interesting from the point of view of 
the logic of science. Directly speaking, no such hypothesis is 
capable of confirmation or of refutation, because we can not 
return into the past to verify by our own experience what then 
happened. Yet indirectly, such hypotheses may lead to predic- 
tions of coming experience. These latter will be subject to con- 
troL Thus, Schliemann's confidence that the legend of Troy had 
a definite historical foundation led to predictions regarding what 
certain excavations would reveal. In a sense somewhat different 
from that which filled Schliemann's enthusiastic mind, these pre- 
dictions proved verifiable. The result has been a considerable 


change in the attitude of historians toward the legend of Troy. 
Geological investigation leads to predictions regarding the order 
of the strata or the course of mineral veins in a district, regard- 
ing the fossils which may be discovered in given formations, and 
so on. These hypotheses are subject to the control of experience. 
The various theories of evolutionary doctrine include many hy- 
potheses capable of confirmation and of refutation by empirical 
tests. Yet, despite all such empirical control, it still remains 
true that whenever a science is mainly concerned with the remote 
past, whether this science be archeology, or geology, or anthro- 
pology, or Old Testament history, the principal theoretical con- 
structions always include features which no appeal to present 
or to accessible future experience can ever definitely test. Hence 
the suspicion with which students of experimental science often 
regard the theoretical constructions of their confreres of the sci- 
ences that deal with the past. The origin of the races of men, 
of man himself, of life, of species, of the planet ; the hypotheses 
of anthropologists, of archeologists, of students of 'higher criti- 
cism' — ^all these are matters which the men of the laboratory 
often regard with a general incredulity as belonging not at all 
to the domain of true science. Yet no one can doubt the im- 
portance and the inevitableness of endeavoring to apply scientific 
method to these regions also. Science needs theories regarding 
the past history of the world. And no one who looks closer into 
the methods of these sciences of past time can doubt that verifi- 
able and unverifiable hypotheses are in all these regions inevitably 
interwoven; so that, while experience is always the guide, the 
attitude of the investigator towards experience is determined by 
interests which have to be partially due to what I should call 
that 'internal meaning,' that human interest in rational theoret- 
ical construction which inspires the scientific inquiry; and the 
theoretical constructions which prevail in such sciences are 
neither unbiased reports of the actual constitution of an external 
reality, nor yet arbitrary constructions of fancy. These con- 
structions in fact resemble in a measure those which M. Poincarfi 
in this book has analyzed in the case of geometry. They are 
constructions molded, but not predetermined in their details, by 
experience. We report facts ; we let the facts speak ; but we, as 


we inyestigate, in the popular phrase, Halk back' to the facts. 
We interpret as well as report Man is not merely made for 
science, but science is made for man. It expresses his deepest 
intellectual needs, as well as his careful observations. It is an 
effort to bring internal meanings into harmony with external 
verifications. It attempts therefore to control, as well as to 
submit, to conceive with rational unity, as well as to accept data. 
Its arts are those directed towards self-possession as well as 
towards an imitation of the outer reality which we find. It 
seeks therefore a disciplined freedom of thought. The discipline 
is as essential as the freedom; but the latter has also its place. 
The theories of science are human, as well as objective, inter- 
nally rational, as well as (when that is possible) subject to ex- 
ternal tests. 

In a field very different from that of the historical sciences, 
namely, in a science of observation and of experiment, which is 
at the same time an organic science, I have been led in the course 
of some study of the history of certain researches to notice the 
existence of a theoretical conception which has proved extremely 
fruitful in guiding research, but which apparently resembles in 
a measure the type of hypotheses of which M. Poincar4 speaks 
when he characterizes the principles of mechanics and of the 
theory of energy. I venture to call attention here to this con- 
ception, which seems to me to illustrate M. Poincare's view of the 
functions of hypothesis in scientific work. 

The modem science of pathology is usually regarded as dating 
from the earlier researches of Virchow, whose * Cellular Path- 
ology' was the outcome of a very careful and elaborate induc- 
tion. Virchow, himself, felt a strong aversion to mere specula- 
tion. He endeavored to keep close to observation, and to relieve 
medical science from the control of fantastic theories, such as 
those of the Naturphilosophen had been. Yet Virchow 's re- 
searches were, as early as 1847, or still earlier, already under the 
guidance of a theoretical presupposition which he himself states 
as follows: **We have learned to recognize," he says, **that dis- 
eases are not autonomous organisms, that they are no entities 
that have entered into the body, that they are no parasites which 
take root in the body, but that they merely show tis the course of 


the vital proeeM$es under mltered ccmdUions" Cdaas sie nnr 
AMauf der Lebensendieiiiiiiigeii anter Teiudcrten Bedingnn- 

gen dAntdkn')- 

The enoTiDoiis importmiiee of this theoredcal presupposition 
for all the earljr so ccc ss c s of modem pmtiiologieal inresligation 
k generalljr recognized by the experts. I do not doubt this 
opinion. It spi>ear8 to be a eommonplsee of tiie history of this 
aeienee. Bnt in Yirchow's later jrears this Tery presupposition 
seemed to some of his contemporaries to be ealled in qaestion by 
the soccesses of recent bacteriology. The qaestion arose whether 
the theoretical foundations of Virchow's pathology had not been 
set aside. And in fact the theoiy of the parasitical origin of 
a vast number of diseased conditions has indeed come upon an 
empirical basis to be generally recognized. Yet to the end of his 
own career Virchow stoutly maintained that in all its essential 
significance his own fundamental principle remained quite un- 
touched by the newer discoreries. And, as a fact, this view 
could indeed be maintained. For if diseases proved to be the 
consequences of the presence of parasites, the diseases them- 
selves, so far as they belonged to the diseased organism, were 
still not the parasites, but were, as before, the reaction of the 
organism to the verdnderie Bedingungen which the presence of 
the parasites entailed. So Virchow could well insist And if 
the famous principle in question is only stated with sufficient 
generality, it amounts simply to saying that if a disease in- 
volves a change in an organism, and if this change is subject to 
law at all, then the nature of the organism and the reaction of 
the organism to whatever it is which causes the disease must be 
underHtood in case the disease is to be understood. 

For this very reason, however, Virchow's theoretical principle 
in its most general form could be neither confirmed nor refuted 
by experience. It would remain empirically irrefutable, so far 
as I can see, even if we should learn that the devil was the 
true cause of all diseases. For the devil himself would then 
simply predetermine the verdnderte Bedingungen to which the 
diseased organism would be reacting. Let bullets or bacteria, 
poisons or compressed air, or the devil be the Bedingungen to 
which a diseased organism reacts, the postulate that Virchow 


states in the passage just quoted will remain irrefutable, if only 
this postulate be interpreted to meet the case. For the principle 
in question merely says that whatever entity it may be, bullet, or 
poison, or devil, that affects the organism, the disease is not that 
entity, but is the resulting alteration in the process of the 

I insist, then, that this principle of Virchow's is no trial sup- 
position, no scientific hypothesis in the narrower sense — capable 
of being submitted to precise empirical tests. It is, on the 
contrary, a very precious leading idea, a theoretical interpre- 
tation of phenomena, in the light of which observations are to be 
made — *a regulative principle' of research. It is equivalent to 
a resolution to search for those detailed connections which link 
the processes of disease to the normal process of the organism. 
Such a search undertakes to find the true unity, whatever that 
may prove to be, wherein the pathological and the normal proc- 
esses are linked. Now without some such leading idea, the cellu- 
lar pathology itself could never have been reached ; because the 
empirical facts in question would never have been observed. 
Hence this principle of Virchow's was indispensable to the 
growth of his science. Yet it was not a verifiable and not a re- 
futable hypothesis. One value of unverifiable and irrefutable 
hyx)otheses of this type lies, then, in the sort of empirical 
inquiries which they initiate, inspire, organize and guide. In 
these inquiries hypotheses in the narrower sense, that is, trial 
propositions which are to be submitted to definite empirical con- 
trol, are indeed everywhere present. And the use of the other 
sort of principles lies wholly in their application to experience. 
Yet without what I have just proposed to call the 'leading ideas' 
of a science, that is, its principles of an unverifiable and irre- 
futable character, suggested, but not to be finally tested, by 
experience, the hypotheses in the narrower sense would lack that 
guidance which, as M. Poincare has shown, the larger ideas of 
science give to empirical investigation. 


I have dwelt, no doubt, at too great length upon one aspect 
only of our author's varied and well-balanced discussion of the 


problems and concepts of scientific theory. Of the hypotheses 
in the narrower sense and of the value of direct empirical control, 
he has also spoken with the authority and the originality which 
belong to his position. And in dealing with the foundations of 
mathematics he has raised one or two questions of great philo- 
sophical import into which I have no time, even if I had the 
right, to enter here. In particular, in speaking of the essence 
of mathematical reasoning, and of the difficult problem of what 
makes possible novel results in the field of pure mathematics, M. 
Poincar6 defends a thesis regarding the office of 'demonstration 
by recurrence' — ^a thesis which is indeed disputable, which has 
been disputed and which I myself should be disposed, so far as 
I at present understand the matter, to modify in some respects, 
even in accepting the spirit of our author's assertion. Yet there 
can be no doubt of the importance of this thesis, and of the fact 
that it defines a characteristic that is indeed fundamental in a 
wide range of mathematical research. The philosophical prob- 
lems that lie at the basis of recurrent proofs and processes are, 
as I have elsewhere argued, of the most fundamental importance. 

These, then, are a few hints relating to the significance of 
our author's discussion, and a few reasons for hoping that our 
own students will profit by the reading of the book as those of 
other nations have already done. 

Of the person and of the life-work of our author a few words 
are here, in conclusion, still in place, addressed, not to the stu- 
dents of his own science, to whom his position is well known, but 
to the general reader who may seek guidance in these pages. 

Jules Henri Poincar6 was born at Nancy, in 1854, the son 
of a professor in the Faculty of Medicine at Nancy. He 
studied at the i^cole Polytechnique and at the i^cole des Mines, 
and later received his doctorate in mathematics in 1879. In 
1883 he began courses of instruction in mathematics at the 
£cole Polytechnique ; in 1886 received a professorship of mathe- 
matical physics in the Faculty of Sciences at Paris; then 
became member of the Academy of Sciences at Paris, in 1887, 
and devoted his life to instruction and investigation in the 
regions of pure mathematics, of mathematical physics and of 
celestial mechanics. His list of published treatises relating to 


yarious branches of his chosen sciences is long; and his ori- 
ginal memoirs have included several momentous investigations, 
which have gone far to transform more than one branch of 
research. His presence at the International Congress of Arts 
and Science in St. Louis was one of the most noticeable features 
of that remarkable gathering of distinguished foreign guests. 
In Poincar6 the reader meets, then, not one who is primarily a 
speculative student of general problems for their own sake, but 
an original investigator of the highest rank in several distinct, 
although interrelated, branches of modem research. The theory 
of functions — ^a highly recondite region of pure mathematics — 
owes to him advances of the first importance, for instance, the 
definition of a new type of functions. The 'problem of the three 
bodies, ' a famous and fundamental problem of celestial mechanics, 
has received from his studies a treatment whose significance has 
been recognized by the highest authorities. His international 
reputation has been confirmed by the conferring of more than one 
important prize for his researches. His membership in the most 
eminent learned societies of various nations is widely extended; 
his volumes bearing upon various branches of mathematics and 
of mathematical physics are used by special students in all parts 
of the learned world ; in brief, he is, as geometer, as analyst and 
as a theoretical physicist, a leader of his age. 

Meanwhile, as contributor to the philosophical discussion of 
the bases and methods of science, M. Poincar^ has long been 
active. When, in 1893, the admirable Revue de Meiaphysique et 
de Morale began to appear, M. Poincar^ was soon found amongst 
the most satisfactory of the contributors to the work of that 
journal, whose office it has especially been to bring philosophy 
and the various special sciences (both natural and moral) into 
a closer mutual understanding. The discussions brought to- 
gether in the present volume are in large part the outcome of 
M. Poincar^'s contributions to the Revue de Meiaphysique et de 
Morale. The reader of M. Poincar^'s book is in presence, then, 
of a great special investigator who is also a philosopher. 



Fob a superficial observer, scientific truth is beyond the possi- 
bility of doubt ; the logic of science is infallible, and if the scien- 
tists are sometimes mistaken, this is only from their mistaking 
its rules. 

''The mathematical verities flow from a small number of self- 
evident propositions by a chain of impeccable reasonings; they 
impose themselves not only on us, but on nature itself. They 
fetter, so to speak, the Creator and only permit him to choose 
between some relatively few solutions. A few experiments then 
will suffice to let us know what choice he has made. From each 
experiment a crowd of consequences will follow by a series of 
mathematical deductions, and thus each experiment will make 
known to us a comer of the universe." 

Behold what is for many people in the world, for scholars get- 
ting their first notions of physics, the origin of scientific certi- 
tude. This is what they suppose to be the role of experimenta- 
tion and mathematics. This same conception, a hundred years 
ago, was held by many savants who dreamed of constructing the 
world with as little as possible taken from experiment. 

On a little more reflection it was perceived how great a place 
hypothesis occupies; that the mathematician can not do without 
it, still less the experimenter. And then it was doubted if all 
these constructions were really solid, and believed that a breath 
would overthrow them. To be skeptical in this fashion is still to 
be superficial. To doubt everything and to believe everything 
are two equally convenient solutions; each saves us from 

Instead of pronouncing a summary condemnation, we ought 
therefore to examine with care the role of hypothesis; we shall 
then recognize, not only that it is necessary, but that usually it is 



le^timate. We shall also see that there are several sorts of hy- 
potheses ; that some are verifiabley and once confirmed by experi- 
ment become fruitful truths; that others, powerless to lead us 
astray, may be useful to us in fixing our ideas; that others, 
finally, are hypotheses only in appearance and are reducible to 
disguised definitions or conventions. 

These last are met with above all in mathematics and the 
related sciences. Thence precisely it is that these sciences get 
their rigor; these conventions are the work of the free activity 
of our mind, which, in this domain, recognizes no obstacle. Here 
our mind can affirm, since it decrees ; but let us understctnd that 
while these decrees are imposed upon our science, which, without 
them, would be impossible, they are not imposed upon nature. 
Are they then arbitrary! No, else were they sterile. Experi- 
ment leaves us our freedom of choice, but it guides us by aiding 
us to discern the easiest way. Our decrees are therefore like 
those of a prince, absolute but wise, who consults his council of 

Some people have been struck by this character of free conven- 
tion recognizable in certain fundamental principles of the 
sciences. They have wished to generalize beyond measure, and, 
at the same time, they have forgotten that liberty is not license. 
Thus they have reached what is called nominalism, and have 
asked themselves if the savant is not the dupe of his own defi- 
nitions and if the world he thinks he discovers is not simply 
created by his own caprice.^ Under these conditions science 
would be certain, but deprived of significance. 

If this were so, science would be powerless. Now every day 
we see it work under our very eyes. That could not be if it 
taught us nothing of reality. Still, the things themselves are 
not what it can reach, as the naive dogmatists think, but only 
the relations between things. Outside of these relations there 
is no knowable reality. 

Such is the conclusion to which we shall come, but for that we 
must review the series of sciences from arithmetic and geometry 
to mechanics and experimental physics. 

i-See Le B07, 'Science et Philosophie, ' Bevue de M^aphysique et de 
Morale, 1901. 


What is the nature of mathematical reasoning f Is is really 
deductivey as is commonly supposed? A deeper analysis shows 
us that it is not, that it partakes in a certain measure of the 
nature of inductive reasoning, and just because of this is it so 
fruitful. None the less does it retain its character of rigor 
absolute; this is the first thing that had to be shown. 

Knowing better now one of the instruments which mathemat- 
ics puts into the hands of the investigator, we had to analyze an- 
other fundamental notion, that of mathematical magnitude. Do 
we find it in nature, or do we ourselves introduce it there f And, 
in this latter case, do we not risk marring everything! Com- 
paring the rough data of our senses with that extremely complex 
and subtile concept which mathematicians call magnitude, we are 
forced to recognize a difference ; this frame into which we wish to 
force everything is of our own construction; but we have not 
made it at random. We have made it, so to speak, by measure 
and therefore we can make the facts fit into it without changing 
what is essential in them. 

Another frame which we impose on the world is space. 
Whence come the first principles of geometry! Are they im- 
posed on us by logic ! Lobachevski has proved not, by creating 
non-Euclidean geometry. Is space revealed to us by our senses ! 
Still no, for the space our senses could show us differs absolutely 
from that of the geometer. Is experience the source of geom- 
etry ? A deeper discussion will show us it is not. We therefore 
conclude that the first principles of geometry are only conven- 
tions ; but these conventions are not arbitrary and if transported 
into another world (that I call the non-Euclidean world and seek 
to imagine), then we should have been led to adopt others. 

In mechanics we should be led to analogous conclusions, and 
should see that the principles of this science, though more di- 
rectly based on experiment, still partake of the conventional 
character of the geometric postulates. Thus far nominalism 
triumphs ; but now we arrive at the physical sciences, properly so 
called. Here the scene changes; we meet another sort of hy- 
potheses and we see their fertility. Without doubt, at first blush, 
the theories seem to us fragile, and the history of science proves 
to us how ephemeral they are; yet they do not entirely perish, 


and of each of them something remains. It is this something 
we most seek to disentangle, since there and there alone is the 
veritable reality. 

The method of the physical sciences rests on the induction 
which makes ns expect the repetition of a phenomenon when the 
circumstances under which it first happened are reproduced* U 
all these circumstances could be reproduced at once, this prin- 
ciple could be applied without fear; but that will never happen; 
some of these circumstances will always be lacking. Are we 
absolutely sure they are unimportant! Evidently not. That 
may be probable, it can not be rigorously certain. Hence the 
important role the notion of probability plays in the physical 
sciences. The calculus of probabilities is therefore not merely 
a recreation or a guide to players of baccarat, and we must seek 
to go deeper with its foundations. Under this head I have been 
able to give only very incomplete results, so strongly does this 
vague instinct which lets us discern probability defy analysis. 

After a study of the conditions under which the physicist 
works, I have thought proper to show him at work. For that I 
have taken instances from the history of optics and of electricity. 
We shall see whence have sprung the ideas of Fresnel, of Max- 
well, and what unconscious hypotheses were made by Ampere 
and the other founders of electrodynamics. 



On the Nature of Mathematical BsASONiNa 

The very possibility of the science of mathematics seems 
an insoluble contradiction. If this science is deductive only in 
appearance, whence does it derive that perfect rigor no one 
dreams of doubting? If, on the contrary, all the propositions it 
enunciates can be deduced one from another by the rules of 
formal logic, why is not mathematics reduced to an immense 
tautology? The syllogism can teach us nothing essentially new, 
and, if everything is to spring from the principle of identity, 
everything should be capable of being reduced to it. Shall we 
then admit that the enunciations of all those theorems which fill 
80 many volumes are nothing but devious ways of saying A is A ? 

Without doubt, we can go back to the axioms, which are at 
the source of all these reasonings. If we decide that these can 
not be reduced to the principle of contradiction, if still less we 
see in them experimental facts which could not partake of mathe- 
matical necessity, we have yet the resource of classing them 
among synthetic a priori judgments. This is not to solve the diflS- 
culty, but only to baptize it ; and even if the nature of synthetic 
judgments were for us no mystery, the contradiction would not 
have disappeared, it would only have moved back ; syllogistic rea- 
soning remains incapable of adding anything to the data given 
it ; these data reduce themselves to a few axioms, and we should 
find nothing else in the conclusions. 

No theorem could be new if no new axiom intervened in its 
demonstration; reasoning could give us only the immediately 



evident verities borrowed from direct intuition ; it would be only 
an intermediary parasite, and therefore should we not have good 
reason to ask whether the whole syllogistic apparatus did not 
serve solely to disguise our borrowing? 

The contradiction will strike us the more if we open any book 
on mathematics ; on every page the author will announce his in- 
tention of generalizing some proposition already known. Does 
the mathematical method proceed from the particular to the gen- 
eral, and, if so, how then can it be called deductive f 

If finally the science of number were purely analytic, or 
could be analytically derived from a small number of Gfynthetic 
judgments, it seems that a mind sufficiently powerful could at 
a glance perceive all its truths; nay more, we might even hope 
that some day one would invent to express them a language suffi- 
ciently simple to have them appear self-evident to an ordinary 

If we refuse to admit these consequences, it must be conceded 
that mathematical reasoning has of itself a sort of creative virtue 
and consequently differs from the syllogism. 

The difference must even be profound. We shall not, for 
example, find the key to the mystery in the frequent use of that 
rule according to which one and the same uniform operation 
applied to two equal numbers will give identical results. 

All these modes of reasoning, whether or not they be reducible 
to the syllogism properly so called, retain the analytic character, 
and just because of that are powerless. 


The discussion is old; Leibnitz tried to prove 2 and 2 make 4; 
let us look a moment at his demonstration. 

I will suppose the number 1 defined and also the operation 
a? + 1 which consists in adding unity to a given number x. 

These definitions, whatever they be, do not enter into the 
course of the reasoning. 

I define then the numbers 2, 3 and 4 by the equalities 

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

In the same way, I define the operation x + 2 by the relation: 


(4) a? + 2= (« 4-1)4-1. 
That presupposed, we have 

2 4-1 4-1 = 3 4- 1 (Definition 2), 

3 4-1 = 4 (Definition 3), 
24-2= (2 4- 1)4-1 (Definition 4), 


24-2 = 4 Q.E.D. 

It can not be denied that this reasoning is purely analytic. 
But ask any mathematician: 'That is not a demonstration prop- 
erly so called,' he will say to you: 'that is a verification.' We 
have confined ourselves to comparing two purely conventional 
definitions and have ascertained their identity ; we have learned 
nothing new. Verification differs from true demonstration pre- 
cisely because it is purely analytic and because it is sterile. It is 
sterile because the conclusion is nothing but the premises trans- 
lated into another language. On the contrary, true demonstration 
is fruitful because the conclusion here is in a sense more general 
than the premises. 

The equality 2 + 2 = 4 is thus susceptible of a verification 
only because it is particular. Every particular enunciation in 
mathematics can always be verified in this same way. But if 
mathematics could be reduced to a series of such verifications, it 
would not be a science. So a chess-player, for example, does not 
create a science in winning a game. There is no science apart 
from the general. 

It may even be said the very object of the exact sciences is to 
spare us these direct verifications. 


Let us, therefore, see the geometer at work and seek to catch 
his 'process. 

The task is not without diflSculty; it does not suflSce to open 
a work at random and analyze any demonstration in it. 

We must first exclude geometry, where the question is com- 
plicated by arduous problems relative to the role of the postu- 
lates, to the nature and the origin of the notion of space. For 
analogous reasons we can not turn to the infinitesimal analysis. 


We must seek mathematical thought where it has remained pure^ 
that is, in arithmetic. 

A choice still is necessary; in the higher parts of the theory 
of numbers, the primitive mathematical notions have already un- 
dergone an elaboration so profound that it becomes difficult to 
analyze them. 

It is, therefore, at the beginning of arithmetic that we must 
expect to find the explanation we seek, but it happens that pre- 
cisely in the demonstration of the most elementary theorems the 
authors of the classic treatises have shown the least precision and 
rigor. We must not impute this to them as a crime; they have 
yielded to a necessity ; beginners are not prepared for real mathe- 
matical rigor ; they would see in it only useless and irksome sub- 
tleties; it would be a waste of time to try prematurely to make 
them more exacting; they must pass over rapidly, but without 
skipping stations, the road traversed slowly by the founders of 
the science. 

Why is so long a preparation necessary to become habituated 
to this perfect rigor, which, it would seem, should naturally im- 
press itself upon all good minds? This is a logical and psy- 
chological problem well worthy of study. 

But we shall not take it up; it is foreign to our purpose; all 
I wish to insist on is that, not to fail of our purpose, we must 
recast the demonstrations of the most elementary theorems and 
give them, not the crude form in which they are left, so as not to 
harass beginners, but the form that will satisfy a skilled 

Definition op Addition. — I suppose already defined the 
operation a; + 1, which consists in adding the number 1 to a 
given number x. 

This definition, whatever it be, does not enter into our sub- 
sequent reasoning. 

We now have to define the operation « -f a, which consists in 
adding the number a to a given number x. 

Supposing we have defined the operation 

a?+(a — 1), 

the operation a; + a will be defined by the equality 
(1) x + a=lx-\' (a — 1)]+1. 


We shall know then what x-\-a \a when we know what 
«-|- (<* — 1) is, and as I have supposed that to start with we 
knew what a?-|-l ^ w® <^*^ define successively and *by recur- 
rence ' the operations a? + 8, a; + 3, etc. 

This definition deserves a moment's attention; it is of a par- 
ticular nature which already distinguishes it from the purely 
logical definition; the equality (1) contains an infinity of dis- 
tinct definitions, each having a meaning only when one knows the 

Pbopebtibs op ADDrnoN. — Assodaiivity. — I say that 

a+(& + c) = (a + &)+c. 
In fact the theorem is true for c = l; it is then written 

o+(& + l) = (o+b)+l, 

which, apart from the difference of notation, is nothing but the 
equality (1), by which I have just defined addition. 
Supposing the theorem true for c=y, I say it will be true for 


In fact, supposing 

(a + &)+7 = a+(& + 7), 

it follows that 

[(a + b)4-7]+l = [a+(& + 7)]+l 

or by definition (1) 

(a+ &) + (7 + 1) =a + (& + 7 4- 1) =a + [6 4- (7 + 1)], 

which shows, by a series of purely analytic deductions, that the 
theorem is true for y + 1. 

Being true for c = 1, we thus see successively that so it is for 
c=2, for c = 3, etc. 

Commutaiivity. — 1° I say that 

a + 1 = 1 + a. 

The theorem is evidently true for a=il; we can verify by 
purely analytic reasoning that if it is true for a=y it will be 
true for a =y + 1 ; for then 

(7 + 1)4-1= (1 + 7) +1 = 1 + (7 + 1); 

now it is true for a = l, therefore it will be true for a = 2, for 
a =3, etc., which is expressed by saying that the enunciated 
proposition is demonstrated by recurrence. 


2** I say that 

The theorem has just been demonstrated for & =: 1 ; it can be 
verified analytically that if it is true for b=fi,it will be true for 

The proposition is therefore established by recurrence. 
Definition op Multiplication. — ^We shall define multiplica- 
tion by the equalities. 

(1) axi = a. 

(2) aXh = [aX (6 — l)] + o. 

Like equality (1), equality (2) contains an infinity of defini- 
tions ; having defined a X !> it enables us to define successively : 
a X 2, a X 3, etc. 

Properties op Multiplication. — Distributivity. — ^I say that 

(a + 6) Xc=(oXc) + (bxc). 

We verify analytically that the equality is true for c = l ; then 
that if the theorem is true for c = y, it will be true for c =y + 1. 
The proposition is, therefore, demonstrated by recurrence. 
Commutativity, — 1° I say that 

a X 1 = 1 X a. 

The theorem is evident for a=l. 

We verify analytically that if it is true for o = o, it will be 

true for o =s o + 1. 

2M say that 

a X ft = & X o. 

The theorem has just been proven for 6 = 1. We could verify 
analytically that if it is true for b=py it will be true for 
b = p + l. 


Here I stop this monotonous series of reasonings. But this 
very monotony has the better brought out the procedure which is 
uniform and is met again at each step. 

This procedure is the demonstration by recurrence. We first 
establish a theorem for n = 1 ; then we show that if it is true of 
w — 1, it is true of n, and thence conclude that it is true for all 
the whole numbers. 


We have just seen how it may be used to demonstrate the rules 
of addition and multiplication, that is to say, the rules of the 
algebraic calculus ; this calculus is an instrument of transforma- 
tion, which lends itself to many more differing combinations than 
joes the simple syllogism; but it is still an instrument purely 
analytic, and incapable of teaching us anything new. If mathe- 
matics had no other instrument, it would therefore be forth- 
with arrested in its development; but it has recourse anew to 
the same procedure, that is, to reasoning by recurrence, and it is 
able to continue its forward march. 

If we look closely, at every step we meet again this mode of 
reasoning, either in the simple form we have just given it, or 
under a form more or less modified. 

Here then we have the mathematical reasoning par excellence, 
and we must examine it more closely. 

The essential characteristic of reasoning by recurrence is that 
it contains, condensed, so to speak, in a single formula, an 
infinity of syllogisms. 

That this may the better be seen, I will state one after another 
these syllogisms which are, if you will allow me the expression, 
arranged in 'cascade.' 

These are of course hypothetical syllogisms. 
The theorem is true of the number 1. 

Now, if it is true of 1, it is true of 2. 

Therefore it is true of 2. 

Now, if it is true of 2, it is true of 3. 

Therefore it is true of 3, and so on. 

We see that the conclusion of each syllogism serves as minor to 
the following. 

Furthermore the majors of all our syllogisms can be reduced 
to a single formula. 

If the theorem is true of n — 1, so it is of n. 

We see, then, that in reasoning by recurrence we confine our- 
selves to stating the minor of the first syllogism, and the general 
formula which contains as particular cases all the majors. 

This never-ending series of syllogisms is thus reduced to a 
phrase of a few lines. 


It is now easy to comprehend why every particular conse- 
quence of a theorem can, as I have explained above, be verified 
by purely analytic procedures. 

If instead of showing that our theorem is true of all num- 
bers, we only wish to show it true of the number 6, for example, 
it will sufSce for us to establish the first 5 syllogisms of our cas- 
cade ; 9 would be necessary if we wished to prove the theorem for 
the number 10; more would be needed for a larger number; but, 
however great this number might be, we should always end 
by reaching it, and the analytic verification would be possible. 

And yet, however far we thus might go, we could never rise 
to the general theorem, applicable to all numbers, which alone 
can be the object of science. To reach this, an infinity of syl- 
logisms would be necessary ; it would be necessary to overleap an 
abyss that the patience of the analyst, restricted to the resources 
of formal logic alone, never could fill up. 

I asked at the outset why one could not conceive of a mind 
sufSciently powerful to perceive at a glance the whole body of 
mathematical truths. 

The answer is now easy; a chess-player is able to combine 
four moves, five moves, in advance, but, however extraordinary 
he may be, he will never prepare more than a finite number of 
them; if he applies his faculties to arithmetic, he will not be 
able to perceive its general truths by a single direct intuition ; to 
arrive at the smallest theorem he can not dispense with the aid 
of reasoning by recurrence, for this is an instrument which 
enables us to pass from the finite to the infinite. 

This instrument is always useful, for, allowing us to overleap 
at a bound as many stages as we wish, it spares us verifications, 
long, irksome and monotonous, which would quickly become im- 
practicable. But it becomes indispensable as soon as we aim at 
the general theorem, to which analytic verification would bring 
us continually nearer without ever enabling us to reach it. 

In this domain of arithmetic, we may think ourselves very far 
from the infinitesimal analysis, and yet, as we have just seen, 
the idea of the mathematical infinite already plays a preponder- 
ant role, and without it there would be no science, because there 
would be nothing general. 



The judgment on which reasoning by recurrence rests can be 
put under other forms; we may say, for example, that in an 
infinite collection of different whole numbers there is always one 
which is less than all the others. 

We can easily pass from one enunciation to the other and thus 
get the illusion of having demonstrated the legitimacy of reason- 
ing by recurrence. But we shall always be arrested, we shall 
always arrive at an undemonstrable axiom which will be in 
reality only the proposition to be proved translated into another 

We can not therefore escape the conclusion that the rule of 
reasoning by recurrence is irreducible to the principle of con- 

Neither can this rule come to us from experience; experience 
could teach us that the rule is true for the first ten or hundred 
numbers; for example, it can not attain to the indefinite series 
of numbers, but only to a portion of this series, more or less long 
but always limited. 

Now if it were only a question of that, the principle of con- 
tradiction would sufiSce ; it would always allow of our developing 
as many i^llogisms as we wished ; it is only when it is a question 
of including an infinity of them in a single formula, it is only 
before the infinite that this principle fails, and there too, experi- 
ence becomes powerless. This rule, inaccessible to analytic 
demonstration and to experience, is the veritable type of the 
S3mthetic a priori judgment. On the other hand, we can not 
think of seeing in it a convention, as in some of the postulates of 

Why then does this judgment force itself upon us with an 
irresistible evidence? It is because it is only the affirmation of 
the power of the mind which knows itself capable of conceiving 
the indefinite repetition of the same act when once this act is 
possible. The mind has a direct intuition of this power, and 
experience can only give occasion for using it and thereby 
becoming conscious of it. 

But, one will say, if raw experience can not legitimatize 
reasoning by recurrence, is it so of experiment aided by indue- 


tion f We see successively that a theorem is true of the number 
1, of the number 2, of the number 3 and so on ; the law is evident, 
we say, and it has the same warranty as every physical law based 
on observations, whose number is very great but limited. 

Here is, it must be admitted, a striking analogy with the usual 
procedures of induction. But there is an essential difference. 
Induction applied to the physical sciences is always uncertain, 
because it rests on the belief in a general order of the universe, 
an order outside of us. Mathematical induction, that is, demon- 
stration by recurrence, on the contrary, imposes itself necessarily 
because it is only the affirmation of a property of the mind itself. 


Mathematicians, as I have said before, always endeavor to 
generalize the propositions they have obtained, and, to seek no 
other example, we have just proved the equality : 

a + l = l + a 
and afterwards used it to establish the equality 

which is manifestly more general. 

Mathematics can, therefore, like the other sciences, proceed 
from the particular to the general. 

This is a fact which would have appeared incomprehensible 
to us at the outset of this study, but which is no longer mys^ 
terious to us, since we have ascertained the analogies between 
demonstration by recurrence and ordinary induction. 

Without doubt recurrent reasoning in mathematics and in- 
ductive reasoning in physics rest on different foundations, but 
their march is parallel, they advance in the same sense, that is 
to say, from the particular to the general. 

Let us examine the case a little more closely. 

To demonstrate the equality 

it suffices to twice apply the rule 

(1) a+l = l + a 
and write 

(2) a-h 2 = a + 1 -h 1 = 1 -ha + 1 = 1 + 1+0 = 2 + a. 


The equality (2) thus deduced in purely analytic way from 
the equality (1) is, however, not simply a particular case of it; 
it is something quite different. 

We can not therefore even say that in the really analytic 
and deductive part of mathematical reasoning we proceed from 
the general to the particular in the ordinary sense of the word. 

The two members of the equality (2) are simply combinations 
more complicated than the two members of the equality (1), and 
analysis only serves to separate the elements which enter into 
these combinations and to study their relations. 

Mathematicians proceed therefore *by construction,' they 'con- 
struct' combinations more and more complicated. Coming back 
then by the analysis of these combinations, of these aggregates, 
80 to speak, to their primitive elements, they perceive the rela- 
tions of these elements and from them deduce the relations of 
the aggregates themselves. 

This is a purely analytical proceeding, but it is not, however, 
a proceeding from the general to the particular, because evi- 
dently the aggregates can not be regarded as more particular 
than their elements. 

Oreat importance, and justly, has been attached to this pro- 
cedure of 'construction,' and some have tried to see in it the 
necessary and sufficient condition for the progress of the exact 

Necessary, without doubt ; but sufficient, no. 

For a construction to be useful and not a vain toil for the 
mind, that it may serve as stepping-stone to one wishing to 
mount, it must first of all possess a sort of unity enabling us to 
see in it something besides the juxtaposition of its elements. 

Or, more exactly, there must be some advantage in considering 
the construction rather than its elements themselves. 

What can this advantage be t 

Why reason on a polygon, for instance, which is always de- 
composable into triangles, and not on the elementary triangles? 

It is because there are properties appertaining to polygons 
of any number of sides and that may be immediately applied to 
any particular polygon. 

Usually, on the contrary, it is only at the cost of the most 



prolonged exertions that they could be fonnd by studying 
directly the relations of the elementary triangles. The knowl- 
edge of the general theorem spares us these efforts. 

A construction, therefore, becomes interesting only when it 
can be ranged beside other analogous constructions, forming spe- 
cies of the same genus. 

If the quadrilateral is something besides the juxtaposition of 
two triangles, this is because it belongs to the genus polygon. 

Moreover, one must be able to demonstrate the properties of 
the genus without being forced to establish them successively for 
each of the species. 

To attain that, we must necessarily mount from the particular 
to the general, ascending one or more steps. 

The analytic procedure 'by construction' does not oblige us 
to descend, but it leaves us at the same level. 

We can ascend only by mathematical induction, which alone 
can teach us something new. Without the aid of this induction, 
different in certain respects from physical induction, but quite 
as fertile, construction would be powerless to create science. 

Observe finally that this induction is possible only if the same 
operation can be repeated indefinitely. That is why the theory 
of chess can never become a science, for the different moves of 
the same game do not resemble one another. 

Mathematical Maqnttude and Expebienge 

To learn what mathematicians understand by a continunmy 
one should not inquire of geometry. The geometer always seeks 
to represent to himself more or less the figures he studies, but 
his representations are for him only instruments; in making 
geometry he uses space just as he does chalk; so too much weight 
should not be attached to non-essentials, often of no more im- 
portance than the whiteness of the chalk. 

The pure analyst has not this rock to fear. He has disen- 
gaged the science of mathematics from all foreign elements, and 
can answer our question: 'What exactly is this continuum about 
which mathematicians reason T Many analysts who reflect on 
their art have answered already; Monsieur Tannery, for example, 
in his Introduction d la thSorie des fonctions d^une variable. 

Let us start from the scale of whole numbers; between two 
consecutive steps, intercalate one or more intermediary steps, 
then between these new steps still others, and so on indefinitely. 
Thus we shall have an unlimited number of terms; these will 
be the numbers called fractional, rational or commensurable. 
But this is not yet enough ; between these terms, which, however, 
are already infinite in number, it is still necessary to intercalate 
others called irrational or incommensurable. A remark before 
going further. The continuum so conceived is only a collection 
of individuals ranged in a certain order, infinite in number, it is 
true, but exterior to one another. This is not the ordinary con- 
ception, wherein is supposed between the elements of the con- 
tinuum a sort of intimate bond which makes of them a whole, 
where the point does not exist before the line, but the line before 
the point Of the celebrated formula, *the continuum is unity 
in multiplicity,' only the multiplicity remains, the unity has 
disappeared. The analysts are none the less right in defining 
their continuum as they do, for they always reason on just this 
as soon as they pique themselves on their rigor. But this is 



enough to apprise us that the veritable mathematical continuum 
is a very different thing from that of the physicists and that of 
the metaphysicians. 

It may also be said perhaps that the mathematicians who are 
content with this definition are dupes of words, that it is neces- 
sary to say precisely what each of these intermediary steps is, to 
explain how they are to be intercalated and to demonstrate that 
it is possible to do it. But that would be wrong ; the only prop- 
erty of these steps which is used in their reasonings^ is that of 
being before or after such and such steps; therefore also this 
alone should occur in the definition. 

So how the intermediary terms should be intercalated need 
not concern us ; on the other hand, no one will doubt the possi- 
bility of this operation, unless from forgetting that possible, in 
the language of geometers, simply means free from contradiction. 

Our definition, however, is not yet complete, and I return to 
it after this over-long digression. 

Definition of Incommensurables. — The mathematicians of 
the Berlin school, Kronecker in particular, have devoted them- 
selves to constructing this continuous scale of fractional and irra- 
tional numbers without using any material other than the whole 
number. The mathematical continuum would be, in this view, 
a pure creation of the mind, where experience would have no 

The notion of the rational number seeming to them to present 
no diflSculty, they have chiefly striven to define the incommen- 
surable number. But before producing here their definition, I 
must make a remark to forestall the astonishment it is sure to 
arouse in readers unfamiliar with the customs of geometers. 

Mathematicians study not objects, but relations between ob- 
jects; the replacement of these objects by others is therefore 
indifferent to them, provided the relations do not change. The 
matter is for them unimportant, the form alone interests them. 

Without recalling this, it would scarcely be comprehensible 
that Dedekind should designate by the name incommensurable 
number a mere symbol, that is to say, something very different 

iWith those contained in the special conventions which serve to define 
addition and of which we shaU speak later. 


from the ordinary idea of a quantity, which should be measurable 
and almost tangible. 

Let us see now what Dedekind's definition is: 

The commensurable numbers can in an infinity of ways be 
partitioned into two classes, such that any number of the first 
dass is greater than any number of the second class. 

It may happen that among the numbers of the first class 
there is one smaller than all the others ; if, for example, we range 
in the first class all numbers greater than 2, and 2 itself, and in 
the second class all numbers less than 2, it is clear that 2 will be 
the least of all numbers of the first class. The number 2 may be 
chosen as symbol of this partition. 

It may happen, on the contrary, that among the numbers of 
the second class is one greater than all the others; this is the 
ease, for example, if the first class comprehends all numbers 
greater than 2, and the second all numbers less than 2, and 2 
itself. Here again the number 2 may be chosen as symbol of this 

But it may equally well happen that neither is there in the 
first class a number less than aU the others, nor in the second 
class a number greater than all the others. Suppose, for ex- 
ample, we put in the first class all commensurable numbers whose 
squares are greater than 2 and in the second all whose squares 
are less than 2. There is none whose square is precisely 2. Evi- 
dently there is not in the first class a number less than all the 
others, for, however near the square of a number may be to 2, 
we can always find a commensurable number whose square is 
still closer to 2. 

In Dedekind's view, the incommensurable number 

V2 or (2)* 

is nothing but the symbol of this particular mode of partition 
of commensurable numbers; and to each mode of partition cor- 
responds thus a number, commensurable or not, which serves as 
its symbol. 

But to be content with this would be to forget too far the 
origin of these symbols ; it remains to explain how we have been 
led to attribute to them a sort of concrete existence, and, besides, 


does not the difiScolty begin even for the fractional numbers 
themselves f Should we have the notion of these numbers if we 
had not previously known a matter that we conceive as infinitely 
divisible, that is to say, a continuum? 

The Physical Continuum. — ^We ask ourselves then if the 
notion of the mathematical continuum is not simply drawn from 
experience. If it were, the raw data of experience, which are 
our sensations, would be susceptible of measurement. We might 
be tempted to believe they really are so, since in these latter days 
the attempt has been made to measure them and a law has even 
been formulated, known as Fechner's law, according to which 
sensation is proportional to the logarithm of the stimulus. 

But if we examine more closely the experiments by which 
it has been sought to establish this law, we shall be led to a 
diametrically opposite conclusion. It has been observed, for ex- 
ample, that a weight A of 10 grams and a weight B of 11 grams 
produce identical sensations, that the weight B is just as indis- 
tinguishable from a weight C of 12 grams, but that the weight A 
is easily distinguished from the weight C. Thus the raw results 
of experience may be expressed by the following relations : 

A = B, B=zC, A<C, 

which may be regarded as the formula of the physical continuum. 

But here is an intolerable discord with the principle of con- 
tradiction, and the need of stopping this has compelled us to 
invent the mathematical continuum. 

We are, therefore, forced to conclude that this notion has 
been created entirely by the mind, but that experience has given 
the occasion. 

We can not believe that two quantities equal to a third are 
not equal to one another, and so we are led to suppose that A is 
different from B and B from C, but that the imperfection of our 
senses has not permitted of our distinguishing them. 

Creation op the Mathematical Continuum. — First Stage. 
So far it would sufSce, in accounting for the facts, to intercalate 
between A and B a few terms, which would remain discrete. 
What happens now if we have recourse to some instrument to 


sapplement the feebleness of our senses, if, for example, we 
make use of a microscope f Terms such as A and B, before indis- 
tingnishable, appear now distinct ; but between A and B, now be- 
come distinct, will be intercalated a new term, D, that we can 
distingmsh neither from A nor from B. Despite the employ-^ 
ment of the most highly perfected methods, the raw results of our 
experience will always present the characteristics of the physical 
continuum with the contradiction which is inherent in it. 

We shall escape it only by incessantly intercalating new terms 
between the terms already distinguished, and this operation must 
be continued indefinitely. We might conceive the stopping of 
this operation if we could imagine some instrument sufSciently 
powerful to decompose the physical continuum into discrete ele- 
ments, as the telescope resolves the milky way into stars. But 
this we can not imagine ; in fact, it is with the eye we observe the 
image magnified by the microscope, and consequently this image 
must always retain the characteristics of visual sensation and 
consequently those of the physical continuum. 

Nothing distinguishes a length observed directly from the 
half of this length doubled by the microscope. The whole is 
homogeneous with the part; this is a new contradiction, or 
rather it would be if the number of terms were supposed finite ; 
in fact, it is clear that the part containing fewer terms than the 
whole could not be similar to the whole. 

The contradiction ceases when the number of terms is regarded 
as infinite ; nothing hinders, for example, considering the aggre- 
gate of whole numbers as similar to the aggregate of even num- 
bers, which, however, is only a part of it ; and, in fact, to each 
whole number corresponds an even number, its double. 

But it is not only to escape this contradiction contained in the 
empirical data that the mind is led to create the concept of a 
continuum, formed of an indefinite number of terms. 

All happens as in the sequence of whole numbers. We have 
the faculty of conceiving that a unit can be added to a collection 
of units ; thanks to experience, we have occasion to exercise this 
faculty and we become conscious of it; but from this moment 
we feel that our power has no limit and that we can count in- 
definitely, though we have never had to count more than a finite 
number of objects. 


Just so, as soon as we have been led to intercalate means 
between two consecutive terms of a series, we feel that this opera- 
tion can be continued beyond all limit, and that there is, so to 
speak, no intrinsic reason for stopping. 

As an abbreviation, let me call a mathematical continuum 
of the first order every aggregate of terms formed according to 
the same law as the scale of commensurable numbers. If we 
afterwards intercalate new steps according to the law of for- 
mation of incommensurable numbers, we shall obtain what we 
will call a continuum of the second order. 

Second Stage. — ^We have made hitherto only the first stride; 
we have explained the origin of continua of the first order ; but it 
is necessary to see why even they are not sufficient and why the 
incommensurable numbers had to be invented. 

If we try to imagine a line, it must have the characteristics 
of the physical continuum, that is to say, we shall not be able 
to represent it except with a certain breadth. Two lines then 
will appear to us under the form of two narrow bands, and, if 
we are content with this rough image, it is evident that if the 
two lines cross they will have a common part. 

But the pure geometer makes a further effort ; without entirely 
renouncing the aid of the senses, he tries to reach the concept of 
the line without breadth, of the point without extension. This 
he can only attain to by regarding the line as the limit toward 
which tends an ever narrowing band, and the point as the limit 
toward which tends an ever lessening area. And then, our two 
bands, however narrow they may be, will always have a common 
area, the smaller as they are the narrower, and whose limit will 
be what the pure geometer calls a point. 

This is why it is said two lines which cross have a point in 
common, and this truth seems intuitive. 

But it would imply contradiction if lines were conceived as 
continua of the first order, that is to say, if on the lines traced 
by the geometer should be found only points having for coordi- 
nates rational numbers. The contradiction would be manifest 
as soon as one affirmed, for example, the existence of straights 
and circles. 

It is clear, in fact, that if the points whose coordinates are 


commensurable were alone regarded as real, the circle inscribed 
in a square and the diagonal of this square would not intersect, 
since the coordinates of the point of intersection are incom- 

That would not yet be sufiScient, because we should get in this 
way only certain incommensurable numbers and not all those 

But conceive of a straight line divided into two rays. Each 
of these rays will appear to our imagination as a band of a cer- 
tain breadth; these bands moreover will encroach one on the 
other, since there must be no interval between them. The com- 
mon part will appear to us as a point which will always remain 
when we try to imagine our bands narrower and narrower, so 
that we admit as an intuitive truth that if a straight is cut into 
two raya their common frontier is a point ; we recognize here the 
conception of Dedekind, in which an incommensurable number 
was regarded as the common frontier of two classes of rational 

Such is the origin of the continuum of the second order, which 
is the mathematical continuum properly so called. 

Resume. — ^In recapitulation, the mind has the faculty of cre- 
ating symbols, and it is thus that it has constructed the mathe- 
matical continuum, which is only a particular system of symbols. 
Its power is limited only by the necessity of avoiding all contra- 
diction ; but the mind only makes use of this faculty if experience 
furnishes it a stimulus thereto. 

In the case considered, this stimulus was the notion of the 
physical continuum, drawn from the rough data of the senses. 
But this notion leads to a series of contradictions from which it 
is necessary successively to free ourselves. So we are forced to 
imagine a more and more complicated system of symbols. That 
at which we stop is not only exempt from internal contradiction 
(it was so already at all the stages we have traversed), but 
neither is it in contradiction with various propositions called in- 
tuitive, which are derived from empirical notions more or less 

Measubable Maonttude. — The magnitudes we have studied 
hitherto are not measurable; we can indeed say whether a given 


one of these magnitudes is greater than another, but not whether 
it is twice or thrice as great. 

So far, I have only considered the order in which our temu 
are ranged. But for most applications that does not suffice. We 
must learn to compare the interval which separates any two 
terms. Only on this condition does the continuum become a 
measurable magnitude and the operations of arithmetic ap- 

This can only be done by the aid of a new and special con- 
vention. We will agree that in such and such a case the interval 
comprised between the terms A and B is equal to the interval 
which separates C and D. For example, at the beginning of oiu 
work we have set out from the scale of the whole numbers and we 
have supposed intercalated between two consecutive steps n 
intermediary steps ; well, these new steps will be by conventios 
regarded as equidistant. 

This is a way of defining the addition of two magnitudes, be- 
cause if the interval AB is by definition equal to the interval CD^ 
the interval AD will be by definition the sum of the intervals 
AB and AC. 

This definition is arbitrary in a very large measure. It is not 
completely so, however. It is subjected to certain conditions 
and, for example, to the rules of commutativity and associativity 
of addition. But provided the definition chosen satisfies these 
rules, the choice is indifferent, and it is useless to particularize it. 

Various Remarks. — ^We can now discuss several important 
questions : 

1** Is the creative power of the mind exhausted by the creation 
of the mathematical continuiunf 

No : the works of Du Bois-Beymond demonstrate it in a striking 

We know that mathematicians distinguish between infinitesi- 
mals of different orders and that those of the second order are 
infinitesimal, not only in an absolute way, but also in relatioi 
to those of the first order. It is not difficult to imagine infinites- 
imals of fractional or even of irrational order, and thus we find 
again that scale of the mathematical continuum which has beei 
dealt with in the preceding pages. 


Farther, there are infinitesimals which are infinitely small in 
relation to those of the first order, and, on the contrary, infinitely 
great in relation to those of order 1 + c, and that however small 
c may be. Here, then, are new terms intercalated in our series, 
and if I may be permitted to revert to the phraseology lately em- 
ployed which is very convenient though not consecrated by usage, 
I shall say that thus has been created a sort of continuum of the 
third order. 

It would be easy to go further, but that would be idle; one 
would only be imagining symbols without possible application, 
and no one will think of doing that. The continuum of the third 
order, to which the consideration of the different orders of infini- 
tesimals leads, is itself not useful enough to have won citizenship, 
and geometers regard it only as a mere curiosity. The mind uses 
its creative faculty only when experience requires it. 

2^ Once in possession of the concept of the mathematical con- 
tinuum, is one safe from contradictions analogous to those which 
gave birth to it? 

No, and I will give an example. 

One must be very wise not to regard it as evident that every 
curve has a tangent ; and in fact if we picture this curve and a 
straight as two narrow bands we can always so dispose them that 
they have a part in common without crossing. If we imagine 
then the breadth of these two bands to diminish indefinitely, this 
common part will always subsist and, at the limit, so to speak, the 
two lines will have a point in common without crossing, that is to 
say, they will be tangent. 

The geometer who reasons in this way, consciously or not, is 
only doing what we have done above to prove two lines which 
cut have a point in common, and his intuition might seem just as 

It would deceive him however. We can demonstrate that 
there are curves which have no tangent, if such a curve is de- 
fined as an analytic continuum of the second order. 

Without doubt some artifice analogous to those we have dis- 
enssed above would have suflBced to remove the contradiction; 
but, as this is met with only in very exceptional cases, it has 
received no further attention. 


Instead of seeking to reconcile intuition with analysis, we have 
been content to sacrifice one of the two, and as analysis must 
remain impeccable, we have decided against intuition. 

The Physical Continuum op Several Dimensions. — ^We 
have discussed above the physical continuum as derived from the 
immediate data of our senses, or, if you wish, from the rough re- 
sults of Fechner's experiments; I have shown that these results 
are summed up in the contradictory formulas 

A=zB, B = C, A<C. 

Let us now see how this notion has been generalized and how 
from it has come the concept of many-dimensional continua. 

Consider any two aggregates of sensations. Either we can 
discriminate them one from another, or we can not, just as in 
Fechner's experiments a weight of 10 grams can be distinguished 
from a weight of 12 grams, but not from a weight of 11 grams. 
This is all that is required to construct the continuum of several 

Let us call one of these aggregates of sensations an element. 
That will be something analogous to the point of the mathe- 
maticians; it will not be altogether the same thing however. 
We can not say our element is without extension, since we can 
not distinguish it from neighboring elements and it is thus 
surrounded by a sort of haze. If the astronomical comparison 
may be allowed, our * elements' would be like nebulae, whereas 
the mathematical points would be like stars. 

^p That being granted, a system of elements will form a con- 
Hinuum if we can pass from any one of them to any other, by a 
V,*- if series of consecutive elements such that each is indistinguish- 
j able from the preceding. This linear series is to the line of the 
^i" ■ mathematician what an isolated element was to the point. 

/ Before going farther, I must explain what is meant by a 
I cut. Consider a continuum C and remove from it certain of its 
1 elements which for an instant we shall regard as no longer be- 
\ longing to this continuum. The aggregate of the elements so 
removed will be called a cut. It may happen that, thanks to this 
cut, C may be subdivided into several distinct continua, the ag- 
t gregate of the remaining elements ceasing to form a unique con- 
/ tinuum. 




There will then be on C two elements, A and B, that must be \ 
regarded as belonging to two distinct continua, and this will be J v ^ 
recognized because it will be impossible to find a linear series j ^ 
of consecutive elements of C, each of these elements indistin- \^ i^ '^ 
guishable from the preceding, the first being A and the last B, I 
without one of the elements of this series being indistinguishable \ . /^ 
from one of the elements of the cut. j ^ a-' 

On the contrary, it may happen that the cut made is insuffi-j 
cient to subdivide the continuum C. To classify the physical! 
continua, we will examine precisely what are the cuts which must \ 
be made to subdivide them. 

If a physical continuum C can be subdivided by a cut reduc- 
ing to a finite number of elements all distinguishable from one 
another (and consequently forming neither a continuum, nor 
several continua), we shall say C is a one-dimensional continuum. 

If, on the contrary, C can be subdivided only by cuts which 
are themselves continua, we shall say C has several dimen- 
sions. If cuts which are continua of one dimension sufiSce, we 
shall say C has two dimensions ; if cuts of two dimensions sufSce, 
we shall say C has three dimensions, and so on. 

Thus is defined the notion of the physical continuum of several 
dimensions, thanks to this very simple fact that two aggregates 
of sensations are distinguishable or indistinguishable. 

The Mathematical Continuum op Several Dimensions. — 
Thence the notion of the mathematical continuum of n dimen- 
sions has sprung quite naturally by a process very like that we 
discussed at the beginning of this chapter. A point of such a 
continuum, you know, appears to us as defined by a system of 
n distinct magnitudes called its coordinates. 

These magnitudes need not always be measurable; there is, 
for instance, a branch of geometry independent of the measure- 
ment of these magnitudes, in which it is only a question of know- 
ing, for example, whether on a curve ABC, the point B is be- 
tween the points A and C, and not of knowing whether the arc 
AB is equal to the arc BC or twice as great. This is what is 
called Analysis Situs. 

This is a whole body of doctrine which has attracted the 


attention of the greatest geometers and where we see flow one 
fram another a series of remarkable theorems. What distin- 
guishes these theorems from those of ordinary geometry is that 
they are purely qualitative and that they would remain true if 
the figures were copied by a draughtsman so awkward as to 
grossly distort the proportions and replace straights by strokes 
more or less curved. 

Through the wish to introduce measure next into the contin- 
uum just defined this continuum becomes space, and geometry is 
born. But the discussion of this is reserved for Part Second. 



The Non-Euclidean Geometries 

Evert conclusion supposes premises ; these premises themselves 
either are self-evident and need no demonstration, or can be 
established only by relying upon other propositions, and since 
we can not go back thus to infinity, every deductive science, and 
in particular geometry, must rest on a certain number of unde- 
monstrable axioms. All treatises on geometry begin, therefore, 
by the enunciation of these axioms. But among these there is a 
distinction to be made: Some, for example, * Things which are 
equal to the same thing are equal to one another, ' are not propo- 
sitions of geometry, but propositions of analysis. I regard them 
as analytic judgments a priori, and shall not concern myself with 

But I must lay stress upon other axioms which are peculiar to 
geometry. Most treatises enunciate three of these explicitly : 

1® Through two points can pass only one straight; 

2** The straight line is the shortest path from one point to 
another ; 

3** Through a given point there is not more than one parallel 
to a given straight. 

Although generally a proof of the second of these axioms is 
omitted, it would be possible to deduce it from the other two and 
from those, much more numerous, which are implicitly admitted 
without enunciating them, as I shall explain further on. 

It was long sought in vain to demonstrate likewise the third 
axiom, known as Euclid^ s Postulate. What vast effort has been 
wasted in this chimeric hope is truly unimaginable. Finally, in 



the first quarter of the nineteenth century, and almost at the 
same time, a Hungarian and a Russian, Bolyai and Lobachevski, 
established irrefutably that this demonstration is impossible ; they 
have almost rid us of inventors of geometries *sans postulatum'; 
since then the Academic des Sciences receives only about one or 
two new demonstrations a year. 

The question was not exhausted; it soon made a great 
stride by the publication of Riemann's celebrated memoir en- 
titled: Ueber die Hypothesen welche der Oeometrie zu Orunde 
liegen. This paper has inspired most of the recent works of which 
I shall speak further on, and among which it is proper to cite 
those of Beltrami and of Helmholtz. 

The Bolyai-Lobachevski Geometby. — If it were possible to 
deduce Euclid's postulate from the other axioms, it is evident 
that in denying the postulate and admitting the other axioms, we 
should be led to contradictory consequences; it would therefore 
be impossible to base on such premises a coherent geometry. 

Now this is precisely what Lobachevski did. 

He assumes at the start that: Through a given point can he 
drawn two parallels to a given straight. 

And he retains besides all Euclid's other axioms. From these 
hypotheses he deduces a series of theorems among which it is 
impossible to find any contradiction, and he constructs a 
geometry whose faultless logic is inferior in nothing to that of 
the Euclidean geometry. 

The theorems are, of course, very different from those to which 
we are accustomed, and they can not fail to be at first a little 

Thus the sum of the angles of a triangle is always less than 
two right angles, and the diflference between this sum and two 
right angles is proportional to the surface of the triangle. 

It is impossible to construct a figure similar to a given figure 
but of different dimensions. 

If we divide a circumference into n equal parts, and draw 
tangents at the points of division, these n tangents will form a 
polygon if the radius of the circle is small enough; but if this 
radius is sufficiently great they will not meet. 

It is useless to multiply these examples; Lobachevski 's propo- 


sitions have no relation to those of Euclid, but they are not less 
logically bound one to another. 

Biemann's Geometby. — Imagine a world uniquely peopled 
by beings of no thickness (height) ; and suppose these infinitely 
flat' animals are all in the same plane and can not get out. Ad- 
mit besides that this world is sufficiently far from others to be 
free from their influence. While we are making hypotheses, it 
costs us no more to endow these beings with reason and believe 
them capable of creating a geometry. In that case, they will cer- 
tainly attribute to space only two dimensions. 

But suppose now that these imaginary animals, while remain- 
ing without thickness, have the form of a spherical, and not of a 
plane, figure, and are all on the same sphere without power to get 
off. What geometry will they construct? First it is clear they 
will attribute to space only two dimensions; what will play for 
them the role of the straight line will be the shortest path from 
one point to another on the sphere, that is to say, an arc of a great 
circle ; in a word, their geometry will be the spherical geometry. 

What they will call space will be this sphere on which they 
must stay, and on which happen all the phenomena they can 
know. Their space will therefore be unbou^ided since on a 
sphere one can always go forward without ever being stopped, 
and yet it will be finite; one can never find the end of it, but one 
can make a tour of it. 

Well, Riemann's geometry is spherical geometry extended to 
three dimensions. To construct it, the German mathematician 
had to throw overboard, not only Euclid's postulate, but also the 
first axiom : Only one straight can pass through two points. 

On a sphere, through two given points we can draw in general 
only one great circle (which, as we have just seen, would play the 
role of the straight for our imaginary beings) ; but there is an 
exception : if the two given points are diametrically opposite, an 
infinity of great circles can be drawn through them. 

In the same way, in Riemann's geometry (at least in one of 
its forms) , through two points will pass in general only a single 
straight; but there are exceptional cases where through two 
points an infinity of straights can pass. 


There is a sort of opposition between Riemann's geometry and 
that of Lobaehevski. 

Thus the sum of the angles of a triangle is : 

Equal to two right angles in Euclid's geometry; 

Less than two right angles in that of Lobaehevski ; 

Greater than two right angles in that of Biemann. 

The number of straights through a given point that can be 
drawn coplanar to a given straight, but nowhere meeting it, is 
equal : 

To one in Euclid's geometry; 

To zero in that of Riemann ; 

To infinity in that of Lobaehevski. 

Add that Biemann 's space is finite, although unbounded, in 
the sense given above to these two words. 

The Surfaces op Constant Cubvatube. — One objection still 
remained possible. The theorems of Lobaehevski and of Bie- 
mann present no contradiction ; but however numerous the con- 
sequences these two geometers have drawn from their hypotheses, 
they must have stopped before exhausting them, since their num- 
ber would be infinite ; who can say then that if they had pushed 
their deductions farther they would not have eventually reached 
some contradiction f 

This diflSculty does not exist for Biemann 's geometry, pro- 
vided it is limited to two dimensions; in fact, as we have seen, 
two-dimensional Riemannian geometry does not differ from spher- 
ical geometry, which is only a branch of ordinary geometry, and 
consequently is beyond all discussion. 

Beltrami, in correlating likewise Lobaehevski 's two-dimen- 
sional geometry with a branch of ordinary geometry, has equally 
refuted the objection so far as it is concerned. 

Here is how he accomplished it. Consider any figure on a 
surface. Imagine this figure traced on a flexible and inextensible 
canvas applied over this surface in such a way that when the 
canvas is displaced and deformed, the various lines of this figure 
can change their form without changing their length. In gen- 
eral, this flexible and inextensible figure can not be displaced 
without leaving the surface ; but there are certain particular sur- 


faces for which such a movement would be possible ; these are the 
surfaces of constant curvature. 

If we resume the comparison made above and imagine beings 
without thickness living on one of these surfaces, they will regard 
as possible the motion of a figure all of whose lines remain con- 
stant in length. On the contrary, such a movement would appear 
absurd to animals without thickness living on a surface of vari- 
able curvature. 

These surfaces of constant curvature are of two sorts: Some 
are of positive curvature, and can be deformed so as to be applied 
over a sphere. The geometry of these surfaces reduces itself 
therefore to the spherical geometry, which is that of Riemann. 

The others are of negative curvature. Beltrami has shown 
that the geometry of these surfaces is none other than that of 
Lobachevski. The two-dimensional geometries of Riemann and 
Lobachevski are thus correlated to the Euclidean geometry. 

Interpretation op Non-Euclidean Geometries. — So van- 
ishes the objection so far as two-dimensional geometries are con- 

It would be easy to extend Beltrami's reasoning to three- 
dimensional geometries. The minds that space of four dimen- 
sions does not repel will see no diflSculty in it, but they are few. 
I prefer therefore to proceed otherwise. 

Consider a certain plane, which I shall call the fundamental 
plane, and construct a sort of dictionary, by making correspond 
each to each a double series of terms written in two columns, just 
as correspond in the ordinary dictionaries the words of two lan- 
guages whose significance is the same : 

Space: Portion of space situated above the fundamental plane. 

Plane: Sphere cutting the fundamental plane orthogonally. 

Straight: Circle cutting the fundamental plane orthogonally. 

Sphere: Sphere. 

Circle: Circle. 

Angle: Angle. 

Distance between two points: Logarithm of the cross ratio of 
these two points and the intersections of the fundamental plane 
with a circle passing through these two points and cutting it 
orthogonally. Etc., Etc. 


Now take Lobachevski's theorems and translate them with 
the aid of this dictionary as we transate a German text with the 
aid of a German-English dictionary. We shall thus, obtain the- 
orems of the ordinary geometry. For example, that theorem of 
Lobachevski : ^the sum of the angles of a triangle is less than two 
right angles' is translated thus: ^'If a curvilinear triangle has 
for sides circle-arcs which prolonged would cut orthogonally the 
fundamental plane, the sum of the angles of this curvilinear tri- 
angle will be less than two right angles." Thus, however far the 
consequences of Lobachevski's hypotheses are pushed, they will 
never lead to «a contradiction. In fact, if two of Lobachevski's 
theorems were contradictory, it would be the same with the trans- 
lations of these two theorems, made by the aid of our dictionary, 
but these translations are theorems of ordinary geometry and no 
one doubts that the ordinary geometry is free from contradiction. 
Whence comes this certainty and is it justified? That is a ques- 
tion I can not treat here because it would require to be enlarged 
upon, but which is very interesting and I think not insoluble. 

Nothing remains then of the objection above formulated. 
This is not all. Lobachevski's geometry, susceptible of a concrete 
interpretation, ceases to be a vain logical exercise and is capal)le 
of applications ; I have not the time to speak here of these appli- 
cations, nor of the aid that Klein and I have gotten from them 
for the integration of linear differential equations. 

This interpretation moreover is not unique, and several dic- 
tionaries analogous to the preceding could be constructed, which 
would enable us by a simple 'translation' to transform Loba- 
chevski's theorems into theorems of ordinary geometry. 

The Implicit Axioms. — Are the axioms explicitly enunciated 
in our treatises the sole foundations of geometry? We may be 
assured of the contrary by noticing that after they are succes- 
sively abandoned there are still left over some propositions com- 
mon to the theories of Euclid, Lobachevski and Riemann. These 
propositions must rest on premises the geometers admit without 
enunciation. It is interesting to try to disentangle them from 
the classic demonstrations. 

Stuart Mill has claimed that every definition contains an 


axiom, because in defining one affirms implicitly the existence 
of the object defined. This is going much too far ; it is rare that 
in mathematics a definition is given without its being followed by 
the demonstration of the existence of the object defined, and 
when this is dispensed with it is generally because the reader 
can easily supply it. It must not be forgotten that the word 
existence has not the same sense when it refers to a mathematical 
entity and when it is a question of a material object. A mathe- 
matical entity exists, provided its definition implies no contradic- 
tioUy either in itself, or with the propositions already admitted. 

But if Stuart Mill's observation can not be applied to all 
definitions, it is none the less just for some of them. The plane 
is sometimes defined as follows : 

The plane is a surface such that the straight which joins any 
two of its points is wholly on this surface. 

This definition manifestly hides a new axiom; it is true we 
might change it, and that would be preferable, but then we 
should have to enunciate the axiom explicitly. 

Other definitions would suggest reflections not less important. 

Such, for example, is that of the equality of two figures ; two 
figures are equal when they can be superposed; to superpose 
them one must be displaced until it coincides with the other ; but 
how shall it be displaced? If we should ask this, no doubt we 
should be told that it must be done without altering the shape 
and as a rigid solid. The vicious circle would then be evident. 

In fact this definition defines nothing; it would have no mean- 
ing for a being living in a world where there were only fluids. 
If it seems clear to us, that is because we are used to the proper- 
ties of natural solids which do not differ much from those of the 
ideal solids, all of whose dimensions are invariable. 

Yet, imperfect as it may be, this definition implies an axiom. 

The possibility of the motion of a rigid figure is not a self- 
evident truth, or at least it is so only in the fashion of Euclid's 
postulate and not as an analytic judgment a priori would be. 

Moreover, in studying the definitions and the demonstrations 
of geometry, we see that one is obliged to admit without proof 
not only the possibility of this motion, but some of its properties 


This is at once seen from the definition of the straight line. 
Many defective definitions have been given, but the true one is 
that which is implied in all the demonstrations where the straight 
line enters: 

''It may happen that the motion of a rigid figure is such that 
all the points of a line belonging to this figure remain motionless 
while all the points situated outside of this line move. Such a 
line will be called a straight line." We have designedly, in this 
enunciation, separated the definition from the axiom it implies. 

Many demonstrations, such as those of the cases of the equality 
of triangles, of the possibility of dropping a perpendicular from 
a point to a straight, presume propositions which are not enun- 
ciated, for they require the admission that it is possible to trans- 
port a figure in a certain way in space. 

The Fourth Geometry. — Among these implicit axioms, there 
is one which seems to me to merit some attention, because when 
it is abandoned a fourth geometry can be constructed as coherent 
as those of Euclid, Lobachevski and Biemann. 

To prove that a perpendicular may always be erected at a 
point A to a straight AB, we consider a straight AC movable 
around the point A and initially coincident with the fixed 
straight AB; and we make it turn about the point A until it 
comes into the prolongation of AB. 

Thus two propositions are presupposed : First, that such a ro- 
tation is possible, and next that it may be continued until the 
two straights come into the prolongation one of the other. 

If the first point is admitted and the second rejected, we are 
led to a series of theorems even stranger than those of Loba- 
chevski and Riemann, but equally exempt from contradiction. 

I shall cite only one of these theorems and that not the most 
singular: A real straight may be perpendicular to itself. 

LiE^s Theorem. — The number of axioms implicitly intro- 
duced in the classic demonstrations is greater than necessary, and 
it would be interesting to reduce it to a minimum. It may first 
be asked whether this reduction is possible, whether the number 
of necessary axioms and that of imaginable geometries are not 


A theorem of Sophus Lie dominates this whole discussion. It 
may be thus enunciated: 

Suppose the following premises are admitted: 

1^ Space has n dimensions; 

2^ The motion of a rigid figure is possible; 

3^ It requires p conditions to determine the position of this 
figure in space. 

The number of geometries compatible with these premises luill 
he limited. 

I may even add that if n is given, a superior limit can be 
assigned to p. 

If therefore the possibility of motion is admitted, there can 
be invented only a finite (and even a rather small) number of 
three-dimensional geometries. 

Biemann's Geometries. — ^Tet this result seems contradicted 
by Biemann, for this savant constructs an infinity of different 
geometries, and that to which his name is ordinarily given is only 
a particular case. 

All depends, he says, on how the length of a curve is defined. 
NW, there is an infinity of ways of defining this length, and each 
of them may be the starting point of a new geometry. 

That is perfectly true, but most of these definitions are incom- 
patible with the motion of a rigid figure, which in the theorem 
of Lie is supposed possible. These geometries of Rieraann, in 
many ways so interesting, could never therefore be other than 
purely analytic and would not lend themselves to demonstrations 
analogous to those of Euclid. 

On the Nature op Axioms. — Most mathematicians regard 
Lobachevski's geometry only as a mere logical curiosity; some of 
them, however, have gone farther. Since several geometries are 
possible, is it certain ours is the true one ? Experience no doubt 
teaches us that the sum of the angles of a triangle is equal to two 
right angles ; but this is because the triangles we deal with are 
too little; the difference, according to Lobachevski, is propor- 
tiozud to the surface of the triangle ; will it not perhaps become 
KQgible when we shall operate on larger triangles or when our 
nteasurements shall become more precise ? The Euclidean geom- 
etry would thus be only a provisional geometry. 


To discuss this opinion, we should first ask ourselves what 
is the nature of the geometric axioms. 

Are they synthetic a priori judgments, as Kant said! 

They would then impose themselves upon us with such force 
that we could not conceive the contrary proposition, nor build 
upon it a theoretic edifice. There would be no non-Euclidean 

To be convinced of it take a veritable synthetic a priori 
judgment, the following, for instance, of which we have seen 
the preponderant role in the first chapter : 

// a theorem is true for the number 1, and if it has been proved 
that it is true of n-\-l provided it is true of n, it u)iU be true of 
all the positive whole numbers. 

Then try to escape from that and, denying this proposition, 
try to found a false arithmetic analogous to non-Euclidean 
geometry — ^it can not be done ; one would even be tempted at first 
blush to regard these judgments as analytic. 

Moreover, resuming our fiction of animals without thickness, 
we can hardly admit that these beings, if their minds are like 
ours, would adopt the Euclidean geometry which would be con- 
tradicted by all their experience. 

Should we therefore conclude that the axioms of geometry are 
experimental verities? But we do not experiment on ideal 
straights or circles; it can only be done on material objects. On 
what then could be based experiments which should serve as 
foundation for geometry? The answer is easy. 

We have seen above that we constantly reason as if the geo- 
metric figures behaved like solids. What geometry would bor- 
row from experience would therefore be the properties of these 
bodies. The properties of light and its rectilinear propagation 
have also given rise to some of the propositions of geometry, 
and in particular those of projective geometry, so that from this 
point of view one would be tempted to say that metric geometry 
is the study of solids, and projective, that of light. 

But a difficulty remains, and it is insurmountable. If geom- 
etry were an experimental science, it would not be an exact 
science, it would be subject to a continual revision. Nay, it 
would from this very day be convicted of error, since we know 
that there is no rigorously rigid solid. 


The axioms of geometry therefore are neither synthetic a 
priori judgments nor experimental facts. 

They are conventions; our choice amon^ all possible conven- 
tions is guided by experimental facts ; but it remains free and is 
limited only by the necessity of avoiding all contradiction. Thus 
it is that the postulates can remain rigorously true even though 
the experimental laws which have determined their adoption are 
only approximative. 

In other words, the ctxioms of geometry (I do not speak of 
those of arithmetic) are merely disguised definitions. 

Then what are we to think of that question : Is the Euclidean 
geometry truet 

It has no meaning. 

As well ask whether the metric system is true and the old 
measures false ; whether Cartesian coordinates are true and polar 
coordinates false. One geometry can not be more true than an- 
other; it can only be more convenient. 

Now, Euclidean geometry is, and will remain, the most con- 
venient : 

1^ Because it is the simplest ; and it is so not only in conse- 
quence of our mental habits, or of I know not what direct in- 
tuition that we may have of Euclidean space ; it is the simplest in 
itself, just as a polynomial of the first degree is simpler than one 
of the second; the formulas of spherical trigonometry are more 
complicated than those of plane trigonometry, and they would 
still appear so to an analyst ignorant of their geometric signifi- 

2® Because it accords sufficiently well with the properties of 
natural solids, those bodies which our hands and our eyes com- 
pare and with which we make our instruments of measure. 


Space and Geometry 

Let us begin by a little paradox. 

Beings with minds like ours, and having the same senses as 
we, but without previous education, would receive from a suitably 
chosen external world impressions such that they would be led 
to construct a geometry other than that of Euclid and to localize 
the phenomena of that external world in a non-Euclidean space, 
or even in a space of four dimensions. 

As for us, whose education has been accomplished by our 
actual world, if we were suddenly transported into this new 
world, we should have no difficulty in referring its phenomena to 
our Euclidean space. Conversely, if these beings were trans- 
ported into our environment, they would be led to relate our 
phenomena to non-Euclidean space. 

Nay more; with a little effort we likewise could do it. A 
person who should devote his existence to it might perhaps attain 
to a realization of the fourth dimension. 

Geometric Space and Perceptual Space. — It is often said 
the images of external objects are localized in space, even that 
they can not be formed except on this condition. It is also said 
that this space, which serves thus as a ready prepared frame for 
our sensations and our representations, is identical with that of 
the geometers, of which it possesses all the properties. 

To all the good minds who think thus, the preceding state- 
ment must have appeared quite extraordinary. But let us see 
whether they are not subject to an illusion that a more profound 
analysis would dissipate. 

What, first of all, are the properties of space, properly so 
called! I mean of that space which is the object of geometry 
and which I shall call geometric space. 

The following are some of the most essential: 

1** It is continuous; 



2° It is infinite; 

3° It has three dimensions; 

4° It is homogeneous, that is to say, all its i>oints are identical 
one with another; 

5^ It is isotropic, that is to say, all the straights which pass 
through the same point are identical one with another. 

Compare it now to the frame of our representations and our 
sensations, which I may call perceptual space. 

Visual Space. — Consider first a purely visual impression, due 
to an image formed on the bottom of the retina. 

A cursory analysis shows us this image as continuous, but as 
possessing only two dimensions; this already distinguishes from 
geometric space what we may call pure visual space. 

Besides, this image is enclosed in a limited frame. 

Finally, there is another difference not less important: this 
pure visvM space is not homogeneous. All the points of the 
retina, aside from the images which may there be formed, do not 
play the same role. The yellow spot can in no way be regarded 
as identical with a point on the border of the retina. In fact, not 
only does the same object produce there much more vivid im- 
pressions, but in every limited frame the point occupying the 
center of the frame will never appear as equivalent to a point 
near one of the borders. 

No doubt a more profound analysis would show us that this 
continuity of visual space and its two dimensions are only an 
illusion ; it would separate it therefore still more from geometric 
space, but we shall not dwell on this remark. 

Sight, however, enables us to judge of distances and conse- 
quently to perceive a third dimension. But every one knows 
that this perception of the third dimension reduces itself to the 
sensation of the effort at accommodation it is necessary to make, 
and to that of the convergence which must be given to the two 
eyes, to perceive an object distinctly. 

These are muscular sensations altogether different from the 
visual sensations which have given us the notion of the first two 
dimensions. The third dimension therefore will not appear to 
us as playing the same role as the other two. What may be 
called complete visual space is therefore not an isotropic space. 


It has, it is true, precisely three dimensions, which means that 
the elements of our visual sensations (those at least which com- 
bine to form the notion of extension) will be completely de- 
fined when three of them are known; to use the language of 
mathematics, they will be functions of three independent 

But examine the matter a little more closely. The third 
dimension is revealed to us in two different ways: by the effort 
of accommodation and by the convergence of the eyes. 

No doubt these two indications are always concordant, there 
is a constant relation between them, or, in mathematical terms, 
the two variables which measure these two muscular sensations 
do not appear to us as independent ; or again, to avoid an appeal 
to mathematical notions already rather refined, we may go back 
to the language of the preceding chapter and enunciate the same 
fact as follows : If two sensations of convergence, A and B, are 
indistinguishable, the two sensations of accommodation, A' and 
B'f which respectively accompany them, will be equally indistin- 

But here we have, so to speak, an experimental fact; a priori 
nothing prevents our supposing the contrary, and if the contrary 
takes place, if these two muscular sensations vary independently 
of one another, we shall have to take account of one more inde- 
pendent variable, and 'complete visual space' will appear to us 
as a physical continuum of four dimensions. 

We have here even, I will add, a fact of external experience. 
Nothing prevents our supposing that a being with a mind like 
ours, having the same sense organs that we have, may be placed 
in a world where light would only reach him after having 
traversed reflecting media of complicated form. The two indi- 
cations which serve us in judging distances would cease to be 
connected by a constant relation. A being who should achieve 
in such a world the education of his senses would no doubt 
attribute four dimensions to complete visual space. 

Tactile Space and Motor Space. — ^'Tactile space' is still 
more complicated than visual space and farther removed from 
geometric space. It is superfluous to repeat for touch the discus- 
sion I have given for sight. 


But apart from the data of sight and touch, there are other 
sensations which contribute as much and more than they to the 
genesis of the notion of space. These are known to every one; 
they accompany all our movements, and are usually called mus- 
cular sensations. 

The corresponding frame constitutes what may be called motor 

Each muscle gives rise to a special sensation capable of aug- 
menting or of diminishing, so that the totality of our muscular 
sensations will depend upon as many variables as we have 
muscles. From this point of view, motor space would have as 
many dimensions as we have mtiscles. 

I know it will be said that if the muscular sensations con- 
tribute to form the notion of space, it is because we have the 
sense of the direction of each movement and that it makes an 
integrant part of the sensation. If this were so, if a muscular 
sensation could not arise except accompanied by this geometric 
9ense of direction, geometric space would indeed be a form im- 
posed upon our sensibility. 

But I perceive nothing at all of this when I analyze my sen- 

What I do see is that the sensations which correspond to move- 
ments in the same direction are connected in my mind by a mere 
association of ideas. It is to this association that what we call 
'the sense of direction' is reducible. This feeling therefore can 
not be found in a single sensation. 

This association is extremely complex, for the contraction of 
the same muscle may correspond, according to the position of the 
limbs, to movements of very different direction. 

Besides, it is evidently acquired; it is, like all associations of 
ideas, the result of a habit; this habit itself results from very 
numerous experiences; without any doubt, if the education of our 
senses had been accomplished in a different environment, where 
we should have been subjected to different impressions, con- 
trary habits would have arisen and our muscular sensations 
would have been associated according to other laws. 

Chabacteristics op Perceptual Space. — Thus perceptual 
space, under its triple form, visual, tactile and motor, is essen- 
tially different from geometric space. 


It is neither homogeneous, nor isotropic ; one can not even say 
that it has three dimensions. 

It is often said that we 'project' into geometric space the 
objects of our external perception; that we 'localize' them. 

Has this a meaning, and if so whatf 

Does it mean that we represent to ourselves external objects in 
geometric space f 

Our representations are only the reproduction of our sensa- 
tions; they can therefore be ranged only in the same frame as 
these, that is to say, in perceptual space. 

It is as impossible for us to represent to ourselves external 
bodies in geometric space, as it is for a painter to paint on a 
plane canvas objects with their three dimensions. 

Perceptual space is only an image of geometric space, an 
image altered in shape by a sort of perspe<Hive, and we can repre- 
sent to ourselves objects only by bringing them under the laws of 
this perspective. 

Therefore we do not represent to ourselves external bodies in 
geometric space, but we reason on these bodies as if they were 
situated in geometric space. 

When it is said then that we 'localize' such and such an object 
at such and such a point of space, what does it meant 

It simply means that we represent to ourselves the movements 
it would be necessary to m>ake to reach that object; and one may 
not say that to represent to oneself these movements, it is neces- 
sary to project the movements themselves in space and that the 
notion of space must, consequently, pre-exist. 

When I say that we represent to ourselves these movements, 
I mean only that we represent to ourselves the muscular sensa- 
tions which accompany them and which have no geometric char- 
acter whatever, which consequently do not at all imply the pre- 
existence of the notion of space. 

Change op State and Change op Position. — ^But, it will 
be said, if the idea of geometric space is not imposed upon our 
mind, and if, on the other hand, none of our sensations can 
furnish it, how could it have come into existence? 

This is what we have now to examine, and it will take some 
time, but I can summarize in a few words the attempt at explana- 
tion that I am about to develop. 


None of our sensations, isolated, could have conducted us to 
ike idea of space; we are led to it only in studying the laws, 
according to which these sensations succeed each other. 

We see first that our impressions are subject to change; but 
among the changes we ascertain we are soon led to make a dis- 

At one time we say that the objects which cause these im- 
pressions have changed state, at another time that they have 
changed position, that they have only been displaced. 

Whether an object changes its state or merely its position, 
this is always translated for us in the same manner: by a modifi- 
cation in an aggregate of impressions. 

How then could we have been led to distinguish between the 
twof It is easy to account for. If there has only been a 
change of position, we can restore the primitive aggregate of 
impressions by making movements which replace us opposite the 
mobile object in the same relative situation. We thus correct 
the modification that happened and we reestablish the initial 
state by an inverse modification. 

If it is a question of sight, for example, and if an object 
changes its place before our eye, we can * follow it with the 
eye' and maintain its image on the same point of the retina by 
appropriate movements of the eyeball. 

These movements we are conscious of because they are volun- 
tary and because they are accompanied by muscular sensations, 
but that does not mean that we represent them to ourselves in 
geometric space. 

So what characterizes change of position, what distinguishes 
it from change of state, is that it can always be corrected in this 

It may therefore happen that we pass from the totality of 
impressions A to the totality B in two different ways : 

1** Involuntarily and without experiencing muscular sensa- 
tions ; this happens when it is the object which changes place ; 

2*" Voluntarily and with muscular sensations; this happens 
when the object is motionless, but we move so that the object has 
relative motion with reference to us. 

If this be so, the passage from the totality A to the totality B 
is only a change of position. 


It follows from this that sight and toach could not have 
given UB the notion of space without the aid of the 'muscular 

Not only could this notion not be derived from a single sen' 
sation or even from a series of sensations, but what is more, an 
immobile being could never have acquired it, since, not being 
able to correct by his movements the effects of the changes of 
position of exterior objects, he would have had no reason what- 
ever to distinguish them from changes of state. Just as little 
could he have acquired it if his motions had not been voluntary 
or were unaccompanied by any sensations. 

Conditions op Compensation. — How is a like compensation 
possible, of such sort that two changes, otherwise independent of 
each other, reciprocally correct each othert 

A mind already familiar with geometry would reason as fol- 
lows: Evidently, if there is to be compensation, the various 
parts of the external object, on the one hand, and the various 
sense organs, on the other hand, must be in the same relative 
poffltioD after the double change. And, for that to be the case, 
the various parts of the external object must likewise have 
retained in reference to each other the same relative pontion, 
and the same must be true of the various parts of our body in 
regard to each other. 

In other words, the external object, in the first change, must 
be displaced as is a rigid solid, nnd so must it be with the whole 
of our body in the second change which corrects the first. 

Under these conditions, oompunsation may take place. 

But we who as yet know nothing of geometry, since for ia"fl| 
notion of space is not yet formed, we can not reason i 
can not foresee a priori whether compensatiou is pOssibl 
experience teaches us that it sometimes happens, and it j 
this experimental fact that we star: to distingoiili t 
state from changes of position. 

Solid Bodies and Gbouets 
there are some which ^tqv 
ceptible of being thus eon 
our own body; these are 



whose form is variable, only exceptionally undergo like displace- 
ments (change of position without change of form). When « 
body changes ita place and its shape, we can no longer, by appro- 
priate movements, bring back our sense-organs into the same 
relative situation with regard to this body; consequently we can 
DO longer reestablish the primitive totality of impressions. 

It is only later, and as a consequence of new experiences, that 
we learn how to decompose the bodies of variable form into 
smaller elements, such that each ia displaced almost in accord- 
ance with the same laws as solid bodies. Thus we distinguish 
'deformations" from other changes of state; in these deforma- 
tions, each element undergoes a mere change of position, which 
can he corrected, but the modification undergone by the aggre- 
gate is more profound and is no longer susceptible of correction 
by a correlative movement. 

8uch a notion is already very complex and must have been 
relatively late in appearing ; moreover it could not have arisen if 
the observation of solid bodies f d not already taught us to dis- 
tiDguish changes of portion. 

Therefore, if there were no solid bodies in nature, there wcndd 
ht no geometry. 

Another remark also deserves a moment's attention. Suppose 
a solid body to occupy successively the positions a. and p; in its 
first position, it wrill produce on us the totality of impressions A, 
lity of impressions B. Let 
viag qualities entirely djffer- 
ilerent color. Suppose it to 
us the totality of im- 
the totality of irn- 

eommon with 

;y B'. The trau- 

that from the 

:e8 which in 


It is simply because they can both be corrected by the same 
correlative movement of our body. 

'Correlative movement' therefore constitutes the sole connec- 
tion between two phenomena which otherwise we never should 
have dreamt of likening. 

On the other hand, our body, thanks to the number of its 
articulations and muscles, may make a multitude of different 
movements; but all are not capable of 'correcting' a modification 
of external objects ; only those will be capable of it in which our 
whole body, or at least all those of our sense-organs which come 
into play, are displaced as a whole, that is, without their relative 
positions varying, or in the fashion of a solid body. 

To summarize: 

1^ We are led at first to distinguish two categories of phe- 
nomena : 

Some, involuntary, unaccompanied by muscular sensations, are 
attributed by us to external objects ; these are external changes ; 

Others, opposite in character and attributed by us to the 
movements of our own body, are internal changes ; 

2** We notice that certain changes of each of these categories 
may be corrected by a correlative change of the other category; 

3** We distinguish among external changes those which have 
thus a correlative in the other category; these we call displace- 
ments; and just so among the internal changes, we distinguish 
those which have a correlative in the first category. 

Thus are defined, thanks to this reciprocity, a particular class 
of phenomena which we call displacements. 

The laws of these phenomena constitute the object of geometry. 

Law op Homogeneity. — The first of these laws is the law of 

Suppose that, by an external change a, we pass from the total- 
ity of impressions A to the totality B, then that this change 
a is corrected by a correlative voluntary movement j8, so that we 
are brought back to the totality A. 

Suppose now that another external change a makes us pass 
anew from the totality A to the totality B. 

Experience teaches us that this change a is, like a, sus- 
ceptible of being corrected by a correlative voluntary movement 


fif and that this moyement p' corresponds to the same mnscnlar 
sensations as the movement p which corrected a. 

This fact is usually enunciated by saying that space is homo- 
geneous and isotropic. 

It may also be said that a movement which has once been pro- 
duced may be repeated a second and a third time, and so on, 
without its properties varying. 

In the first chapter, where we discussed the nature of mathe- 
matical reasoning, we saw the importance which must be 
attributed to the possibility of repeating indefinitely the same 

It is from this repetition that mathematical reasoning gets its 
I>ower; it is, therefore, thanks to the law of homogeneity, that it 
has a hold on the geometric facts. 

For completeness, to the law of homogeneity should be added 
a multitude of other analogous laws, into the details of which I 
do not wish to enter, but which mathematicians sum up in a word 
by saying that displacements form 'a group.' 

The Non-Eucudean World. — If geometric space were a 
frame imposed on each of our representations, considered indi- 
vidually, it would be impossible to represent to ourselves an 
image stripped of this frame, and we could change nothing of 
our geometry. 

But this is not the ease ; geometry is only the resume of the 
laws according to which these images succeed each other. Noth- 
ing then prevents us from imagining a series of representations, 
similar in all points to our ordinary representations, but suc- 
ceeding one another according to laws different from those to 
which we are accustomed. 

We can conceive then that beings who received their educa- 
tion in an environment where these laws were thus upset might 
have a geometry very different from ours. 

Suppose, for example, a world enclosed in a great sphere and 
subject to the following laws: 

The temperature is not uniform; it is greatest at the center, 
and diminishes in proportion to the distance from the center, to 
sink to absolute zero when the sphere is reached in which this 
world is enclosed. 


To specify still more precisely the law in accordance with 
which this temperature varies: Let B be the radius of the lim- 
iting sphere; let r be the distance of the point considered from 
the center of this sphere. The absolute temperature shall be 
proportional to J2* — r*. 

I shall further suppose that, in this world, all bodies have 
the same coefScient of dilatation, so that the length of any rule 
is proportional to its absolute temperature. 

Finally, I shall suppose that a body transported from one 
point to another of different temperature is put immediately into 
thermal equilibrium with its new environment. 

Nothing in these hypotheses is contradictory or unimaginable. 

A movable object will then become smaller and smaller in pro- 
portion as it approaches the limit-sphere. 

Note first that, though this world is limited from the point 
of view of our ordinary geometry, it will appear infinite to its 

In fact, when these try to approach the limit-sphere, they cool 
off and become smaller and smaller. Therefore the steps they 
take are also smaller and smaller, so that they can never reach the 
limiting sphere. 

If, for us, geometry is only the study of the laws according 
to which rigid solids move, for these imaginary beings it will be 
the study of the laws of motion of solids distorted hy the differ- 
ences of temperature just spoken of. 

No doubt, in our world, natural solids likewise undergo varia- 
tions of form and volume due to warming or cooling. But we 
neglect these variations in laying the foundations of geometry, 
because, besides their being very slight, they are irregular and 
consequently seem to us accidental. 

In our hypothetical world, this would no longer be the case, 
and these variations would follow regular and very simple laws. 

Moreover, the various solid pieces of which the bodies of its 
inhabitants would be composed would undergo the same, varia- 
tions of form and volume. 

I will make still another hypothesis; I will suppose light 
traverses media diversely refractive and such that the index of 
refraction is inversely proportional to J2* — r*. It is easy to 


see that, under these conditions, the rays of light would not be 
rectilinear, but circular. 

To justify what precedes, it remains for me to show that 
certain changes in the position of external objects can be cor- 
reded by correlative movements of the sentient beings inhabit- 
ing this imaginary world, and that in such a way as to restore the 
primitive aggregate of impressions experienced by these sentient 

Suppose in fact that an object is displaced, undergoing de- 
formation, not as a rigid solid, but as a solid subjected to unequal 
dilatations in exact conformity to the law of temperature above 
supposed. Permit me for brevity to call such a movement a 
nan-Euclidean displacement. 

If a sentient being happens to be in the neighborhood, his 
impressions will be modified by the displacement of the object, 
but he can reestablish them by moving in a suitable manner. It 
suffices if finally the aggregate of the object and the sentient 
being, considered as forming a single body, has undergone one of 
those particular displacements I have just called non-Euclidean. 
This is possible if it be supposed that the limbs of these beings 
dilate according to the same law as the other bodies of the world 
they inhabit. 

Although from the point of view of our ordinary geometry 
there is a deformation of the bodies in this displacement and 
their various parts are no longer in the same relative position, 
nevertheless we shall see that the impressions of the sentient 
being have once more become the same. 

In fact, though the mutual distances of the various parts may 
have varied, yet the parts originally in contact are again in 
contact. Therefore the tactile impressions have not changed. 

On the other hand, taking into account the hypothesis made 
above in regard to the refraction and the curvature of the rays 
of light, the visual impressions will also have remained the same. 

These imaginary beings will therefore like ourselves be led 
to classify the phenomena they witness and to distinguish among 
them the * changes of position' susceptible of correction by a cor- 
relative voluntary movement. 

If they construct a geometry, it will not be, as ours is, the 


study of the moyements of our rigid solids ; it will be the study 
of the changes of position which they will thus have distin- 
guished and which are none other than the 'non-Euclidean dis- 
placements'; t^ tvUl he non-EucUdean geometry. 

Thus beings like ourselves, educated in such a world, would 
not have the same geometry as ours. 

The World op Four Dimensions. — ^We can represent to our- 
selves a four-dimensional world just as well as a non-Euclidean. 

The sense of sight, even with a single eye, together with the 
muscular sensations relative to the movements of the eyeball, 
would sufSce to teach us space of three dimensions. 

The images of external objects are painted on the retina, which 
is a two-dimensional canvas; they are perspectives. 

But, as eye and objects are movable, we see in succession vari- 
ous perspectives of the same body, taken from different points 
of view. 

At the same time, we find that the transition from one per- 
spective to another is often accompanied by muscular sensations. 

If the transition from the perspective A to the perspective 
B, and that from the perspective A' to the perspective B' are 
accompanied by the same muscular sensations, we liken them one 
to the other as operations of the same nature. 

Studying then the laws according to which these operations 
combine, we recognize that they form a group, which has the 
same structure as that of the movements of rigid solids. 

Now, we have seen that it is from the properties of this group 
we have derived the notion of geometric space and that of three 

We understand thus how the idea of a space of three dimen- 
sions could take birth from the pageant of these perspectives, 
though each of them is of only two dimensions, since they follow 
one another according to certain laws. 

Well, just as the perspective of a three-dimensional figure 
can be made on a plane, we can make that of a four-dimensional 
figure on a picture of three (or of two) dimensions. To a 
geometer this is only child's play. 

We can even take of the same figure several perspectives from 
several different points of view. 


We can easily represent to ourselves these perspectives, since 
thej are of only three dimensions. 

Imagine that the various perspectives of the same object suc- 
ceed one another, and that the transition from one to the other 
is accompanied by muscular sensations. 

We shall of course consider two of these transitions as two 
operations of the same nature when they are associated with the 
same muscular sensations. 

Nothing then prevents us from imagining that these opera- 
tions combine according to any law we choose, for example, so as 
to form a group with the same structure as that of the move- 
ments of a rigid solid of four dimensions. 

Here there is nothing unpicturable, and yet these sensations 
are precisely those which would be felt by a being possessed of 
a two-dimensional retina who could move in space of four dimen- 
sions. In this sense we may say the fourth dimension is 

CONCLUSIONS. — ^We see that experience plays an indispensable 
role in the genesis of geometry ; but it would be an error thence 
to conclude that geometry is, even in part, an experimental 

If it were experimental, it would be only approximative and 
provisional. And what rough approximation! 

Geometry would be only the study of the movements of solids ; 
but in reality it is not occupied with natural solids, it has for 
object certain ideal solids, absolutely rigid, which are only a 
simplified and very remote image of natural solids. 

The notion of these ideal solids is drawn from all parts of our 
mind, and experience is only an occasion which induces us to 
bring it forth from them. 

The object of geometry is the study of a particular * group'; 
but the general group concept pre-exists, at least potentially, in 
our minds. It is imposed on us, not as form of our sense, but as 
form of our understanding. 

Only, from among all the possible groups, that must be chosen 
which will be, so to speak, the standard to which we shall refer 
natural phenomena. 

Experience guides us in this choice without forcing it upon 


us; it tells us not which is the truest geometry, but which is the 
most convenient. 

Notice that I have been able to describe the fantastic worlds 
above imagined without ceasing to employ the language of ordi- 
nary geometry. 

And, in fact, we should not have to change it if transported 

Beings educated there would doubtless find it more convenient 
to create a geometry different from ours, and better adapted to 
their impressions. As for us, in face of the same impressions, it 
is certain we should find it more convenient not to change our 

Experience and Qeometby 

1. Already in the preceding pages I have several times tried 
to show that the principles of geometry are not experimental 
facts and that in particular Euclid's postulate can not be proven 

However decisive appear to me the reasons already given, I 
believe I should emphasize this point because here a false idea 
is profoundly rooted in many minds. 

2. If we construct a material circle, measure its radius and 
circumference, and see if the ratio of these two lengths is equal 
to ir, what shall we have done T We shall have made an experi- 
ment on the properties of the matter with which we constructed 
this round thing, and of that of which the measure used was made. 

3. Qeometry and Astronomy. — The question has also been 
put in another way. If Lobachevski's geometry is true, the paral- 
lax of a very distant star will be finite; if Riemann's is true, it 
will be negative. These are results which seem within the reach 
of experiment, and there have been hopes that astronomical obser- 
vations might enable us to decide between the three geometries. 

But in astronomy * straight line' means simply *path of a ray 
of light. ' 

If therefore negative parallaxes were found, or if it were 
demonstrated that all parallaxes are superior to a certain limit, 
two courses would be open to us; we might either renounce 
Euclidean geometry, or else modify the laws of optics and sup- 
pose that light does not travel rigorously in a straight line. 

It is needless to add that all the world would regard the latter 
solution as the more advantageous. 

The Euclidean geometry has, therefore, nothing to fear from 
fresh experiments. 

4. Is the position tenable, that certain phenomena, possible 
in Euclidean space, would be impossible in non-Euclidean space, 

7 81 


SO that ezperienee, in eetsbliBhing these phenomena, woald di- 
rectly contradict the non-Euclidean hypothesis t For my part I 
think no such question can be put. To my mind it is precisely 
equivalent to the following, whose absurdity is patent to all eyes: 
are there lengths expressible in meters and centimeters, but which 
can not be measured in fathoms, feet and inches, so that experi- 
ence, in ascertaining the existence of these lengths, would directly 
contradict the hypothesis that there are fathoms divided into 
six feet I 

Examine the question more closely. I suppose that the straight 
line possesses in Euclidean space any two properties which I 
shall call A and B ; that in non-Euclidean space it still possesses 
the property A, but no longer has the property B ; finally I sup* 
pose that in both Euclidean and non-Euclidean space the straight 
line is the only line having the property A. 

If this were so, experience would be capable of deciding between 
the hypothesis of Euclid and that of Lobacbevski. It woold be 
ascertained that a definite concrete object, accessible to experi- 
ment, for example, a pencil of rays of light, possesBea the proper^ 
A ; we should conclude that it is rectilinear, and then inrectitgatt 
whether or not it has the property B. W^M 

But this is not so; no property exists which, like this propeHI^^ 
A, can be an absolute criterion enabling us to recognize the 
straight line and to distinguish it from every other line. 

Shall we say, for instance: "the following is such a pre 
the straight line is a line such that a figure of which 1 
forma a part can be moved without the mutual d 
points varying and so that all points oF this line remain t 

This, in fact, is a property which, in Euclidean or D 
can space, belongs to the straight and belongs only t 
how shall we ascertain experimentally whether it h 
or that concrete object? It will be necessary i 
tances, and how slmll one know i 
which I have measured 'with my m 
sents the abstract distance T 

We have only pushed twfl^ 

In reality the prop«t7 } 
the straight line al<ni^ i* 


distance. For it to serve as absolute criterion, we should have 
to be able to establish not only that it doea not also belong to a 
line other than the straight and to distance, but in addition that 
it does not belong to a line other than the straight and to a 
ma^tude other than distance. Now thia is not true. 

It is therefore impossible to imagine a concrete experiment 
which can be interpreted in the Euclidean system and not in the 
Lobachevskian system, so that I may conclude : 

No experience will ever be in contradiction to Euclid's pos- 
tulate; nor, on the other hand, will any experience ever contra- 
dict the postulate of Lobachevski. 

5. But it is not enough that the Euclidean (or non-Euclidean) 
geometry can never be directly contradicted by experience. Might 
it not happen that it can accord with experience only by violating 
the principle of sufficient reason or that of the relativity of space T 

I will explain myself: consider any material system; we shall 
have to regard, on the one hand, 'the state' of the various bodies 
of this system (for instance, their temperature, their electric 
potential, etc.), and, on the other hand, their position in space; 
and among the data which enable us to define this position we 
aball, moreover, distinguish the mutual distances of these bodies, 
which define their relative positions, from the conditions which 
define the absolute position of the system and its absolute orien- 
tation in sjwce. 

Til.' ■ i ■ I'i i ■; ■:; i :! i '. 'ippea in this system 

will ' : their mutual dis- 

lani!' - ' . ■ •■■. ity of space, they 

H-ill not Jtj- ' tionof the 

In ..i»>-r- '• . iiitiml dis- 

tail''-' .r iiL'se 


the non-Euclidean hypothesis. Well, we have made a series of 
experiments ; we have interpreted them on the Euclidean hyjxoth- 
esis, and we have recognized that these experiments thus inter- 
preted do not violate this 'law of relativity.' 

We now interpret them on the non-Euclidean hypothesis: 
this is always possible ; only the non-Euclidean distances of our 
different bodies in this new interpretation will not generally be the 
same as the Euclidean distances in the primitive interpretation. 

Will our experiments, interpreted in this new manner, still 
be in accord with our 'law of relativity'! And if there were 
not this accord, should we not have also the right to say experi- 
ence bad proven the falsity of the non-Euclidean geometry? 

It is easy to see that this is an idle fear; in fact, to apply 
the law of relativity in all rigor, it must be applied to the entire 
universe. For if only a part of this universe were considered, 
and if the absolute position of this part happened to vary, the 
distances to the other bodies of the universe would likewise vary, 
their influence on the part of the universe considered would con- 
sequently augment or diminish, which might modify the laws 
of the phenomena happening there. 

But if our system is the entire universe, experience is power- 
less to give information about its absolute position and orienta- 
tion in space. All that our instruments, however perfected they 
may be, can tell us will be the state of the various parts of the 
tmiverse and their mutual distances. 

So our law of relativity may be thus enunciated : 

The readings we shall be able to make on our instruments at 
any instant will depend only on the readings we could have made 
on these same instruments at the initial instant. 

Now such an enunciation is independent of every interpreta- 
tion of experimental facts. If the law is true in the Euclidean 
interpretation, it will also be true in the non-Euclidean interpre- 

Allow me here a short digression. I have sx)oken above of 
the data which define the position of the various bodies of the 
system ; I should likewise have spoken of those which define their 
velocities; I should then have had to distinguish the velocities 
with which the mutual distances of the different bodies vary; 


and, on the other hand, the velocities of translation and rotation 
of the system, that is to say, the velocities (with which its absolute 
position land orientation vary. 

To fully satisfy the mind, the law of relativity should be 
expressible thus : 

The state of bodies and their mutual distances at any instant, 
as well as the velocities with which these distances vary at this 
same instant, will depend only on the state of those bodies and 
their mutual distances at the initial instant, and the velocities 
with which these distances vary at this initial instant, but they 
will not depend either upon the absolute initial position of the 
system, or upon its absolute orientation, or upon the velocities 
with which this absolute position and orientation varied at the 
initial instant. 

Unhappily the law thus enunciated is not in accord with ex- 
periments, at least as they are ordinarily interpreted. 

Suppose a man be transported to a planet whose heavens were 
always covered with a thick curtain of clouds, so that he could 
never see the other stars ; on that planet he would live as if it 
were isolated in space. Yet this man could become aware that it 
turned, either by measuring its oblateness (done ordinarily by 
the aid of astronomic observations, but capable of being done by 
purely geodetic means) , or by repeating the experiment of Fou- 
cault's pendulum. The absolute rotation of this plcmet could 
therefore be made evident. 

That is a fact which shocks the philosopher, but which the 
physicist is compelled to accept. 

We know that from this fact Newton inferred the existence 
of absolute space ; I myself am quite unable to adopt this view. 
I shall explain why in Part III. For the moment it is not my 
intention to enter upon this diflSculty. 

Therefore I must resign myself, in the enunciation of the law 
of relativity, to including velocities of every kind among the data 
which define the state of the bodies. 

However that may be, this difficulty is the same for Euclid's 
geometry as for Lobachevski's; I therefore need not trouble my- 
self with it, and have only mentioned it incidentally. 


What is important is the conclusion: experiment can not de- 
cide between Euclid and Lobachevski. 

To sum up, whichever way we look at it, it is impossible to 
discover in geometric empiricism a rational meaning. 

6. Experiments only teach us the relations of bodies to one 
another; none of them bears or can bear on the relations of bodies 
with space, or on the mutual relations of different parts of space. 

"Yes," you reply, "a single experiment is insuflScient, be- 
cause it gives me only a single equation with several unknowns ; 
but when I shall have made enough experiments I shall have 
equations enough to calculate all my unknowns." 

To know the height of the mainmast does not sufSce for calcu- 
lating the age of the captain. When you have measured every 
bit of wood in the ship you will have many equations, but 
you will know his age no better. All your measurements bear- 
ing only on your bits of wood can reveal to you nothing except 
concerning these bits of wood. Just so your experiments, how- 
ever numerous they may be, bearing only on the relations of 
bodies to one another, will reveal to us nothing about the mutual 
relations of the various parts of space. 

7. Will you say that if the experiments bear on the bodies, 
they bear at least upon the geometric properties of the bodies? 
But, first, what do you understand by geometric properties of 
the bodies? I assume that it is a question of the relations of the 
bodies with space ; these properties are therefore inaccessible to 
experiments which bear only on the relations of the bodies to one 
another. This alone would suffice to show that there can be no 
question of these properties. 

StiU let us begin by coming to an understanding about the 
sense of the phrase: geometric properties of bodies. When I 
say a body is composed of several parts, I assume that I do not 
enunciate therein a geometric property, and this would remain 
true even if I agreed to give the improper name of points to the 
smallest parts I consider. 

When I say that such a part of such a body is in contact 
with such a part of such another body, I enunciate a proposition 
which concerns the mutual relations of these two bodies and not 
their relations with space. 


I suppose you will grant me these are not geometric properties; 
at least I am sure you will grant me these properties are inde- 
pendent of all knowledge of metric geometry. 

This presupposed, I imagine that we have a solid body formed 
of eight slender iron rods, OA, OB, OC, OD, OE, OF, 00, OH, 
united at one of their extremities 0. Let us besides have a second 
«olid body, for example a bit of wood, to be marked with three 
little flecks of ink which I shall call a, p, y. I further suppose it 
ascertained that apy may be brought into contact with AOO (I 
mean a with A, and at the same time fi with O and y with 0), 
then that we may bring successively into contact aPy with BOO, 
COO, DOO, EOO, FOO, then with AHO, BEO, CEO, DEO, 
EEO, FEO, then ay successively with AB, BC, CD, DE, EF, FA. 

These are determinations we may make without having in 
advance any notion about form or about the metric properties of 
space. They in no wise bear on the 'geometric properties of 
bodies.' And these determinations will not be possible if the 
bodies experimented upon move in accordance with a group 
having the same structure as the Lobachevskian group (I mean 
according to the same laws as solid bodies in LobachevsM's geom- 
etry). They suffice therefore to prove that these bodies move in 
accordance with the Euclidean group, or at least that they do 
not move according to the Lobachevskian group. 

That they are compatible with the Euclidean group is easy 
to see. For they could be made if the body apy was a rigid 
solid of our ordinary geometry presenting the form of a right- 
angled triangle, and if the points ABCDEFOE were the summits 
of a polyhedron formed of two regular hexagonal pyramids of our 
ordinary geometry, having for common base ABCDEF and for 
apices the one and the other E. 

Suppose now that in place of the preceding determination it 
is observed that as above aPy can be successively applied to AOO, 
then that ap (and no longer ay) can be successively applied to 
AB, BC, CD, DE, EF and FA. 

These are determinations which could be made if non-Euclid- 
ean geometry were true, if the bodies aPy and OABCDEFOE 
were rigid solids, and if the first were a right-angled triangle 


and the second a double regular hexagonal pyramid of snitaible 

Therefore these new determinations are not possible if the 
bodies move according to the Euclidean group ; but they become 
so if it be supposed that the bodies move according to the Loba- 
chevskian group. They would suffice, therefore (if one made 
them), to prove that the bodies in question do not move accord- 
ing to the Euclidean group. 

Thus, without making any hypothesis about form, about the 
nature of space, about the relations of bodies to space, and with- 
out attributing to bodies any geometric property, I have made 
observations which have enabled me to show in one case that 
the bodies experimented upon move according to a group whose 
structure is Euclidean, in the other case that they move according 
to a group whose structure is Lobachevskian. 

And one may not say that the first aggregate of determinations 
would constitute an experiment proving that space is Euclidean, 
and the second an experiment proving that space is non-Euclidean. 

In fact one could imagine (I say imagine) bodies moving so 
as to render possible the second series of determinations. And 
the proof is that the first mechanician met could construct such 
bodies if he cared to take the pains and make the outlay. You 
will not conclude from that, however, that space is non-Euclidean. 

Nay, since the ordinary solid bodies would continue to exist 
when the mechanician had constructed the strange bodies of which 
I have just spoken, it would be necessary to conclude that space is 
at the same time Euclidean and non-Euclidean. 

Suppose, for example, that we have a great sphere of radius B 
and that the temperature decreases from the center to the surface 
of this sphere according to the law of which I have spoken in 
describing the non-Euclidean world. 

"We might have bodies whose expansion would be negligible 
and whioh would act like ordinary rigid solids ; and, on the other 
hand, bodies very dilatable and which would act like non-Euclidean 
solids. We might have two double pyramids OABCDEFOH and 
O'A'B'C'D'E'F'G'W and two triangles afiy and a'p^y'. The first 
double pyramid might be rectilinear and the second curvilinear;. 


the triangle aPy might be made of inexpansible matter and the 
other of a very dilatable matter. 

It would then be possible to make the first observations with 
the double pyramid OAH and the triangle aPy, and the second 
with the double pyramid O'A'H' and the triangle a'fify. And 
then experiment would seem to prove first that the Euclidean 
geometry is true and then that it is false. 

Experiments therefore have a bearing, not on space, hut on 


8. To complete the matter, I ought to speak of a very delicate 
question, which would require long development; I shall confine 
myself to summarizing here what I have expounded in the Revue 
de MStaphysique et de Morale and in The Monist. When we 
say space has three dimensions, what do we mean t 

We have seen the importance of those 'internal changes' 
revealed to us by our muscular sensations. They may serve to 
characterize the various attitudes of our body. Take arbitrarily 
as origin one of these attitudes A. When we pass from this 
initial attitude to any other attitude B, we feel a series of mus- 
cular sensations, and this series 8 will define B. Observe, how- 
ever, that we shall often regard two series 8 and 8' as defining 
the same attitude B (since the initial and final attitudes A and B 
remaining the same, the intermediary attitudes and the corre- 
sponding sensations may differ). How then shall we recognize 
the equivalence of these two series ! Because they may serve to 
compensate the same external change, or more generally because, 
when it is a question of compensating an external change, one 
of the series can be replaced by the other. Among these series, 
we have distinguished those which of themselves alone can com- 
pensate an external change, and which we have called 'displace- 
ments.' As we can not discriminate between two displacements 
which are too close together, the totality of these displacements 
presents the characteristics of a physical continuum ; experience 
teaches us that they are those of a physical continuum of six 
dimensions; but we do not yet know how many dimensions 
space itself has, we must first solve another question. 

What is a point of space! Everybody thinks he knows, but 


that is an illusion. What we see when we try to represent to our- 
selves a point of space is a black speck on white paper, a speck of 
chalk on a blackboard, always an object. The question should 
therefore be understood as follows : 

What do I mean when I say the object B is at the same point 
that the object A occupied just now t Or further, what criterion 
will enable me to apprehend thist 

I mean that, although I have not budged (which my muscular 
sense tells me) , my first finger which just now touched the object A 
touches at present the object B. I could have used other criteria ,* 
for instance another finger or the sense of sight. But the first 
criterion is sufficient; I know that if it answers yes, all the other 
criteria will give the same response. I know it by experience, I 
can not know it a priori. For the same reason I say that touch 
can not be exercised at a distance ; this is another way of enunci- 
ating the same experimental fact. And if, on the contrary, I say 
that sight acts at a distance, it means that the criterion furnished 
by sight may respond yes while the others reply no. 

And in fact, the object, although moved away, may form its 
image at the same point of the retina. Sight responds yes, the 
object has remained at the same point and touch answers no, 
because my finger which just now touched the object, at present 
touches it no longer. If experience had shown us that one finger 
may respond no when the other says yes, we should likewise 
say that touch acts at a distance. 

In short, for each attitude of my body, my first finger deter- 
mines a point, and this it is, and this alone, which defines a point 
of space. 

To each attitude corresponds thus a point ; but it often happens 
that the same point corresponds to several different attitudes (in 
this case we say our finger has not budged, but the rest of the 
body has moved). We distinguish, therefore, among the changes 
of attitude those where the finger does not budge. How are we 
led thereto f It is because often we notice that in these changes 
the object which is in contact with the finger remains in contact 
with it. 

Range, therefore, in the same class all the attitudes obtainable 
from each other by one of the changes we have thus distinguished. 


To all the attitudes of the class will correspond the same point 
of space. Therefore to each class will correspond a point and to 
each point a class. But one may say that what experience arrives 
at is not the point, it is this class of changes or, better, the cor- 
responding class of muscular sensations. 

And when we say space has three dimensions, we simply mean 
that the totality of these classes appears to us with the character- 
istics of a physical continuum of three dimensions. 

One might be tempted to conclude that it is experience which 
has taught us how many dimensions space has. But in reality 
here also our experiences have bearing, not on space, but on our 
body and its relations with the neighboring objects. Moreover 
they are excessively crude. 

In our mind pre-existed the latent idea of a certain number 
of groups — ^those whose theory Lie has developed. Which group 
shall we choose, to make of it a sort of standard with which to com- 
pare natural phenomena? And, this group chosen, which of its 
sub-groups shall we take to characterize a point of space t Ex- 
perience has guided us by showing us which choice best adapts 
itself to the properties of our body. But its role is limited to that. 

Ancestral Experience 

It has often been said that if individual experience could 
not create geometry the same is not true of ancestral experience. 
But what does that meant Is it meant that we could not experi- 
mentally demonstrate Euclid's postulate, but that our ancestors 
have been able to do it f Not in the least. It is meant that by 
natural selection our mind has adapted itself to the conditions of 
the external world, that it has adopted the geometry most advan- 
tageous to the species: or in other words the most convenient. 
This is entirely in conformity with our conclusions ; geometry is 
not true, it is advantageous. 




The Classic Mechanics 

The English teach mechanics as an experimental science; on 
the continent it is always expounded as more or less a deductive 
and a priori science. The English are right, that goes without 
saying; but how could the other method have been persisted in 
so longf Why have the continental savants who have sought to 
get out of the ruts of their predecessors been usually unable to 
free themselves completely ! 

On the other hand, if the principles of mechanics are only of 
experimental origin, are they not therefore only approximate and 
provisional! Might not new experiments some day lead us to 
modify or even to abandon them ? 

Such are the questions which naturally obtrude themselves, 
and the diflSculty of solution comes principally from the fact 
that the treatises on mechanics do not clearly distinguish between 
what is experiment, what is mathematical reasoning, what is con- 
vention, what is hypothesis. 

That is not all : 

1** There is no absolute space and we can conceive only of 
relative motions ; yet usually the mechanical facts are enunciated 
as if there were an absolute space to which to refer them. 

2° There is no absolute time; to say two durations are equal 
is an assertion which has by itself no meaning and which can 
acquire one only by convention. 

3° Not only have we no direct intuition of the equality of 
two durations, but we have not even direct intuition of the 




fiimnltaneity of two events occurring in different places: this I 
liave explained in an article entitled La mesure du iemps.^ ' 

4** Finally, our Euclidean geometry is itself only a sort of 
convention of language; mechanical facts might be enunciated 
with reference to a non-Euclidean space which would be a guide 
less convenient than, but just as legitimate as, our ordinary space ; 
the enunciation would thus become much more complicated, but it 
would remain possible. 

Thus absolute space, absolute time, geometry itself, are not 
conditions which impose themselves on mechanics ; all these things 
are no more antecedent to mechanics than the French language is 
logically antecedent to the verities one expresses in French. 

We might try to enunciate the fundamental laws of mechanics 
in a language independent of all these conventions; we should 
thus without doubt get a better idea of what these laws are in 
themselves ; this is what M. Andrade has attempted to do, at least 
in part, in his Leqons de mecanique physique. 

The enunciation of these laws would become of course much 
more complicated, because all these conventions have been devised 
expressly to abridge and simplify this enunciation. 

As for me, save in what concerns absolute space, I shall ignore 
all these difficulties ; not that I fail to appreciate them, far from 
that; but we have sufficiently examined them in the first two 
parts of the book. 

I shall therefore admit, provisionally, absolute time and Eu- 
clidean geometry. 

The Principle op Inertia. — ^A body acted on by no force can 
only move uniformly in a straight line. 

Is this a truth imposed a priori upon the mindf If it were 
80, how should the Greeks have failed to recognize it? How could 
they have believed that motion stops when the cause which gave 
birth to it ceases ! Or again that every body if nothing prevents, 
will move in a circle, the noblest of motions? 

If it is said that the velocity of a body can not change if there 
is no reason for it to change, could it not be maintained just as 
well that the position of this body can not change, or that the 

^Eevue de M4taphysique et de Morale, t. YI., pp. 1-13 (January, 1898). 


curvature of its trajectory can not change, if no external canse 
intervenes to modify themt 

Is the principle of inertia, which is not an a priori truth, 
therefore an experimental factt But has any one ever experi- 
mented on bodies withdrawn from the action of every force t and, 
if so, how was it known that these bodies were subjected to no 
force t The example ordinarily cited is that of a baU rolling a 
very long time on a marble table ; but why do we say it is sub- 
jected to no force t Is this because it is too remote from all other 
bodies to experience any appreciable action from themt Yet it 
is not farther from the earth than if it were thrown freely into 
the air ; and every one knows that in this case it would experience 
the influence of gravity due to the attraction of the earth. 

Teachers of mechanics usually pass rapidly over the example 
of the ball ; but they add that the principle of inertia is verified 
indirectly by its consequences. They express themselves badly; 
they evidently mean it is possible to verify various consequences 
of a more general principle, of which that of inertia is only a 
particular case. 

I shall propose for this general principle the following enun- 
ciation : 

The acceleration of a body depends only upon the position 
of this body and of the neighboring bodies and upon their 

Mathematicians would say the movements of all the material 
molecules of the universe depend on differential equations of the 
second order. 

To make it clear that this is really the natural generalization 
of the law of inertia, I shall beg you to permit me a bit of fiction. 
The law of inertia, as I have said above, is not imposed upon us 
a priori; other laws would be quite as compatible with the prin- 
ciple of suflScient reason. If a body is subjected to no force, in 
lieu of supposing its velocity not to change, it might be supposed 
that it is its position or else its acceleration which is not to change. 

Well, imagine for an instant that one of these two hypothetical 
laws is a law of nature and replaces our law of inertia. What 
would be its natural generalization? A moment's thought will 
show us. 


In the first case, we must suppose that the velocity of a body 
depends only upon its position and upon that of the neighboring 
bodies; in the second case that the change of acceleration of a 
body depends only upon the position of this body and of the 
neighboring 'bodies, upon their velocities and upon their acceler- 

Or to speak the language of mathematics, the differential 
equations of motion would be of the first order in the first case, 
and of the third order in the second case. 

Let us slightly modify our fiction. Suppose a world analogous 
to our solar system, but where, by a strange chance, the orbits of 
all the planets are without eccentricity and without inclination. 
Suppose further that the masses of these planets are too slight 
for their mutual perturbations to be sensible. Astronomers in- 
habiting one of these planets could not fail to conclude that the 
orbit of a star can only be circular and parallel to a certain plane ; 
the position of a star at a given instant would then suffice to de- 
termine its velocity and its whole path. The law of inertia which 
they would adopt would be the first of the two hypothetical laws 
I have mentioned. 

Imagine now that this system is some day traversed with great 
velocity by a body of vast mass, coming from distant constella- 
tions. All the orbits would be profoundly disturbed. Still our 
astronomers would not be too greatly astonished ; they would very 
well divine that this new star was alone to blame for all the 
mischief. *'But," they would say, *'when it is gone, order will 
of itself be reestablished ; no doubt the distances of the planets 
from the sun will not revert to what they were before the cata- 
clysm, but when the perturbing star is gone, the orbits will again 
become circular." 

It would only be when the disturbing body was gone and when 
nevertheless the orbits, in lieu of again becoming circular, became 
elliptic, that these astronomers would become conscious of their 
error and the necessity of remaking all their mechanics. 

I have dwelt somewhat upon these hypotheses because it seems 
to me one can clearly comprehend what our generalized law of 
inertia really is only in contrasting it with a contrary hypothesis. 

Well, now, has this generalized law of inertia been verified by 


experiment, or can it bef When Newton wrote the Prindpia 
he quite regarded this truth as experimentally acquired and dem- 
onstrated. It was so in his eyes, not only through the anthropo- 
morphism of which we shall speak further on, but through the 
work of Galileo. It was so even from Kepler's laws themselves; 
in accordance with these laws, in fact, the path of a planet is 
completely determined by its initial position and initial velocity; 
this is just what our generalized law of inertia requires. 

For this principle to be only in appearance true, for one to 
have cause to dread having some day to replace it by one of the 
analogous principles I have just now contrasted with it, would be 
necessary our having been misled by some amazing chance, like 
that which, in the fiction above developed, led into error our 
imaginary astronomers. 

Such a hypothesis is too unlikely to delay over. No one will 
believe that such coincidences can happen; no doubt the prob- 
ability of two eccentricities being both precisely null, to within 
errors of observation, is not less than the probability of one being 
precisely equal to 0.1, for instance, and the other to 0.2, to within 
errors of observation. The probability of a simple event is not 
less than that of a complicated event ; and yet, if the first happens, 
we shall not consent to attribute it to chance ; we should not believe 
that nature had acted expressly to deceive us. The hypothesis of 
an error of this sort being discarded, it may therefore be admitted 
that in so far as astronomy is concerned, our law has been veri- 
fied by experiment. 

But astronomy is not the whole of physics. 

May we not fear lest some day a new experiment should come 
to falsify the law in some domain of physics t An experimental 
law is always subject to revision; one should always expect to see 
it replaced by a more precise law. 

Yet no one seriously thinks that the law we are speaking of 
will ever be abandoned or amended. Whyt Precisely because 
it can never be subjected to a decisive test. 

First of all, in order that this trial should be complete, it 
would be necessary that after a certain time all the bodies in the 
universe should revert to their initial positions with their initial 


velocities. It might then be seen whether, starting from this 
moment, they would resume their original paths. 

But this test is impossible, it can be only partially applied, 
and, however well it is made, there will always be some bodies 
which will not revert to their initial positions ; thus every deroga- 
tion of the law will easily find its explanation. 

This is not all ; in astronomy we see the bodies whose motions 
we study and we usually assume that they are not subjected to the 
action of other invisible bodies. Under these conditions our law 
must indeed be either verified or not verified. 

But it is not the same in physics ; if the physical phenomena 
are due to motions, it is to the motions of molecules which we do 
not see. If then the acceleration of one of the bodies we see 
appears to us to depend on something else besides the positions 
or velocities of other visible bodies or of invisible molecules whose 
existence we have been previously led to admit, nothing prevents 
our supposing that this something else is the position or the 
velocity of other molecules whose presence we have not before 
suspected. The law will find itself safeguarded. 

Permit me to employ mathematical language a moment to 
express the same thought under another form. Suppose we ob- 
serve n molecules and ascertain that their 3n coordinates satisfy 
a system of 3n differential equations of the fourth order (and 
not of the second order as the law of inertia would require) . We 
know that by introducing 3n auxiliary variables, a system of 3n 
equations of the fourth order can be reduced to a system of 6n 
equations of the second order. If then we suppose these 3n 
auxiliary variables represent the coordinates of n invisible mole- 
cules, the result is again in conformity with the law of inertia. 

To sum up, this law, verified experimentally in some particular 
cases, may unhesitatingly be extended to the most general cases, 
since we know that in these general cases experiment no longer 
is able either to confirm or to contradict it. 

The Law op Acceleration. — The acceleration of a body is 
equal to the force acting on it divided by its mass. Can this law 
be verified by experiment! For that it would be necessary to 



measure the three magnitudes which figure in the enunciation: 
acceleration, force and mass. 

I assume that acceleration can be measured, for I pass over 
the difSculty arising from the measurement of time. But how 
measure force, or mass t We do not even know what they are. 

What is mass? According to Newton, it is the product of the 
volume by the density. According to Thomson and Tait, it would 
be better to say that density is the quotient of the mass by the 
volume. What is force f It is, replies Lagrange, that which 
moves or tends to move a body. It is, Kirchhoff will say, the 
product of the mass by the acceleration. But then, why not say 
the mass is the quotient of the force by the acceleration t 

These di£Sculties are inextricable. 

When we say force is the cause of motion, we talk metaphysics, 
and this definition, if one were content with it, would be abso- 
lutely sterile. For a definition to be of any use, it must teach us 
to measure force ; moreover that sufSces ; it is not at all necessary 
that it teach us what force is in itself, nor whether it is the cause 
or the effect of motion. 

We must therefore first define the equality of two forces. 
When shall we say two forces are equal! It is, we are told, 
when, applied to the same mass, they impress upon it the same 
acceleration, or when, opposed directly one to the other, they pro- 
duce equilibrium. This definition is only a sham. A force applied 
to a body can not be uncoupled to hook it up to another body, 
as one uncouples a locomotive to attach it to another train. It 
is therefore impossible to know what acceleration such a force, 
applied to such a body, would impress upon such another body, 
if it were applied to it. It is impossible to know how two forces 
which are not directly opposed would act, if they were directly 

It is this definition we try to materialize, so to speak, when 
we measure a force with a dynamometer, or in balancing it with 
a weight. Two forces F and F\ which for simplicity I will sup- 
pose vertical and directed upward, are applied respectively to two 
bodies C and C ; I suspend the same heavy body P first to the 
body C, then to the body C ; if equilibrium is produced in both 
cases, I shall conclude that the two forces F and F' are equal to 


one another, since they are each equal to the weight of the body P. 

But am I sure the body P has retained the same weight when 
I have transported it from the first body to the second t Far from 
it; I am sure of the contrary; I know the intensity of gravity 
varies from one point to another, and that it is stronger, for 
instance, at the pole than at the equator. No doubt the difference 
is very slight and, in practise, I shall take no account of it; but 
a properly constructed definition should have mathematical 
rigor; this rigor is lacking. What I say of weight would evi- 
dently apply to the force of the resiliency of a dynamometer,, 
which the temperature and a multitude of circumstances may 
cause to vary. i 

This is not aU; we can not say the weight of the body P 
may be applied to the body C and directly balance the force P. 
What is applied to the body C is the action A of the body P on 
the body C ; the body P is submitted on its part, on the one hand, 
to its weight; on the other hand, to the reaction B of the body C 
on P. Finally, the force F is equal to the force A, since it balances 
it; the force A is equal to 2?, in virtue of the principle of the 
equality of action and reaction ; lastly, the force B is equal to the 
weight of P, since it balances it It is from these three equalities 
we deduce as consequence the equality of P and the weight of P. 

We are therefore obliged in the definition of the equality of 
the two forces to bring in the principle of the equality of action 
and reaction; on this account, this principle must no longer be 
regarded as an experimental law, but as a definition. 

For recognizing the equality of two forces here, we are then 
in possession of two rules : equality of two forces which balance ; 
equality of action and reaction. But, as we have seen above, 
these two rules are insuflScient ; we are obliged to have recourse to 
a third rule and to assume that certain forces, as, for instance, the 
weight of a body, are constant in magnitude and direction. But 
this third rule, as I have said, is an experimental law ; it is only 
approximately true ; it is a bad defimtion. 

We are therefore reduced to Kirchhoff's definition; force is 
equal to the m^iss multipled by the acceleration. This 'law of 
Newton' in its turn ceases to be regarded as an experimental law, 
it is now only a definition. But this definition is still insufficient, 


for we do not know what mass is. It enables us donbUess to cal- 
culate the relation of two forces applied to the same body at dif- 
ferent instants; it teaches us nothing about the relation of two 
forces applied to two different bodies. 

To complete it, it is necessary to go back anew to Newton's 
third law (equality of action and reaction), regarded again, not 
as an experimental law, but as a definition. Two bodies A and B 
act one upon the other; the acceleration of A multiplied by the 
mass of A is equal to the action of B upon A ; in the same way, 
the product of the acceleration of B by its mass is equal to the 
reaction of A upon B, As, by definition, action is equal to reac- 
tion, the masses of A and B are in the inverse ratio of their 
accelerations. Here we have the ratio of these two masses defined, 
and it is for experiment to verify that this ratio is constant. 

That would be all very well if the two bodies A and B alone 
were present and removed from the action of the rest of the 
world. This is not at all the case ; the acceleration of A is not due 
merely to the action of B, but to that of a multitude of other 
bodies C, D, . . . To apply the preceding rule, it is therefore 
necessary to separate the acceleration of A into many components, 
and discern which of these components is due to the action of B. 

This separation would still be possible, if we should assume 
that the action of C upon A is simply adjoined to that of B 
upon A, without the presence of the body C modifying the action 
of B upon A ; or the presence of B modifying the action of C 
upon ^ ; if we should assume, consequently, that any two bodies 
attract each other, that their mutual action is along their join 
and depends only upon their distance apart; if, in a word, we 
assume the hypothesis of central forces. 

You know that to determine the masses of the celestial bodies 
we use a wholly different principle. The law of gravitation 
teaches us that the attraction of two bodies is proportional to 
their masses ; if r is their distance apart m and m' their masses, 
k a constant, their attraction will be kmm'/r^. 

What we are measuring then is not mass, the ratio of force to 
acceleration, but the attracting mass; it is not the inertia of the 
body, but its attracting force. 

This is an indirect procedure, whose employment is not theo- 


retically indispensable. It might very well have been that attrac- 
tion was inversely proportional to the square of the distance with- 
ont being proportional to the product of the masses, that it was 
equal to //r^, but without our having / = kmm'. 

If it were so, we could nevertheless, by observation of the 
relative motions of the heavenly bodies, measure the masses of 
these bodies. 

But have we the right to admit the hypothesis of central 
forces t Is this hypothesis rigorously exact t Is it certain it 
will never be contradicted by experiment t Who would dare 
affirm that t And if we must abandon this hypothesis, the whole 
edifice so laboriously erected will crumble. 

We have no longer the right to speak of the component of 
the acceleration of A due to the action of B. We have no means 
of distinguishing it from that due to the action of C or of another 
body. The rule for the measurement of masses becomes in- 

What remains then of the principle of the equality of action 
and reaction! If the hypothesis of central forces is rejected, 
this principle should evidently be enunciated thus : the geometric 
resultant of all the forces applied to the various bodies of a 
fifystem isolated from all external action will be null. Or, in 
other words, the motion of the center of gravity of this system 
will he rectilinear and uniform. 

There it seems we have a means of defining mass ; the position 
of the center of gravity evidently depends on the values attrib- 
uted to the masses ; it will be necessary to dispose of these values 
in such a way that the motion of the center of gravity may be 
rectilinear and uniform; this will always be possible if Newton's 
third law is true, and possible in general only in a single way. 

But there exists no system isolated from all external action; 
all the parts of the universe are subject more or less to the action 
of all the other parts. The law of the motion of the center of 
gravity is rigorously true only if applied to the entire universe. 

But then, to get from it the values of the masses, it would be 
necessary to observe the motion of the center of gravity of the 
universe. The absurdity of this consequence is manifest; we 
know only relative motions ; the motion of the center of gravity 
of the universe will remain for us eternally unknown. 


Therefore nothing remains and our efforts have been fmiileas; 
we are driven to the following definition, which is only an 
avowal of powerlessness: masses a/re coefficients it is convenient 
to introduce into calculations. 

We could reconstruct all mechanics by attributing different 
values to all the masses. This new mechanics would not be in 
contradiction either with experience or with the general prin- 
ciples of dynamics (principle of inertia, proportionality of 
forces to masses and to accelerations, equality of action and 
reaction, rectilinear and uniform motion of the center of gravis, 
principle of areas). 

Only the equations of this new mechanics would be less simple. 
Let us understand clearly : it would only be the first terms which 
would be less simple, that is those experience has already made us 
acquainted with; perhaps one could alter the masses by small 
quantities without the complete equations gaining or losing in 

Hertz has raised the question whether the principles of me- 
chanics are rigorously true. **In the opinion of many physi- 
cists," he says, **it is inconceivable that the remotest experience 
should ever change anything in the immovable principles of 
mechanics; and yet, what comes from experience may always 
be rectified by experience.'' After what we have just said, these 
fears will appear groundless. 

The principles of dynamics at first appeared to us as experi- 
mental truths; but we have been obliged to use them as defini- 
tions. It is hy definition that force is equal to the product of 
mass by acceleration; here, then, is a principle which is hence- 
forth beyond the reach of any further experiment. It is in the 
same way by definition that action is equal to reaction. 

But then, it will be said, these unverifiable principles are abso- 
lutely devoid of any significance ; experiment can not contradict 
them; but they can teach us nothing useful; then what is the 
use of studying dynamics! 

This over-hasty condemnation would be unjust. There is not 
in nature any system perfectly isolated, perfectly removed from 
all external action ; but there are systems almost isolated. 

If such a system be observed, one may study not only the 


relative motion of its various parts one in reference to another, 
but also the motion of its center of gravity in reference to the 
other parts of the universe. We ascertain then that the motion 
of this center of gravity is almost rectilinear and unif orm, in 
conformity with Newton's third law. 

That is an experimental truth, but it can not be invalidated 
by experience; in fact, what would a more precise experiment 
teach ust It would teach us that the law was only almost true; 
but that we knew already. 

We can now understand how experience has been able to serve 
as basis for the principles of mechanics and yet wUl never be 
able to contradict them. 

Anthropomorphio Mechanics. — **Kirchhoff," it will be said, 
''has only acted in obedience to the general tendency of mathe- 
maticians toward nominalism ; from this his ability as a physicist 
has not saved him. He wanted a definition of force, and he 
took for it the first proposition that presented itself; but we 
need no definition of force : the idea of force is primitive, irre- 
ducible, indefinable; we know all that it is, we have a direct 
intuition of it. This direct intuition comes from the notion of 
effort, which is familiar to us from infancy." 

But first, even though this direct intuition made known to 
us the real nature of force in itself, it would be insufficient as a 
foundation for mechanics; it would besides be wholly useless. 
What is of importance is not to know what force is, but to know 
how to measure it. 

Whatever does not teach us to measure it is as useless to 
mechanics as is, for instance, the subjective notion of warmth 
and cold to the physicist who is studying heat. This subjective 
notion can not be translated into numbers, therefore it is of no 
use; a scientist whose skin was an absolutely bad conductor of 
heat and who, consequently, would never have felt either sensa- 
tions of cold or sensations of warmth, could read a thermometer 
just as well as any one else, and that would suffice him for con- 
structing the whole theory of heat. 

Now this immediate notion of effort is of no use to us for 
measuring force ; it is dear, for instance, that I should feel more 


fatigue in lifting a weight of fifty kilos than a man accnstomed 
to carry burdens. 

But more than that: this notion of effort does not teach us 
the real nature of force; it reduces itself finally to a remem- 
brance of muscular sensations, and it will hardly be maintained 
that the sun feels a muscular sensation when it draws the earth. 

All that can there be sought is a symbol, less precise and less 
convenient than the arrows the geometers use, but just as remote 
from the reality. 

Anthropomorphism has played a considerable historic role in 
the genesis of mechanics; perhaps it will still at times furnish 
a symbol which will appear convenient to some minds ; but it can 
not serve as foundation for anything of a truly scientific or 
philosophic character. 

'The School op the Thread.' — ^M. Andrade, in his Legons 
de mechanique physique, has rejuvenated anthropomorphic me- 
chanics. To the school of mechanics to which Kirchhoff belongs, 
he opposes that which he bizarrely calls the school of the thread. 

This school tries to reduce everything to ''the consideration 
of certain material systems of negligible mass, envisaged in the 
state of tension and capable of transmitting considerable efforts 
to distant bodies, systems of which the ideal type is the thread." 

A thread which transmits any force is slightly elongated under 
the action of this force; the direction of the thread tells us the 
direction of the force, whose magnitude is measured by the 
elongation of the thread. 

One may then conceive an experiment such as this. A body 
A is attached to a thread ; at the other extremity of the thread 
any force acts which varies until the thread takes an elongation 
a; the acceleration of the body A is noted; A is detached and 
the body B attached to the same thread; the same force or 
another force acts anew, and is made to vary until the thread 
takes again the elongation a ; the acceleration of the body B is 
noted. The experiment is then renewed with both A and B, 
but so that the thread takes the elongation p. The four observed 
accelerations should be proportional. We have thus an experi- 
mental verification of the law of acceleration above enunciated. 

Or still better, a body is submitted to the simultaneous action 


of several identical threads in equal tension, and by experiment 
it is sought what must be the orientations of all these threads that 
the body may remain in equilibrium. We have then an experi- 
mental verification of the law of the composition of forces. 

But, after all, what have we done! We have defined the 
force to which the thread is subjected by the deformation under- 
gone by this thread, which is reasonable enough ; we have further 
assumed that if a body is attached to this thread, the effort trans- 
mitted to it by the thread is equal to the action this body exercises 
on this thread ,* after all, we have therefore used the principle of 
the equality of action and reaction, in considering it, not as an 
experimental truth, but as the very definition of force. 

This definition is just as conventional as Kirchhoff's, but far 
less generaL 

All forces are not transmitted by threads (besides, to be able 
to compare them, they would all have to be transmitted by iden- 
tical threads). Even if it should be conceded that the earth is 
attached to the sun by some invisible thread, at least it would be 
admitted that we have no means of measuring its elongation. 

Nine times out of ten, consequently, our definition would be at 
fault ; no sort of sense could be attributed to it, and it would be 
necessary to fall back on Kirchhoff 's. 

Why then take this d6tourt You admit a certain definition 
of force which has a meaning only in certain particular cases. 
In these cases you verify by experiment that it leads to the law 
of acceleration. On the strength of this experiment, you then 
take the law of acceleration as a definition of force in all the 
other cases. 

Would it not be simpler to consider the law of acceleration as 
a definition in all cases, and to regard the experiments in ques- 
tion, not as verifications of this law, but as verifications of the 
principle of reaction, or as demonstrating that the deformations 
of an elastic body depend only on the forces to which this body is 
subjected t 

And this is without taking into account that the conditions 
under which your definition could be accepted are never fulfilled 
except imperfectly, that a thread is never without mass, that it 
is never removed from every force except the reaction of the 
bodies attached to its extremities. 


Andrade's ideas are nevertheless very interesting; if th^ 
do not satisfy our logical craving, they make us understand 
better the historic genesis of the fundamental ideas of mechanics. 
The reflections they suggest show us how the human mind has 
raised itself from a naive anthropomorphism to the present con- 
ceptions of science. 

We see at the start a very particular and in sum rather crude 
experiment ; at the finish, a law perfectly general, perfectly pre- 
cise, the certainty of which we regard as absolute. This cer- 
tainty we ourselves have bestowed upon it voluntarily, so to 
speak, by looking upon it as a convention. 

Are the law of acceleration, the rule of the composition of 
forces then only arbitrary conventions t Conventions, yes; arbi- 
trary, no ; they would be if we lost sight of the experiments which 
led the creators of the science to adopt them, and which, imper- 
fect as they may be, suffice to justify them. It is well that from 
time to time our attention is carried back to the experimental 
origin of these conventions. 

Belativb Motion and Absolute Motion 

The Principle of Relative Motion. — The attempt has some- 
times been made to attach the law of acceleration to a more 
general principle. The motion of any system must obey the 
same laws, whether it be referred to fixed axes, or to movable 
axes carried along in a rectilinear and uniform motion. This is 
the principle of relative motion, which forces itself upon us for 
two reasons: first, the commonest experience confirms it, and 
second, the contrary hypothesis is singularly repugnant to the 

Assume it then, and consider a body subjected to a force; 
the relative motion of this body, in reference to an observer 
moving with a uniform velocity equal to the initial velocity of 
the body, must be identical to what its absolute motion would be 
if it started from rest. We conclude hence that its acceleration 
can not depend upon its absolute velocity ; the attempt has even 
been made to derive from this a demonstration of the law of 

There long were traces of this demonstration in the regula- 
tions for the degree B. ^s Sc. It is evident that this attempt is 
idle. The obstacle which prevented our demonstrating the law 
of acceleration is that we had no definition of force ; this obstacle 
subsists in its entirety, since the principle invoked has not fur- 
nished us the definition we lacked. 

The principle of relative motion is none the less highly inter- 
esting and deserves study for its own sake. Let us first try to 
enunciate it in a precise manner. 

We have said above that the accelerations of the different 
bodies forming part of an isolated system depend only on their 
relative velocities and positions, and not on their absolute veloc- 
ities and positions, provided the movable axes to which the rela- 
tive motion is referred move uniformly in a straight line. Or, if 



we prefer, their acceleratioiis depend only on the differences o£ 
their velocities and the differences of their coordinates, and not 
on the absolute values of these velocities and coordinates. 

If this principle is true for relative accelerations, or rather 
for differences of acceleration, in combining it with the law of 
reaction we shall thence deduce that it is still true of absolute 

It then remains to be seen how we may demonstrate that the 
differences of the accelerations depend only on the differences 
of the velocities and of the coordinates, or, to speak in math- 
ematical language, that these differences of coordinates satisfy 
differential equations of the second order. 

Can this demonstration be deduced from experiments or from 
a priori considerations t 

Recalling what we have said above, the reader can answer for 

Thus enunciated, in fact, the principle of relative motion 
singularly resembles what I called above the generalized principle 
of inertia ; it is not altogether the same thing, since it is a ques- 
tion of the differences of coordinates and not of the coordinates 
themselves. The new principle teaches us therefore something 
more than the old, but the same discussion is applicable and 
would lead to the same conclusions; it is unnecessary to return 
to it. 

Newton's Argument. — Here we encounter a very important 
and even somewhat disconcerting question. I have said the prin- 
ciple of relative motion was for us not solely a result of experi- 
ment and that a priori every contrary hypothesis would be re- 
pugnant to the mind. 

But then, why is the principle true only if the motion of the 
movable axes is rectilinear and uniform! It seems that it ought 
to impose itself upon us with the same force, if this motion is 
varied, or at any rate if it reduces to a uniform rotation. Now, 
in these two cases, the principle is not true. I will not dwell 
long on the case where the motion of the axes is rectilinear with- 
out being uniform; the paradox does not bear a moment's exam- 
ination. If I am on board, and if the train, striking any ob- 


stade, stops suddenly, I shall be thrown against the seat in front 
of me, although I have not been directly subjected to any force. 
There is nothing mysterious in that; if I have undergone the 
action of no external force, the train itself has experienced an 
external impact. There can be nothing paradoxical in the rela- 
tive motion of two bodies being disturbed when the motion of 
one or the other is modified by an external cause. 

I will pause longer on the case of relative motions referred to 
axes which rotate uniformly. If the heavens were always 
covered with clouds, if we had no means of observing the stars, 
we nevertheless might conclude that the earth turns round; we 
could learn this from its flattening or again by the Foucault pen- 
dulum experiment. 

And yet, in this case, would it have any meaning, to say the 
earth turns round! If there is no absolute space, can one turn 
without turning in reference to something else! and, on the other 
hand, how could we admit Newton's conclusion and believe in 
absolute space t 

But it does not su£5ce to ascertain that all possible solutions 
are equally repugnant to us ; we must analyze, in each case, the 
reasons for our repugnance, so as to make our choice intelli- 
gently. The long discussion which follows will therefore be 

Let us resume our fiction: thick clouds hide the stars from 
men, who can not observe them and are ignorant even of their 
existence; how shall these men know the earth turns round! 

Even more than our ancestors, no doubt, they will regard the 
ground which bears them as fixed and immovable; they will 
await much longer the advent of a Copernicus. But in the end 
the Copernicus would come — ^how! 

The students of mechanics in this world would not at first be 
confronted with an absolute contradiction. In the theory of 
relative motion, besides real forces, two fictitious forces are met 
which are called ordinary and compound centrifugal force. Our 
imaginary scientists could therefore explain everything by re- 
garding these two forces as real, and they would not see therein 
any contradiction of the generalized principle of inertia, for 
these forces would depend, the one on the relative positions of 


the various parts of the system, as real attractions do, the other 
on their relative velocities, as real frictions do. 

Many difSculties, however, would soon awaken their attention; 
if they succeeded in realizing an isolated system, the center of 
gravity of this system would not have an almost rectilinear path. 
They would invoke, to explain this fact, the centrifugal forces 
which they would regard as real, and which they would attribute 
no doubt to the mutual actions of the bodies. Only they would 
not see these forces become null at great distances, that is to say 
in proportion as the isolation was better realized; far from it; 
centrifugal force increases indefinitely with the distance. 

This difSculty would seem to them already sufSciently great; 
and yet it would not stop them long ; they would soon imagine 
some very subtile medium, analogous to our ether, in which all 
bodies would be immersed and which would exert a repellent 
action upon them. 

But this is not all. Space is symmetric, and yet the laws of 
motion would not show any synunetry; they would have to dis- 
tinguish between right and left. It would be seen for instance 
that cyclones turn always in the same sense, whereas by reason 
of symmetry these winds should turn indifferently in one sense 
and in the other. If our scientists by their labor had succeeded 
in rendering their universe perfectly symmetric, this symmetry 
would not remain, even though there was no apparent reason 
why it should be disturbed in one sense rather than in the other. 

They would get themselves out of the diflBculty doubtless, they 
would invent something which would be no more extraordinary 
than the glass spheres of Ptolemy, and so it would go on, com- 
plications accumulating, until the long-expected Copernicus 
sweeps them all away at a single stroke, saying: It is much 
simpler to assume the earth turns round. 

And just as our Copernicus said to us : It is more convenient 
to suppose the earth turns round, since thus the laws of astron- 
omy are expressible in a much simpler language ; this one would 
say: It is more convenient to suppose the earth turns round, 
since thus the laws of mechanics are expressible in a much 
simpler language. 

This does not preclude maintaining that absolute space, that 


is to say the mark to which it would be necessary to refer the 
earth to know whether it really moves, has no objective existence. 
Hence, this afSnration; 'the earth turns round' has no mean- 
ing, since it can be verified by no experiment; since such an 
experiment, not only could not be either realized or dreamed by 
the boldest Jules Verne, but can not be conceived of without con- 
tradiction; or rather these two propositions; 'the earth turns 
round,' and, 'it is more convenient to suppose the earth turns 
round' have the same meaning; there is nothing more in the one 
than in the other. 

Perhaps one will not be content even with that, and will find 
it already shocking that among all the hypotheses, or rather 
all the conventions we can make on this subject, there is one more 
convenient than the others. 

But if it has been admitted without difficulty when it was a 
question of the laws of astronomy, why should it be shocking in 
that which concerns mechanics ? 

We have seen that the coordinates of bodies are determined 
by differential equations of the second order, and that so are the 
differences of tJiese coordinates. This is what we have called 
the generalized principle of inertia and the principle of relative 
motion. If the distances of these bodies were determined like- 
wise by equations of the second order, it seems that the mind 
ought to be entirely satisfied. In what ineasure does the mind 
get this satisfaction and why is it not content with it! 

To account for this, we had better take a simple example. 
I suppose a sj'stem analogous to our solar systv^m, but where one 
can not perceive fixed stars foreign to this system, so that astron- 
omers can observe only the mutual distances of the planets and 
the sun, and not the absolute longitudes of the planets. If we 
deduce directly from Newton's law the differential equations 
which define the variation o£ these distances, these equations will 
not be of the second order. I mean that if. besides Newton's law, 
one knew the initial values of these distances and of their de- 
rivatives with respect to tlie time, that would not suffice to deter- 
mine the valnes of these same distances at a subsequent instant. 
There would still be lacking one datum, and this datum might be 
ior instance what astronomers call the area-constant. 


But here two different points of view may be taken ; we may 
distinguish two sorts of constants. To the eyes of the physicist 
the world reduces to a series of phenomena, depending, on the 
one handy solely upon the initial phenomena ; on the other hand, 
upon the laws which bind the consequents to the antecedents. 
If then observation teaches us that a certain quantity is a con- 
stant, we shall have the choice between two conceptions. 

Either we shall assume that there is a law requiring this 
quantity not to vary, but that by chance, at the beginning of 
the ages, it had, rather than another, this value it has been 
forced to keep ever since. This quantity might then be called 
an accidental constant. 

Or else we shall assume, on the contrary, that there is a law 
of nature which imposes upon this quantity such a value and 
not such another. 

We shall then have what we may call an essential constant. 

For example, in virtue of Newton's laws, the duration of the 
revolution of the earth must be constant. But if it is 366 
sidereal days and something over, and not 300 or 400, this is in 
consequence of I know not what initial chance. This is an 
accidental constant. If, on the contrary, the exponent of the 
distance which figures in the expression of the attractive force is 
equal to — 2 and not to — 3, this is not by chance, but because 
Newton's law requires it. This is an essential constant. 

I know not whether this way of giving chance its part is 
legitimate in itself, and whether this distinction is not somewhat 
artificial ; it is certain at least that, so long as nature shall have 
secrets, this distinction will be in application extremely arbitrary 
and always precarious. 

As to the area-constant, we are accustomed to regard it as 
accidental. Is it certain our imaginary astronomers would do 
the same? If they could have compared two different solar 
systems, they would have the idea that this constant may have 
several different values; but my very supposition in the begin- 
ning was that their system should appear as isolated, and that 
they should observe no star foreign to it. Under these condi- 
tions, they would see only one single constant which would have 
a single value absolutely invariable; they would be led without 
any doubt to regard it as an essential constant. 


A word in passing to forestall an objection: the inhabitants 
of this imaginary world could neither observe nor define the 
area-constant as we do, since the absolute longitudes escape them; 
that would not preclude their being quickly led to notice a cer- 
tain constant which would introduce itself naturally into their 
equations and which would be nothing but what we call the area- 

But then see what would happen. If the area-constant is 
regarded as essential, as depending upon a law of nature, to cal- 
culate the distances of the planets at any instant it will suffice 
to know the initial values of these distances and those of their 
first derivatives. From this new point of view, the distances will 
be determined by differential equations of the second order. 

Yet would the mind of these astronomers be completely satis- 
fied T I do not believe so; first, they would soon perceive that 
in differentiating their equations and thus raising their order, 
these equations became much simpler. And above all they would 
be struck by the difficulty which comes from symmetry. It 
would be necessary to assume different laws, according as the 
aggregate of the planets presented the figure of a certain polyhe- 
dron or of the symmetric polyhedron, and one would escape from 
this consequence only by regarding the area-constant as acci- 

I have taken a very special example, since I have supposed 
astronomers who did not at all consider terrestrial mechanics, 
and whose view was limited to the solar system. Our universe is 
more extended than theirs, as we have fixed stars, but still it too 
is limited, and so we might reason on the totality of our universe 
as the astronomers on their solar system. 

Thus we see that finally we should be led to conclude that the 
equations which define distances are of an order superior to the 
second. Why should we be shocked at that, why do we find it 
perfectly natural for the series of phenomena to depend upon 
the initial values of the first derivatives of these distances, while 
we hesitate to admit that they may depend on the initial values of 
the second derivatives? This can only be because of the habits 
of mind created in us by the constant study of the generalized 
principle of inertia and its consequences. 


<tf dte 



«f t_ I II law II BAcr it 






c -^ ad -s! 

ka?v aacy^hfi^ cf tkx bodr bat ill 

vf OCT 

ibsj ixn errai ns or vocid ^t« cirin ^s fiaiaeil y, this 

Energy and Thermodynamics 

Energetics. — The difficulties inherent in the classic mechan- 
ics have led certain minds to prefer a new system they call 

Energetics took its rise as an outcome of the discovery of the 
principle of the conservation of energy. Helmholtz gave it its 
final form. 

It begins by defining two quantities which play the funda- 
mental role in this theory. They are kinetic energy, or vis viva, 
and potential energy. 

All the changes which bodies in nature can undergo are regu- 
lated by two experimental laws: 

1^ The sum of kinetic energy and potential energy is con- 
stant. This is the principle of the conservation of energy. 

2° If a system of bodies is at A at the time ^o ^^^ &t B at 
the time t^, it always goes from the first situation to the second 
in such a way that the mean value of the difference between the 
two sorts of energy, in the interval of time which separates the 
two epochs ^0 <^d ^1) ^^7 ^^ ^ small as possible. 

This is Hamilton's principle, which is one of the forms of the 
principle of least action. 

The energetic theory has the following advantages over the 
classic theory: 

1° It is less incomplete; that is to say, Hamilton's principle 
and that of the conservation of energy teach us more than the 
fundamental principles of the classic theory, and exclude certain 
motions not realized in nature and which would be compatible 
with the classic theory : 

2° It saves us the hypothesis of atoms, which it was almost 
impossible to avoid with the classic theory. 

But it raises in its turn new difficulties : 

The definitions of the two sorts of energy would raise diffi- 
culties almost as great as those of force and mass in the first 



STstem. Yet they may be gotten over more easily, at least in 
the simplest cases. 

Suppose an isolated system formed of a certain number of 
material points; suppose these points subjected to forces depend- 
ing only on their relative position and their mutual distances, 
and independent of their velocities. In virtue of the principle 
of the conservation of energy, a function of forces must exist. 

In this simple case the enunciation of the principle of the 
conservation of energy is of extreme simplicity. A certain quan- 
tity, accessible to experiment, must remain constant. This quan- 
tity is the sum of two terms ; the first depends only on the posi- 
tion of the material points and is independent of their velocities; 
the second is proportional to the square of these velocities. This 
resolution can take place only in a single way. 

The first of these terms, which I shall call U, will be the 
potential energy; the second, which I shall call T, will be the 
kinetic energy. 

It is true that ii T+U ib a constant, so is any function of 

</>(T+ U), 

But this function <I>{T^U) will not be the sum of two terms the 
one independent of the velocities, the other proportional to the 
square of these velocities. Among the functions which remain 
constant there is only one which enjoys this property, that is 
T+ 17 (or a linear function ot T-^-Uy which comes to the same 
thing, since this linear function may always be reduced to T -j- 17 
by change of unit and of origin). This then is what we shall 
call energy ; the first term we shall call potential energy and the 
second kinetic energy. The definition of the two sorts of energy 
can therefore be carried through without any ambiguity. 

It is the same with the definition of the masses. Kinetic 
energy, or vis viva, is expressed very simply by the aid of the 
masses and the relative velocities of all the material points with 
reference to one of them. These relative velocities are accessible 
to observation, and, when we know the expression of the kinetic 
energy as function of these relative velocities, the coefScients of 
this expression will give us the masses. 


Thus, in this simple case, the fundamental ideas may be de- 
fined without difSculty. But the difficulties reappear in the 
more complicated cases and, for instance, if the forces, in lieu 
of depending only on the distances, depend also on the velocities. 
For example, Weber supposes the mutual action of two electric 
molecules to depend not only on their distance, but on their veloc- 
ity and their acceleration. If material points should attract each 
other according to an analogous law, V would depend on the 
velocity, and might contain a term proportional to the square of 
the velocity. 

Among the terms proportional to the squares of the velocities^ 
how distinguish those which come from T or from 17 T Conse- 
quently, how distinguish the two parts of energy T 

But still more; how define energy itself T We no longer have 
any reason to take as definition T^TJ rather than any other 
function of T + U, when the property which characterized T + Z7 
has disappeared, that, namely, of being the sum of two terms of 
a particular form. 

But this is not all; it is necessary to take account, not only 
of mechanical energy properly so called, but of the other forms 
of energy, heat, chemical energy, electric energy, etc. The prin- 
ciple of the conservation of energy should be written : 

r-f i7 + o=coii8t. 

where T would represent the sensible kinetic energy, TJ the poten- 
tial energy of position, depending only on the position of the 
bodies, Q the internal molecular energy, under the thermal, 
chemic or electric form. 

All would go well if these three terms were absolutely distinct, 
if T were proportional to the square of the velocities, TJ inde- 
pendent of these velocities and of the state of the bodies, Q inde- 
pendent of the velocities and of the positions of the bodies and 
dependent only on their internal state. 

The expression for the energy could be resolved only in one 
single way into three terms of this form. 

But this is not the case ; consider electrified bodies ; the electro- 
static energy due to their mutual action will evidently depend 
upon their charge, that is to say, on their state ; but it will equally 


depend upon their position. If these bodies are in motion, they 
will act one upon another eleetrodynamically and the electro- 
dynamic energy will depend not only upon their state and their 
position, but upon their velocities. 

We therefore no longer have any means of making the sepa- 
ration of the terms which should make part of T, of U and of Q, 
and of separating the three parts of energy. 

If (T+U + Q) is constant so is any function <^(T + 17 -fQ). 

If T + J7 + P were of the particular form I have above 
considered, no ambiguity would result; among the functions 
^(T + J7 + Q) which remain constant, there would only be one 
of this particular form, and that I should convene to call energy. 

But as I have said, this is not rigorously the case; among 
the functions which remain constant, there is none which can 
be put rigorously under this particular form ; hence, how choose 
among them the one which should be called energy T We no 
longer have anything to guide us in our choice. 

There only remains for us one enunciation of the principle of 
the conservation of enei^: There is something which remains 
constant. Under this form it is in its turn out of the reach of 
experiment and reduces to a sort of tautology. It is clear that if 
the world is governed by laws, there will be quantities which will 
remain constant. Like Newton's laws, and, for an analogous 
reason, the principle of the conservation of energy, founded on 
experiment, could no longer be invalidated by it. 

This discussion shows that in passing from the classic to the 
energetic system progress has been made ; but at the same time 
it shows this progress is insufScient. 

Another objection seems to me still more grave: the prin- 
ciple of least action is applicable to reversible phenomena; but it 
is not at all satisfactory in so far as irreversible phenomena are 
concerned ; the attempt by Helmholtz to extend it to this kind of 
phenomena did not succeed and could not succeed ; in this regard 
everything remains to be done. The very statement of the prin- 
ciple of least action has something about it repugnant to the mind. 
To go from one point to another, a material molecule, acted upon 
by no force, but required to move on a surface, will take the 
geodesic line, that is to say, the shortest path. 


This molecule seems to know the point whither it is to go, to 
foresee the time it would take to reach it by such and such 
a route, and then to choose the most suitable path. The state- 
ment presents the molecule to us, so to speak, as a living and 
free being. Clearly it would be better to replace it by an enun- 
ciation less objectionable, and where, as the philosophers would 
say, final causes would not seem to be substituted for efficient 

Thermodynamics.^ — The role of the two fundamental prin- 
ciples of thermodynamics in all branches of natural philosophy 
becomes daily more important. Abandoning the ambitious the- 
ories of forty years ago, which were encumbered by molecular 
hypotheses, we are trying to-day to erect upon thermodynamics 
alone the entire edifice of mathematical physics. Will the two 
principles of Mayer and of Clausius assure to it foundations 
solid enough for it to last some timet No one doubts it; but 
whence comes this confidence T 

An eminent physicist said to me one day d propos of the law 
of errors: "All the world believes it firmly, because the mathe- 
maticians imagine that it is a fact of observation, and the ob- 
servers that it is a theorem of mathematics." It was long so for 
the principle of the conservation of energy. It is no longer so 
to-day ; no one is ignorant that this is an experimental fact. 

But then what gives us the right to attribute to the principle 
itself more generality and more precision than to the experiments 
which have served to demonstrate it? This is to ask whether 
it is legitimate, as is done every day, to generalize empirical 
data, and I shall not have the presumption to discuss this ques- 
tion, after so many philosophers have vainly striven to solve 
it. One thing is certain; if this power were denied us, science 
could not exist or, at least, reduced to a sort of inventory, to 
the ascertaining of isolated facts, it would have no value for us, 
since it could give no satisfaction to our craving for order and 
harmony and since it would be at the same time incapable of 
foreseeing. As the circumstances which have preceded any fact 
will probably never be simultaneously reproduced, a first general- 

1 The following lines are a partial reproduction of the preface of mj 
book Thermodynamigue, 


ization is already necessary to foresee whether this fact will be 
reproduced again after the least of these circumstances shall 
be changed. 

But every proposition may be generalized in an infinity of 
ways. Among all the generalizations possible, we must choose, 
and we can only choose the simplest. We are therefore led to act 
as if a simple law were, other things being equal, more probable 
than a complicated law. 

Half a century ago this was frankly confessed, and it was 
proclaimed that nature loves simplicity; she has since too often 
given us the lie. To-day we no longer confess this tendency, 
and we retain only so much of it as is indispensable if science 
is not to become impossible. 

In formulating a general, simple and precise law on the basis 
of experiments relatively few and presenting certain divergences, 
we have therefore only obeyed a necessity from which the human 
mind can not free itself. 

But there is something more, and this is why I dwell upon 
the point. 

No one doubts that Mayer's principle is destined to survive 
all the particular laws from which it was obtained, just as New- 
ton's law has survived Kepler's laws, from which it sprang, 
and which are only approximative if account be taken of 

Why does this principle occupy thus a sort of privileged place 
among all the physical lawsT There are many little reasons 
for it. 

First of all it is believed that we could not reject it or even 
doubt its absolute rigor without admitting the possibility of per- 
petual motion ; of course we are on our guard at such a prospect, 
and we think ourselves less rash in afiSrming Mayer's principle 
than in denying it. 

That is perhaps not wholly accurate ; the impossibility of per- 
petual motion implies the conservation of energy only for re- 
versible phenomena. 

The imposing simplicity of Mayer's principle likewise con- 
tributes to strengthen our faith. In a law deduced immediately 
from experiment, like Mariotte's, this simplicity would rather 


seem to us a reason for distrust; but here this is no longer the 
case; we see elements, at first sight disparate, arrange them- 
selves in an unexpected order and form a harmonious whole ; and 
we refuse to believe that an unforeseen harmony may be a 
simple effect of chance. It seems that our conquest is the dearer 
to us the more effort it has cost us, or that we are the surer of 
having wrested her true secret from nature the more jealously 
she has hidden it from us. 

But those are only little reasons; to establish Mayer's law as 
an absolute principle, a more profound discussion is necessary. 
But if this be attempted, it is seen that this absolute principle is 
not even easy to state. 

In each particular case it is clearly seen what energy is and at 
least a provisional definition of it can be given; but it is im- 
jKMSsible to find a general definition for it. 

If we try to enunciate the principle in all its generality and 
apply it to the universe, we see it vanish, so to speak, and nothing 
is left but this : There is something which remains constant. 

But has even this any meaning T In the determinist hypoth- 
esis, the state of the universe is determined by an extremely great 
number n of parameters which I shall call x^, X2, > . . Xn. As 
soon as the values of these n parameters at any instant are 
known, their derivatives with respect to the time are likewise 
known and consequently the values of these same parameters at 
a preceding or subsequent instant can be calculated. In other 
words, these n parameters satisfy n differential equations of the 
first order. 

These equations admit of n — 1 integrals and consequently 
there are n — 1 functions of x^, Xz, ... Xn, which remain 
constant. If then we say there is something which remains 
constant, we only utter a tautology. We should even be puzzled 
to say which among all our integrals should retain the name of 

Besides, Mayer's principle is not understood in this sense 
when it is applied to a limited system. It is then assumed that 
p of our parameters vary independently, so that we only have 
n — p relations, generally linear, between our n parameters and 
their derivatives. 


To simplify the enunciation, suppose that the sum of the 
work of the external forces is null, as well as that of the quan- 
tities of heat given off to the outside. Then the signification 
of our principle will be: 

There is a combination of these n — p relations whose first 
member is an exact differential; and then this differential vanish- 
ing in virtue of our n — p relations, its integral is a constant 
and this integral is called energy. 

But how can it be possible that there are several parameters 
whose variations are independent T That can only happen under 
the influence of external forces (although we have supposed, for 
simplicity, that the algebraic sum of the effects of these forces 
is null). In fact, if the system were completely isolated from 
all external action, the values of our n parameters at a given 
instant would suffice to determine the state of the system at any 
subsequent instant, provided always we retain the determinist 
hypothesis; we come back therefore to the same difficulty as 

If the future state of the system is not entirely determined by 
its present state, this is because it depends besides upon the 
state of bodies external to the system. But then is it probable 
that there exist between the parameters x, which define the state 
of the system, equations independent of this state of the external 
bodies ? and if in certain cases we believe we can find such, is this 
not solely in consequence of our ignorance and because the influ- 
ence of these bodies is too slight for our experimenting to 
detect it? 

If the system is not regarded as completely isolated, it is 
probable that the rigorously exact expression of its internal 
energy will depend on the state of the external bodies. Again, 
I have above supposed the sum of the external work was null, 
and if we try to free ourselves from this rather artificial restric- 
tion, the enunciation becomes still more difficult. 

To formulate Mayer's principle in an absolute sense, it is 
therefore necessary to extend it to the whole universe, and then 
we find ourselves face to face with the very difficulty we sought 
to avoid. 

In conclusion, using ordinary language, the law of the con- 


senration of energy can have only one signification, which is 
that there is a property conunon to all the possibilities; but on 
the determinist hypothesis there is only a single possibility, and 
then the law has no longer any meaning. 

On the indeterminist hypothesis, on the contrary, it would 
have a meaning, eveh if it were taken in an absolute sense; it 
would appear as a limitation imposed upon freedom. 

But this word reminds me that I am digressing and am on 
the point of leaving the domain of mathematics and physics. I 
check myself therefore and will stress of all this discussion only 
one impression, that Mayer's law is a form flexible enough for 
us to put into it almost whatever we wish. By that I do not mean 
it corresponds to no objective reality, nor that it reduces itself 
to a mere tautology, since, in each particular case, and provided 
one does not try to push to the absolute, it has a perfectly clear 

This flexibility is a reason for believing in its permanence, 
and as, on the other hand, it will disappear only to lose itself 
in a higher harmony, we may work with confidence, supporting 
ourselves upon it, certain beforehand that our labor will not be 

Almost everything I have just said applies to the principle 
of Clausius. What distinguishes it is that it is expressed by 
an inequality. Perhaps it will be said it is the same with all 
physical laws, since their precision is always limited by errors 
of observation. But they at least claim to be first approxima- 
tions, and it is hoped to replace them little by little by laws more 
and more precise. If, on the other hand, the principle of Clau- 
sius reduces to an inequality, this is not caused by the imper- 
fection of our means of observation, but by the very nature of 
the question. 

General Conclusions on Pabt Thibd 

The principles of mechanics, then, present themselves to us 
under two different aspects. On the one hand, they are truths 
founded on experiment and approximately verified so far as 
eoncems almost isolated systems. On the other hand, they are 


postulates applicable to the totality of the universe and reg 
as rigorously true. 

If these postulates possess a generality and a certainty 
are lacking to the experimental verities whence they are d 
this is because they reduce in the last analysis to a mer 
vention which we have the right to make, because we are c 
beforehand that no experiment can ever contradict it. 

This convention, however, is not absolutely arbitrary; i 
not spring from our caprice ; we adopt it because certain e 
ments have shown us that it would be convenient. 

Thus is explained how experiment could make the prio 
of mechanics, and yet why it can not overturn them. 

Compare with geometry: The fundamental propositi^ 
geometry, as for instance Euclid's postulate, are nothing 
than conventions, and it is just as unreasonable to ii 
whether they are true or false as to ask whether the metri 
tem is true or false. 

Only, these conventions are convenient, and it is certain e 
ments which have taught us that. 

At first blush, the analogy is complete; the role of e 
ment seems the same. One will therefore be tempted tc 
Either mechanics must be regarded as an experimental sc 
and then the same must hold for geometry ; or else, on th 
trary, geometry is a deductive science, and then one may \ 
much of mechanics. 

Such a conclusion would be illegitimate. The experi 
which have led us to adopt as more convenient the fundan 
conventions of geometry bear on objects which have nothi 
common with those geometry studies ; they bear on the prop 
of solid bodies, on the rectilinear propagation of light, 
are experiments of mechanics, experiments of optics; the 
not in any way be regarded as experiments of geometry, 
even the principal reason why our geometry seems conv< 
to us is that the different parts of our body, our eye, our \ 
have the properties of solid bodies. On this account, our f 
mental experiments are preeminently physiological experii 
which bear, not on space which is the object the geometer 


stady, but on his body, that is to say, on the instrument he must 
use for this study. 

On the contrary, the fundamental conventions of mechanics, 
and the experiments which prove to us that they are convenient, 
bear on exactly the same objects or on analogous objects. The 
conventional and general principles are the natural and direct 
generalization of the experimental and particular principles. 

Let it not be said that thus I trace artificial frontiers between 
the sciences ; that if I separate by a barrier geometry properly 
so called from the study of solid bodies, I could just as well erect 
one between experimental mechanics and the conventional me> 
chanics of the general principles. In fact, who does not see that 
in separating these two sciences I mutilate them both, and that 
what will remain of conventional mechanics when it shall be 
isolated will be only a very small thing and can in no way be com- 
pared to that superb body of doctrine called geometry! 

One sees now why the teaching of mechanics should remain 

Only thus can it make us comprehend the genesis of the science, 
and that is indispensable for the complete understanding of the 
science itself. 

Besides, if we study mechanics, it is to apply it; and we can 
apply it only if it remains objective. Now, as we have seen, what 
the principles gain in generality and certainty they lose in objec- 
tivity. It is, therefore, above all with the objective side of the 
principles that we must be familiarized early, and that can be 
done only by going from the particular to the general, instead of 
the inverse. 

The principles are conventions and disguised definitions. Yet 
they are drawn from experimental laws; these laws have, so 
to speak, been exalted into principles to which our mind attri- 
butes an absolute value. 

Some philosophers have generalized too far; they believed the 
principles were the whole science and consequently that the whole 
adence was conventional. 

This paradoxical doctrine, called nominalism, will not bear 


How can a law become a principle T It expressed a relation 
between two real terms A and B. But it was not rigorooaly tme, 
it was only approximate. We introduce arbitrarily an inter- 
mediary term C more or less fictitious, and C is by definition that 
which has with A exactly the relation expressed by the law. 

Then our law is separated into an absolute and rigorous prin- 
ciple which expresses the relation of A to C and an experimental 
law, approximate and subject to revision, which expresses the 
relation ot C io B. It is clear that, however far this partition is 
pushed, some laws will always be left remaining. 

We go to enter now the domain of laws properly so called. 



Hypotheses in Physics 

The RdiiE of Experiment and Qenebauzation. — Experiment 
is the sole source of truth. It alone can teach us anything new ; 
it alone can give us certainty. These are two points that can not 
be questioned. 

But then, if experiment is everything, what place will remain 
for mathematical physics T What has experimental physics to do 
with such an aid, one which seems useless and perhaps even 
dangerous T 

And yet mathematical physics exists, and has done unquestion- 
able service. We have here a fact that must be explained. 

The explanation is that merely to observe is not enough. We 
must use our observations, and to do that we must generalize. 
This is what men always have done ; only as the memory of past 
errors has made them more and more careful, they have observed 
more and more, and generalized less and less. 

Every age has ridiculed the one before it, and accused it of 
having generalized too quickly and too naively. Descartes pitied 
the lonians; Descartes, in his turn, makes us smile. No doubt 
our children will some day laugh at us. 

But can we not then pass over immediately to the goal? Is 
not this the means of escaping the ridicule that we foresee 1 Can 
we not be content with just the bare experiment? 

No, that is impossible; it would be to mistake utterly the 
tme nature of science. The scientist must set in order. Science 
is built up with facts, as a house is with stones. But a collection 
of facts is no more a science than a heap of stones is a house. 



And above all the scientist most foresee. Garlyle has some- 
where said something like this: '^ Nothing but facts are of im- 
portance. John Lackland passed by here. Here is something 
that is admirable. Here is a reality for which I wonld give all 
the theories in the world." Garlyle was a fellow countryman of 
Bacon ; but Bacon would not have said that. That is the language 
of the historian. The physicist would say rather: *'John Lack- 
land passed by here; that makes no difference to me, for he 
never will pass this way again." 

We all know that there are good experiments and poor ones. 
The latter will accumulate in vain ; though one may have made a 
hundred or a thousand, a single piece of work by a true master, 
by a Pasteur, for example, will sufSce to tumble them into oblivion. 
Bacon would have well understood this ; it is he who invented the 
phrase Experimentum crucis. But Garlyle would not have under- 
stood it. A fact is a fact. A pupil has read a certain number on 
his thermometer; he has taken no precaution; no matter, he has 
read it, and if it is only the fact that counts, here is a reality of 
the same rank as the peregrinations of King John Lackland. Why 
is the fact that this pupil has made this reading of no interest, 
while the fact that a skilled physicist had made another reading 
might be on the contrary very important! It is because from the 
first reading we could not infer anything. What then is a good 
experiment? It is that which informs us of something besides 
an isolated fact ; it is that which enables us to foresee, that is, that 
which enables us to generalize. 

For without generalization foreknowledge is impossible. The 
circumstances under which one has worked will never reproduce 
themselves all at once. The observed action then will never recur ; 
the only thing that can be afSrmed is that under analogous cir- 
cumstances an analogous action will be produced. In order to 
foresee, then, it is necessary to invoke at least analogy, that is to 
say, already then to generalize. 

No matter how timid one may be, still it is necessary to inter- 
polate. Experiment gives us only a certain number of isolated 
I>oints. We must unite these by a continuous line. This is a 
veritable generalization. But we do more ; the curve that we shall 
trace will pass between the observed points and near these points ; 


it will not pass through these points themselves. Thus one does 
ZK>t restrict himself to generalizing the experiments, but corrects 
them ; and the physicist who should try to abstain from these cor- 
rections and really be content with the bare experiment, would be 
forced to enunciate some very strange laws. 

The bare facts, then, would not be enough for us; and that is 
why we must have science ordered, or rather organized. 

It is often said experiments must be made without a pre- 
conceived idea. That is impossible. Not only would it make 
all experiment barren, but that would be attempted which could 
not be done. Every one carries in his mind his own conception 
of the world, of which he can not so easily rid himself. We must, 
for instance, use language ; and our language is made up only of 
preconceived ideas and can not be otherwise. Only these are 
unconscious preconceived ideas, a thousand times more dangerous 
than the others. 

Shall we say that if we introduce others, of which we are 
fully conscious, we shall only aggravate the evil? I think not. 
I believe rather that they will serve as counterbalances to each 
other — I was going to say as antidotes ; they will in general accord 
ill with one another — they will come into conflict with one another, 
and thereby force us to regard things under different aspects. 
This is enough to emancipate us. He is no longer a slave who 
can choose his master. 

Thus, thanks to generalization, each fact observed enables us 
to foresee a great many others ; only we must not forget that the 
first alone is certain, that all others are merely probable. No 
matter how solidly founded a prediction may appear to us, we are 
never absolutely sure that experiment will not contradict it, if 
we undertake to verify it. The probability, however, is often so 
great that practically we may be content with it. It is far better 
to foresee even without certainty than not to foresee at all. 

One must, then, never disdain to make a verification when 
opportunity offers. But all experiment is long and difficult ; the 
workers are few ; and the number of facts that we need to foresee 
is immense. Compared with this mass the number of direct verifi- 
cations that we can make will never be anything but a negligible 


Of this few tliat we can directly attain, we mnst make the best 
nse ; it is very necessary to get from every experiment the greatest 
IKMSsible number of predictions, and with the highest possible 
degree of probability. The problem is, so to sp^ik, to increase 
the yield of the scientific machine. 

Let XLS compare science to a library that ought to grow continu- 


ally. The librarian has at his disposal for his purchases only 
insufficient funds. He ought to make an effort not to waste them. 

It is experimental physics that is entrusted with the purchases. 
It alone, then, can enrich the library. 

As for mathematical physics, its task will be to make out the 
catalogue. If the catalogue is well made, the library will not be 
any richer, but the reader will be helped to use its riches. 

And even by showing the librarian the gaps in his collections, 
it will enable him to make a judicious use of his funds ; which is all 
the more important because these funds are entirely inadequate. 

Such, then, is the role of mathematical physics. It must direct 
generalization in such a manner as to increase what I just now 
called the yield of science. By what means it can arrive at this, 
and how it can do it without danger, is what remains for us to 

Thb Unity op Nature. — ^Let us notice, first of all, that every 
generalization implies in some measure the belief in the unity 
and simplicity of nature. As to the unity there can be no diffi- 
culty. If the different parts of the universe were not like the 
members of one body, they would not act on one another, they 
would know nothing of one another ; and we in particular would 
know only one of these parts. We do not ask, then, if nature is 
one, but how it is one. 

As for the second point, that is not such an easy matter. It is 
not certain that nature is simple. Can we without danger act 
as if it were ! 

There was a time when the simplicity of Mariotte's law was 
an argument invoked in favor of its accuracy ; when Fresnel him- 
self, after having said in a conversation with Laplace that nature 
was not concerned about analytical difficulties, felt himself 
obliged to make explanations, in order not to strike too hard 
at prevailing opinion. 


To-day ideas have greatly changed ; and yet, those who do not 
believe that natural laws have to be simple, are still often obliged 
to act as if they did. They could not entirely avoid this neceesity 
without making impossible all generalization, and consequently 
all science. 

It is clear that any fact can be generalized in an infinity of 
■ways, and it is a question of choice. The choipe can be guided 
only by considerations of amplieity. Let us take the most com- 
monplace case, that of interpolation. We pass a continuous line, 
AS regular as possible, between the points given by observation. 
Why do we avoid points making angles and too abrupt turns T 
Why do we not make our curve describe the most capricious zig- 
aagsl It is because we know beforehand, or believe we know, that 
the law to be expressed can not be so complicated as all that. 

We may calculate the mass of Jupiter from either the move- 
ments of its satellites, or the perturbations of the major planets, 
or those of the minor planets. If we take the averages of the 
determinations obtained by these three methods, we find three 
numbers very close together, but different. We might interpret 
this result by supposing that the coefficient of gravitation is not 
the same in the tliree eases. The observations would certainly be 
maeh better represented. Why do we reject this interpretation t 
Not because it is absurd, but because it is needlessly complicated. 
We shall only accept it when we are forced to, and that is not yet. 

To sum up, ordinarily every law is held to be simple till the 
contrary is proved. 

This custom is imposed upon physicists by the causes that I 
have just explained. But how shall we justify it in the presence 
of discoveries that show us every day new details that are richer 
and more complex V How shall we even reconcile it with the 
belief in the unity of nature! For if everything depends on 
rveiything, relationships where flo many diverse factors enter can 
DO longer be smple. 

If we study the history of science, we see happen two invetse 
phenomena, so to speak. Sometimes simplicity hides under com- 
plex appearances ; sometimes it is the simplicity which is appar- 
ent, and which disguises extremely complicated realities. 

What is more complicated than the confused movements of 



the planets T What simpler than Newton's lawT Here nature, 
making sport, as Fresnel said, of analytical difficulties, employs 
only simple means, and by combining them produces I know not 
what inextricable tangle. Here it is the hidden simplicity which 
must be discovered. 

Examples of the opposite abound. In the kinetic theory of 
gases, one deals with molecules moving with great velocities, 
whose paths, altered by incessant collisions, have the most capri- 
cious forms and traverse space in every direction. The observable 
result is Mariotte's simple law. Every individual fact was com- 
plicated. The law of great numbers has reestablished simplicity 
in the average. Here the simplicity is merely apparent, and only 
the coarseness of our senses prevents our perceiving the complexity. 

Many phenomena obey a law of proportionality. But why! 
Because in these phenomena there is something very small. The 
simple law observed, then, is only a result of the general ana- 
lytical rule that the infinitely small increment of a function is 
proportional to the increment of the variable. As in reality our 
increments are not infinitely small, but very small, the law of 
proportionality is only approximate, and the simplicity is only 
apparent. What I have just said applies to the rule of the super- 
position of small motions, the use of which is so fruitful, and 
which is the basis of optics. 

And Newton's law itself? Its simplicity, so long undetected, 
is perhaps only apparent. Who knows whether it is not due to 
some complicated mechanism, to the impact of some subtile matter 
animated by irregular movements, and whether it has not become 
simple only through the action of averages and of great num- 
bers? In any ease, it is difficult not to suppose that the true law 
contains complementary terms, which would become sensible at 
small distances. If in astronomy they are negligible as modify- 
ing Newton's law, and if the law thus regains its simplicity, it 
would be only because of the immensity of celestial distances. 

No doubt, if our means of investigation should become more 
and more penetrating, we should discover the simple under the 
complex, then the complex under the simple, then again the simple 
under the complex, and so on, without our being able to foresee 
what wiU be the last term. 


We must stop somewhere, and that science may be possible, we 
must stop when we have found simplicity. This is the only ground 
on which we can rear the edifice of our generalizations. But 
this simplicity being only apparent, will the ground be firm 
enough? This is what must be investigated. 

For that purpose, let us see what part is played in our gener- 
alizations by the belief in simplicity. We have verified a simple 
law in a good many particular cases ; we refuse to admit that this 
agreement, so often repeated, is simply the result of chance, and 
conclude that the law must be true in the general case. 

Kepler notices that a planet's positions, as observed by Tycho, 
are all on one ellipse. Never for a moment does he have the 
thought that by a strange play of chance Tycho never observed 
the heavens except at a moment when the real orbit of the planet 
happened to cut this ellipse. 

What does it matter then whether the simplicity be real, or 
whether it covers a complex reality? Whether it is due to the 
influence of great numbers, which levels down individual diflfer- 
ences, or to the greatness or smallness of certain quantities, which 
allows us to neglect certain terms, in no case is it due to chance. 
This simplicity, real or apparent, always has a cause. We can 
always follow, then, the same course of reasoning, and if a simple 
law has been observed in several particular cases, we can legiti- 
mately suppose that it will still be true in analogous cases. To 
refuse to do this would be to attribute to chance an inadmis- 
sible role. 

There is, however, a diflFerence. If the simplicity were real 
and essential, it would resist the increasing precision of our means 
of measure. If then we believe nature to be essentially simple, 
we must, from a simplicity that is approximate, infer a simplicity 
that is rigorous. This is what was done formerly; and this is 
what we no longer have a right to do. 

The simplicity of Kepler's laws, for example, is only apparent. 
That does not prevent their being applicable, very nearly, to all 
systems analogous to the solar system ; but it does prevent their 
being rigorously exact. 

The RdLE op Hypothesis. — All generalization is a hypothesis. 
Hypothesis, then, has a necessary role that no one has ever con- 


tested. Only, it ought always, as soon as possible and as often 
as possible, to be subjected to verification. And, of course, if it 
does not stand this test, it ought to be abandoned without reserve. 
This is what we generally do, but sometimes with rather an ill 

Well, even this ill humor is not justified. The physicist who 
has just renounced one of his hypotheses ought, on the contraryi 
to be full of joy ; for he has found an unexpected opportunity ' 
for discovery. His hypothesis, I imagine, had not been adopted 
without consideration ; it took account of all the known factors 
that it seemed could enter into the phenomenon. If the test does 
not support it, it is because there is something unexpected and 
extraordinary ; and beoause there is going to be something found 
that is unknown and new. 

Has the discarded hypothesis, then, been barren T Far from 
that, it may be said it has rendered more service than a true 
hypothesis. Not only has it been the occasion of the decisive 
experiment, but, without having made the hypothesis, the experi- 
ment would have been made by chance, so that nothing would 
have been derived from it. One would have seen nothing ex- 
traordinary ; only one fact the more would have been catalogued 
without deducing from it the least consequence. 

Now on what condition is the use of hypothesis without danger T 

The firm determination to submit to experiment is not enough ; 
there are still dangerous hypotheses; first, and above all, those 
which are tacit and unconscious. Since we make them without 
knowing it, we are powerless to abandon them. Here again, then, 
is a service that mathematical physics can render us. By the 
precision that is characteristic of it, it compels us to formulate 
all the hypotheses that we should make without it, but uncon- 

Let us notice besides that it is important not to multiply 
hypotheses beyond measure, and to make them only one after the 
other. If we construct a theory based on a number of hypotheses, 
and if experiment condemns it, which of our premises is it neces- 
sary to change 1 It will be impossible to know. And inversely, 
if the experiment succeeds, shall we believe that we have demon- 


strated all the hypotheses at onceT Shall we believe that with 
one single equation we have determined several unknowns T 

We must equally take care to distinguish between the different 
kinds of hypotheses. There are first those which are perfectly 
natural and from which one can scarcely escape. It is difficult 
not to suppose that the influence of bodies very remote is quite 
negligible, that small movements follow a linear law, that the 
effect is a continuous function of its cause. I will say as much 
of the conditions imposed by symmetry. All these hypotheses 
form, as it were, the common basis of all the theories of mathe- 
matical physics. They are the last that ought to be abandoned. 

There is a second class of hypotheses, that I shall term neutral. 
In most questions the analyst assumes at the beginning of his 
calculations either that matter is continuous or, on the contrary, 
that it is formed of atoms. He might have made the opposite 
assumption without changing his results. He would only have 
had more trouble to obtain them ; that is all. If, then, experiment 
confirms his conclusions, will he think that he has demonstrated, 
for instance, the real existence of atoms T 

In optical theories two vectors are introduced, of which one 
is regarded as a velocity, the other as a vortex. Here again is 
a neutral hypothesis, since the same conclusions would have been 
reached by taking precisely the opposite. The success of the 
experiment, then, can not prove that the first vector is indeed a 
velocity ; it can only prove one thing, that it is a vector. This 
is the only hypothesis that has really been introduced in the 
premises. In order to give it that concrete appearance which the 
weakness of our minds requires, it has been necessary to consider 
it either as a velocity or as a vortex, in the same way that it has 
been necessary to represent it by a letter, either z or y. The 
result, however, whatever it may be, will not prove that it was 
right or wrong to regard it as a velocity any more than it will 
prove that it was right or wrong to call it x and not y. 

These neutral hypotheses are never dangerous, if only their 
character is not misunderstood. They may be useful, either as 
devices for computation, or to aid our understanding by concrete 
images, to fix our ideas as the saying is. There is, then, no occa- 
sion to exclude them. 


The hypotheses of the third class are the real generalizations. 
They are the ones that experiment must confirm or invalidate. 
Whether verified or condemned, they will always be froitfoL 
But for the reasons that I have set forth, they will only be fruit- 
ful if they are not too numerous. 

Origin op Mathematical Physics. — ^Let us penetrate further, 
and study more closely the conditions that have permitted the 
development of mathematical physics. We observe at once that 
the efforts of scientists have always aimed to resolve the complex 
phenomenon directly given by experiment into a very large num- 
ber of elementary phenomena. 

This is done in three different ways : first, in time. Instead of 
embracing in its entirety the progressive development of a 
phenomenon, the aim is simply to connect each instant with the 
instant immediately preceding it. It is admitted that the actual 
state of the world depends only on the immediate past, without 
being directly influenced, so to speak, by the memory of a distant 
past. Thanks to this postulate, instead of studying directly the 
whole succession of phenomena, it is possible to confine ourselves 
to writing its * differential equation.' For Kepler's laws we sul> 
stitute Newton's law. 

Next we try to analyze the phenomenon in space. What ex- 
periment gives us is a confused mass of facts presented on a 
stage of considerable extent. We must try to discover the ele- 
mentary phenomenon, which will be, on the contrary, localized in 
a very small region of space. 

Some examples will perhaps make my thought better under- 
stood. If we wished to study in all its complexity the distribu- 
tion of temperature in a cooling solid, we should never succeed. 
Everything becomes simple if we reflect that one point of the 
solid can not give up its heat directly to a distant point ; it will 
give up its heat only to the points in the immediate neighbor- 
hood, and it is by degrees that the flow of heat can reach other 
parts of the solid. The elementary phenomenon is the exchange 
of heat between two contiguous points. It is strictly localized, 
and is relatively simple, if we admit, as is natural, that it is not 
influenced by the temperature of molecules whose distance is 


I bend a rod. It is going to take a very complicated form, 
the direct study of which would be impossible. But I shall be 
able, however, to attack it, if I observe that its flexure is a result 
only of the deformation of the very small elements of the rod, and 
that the deformation of each of these elements depends only on 
the forces that are directly applied to it, and not at all on those 
which may act on the other elements. 

In all these examples, which I might easily multiply, we 
admit that there is no action at a distance, or at least at a great 
distance. This is a hypothesis. It is not always true, as the 
law of gravitation shows us. It must, then, be submitted to veri- 
fication. If it is confirmed, even approximately, it is precious, 
for it will enable us to make mathematical physics, at least by 
successive approximations. 

If it does not stand the test, we must look for something else 
analogous; for there are still other means of arriving at the 
elementary phenomenon. If several bodies act simultaneously, 
it may happen that their actions are independent and are simply 
added to one another, either as vectors or as scalars. The ele- 
mentary phenomenon is then the action of an isolated body. Or 
again, we have to deal with small movements, or more generally 
with small variations, which obey the well-known law of super- 
position. The observed movement will then be decomposed into 
simple movements, for example, sound into its harmonics, white 
light into its monochromatic components. 

When we have discovered in what direction it is advisable to 
look for the elementary phenomenon, by what means can we 
reach it? 

First of all, it will often happen that in order to detect it, 
or rather to detect the part of it useful to us, it will not be neces- 
sary to penetrate the mechanism ; the law of great numbers will 

Let us take again the instance of the propagation of heat. 
Every molecule emits rays toward every neighboring molecule. 
According to what law, we do not need to know. If we should 
make any supposition in regard to this, it would be a neutral 
hypothesis and consequently useless and incapable of verification. 
And, in fact, by the action of averages and thanks to the sym- 


metry of the medium, all the differences are leveled down, and 
whatever hypothesis may be made, the result is always the same. 

The same circumstance is presented in the theory of electricity 
and in that of capillarity. The neighboring molecules attract 
and repel one another. We do not need to know according to 
what law; it is enough for us that this attraction is sensible only 
at small distances, that the molecules are very numerous, that 
the medium is symmetrical, and we shall only have to let the law 
of great numbers act. 

Here again the simplicity of the elementary phenomenon 
was hidden under the complexity of the resultant observable phe- 
nomenon ; but, in its turn, this simplicity was only apparent, and 
concealed a very complex mechanism. 

The best means of arriving at the elementary phenomenon 
would evidently be experiment. We ought by experimental con- 
trivance to dissociate the complex sheaf that nature offers to our 
researches, and to study with care the elements as much isolated 
as possible. For example, natural white light would be decom- 
posed into monochromatic lights by the aid of the prism, and 
into polarized light by the aid of the polarizer. 

Unfortunately that is neither always possible nor always suflS- 
cient, and sometimes the mind must outstrip experiment. I shall 
cite only one example, which has always struck me forcibly. 

If I decompose white light, I shall be able to isolate a small part 
of the spectrum, but however small it may be, it will retain a 
certain breadth. Likewise the natural lights, called monochrO' 
matic, give us a very narrow line, but not, however, infinitely 
narrow. It might be supposed that by studying experimentally 
the properties of these natural lights, by working with finer and 
finer lines of the spectrum, and by passing at last to the limit, so 
to speak, we should succeed in learning the properties of a light 
strictly monochromatic. 

That would not be accurate. Suppose that two rays emanate 
from the same source, that we polarize them first in two perpen- 
dicular planes, then bring them back to the same plane of polari- 
zation, and try to make them interfere. If the light were strictly 
monochromatic, they would interfere. With our lights, which 
are nearly monochromatic, there will be no interference, and 


that no matter how narrow the line. In order to be otherwise 
it would have to be several million times as narrow as the finest 
known lines. 

Here, then, the passage to the limit would have deceived us. 
The mind must outstrip the experiment, and if it has done so 
with success, it is because it has allowed itself to be guided by the 
instinct of simplicity. 

The knowledge of the elementary fact enables us to put the 
problem in an equation. Nothing remains but to deduce from 
this by combination the complex fact that can be observed and 
verified. This is what is called integration, and is the business 
of the mathematician. 

It may be asked why, in physical sciences, generalization so 
readily takes the mathematical form. The reason is now easy to 
see. It is not only because we have numerical laws to express ; it 
is because the observable phenomenon is due to the superposition 
of a great number of elementary phenomena all alike. Thus 
quite naturally are introduced differential equations. 

It is not enough that each elementary phenomenon obeys sim- 
ple laws ; all those to be combined must obey the same law. Then 
only can the intervention of mathematics be of use ; mathematics 
teaches us in fact to combine like with like. Its aim is to learn 
the result of a combination without needing to go over the com- 
bination piece by piece. If we have to repeat several times the 
same operation, it enables us to avoid this repetition by telling us 
in advance the result of it by a sort of induction. I have ex- 
plained this above, in the chapter on mathematical reasoning. 

But, for this, all the operations must be alike. In the opposite 
case, it would evidently be necessary to resign ourselves to doing 
them in reality one after another, and mathematics would become 

It is then thanks to the approximate homogeneity of the 
matter studied by physicists, that mathematical physics could be 

In the natural sciences, we no longer find these conditions: 
homogeneity, relative independence of remote parts, simplicity 
of the elementary fact ; and this is why naturalists are obliged 
to resort to other methods of generalization. 


The Theories of Modebn Physics 

Meaning op Physical Theories. — The laity are stmck to 
see how ephemeral scientific theories are. After some years of 
prosperity, they see them successively abandoned ; they see ruins 
accumulate upon ruins ; they foresee that the theories fashionable 
to-day will shortly succumb in their turn and hence they con- 
clude that these are absolutely idle. This is what they call the 
bankruptcy of science. 

Their scepticism is superficial ; they give no account to them- 
selves of the aim and the role of scientific theories; otherwise 
they would comprehend that the ruins may still be good for 

No theory seemed more solid than that of Fresnel which 
attributed light to motions of the ether. Yet now Maxwell's 
is preferred. Does this mean the work of Fresnel was in vainT 
No, because the aim of Fresnel was not to find out whether 
there is really an ether, whether it is or is not formed of atoms, 
whether these atoms really move in this or that sense ; his object 
was to foresee optical phenomena. 

Now, Fresnel's theory always permits of this, to-day as well 
as before Maxwell. The diflFerential equations are always true; 
they can always be integrated by the same procedures and the 
results of this integration always retain their value. 

And let no one say that thus we reduce physical theories to 
the role of mere practical recipes; these equations express rela- 
tions, and if the equations remain true it is because these rela- 
tions preserve their reality. They teach us, now as then, that 
there is such and such a relation between some thing and some 
other thing; only this something formerly we called motion; we 
now call it electric current. But these appellations were only 
images substituted for the real objects which nature will eternally 
hide from us. The true relations between these real objects are 
the only reality we can attain to, and the only condition is that 



the same relations exist between these objects as between the 
images by which we are forced to replace them. If these rela- 
tions are known to us, what matter if we deem it convenient 
to replace one image by another. 

That some periodic phenomenon (an electric oscillation, for 
instance) is really due to the vibration of some atom which, act- 
ing like a pendulum, really moves in this or that sense, is neither 
certain nor interesting. But that between electric oscillation, 
the motion of the pendulum and all periodic phenomena there 
exists a close relationship which corresponds to a profound real- 
ity ; that this relationship, this similitude, or rather this parallel- 
ism extends into details ; that it is a consequence of more general 
principles, that of energy and that of least action; this is what 
we can affirm; this is the truth which will always remain the 
same under all the costumes in which we may deem it useful to 
deck it out. 

Numerous theories of dispersion have been proposed; the 
first was imperfect and contained only a small part of truth. 
Afterwards came that of Helmholtz ; then it was modified in vari- 
ous ways, and its author himself imagined another founded on 
the principles of Maxwell. But, what is remarkable, all the sci- 
entists who came after Helmholtz reached the same equations, 
starting from points of departure in appearance very widely 
separated. I will venture to say these theories are all true at 
the same time, not only because they make us foresee the same 
phenomena, but because they put in evidence a true relation, that 
of absorption and anomalous dispersion. What is true in the 
premises of these theories is what is common to all the authors; 
this is the affirmation of this or that relation between certain 
things which some call by one name, others by another. 

The kinetic theory of gases has given rise to many objections, 
which we could hardly answer if we pretended to see in it the 
absolute truth. But all these objections will not preclude its 
having been useful, and particularly so in revealing to us a 
relation true and but for it profoundly hidden, that of the 
gaseous pressure and the osmotic pressure. In this sense, then, 
it may be said to be true. 

When a physicist finds a contradiction between two theories 


equally dear to him, he sometimes says: "We will not bother 
about that, but hold firmly the two ends of the chain, though the 
intermediate links are hidden from us." This argument of an 
embarrassed theologian would be ridiculous if it were necessary 
to attribute to physical theories the sense the laity give them. 
In case of contradiction, one of them at least must then be re- 
garded as false. It is no longer the same if in them be sought 
only what should be sought. May be they both express true 
relations and the contradiction is only in the images wherewith 
we have clothed the reality. 

To those who find we restrict too much the domain accessible 
to the scientist, I answer: These questions which we interdict 
to you and which you regret, are not only insoluble, they are 
illusory and devoid of meaning. 

Some philosopher pretends that all physics may be explained 
by the mutual impacts of atoms. If he merely means there are 
between physical phenomena the same relations as between the 
mutual impacts of a great number of balls, well and good, that 
is verifiable, that is perhaps true. But he means something 
more ; and we think we understand it because we think we know 
what impact is in itself; whyt Simply because we have often 
seen games of billiards. Shall we think Qod, contemplating his 
work, feels the same sensations as we in watching a billiard 
match? If we do not wish to give this bizarre sense to his asser- 
tion, if neither do we wish the restricted sense I have just ex- 
plained, which is good sense, then it has none. 

Hypotheses of this sort have therefore only a metaphorical 
sense. The scientist should no more interdict them than the poet 
does metaphors; but he ought to know what they are worth. 
They may be useful to give a certain satisfaction to the mind, 
and they will not be injurious provided they are only indifferent 

These considerations explain to us why certain theories, sup- 
posed to be abandoned and finally condemned by experiment, 
suddenly arise from their ashes and recommence a new life. 
It is because they expressed true relations; and because they 
had not ceased to do so when, for one reason or another, we 
felt it necessary to enunciate the same relations in another 
language. So they retained a sort of latent life. 


Scarcely fifteen years ago was there anything more ridicnlons, 
more naively antiquated, than Coulomb 's fluids t And yet here 
they are reappearing under the name of electrons. Wherein do 
these permanently electrified molecules differ from Coulomb's 
electric molecules? It is true that in the electrons the electricity 
is supported by a little, a very little matter ; in other words, they 
have a mass (and yet this is now contested) ; but Coulomb did 
not deny mass to his fluids, or, if he did, it was only with reluc- 
tance. It would be rash to affirm that the belief in electrons 
will not again suffer eclipse ; it was none the less curious to note 
this unexpected resurrection. 

But the most striking example is Camot's principle. Camot 
set it up starting from false hypotheses ; when it was seen that 
heat is not indestructible, but may be transformed into work, his 
ideas were completely abandoned ; afterwards Clausius returned 
to them and made them finally triumph. Camot's theory, under 
its primitive form, expressed, aside from true relations, other 
inexact relations, dihris of antiquated ideas ; but the presence of 
these latter did not change the reality of the others. Clausius 
had only to discard them as one lops off dead branches. 

The result was the second fundamental law of thermodynamics. 
There were always the same relations ; though these relations no 
longer subsisted, at least in appearance, between the same ob- 
jects. This was enough for the principle to retain its value. 
And even the reasonings of Camot have not perished because 
of that ; they were applied to a material tainted with error ; but 
their form (that is to say, the essential) remained correct. 

What I have just said illuminates at the same time the role 
of general principles such as the principle of least action, or that 
of the conservation of energy. 

These principles have a very high value; they were obtained 
in seeking what there was in common in the enunciation of nu- 
merous physical laws; they represent therefore, as it were, the 
quintessence of innumerable observations. 

However, from their very generality a consequence results to 
which I have called attention in Chapter VIII., namely, that 
they can no longer be verified. As we can not give a general 
definition of energy, the principle of the conservation of energy 


signifies simply that there is something which remains constant 
Well, whatever be the new notions that future experiments shall 
give us about the world, we are sure in advance that there will 
be something there which will remain constant and which may 
be called energy. 

Is this to say that the principle has no meaning and vanishes 
in a tautology t Not at all ; it signifies that the different things 
to which we give the name of energy are connected by a true kin- 
ship ; it aflBrms a real relation between them. But then if this 
principle has a meaning, it may be false ; it may be that we have 
not the right to extend indefinitely its applications, and yet it is 
certain beforehand to be verified in the strict acceptation of the 
term ; how then shall we know when it shall have attained all the 
extension which can legitimately be given it 1 Just simply when 
it shall cease to be useful to us, that is, to make us correctly fore- 
see new phenomena. We shall be sure in such a case that the 
relation aflBrmed is no longer real; for otherwise it would be 
fruitful; experiment, without directly contradicting a new ex- 
tension of the principle, will yet have condemned it. 

Physics and JVIechanism. — Most theorists have a constant 
predilection for explanations borrowed from mechanics or dy- 
namics. Some would be satisfied if they could explain all phe- 
nomena by motions of molecules attracting each other according 
to certain laws. Others are more exacting ; they would suppress 
attractions at a distance ; their molecules should follow rectilinear 
paths from which they could be made to deviate only by impacts. 
Others again, like Hertz, suppress forces also, but suppose their 
molecules subjected to geometric attachments analogous, for in- 
stance, to those of our linkages ; they try thus to reduce dynamics 
to a sort of kinematics. 

In a word, all would bend nature into a certain form outside 
of which their mind could not feel satisfied. Will nature be 
sufficiently flexible for that? 

We shall examine this question in Chapter XII., d propos of 
Maxwell's theory. Whenever the principles of energy and of 
least action are satisfied, we shall see not only that there is always 
one possible mechanical explanation, but that there is always an 
infinity of them. Thanks to a well-known theorem of Eonig's on 


linkages, it could be shown that we can, in an infinity of ways, 
explain everything by attachments after the manner of Hertz, or 
also by central forces. Without doubt it could be demonstrated 
just as easily that everything can always be explained by simple 

For that, of course, we need not be content with ordinary 
matter, with that which falls under our senses and whose motions 
we observe directly. Either we shall suppose that this common 
matter is formed of atoms whose internal motions elude us, the 
displacement of the totality alone remaining accessible to our 
senses. Or else we shall imagine some one of those subtile fluids 
which under the name of ether or under other names, have at all 
times played so great a role in physical theories. 

Often one goes further and regards the ether as the sole 
primitive matter or even as the only true matter. The more 
moderate consider common matter as condensed ether, which is 
nothing startling; but others reduce still further its importance 
and see in it nothing more than the geometric locus of the ether's 
singularities. For instance, what we call matter is for Lord 
Kelvin only the locus of points where the ether is animated by 
vortex motions; for Biemann, it was the locus of points where 
ether is constantly destroyed; for other more recent authors, 
Wiechert or Larmor, it is the locus of points where the ether 
undergoes a sort of torsion of a very particular nature. If the 
attempt is made to occupy one of these points of view, I ask 
myself by what right shall we extend to the ether, under pretext 
that this is the true matter, mechanical properties observed in 
ordinary matter, which is only false matter. 

The ancient fluids, caloric, electricity, etc., were abandoned 
when it was perceived that heat is not indestructible. But they 
were abandoned for another reason also. In materializing them, 
their individuality was, so to speak, emphasized, a sort of abyss 
was opened between them. This had to be filled up on the coming 
of a more vivid feeling of the unity of nature, and the perception 
of the intimate relations which bind together all its parts. Not 
only did the old physicists, in multiplying fluids, create entities 
unnecessarily, but they broke real ties. 

It is not su£Bcient for a theory to affirm no false relations, it 
must not hide true relations. 


And does our ether really exist f We know the origin of our 
belief in the ether. If light reaches us from a distant star, dur- 
ing several years it was no longer on the star and not yet on the 
earth ; it must then be somewhere and sustained, so to speak, by 
some material support. 

The same idea may be expressed under a more mathematical 
and more abstract form. What we ascertain are the changes un- 
dergone by material molecules; we see, for instance, that our 
photographic plate feels the consequences of phenomena of which 
the incandescent mass of the star was the theater several years 
before. Now, in ordinary mechanics the state of the system 
studied depends only on its state at an instant immediately an- 
terior; therefore the system satisfies differential equations. On 
the contrary, if we should not believe in the ether, the state of the 
material universe would depend not only on the state immedi- 
ately preceding, but on states much older; the system would 
satisfy equations of finite differences. It is to escape this deroga- 
tion of the general laws of mechanics that we have invented the 

That would still only oblige us to fill up, with the ether, the 
interplanetary void, but not to make it penetrate the bosom of 
the material media themselves. Fizeau's experiment goes fur- 
ther. By the interference of rays which have traversed air or 
water in motion, it seems to show us two different media inter- 
penetrating and yet changing place one with regard to the other. 

We seem to touch the ether with the finger. 

Yet experiments may be conceived which would make us touch 
it still more nearly. Suppose Newton's principle, of the equality 
of action and reaction, no longer true if applied to matter alone, 
and that we have established it. The geometric sum of all the 
forces applied to all the material molecules would no longer be 
null. It would be necessary then, if we did not wish to change 
all mechanics, to introduce the ether, in order that this action 
which matter appeared to experience should be counterbalanced 
by the reaction of matter on something. 

Or again, suppose we discover that optical and electrical 
phenomena are influenced by the motion of the earth. We should 
be led to conclude that these phenomena might reveal to us not 


only the relative motions of material bodies, but what would 
seem to be their absolute motions. Again, an ether would be 
necessary, that these so-called absolute motions should not be 
their displacements with regard to a void space, but their dis- 
placements with regard to something concrete. 

Shall we ever arrive at that? I have not this hope, I shall 
soon say why, and yet it is not so absurd, since others have 
had it. 

For instance, if the theory of Lorentz, of which I shall speak 
in detail further on in Chapter XIII., were true, Newton's prin- 
ciple would not apply to matter alone, and the difference would 
not be very far from being accessible to experiment. 

On the other hand, many researches have been made on the 
influence of the earth's motion. The results have always been 
negative. But these experiments were undertaken because the 
outcome was not sure in advance, and, indeed, according to the 
ruling theories, the compensation would be only approximate, 
and one might expect to see precise methods give positive results. 

I believe that such a hope is illusory; it was none the less 
interesting to show that a success of this sort would open to us, 
in some sort, a new world. 

And now I must be permitted a digression ; I must explain, in 
fact, why I do not believe, despite Lorentz, that more precise 
observations can ever put in evidence anything else than the rela- 
tive displacements of material bodies. Experiments have been 
made which should have disclosed the terms of the first order; 
the results have been negative; could that be by chance t No 
one has assumed that ; a general explanation has been sought, and 
Lorentz has found it; he has shown that the terms of the first 
order must destroy each other, but not those of the second. Then 
more precise experiments were made; they also were negative; 
neither could this be the effect of chance; an explanation was 
necessary; it was found; they always are found; of hypotheses 
there is never lack. 

But this is not enough ; who does not feel that this is still to 
leave to chance too great a role? Would not that also be a 
chance, this singular coincidence which brought it about that a 
certain circumstance should come just in the nick of time to 


destroy the terms of the first order, and that another circTun- 
stance, wholly different, but just as opportnne, should take upon 
itself to destroy those of the second order ? No, it is necessary to 
find an explanation the same for the one as for the other, and 
then everything leads us to think that this explanation vdll 
hold good equally well for the terms of higher order, and that the 
mutual destruction of these terms will be rigorous and absolute. 

Present State op the Science. — ^In the history of the de- 
velopment of physics we distinguish two inverse tendencies. 

On the one hand, new bonds are continually being discovered 
between objects which had seemed destined to remain forever 
unconnected; scattered facts cease to be strangers to one another; 
they tend to arrange themselves in an imposing i^mthesis. 
Science advances toward unity and simplicity. 

On the other hand, observation reveals to us every day new 
phenomena ; they must long await their place and sometimes, to 
make one for them, a comer of the edifice must be demolished. 
In the known phenomena themselves, where our crude senses 
showed us uniformity, we perceive details from day to day more 
varied; what we believed simple becomes complex, and science 
appears to advance toward variety and complexity. 

Of these two inverse tendencies, which seem to triumph turn 
about, which will wint If it be the first, science is possible; 
but nothing proves this a priori, and it may well be feared that 
after having made vain efforts to bend nature in spite of herself 
to our ideal of unity, submerged by the ever-rising flood of our 
new riches, we must renounce classifying them, abandon our 
ideal, and reduce science to the registration of innumerable 

To this question we can not reply. All we can do is to ob- 
serve the science of to-day and compare it with that of yesterday. 
Prom this examination we may doubtless draw some encourage- 

Half a century ago, hope ran high. The discovery of the 
conservation of energy and of its transformations had revealed to 
us the unity of force. Thus it showed that the phenomena of 
heat could be explained by molecular motions. What was the 
nature of these motions was not exactly known, but no one 


doubted that it soon would be. For light, the task seemed com- 
pletely accomplished. In what concerns electricity, things were 
less advanced. Electricity had just annexed magnetism. This 
was a considerable step toward unity, and a decisive step. 

But how should electricity in its turn enter into the general 
unity, how should it be reduced to the universal mechanism t 

Of that no one had any idea. Yet the possibility o£ this reduc- 
tion was doubted by none, there was faith. Finally, in what 
concerns the molecular properties of material bodies, the reduc- 
tion seemed still easier, but all the detail remained hazy. In 
a word, the hopes were vast and animated, but vague. To-day, 
what do we see t First of all, a prime progress, immense prog- 
ress. The relations of electricity and light are now known ; the 
three realms, of light, of electricity and of magnetism, previously 
separated, form now but one ; and this annexation seems final. 

This conquest, however, has cost us some sacrifices. The optical 
phenomena subordinate themselves as particular cases under the 
electrical phenomena ; so long as they remained isolated, it was 
easy to explain them by motions that were supposed to be known 
in all their details, that was a matter of course; but now an 
explanation, to be acceptable, must be easily capable of extension 
to the entire electric domain. Now that is a matter not without 

The moat satisfactory theory we have is that of Lorentz, which, 
as we shall see in the last chapter, explains electric currents by 
the motions of little electrified particles ; it is unquestionbly the 
one which best esplains the known facts, the one which illumi- 
nates the greatest number of true relations, tlie one of which most 
traces will be found in the final construction. Nevertheless, it 
still has a serious defect, which I have indicated above; it is 
contrary to Newton's law of the equality of action and reaction; 
or rather, this principle, in the eyes of Lorentz, would not be 
applicable to matter alone ; for it to be true, it would be necessary 
to take account of the action of the ether on matter and of tb« 
reaction of matter on the ether. 

Now, from what we know at present, it seems probable that 
things do not happen in this way. 

However that may be, thanks to horentz, Fizeau's results on 


the optics of moving bodies, the laws of normal and anomalous dis- 
persion and of absorption find themselves linked to one another 
and to the other properties of the ether by bonds which beyond 
any doubt will never more be broken. See the facility with which 
the new Zeeman effect has found its place already and has even 
aided in classifying Faraday's magnetic rotation which had de- 
fied Maxwell's efforts; this facility abundantly proves that the 
theory of Lorentz is not an artificial assemblage destined to fall 
asunder. It will probably have to be modified, but not destroyed. 

But Lorentz had no aim beyond that of embracing in one 
totality all the optics and electrodynamics of moving bodies; he 
never pretended to give a mechanical explanation of them. Lar- 
mor goes further; retaining the theory of Lorentz in essentials, 
he grafts upon it, so to speak, MacGullagh's ideas on the direction 
of the motions of the ether. 

According to him, the velocity of the ether would have the 
same direction and the same magnitude as the magnetic force. 
However ingenious this attempt may be, the defect of the theory 
of Lorentz remains and is even aggravated. With Lorentz, we do 
not know what are the motions of the ether ; thanks to this igno- 
rance, we may suppose them such that, compensating those of 
matter, they reestablish the equality of action and reaction. 
With Larmor, we know the motions of the ether, and we can 
ascertain that the compensation does not take place. 

If Larmor has failed, as it seems to me he has, does tiiat mean 
that a mechanical explanation is impossible? Far from it: I 
have said above that when a phenomenon obeys the two principles 
of energy and of least action, it admits of an infinity of mechan- 
ical explanations ; so it is, therefore, with the optical and electrical 

But this is not enough: for a mechanical explanation to be 
good, it must be simple ; for choosing it among all which are pos- 
sible, there should be other reasons besides the necessity of mak- 
ing a choice. Well, we have not as yet a theory satisfying this 
condition and consequently good for something. Must we lament 
this? That would be to forget what is the goal sought; this is 
not mechanism ; the true, the sole aim is unity. 

We must therefore set bounds to our ambition ; let us not try 


to formulate a mechanical explanation; let us be content with 
showing that we could always find one if we wished to. In this 
regard we have been successful ; the principle of the conservation 
of energy has received only confirmations ; a second principle has 
come to join it, that of least action, put under the form which is 
suitable for physics. It also has always been verified, at least 
in so far as concerns reversible phenomena which thus obey the 
equations of Lagrange, that is to say, the most general laws of 

Irreversible phenomena are much more rebellious. Yet these 
also are being coordinated, and tend to come into unity ; the light 
which has illuminated them has come to us from Gamot's prin- 
ciple. Long did thermodynamics confine itself to the study of 
the dilatation of bodies and their changes of state. For some time 
past it has been growing bolder and has considerably extended 
its domain. We owe to it the theory of the galvanic battery, and 
that of the thermoelectric phenomena ; there is not in all physics 
a comer that it has not explored, and it has attacked chemistry 

Everywhere the same laws reign ; everywhere, under the diver- 
sity of appearances, is found again Gamot's principle; every- 
where also is found that concept so prodigiously abstract of 
entropy, which is as universal as that of energy and seems like it 
to cover a reality. Radiant heat seemed destined to escape it; but 
recently we have seen that submit to the same laws. 

In this way fresh analogies are revealed to us, which may 
often be followed into detail; ohmic resistance resembles the 
viscosity of liquids ; hysteresis would resemble rather the friction 
of solids. In all cases, friction would appear to be the type which 
the most various irreversible phenomena copy, and this kinship 
is real and profound. 

Of these phenomena a mechanical explanation, properly so 
called, has also been sought. They hardly lent themselves to it. 
To find it, it was necessary to suppose that the irreversibility is 
only apparent, that the elementary phenomena are reversible and 
obey the known laws of dynamics. But the elements are extremely 
numerous and blend more and more, so that to our crude sight all 
appears to tend toward uniformity, that is, everything seems to 


go forward in the same sense without hox>e of return. The ap- 
parent irreversibility is thns only an effect of the law of greaX 
numbers. But, only a being with infinitely subtile senses, like 
Maxwell's imaginary demon, could disentangle this inextricable 
skein and turn back the course of the universe. 

This conception, which attaches itself to the kinetic theory 
of gases, has cost great efforts and has not, on the whole, been 
fruitful ; but it may become so. This is not the place to examine 
whether it does not lead to contradictions and whether it is in 
conformity with the true nature of things. 

We signalize, however, M. Grouy 's original ideas on the Brownian 
movement. According to this scientist, this singular motion 
should escape Camot's principle. The particles which it puts in 
swing would be smaller than the links of that so compacted skein; 
they would therefore be fitted to disentangle them and hence to 
make the world go backward. We should almost see Maxwell's 
demon at work. 

To summarize, the previously known phenomena are better and 
better classified, but new phenomena come to claim their place; 
most of these, like the Zeeman effect, have at once found it. 

But we have the cathode rays, the X-rays, those of uranium 
and of radium. Herein is a whole world which no one suspected. 
How many unexpected guests must be stowed away 1 

No one can yet foresee the place they will occupy. But I do 
not believe they will destroy the general unity ; I think they will 
rather complete it. On the one hand, in fact, the new radiations 
seem connected with the phenomena of luminescence; not only 
do they excite fluorescence, but they sometimes take birth in the 
same conditions as it. 

Nor are they without kinship with the causes which produce 
the electric spark under the action of the ultra-violet light. 

Finally, and above all, it is believed that in all these phenomena 
are found true ions, animated, it is true, by velocities incom- 
parably greater than in the electrolytes. 

That is all very vague, but it will all become more precise. 

Phosphorescence, the action of light on the spark, these were 
regions rather isolated, and consequently somewhat neglected by 
investigators. One may now hope that a new path wiU be con- 


stracted which will facilitate their communications with the rest 
of science. 

Not only do we discover new phenomena, but in those we 
thought we knew, unforeseen aspects reveal themselves. In the 
free ether, the laws retain their majestic simplicity ; but matter, 
properly so called, seems more and more complex; all that is 
said of it is never more than approximate, and at each instant 
our formulas require new terms. 

Nevertheless the frames are not broken ; the relations that we 
have recognized between objects we thought simple still subsist 
between these same objects when we know their complexity, and 
it is that alone which is of importance. Our equations become, it 
is true, more and more complicated, in order to embrace more 
closely the complexity of nature ; but nothing is changed in the 
relations which permit the deducing of these equations one from 
another. In a word, the form of these equations has persisted. 

Take, for example, the laws of reflection : Fresnel had estab- 
lished them by a simple and seductive theory which experiment 
seemed to confirm. Since then more precise researches have 
proved that this verification was only approximate; they have 
shown everywhere traces of elliptic polarization. But, thanks to 
the help that the first approximation gave us, we found forthwith 
the cause of these anomalies, which is the presence of a transition 
layer; and Fresnel 's theory has subsisted in its essentials. 

But there is a reflection we can not help making: All these 
relations would have remained unperceived if one had at first 
suspected the complexity of the objects they connect. It has long 
been said: If Tycho had had instruments ten times more pre- 
cise neither Kepler, nor Newton, nor astronomy would ever have 
been. It is a misfortune for a science to be bom too late, when 
the means of observation have become too perfect. This is to-day 
the case with physical chemistry; its founders are embarrassed 
in their general grasp by third and fourth decimals ; happily they 
are men of a robust faith. 

The better one knows the properties of matter the more one 
sees continuity reign. Since the labors of Andrews and of van der 
Wals, we get an idea of how the passage is made from the liquid 
to the gaseous state and that this passage is not abrupt. Similarly, 


there is no gap between the liquid and solid states^ and in tiie 
proceedings of a recent congress is to be seen, alongside of a work 
on the rigidity of liquids, a memoir on the flow of solids. 

By this tendency no doubt simplicity loses ; some phenomenon 
was formerly represented by several straight lines, now these 
straights must be joined by curves more or less complicated. In 
compensation unity gains notably. Those cut-off categories quieted 
the mind, but they did not satisfy it. 

Finally the methods of physics have invaded a new domain, 
that of chemistry; physical chemistry is bom. It is still very 
young, but we already see that it will enable us to connect such 
phenomena as electrolysis, osmosis and the motions of ions. 

From this rapid exposition, what shall we conclude ? 

Everything considered, we have approached unity; we have 
not been as quick as was hoped fifty years ago, we have not always 
taken the predicted way; but, finally, we have gained ever so 
much ground. 

The Calculus op Probabilities 

Doubtless it will be astonishing to find here thoughts about 
the calculus of probabilities. What has it to do with the method 
of the physical sciences t And yet the questions I shall raise with- 
out solving present themselves naturally to the philosopher who 
is thinking about physics. So far is this the case that in the 
two preceding chapters I have often been led to use the words 
' probability ' and ' chance. ' 

'Predicted facts,' as I have said above, 'can only be probable.' 
''However solidly founded a prediction may seem to us to be, 
we are never absolutely sure that experiment will not prove it 
false. But the probability is often so great that practically we 
may be satisfied with it." And a little further on I have added : 
"See what a role the belief in simplicity plays in our generaliza- 
tions. We have verified a simple law in a great number of par- 
ticular cases; we refuse to admit that this coincidence, so often 
repeated, can be a mere effect of chance. ..." 

Thus in a multitude of circumstances the physicist is in the 
same position as the gambler who reckons up his chances. As 
often as he reasons by induction, he requires more or less con- 
sciously the calculus of probabilities, and this is why I am obliged 
to introduce a parenthesis, and interrupt our study of method in 
the physical sciences in order to examine a little more closely the 
value of this calculus, and what confidence it merits. 

The very name calculus of probabilities is a paradox. Prob- 
ability opposed to certainty is what we do not know, and how can 
we calculate what we do not know? Yet many eminent savants 
have occupied themselves with this calculus, and it can not be 
denied that science has drawn therefrom no small advantage. 

How can we explain this apparent contradiction ? 

Has probability been defined? Can it even be defined? And 
if it can not, how dare we reason about it t The definition, it will 



be said, is very simple : the probability of an event is the ratio of 
the number of eases favorable to this event to the total number of 
possible cases. 

A simple example will show how incomplete this definition is. 
I throw two dice. What is the probability that one of the two 
at least turns up a sixf Each die can turn up in six different 
ways; the number of possible cases is 6 X 6 = 36; the number 
of favorable cases is 11 ; the probability is 11/36. 

That is the correct solution. But could I not just as well say: 
The points which turn up on the two dice can form 6 X 7/2=21 
different combinations f Among these combioationa 6 are favor- 
able ; the probability is 6/21. 

Now why is the first method of enumerating the i)0ssible cases 
more legitimate than the second f In any case it is not our 
definition that tells us. 

We are therefore obliged to complete this definition by saying : 
' ... to the total number of possible cases provided these cases 
are equally probable.' So, therefore, we are reduced to defining 
the probable by the probable. 

How can we know that two possible cases are eqvally probable f 
Will it be by a convention 1 If we place at the beginning of each 
problem an explicit convention, well and good. We shall then 
have nothing to do but apply the rules of arithmetic and of 
algebra, and we shall complete our calculation without our result 
leaving room for doubt. But if we wish to make the slightest 
application of this result, we must prove our convention was 
legitimate, and we shall find ourselves in the presence of the very 
difficulty we thought to escape. 

Will it be said that good sense suffices to show us what con- 
vention should be adopted 1 Alas ! M. Bertrand has amused him- 
self by discussing the following simple problem: "What is the 
probability that a chord of a circle may be greater than the side 
of the inscribed equilateral triangle?" The illustrious geometer 
successively adopted two conventions which good sense seemed 
equally to dictate and with one he found 1/2, with the other 1/3. 

The conclusion which seems to follow from all this is that the 
calculus of probabilities is a useless science, and that the obscure 


instinct which we may call good sense, and to which we are wont 
to appeal to legitimatize our conventions, must be distrusted. 

But neither can we subscribe to this conclusion; we can not 
do without this obscure instinct. Without it science would be 
impossible, without it we could neither discover a law nor apply 
it. Have we the right, for instance, to enunciate Newton's lawt 
Without doubt, numerous observations are in accord with it ; but 
is not this a simple effect of chance t Besides how do we know 
whether this law, true for so many centuries, will still be true 
next year! To this objection, you will find nothing to reply, 
except: 'That is very improbable.' 

But grant the law. Thanks to it, I believe myself able to 
calculate the position of Jupiter a year from now. Have I the 
right to believe this! Who can tell if a gigantic mass of enor- 
mous velocity will not between now and that time pass near the 
aolar system, and produce unforeseen perturbations! Here again 
the only answer is: 'It is very improbable.' 

Prom this point of view, all the sciences would be only uncon- 
scious applications of the calculus of probabilities. To condemn 
this calculus would be to condemn the whole of science. 

I shall dwell lightly on the scientific problems in which the 
intervention of the calculus of probabilities is more evident. In 
the forefront of these is the problem of interpolation, in which, 
knowing a certain number of values of a function, we seek to 
divine the intermediate values. 

I shall likewise mention: the celebrated theory of errors of 
observation, to which I shall return later; the kinetic theory of 
g^es, a well-known hypothesis, wherein each gaseous molecule is 
supposed to describe an extremely complicated trajectory; but in 
which, through the effect of great numbers, the mean phenomena, 
alone observable, obey the simple laws of Mariotte and Gay- 

All these theories are based on the laws of great numbers, and 
the calculus of probabilities would evidently involve them in its 
ruin. It is true that they have only a particular interest, and 
that, save as far as interpolation is concerned, these are sacrifices 
to which we might readily be resigned. 

But, as I have said above, it would not be only these partial 


sacrifices that would be in question ; it would be the legitimacy of 
the whole of science that would be challenged. 

I quite see that it might be said: ''We are ignorant, and yet 
we must act. For action, we have not time to devote ourselves 
to an inquiry sufficient to dispel our ignorance. Besides, such an 
inquiry would demand an infinite time. We must therefore decide 
without knowing ; we are obliged to do so, hit or miss, and we must 
follow rules without quite believing them. What I know is not 
that such and such a thing is true, but that the best course for me 
is to act as if it were true." The calculus of probabilities, and 
consequently science itself, would thenceforth have merely a prac- 
tical value. 

Unfortunately the difficulty does not thus disappear. A gam- 
bler wants to try a coup; he asks my advice. If I give it to him, 
I shall use the calculus of probabilities, but I shall not guarantee 
success. This is what I shall call subjective probability. In this 
case, we might be content with the explanation of which I have 
just given a sketch. But suppose that an observer is present at 
the game, that he notes all its coupsy and that the game goes on a 
long time. When he makes a summary of his book, he will find 
that events have taken place in conformity with the laws of the 
calculus of probabilities. This is what I shall call objective 
probability, and it is this phenomenon which has to be explained. 

There are numerous insurance companies which apply the rules 
of the calculus of probabilities, and they distribute to their share- 
holders dividends whose objective reality can not be contested. 
To invoke our ignorance and the necessity to act does not suffice 
to explain them. 

Thus absolute skepticism is not admissible. We may distrust, 
but we can not condemn en bloc. Discussion is necessary. 

I. Classification op the Problems op Probability. — In 
order to classify the problems which present themselves d propos 
of probabilities, we may look at them from many different points 
of view, and, first, from the point of vieiv of generality. I have 
said above that probability is the ratio of the number of favorable 
cases to the number of possible cases. What for want of a better 
term I call the generality will increase with the number of pos- 


sible cases. This number may be finite, as, for instance, if we 
take a throw of the dice in which the number of possible cases is 
36. That is the first degree of generality. 

But if we ask, for example, what is the probability that a 
point within a circle is within the inscribed square, there are as 
many possible cases as there are points in the circle, that is to 
say, an infinity: This is the second degree of generality. Gener- 
ality can be pushed further still. We may ask the probability that 
a function will satisfy a given condition. There are then as many 
possible cases as one can imagine different functions. This is the 
third degree of generality, to which we rise, for instance, when 
we seek to find the most probable law in conformity with a finite 
number of observations. 

We may place ourselves at a point of view wholly different. 
H we were not ignorant, there would be no probability, there 
would be room for nothing but certainty. But our ignorance can 
not be absolute, for then there would no longer be any probability 
at all, since a little light is necessary to attain even this uncertain 
science. Thus the problems of probability may be classed accord- 
ing to the greater or less depth of this ignorance. 

In mathematics even we may set ourselves problems of prob- 
ability. What is the probability that the fifth decimal of a log- 
arithm taken at random from a table is a *9't There is no 
hesitation in answering that this probability is 1/10; here we 
possess all the data of the problem. We can calculate our loga- 
rithm without recourse to the table, but we do not wish to give 
ourselves the trouble. This is the first degree of ignorance. 

In the physical sciences our ignorance becomes greater. The 
state of a system at a given instant depends on two things : Its 
initial state, and the law according to which that state varies. If 
we know both this law and this initial state, we shall have then 
only a mathematical problem to solve, and we fall back upon the 
first degree of ignorance. 

But it often happens that we know the law, and do not know 
the initial state. It may be asked, for instance, what is the 
present distribution of the minor planets t We know that from 
all time they have obeyed the laws of Kepler, but we do not know 
what was their initial distribution. 


In the kmetic theory of gases, we assume that the gaseous 
molecules follow rectilinear trajectories, and obey the laws of 
impact of elastic bodies. But, as we know nothing of their initial 
velocities, we know nothing of their present velocities. 

The calculus of probabilities only enables us to predict the 
mean phenomena which will result from the combination of these 
velocities. This is the second degree of ignorance. 

Finally it is possible that not only the initial conditions but 
the laws themselves are unknown. We then reach the third degree 
of ignorance and in general we can no longer affirm anything at 
all as to the probability of a phenomenon. 

It often happens that instead of trying to guess an event, by 
means of a more or less imperfect knowledge of the law, the 
events may be known and we want to find the law ; or that instead 
of deducing effects from causes, we wish to deduce the causes 
from the effects. These are the problems called probability of 
causes, the most interesting from the point of view of their sci- 
entific applications. 

I play 6carte with a gentleman I know to be perfectly honest 
He is about to deal. What is the probability of his turning up 
the kingt It is 1/8. This is a problem of the probability of 

I play with a gentleman whom I do not know. He has dealt 
ten times, and he has turned up the king six times. What is 
the probability that he is a sharper! This is a problem in the 
probability of causes. 

It may be said that this is the essential problem of the experi- 
mental method. I have observed n values of x and the corres- 
ponding values of y. I have found that the ratio of the latter to 
the former is practically constant. There is the event, what is 
the cause? 

Is it probable that there is a general law according to which y 
would be proportional to re, and that the smaU divergencies are 
due to errors of observation ? This is a type of question that one 
is ever asking, and which we unconsciously solve whenever we are 
engaged in scientific work. 

I am now going to pass in review these different categories of 

problems, discussing in snccession what I have called above sub- 
jective and objective probability. 

II, Probability in Mathematics. — The impossibility of squar- 
ing the circle has been proved since 1882; but even before 
that date all geometers considered that impossibility as so 
'probable,' that the Academy of Sciences rejected without exami- 
nation the alas! too numerous memoirs on this subject, that some 
unhappy madmen sent in every year. 

Was the Academy wrong I Evidently not, and it knew well 
that in acting thus it did sot run the least risk of stilling a dis- 
covery of moment. The Academy could not have proved that it 
was right; but it knew quite well that its instinct was not mis- 
taken. If you had asked the Academicians, they would have 
answered: "We have compared the probability that an unknown 
savant should have found out what has been vainly sought for so 
long, with the probability that there is one madman the more 
on the earth; the second appears to us the greater." These are 
very good reasons, but there is nothing mathematical about them ; 
Ibey are purely psychological. 

And if you had pressed them further they would have added: 
"Why do you suppose a particular value of a transcendental 
function to be an algebraic number i and if jt were a root of aa 
algebraic equation, why do yon suppose this root to be a period of 
the function sin 2x, and not the same about the other roots of this 
same equation?" To sum up, they would have invoked the prin- 
ciple of sufficient reason in its vaguest form. 

But what could they deduce from itT At most a rule of con- 
duct for tile employment of their time, more usefully spent at 
their ordinary work than in reading a lucubration that inspired 
in them a legitimate distrust. But what I call above objective 
probabili^ has nothing in common with this first problem. 

It ia otherwise with the second problem. 

Consider the first 10,000 logarithms that we find in a table. 
Among these 10,000 logarithms I take one at random. What is 
the probabQity that its third decimal is an even number? Ton 
will not hesitate to answer 1/2; and in fact if you pick out in a 
table the third decimals of these 10,000 numbers, you will find 
nearly as many even digits aa odd. 


Or if you prefer, let us write 10,000 numbers corresponding 
to our 10,000 logarithms, each of these numbers being +1 ^ 
the third decimal of the corresponding logarithm is even, and 
— 1 if odd. Then take the mean of these 10,000 numbers. 

I do not hesitate to say that the mean of these 10,000 numbers 
is probably 0, and if I were actually to calculate it I should 
verify that it is extremely small. 

But even this verification is needless. I might have rigorously 
proved that this mean is less than 0.003. To prove this result, I 
should have had to make a rather long calculation for which there 
is no room here, and for which I confine myself to citing an article 
I published in the Revue gSnerale des Sciences, April 15, 1899. 
The only point to which I wish to call attention is the following: 
in this calcula^tion, I should have needed only to rest my case on 
two facts, to wit, that the first and second derivatives of the log- 
arithm remain, in the interval considered, between certain limits. 

Hence this important consequence that the property is true not 
only of the logarithm, but of any continuous function whatever, 
since the derivatives of every continuous function are limited. 

If I was certain beforehand of the result, it is first, because I 
had often observed analogous facts for other continuous func- 
tions; and next, because I made in my mind, in a more or less 
unconscious and imperfect manner, the reasoning which led me to 
the preceding inequalities, just as a skilled calculator before 
finishing his multiplication takes into account what it should 
eome to approximately. 

And besides, since what I call my intuition was only an in- 
complete summary of a piece of true reasoning, it is clear why 
observation has confirmed my predictions, and why the objective 
probability has been in agreement with the subjective probability. 

As a third example I shall choose the following problem: A 
number u is taken at random, and n is a given very large integer. 
What is the probable value of sin nu t This problem has no mean- 
ing by itself. To give it one a convention is needed. We shall 
agree that the probability for the number u to lie between a and 
o + da is equal to <^(a) da ; that it is therefore proportional to the 
infinitely small interval da, and equal to this multiplied by a 
function <^(a) depending only on a. As for this function, I 


choose it arbitrarily^ but I must assume it to be continuous. The 
value of sin nu remaining the same when u increases by 2^, I may 
without loss of generality assume that u lies between and 2ir, 
and I shall thus be led to suppose that <^(a) is a periodic function 
whose period is 2ir. 

The probable value sought is readily expressed by a simple 
integral, and it is easy to show that this integral is less than 


Mjfc being the maximum value of the k^ derivative of if>{u). We 
see then that if the k^ derivative is finite, our probable value will 
tend toward when n increases indefinitely, and that more rapidly 
than 1/n^K 

The probable value of sin nu when n is very large is therefore 
naught. To define this value I required a convention; but the 
result remains the same whatever thai convention may be. I 
have imposed upon myself only slight restrictions in assuming 
that the function <^(a) is continuous and periodic, and these hy- 
potheses are so natural that we may ask ourselves how they can 
be escaped. 

Examination of the three preceding examples, so different in 
all respects, has already given us a glimpse, on the one hand, 
of the role of what philosophers call the principle of sufficient 
reason, and, on the other hand, of the importance of the fact that 
certain properties are common to all continuous functions. The 
study of probability in the physical sciences will lead us to the 
same result. 

III. Probability in the Physical Sciences. — ^We come now 
to the problems connected with what I have called the second 
degree of ignorance, those, namely, in which we know the law, 
but do not know the initial state of the system. I could multiply 
examples, but will take only one. What is the probable present 
distribution of the minor planets on the zodiac t 

We know they obey the laws of Kepler. We may even, with- 
out at all changing the nature of the problem, suppose that their 
orbits are all circular, and situated in the same plane, and that we 
know this plane. On the other hand, we are in absolute ignorance 
as to what was their initial distribution. However, we do not 


hesitate to affirm that their distribution is now nearly nnifomu 

Let b be the longitade of a minor planet in the initial epoch, 
that is to say y the epoch zero. Let a be its mean motion. Its longi- 
tude at the present epoch, that is to say, at the epoch ty will be 
at + b. To say that the present distribution is uniform is to say 
that the mean value of the sines and cosines of multiples otat-\'b 
is zero. Why do we assert this ? 

Let us represent each minor planet by a point in a plane, to 
wit, by a point whose coordinates are precisely a and b. All 
these representative points will be contained in a certain region 
of the plane, but as they are very numerous, this region will 
appear dotted with points. We know nothing else about the dis- 
tribution of these points. 

What do we do when we wish to apply the calculus of proba- 
bilities to such a question! What is the probability that one or 
more representative points may be found in a certain portion of 
the plane t In our ignorance, we are reduced to making an arbi- 
trary hypothesis. To explain the nature of this hypothesis, allow 
me to use, in lieu of a mathematical formula, a crude but con- 
crete image. Let us suppose that over the surface of our plane 
has been spread an imaginary substance, whose density is vari- 
able, but varies continuously. We shall then agree to say that the 
probable number of representative points to be found on a portion 
of the plane is proportional to the quantity of fictitious matter 
found there. If we have then two regions of the plane of the 
same extent, the probabilities that a representative point of one 
of our minor planets is found in one or the other of these regions 
will be to one another as the mean densities of the fictitious matter 
in the one and the other region. 

Here then are two distributions, one real, in which the repre- 
sentative points are very numerous, very close together, but dis- 
crete like the molecules of matter in the atomic hypothesis; the 
other remote from reality, in which our representative points are 
replaced by continuous fictitious matter. We know that the latter 
can not be real, but our ignorance forces us to adopt it. 

If again we had some idea of the real distribution of the 
representative points, we could arrange it so that in a region 



of some extent the density of thia imaginary continuous matter 
would be nearly proportional to tlie number of the representative 
points, or, if you wish, to the number of atoms which are con- 
tained in that region. Even that is impossible, and our ignorance 
is BO great that we are forced to choose arbitrarily the function 
which de6nes the density of our imaginary matter. Only we shall 
be forced to a hypothesis from which we can hardly get away, 
we shall suppose that this function is continuous. That is suf- 
ficient, as we shall see, to enable us to reach a conclusion. 

What is at the instant t the probable distribution of the minor 
planetst Or rather what is the probable value of the sine of the _ 
longitude at the instant (, that is to say of sin (ot + 6) I Wtf] 
made at the outset an arbitrary convention, but if we adopt itM 
this probable value is entirely defined. Divide the plane into ele- 
ments of surface. Consider the value of sin (o( -\-h) at the cen- 
ter of each of these elements; multiply this value by the surface 
of the element, and by the corresponding density of the imaginary 
matter. Take then the sum for alt the elements of the plane. 
This 8um, by definition, will be the probable mean value we seek, 
which will thus be expressed by a double integral. It may be 
thought at first that this mean value depends on the choice of the 
function which defines the density of the imaginary matter, and 
that, as this function ^ is arbitrary, we can, according to the 
arbitrary choice which we make, obtain any mean value. This 
is not so, 

A simple calculation shows that our double integral decreases 
veiy rapidly when ( increases. Thus I could not quite tell what 
liypothesis to make as to the probability of this or that initial 
distribution 1 but whatever the hypothesis made, the result will 
be the same, and this gets me out of my difficulty. 

Whatever be the function <^, the mean value tends toward zero 
as ( increases, and as the minor planets have certainly accom- 
plished a very great number of revolutions, I may assert that this 
mean value is very small. 

I may clioose ^ as I wish, save always on« restriction: thia 
function must be continuous ; and, in fact, from the poiat of v 
of subjective probability, the choice of a discontinuous functiol 
would have been unreasonable. For instance, what reason coal4 


I have for supposing that the initial longitude might be exactly 
0®, but that it could not lie between 0® and 1® t 

But the diflBeulty reappears if we take the point of view of 
objective probability, if we pass from our imaginary distribution 
in which the fictitious matter was supposed continuous, to the real 
distribution in which our representative points form, as it were, 
discrete atoms. 

The mean value of sin (a^ + &) will be represented quite 
simply by 

Izsin (a« + d), 

n being the number of minor planets. In lieu of a double integral 
referring to a continuous function, we shall have a sum of dis- 
crete terms. And yet no one will seriously doubt that this mean 
value is practically very small. 

Our representative points being very close together, our dis- 
crete sum will in general differ very little from an integral. 

An integral is the limit toward which a sum of terms tends 
when the number of these terms is indefinitely increased. If the 
terms are very numerous, the sum will differ very little from 
its limit, that is to say from the integral, and what I said of this 
latter will still be true of the sum itself. 

Nevertheless, there are exceptions. If, for instance, for all 
the minor planets, 

6 = ^— ot, 

the longitude for all the planets at the time t would be 7r/2, and 
the mean value would evidently be equal to unity. For this to 
be the ease, it would be necessary that at the epoch 0, the minor 
planets must have all been lying on a spiral of peculiar form, with 
its spires very close together. Every one will admit that such an 
initial distribution is extremely improbable (and, even supposing 
it realized, the distribution would not be uniform at the present 
time, for example, on January 1, 1913, but it would become so 
a few years later) . 

Why then do we think this initial distribution improbable t 
This must be explained, because if we had no reason for rejecting 


as improbable this absurd hypothesis everything would break 
down, and we could no longer make any affirmation about the 
probability of this or that present distribution. 

Once more we shall invoke the principle of sufficient reason to 
which we must always recur. We might admit that at the begin- 
ning the planets were distributed almost in a straight line. We 
might admit that they were irregularly distributed. But it seems 
to us that there is no sufficient reason for the unknown cause that 
gave them birth to have acted along a curve so regular and yet so 
complicated, which would appear to have been expressely chosen 
so that the present distribution would not be uniform. 

rV. RouoE ET Nom. — The questions raised by games of chance, 
such as roulette, are, fundamentally, entirely analogous to those 
we have just treated. For example, a wheel is partitioned into 
a great number of equal subdivisions, alternately red and black. 
A needle is whirled with force, and after having made a great 
number of revolutions, it stops before one of these subdivisions. 
The probability that this division is red is evidently 1/2. The 
needle describes an angle 0, including several complete revolu- 
tions. I do not know what is the probability that the needle may 
be whirled with a force such that this angle should lie between d 
and 6 -{-dO ; but I can make a convention. I can suppose that this 
probability is 4>{6)dB. As for the function <^(^), I can choose it 
in an entirely arbitrary manner. There is nothing that can guide 
me in my choice, but I am naturally led to suppose this function 

Let € be the length (measured on the circumference of radius 
1) of each red and black subdivision. We have to calculate the 
integral of <t>{0)d$, extending it, on the one hand, to all the red 
divisions, and, on the other hand, to all the black divisions, and to 
compare the results. 

Consider an interval 2c, comprising a red division and a black 
division which follows it. Let M and m be the greatest and least 
values of the function <f>{0) in this interval. The integral extended 
to the red divisions will be smaller than 2Mc ; the integral extended 
to the black divisions will be greater than ^m€\ the diflPerence 
will therefore be less than 2(M — m)€. But, if the function $ 
is supposed continuous; if, besides, the interval c is very 


small with respect to the total angle described by the needle, 
the difference M — m will be very small. The difference of the 
two integrals will therefore be very small, and the probability 
will be very nearly 1/2. 

We see that without knowing anything of the function 0, I 
must act as if the probability were 1/2. We understand, on the 
other hand, why, if, placing myself at the objective point of 
view, I observe a certain number of coups, observation will give 
me about as many black coups as red. 

All players know this objective law; but it leads them into a 
remarkable error, which has been often exposed, but into which 
they always fall again. When the red has won, for instance, six 
times running, they bet on the black, thinking they are playing a 
safe game ; because, say they, it is very rare that red wins seven 
times running. 

In reality their probability of winning remains 1/2. Observa- 
tion shows, it is true, that series of seven consecutive reds are very 
rare, but series of six reds followed by a black are just as rare. 

They have noticed the rarity of the series of seven reds; if 
they have not remarked the rarity of six reds and a black, it is 
only because such series strike the attention less. 

V. The ProbabiltITy op Causes. — ^We now come to the prob- 
lems of the probability of causes, the most important from the 
point of view of scientific applications. Two stars, for instance, 
are very close together on the celestial sphere. Is this apparent 
contiguity a mere effect of chance? Are these stars, although on 
almost the same visual ray, situated at very different distances 
from the earth, and consequently very far from one another! 
Or, perhaps, does the apparent correspond to a real contiguity! 
This is a problem on the probability of causes. 

I recall first that at the outset of all problems of the proba- 
bility of effects that have hitherto occupied us, we have always 
had to make a convention, more or less justified. And if in most 
cases the result was, in a certain measure, independent of this 
convention, this was only because of certain hypotheses which 
permitted us to reject a priori discontinuous functions, for ex- 
ample, or certain absurd conventions. 

We shall find something analogous when we deal with the 

' |»*obBbiUty of causes. An effect may be produced by the cause 
A or by the cause B, The effect has just been observed. We 
ask the probability that it is due to the cause A. This is an a 
posteriori probability of cause. But I could not calculate it, if 
a convention more or less justified did not tell me in advance 
irbat is the a priori probability for the cause A to come into 
play; I mean the probability of this event for some one who had 
not observed the effect. 

The better to explain myself I go back to the example of the 
game of ^carte mentioned above. My adversary deals for the 
first time and he turns up a king. What is the probability that he 
is a sharper? The formulas ordinarily taught give 8/9, a result 
evidently rather surprising. If we look at it closer, we see that 
the calculation is made as if, before sittirtg down at the tahle, I 
had considered that there was one chance in two that my adver- 
sary was not honest. An absurd hypothesis, because in that case 
I should have certainly not played with him, and this explains 
the absurdity of the conclusion. 

The convention about the a priori probability was unjustitied, 
and that is why the calculation of the a posteriori probnbility led 
me to an inadmissible result. We see the importance of this pre- 
liminary convention. I shall even add that if none were made, 
the problem of the a posteriori probability would have no mean- 
ii^. It must always be made either explicitly or tacitly. 

Pass to an example of a more scientific character. I wish to 
determine an experimental law. This law, when 1 know it, can 
be represented by a curve. I make a certain number of isolated 
observations; each of these will be represented by a point. When 
I have obtained these different points, I draw a curve between 
them, striving to pass as near to them as possible and yet preserve 
for my curve a regular form, without angular points, or inflec- 
tions too accentuated, or brusque variation of the radius of curva- 
ture. This curve will represent for me the probable law, and I 
assnme not only that it will tell me the values of the function 
intermediate between those which have been observed, but also 
that it will give me the observed values themselves more exactly J 
than direct observation. This is why I make it pass near t 
I, and not through the points themselves. 


Here is a problem in the probability of causes. The effects 
are the measurements I have recorded ; they depend on a combina- 
tion of two causes: the true law of the phenomenon and the 
errors of observation. Knowing the effects, we have to seek the 
probability that the phenomenon obeys this law or that, and that 
the observations have been affected by this or that error. The 
most probable law then corresponds to the curve traced, and the 
most probable error of an observation is represented by the dis- 
tance of the corresponding point from this curve. 

But the problem would have no meaning if, before any obser- 
vation, I had not fashioned an a priori idea of the probability of 
this or that law, and of the chances of error to which I am exposed. 

If my instruments are good (and that I knew before making 
the observations), I shall not permit my curve to depart much 
from the points which represent the rough measurements. If 
they are bad, I may go a little further away from them in order 
to obtain a less sinuous curve ; I shall sacrifice more to regularity. 

Why then is it that I seek to trace a curve without sinuosities! 
It is because I consider a priori a law represented by a continu- 
ous function (or by a function whose derivatives of high order 
are small), as more probable than a law not satisfying these con- 
ditions. Without this belief, the problem of which we speak 
would have no meaning; interpolation would be impossible; no 
law could be deduced from a finite number of observations; 
science would not exist. 

Fifty years ago physicists considered, other things being equal, 
a simple law as more probable than a complicated law. They 
even invoked this principle in favor of Mariotte's law as against 
the experiments of Eegnault. To-day they have repudiated this 
belief; and yet, how many times are they compelled to act as 
though they still held it ! However that may be, what remains of 
this tendency is the belief in continuity, and we have just seen 
that if this belief were to disappear in its turn, experimental 
science would become impossible. 

VI. The Theory op Errors. — ^We are thus led to speak of 
the theory of errors, which is directly connected with the problem 
of the probability of causes. Here again we find effects, to wit, 
a certain number of discordant observations, and we seek to 


divine the causes, which are, on the one hand, the real value of the 
quantity to be measured; on the other hand, the error made in 
each isolated observation. It is necessary to calculate what is 
a posteriori the probable magnitude of each error, and conse- 
quently the probable value of the quantity to be measured. 

But as I have just explained, we should not know how to un- 
dertake this calculation if we did not admit a priori, that is to 
say, before all observation, a law of probability of errors. Is 
there a law of errors t 

The law of errors admitted by all calculators is Gauss's law, 
which is represented by a certain transcendental curve known 
under the name of 'the bell.' 

But first it is proper to recall the classic distinction between 
systematic and accidental errors. If we measure a length with 
too long a meter, we shall always find too small a number, and 
it will be of no use to measure several times; this is a systematic 
error. If we measure with an accurate meter, we may, however, 
make a mistake ; but we go wrong, now too much, now too little, 
and when we take the mean of a great number of measurements, 
the error will tend to grow small. These are accidental errors. 

It is evident from the first that systematic errors can not 
satisfy Gauss's law; but do the accidental errors satisfy itt A 
great number of demonstrations have been attempted ; almost all 
are crude paralogisms. Nevertheless, we may demonstrate 
Gauss's law by starting from the following hypotheses: the error 
committed is the result of a great number of partial and inde- 
pendent errors; each of the partial errors is very little and 
besides, obeys any law of probability, provided that the prob- 
ability of a positive error is the same as that of an equal negative 
error. It is evident that these conditions will be often but not 
always fulfilled, and we may reserve the name of accidental for 
errors which satisfy them. 

We see that the method of least squares is not legitimate in 
every case; in general the physicists are more distrustful of it 
than the astronomers. This is, no doubt, because the latter, be- 
sides the systematic errors to which they and the physicists are 
subject alike, have to contend with an extremely important source 
of error which is wholly accidental ; I mean atmospheric undula- 


lions. So it is very curious to hear a physicist discuss with an 
astronomer about a method of observation. The physicist, per- 
suaded that one good measurement is worth more than many 
bad ones, is before all concerned with eliminating by dint of 
precautions the least rystematic errors, and the astronomer says 
to him: 'But thus you can observe only a small number of stars; 
the accidental errors will not disappear. ' 

What should we conclude f Must we continue to use the 
method of least squares f We must distinguish. We have elimi- 
nated all the systematic errors we could suspect ; we know well 
there are still others, but we can not detect them; yet it is 
necessary to make up our mind and adopt a definitive value 
which will be regarded as the probable value ; and for that it is 
evident the best thing to do is to apply Gauss's method. We 
have only applied a practical rule referring to subjective prob- 
ability. There is nothing more to be said. 

But we wish to go farther and affirm that not only is the 
probable value so much, but that the probable error in the re- 
sult is so much. This is absolutely illegitimate; it would be true 
only if we were sure that all the systematic errors were elimi- 
nated, and of that we know absolutely nothing. We have two 
series of observations ; by applying the rule of least squares, we 
find that the probable error in the first series is twice as small 
as in the second. The second series may, however, be better than 
the first, because the first perhaps is affected by a large system- 
atic error. All we can say is that the first series is probably 
better than the second, since its accidental error is smaller, and 
we have no reason to affirm that the systematic error is greater 
for one of the series ^han for the other, our ignorance on this 
point being absolute. 

VII. Conclusions. — In the lines which precede, I have set 
many problems without solving any of them. Yet I do not regret 
having written them, because they will perhaps invite the reader 
to reflect on these delicate questions. 

However that may be, there are certain points which seem 
well established. To undertake any calculation of probability, 
and even for that calculation to have any meaning, it is neces- 


sary to admit, as point of departure, a hypothesis or convention 
which has always something arbitrary about it. In the choice 
of this convention, we can be guided only by the principle of 
sufBcient reason. Unfortunately this principle is very vague 
and very elastic, and in the cursory examination we have just 
made, we have seen it take many different forms. The form un- 
der which we have met it most often is the belief in continuity, a 
belief which it would be difficult to justify by apodeictic reason- 
ing, but without which all science would be impossible. Finally 
the problems to which the calculus of probabilities may be applied 
with profit are those in which the result is independent of the 
hypothesis made at the outset, provided only that this hypothesis 
satisfies the condition of continuity. 

Optics and Electricity 

Fresnel's Theory. — The best example^ that can be chosen 
of physics in the making is the theory of light and its relations to 
the theory of electricity. Thanks to Fresnel, optics is the best 
developed part of physics; the so-called wave-theory forms a 
whole truly satisfying to the mind. We must not, however, ask 
of it what it can not give us. 

The object of mathematical theories is not to reveal to us the 
true nature of things ; this would be an unreasonable pretension. 
Their sole aim is to coordinate the physical laws which experi- 
ment reveals to us, but which, without the help of mathematics, 
we should not be able even to state. 

It matters little whether the ether really exists; that is the 
affair of metaphysicians. The essential thing for us is that 
everything happens as if it existed, and that this hypothesis is 
convenient for the explanation of phenomena. After all, have 
we any other reason to believe in the existence of material 
objects? That, too, is only a convenient hypothesis; only this 
will never cease to be so, whereas, no doubt, some day the ether 
will be thrown aside as useless. But even at that day, the laws 
of optics and the equations which translate them analytically 
will remain true, at least as a first approximation. It will always 
be useful, then, to study a doctrine that unites all these equations. 

The undulatory theory rests on a molecular hypothesis. For 
those who think they have thus discovered the cause under the 
law, this is an advantage. For the others it is a reason for dis- 
trust. But this distrust seems to me as little justified as the 
illusion of the former. 

These hypotheses play only a secondary part. They might be 
sacrificed. They usually are not, because then the explanation 
would lose in clearness ; but that is the only reason. 

1 This chapter is a partial reproduction of the prefaces of two of my 
works: TMorie math&matique de la lumi^e (Paris, Naud, 1889), and Eleo- 
iriciU et optigue (Paris, Naud, 1901). 



In fact, if we looked closer we should see that only two things 
are borrowed from the molecular hypotheses : the principle of the 
consecration of energy, and the linear form of the equations, 
which is the general law of small movements, as of all small 

This explains why most of Fresnel's conclusions remain un- 
changed when we adopt the electromagnetic theory of light. 

Maxwell's Theory. — ^Maxwell, we know, connected by a 
close bond two parts of physics until then entirely foreign to one 
another, optics and electricity. By blending thus in a vaster 
whole, in a higher harmony, the optics of Fresnel has not ceased 
to be alive. Its various parts subsist, and their mutual relations 
are stiU the same. Only the language we used to express them 
has changed ; and, on the other hand, MaxweU has revealed to us 
other relations, before unsuspected, between the different parts 
of optics and the domain of electricity. 

When a French reader first opens Maxwell's book, a feeling 
of uneasiness and often even of mistrust mingles at first with his 
admiration. Only after a prolonged acquaintance and at the 
cost of many efforts does this feeling disappear. There are even 
some eminent minds that never lose it. 

Why are the English scientist's ideas with such difficulty 
acclimatized among usT It is, no doubt, because the education 
received by the majority of enlightened Frenchmen predisposes 
them to appreciate precision and logic above every other quality. 

The old theories of mathematical physics gave us in this re- 
spect complete satisfaction. All our masters, from Laplace to 
Cauchy, have proceeded in the same way. Starting from clearly 
stated hypotheses, they deduced all their consequences with 
mathematical rigor, and then compared them with experiment. 
It seemed their aim to give every branch of physics the same pre- 
cision as celestial mechanics. 

A mind accustomed to admire such models is hard to suit with 
a theory. Not only will it not tolerate the least appearance of 
contradiction, but it will demand that the various parts be 
logically connected with one another, and that the number of 
distinct hypotheses be reduced to minimum. 

This is not all ; it will have still other demands, which seem to 


me less reasonable. Behind the matter which our senses can 
reach, and which experiment tells us of, it will desire to see 
another, and in its eyes the only real, matter, which wiU have 
only purely geometric properties, and whose atoms wiU be noth- 
ing but mathematical points, subject to the laws of dynamics 
alone. And yet these atoms, invisible and without color, it will 
seek by an unconscious contradiction to represent to itself and 
consequently to identify as closely as possible with common 

Then only will it be fully satisfied and imagine that it has 
penetrated the secret of the universe. If this satisfaction is de- 
ceitful, it is none the less difficult to renounce. 

Thus, on opening Maxwell, a Frenchman expects to find a 
theoretical whole as logical and precise as the physical optics 
based on the hypothesis of the ether; he thus prepares for him- 
self a disappointment which I should Uke to spare the reader by 
informing him immediately of what he must look for in Maxwell, 
and what he can not find there. 

Maxwell does not give a mechanical explanation of electricity 
and magnetism; he confines himself to demonstrating that such 
an explanation is possible. 

He shows also that optical phenomena are only a special case 
of electromagnetic phenomena. From every theory of electri- 
city, one can therefore deduce immediately a theory of light. 

The converse unfortunately is not true; from a complete ex- 
planation of light, it is not always easy to derive a complete ex- 
planation of electric phenomena. This is not easy, in particular, 
if we wish to start from Fresnel's theory. Doubtless it would 
not be impossible ; but nevertheless we must ask whether we are 
not going to be forced to renounce admirable results that we 
thought definitely acquired. That seems a step backward; and 
many good minds are not willing to submit to it. 

When the reader shall have consented to limit his hopes, he 
will still encounter other difficulties. The English scientist does 
not try to construct a single edifice, final and well ordered; he 
seems rather to erect a great number of provisional and inde- 
pendent constructions, between which communication is difficult 
and sometimes impossible. 



Take as example the chapter in which he explains electrostatie 
attractionB by pressures and tensions in the dielectric medium. 
Thia chapter might be omitted without making thereby the rest 
of the book less clear or complete ; and, on the other hand, it con- 
tains a theory complete in itself which one could understand with- 
out having read a single line that precedes or follows. But it 
is not only independent of the rest of the work; it is difGcult to 
reconcile with the fundamental ideas of the book. Maxwell does 
not even attempt thia reconciliation; he merely says: "I have 
not been able to make the nest step, namely, to account by 
mechanical considerations for these stresses in the dielectric." 

This example will suEBce to make my thought understood ; I 
could cite many others. Thua who would suspect, in reading 
the pages devoted to magnetic rotary polarization, that there is 
an identity between optical and magnetic phenomena! 

One must not then datter himself that he can avoid all con- 
tradiction; to that it is necessary to be resigned. In fact, two 
contradictory theories, provided one does not mingle them, and 
if one does not seek in them the basis of things, may both be 
useful instruments of research; and perhaps the reading of 
Maxwell would be less suggestive if he had not opened up to us 
so many new and divergent paths. 

The fundamental idea, however, is thus a little obscured. So 
far ie this the case that in the majority of popularized versions 
it is the only point completely left aside. 

I feel, then, that the better to make its importance stand out. 
I ought to explain in what this fundamental idea consists. Bnt 
for that a short digression is necessary. 

The Mechanical Explanation op Physical Phenouena. — 
Th«re is in every physical phenomenon a certain number of 
parameters which experiment reaches directly and allows us to 
measure. I shall call these the parameters q. 

Observation then teaches us the laws of the variations of these 
parameters; and these laws can generally be put in the form 
of differential equations, which connect the parameters q with the 

What is it necessary to do to give a mechanical interpretation 
of Baeb a phenomenon! 


One will try to explain it either by the motions of ordinary 
matter, or by those of one or more hypothetical fluids. 

These fluids will be considered as formed of a very great num- 
ber of isolated molecules m. 

When shall we say, then, that we have a complete mechanical 
explanation of the phenomenon f It will be, on the one hand, 
when we know the differential equations satisfied by the coordi- 
nates of these hypothetical molecules m, equations which, more- 
over, must conform to the principles of dynamics; and, on the 
other hand, when we know the relations that define the coordi- 
nates of the molecules m as functions of the parameters q acces- 
sible to experiment. 

These equations, as I have said, must conform to the prin- 
ciples of dynamics, and, in particular, to the principle of the 
conservation of energy and the principle of least action. 

The first of these two principles teaches us that the total energy 
is constant and that this energy is divided into two parts : 

1® The kinetic energy, or vis viva, which depends on the 
masses of the hypothetical molecules m, and their velocities, and 
which I shall call T. 

2° The potential energy, which depends only on the coordi- 
nates of these molecules and which I shall call U. It is the sum 
of the two energies T and U which is constant. 

"What now does the principle of least action tell us? It tells 
us that to pass from the initial position occupied at the instant ^o 
to the final position occupied at the instant ^i, the system must 
take such a path that, in the interval of time that elapses be- 
tween the two instants t^ and ^i, the average value of *the 
action' (that is to say, of the differ erice between the two energies 
T and U) shall be as small as possible. 

If the two functions T and U are known, this principle suffices 
to determine the equations of motion. 

Among all the possible ways of passing from one position to 
another, there is evidently one for which the average value of 
the action is less than for any other. There is, moreover, only 
one; and it results from this that the principle of least action 
suffices to determine the path followed and consequently the 
equations of motion. 


Thus we obtain what are called the equations of Lagrange. 

In these equations, the independent variables are the coordi- 
nates of the hypothetical molecules m; but I now suppose that 
one takes as variables the parameters q directly accessible to ex- 

The two parts of the energy must then be expressed as func- 
tions of the parameters q and of their derivatives. They will 
evidently appear under this form to the experimenter. The 
latter will naturally try to define the potential and the kinetic 
energy by the aid of quantities that he can directly observe.^ 

That granted, the system will always go from one position to 
another by a path such that the average action shall be a mini- 

It matters little that T and U are now expressed by the aid 
of the parameters q and their derivatives ; it matters little that it 
is also by means of these parameters that we define the initial and 
final positions ; the principle of least action remains always true. 

Now here again, of all the paths that lead from one position 
to another, there is one for which the average action is a mini- 
mum, and there is only one. The principle of least action 
suffices, then, to determine the differential equations which de- 
fine the variations of the parameters q. 

The equations thus obtained are another form of the equa- 
tions of Lagrange. 

To form these equations we need to know neither the relations 
that connect the parameters q with the coordinates of the 
hypothetical molecules, nor the masses of these molecules, nor 
the expression of 17 as a function of the coordinates of these 

All we need to know is the expression of Z7 as a function of 
the parameters, and that of T as a function of the parameters q 
and their derivatives, that is, the expressions of the kinetic and 
of the potential energy as functions of the experimental data. 

Then we shall have one of two things: either for a suitable 

s We add that U wiU depend only on the parameters q, that T will depend 
on the parameters q and their derivatives with respect to the time and wiU 
be a homogeneous polynomial of the second degree with respect to these 


choice of the functions T and U, the equations of Lagrange, con- 
structed as we have just said, will be identical with the differ- 
ential equations deduced from experiments; or else there will 
exist no functions T and U, for which this agreement takes place. 
In the latter case it is clear that no mechanical explanation i^ 

The necessary condition for a mechanical explanation to be 
possible is therefore that we can choose the functions T and U 
in such a way as to satisfy the principle of least action, which in- 
volves that of the conservation of energy. 

This condition, moreover, is sufficient. Suppose, in fact, that 
we have found a function Z7 of the parameters q, which repre- 
sents one of the parts of the energy; that another part of the 
energy, which we shall represent by T, is a function of the 
parameters q and their derivatives, and that it is a homogeneous 
polynomial of the second degree with respect to these derivatives; 
and finally that the equations of Lagrange, formed by means of 
these two functions, T and U, conform to the data of the 

What is necessary in order to deduce from this a mechanical 
explanation? It is necessary that U can be regarded as the po- 
tential energy of a system and T as the vis viva of the same 

There is no diflSculty as to U, but can T be regarded as the 
vis viva of a material system ? 

It is easy to show that this is always possible, and even in 
an infinity of ways. I will confine myself to referring for more 
details to the preface of my work, 'Electricity et optique.' 

Thus if the principle of least action can not be satisfied, no 
mechanical explanation is possible ; if it can be satisfied, there is 
not only one, but an infinity, whence it follows that as soon as 
there is one there is an infinity of others. 

One more observation. 

Among the quantities that experiment gives us directly, we 
shall regard some as functions of the coordinates of our hypo- 
thetical molecules; these are our parameters g. We shall look 
upon the others as dependent not only on the coordinates, but on 
the velocities, or, what comes to the same thing, on the derivatives 

L '4iCtiie parameters g, or as combinations of these parameters and 
their derivatives. 

And then a question presents itself: among all these quantities 
measured experimentally, which shall we choose to represent the 
parameters gT Which sliall we prefer to regard aa the deriva- 
tives of these parameters'? This choice remains arbitrary to a 
very large extent ; but, for a mechanical explanation to be possi- 
ble, it suffices if we can make tlie choice in such a way as to 
accord with the prinei]jle of least action. 

And then Maxwell asked himself whether he could make this 
choice and that of the two energies T and U, in such a way 
that the electrical phenomena would satisfy this principle. Ex- 
periment shows us that the energy of an electromagnetic field is 
decomposed into two parts, the electrostatic energy and the elee- 
trodynamic energy. Maxwell observed that if we regard the 
first as representing the potential energy U, the second as repre- 
senting the kinetic energy T; if, moreover, the electrostatic 
charges of the conductors are considered as parameters q and 
the intensities of the currents aa the derivatives of other para- 
meters q ; under these conditions, I say. Maxwell observed that the 
electric phenomena satisfy the principle of least action. Thence- 
forth he was certain of the possibility of a mechanical ex- 

If he had explained this idea at the beginning of his book 
instead of relegating it to an obscure part of the second volume, 
it would not have escaped thp majority of readers. 

If, then, a phenomenon admits of a complete mechanical ex- 
planation, it will admit of an infinity of others, that will render 
an acoount equally well of all the particulars revealed by ex- 

And this is confirmed by the history of every branch of 
physics ; in optics, for instance, Presnel believed vibration to be 
perpendicular to the plane of polarization; Neumann regarded 
it as parallel to this plane. An ' experimentnm erucis' has long 
been sought which would enable us to decide between these two 
theories, but it has not been found. 

In the same way, without leaving the domain of electricity, 
we may ascertain that the theory of two fluids and that of the 


single fluid both acconnt in a fashion equally satisfactory for all 
the observed laws of electrostatics. 

All these facts are easily explicable, thanks to the properties 
of the equations of Lagrange which I have just recalled. 

It is easy now to comprehend what is Maxwell's fundamental 

To demonstrate the possibility of a mechanical explanation of 
electricity, we need not preoccupy ourselves with finding this 
explanation itself; it suffices us to know the expression of the 
two functions T and U, which are the two parts of energy, to 
form with these two functions the equations of Lagrange and 
then to compare these equations with the experimental laws. 

Among all these possible explanations, how make a choice for 
which the aid of experiment fails usf A day will come perhaps 
when physicists will not interest themselves in these questions, 
inaccessible to x>ositive methods, and will abandon them to the 
metaphysicians. This day has not yet arrived; man does not 
resign himself so easily to be forever ignorant of the foundation 
of things. 

Our choice can therefore be further guided only by considera- 
tions where the part of personal appreciation is very great ; there 
are, however, solutions that all the world will reject because of 
their whimsicaUty, and others that all the world wiU prefer be- 
cause of their simplicity. 

In what concerns electricity and magnetism, Maxwell abstains 
from making any choice. It is not that he systematically dis- 
dains all that is unattainable by positive methods; the time he 
has devoted to the kinetic theory of gases sufficiently proves that. 
I will add that if, in his great work, he develops no complete 
explanation, he had previously attempted to give one in an article 
in the Philosophical Magazine. The strangeness and the com- 
plexity of the hypotheses he had been obliged to make had led 
him afterwards to give this up. 

The same spirit is found throughout the whole work. What 
is essential, that is to say what must remain common to all 
theories, is made prominent; all that would only be suitable to 
a particular theory is nearly always passed over in silence. Thus 
the reader finds himself in the presence of a form almost devoid 


of matter, which he is at first tempted to take for a fugitive 
shadow not to be grasped. But the efforts to which he is thus 
condemned force him to think and he ends by comprehending 
what was often rather artificial in the theoretic constructs he 
had previously only wondered at. 


The history of electrodynamics is particularly instructive from 
our point of view. 

Ampere entitled his immortal work, 'Thdorie des ph6nom^nes 
^lectrodynamiques, uniquement fond^ sur Inexperience/ He 
therefore imagined that he had made no hypothesis, but he had 
made them, as we shall soon see; only he made them without 
being conscious of it. 

His successors, on the other hand, perceived them, since their 
attention was attracted by the weak points in Ampere's solution. 
They made new hypotheses, of which this time they were fully 
conscious ; but how many times it was necessary to change them 
before arriving at the classic system of to-day which is perhaps 
not yet final ; this we shall see. 

I. Amp&ke's Theory. — ^When Ampere studied experimentally 
the mutual actions of currents, he operated and he only could 
operate with closed currents. 

It was not that he denied the possibility of open currents. 
If two conductors are charged with positive and negative elec- 
tricity and brought into communication by a wire, a current is 
established going from one to the other, which continues until the 
two potentials are equal. According to the ideas of Ampere's 
time this was an open current; the current was known to go 
from the first conductor to the second, it was not seen to return 
from the second to the first. 

So Ampere considered as open currents of this nature, for ex- 
ample, the currents of discharge of condensers; but he could 
not make them the objects of his experiments because their 
duration is too short. 

Another sort of open current may also be imagined. I sup- 
pose two conductors, A and B, connected by a wire AMB. Small 
conducting masses in motion first come in contact with the 



conductor B, take from it an electric charge, leave contact with 
B and move along the path BNA, and, transporting with them 
their charge, come into contact with A and give to it their charge, 
which returns then to B along the wire AMB, 

Now there we have in a sense a closed circuit, since the elec- 
tricity describes the closed circuit BNAMB; but the two parts 
of this current are very different. In the wire AMB, the elec- 
tricity is displaced through a fixed conductor, like a voltaic cur- 
rent, overcoming an ohmic resistance and developing heat; we 
say that it is displaced by conduction. In the part BNA, the 
electricity is carried by a moving conductor ; it is said to be dis- 
placed by convection. 

If then the current of convection is considered as altogether 
analogous to the current of conduction, the circuit BNAMB is 
closed; if, on the contrary, the convection current is not 'a true 
current,' and, for example, does not act on the magnet, there 
remains only the conduction current AMB, which is open. 

For example, if we connect by a wire the two poles of a Holtz 
machine, the charged rotating disc transfers the electricity by 
convection from one pole to the other, and it returns to the first 
pole by conduction through the wire. 

But currents of this sort are very difficult to produce with ap- 
preciable intensity. With the means at Ampere's disposal, we 
may say that this was impossible. 

To sum up, Ampere could conceive of the existence of two 
kinds of open currents, but he could operate on neither because 
they were not strong enough or because their duration was too 

Experiment therefore could only show him the action of a 
closed current on a closed current, or, more accurately, the action 
of a closed current on a portion of a current, because a current 
can be made to describe a closed circuit composed of a moving 
part and a fixed part. It is possible then to study the displace- 
ments of the moving part under the action of another closed 

On the other hand, Ampfere had no means of studying the 
action of an open current, either on a closed current or another 
open current. 


1. The Case of Closed Currents. — ^In the case of the mntaal 
action of two closed currents, experiment revealed to Ampere re- 
markably simple laws. 

I recall rapidly here those which will be useful to us in the 
sequel : 

V If the intensity of the currents is kept constant, and if 
the two circuits, after having undergone any deformations and 
displacements whatsoever, return finally to their initial positions, 
the total work of the electrodynamic actions will be nulL 

In other words, there is an electrodynamic potential of the 
two circuits, proportional to the product of the intensities, and 
depending on. the form and relative position of the circuits; the 
work of the electrodynamic actions is equal to the variation of 
this potential: 

2^ The action of a closed solenoid is nulL 

3° The action of a circuit C on another voltaic circuit C de- 
pends only on the 'magnetic field' developed by this circuit. At 
each point in space we can in fact define in magnitude and direc- 
tion a certain force called magnetic force, which enjoys the fol- 
lowing properties : 

(a) The force exercised by C on a magnetic pole is applied to 
that pole and is equal to the magnetic force multiplied by the 
magnetic mass of that pole ; 

(b) A very short magnetic needle tends to take the direction 
of the magnetic force, and the couple to which it tends to reduce 
is proportional to the magnetic force, the magnetic moment of 
the needle and the sine of the dip of the needle ; 

(c) If the circuit C is displaced, the work of the electrody- 
namic action exercised by C on C will be equal to the increment 
of the 'flow of magnetic force' which passes through the circuit. 

2. Action of a Closed Current on a Portion of Current. — 
Ampere not having been able to produce an open current, prop- 
erly so called, had only one way of studying the action of a 
closed current on a portion of current. 

This was by operating on a circuit C composed of two parts, 
the one fixed, the other movable. The movable part was, for 
instance, a movable wire ap whose extremities a and p could 


dide along a fixed wire. In one of the positions of the movable 
wire, the end a rested on the A of the fixed wire and the extrem- 
ity p on the point B of the fixed wire. The current circulated 
from a to p^ that is to say, from Aio B along the movable wire, 
and then it returned from B io A along the fixed wire. This 
current was therefore closed. 

In a second position, the movable wire having slipped, the ex- 
tremity a rested on another point A' of the fixed wire, and the 
extremity p on another point B' of the fixed wire. The current 
circulated then from a to Pj that is to say from A' to B' along the 
movable wire, and it afterwards returned from B' to B, then from 
BXo Aj then finally from A to A'^ always following the fixed wire. 
The current was therefore also closed. 

If a like current is subjected to the action of a closed current 
C, the movable part will be displaced just as if it were acted 
upon by a force. Ampere assumes that the apparent force to 
which this movable part AB seems thus subjected, representing 
the action of the C on the portion ap of the current, is the same 
Bsitap were traversed by an open current, stopping at a and p, 
in place of being traversed by a closed current which after arriv- 
ing at p returns to a through the fixed part of the circuit. 

This hypothesis seems natural enough, and Ampere made it 
unconsciously ; nevertheless it is not necessary, since we shall see 
further on that Helmholtz rejected it. However that may be, it 
permitted Ampere, though he had never been able to produce an 
open current, to enunciate the laws of the action of a closed cur- 
rent on an open current, or even on an element of current. 

The laws are simple: 

1^ The force which acts on an element of current is applied 
to this element; it is normal to the element and to the magnetic 
force, and proportional to the component of this magnetic force 
which is normal to the element. 

2° The action of a closed solenoid on an element of current is 

But the electrodynamic potential has disappeared, that is to 
say that, when a closed current and an open current, whose in- 
tensities have been maintained constant, return to their initial 
positions, the total work is not null. 


3. Continuous Rotations. — ^Among electrodynamic experiments, 
the most remarkable are those in which continuous rotations are 
produced and which are sometimes called unipolar induction ex- 
periments. A magnet may turn about its axis; a current passes 
j5rst through a fixed wire, enters the magnet by the pole N, for 
example, passes through half the magnet, emerges by a sliding 
contact and reenters the fixed wire. 

The magnet then begins to rotate continuously without being 
able ever to attain equilibrium; this is Faraday's experiment 

How is it possible? If it were a question of two circuits of 
invariable form, the one C fixed, the other C movable about an 
axis, this latter could never take on continuous rotation ; in fact 
there is an electrodynamic potential; there must therefore be 
necessarily a position of equilibrium when this potential is a 

Continuous rotations are therefore possible only when the cir- 
cuit C is composed of two parts: one fixed, the other movable 
about an axis, as is the case in Faraday's experiment. Here 
again it is convenient to draw a distinction. The passage from 
the fixed to the movable part, or inversely, may take place either 
by simple contact (the same point of the movable part remaining 
constantly in contact with the same point of the fixed part) , or by 
a sliding contact (the same point of the movable part coming 
successively in contact with diverse points of the fixed part). 

It is only in the second case that there can be continuous rota- 
tion. This is what then happens: The system tends to take a 
position of equilibrium; but, when at the point of reaching that 
position, the sliding contact puts the movable part in communi- 
cation with a new point of the fixed part; it changes the con- 
nections, it changes therefore the conditions of equilibrium, so 
that the position of equilibrium fleeing, so to say, before the 
system which seeks to attain it, rotation may take place indefi- 

Ampere assumes that the action of the circuit on the movable 
part of C is the same as if the fixed part of C did not exist, and 
therefore as if the current passing through the movable part were 



He concludes therefore that the action of a closed on an open 
cnrreot, or inversely that of an open current on a closed current, 
may g^ve rbe to a continuoas rotation. 

But this conclusion depends on the hypotJiesis I have enun- 
ciated and which, as I said above, is not admitted by Helmholtz. 

4. Mutual Action of Two Open Currents, — In what concerns 
the mutual actions of two open currents, and in particular that 
of two elements of current, all experiment breaks down. Am- 
pere has recourse to hypothesis. He supposes : 

1" That the mutual action of two elements reduces to a force 
acting along their join; 

2" That the action of two closed currents is the resultant of 
the mutual actions of their diverse elements, which are besides 
the same as if these elements were isolated. 

What is remarkable is that here again Ampere makes these 
hypotheses unconsciously. 

However that may be, these two hypotheses, together with the 
experiments on closed currents, suffice to determine completely 
the law of the mutual action of two elements. But then most 
of the simple laws we have met in the case of closed currents are 
no longer true. 

In the first place, there is no electrodynamic potential ; nor wag 
there any, as we have seen, in tlie case of a closed current acting 
on an open current. 

Next there is, properly speaking, no magnetic force. 

And, in fact, we have given above three different definitions 
of this force: 

1° By the action on a magnetic pole; 

2° By the director couple which orientates the magnetic 

3° By the action on an element of current. 

But in the case which now occupies us, not only these three 
de&utions are no longer in harmony, but each has lost its mean- 
ing, and in fact: 

1° A magnetic pole is no longer acted upon simply by a ungle 
force applied to this pole. We have seen in fact that the force 
due to the action of an element of current on a pole is not applied 
to the pole, but to the element ; it may moreover be replaced by 
a force applied to the pole and by a couple ; 


2^ The couple which acts on the magnetic needle is no longer 
a simple director couple, for its moment with respect to the axis 
of the needle is not null. It breaks up into a director couple, 
properly so called, and a supplementary couple which tends to 
produce the continuous rotation of which we have above spoken; 

3"* Finally the force acting on an element of current is not 
normal to this element. 

In other words, the unity of th6 magnetic force has disap 

Let us see in what this unity consists. Two systems which 
exercise the same action on a magnetic pole will exert also the 
same action on an indefinitely small magnetic needle, or on an 
element of current placed at the same point of space as this pole. 

Well, this is true if these two systems contain only closed 
currents ; this would no longer be true if these two systems con- 
tained open currents. 

It su£5ces to remark, for instance, that, if a magnetic pole is 
placed at A and an element at B, the direction of the element 
being along the prolongation of the sect AB^ this element which 
will exercise no action on this pole will, on the other hand, exer- 
cise an action either on a magnetic needle placed at the point A, 
or on an element of current placed at the point A. 

5. Induction. — ^We know that the discovery of electrodynamic 
induction soon followed the immortal work of AmpSre. 

As long as it is only a question of closed currents there is no 
difficulty, and Helmholtz has even remarked that the principle of 
the conservation of energy is sufficient for deducing the laws 
of induction from the electrodynamic laws of Ampfere. But 
always on one condition, as Bertrand has well shown; that we 
make besides a certain number of hypotheses. 

The same principle again permits this deduction in the case of 
open currents, although of course we can not submit the result 
to the test of experiment, since we can not produce such currents. 

If we try to apply this mode of analysis to Ampere's theory 
of open currents, we reach results calculated to surprise us. 

In the first place, induction can not be deduced from the 
variation of the magnetic field by the formula well known to 
savants and practicians, and, in fact, as we have said, properly 
speaking there is no longer a magnetic field. 


Bat, further; if a circuit C is subjected to the induction of a 
variable voltaic system 8, if this system 8 be displaced and de- 
formed in any way whatever, so that the intensity of the currents 
of this system varies according to any law whatever, but that 
after these variations the system finally returns to its initial sit- 
uation, it seems natural to suppose that the mean electromotive 
force induced in the circuit C is null. 

This is true if the circuit C is closed and if the system 8 con- 
tains only closed currents. This would no longer be true, if one 
accepts the theory of AmpSre, if there were open currents. So 
that not only induction will no longer be the variation of the 
flow of magnetic force, in any of the usual senses of the word, but 
it can not be represented by the variation of anything whatever. 

11. Theory op Helmholtz. — I have dwelt upon the conse- 
quences of Ampere's theory, and of his method of explaining 
open currents. 

It is difficult to overlook the paradoxical and artificial char- 
acter of the propositions to which we are thus led. One can not 
help thinking 'that can not be so.' 

We understand therefore why Helmholtz was led to seek some- 
thing else. 

Helmholtz rejects Ampere's fundamental hypothesis, to wit, 
that the mutual action of two elements of current reduces to a 
force along their join. He assumes that an element of current is 
not subjected to a single force, but to a force and a couple. It is 
just this which gave rise to the celebrated polemic between Ber- 
trand and Helmholtz. 

Helmholtz replaces Ampere's hypothesis by the following: two 
elements always admit of an electrodynamic potential depend- 
ing solely on their position and orientation ; and the work of the 
forces that they exercise, one on the other, is equal to the varia- 
tion of this potential. Thus Helmholtz can no more do without 
hypothesis than Ampere ; but at least he does not make one with- 
out explicitly announcing it. 

In the case of closed currents, which are alone accessible to 
experiment, the two theories agree. 

In all other cases they differ. 

In the first place, contrary to what Ampere supposed, the force 


which seems to act on the movable portion of a closed current 
is not the same as would act upon this movable portion if it 
were isolated and constituted an open current. 

Let us return to the circuit C, of which we spoke above, and 
which was formed of a movable wire ap sliding on a fixed wire. 
In the only experiment that can be made, the movable portion afi 
is not isolated, but is part of a closed circuit. When it passes 
from AB to A'B', the total electrodynamic potential varies for 
two reasons: 

l"* It undergoes a first increase because the potential of A'V 
with respect to the circuit C is not the same as that of AB\ 

2^ It takes a second increment because it must be increased 
by the potentials of the elements AA', BB' with respect to C. 

It is this double increment which represents the work of the 
force to which the portion AB seems subjected. 

If, on the contrary, ap were isolated, the potential would 
undergo only the first increase, and this first increment alone 
would measure the work of the force which acts on AB. 

In the second place, there could be no continuous rotation 
without sliding contact, and, in fact, that, as we have seen i 
propos of closed currents, is an immediate consequence of the 
existence of an electrodynamic potential. 

In Faraday's experiment, if the magnet is fixed and if the 
part of the current exterior to the magnet runs along a movable 
wire, that movable part may undergo a continuous rotation. 
But this does not mean to say that if the contacts of the wire 
with the magnet were suppressed, and an open current were to 
run along the Avire, the wire would still take a movement of con- 
tinuous rotation. 

I have just said in fact that an isolated element is not acted 
upon in the same way as a movable element making part of a 
closed circuit. 

Another diflference: The action of a closed solenoid on a 
closed current is null according to experiment and according to 
the two theories. Its action on an open current would be null 
according to AmpSre; it would not be null according to Helm- 
holtz. From this follows an important consequence. We have 
given above three definitions of magnetic force. The third has 


no meaning here aince an element of current is no longer acted 
upon by a single force. No more has the first any meaning. 
What, in fact, is a magnetic pole? It is the extremity of an 
indefinite linear magnet. This magnet may be replaced by an 
indefinite solenoid. For the definition of magnetic force to have 
any meaning, it would be necessary that the action exercised by 
an open current on an indefinite solenoid should depend only on 
the position of the extremity of this solenoid, that is to say, that 
the action on a closed solenoid should be null. Now we have 
just seen that such is not tlie case. 

On the other hand, nothing prevents our adopting the second 
definition, which is founded on the measurement of the director 
couple which tends to orientate the magnetic needle. 

But if it is adopted, neither the effects of induction nor the 
electrodynamic effects will depend solely on the distribution of 
the lines of force in this magnetic field. 

1X1. Difficulties Raised by These Theories. — The theory 
of Helmholtz is in advance of that of Ampere ; it Js necessary, 
however, that all the difficulties should be smoothed away. In 
the one as in the other, the phrase 'magnetic field' has no mean- 
ing, or, if we give it one, by a more or less artificial convention, 
the ordinary laws so familiar to all electricians no longer apply i 
thus the electromotive force induced in a wire is no longer 
measured by the number of lines of force met by this wire. 

And our repugnance does not come alone from the difficulty 
of renouncing inveterate habits of language iind of thought. 
There is something more. If we do not believe in action at a di»- 
tanee, electrodynamic phenomena must be explained by a modi- 
fication of the medium. It is precisely this modification that we 
call 'magnetic field.' And then the electrodynamic effects must 
depend only on this field. 

All these difficulties arise from the hypothesis of open currents. 

rV. Maxwell's Theoby. — Such were the difficulties raised 
by the dominant theories when Maxwell appeared, who with a 
ftroke of the pen made them all vanish. To his mind, in fact, 
all currents are closed currents. Maxwell assumes that if in 
8 dielectric the electric field happens to vary, this dielectric 
becomes the seat of a particular phenomenon, acting on the gal- 


vanometer like a current, and which he calls current of dis- 

If then two conductors bearing contrary charges are put in 
communication by a wire, in this wire during the discharge there 
is an open current of conduction; but there are produced at the 
same time in the surrounding dielectric, currents of displacement 
which close this current of conduction. 

We know that Maxwell's theory leads to the explanation of 
optical phenomena, which would be due to extremely rapid elec- 
trical oscillations. 

At that epoch such a conception was only a bold hypothesis, 
which could be supported by no experiment. 

At the end of twenty years, Maxwell's ideas received the con- 
firmation of experiment. Hertz succeeded in producing sys- 
tems of electric oscillations which reproduce all the properties 
of light, and only differ from it by the length of their wave ; that 
is to say as violet differs from red. In some measure he made 
the synthesis of light. 

It might be said that Hertz has not demonstrated directly 
Maxwell's fundamental idea, the action of the current of dis- 
placement on the galvanometer. This is true in a sense. What 
he has shown in sum is that electromagnetic induction is not 
propagated instantaneously as was supposed; but with the speed 
of light. 

But to suppose there is no current of displacement, and induc- 
tion is propagated with the speed of light ; or to suppose that the 
currents of displacement produce effects of induction, and that 
the induction is propagated instantaneously, comes to the samA 

This can not be seen at the first glance, but it is proved by an 
analysis of which I must not think of giving even a summary 

V. Rowland's Experiment. — But as I have said above, there 
are two kinds of open conduction currents. There are first the 
currents of discharge of a condenser or of any conductor what- 

There are also the cases in which electric discharges describe 

a closed contour, being displaced by conduction in one part of 
the circuit and by convection in the other part. 

For open currents of the first sort, the question might be con- 
sidered as solved; they were closed by the currents of displace- 

For open currents of the second sort, the solution appeared 
still more simple. It seemed that if the current were closed, it 
could only be by the current of convection itself. For that it 
snfficed to assume that a 'convection current,' that is to say a 
charged conductor in motion, could act on the galvanometer. 

But experimental confirmation was lacking. It appeared diffi- 
cult in fact to obtain a sufReient intensity even by augmenting as 
much as possible the charge and the velocity of the conductors. It 
was Rowland, an extremely skillful experimenter, who first tri- 
umphed over these difficulties. A disc received a strong electro- 
static charge and a very great speed of rotation. An astatic mag- 
netic system placed beside the disc underwent deviations. 

The experiment was made twice by Rowland, once in Berlin, 
once in Baltimore. It was afterwards repeated by Himetedt. 
These physicists even announced that they had succeeded in mak- 
ing quantitative measurements. 

In fact, for twenty years Rowland's law was admitted without 
objection by all physicists. Besides everything seemed to confirm 
it. The spark certainly does produce a magnetic effect. Now does 
it not aeem probable that the discharge by spark is due to particles 
taken from one of the electrodes and transferred to the other elec- 
trode with their charge t Is not the very spectrum of the spark, 
in which we recognize the lines of the raetal of the electrode, a 
proof of itt The spark would then be a veritable current of 

On the other hand, it is also admitted that in an electrolyte 
tfas electricity is carried by the iona in motion. The current in 
•n electrolj^e would therefore be also a current of convection; 
BOW, it acts on the magnetic needle. 

^nie Bame for cathode rays. Crookes attributed these raya 
to a very subtile matter charged with electricity and movi 
irith a very great velocity. He regarded them, in other, 
words, as currents of convection. Now these cathode rays 

■ayi ^^^H 

ate ^^^H 


deviated by the magnet. In virtue of the principle of action and 
reaction, they should in turn deviate the magnetic needle. It is 
true that Hertz believed he had demonstrated that the cathode 
rays do not carry electricity, and that they do not act on the 
magnetic needle. But Hertz was mistaken. First of all, Perrin 
succeeded in collecting the electricity carried by these rays, elec- 
tricity of which Hertz denied the existence ; the Oerman scientist 
appears to have been deceived by effects due to the action of 
X-rays, which were not yet discovered. Afterwards, and quite 
recently, the action of the cathode rays on the magnetic needle 
has been put in evidence. 

Thus all these phenomena regarded as currents of convection, 
sparks, electrolytic currents, cathode rays, act in the same manner 
on the galvanometer and in conformity with Bowland's law. 

VI. Theory op Lokentz. — ^We soon went further. Accord- 
ing to the theory of Lorentz, currents of conduction themselves 
would be true currents of convection. Electricity would remain 
inseparably connected with certain material particles called elec- 
trons. The circulation of these electrons through bodies would 
produce voltaic currents. And what would distinguish con- 
ductors from insulators would be that the one could be traversed 
by these electrons while the others would arrest their movements. 

The theory of Lorentz is very attractive. It gives a very 
simple explanation of certain phenomena which the earlier the- 
ories, even Maxwell's in its primitive form, could not explain in a 
satisfactory way; for example, the aberration of light, the par- 
tial carrying away of luminous waves, magnetic polarization and 
the Zeeman effect. 

Some objections still remained. The phenomena of an elec- 
tric system seemed to depend on the absolute velocity of transla- 
tion of the center of gravity of this system, which is contrary to 
the idea we have of the relativity of space. Supported by M. 
Cremieu, M. Lippmann has presented this objection in a striking 
form. Imagine two charged conductors with the same velocity 
of translation; they are relatively at rest. However, each of 
them being equivalent to a current of convection, they ought to 
attract one another, and by measuring this attraction we could 
measure their absolute velocity. 


"No!" replied the partisans of Lorentz. ''What we could 
measure in tbat way is not their absolute velocity, but their rela- 
tive velocity wiih respect to the ether, so that the principle of 
relativity is safe." 

Whatever there may be in these latter objections, the edifice of 
electrodynamics, at least in its broad lines, seemed definitively 
constructed. Everything was presented under the most satis- 
factory aspect. The theories of Ampere and of Helmholtz, made 
for open currents which no longer existed, seemed to have no 
longer anything but a purely historic interest, and the inextricable 
complications to which these theories led were almost forgotten. 

This quiescence has been recently disturbed by the experi- 
ments of M. Cr^mieu, which for a moment seemed to contradict 
the result previously obtained by Eowland. 

But fresh researches have not confirmed them, and the theory 
of Lorentz has victoriously stood the test. 

The history of these variations will be none the less instruct- 
ive; it will teach us to what pitfalls the scientist is exposed, and 
how he may hope to escape them. 



1. Does the Scientist create Science f — ^Professor Rados of Buda- 
pest in his report to the Hungarian Academy of Science on the 
award to Poincar^ of the Bolyai prize of ten thousand crowns, 
speaking of him as unquestionably the most powerful investiga- 
tor in the domain of mathematics and mathematical physics, 
characterized him as the intuitive genius drawing the inspiration 
for his wide-reaching researches from the exhaustless fountain 
of geometric and physical intuition, yet working this inspira* 
tion out in detail with marvelous logical keenness. With his 
brilliant creative genius was combined the capacity for sharp 
and successful generalization, pushing far out the boundaries of 
thought in the most widely different domains, so that his works 
must be ranked with the greatest mathematical achievements of 
all time. *' Finally, '* says Bados, *' permit me to make especial 
mention of his intensely interesting book, 'The Value of Science,' 
in which he in a way has laid down the scientist's creed." Now 
what is this creed f 

Sense may act as stimulus, as suggestive, yet not to awaken a 
dormant depiction, or to educe the conception of an archetypal 
form, but rather to strike the hour for creation, to sununon to 
work a sculptor capable of smoothing a Venus of Milo out of the 
formless clay. Knowledge is not a gift of bare experience, nor 
even made solely out of experience. The creative activity of 
mind is in mathematics particularly clear. The axioms of geom- 
etry are conventions, disguised definitions or unprovable hy- 
potheses precreated by auto-active animal and human minds. 
Bertrand Russell says of projective geometry: **It takes nothing 
from experience, and has, like arithmetic, a creature of the pure 
intellect for its object. It deals with an object whose properties 
are logically deduced from its definition, not empirically dis- 
covered from data." Then does the scientist create science? 
This is a question Poincare here dissects with a master hand. 

The physiologic-psychologic investigation of the space problem 



must give the meaning of the words geometric fact, geometric 
reality. Poincarg here subjects to the most successful analysis 
ever made the tridimensionality of our space. 

2. The Mind Dispelling Optical Illusions. — ^Actual perception 
of spatial properties is accompanied by movements correspond- 
ing to its character. In the case of optical illusions, with the so- 
called false perceptions eye-movements are closely related. But 
though the perceived object and its environment remain constant, 
the sufficiently powerful mind can, as we say, dispel these illu- 
sions, the perception itself being creatively changed. Photo- 
graphs taken at intervals during the presence of these optical 
illusions, during the change, perhaps gradual and unconscious, 
in the perception, and after these illusions have, as the phrase is, 
finally disappeared, show quite clearly that changes in eye- 
movements corresponding to those internally created in percep- 
tion itself successively occur. What is called accuracy of move- 
ment is created by what is called correctness of perception. The 
higher creation in the perception is the determining cause of an 
improvement, a precision in the motion. Thus we see correct per- 
ception in the individual helping to make that cerebral organiza- 
tion and accurate motor adjustment on which its possibility and 
permanence seem in so far to depend. So-called correct percep- 
tion is connected with a long-continued process of perceptual 
education motived and initiated from within. How this may 
take place is here illustrated at length by our author. 

3. Euclid not Necessary. — Geometry is a construction of the 
intellect, in application not certain but convenient. As Schiller 
says, when we see these facts as clearly as the development of 
metageometry has compelled us to see them, we must surely con- 
fess that the Kantian account of space is hopelessly and demon- 
strably antiquated. As Royce says in * Kant's Doctrine of the 
Basis of Mathematics, ' * * That very use of intuition which Kant 
regarded as geometrically ideal, the modem geometer regards 
as scientifically defective, because surreptitious. No mathemat- 
ical exactness without explicit proof from assumed principles — 
such is the motto of the modem geometer. But suppose the 
reasoning of Euclid purified of this comparatively surreptitious 


appeal to intuition. Suppose that the principles of geometry are 
made quite explicit at the outset of the treatise, as Fieri and 
Hilbert or Professor Halsted or Dr. Yeblen makes his principles 
explicit in his recent treatment of geometry. Then, indeed, geom- 
etry becomes for the modem mathematician a purely rational 
scienee. But very few students of the logic of mathematics at the 
present time can see any warrant in the analysis of geometrical 
truth for regarding just the Euclidean system of principles as 
possessing any discoverable necessity." Yet the environmental 
and perhaps hereditary premiums on Euclid still make even the 
scientist think Euclid most convenient. 

4. Without Hypotheses, no Science. — ^Nobody ever observed an 
equidistantial, but also nobody ever observed a straight line. 
Emerson's Uriel 

''Gave hlB sentiment divine 
Against the being of a line. 
Line in Nature is not found. ' ' 

Clearly not, being an eject from man's mind. What is called 'a 
knowledge of facts' is usually merely a subjective realization that 
the old hypotheses are still sufficiently elastic to serve in some 
domain; that is, with a sufficiency of conscious or unconscious 
omissions and doctorings and fudgings more or less wilful. In 
the present book we see the very foundation rocks of science, the 
conservation of energy and the indestructibility of matter, beat- 
ing against the bars of their cages, seemingly anxious to take 
wing away into the empyrean, to chase the once divine parallel 
postulate broken loose from Euclid and Kant. 

5. What Outcome? — ^What now is the definite, the permanent 
outcome ? What new islets raise their f ronded palms in air within 
thought's musical domain? Over what age-gray barriers rise the 
fragrant floods of this new spring-tide, redolent of the wolf- 
haunted forest of Transylvania, of far Erd^ly's plunging river, 
llaros the bitter, or broad mother Volga at Kazan ? What victory 
heralded the great rocket for which young Lobachevski, the 
widow's son, was cast into prison? What severing of age-old 
mental fetters symbolized young Bolyai's cutting-oflf with his 



Dam&scos blade the apikes driven into his door-post, and atrew- 
ing over the aod the tliirteen Anatrian cavalry officerat This 
boob b7 the greatest mathematician of our time gives weightiest 
and most charming answer. 

QEOBas Bruce Hai£ted. 


The search for truth should be the goal of our activities ; it is 
the sole end worthy of them. Doubtless we should first bend our 
efforts to assuage human suffering, but whyf Not to suffer is a 
negative ideal more surely attained by the annihilation of the 
world. If we wish more and more to free man from material 
cares, it is that he may be able to employ the liberty obtained in 
the study and contemplation of truth. 

But sometimes truth frightens ns. And in fact we know that it 
is sometimes deceptive, that it is a phantom never showing itself 
for a moment except to ceaselessly fiee, that it must be pursued 
further and ever further without ever being attained. Yet to 
work one must stop, as some Qreek, Aristotle or another, has said. 
We also know how cruel the truth often is, and we wonder 
whether illusion is not more consoling, yea, even more bracing, 
for illusion it is which gives confidence. When it shall have 
vanished, will hope remain and shall we have the courage to 
achieve? Thus would not the horse harnessed to his treadmill 
refuse to go, were his eyes not bandaged? And then to seek 
truth it is necessary to be independent, wholly independent. If, 
on the contrary, we wish to act, to be strong, we should be united. 
This is why many of us fear truth; we consider it a cause of 
weakness. Yet truth should not be feared, for it alone is beautiful. 

When I speak here of truth, assuredly I refer first to scientific 
truth ; but I also mean moral truth, of which what we call justice 
is only one aspect. It may seem that I am misusing words, that 
I combine thus under the same name two things having nothing 
in common ; that scientific truth, which is demonstrated, can in no 
way be likened to moral truth, which is felt. And yet I can not 
separate them, and whosoever loves the one can not help loving 
the other. To find the one, as well as to find the other, it is neces- 
sary to free the soul completely from prejudice and from passion ; 
it is necessary to attain absolute sincerity. These two sorts of 



truth when discovered give the same joy; each when perceived 
beams with the same splendor, so that we must see it or close our 
eyes. Lastly, both attract us and flee from us; they are never 
fixed : when we think to have reached them, we find that we have 
still to advance, and he who pursues them is condemned never to 
know repose. It must be added that those who fear the one will 
also fear the other; for they are the ones who in everything are 
concerned above all with consequences. In a word, I liken the 
two truths, because the same reasons make ns love them and 
because the same reasons make ns fear them. 

If we ought not to fear moral truth, still less should we dread 
scientific truth. In the first place it can not confiict with ethics. 
Ethics and science have their own domains, which touch but do 
not interpenetrate. The one shows ns to what goal we should 
aspire, the other, given the goal, teaches us how to attain it. So 
they can never conflict since they can never meet. There can no 
more be immoral science than there can be scientific morals. 

But if science is feared, it is above all because it can not give us 
happiness. Of course it can not. We may even ask whether the 
beast does not suffer less than man. But can we regret that 
earthly paradise where man brute-like was really immortal in 
knowing not that he must die ? When we have tasted the apple, 
no suffering can make us forget its savor. We always come back 
to it. Could it be otherwise? As well ask if one who has seen 
and is blind will not long for the light. Man, then, can not be 
happy through science, but to-day he can much less be happy 
without it. 

But if truth be the sole aim worth pursuing, may we hope to 
attain it? It may well be doubted. Readers of my little book 
* Science and Hypothesis' already know what I think about the 
question. The truth we are permitted to glimpse is not alto- 
gether what most men call by that name. Does this mean that 
our most legitimate, most imperative aspiration is at the same 
time the most vain? Or can we, despite all, approach truth on 
some side ? This it is which must be investigated. 

In the first place, what instrument have we at our disposal for 
this conquest? Is not human intelligence, more specificaUy the 


intelligence of the scientist, susceptible of infinite variation f 
Volumes could be written without exhausting this subject ; I, in 
a few brief pages, have only touched it lightly. That the geom- 
eter's mind is not like the physicist's or the naturalist's, aU the 
world would agree; but mathematicians themselves do not re- 
semble each other; some recognize only implacable logic, others 
appeal to intuition and see in it the only source of discovery. 
And this would be a reason for distrust. To minds so unlike can 
the mathematical theorems themselves appear in the same light t 
Truth which is not the same for all, is it truth f But looking 
at things more closely, we see how these very different workers 
collaborate in a common task which could not be achieved without 
their cooperation. And that already reassures us. 

Next must be examined the frames in which nature seems en- 
closed and which are called time and space. In 'Science and 
Hypothesis' I have already shown how relative their value is; 
it is not nature which imposes them upon us, it is we who impose 
them upon nature because we find them convenient. But I have 
spoken of scarcely more than space, and particularly quanti- 
tative space, so to say, that is of the mathematical relations whose 
aggregate constitutes geometry. I should have shown that it is 
the same with time as with space and still the same with 'qualita- 
tive space'; in particular, I should have investigated why we 
attribute three dimensions to space. I may be pardoned then for 
taking up again these important questions. 

Is mathematical analysis, then, whose principal object is the 
study of these empty frames, only a vain play of the mind ? It 
can give to the physicist only a convenient language ; is this not 
a mediocre service, which, strictly speaking, could be done with- 
out ; and even is it not to be feared that this artificial language 
may be a veil interposed between reality and the eye of the 
physicist ? Far from it ; without this language most of the inti- 
mate analogies of things would have remained forever unknown 
to us ; and we should forever have been ignorant of the internal 
harmony of the world, which is, we shall see, the only true 
objective reality. 

The best expression of this harmony is law. Law is one of the 


most recent conquests of the human mind; there still are people 
who live in the presence of a perpetual miracle and are not 
astonished at it. On the contrary, we it is who should be aston- 
ished at nature's regularity. Men demand of their gods to prove 
their existence by miracles ; but the eternal marvel is that there 
are not miracles without cease. The world is divine because it is 
a harmony. If it were ruled by caprice, what could prove to us 
it was not ruled by chance ? 

This conquest of law we owe to astronomy, and just this makes 
the grandeur of the science rather than the material grandeur of 
the objects it considers. It was altogether natural, then, that 
celestial mechanics should be the first model of mathematical 
physics; but since then this science has developed; it is still 
developing, even rapidly developing. And it is already neces- 
sary to modify in certain points the scheme from which I drew 
two chapters of ' Science and Hypothesis. ' In an address at the 
St. Louis exposition, I sought to survey the road traveled; the 
result of this investigation the reader shall see farther on. 

The progress of science has seemed to imperil the best estab- 
lished principles, those even which were regarded as fundamental. 
Yet nothing shows they will not be saved ; and if this comes about 
only imperfectly, they will still subsist even though they are 
modified. The advance of science is not comparable to the changes 
of a city, where old edifices are pitilessly torn down to give place 
to new, but to the continuous evolution of zoologic types which 
develop ceaselessly and end by becoming unrecognizable to the 
common sight, but where an expert eye finds always traces of the 
prior work of the centuries past. One must not think then that 
the old-fashioned theories have been sterile and vain. 

Were we to stop there, we should find in these pages some 
reasons for confidence in the value of science, but many more for 
distrusting it ; an impression of doubt would remain ; it is need- 
ful now to set things to rights. 

Some people have exaggerated the role of convention in science ; 
they have even gone so far as to say that law, that scientific fact 
itself, was created by the scientist. This is going much too far 
in the direction of nominalism. No, scientific laws are not arti- 


ficial creations; we have no reason to regard them as accidental, 
though it be impossible to prove they are not. 

Does the harmony the human intelligence thinks it discovers 
in nature eidst outside of this intelligence f No, beyond doubt 
a reality completely independent of the mind which conceives it, 
sees or feels it, is an impossibility. A world as exterior as that, 
even if it existed, would for us be forever inaccessible. But what 
we call objective reality is, in the last analysis, what is common 
to many thinking beings, and could be common to all ; this com- 
mon part, we shall see, can only be the harmony expressed by 
mathematical laws. It is this harmony then which is the sole 
objective reality, the only truth we can attain ; and when I add 
that the universal harmony of the world is the source of all 
beauty, it will be understood what price we should attach to the 
slow and difficult progress whieh little by little enables us to know 
it better. 





Intuition and Logic in Mathematics 


It is impossible to study the works of the great matheniaticiaiis» 
or even those of the lesser, without noticing and distinguishing 
two opposite tendencies, or rather two entirely different kinds of 
minds. The one sort are above all preoccupied with logic; to 
read their works, one is tempted to believe they have advanced 
only step by step, after the manner of a Vauban who pushes 
on his trenches against the place besieged, leaving nothing to 
chance. The other sort are guided by intuition and at the first 
stroke make quick but sometimes precarious conquests, like bold 
cavalrymen of the advance guard. 

The method is not imposed by the matter treated. Though one 
often says of the first that they are analysts and calls the others 
geometers, that does not prevent the one sort from remaining 
analysts even when they work at geometry, while the others are 
still geometers even when they occupy themselves with pure 
analysis. It is the very nature of their mind which makes them 
logicians or intuitionalists, and they can not lay it aside when 
they approach a new subject. 

Nor is it education which has developed in them one of the two 
tendencies and stifled the other. The mathematician is bom, not 
made, and it seems he is bom a geometer or an analyst. I should 
like to cite examples and there are surely plenty; but to accentu- 
ate the contrast I shall begin with an extreme example, taking the 
liberty of seeking it in two living mathematicians. 




M. M^ray wants to prove that a binomial equation alwaye haa 
a root, or, in ordinary words, that an angle may always be sub- 
divided. If there is any truth that we think we know by direct 
intuition, it is this. Wlio eould doubt that an angle may always 
be divided into any number of equal parts T M, Meray does not 
look at it that way; in his eyes tJiis proposition is not at all 
evident and to prove it he needs several pages. 

On the other hand, look at Professor Klein : he is studying one 
of the most abstract questions of the theory of functions : to deter- 
mine whether on a given Kiemann surface there always exists a 
function admitting of given singularities. What does the cele- 
brated Qerman geometer doT He replaces his Riemann surface 
by a metallic surface whose electric conductivity varies according 
to certain laws. He connects two of its points with the two pole 
of a battery. The current, says he, must pass, and the distribn*! 
lion of this current on the surface will define a function whose ' 
singularities will be precisely those called for by the enunciation. 

Doubtless Professor Klein wcU knows he has given here only 
a sketch; nevertheless he has not hesitated to publish it; and he 
would probably believe he finds in it, if not a rigorous demon- 
stration, at least a kind of moral certainty. A logician would 
have rejected with horror such a conception, or rather he would 
not have had to reject it, because in his mind it would never have 

Again, permit me to compare two men, the honor of French 
science, who have recently been taken from us, but who both 
entered long ago into immortality. I speak of M, Bertrand and J 
il. Hermite. They were scholars of the same school at the s 
time; they bad the same education, were under the same inflo- " 
ences; and yet what a difference 1 Not only does it blaze forth 
in their writings ; it is in their teaching, in their way of speaking, 
in their very look. In the memory of all their pupils these two 
faees are stamped in deathless lines; for all who have had the 
pleasure of following their teaching, this remembrance is still 
fresh 1 it is easy for us to evoke it. 

While speaking, ^I. Bertrand is always in motion ; now he seems 
in combat with some outside enemy, now he outlines with a gesture 
of the hand the figures he studies. Plainly he sees and he is 


eager to paint, this is why he calls gesture to his aid. With M. 
Hermite, it is just the opposite; his eyes seem to shun contact 
with the world ; it is not without, it is within he seeks the vision 
of truth. 

Among the Qerman geometers of this century, two names above 
all are illustrious, those of the two scientists who founded the 
general theory of functions, Weierstrass and Biemann. Weier- 
strass leads everything back to the consideration of series and 
their analytic transformations; to express it better, he reduces 
analysis to a sort of prolongation of arithmetic ; you may turn 
through all his books without finding a figure. Biemann, on the 
contrary, at once calls geometry to his aid; each of his concei>- 
tions is an image that no one can forget, once he has caught its 

More recently. Lie was an intuitionalist; this might have been 
doubted in reading his books, no one could doubt it after talking 
¥dth him ; you saw at once that he thought in pictures. Madame 
Eovalevski was a logician. 

Among our students we notice the same differences; some prefer 
to treat their problems 'by analysis,' others 'by geometry.* The 
first are incapable of 'seeing in space,' the others are quickly 
tired of long calculations and become perplexed. 

The two sorts of minds are equally necessary for the progress 
of science ; both the logicians and the intuitionalists have achieved 
great things that others could not have done. Who would ven- 
ture to say whether he preferred that Weierstrass had never 
written or that there had never been a Biemann t Analysis and 
synthesis have then both their legitimate roles. But it is inter- 
esting to study more closely in the history of science the part 
which belongs to each. 


Strange! If we read over the works of the ancients we are 
tempted to class them all among the intuitionalists. And yet 
nature is always the same ; it is hardly probable that it has begun 
in this century to create minds devoted to logic. If we could put 
ourselves into the flow of ideas which reigned in their time, we 
should recognize that many of the old geometers were in tendency 



analysts. Euclid, for example, erected a scientific stractore 
wherein his contemporaries could find no fault. In this vast 
oonstructiony of which each piece however is due to intuition, we 
may still to-day, without much effort, recognize the work of a 

It is not minds that have changed, it is ideas; the intuitional 
minds have remained the same; but their readers have required 
of them greater concessions. 

What is the -cause of this evolution t It is not hard to find. 
Intuition can not give us rigor, nor even certainty; this has been 
recognized more and more. Let us cite some examples. We know 
there exist continuous functions lacking derivatives. Nothing is 
more shocking to intuition than this proposition which is imposed 
upon us by logic. Our fathers would not have failed to say : ''It 
is evident that every continuous function has a derivative, since 
every curve has a tangent.'* 

How can intuition deceive us on this point f It is because when 
we seek to imagine a curve we can not represent it to ourselves 
without width ; just so, when we represent to ourselves a straight 
line, we see it under the form of a rectilinear band of a certain 
breadth. We well know these lines have no width; we try to 
imagine them narrower and narrower and thus to approach the 
limit; so we do in a certain measure, but we shall never attain 
this limit. And then it is clear we can always picture these two 
narrow bands, one straight, one curved, in a position such that 
they encroach slightly one upon the other without crossing. We 
shall thus be led, unless warned by a rigorous analysis, to con- 
clude that a curve always has a tangent. 

I shall take as second example Dirichlet's principle on which 
rest so many theorems of mathematical physics ; to-day we estab- 
lish it by reasoning very rigorous but very long; heretofore, on 
the contrary, we were content with a very summary proof. A 
certain integral depending on an arbitrary function can never 
vanish. Hence it is concluded that it must have a minimum. The 
flaw in this reasoning strikes us immediately, since we use the 
abstract term function and are familiar with all the singularities 
functions can present when the word is understood in the most 
general sense. 


But it would not be the same had we used concrete images, 
had we, for example, considered this function as an electric poten- 
tial ; it would have been thought legitimate to affirm that electro- 
static equilibrium can be attained. Yet perhaps a physical com- 
parison would have awakened some vague distrust. But if care 
had been taken to translate the reasoning into the language of 
geometry, intermediate between that of analysis and that of 
physics, doubtless this distrust would not have been produced, 
and perhaps one might thus, even to-day, still deceive many 
readers not forewarned. 

Intuition, therefore, does not give us certainty. This is why 
the evolution had to happen ; let us now see how it happened. 

It was not slow in being noticed that rigor could not be intro- 
duced in the reasoning unless first made to enter into the defini- 
tions. For the most part the objects treated of by mathemati- 
cians were long ill defined; they were supposed to be known 
because represented by means of the senses or the imagination; 
but one had only a crude image of them and not a precise idea 
on which reasoning could take hold. It wists there first that the 
logicians had to direct their efforts. 

So, in the case of incommensurable numbers. The vague idea 
of continuity, which we owe to intuition, resolved itself into a 
complicated system of inequalities referring to whole numbers. 

By that means the difficulties arising from passing to the limit, 
or from the consideration of infinitesimals, are finally removed. 
To-day in analysis only whole numbers are left or systems, finite 
or infinite, of whole numbers bound together by a net of equality 
or inequality relations. Mathematics, as they say, is arithmetized. 


A first question presents itself. Is this evolution ended ? Have 
we finally attained absolute rigor? At each stage of the evolu- 
tion our fathers also thought they had reached it. If they deceived 
themselves, do we not likewise cheat ourselves? 

We believe that in our reasonings we no longer appeal to 
intuition ; the philosophers will tell us this is an illusion. Pure 
logic could never lead us to anything but tautologies; it could 


create nothing new; not from it alone can any science issue. In 

one sense these philosopers are right; to make arithmetic, as to 

make geometry, or to make any science, something else than pure 

logic is necessary. To designate this something else we have no 

word other than intuition. But how many different ideas are 

hidden under this same wordf 

Compare these four axioms: (1) Two quantities equal to a 
third are equal to one another; (2) if a theorem is true of the 
number 1 and if we prove that it is true of n + 1 if true for n, 
then will it be true of all whole numbers; (3) if on a straight 
the point C is between A and B and the point D between A and 
C, then the point D will be between A and B ; (4) through a given 
point there is not more than one parallel to a given straight. 

All four are attributed to intuition, and yet the first is the 
enunciation of one of the rules of formal logic ; the second is a 
real synthetic a priori judgment, it is the foundation of rigorous 
mathematical induction ; the third is an appeal to the imagina- 
tion; the fourth is a disguised definition. 

Intuition is not necessarily founded on the evidence of the 
senses ; the senses would soon become powerless ; for example, we 
can not represent to ourselves a chiliagon, and yet we reason by 
intuition on polygons in general, which include the chiliagon as 
a particular case. 

You know what Poncelet understood by the principle of con- 
tinuity. What is true of a real quantity, said Poncelet, should 
be true of an imaginary quantity ; what is true of the hyperbola 
whose asymptotes are real, should then be true of the ellipse 
whose asymptotes are imaginary. Poncelet was one of the most 
intuitive minds of this century; he was passionately, almost 
ostentatiously, so ; he regarded the principle of continuity as one 
of his boldest conceptions, and yet this principle did not rest on 
the evidence of the senses. To assimilate the hyperbola to the 
ellipse was rather to contradict this evidence. It was only a sort 
of precocious and instinctive generalization which, moreover, I 
have no desire to defend. 

We have then many kinds of intuition ; first, the appeal to the 
senses and the imagination; next generalization by induction, 
copied, so to speak, from the procedures of the experimental sci- 


ences; finally, we have the intuition of pnre number, whence 
arose the second of the axioms just enunciated, which is able to 
create the real mathematical reasoning. I have shown above by 
examples that the first two can not give us certainty ; but who will 
seriously doubt the third, who will doubt arithmetic t 

Now in the anal3rsis of to-day, when one cares to take the 
trouble to be rigorous, there can be nothing but EQrllogisms or 
appeals to this intuition of pure number, the only intuition which 
can not deceive us. It may be said that to^ay absolute rigor is 


The philosophers make still another objection : ''What you gain 
in rigor, ' ' they say, ' ' you lose in objectivity. You can risetoward 
your logical ideal only by cutting the bonds which attach you to 
reality- Your science is infallible, but it can only remain so by 
imprisoning itself in an ivory tower and renouncing all relation 
with the external world. From this seclusion it must go out when 
it would attempt the slightest application." 

For example, I seek to show that some property pertains to 
some object whose concept seems to me at first indefinable, be- 
cause it is intuitive. At first I fail or must content myself with 
approximate proofs; finally I decide to give to my object a pre- 
cise definition, and this enables me to establish this property in an 
irreproachable manner. 

'*And then," say the philosophers, **it still remains to show 
that the object which corresponds to this definition is indeed the 
same made known to you by intuition ; or else that some real and 
concrete object whose conformity with your intuitive idea you 
believe you immediately recognize corresponds to your new defi- 
nition. Only then could you affirm that it has the property in 
question. You have only displaced the difficulty." 

That is not exactly so ; the difficulty has not been displaced, it 
has been divided. The proposition to be established was in reality 
composed of two different truths, at first not distinguished. The 
first was a mathematical truth, and it is now rigorously estab- 
lished. The second was an experimental verity. Experience alone 
can teach us that some real and concrete object corresponds or 


does not correspond to some abstract definition. This second 
verity is not mathematically demonstrated, but neither can it be, 
no more than can the empirical laws of the physical and natural 
sciences. It would be unreasonable to ask more. 

Well, is it not a great advance to have distinguished what long 
was wrongly confused f Does this mean that nothing is left of 
this objection of the philosophers? That I do not intend to say; 
in becoming rigorous, mathematical science takes a character so 
artificial as to strike every one ; it forgets its historical origins ; 
we see how the questions can be answered, we no longer see how 
and why they are put. 

This shows us that logic is not enough; that the science of 
demonstration is not all science and that intuition must retain its 
role as complement, I was about to say as counterpoise or as 
antidote of logic. 

I have already had occasion to insist on the place intuition 
should hold in the teaching of the mathematical sciences. With- 
out it young minds could not make a beginning in the under- 
standing of mathematics; they could not learn to love it and 
would see in it only a vain logomachy ; above all, without intui- 
tion they would never become capable of applying mathematics. 
But now I wish before all to speak of the role of intuition in 
science itself. If it is useful to the student it is still more so to 
the creative scientist. 

We seek reality, but what is reality! The physiologists tell us 
that organisms are formed of cells; the chemists add that cells 
themselves are formed of atoms. Does this mean that these atoms 
or these cells constitute reality, or rather the sole reality! The 
way in which these cells are arranged and from which results the 
unity of the individual, is not it also a reality much more inter- 
esting than that of the isolated elements, and should a naturalist 
who had never studied the elephant except by means of the micro- 
scope think himself sufficiently acquainted with that animal f 

Well, there is something analogous to this in mathematics. The 
logician cuts up, so to speak, each demonstration into a very great 
number of elementary operations ; when we have examined these 




operations one after the other and ascertained that each is correct, 
are we to think we have grasped the real meaning of the demon- 
stration f Shall we have understood it even when, by an effort of 
memory, we have become able to repeat this proof by reproducing 
all these elementary operations in just the order in which the 
inventor had arranged them Y Evidently not ; we shall not yet 
possess the entire reality ; that I know not what, which makes the 
unity of the demonstration, will completely elude us. 

Pure analysis puts at our disposal a multitude of procedures 
whose infallibility it guarantees; it opens to us a thousand dif- 
ferent ways on which we can embark in all confidence; we are 
assured of meeting there no obstacles; but of all these ways, 
which will lead us most promptly to our goalf Who shall tell 
us which to choose f We need a faculty which makes us see the 
the end from afar, and intuition is this faculty. It is necessary 
to the explorer for choosing his route ; it is not less so to the one 
following his trail who wants to know why he chose it* 

If you are present at a game of chess, it will not suffice, for the 
understanding of the game, to know the rules for moving the 
pieces. That will only enable you to recognize that each move has 
been made conformably to these rules, and this knowledge will 
truly have very little value. Yet this is what the reader of a 
book on mathematics would do if he were a logician only. To 
understand the game is wholly another matter; it is to know why 
the player moves this piece rather than that other which he could 
have moved without breaking the rules of the game. It is to 
perceive the inward reason which makes of this series of succes- 
sive moves a sort of organized whole. This faculty is still more 
necessary for the player himself, that is, for the inventor. 

Let us drop this comparison and return to mathematics. For 
example, see what has happened to the idea of continuous func- 
tion. At the outset this was only a sensible image, for example, 
that of a continuous mark traced by the chalk on a blackboard. 
Then it became little by little more refined ; ere long it was used 
to construct a complicated system of inequalities, which repro- 
duced, so to speak, all the lines of the original image ; this con- 
struction finished, the centering of the arch, so to say, was 
removed, that crude representation which had temporarily served 


as gapport and which was afterward useless was rejected; there 
remained only the construction itself, irreproachable in the eyes 
of the logician. And yet if the primitive image had totally dis- 
appeared from our recollection, how could we divine by what 
caprice all these inequalities were erected in this fashion one 
upon another? 

Perhaps you think I use too many comparisons ; yet pardon still 
another. You have doubtless seen those delicate assemblages of 
silicious needles which form the skeleton of certain sponges. 
When the organic matter has disappeared, there remains only a 
frail and elegant lace-work. True, nothing is there except silica, 
but what is interesting is the form this silica has taken, and we 
could not understand it if we did not know the living sponge 
which has given it precisely this form. Thus it is that the old 
intuitive notions of our fathers, even when we have abandoned 
them, still imprint their form upon the logical constructions we 
have put in their place. 

This view of the aggregate is necessary for the inventor ; it is 
equally necessary for whoever wishes really to comprehend the 
inventor. Can logic give it to us Y No ; the name mathematicians 
give it would suffice to prove this. In mathematics logic is called 
analysis and analysis means division, dissection. It can have, 
therefore, no tool other than the scalpel and the microscope. 

Thus logic and intuition have each their necessary role. Each 
is indispensable. Logic, which alone can give certainty, is the 
instrument of demonstration; intuition is the instrument of 


But at the moment of formulating this conclusion I am seized 
with scruples. At the outset I distinguished two kinds of mathe- 
matical minds, the one sort logicians and analysts, the others 
intuitionalists and geometers. Well, the analysts also have been 
inventors. The names I have just cited make my insistence on 
this unnecessary. 

Here is a contradiction, at least apparently, which needs expla- 
nation. And first, do you think these logicians have always pro- 
ceeded from the general to the particular, as the rules of formal 


logic would seem to require of themf Not thus could they have 
extended the boundaries of science; scientific conquest is to be 
made only by generalization. 

In one of the chapters of ' Science and Hypothesis, * I have had 
occasion to study the nature of mathematical reasoning, and I 
have shown how this reasoning, without ceasing to be absolutely 
rigorous, could lift us from the particular to the general by a 
procedure I have called mathematical induction. It is by this 
procedure that the analysts have made science progress, and if we 
examine the detail itself of their demonstrations, we shall find it 
there at each instant beside the classic syllogism of Aristotle. 
We, therefore, see already that the analysts are not simply 
makers of syllogisms after the fashion of the scholastics. 

Besides, do you think they have always marched step by step 
with no vision of the goal they wished to attain f They must have 
divined the way leading thither, and for that they needed a guide. 
This guide is, first, analogy. For example, one of the methods of 
demonstration dear to analysts is that founded on the employ- 
ment of dominant functions. We know it has already served to 
solve a multitude of problems; in what consists then the role of 
the inventor who wishes to apply it to a new problem f At the 
outset he must recognize the analogy of this question with those 
which have already been solved by this method; then he must 
perceive in what way this new question differs from the others, 
and thence deduce the modifications necessary to apply to the 

But how does one perceive these analogies and these differences Y 
In the example just cited they are almost always evident, but I 
could have found others where they would have been much more 
deeply hidden ; often a very uncommon penetration is necessary 
for their discovery. The analysts, not to let these hidden analo- 
gies escape them, that is, in order to be inventors, must, without 
the aid of the senses and imagination, have a direct sense of what 
constitutes the unity of a piece of reasoning, of what makes, so 
to speak, its soul and inmost life. 

When one talked with M. Hermite, he never evoked a sensuous 
image, and yet you soon perceived that the most abstract entities 
were for him like living beings. He did not see them, but he per- 


ceived that they are not an artificial assemblage, and that they 
have some principle of internal unity. 

But, one will say, that still is intuition. Shall we conclude that 
the distinction made at the outset was only apparent, that there is 
only one sort of mind and that all the mathematicians are intui- 
tionalists, at least those who are capable of inventing f 

No, our distinction corresponds to something real. I have said 
above that there are many kinds of intuition. I have said how 
much the intuition of pure number, whence comes rigorous mathe- 
matical induction, differs from sensible intuition to which the 
imagination, properly so called, is the principal contributor. 

Is the abyss which separates them less profound than it at first 
appeared? Could we recognize with a little attention that this 
pure intuition itself could not do without the aid of the senses f 
This is the affair of the psychologist and the metaphysician and 
I shall not discuss the question. But the thing's being doubtful 
is enough to justify me in recognizing and affirming an essen- 
tial difference between the two kinds of intuition ; they have not 
the same object and seem to call into play two different faculties 
of our soul ; one would think of two search-lights directed upon 
two worlds strangers to one another. 

It is the intuition of pure number, that of pure logical forms, 
which illumines and directs those we have called analysts. This 
it is which enables them not alone to demonstrate, but also to 
invent. By it they perceive at a glance the general plan of a 
logical edifice, and that too without the senses appearing to inter- 
vene. In rejecting the aid of the imagination, which, as we have 
seen, is not always infallible, they can advance without fear of 
deceiving themselves. Happy, therefore, are those who can do 
without this aid ! We must admire them ; but how rare they are ! 

Among the analysts there will then be inventors, but they will 
be few. The majority of us, if we wished to see afar by pure intu- 
ition alone, would soon feel ourselves seized with vertigo. Our 
weakness has need of a staff more solid, and, despite the excep- 
tions of which we have just spoken, it is none the less true that 
sensible intuition is in mathematics the most usual instrument of 

Apropos of these reflections, a question comes up that I have 


not the time either to solve or even to enuxieiate with the derelop- 
mentA it would admit of. Is there room for a new disdnctioii, for 
distingaishing among the analysts those who above all nse poie 
intuition and those who are first of all preoccupied with formal 
logic f 

M. Ilf-rEiite, for example, whcHn I have just cited, can not be 
classed among the geometers who make use of the sensible intui- 
tion ; but neither is he a logician, properly so called. He does not 
conceal his aversion to purely deductive procedures which start 
from the general and end in the particular. 

The Measure of Time 


So long as we do not go outside the domain of consciousness, 
the notion of time is relatively dear. Not only do we distinguish 
without difficulty present sensation from the remembrance of past 
sensations or the anticipation of future sensations, but we know 
perfectly well what we mean when we say that of two conscious 
phenomena which we remember, one was anterior to the other; 
or that, of two foreseen conscious phenomena, one will be ante- 
rior to the other. 

When we say that two conscious facts are simultaneous, we 
mean that they profoundly interpenetrate, so that analysis can 
not separate them without mutilating them. 

The order in which we arrange conscious phenomena does not 
admit of any arbitrariness. It is imposed upon us and of it 
we can change nothing. 

I have only a single observation to add. For an aggregate of 
sensations to have become a remembrance capable of classifica- 
tion in time, it must have ceased to be actual, we must have 
lost the sense of its infinite complexity, otherwise it would have 
remained present. It must, so to speak, have crystallized around 
a center of associations of ideas which will be a sort of label. It 
is only when they thus have lost all life that we can classify our 
memories in time as a botanist arranges dried flowers in his 

But these labels can only be finite in number. On that score, 
psychologic time should be discontinuous. Whence comes the 
feeling that between any two instants there are others! We 
arrange our recollections in time, but we know that there remain 
empty compartments. How could that be, if time were not a 
form pre-existent in our minds f How could we know there were 
empty compartments, if these compartments were revealed to us 
only by their content! 




But that is not all ; into this form we wish to put not only the 
phenomena of our own consciousness, but those of which other 
consciousnesses are the theater. But more, we wish to put there 
physical facts, these I know not what with which we people space 
and which no consciousness sees directly. This is necessary be- 
cause without it science could not exist. In a word, psychologic 
time is given to us and must needs create scientific and physical 
time. There the difficulty begins, or rather the difficulties, for 
there are two. 

Think of two consciousnesses, which are like two worlds im- 
penetrable one to the other. By what right do we strive to put 
them into the same mold, to measure them by the same standard! 
Is it not as if one strove to measure length with a gram or 
weight with a meter f And besides, why do we speak of measur- 
ing f We know perhaps that some fact is anterior to some other, 
but not hy how much it is anterior. 

Therefore two difficulties: (1) Can we transform psychologic 
time, which is qualitative, into a quantitative timef (2) Can 
we reduce to one and the same measure facts which transpire in 
different worlds! 


The first difficulty has long been noticed ; it has been the sub- 
ject of long discussions and one may say the question is settled. 
We have not a direct intuition of the eqvMity of two intervals 
of time. The persons who believe they possess this intuition are 
dupes of an illusion. When I say, from noon to one the same 
time passes as from two to three, what meaning has this affir- 

The least reflection shows that by itself it has none at all. It 
will only have that which I choose to give it, by a definition which 
will certainly possess a certain degree of arbitrariness. Psy- 
chologists could have done without this definition ; physicists and 
astronomers could not ; let us see how they have managed. 

To measure time they use the pendulum and they suppose by 
definition that all the beats of this pendulum are of equal dura- 
tion. But this is only a first approximation; the temperature, 
the resistance of the air, the barometric pressure, make the pace 


of the pendulum vary. If we could escape these sources of error, 
we should obtain a much closer approximation, but it would still 
be only an approximation. New causes, hitherto neglected, elec- 
tric, magnetic or others, would introduce minute perturbations. 

In fact, the best chronometers must be corrected from time to 
time, and the corrections are made by the aid of astronomic 
observations; arrangements are made so that the sidereal clock 
marks the same hour when the same star passes the meridian* 
In other words, it is the sidereal day, that is, the duration of the 
rotation of the earth, which is the constant unit of time. It is 
supposed, by a new definition substituted for that based on the 
beats of the pendulum, that two complete rotations of the earth 
about its axis have the same duration. 

However, the astronomers are still not content with this defi- 
nition. Many of them think that the tides act as a check on our 
globe, and that the rotation of the earth is becoming slower and 
slower. Thus would be explained the apparent acceleration of 
the motion of the moon, which would seem to be going more 
rapidly than theory permits because our watch, which is the 
earth, is going slow. 


All this is unimportant, one will say ; doubtless our instruments 
of measurement are imperfect, but it suflSces that we can conceive 
a perfect instrument. This ideal can not be reached, but it is 
enough to have conceived it and so to have put rigor into the 
definition of the unit of time. 

The trouble is that there is no rigor in the definition. When 
we use the pendulum to measure time, what postulate do we 
implicitly admit? It is that the duration of two identical phe- 
nomena is the same; or, if you prefer, that the same causes take 
the same time to produce the same effects. 

And at first blush, this is a good definition of the equality of 
two durations. But take care. Is it impossible that experiment 
may some day contradict our postulate? 

Let me explain myself. I suppose that at a certain place in the 
world the phenomenon a happens, causing as consequence at the 
end of a certain time the effect a\ At another place in the world 


very far away from the first, happens the phenomenon )8, which 
causes as consequence the effect p'. The phenomena a and p are 
simultaneous, as are also the effects a' and p'. 

Later, the phenomenon a is reproduced under approximately 
the same conditions as before, and simultaneously the phenom- 
enon p is also reproduced at a very distant place in the world 
and almost under the same circumstances. The effects cf and fi^ 
also take place. Let us suppose that the effect a' happens per- 
ceptibly before the effect p'. 

If experience made us witness such a sight, our postulate 
would be contradicted. For experience would tell us that the 
first duration aa' is equal to the first duration pp' and that the 
second duration aa' is less than the second duration pp'. On the 
other hand, our postulate would require that the two durations 
aa' should be equal to each other, as likewise the two durations 
pp'. The equality and the inequality deduced from experience 
would be incompatible with the two equalities deduced from the 

Now can we affirm that the hypotheses I have just made are 
absurd? They are in no wise contrary to the principle of con- 
tradiction. Doubtless they could not happen without the prin- 
ciple of sufficient reason seeming violated. But to justify a 
definition so fundamental I should prefer some other guarantee. 

But that is not all. In physical reality one cause does not pro- 
duce a given effect, but a multitude of distinct causes contribute 
to produce it, without our having any means of discriminating 
the part of each of them. 

Physicists seek to make this distinction ; but they make it only 
approximately, and, however they progress, they never will 
make it except approximately. It is approximately true that the 
motion of the pendulum is due solely to the earth's attraction; 
but in all rigor every attraction, even of Sirius, acts on the pen- 

Under these conditions, it is clear that the causes which have 
produced a certain effect will never be reproduced except ap- 
proximately. Then we should modify our postulate and our 


definition. Instead of saying: 'The same causes take the same 
time to produce the same effects, * we should say : * Causes almost 
identical take almost the same time to produce almost the same 

Our definition therefore is no longer anything but approxi- 
mate. Besides, as M. Calinon very justly remarks in a recent 
memoir :^ 

One of the circunuitancee of any phenomenon is the velocity of the earth's 
rotation; if this velocity of rotation varies, it constitutes in the reproduction 
of this phenomenon a circumstance which no longer remains the same. But 
to suppose this velocity of rotation constant is to suppose that we know how 
to measure time. 

Our definition is therefore not yet satisfactory; it is certainly 
not that which the astronomers of whom I spoke above implicitly 
adopt, when they afiBlrm that the terrestrial rotation is slowing 

What meaning according to them has this affirmation f We 
can only understand it by analyzing the proofs they give of their 
proposition. They say first that the friction of the tides pro- 
ducing heat must destroy vis viva. They invoke therefore the 
principle of vis viva, or of the conservation of energy. 

They say next that the secular acceleration of the moon, cal- 
culated according to Newton's law, would be less than that de- 
duced from observations unless the correction relative to the 
slowing down of the terrestrial rotation were made. They invoke 
therefore Newton's law. In other words, they define duration 
in the following way: time should be so defined that Newton's 
law and that of vis viva may be verified. Newton's law is an 
experimental truth ; as such it is only approximate, which shows 
that we still have only a definition by approximation. 

If now it be supposed that another way of measuring time is 
adopted, the experiments on which Newton's law is founded 
would none the less have the same meaning. Only the enun- 
ciation of the law would be different, because it would be trans- 
lated into another language; it would evidently be much less 
simple. So that the definition implicitly adopted by the astron- 
omers may be summed up thus : Time should be so defined that 

1 Etude sur les diverses grandeurs, Paris, Gauthier-Yillars, 1897. 


the equations of mechanics may be as simple as possible. In 
other words, there is not one way of measuring time more true 
than another; that which is generally adopted is only more 
convenient. Of two watches, we have no right to say that the 
one goes true, the other wrong; we can only say that it is ad- 
vantageous to conform to the indications of the first. 

The diflSculty which has just occupied us has been, as I have 
said, often pointed out; among the most recent works in which 
it is considered, I may mention, besides M. Calinon's little book, 
the treatise on mechanics of Andrade. 


The second diflSculty has up to the present attracted much 
less attention; yet it is altogether analogous to the preceding; 
and even, logically, I should have spoken of it first. 

Two psychological phenomena happen in two different con- 
sciousnesses; when I say they are simultaneous, what do I meanf 
When I say that a physical phenomenon, which happens outside 
of every consciousness, is before or after a psychological phenom- 
enon, what do I mean 1 

In 1572, Tycho Brahe noticed in the heavens a new star. An 
immense conflagration had happened in some far distant heavenly 
body; but it had happened long before; at least two hundred 
years were necessary for the light from that star to reach our 
earth. This conflagration therefore happened before the discov- 
ery of America. Well, when I say that ; when, considering this 
gigantic phenomenon, which perhaps had no witness, since the 
satellites of that star were perhaps uninhabited, I say this phe- 
nomenon is anterior to the formation of the visual image of the 
isle of Espanola in the consciousness of Christopher Columbus, 
what do I mean? 

A little reflection is suflScient to understand that all these 
aflBrmations have by themselves no meaning. They can have one 
only as the outcome of a convention. 


We should first ask ourselves how one could have had the idea 
of putting into the same frame so many worlds impenetrable to 


one another. We should like to represent to ourselves the ex- 
ternal universe, and only by so doing could we feel that we un- 
derstood it. We know we never can attain this representation : 
our weakness is too great. But at least we desire the ability to 
conceive an infinite intelligence for which this representation 
could be possible, a sort of great consciousness which should see 
all, and which should classify all in its time, as we classify, in 
our time, the little we see. 

This hypothesis is indeed crude and incomplete, because this 
supreme intelligence would be only a demigod; infinite in one 
sense, it would be limited in another, since it would have only an 
imperfect recollection of the past; and it could have no other, 
since otherwise all recollections would be equally present to it 
and for it there would be no time. And yet when we speak of 
time, for all which happens outside of us, do we not uncon- 
sciously adopt this hypothesis; do we not put ourselves in the 
place of this imperfect god; and do not even the atheists put 
themselves in the place where god would be if he existed f 

What I have just said shows us, perhaps, why we have tried 
to put all physical phenomena into the same frame. But that 
can not pass for a definition of simultaneity, since this hypo- 
thetical intelligence, even if it existed, would be for us impene- 
trable. It is therefore necessary to seek something else. 


The ordinary definitions which are proper for psychologic time 
would suflBce us no more. Two simultaneous psychologic facts 
are so closely bound together that analysis can not separate with- 
out mutilating them. Is it the same with two physical facts 1 Is 
not my present nearer my past of yesterday than the present of 
Sirius 1 

It has also been said that two facts should be regarded as 
simultaneous when the order of their succession may be inverted 
at will. It is evident that this definition would not suit two 
physical facts which happen far from one another, and that, in 
what concerns them, we no longer even understand what this 
reversibility would be; besides, succession itself must first be 



Let us then seek to give an account of what is understood by 
simultaneity or antecedence, and for this let us analyze some 

I write a letter; it is afterward read by the friend to whom I 
have addressed it. There are two facts which have had for their 
theater two different consciousnesses. In writing this letter I 
have had the visual image of it, and my friend has had in his turn 
this same visual image in reading the letter. Though these two 
facts happen in impenetrable worlds, I do not hesitate to regard 
the first as anterior to the second, because I believe it is its cause. 

I hear thunder, and I conclude there has been an electric dis- 
charge; I do not hesitate to consider the physical phenomenon 
as anterior to the auditory image perceived in my consciousness, 
because I believe it is its cause. 

Behold then the rule we follow, and the only one we can follow : 
when a phenomenon appears to us as the cause of another, we 
regard it as anterior. It is therefore by cause that we define 
time; but most often, when two facts appear to us bound by a 
constant relation, how do we recognize which is the cause and 
which the effect ? We assume that the anterior fact, the antece- 
dent, is the cause of the other, of the consequent. It is then by 
time that we define cause. How save ourselves from this petitio 

We say now post hoc, ergo propter hoc; now propter hoc, ergo 
post hoc; shall we escape from this vicious circlet 

Let us see, not how we succeed in escaping, for we do not 
completely succeed, but how we try to escape. 

I execute a voluntary act A and I feel afterward a sensation D, 
which I regard as a consequence of the act A ; on the other hand, 
for whatever reason, I infer that this consequence is not imme- 
diate, but that outside my consciousness two facts B and C, which 
I have not witnessed, have happened, and in such a way that 
B is the effect of A, that C is the effect of 5, and D of C. 

But why? If I think I have reason to regard the four facts 
A, B, C, D, as bound to one another by a causal connection, why 


rejige them in the causal order A B C D, and at the same time 
in the chronologic order A B C D, rather than in any other 

I clearly see that in the act A I have the feeling of having 
been active, while in undergoing the sensation D I have that of 
having been passive. This is why I regard A as the initial cause 
and D as the ultimate effect; this is why I put A at the beginning 
of the chain and D at the end; but why put B before C rather 
than C before B1 

If this question is put, the reply ordinarily is : we know that it 
is B which is the cause of C because we always see B happen 
before C. These two phenomena, when witnessed, happen in a 
certain order; when analogous phenomena happen without wit- 
ness, there is no reason to invert this order. 

Doubtless, but take care ; we never know directly the physical 
phenomena B and C. What we know are sensations B' and C 
produced respectively by B and C. Our consciousness tells us 
immediately that B' precedes C and we suppose that B and C 
succeed one another in the same order. 

This rule appears in fact very natural, and yet we are often 
led to depart from it. We hear the sound of the thunder only 
some seconds after the electric discharge of the cloud. Of two 
flashes of lightning, the one distant, the other near, can not the 
first be anterior to the second, even though the sound of the 
second comes to us before that of the first! 

Another difBculty; have we really the right to speak of the 
cause of a phenomenon ! If all the parts of the universe are inter- 
chained in a certain measure, any one phenomenon will not be 
the effect of a single cause, but the resultant of causes infinitely 
numerous; it is, one often says, the consequence of the state of 
the universe a moment before. How enunciate rules applicable 
to circumstances so complex f And yet it is only thus that these 
rules can be general and rigorous. 

Not to lose ourselves in this infinite complexity, let us make a 
simpler hypothesis. Consider three stars, for example, the sun, 
Jupiter and Saturn ; but, for greater simplicity, regard them as 


reduced to material points and isolated from the rest of the 
world. The positions and the velocities of three bodies at a 
given instant sufSce to determine their positions and velocities at 
the following instant, and consequently at any instant. Their 
positions at the instant t determine their positions at the instant 
/ + fc as weU as their positions at the instant t — h. 

Even more; the position of Jupiter at the instant t, together 
with that of Saturn at the instant t + a, determines the position 
of Jupiter at any instant and that of Saturn at any instant 

The aggregate of positions occupied by Jupiter at the instant 
t + e and Saturn at the instant ^ + a + e is bound to the aggre- 
gate of positions occupied by Jupiter at the instant t and Saturn 
at the instant / -f a, by laws as precise as that of Newton, though 
more complicated. Then why not regard one of these aggre- 
gates as the cause of the other, which would lead to considering 
as simultaneous the instant t of Jupiter and the instant ^ -j- a of 
Saturn t 

In answer there can only be reasons, very strong, it is true, of 
convenience and simplicity. 


But let us pass to examples less artificial; to understand the 
definition implicitly supposed by the savants, let us watch them at 
work and look for the rules by which they investigate simul- 

I will take two simple examples, the measurement of the 
velocity of light and the determination of longitude. 

When an astronomer tells me that some stellar phenomenon, 
which his telescope reveals to him at this moment, happened, 
nevertheless, fifty years ago, I seek his meaning, and to that 
end I shall ask him first how he knows it, that is, how he has 
measured the velocity of light. 

He has begun by supposing that light has a constant velocity, 
and in particular that its velocity is the same in all directions. 
That is a postulate without which no measurement of this veloc- 
ity could be attempted. This postulate could never be verified 
directly by experiment; it might be contradicted by it if the 
results of different measurements were not concordant. We 


should think ourselves fortunate that this contradiction has 
not happened and that the slight discordances which may happen 
can be readily explained. 

The postulate, at all events, resembling the principle of suffi- 
cient reason, has been accepted by everybody ; what I wish to em- 
phasize is that it furnishes us with a new rule for the investi- 
gation of simultaneity, entirely different from that which we 
have enunciated above. 

This postulate assumed, let us see how the velocity of light has 
been measured. You know that Boemer used eclipses of the 
satellites of Jupiter, and sought how much the event fell behind 
its prediction. But how is this prediction madef It is by the 
aid of astronomic laws; for instance Newton's law. 

Could not the observed facts be just as weU explained if we at- 
tributed to the velocity of light a little different value from that 
adopted, and supposed Newton's law only approximate? Only 
this would lead to replacing Newton's law by another more com- 
plicated. So for the velocity of light a value is adopted, such 
that the astronomic laws compatible with this value may be as 
simple as possible. When navigators or geographers determine 
a longitude, they have to solve just the problem we are discuss- 
ing; they must, without being at Paris, calculate Paris time. 
How do they accomplish itt They carry a chronometer set for 
Paris. The qualitative problem of simultaneity is made to de- 
pend upon the quantitative problem of the measurement of 
time. I need not take up the difficulties relative to this latter 
problem, since above I have emphasized them at length. 

Or else they observe an astronomic phenomenon, such as an 
eclipse of the moon, and they suppose that this phenomenon is 
perceived simultaneously from aU points of the earth. That is 
not altogether true, since the propagation of light is not instan- 
taneous; if absolute exactitude were desired, there would be a 
correction to make according to a complicated rule. 

Or else finally they use the telegraph. It is clear first that the 
reception of the signal at Berlin, for instance, is after the send- 
ing of this same signal from Paris. This is the rule of cause and 
effect analyzed above. But how much after t In general, the 
duration of the transmission is neglected and the two events are 


regarded as simultaneous. But, to be rigorous, a little correc- 
tion would still have to be made by a complicated calculation; 
in practise it is not made, because it would be well within the 
errors of observation; its theoretic necessity is none the less 
from our point of view, which is that of a rigorous definition. 
From this discussion, I wish to emphasize two things: (1) The 
rules applied are exceedingly various. (2) It is difficult to sep- 
arate the qualitative problem of simultaneity from the quanti- 
tative problem of the measurement of time ; no matter whether a 
chronometer is used, or whether account must be taken of a 
velocity of transmission, as that of light, because such a velocity 
could not be measured without measuring a time. 


To conclude : We have not a direct intuition of simultaneityi 
nor of the equality of two durations. If we think we have this 
intuition, this is an illusion. We replace it by the aid of certain 
rules which we apply almost always without taking count of 

But what is the nature of these rules t No general rule, no 
rigorous rule ,* a multitude of little rules applicable to each par- 
ticular case. 

These rules are not imposed upon us and we might amuse our- 
selves in inventing others ; but they could not be cast aside with- 
out greatly complicating the enunciation of the laws of physics, 
mechanics and astronomy. 

We therefore choose these rules, not because they are true, 
but because they are the most convenient, and we may recapitu- 
late them as foUows: " The simultaneity of two events, or the 
order of their succession, the equality of two durations, are to be 
so defined that the enunciation of the natural laws may be as 
simple as possible. In other words, aU these rules, all these 
definitions are only the fruit of an unconscious opportunism.*' 

The Notion op Space 

1. Introduction 

In the articles I have heretofore devoted to space I have above 
all emphasized the problems raised by non-Euclidean geometry, 
while leaving almost completely aside other questions more diffi- 
cult of approach, such as those which pertain to the number of 
dimensions. All the geometries I considered had thus a common 
basis, that tridimensional continuum which was the same for all 
and which differentiated itself only by the figures one drew in 
it or when one aspired to measure it. 

In this continuum, primitively amorphous, we may imagine a 
network of lines and surfaces, we may then convene to regard 
the meshes of this net as equal to one another, and it is only 
after this convention that this continuum, become measurable, 
becomes Euclidean or non-Euclidean space. From this amor- 
phous continuum can therefore arise indifferently one or the 
other of the two spaces, just as on a blank sheet of paper may 
be traced indifferently a straight or a circle. 

In space we know rectilinear triangles the sum of whose angles 
is equal to two right angles; but equally we know curvilinear 
triangles the sum of whose angles is less than two right angles. 
The existence of the one sort is not more doubtful than that of 
the other. To give the name of straights to the sides of the first 
is to adopt Euclidean geometry ; to give the name of straights to 
the sides of the latter is to adopt the non-Euclidean geometry. 
So that to ask what geometry it is proper to adopt is to ask, to 
what line is it proper to give the name straight t 

It is evident that experiment can not settle such a question; 
one would not ask, for instance, experiment to decide whether I 
should call AB or CD a straight. On the other hand, neither 
can I say that I have not the right to give the name of straights 
to the sides of non-Euclidean triangles because they are not in 



conformity with the eternal idea of straight which I have by 
intuition. I grant, indeed, that I have the intuitive idea of the 
side of the Euclidean triangle, but I have equally the intuitive 
idea of the side of the non-Euclidean triangle. Why should I 
have the right to apply the name of straight to the first of these 
ideas and not to the second f Wherein does this syllable form 
an integrant part of this intuitive idea t Evidently when we say 
that the Euclidean straight is a true straight and that the non- 
Euclidean straight is not a true straight, we simply mean that 
the first intuitive idea corresponds to a more noteworthy object 
than the second. But how do we decide that this object is more 
noteworthy? This question I have investigated in 'Science and 

It is here that we saw experience come in. If the Euclidean 
straight is more noteworthy than the non-Euclidean straight, it 
is so chiefly because it differs little from certain noteworthy 
natural objects from which the non-Euclidean straight differs 
greatly. But, it wiU be said, the definition of the non-Euclidean 
straight is artificial; if we for a moment adopt it, we shall see 
that two circles of different radius both receive the name of 
non-Euclidean straights, while of two circles of the same radius 
one can satisfy the definition without the other being able to sat- 
isfy it, and then if we transport one of these so-called straights 
without deforming it, it will cease to be a straight. But by 
what right do we consider as equal these two figures which the 
Euclidean geometers call two circles with the same radius t It is 
because by transporting one of them without deforming it we 
can make it coincide with the other. And why do we say this 
transportation is effected without deformation t It is impossible 
to give a good reason for it. Among all the motions conceiv- 
able, there are some of which the Euclidean geometers say that 
they are not accompanied by deformation ; but there are others of 
which the non-Euclidean geometers would say that they are not 
accompanied by deformation. In the first, called Euclidean mo- 
tions, the Euclidean straights remain Euclidean straights and the 
non-Euclidean straights do not remain non-Euclidean straights; 
in the motions of the second sort, or non-Euclidean motions, 
the non-Euclidean straights remain non-Euclidean straights 

and the Euclidean straights do not remain Euclidean 
straights. It has, therefore, not been demonstrated that it was 
unreasonable to call straights the sides of non-Euclidean tri- 
angles; it has only been shown that that would be unreasonable 
if one continued to call the Euclidean motions motions witJiout 
deformation; but it has at the same time been shown that it 
would be just as unreasonable to call straights the sides of Eu- 
clidean triangles if the non-Euclidean motions were called mo- 
tions without deformation. 

Now when we say that the Euclidean motions are the (rue 
motions without deformation, what do we meanT We simply 
mean that they are more noteworthy than the others. And why 
are they more noteworthy! It is because certain noteworthy 
natural bodies, the solid bodies, undergo motions almost similar. 

And then when we ask: Can one imagme non-Euclidean space t 
that means : Can we imagine a world where there would he note- 
worthy natural objects affecting almost the form of non-Euclid- 
ean straights, and noteworthy natural bodies frequently under- 
going motions almost similar to the non-EucUdean motions T I 
have shown in 'Science and Hypothesis' that to this question we 
must answer yes. 

It has often been observed that if all the bodies in the universe 
were dilated simultaneously and in the same proportion, we 
shoold have no means of perceiving it, since all our measuring 
instruments would grow at the same time as the objects them- 
selves which they sen'e to measure. The world, after this dila- 
tation, would continue on its course without anything appris- 
ing us of so considerable an event. In other words, two worlds 
similar to one another (understanding the word similitude in 
the sense of Euclid, Book VI,) would be absolutely indistin- 
guishable. But more; worlds will be indistinguishable not only 
if they are equal or similar, that is, if we can pass from one to 
the other by changing the axes of coordinates, or by changing 
the scale to which lengths are referred; but they will still b« 
indistinguishable if we can pass from one to the other by any 
'point-transformation' whatever. I will explain my meaning. I 
suppose that to each point of one corresponds one point of the 
other and only one, and inversely; and besides that the coordi- 



nates of a point are eontinuous fonctionsy otherwise altogetker 
arbitrary, of the corresponding point I suppose besides that to 
each object of the first world corresponds in the second an object 
of the same nature placed precisely at the corresponding x>oint 
I suppose finally that this correspondence fulfilled at the initial 
instant is maintained indefinitely. We should have no means 
of distinguishing these two worlds one from the other. The rela- 
tivity of space is not ordinarily understood in so broad a sense; 
it is thus, however, that it would be proper to understand it. 

If one of these universes is our Euclidean world, what its in- 
habitants will call straight will be our Euclidean straight; but 
what the inhabitants of the second world will call straight will 
be a curve which will have the same properties in relation to the 
world they inhabit and in relation to the motions that they will 
call motions without deformation. Their geometry wiD, there- 
fore, be Euclidean geometry, but their straight will not be our 
Euclidean straight. It will be its transform by the point-trans- 
formation which carries over from our world to theirs. The 
straights of these men will not be our straights, but they will 
have among themselves the same relations as our straights to one 
another. It is in this sense I say their geometry wiU be ours. 
If then we wish after all to proclaim that they deceive them- 
selves, that their straight is not the true straight, if we still are 
unwilling to admit that such an afiSrmation has no meaning, at 
least we must confess that these people have no means whatever 
of recognizing their error. 

2. Qualitative Oeometry 

All that is relatively easy to understand, and I have already so 
often repeated it that I think it needless to expatiate further on 
the matter. Euclidean space is not a form imposed upon our 
sensibility, since we can imagine non-Euclidean space; but the 
two spaces, Euclidean and non-Euclidean, have a common basis, 
that amorphous continuum of which I spoke in the beginning. 
From this continuum we can get either Euclidean space or 
Lobachevskian space, just as we can, by tracing upon it a proper 
graduation, transform an ungraduated thermometer into a Fahr- 
enheit or a Reaumur thermometer. 

And then comes a question : Ib not this amorphous continuum, 
that our analysis has allowed to survive, a form imposed upon 
oar sensibility T If so, we should have enlarged the prison in 
which this sensibility is confined, but it would always be a 

This continuum has a certain number of properties, exempt 
from all idea of measurement. The study of these properties is 
the object of a science which has' been cultivated by many great 
geometers and in particular by Riemann and Betti and which 
has received the name of analysts situs. In this science abstrac- 
tion is made of every quantitative idea and, for example, if we 
ascertain that on a line the point B is between the points A and 
C, we shall be content with this ascertainment and shall not 
trouble to know whether the line ABC is straight or curved, nor 
whether the length AB is equal to the length BC, or whether it 
ia twice as great. 

The theorems of analysis situs have, therefore, this peculiarity, 
that they would remain true if the figures were copied by an 
inexpert draftsman who should grossly change all the propor- 
tions and replace the straights by lines more or less sinuous. In 
mathematical terms, they are not altered by any 'poinMrans- 
formation' whatsoever. It has often been said that metric geom- 
etry was quantitative, while projective geometry was purely qual- 
itative. That is not altogether true. The straight is still dis- 
tinguished from other lines by properties which remain quanti- 
tative in some respects. The real qualitative geometry is, there- 
fore, analysis situs. 

The same questions wtiich came up apropos of the truths of 
Euclidean geometry, come up anew apropos of the theorems of 
aualyds situs. Are they obtainable by deductive reasoning! 
Are they disguised conventions! Are they experimental veri- 
tieat Are they the characteristics of a form imposed either 
upon our sensibility or upon our understanding t 

I wish simply to observe that the last two solutions exclude 
each other. We can not admit at the same time that it is impos- 
sihle to imagine space of four dimensions and that experience 
proves to us that space has three dimensions. The experimenter 
pnta to nature a question : Is it this or that f and he can not put 


it without imagining the two terms of the alternative. If it were 
impossible to imagine one of these terms, it would be futile and 
besides impossible to consult experience. There is no need of ob- 
servation to know that the hand of a watch is not marking the 
hour 15 on the dial, because we know beforehand that there are 
only 12, and we could not look at the mark 15 to see if the hand 
is there, because this mark does not exist. 

Note likewise that in analysis situs the empiricists are disem- 
barrassed of one of the gravest objections that can be leveled 
against them, of that which renders absolutely vain in advance 
all their efforts to apply their thesis to the verities of Euclidean 
geometry. These verities are rigorous and all experimentation 
can only be approximate. In analysis situs approximate exper- 
iments may suffice to give a rigorous theorem and, for instance, 
if it is seen that space can not have either two or less than two 
dimensions, nor four or more than four, we are certain that it has 
exactly three, since it could not have two and a half or three 
and a half. 

Of all the theorems of analysis situs, the most important is 
that which is expressed in sajdng that space has three dimen- 
sions. This it is that we are about to consider, and we shall put 
the question in these terms: "When we say that space has three 
dimensions, what do we meant 

3. The Physical Continuum of Several Dimensions 

I have explained in 'Science and Hypothesis' whence we 
derive the notion of physical continuity and how that of mathe- 
matical continuity has arisen from it. It happens that we are 
capable of distinguishing two impressions one from the other, 
while each is indistinguishable from a third. Thus we can read- 
ily distinguish a weight of 12 grams from a weight of 10 grams, 
while a weight of 11 grams could be distinguished from neither 
the one nor the other. Such a statement, translated into sym- 
bols, may be written : 

A=B, B = C, A<C, 

This would be the formula of the physical continuum, as crude 
experience gives it to us, whence arises an intolerable contradic- 


tion that has been obviated by the introduction of the mathe- 
matical continuum. This is a scale of which the steps (com- 
mensurable or incommensurable numbers) are infinite in number 
but are exterior to one another, instead of encroaching on one 
another as do the elements of the physical continuum, in con- 
formity with the preceding formula. 

The physical continuum is, so to speak, a nebula not resolved ; 
the most perfect instruments could not attain to its resolution. 
Doubtless if we measured the weights with a good balance instead 
of judging them by the hand, we could distinguish the weight of 
11 grams from those of 10 and 12 grams, and our formula would 
become : 

A<B, B<C, A<C. 

But we should always find between A and B and between B 
and C new elements D and £, such that 

A=D, D = B, A<B; B = E, B = C, B<C, 

and the difficulty would only have receded and the nebula would 
always remain unresolved ; the mind alone can resolve it and the 
mathematical continuum it is which is the nebula resolved into 

Yet up to this point we have not introduced the notion of the 
number of dimensions. What is meant when we say that a math- 
ematical continuum or that a physical continuum has two or 
three dimensions? 

First we must introduce the notion of cut, studying first phys- 
ical continua. We have seen what characterizes the physical con- 
tinuum. Each of the elements of this continuum consists of a 
manifold of impressions ; and it may happen either that an ele- 
ment can not be discriminated from another element of the same 
continuum, if this new element corresponds to a manifold of 
impressions not sufficiently different, or, on the contrary, that 
the discrimination is possible; finally it may happen that two 
elements indistinguishable from a third may, nevertheless, be 
distinguished one from the other. 

That postulated, if A and B are two distinguishable elements of 
a continuum C, a series of elements may be found, E^, E^, • • • , Eny 
all belonging to this same continuum C and such that each of 


them is indistin^shable from the preceding, that E^ is indis- 
tinguishable from A, and En indistinguishable from B. There- 
fore we can go from A to B by a continuous route and without 
quitting C. If this condition is fulfilled for any two elements 
A and B of the continuum C, we may say that this continuum C 
is all in one piece. Now let us distinguish certain of the elements 
of C which may either be all distinguishable from one another, 
or themselves form one or several continua. The assemblage of 
the elements thus chosen arbitrarily among all those of C will 
form what I shall call the cut or the cuts. 

Take on C any two elements A and B, Either we can also find 
a series of elements E^, E^, • • •, En, such : (1) that they all belong 
to C; (2) that each of them is indistinguishable from the follow- 
ing, E^ indistinguishable from A and En from B ; (3) and besides 
that none of the elements E is indistinguishable from any element 
of the cut. Or else, on the contrary, in each of the series E^^ E^, 
"-y En satisfying the first two conditions, there will be an element 
E indistinguishable from one of the elements of the cut. In the 
first case we can go from A to 5 by a continuous route without 
quitting C and without meeting the cuts; in the second case that 
is impossible. 

If then for any two elements A and B of the continuum C, it is 
always the first case which presents itself, we shall say that C 
remains all in one piece despite the cuts. 

Thus, if we choose the cuts in a certain way, otherwise arbi- 
trary, it may happen either that the continuum remains all in one 
piece or that it does not remain all in one piece; in this latter 
hypothesis we shall then say that it is divided by the cuts. 

It will be noticed that all these definitions are constructed in 
setting out solely from this very simple fact, that two manifolds 
of impressions sometimes can be discriminated, sometimes can 
not be. That postulated, if, to divide a continuum, it suffices to 
consider as cuts a certain number of elements all distinguishable 
from one another, we say that this continuum is of one dimen- 
sion; if, on the contrary, to divide a continuum, it is necessary to 
consider as cuts a system of elements themselves forming one or 
several continua, we shall say that this continuum is of several 

If to divide a continuum C, cute forming one or several con- 
tinus of one dimension suffice, we shall say that C is a continuum 
of two dimensions; if cuts sufBce wbicli form one or several con- 
tinua of two dimensions at most, we shall say that (7 is a con- 
tinuum of three dimensions; and so on. 

To justify tliia definition it is proper to see whether it is in this 
way that geometers introduce the notion of three dimensions at 
the beginning of their works. Now, what do we see! Usually 
they begin by defining surfaces as the boundaries of solids or 
pieces of space, lines as the boundaries of surfaces, points as the 
boundaries of lines, and they affirm that the same procedure can 
not be pushed further. 

This is just the idea given above : to divide space, cuts that are 
called surfaces are necessary; to divide surfaces, cuts that are 
called lines are necessary; to divide lines, cuts that are called 
points are necessary; we can go no further, the point can not be 
divided, so the point is not a continuum. Then lines which can be 
divided by cuts which are not continua will be continua of one 
dimension; surfaces which can be divided by continuous cuts of 
one dimension will be continua of two dimensions ; finally, space 
which can be divided by continuous cuts of two dimensions will 
be a continuum of three dimensions. 

Thus the definition I have just given does not differ essentially 
from the usual definitions; I have only endeavored to give it a 
form applicable not to the mathematical continuum, but to the 
physical continuum, which alone is susceplible of representation, 
and yet to retain all its precision. Moreover, we see that this 
definition applies not alone to space ; that in all which falls under 
our senses we find the characteristics of the physical continuum, 
which would allow of the same classification; that it would be 
easy to find there esamples of continua of four, of five, dimen- 
sions, in the sense of the preceding definition; such examples 
occur of themselves to the mind. 

I should explain finally, if I had the time, that this science, 
of which I spoke above and to which Riemann gave the name of 
atulysis situs, teaches us to make distinctions among continua of 
th« same number of dimensions and that the classification of these 
continua rests also on the consideration of cuts. 


From this notion has arisen that of the mathematical con- 
tinuum of several dimensions in the same way that the physical 
continuum of one dimension engendered the mathematical con- 
tinuum of one dimension. The formula 

A>C, A = B, B = C, 

which summed up the data of crude experience, implied an in- 
tolerable contradiction. To get free from it, it was necessary to 
introduce a new notion while still respecting the essential char- 
acteristics of the physical continuum of several dimensions. The 
mathematical continuum of one dimension admitted of a scale 
whose divisions, infinite in number, corresponded to the different 
values, commensurable or not, of one same magnitude. To have 
the mathematical continuum of n dimensions, it will suffice to 
take n like scales whose divisions correspond to different values 
of n independent magnitudes called coordinates. We thus shall 
have an image of the physical continuum of n dimensions, and 
this image will be as faithful as it can be after the determina- 
tion not to allow the contradiction of which I spoke above. 

4. The Notion of Point 

It seems now that the question we put to ourselves at the start 
is answered. When we say that space has three dimensions, it 
will be said, we mean that the manifold of points of space satis- 
fies the definition we have just given of the physical continuum 
of three dimensions. To be content with that would be to sup- 
pose that we know what is the manifold of points of space, or even 
one point of space. 

Now that is not as simple as one might think. Every one 
believes he knows what a point is, and it is just because we know 
it too well that we think there is no need of defining it. Surely 
we can not be required to know how to define it, because in going 
back from definition to definition a time must come when we must 
stop. But at what moment should we stop ? 

We shall stop first when we reach an object which falls under 
our senses or that we can represent to ourselves; definition then 
will become useless; we do not define the sheep to a child; we 
say to him : See the sheep. 

So, then, we should ask ourselves if it ia possible to represent 
to ourselves a point of space. Those who answer yes do not refleet 
that they represent to themselves in reality a white spot made 
with the clialk on a blackboan.1 or a black spot made with a pen 
on white paper, and that they can represent to themselves only 
an object or rather the impressions that this object made on their 

When they try to represent to themselves a point, they repre- 
sent the impressions that very little objects made them feeL It 
is needless to add that two different objects, though both very 
little, may produce extremely different impressions, but I 
shall not dwell on this difficulty, which would still require some 

But it is not a question of that ; it does not suffice to represent 
one point, it is necessary to represent a certain point and to have 
the means of distingruishing it from an other point. And in fact, 
that we may be able to apply to a continuum the rule 1 have above 
expounded and by which one may recognize the niunber of its 
diuiensions, we must rely upon the fact that two elements of this 
continuum sometimes can and sometimes cannot be distinguished. 
It is necessary therefore that we should in certain eases know how 
to represent to ourselves a specific element and to cUstinguiah it 
from an other element. 

The question ia to know whether the point that I represented 
to myself an hour ago is the same as this that I now represent 
to myself, or whether it is a different point. In other words, 
how do we know whether the point occupied by the object A at 
the instant a is the same as the point occupied by the object B at 
the instant jff, or still better, what this means I 

I am seated in my room ; an object is placed on my table ; dur- 
ing a second I do not move, do one touches the object. I am 
tempted to say that the point A which this object occupied at the 
beginnint; of this second is idcntienl with the poiot B which it 
occupies at its end. Not at all ; from the point ,4 to the point B 
is 30 kilometers, because the object has been carried along in the 
motion of the earth. We can not know whether an object, be it 
large or small, has not changed its absolute position in space, 
and not only can we not affirm it. but this affirmation has no 



meaning and in any case can not correspond to any representation. 

But then we may ask ourselves if the relative position of an 
object with regard to other objects has changed or not^ and first 
whether the relative position of this object with regard to our 
body has changed. If the impressions this object makes upon us 
have not changed, we shall be inclined to judge that neither has 
this relative position changed; if they have changed, we shall 
judge that this object has changed either in state or in relative 
position. It remains to decide which of the two. I have explained 
in 'Science and Hypothesis' how we have been led to distinguish 
the changes of position. Moreover, I shall return to that further 
on. We come to know, therefore, whether the relative position 
of an object with regard to our body has or has not remained 
the same. 

If now we see that two objects have retained their relative posi- 
tion with regard to our body, we conclude that the relative posi- 
tion of these two objects with regard to one another has not 
changed ; but we reach this conclusion only by indirect reasoning. 
The only thing that we know directly is the relative position of 
the objects with regard to our body. A fortiori it is only by 
indirect reasoning that we think we know ("and, moreover, this 
belief is delusive) whether the absolute position of the object has 

In a word, the system of coordinate axes to which we naturally 
refer all exterior objects is a system of axes invariably bound to 
our body, and carried around with us. 

It is impossible to represent to oneself absolute space ; when I 
try to represent to myself simultaneously objects and myself in 
motion in absolute space, in reality I represent to myself my own 
self montionless and seeing move around me different objects and 
a man that is exterior to me, but that I convene to call me. 

Will the diflSculty be solved if we agree to refer everything to 
these axes bound to our body? Shall we know then what is a 
point thus defined by its relative position with regard to our- 
selves? Many persons will answer yes and will say that they 
^localize' exterior objects. 

What does this mean ? To localize an object simply means to 
represent to oneself the movements that would be necessary to 


reach it. I will explain myself. It is not a question of repre- 
senting the movements themselves in space, but solely of repre- 
senting to oneself the muscular sensations which accompany these 
movements and which do not presuppose the preexistence of the 
notion of space. 

If we suppose two different objects which successively occupy 
the same relative position with regard to ourselves, the impres- 
sions that these two objects make upon us will be very different; 
if we localize them at the same point, this is simply because it is 
necessary to make the same movements to reach them ; apart from 
that, one can not just see what they could have in common. 

But, given an object, we can conceive many different series of 
movements which equally enable us to reach it. If then we repre- 
sent to ourselves a point by representing to ourselves the series 
of muscular sensations which accompany the movements which 
enable us to reach this point, there will be many ways entirely 
different of representing to oneself the same point. If one is not 
satisfied with this solution, but wishes, for instance, to bring in 
the visual sensations along with the muscular sensations, there 
will be one or two more ways of representing to oneself this same 
point and the difficulty will only be increased. In any case the 
following question comes up: Why do we think that all these 
representations so different from one another still represent the 
same point? 

Another remark: I have just said that it is to our own body 
that we naturally refer exterior objects ; that we carry about every- 
where with us a system of axes to which we refer all the points 
of space, and that this system of axes seems to be invariably 
bound to our body. It should be noticed that rigorously we could 
not speak of axes invariably bound to the body unless the dif- 
ferent parts of this body were themselves invariably bound to 
one another. As this is not the case, we ought, before referring 
exterior objects to these fictitious axes, to suppose our body 
brought back to the initial attitude. 

5. The Notion of Displacement 

I have shown in 'Science and Hypothesis' the preponderant 
role played by the movements of our body in the genesis of the 


notion of space. For a being completely immovable there would 
be neither space nor geometry ; in vain would exterior objects be 
displaced about him, the variations which these displacements 
would make in his impressions would not be attributed by this 
being to changes of position, but to simple changes of state; 
this being would have no means of distinguishing these two sorts 
of changes, and this distinction, fundamental for us, would have 
no meaning for him. 

The movements that we impress upon our members have as 
effect the varying of the impressions produced on our senses by 
external objects ; other causes may likewise make them vary ; but 
we are led to distinguish the changes produced by our own 
motions and we easily discriminate them for two reasons: (1) 
because they are voluntary; (2) because they are accompanied 
by muscular sensations. 

So we naturally divide the changes that our impressions may 
undergo into two categories to which perhaps I have given an 
inappropriate designation: (1) the internal changes, which are 
voluntary and accompanied by muscular sensations; (2) the 
external changes, having the opposite characteristics. 

We then observe that among the external changes are some 
which can be corrected, thanks to an internal change which brings 
everything back to the primitive state ; others can not be corrected 
in this way (it is thus that, when an exterior object is displaced, 
we may then by changing our own position replace ourselves 
as regards this object in the same relative position as before, so 
as to reestablish the original aggregate of impressions; if this 
object was not displaced, but changed its state, that is impos- 
sible). Thence comes a new distinction among external changes: 
those which may be so corrected we call changes of position; 
and the others, changes of state. 

Think, for example, of a sphere with one hemisphere blue and 
the other red ; it first presents to us the blue hemisphere, then it 
so revolves as to present the red hemisphere. Now think of a 
spherical vase containing a blue liquid which becomes red in 
consequence of a chemical reaction. In both cases the sensation 
of red has replaced that of blue ; our senses have experienced the 
same impressions which have succeeded each other in the same 


order, and yet these two changes are regarded by iib as very 
different ; the first is a displacement, the second a change of state. 
"Wbyl Because in the first case it is sufficient for me to go around 
the sphere to place myself opposit« the blue hemisphere and 
reestablish tlie original blue sensation. 

Still more ; if the two hemispheres, in place of being red and 
blue, had been yellow and green, how sliould I liave interpreted ] 
the revolution of the splicre ? Before, the red succeeded the blue, 
now the green succeeds the yellow; and yet I say that the two 
^heres have undergone the same revolution, that each has turned 
about its axis ; yet I can not say that the green is to yellow as 
the red is to blue; how then am I led to decide that the two 
spheres have undei^one the same displacement T Evidently be- 
eause, in one case as in the other, I am able to reestablish the 
original sensation by going around the sphere, by making the 
same movements, and I know that 1 have made the same move- 
ments because I have felt the same muscular sensations; to know 
it, I do not need, therefore, to know geometry in advance and to 
represent to myself the movements of my body in geometric space. 

Another example: An object is displaced before my eye; its 
image was first formed at the center of the retina; then it is 
formed at the border; the old sensation was carried to me by a 
nerve fiber ending at the center of the retina ; the new sensation 
is carried to me by another nerve fiber starting from the border 
■ of the retina; these two sensations are qualitatively different; 
otherwise, how could I distinguish them I 

Why then am I led to decide that these two sensations, quali- 
tatively different, represent the same image, which has been dis- 
plac«d t It is because I can follow the object vnth th-e eye and by 
a displacement of the eye, voluntary and aocompanied by muscu- 
lar sensations, bring back the image to the center of the retina 
and reestablish the primitive sensation. 

I suppose that the image of a red object has gone from the 
center A to the border B of the retina, then that the image of a 
bine object goes in its turn from the center A to the border B 
of the retina ; I shall decide that these two objects have under- 
gone the same displacement. Why! Because in botli cases I 
■hall have been able to reestablish the primitive sensation, and 


that to do it I shall have had to execute the same movement of 
the eye, and I shall know that my eye has executed the same 
movement because I shall have felt the same muscular sensations. 

If I could not move my eye, should I have any reason to sup- 
pose that the sensation of red at the center of the retina is to the 
sensation of red at the border of the retina as that of blue at the 
center is to that of blue at the border? I should only have four 
sensations qualitatively different, and if I were asked if they 
are connected by the proportion I have just stated, the question 
would seem to me ridiculous, just as if I were asked if there is an 
analogous proportion between an auditory sensation, a tactile 
sensation and an olfactory sensation. 

Let us now consider the internal changes, that is, those which 
are produced by the voluntary movements of our body and which 
are accompanied by muscular changes. They give rise to the 
two following observations, analogous to those we have just made 
on the subject of external changes. 

1. I may suppose that my body has moved from one point to 
another, but that the same attitude is retained ; all the parts of 
the body have therefore retained or resumed the same relative 
situation, although their absolute situation in space may have 
varied. I may suppose that not only has the position of my body 
changed, but that its attitude is no longer the same, that, for 
instance, my arms which before were folded are now stretched out. 

I should therefore distinguish the simple changes of position 
without change of attitude, and the changes of attitude. Both 
would appear to me under form of muscular sensations. How 
then am I led to distinguish them ? It is that the first may serve 
to correct an external change, and that the others can not, or at 
least can only give an imperfect correction. 

This fact I proceed to explain as I would explain it to some one 
who already knew geometry, but it need not thence be concluded 
that it is necessary already to know geometry to make this dis- 
tinction; before knowing geometry I ascertain the fact (experi- 
mentally, so to speak), without being able to explain it. But 
merely to make the distinction between the two kinds of change, 
I do not need to explain the fact, it suflBces me to ascertain it. 

However that may be, the explanation is easy. Suppose that 

an exterior object is displaced ; if we wish the different parts of 
our body to resume with regard to this object their initial relative 
position, it is necessary that these different parts should have 
resumed likewise their initial relative position with regard to 
one another. Only the internal changes which satisfy this latter 
condition will be capable of correcting the external change pro- 
duced by the displacement of that object. If, therefore, the 
relative position of my eye with regard to my finger has changed, 
I shall still be able to replace the eye in its initial relative situa- 
tion with regard to the object and reestablish thus the primitive 
visual sensations, but then the relative position of the finger with 
regard to the object will have changed and the tactile gensatianEi 
will not be reestablished. 

2. We ascertain likewise that the same external change raay be 
corrected by two internal changes correHponding to different 
inascular sensations. Here again I can ascertain this without 
knowing geometry; and I have no need of anything else; but I 
proceed to give the explanation of the fact, employing geometrical 
language. To go from the position A to the position B I may 
take several routes. To the first of these routes will correspond 
a series S of muscular sensations; to a second route will corre- 
spond another series S", of muscular sensations which generally 
will be completely different, since other muscles will be used. 

How am I led to regard these two series S and S" as corre- 
sponding to the same displacement AB t It is because these two 
series are capable of correcting the same external change. Apart 
from that, they have nothing in common. 

Let us now consider two external changes : a and ^, which shall 
be, for instance, the rotation of a sphere half blue, half red, and 
that of a sphere half yellow, half green ; these two changes have 
nothing in common, since the one is for us the passing of blue 
into red and the other the passing of yellow into green. Con- 
sider, on the other hand, two series of internal changes S and S"; 
like the others, they will have nothing in common. And yet I say 
that a and ^ correspond to the some displacement, and that >S^ and 
8" correspond also to the same displacement. WhyT Simply 
because S can correct a as well as /9 and because a con be cor- 
i by S" as well as by S. And then a question su^^sts itself : 


If I have ascertained that 8 corrects a and p and that 8" corrects 
a, am I certain that 8" likewise corrects pi Experiment alone 
can teach us whether this law is verified. If it were not verified, 
at least approximately, there would be no geometry, there wonld 
be no space, because we should have no more interest in classi- 
fying the internal and external changes as I have just done, and, 
for instance, in distinguishing changes of state from changes of 

It is interesting to see what has been the role of experience in 
all this. It has shown me that a certain law is approximately 
verified. It has not told me how space is, and that it satis- 
fies the condition in question. I knew, in fact, before all exi>eri- 
ence, that space satisfied this condition or that it would not be; 
nor have I any right to say that experience told me that geometry 
is possible ; I very well see that geometry is possible, since it does 
not imply contradiction ; experience only tells me that geometry 
is useful. 

6. Vistuil 8pace 

Although motor impressions have had, as I have just explained, 
an altogether preponderant influence in the genesis of the notion 
of space, which never would have taken birth without them, it 
will not be without interest to examine also the role of visual 
impressions and to investigate how many dimensions 'visual 
space' has, and for that purpose to apply to these impressions the 
definition of § 3. 

A first difficulty presents itself : consider a red color sensation 
affecting a certain point of the retina ; and on the other hand a 
blue color sensation affecting the same point of the retina. It is 
necessary that we have some means of recognizing that these two 
sensations, qualitatively different, have something in common. 
Now, according to the considerations expounded in the preceding 
paragraph, we have been able to recognize this only by the move- 
ments of the eye and the observations to which they have given 
rise. If the eye were immovable, or if we were unconscious of 
its movements, we should not have been able to recognize that 
these two sensations, of different quality, had something in com- 
mon ; we should not have been able to disengage from them what 

gives them a geometric character. The visual sensations, without 
the muscular sensations, would have nothing geometric, so that 
it may be said there is no pure visual space. 

To do away with this difficulty, consider only sensations of the 
same nature, red sensations, for instance, differing one from 
another only as regards the point of the retina that they affect. 
It is clear that I have no reason for making such an arbitrary 
choice among all the possible visual sensations, for the purpose 
of uniting in the same class all the sensations of the same color, 
whatever may be the point of the retina affected, I should never 
have dreamt of it, had I not before learned, by the means we 
have just seen, to distinguish changes of state from changes of 
position, that is, if my eye were immovable. Two sensations of 
the same color affecting two different parts of the retina would 
have appeared to me as qualitatively distinct, just as two sensa- 
tions of different color. 

la restricting myself to red sensations, I therefore impose upon 
myself an artificial limitation and I neglect systematically one 
whole side of the question ; but it is only by this artifice that I am 
able to analyze visual space without mingling any motor sensation. 

Imagine a line traced on the retina and dividing in two its 
surface; and set apart the red sensations affecting a point of this 
line, or those differing from them too little to be distinguished 
from them. The aggregate of these sensations will form a sort of 
cnt that I shall call C, and it is clear that this cut suffices to 
divide the manifold of possible red sensations, and that if I take 
two red sensations affecting two points situated on one side and 
the other of the line, I can not pass from one of these sensations to 
the other in a continuous way without passing at a certain 
moment through a sensation belonging to the cut. 

If, therefore, the cut has n dimensions, the total manifold of my 
I red sensations, or if you wish, the whole visual space, will have 


Now, I distinguish the red sensations affecting a point of the 
cut C. The assemblage of these sensations will form a new cut 
C. It is clear that this will divide the cut C, always giving to the 
divide the same meaning. 


If, therefore, the cut C has n dimensions, the cut C will have 
n + 1 and the whole of visual space n + 2. 

If all the red sensations affecting the same point of the retina 
were regarded as identical, tiie cut C reducing to a single ele- 
ment would have dimension^ and visual space would have 2. 

And yet most often it is said that the eye gives us the sense of 
a third dimension, and enables us in a certaiil measure to recog- 
nize the distance of objects. When we seek to analyze this feel- 
ing, we ascertain that it reduces either to the consciousness of the 
convergence of the eyes, or to that of the effort of accommodation 
which the ciliary muscle makes to focus the image. 

Two red sensations affecting the same point of the retina will 
therefore be regarded as identical only if they are accompanied 
by the same sensation of convergence and also by the same sensa- 
tion of effort of accommodation or at least by sensations of 
convergence and accommodation so slightly different as to be 

On this account the cut C is itself a continuum and the cut C 
has more than one dimension. 

But it happens precisely that experience teaches us that when 
two visual sensations are accompanied by the same sensation of 
convergence, they are likewise accompanied by the same sensa- 
tion of accommodation. If then we form a new cut C" with all 
those of the sensations of the cut C", which are accompanied by a 
certain sensation of convergence, in accordance with the preced- 
ing law they will all be indistinguishable and may be regarded 
as identical. Therefore C" will not be a continuum and will 
have dimension ; and as C" divides C" it will thence result that 
C has one, C two and the whole visual space three dimeyisions. 

But would it be the same if experience had taught us the con- 
trary and if a certain sensation of convergence were not always 
accompanied by the same sensation of accommodation? In this 
case two sensations affecting the same point of the retina and 
accompanied by the same sense of convergence, two sensations 
which consequently would both appertain to the cut C", could 
nevertheless be distinguished since they would be accompanied by 
two different sensations of accommodation. Therefore C" would 
be in its turn a continuum and would have one dimension (at 


least) ; then C would have two, C three and the whole vistial 
space would have four dimensions. 

Will it then be said that it is experience which teaches us that 
space has three dimensions, since it is in setting out from an 
experimental law that we have conie to attribute three to it ? But 
we have therein performed, so to speak, only an experiment in 
physiology ; and as also it would suffice to fit over the eyes glasses 
of suitable construction to put an end to the accord between the 
feelings of convergence and of accommodation, are we to say that 
putting on spectacles is enough to make space have four dimen- 
sions and that the optician who constructed them has given one 
more dimension to space ? Evidently not ; all we can say is that 
experience has taught us that it is convenient to attribute three 
dimensions to space. 

But visual space is only one part of space, and in even the 
notion of this space there is something artificial, as I have ex- 
plained at the beginning. The real space is motor space and this 
it is that we shall examine in the following chapter. 


Space and its Three Dimensions 

1. The Oroup of Displacements 

Let us sum up briefly the results obtained. We proposed to 
investigate what was meant in saying that space has three dimen- 
sions and we have asked first what is a physical continuum and 
when it may be said to have n dimensions. If we consider dif- 
ferent systems of impressions and compare them with one another, 
we often recognize that two of these systems of impressions are 
indistinguishable (which is ordinarily expressed in saying that 
they are too close to one another, and that our senses are too 
crude, for us to distinguish them) and we ascertain besides that 
two of these systems can sometimes be discriminated from one 
another though indistinguishable from a third system. In that 
case we say the manifold of these systems of impressions forms 
a physical continuum C. And each of these systems is called an 
element of the continuum C 

How many dimensions has this continuum? Take first two 
elements A and B of C, and suppose there exists a series S of 
elements, all belonging to the continuum C, of such a sort that A 
and B are the two extreme terms of this series and that each 
term of the series is indistinguishable from the preceding. If 
such a series 2 can be found, we say that A and B. are joined to 
one another; and if any two elements of C are joined to one 
another, we say that C is all of one piece. 

Now take on the continuum C a certain number of elements in 
a way altogether arbitrary. The aggregate of these elements will 
be called a cut. Among the various series 5 which join A to B, 
we shall distinguish those of which an element is indistinguish- 
able from one of the elements of the cut (we shall say that these 
are they which c\it the cut) and those of which all the elements 
are distinguishable from all those of the cut. If all the series S 
which join A to B cut the cut, we shall say that A and B are 




' separated by the cut, and that the cut divides G. If we can not 
find on C two elements which are separated by the cut, we shall 
say that the cut does not divide C. 

These definitions laid down, if the continuum C can be divided 
by cuts which do not themselves form a continuum, this con- 
tinuum C has only one dimension; in the contraiy ease it has 
several. If a cut forming a continuum of 1 dimension suffices 
to divide C, C will have 2 dimensions; if a cut forming a con- 
tinuum of 2 dimensions suffices, C will have 3 dimensions, et<:. 
Thanks to these definitions, we can always recognize how many 
dimensions any physical continuum has. It only remains to find 
a physical continuum which is, so to speak, equivalent to space, 
of such a sort that to every point of space corresponds an ele- 
ment of this continuum, and that to points of space very near one 
another correspond indistinguishable elements. Space will have 
then as many dimensions as this continuum. 

The intermediation of this physical continuum, capable of 
representation, is indispensable; because we can not represent 
space to ourselves, and that for a multitude of reasons. Space 
ia a matliemattcal continuum, it is infinite, and we can represent 
to ourselves only physical continua and finite objects. The dif- 
ferent elements of space, which we call points, are all alike, and, 
to apply our definition, it ia necessary that we know how to dis- 
tinguish the elements from one another, at least if they are not 
too close. Finally absolute space is nonsense, and it is necessary 
for as to begin by referring space to a system of axes invariably 
boond to our body (which we must always suppose put back in 
tile initial attitude). 

Then I have sought to form with our visual sensations a phys- 
ical continuum equivalent to apace ; that certainly is easy and this 
example ia particularly appropriate for the discussion of the 
number of dimensions; this discussion has enabled us to see in 
what measure it is allowable to say that 'visual space' has three 
dimensions. Only this solution is incomplete and artificial. I 
have explained why, and it is not on visual space, but on motor 
space that it is necessary to bring our efforts to bear, I have then 
recalled what is the origin of the distinction we make between 


changes of position and changes of state. Among the changes 
which occur in our impressions, we distinguish, first the ifUermU 
changes, voluntary and accompanied hy muscular sensations, and 
the external changes, having opposite characteristics. We ascer- 
tain that it may happen that an external change may be corrected 
by an internal change which reestablishes the primitive sensa- 
tions. The external changes, capable of being corrected by an 
internal change are called changes of position, those not capable 
of it are called changes of staie. The internal changes capable 
of correcting an external change are called displacements of the 
whole body; the others are called changes of attitude. 

Now let a and p be two external changes, a' and fi' two internal 
changes. Suppose that a may be corrected either by a' or by )9', 
and that a' can correct either aor p; experience tells us then that 
fi' can likewise correct p. In this case we say that a and p cor- 
respond to the same displacement and also that of and p' cor- 
respond to the same displacement. That postulated, we can 
imagine a physical continuum which we shall call the continuum 
or group of displacements and which we shall define in the fol- 
lowing manner. The elements of this continuum shall be the in- 
ternal changes capable of correcting an external change. Two of 
these internal changes a' and p' shall be regarded as indis- 
tinguishable : (1) if they are so naturally, that is, if they are 
too close to one another; (2) if a' is capable of correcting 
the same external change as a third internal change natu- 
rally indistinguishable from p'. In this second case, they will 
be, so to speak, indistinguishable by convention, I mean by agree- 
ing to disregard circumstances which might distinguish them. 

Our continuum is now entirely defined, since we know its ele- 
ments and have fixed under what conditions they may be re- 
garded as indistinguishable. We thus have all that is necessary 
to apply our definition and determine how many dimensions this 
continuum has. We shall recognize that it has six. The con- 
tinuum of displacements is, therefore, not equivalent to space, 
since the number of dimensions is not the same ; it is only related 
to space. Now how do we know that this continuum of displace- 
ments has six dimensions? We know it by experience. 

It would be easy to describe the experiments by which we 

could arrive at this reBult. It would be seeu that in this con- 
tinuuni cuts can be made which divide it and which are con- 
tinua; that these cuts themselves can be divided by other cuts 
of the second order which yet are eontinua, and that this would 
stop only after cuts of the sixth order which would no longer be 
eontinua. From our definitions that would mean that the group 
of displaeeiiients has six dimensions. 

That would be easy, I have said, but that would be rather long; 
and would it not be a little superficial J This group of displace- 
ments, we have seen, is related to space, and space coiUd be de- 
duced from it, but it is not equivalent to space, since it has not 
the same number of dimensions; and when we shall have shown 
how the notion of this continuum can be formed and how that of 
space may be deduced from it, it might always be asked why 
space of three dimensions is much more familiar to us than this 
continuum of six dimensions, and consequently doubted whether 
it was by this detour that the notion of space was formed in the 
human mind. 

2. Identity of Two Points 

"What is a point! How do we know whether two points of 
space are identical or different T Or, in other words, when I say : 
The object A occupied at the instant a the point which the object 
B occupies at the instant fi, what does that meant 

Such is the problem we set ourselves in the preceding chapter, 
§4 ^ I have explained it, it is not a question of comparing the 
positions of the objects A and B in absolute space ; the question 
then would manifestly have no meaning. It is a question of 
comparing the positions of these two objects with regard to axes 
invariably bound to my body, supposing always this body re- 
placed in the same attitude. 

I suppose that between the instants a and fi I have moved 
neither my body nor my eye, as 1 know from my muscular sense. 
Nop have I moved either my head, my arm or my hand. I ascer- 
tain that at the instant a impressions that I attributed to the 
object A were transmitted to me, some by one of the fibers of 
my optic neri-e, the others by one of the sensitive tactile nerves 
of my finger; I ascertain that at the instant P other impressions 
I Wfaieb I attributf> to the object B are transmitted to me, some by 


this same fiber of the optic nerve, the others by this same tactile 

Here I must pause for an explanation ; how am I told that this 
impression which I attribute to A, and that which I attribute to 
B, impressions which are qualitatively different, are transmitted 
to me by the same nerve ? Must we suppose, to take for example 
the visual sensations, that A produces two simultaneous sensa- 
tions, a sensation purely luminous a and a colored sensation a', 
that B produces in the same way simultaneously a luminous sen- 
sation b and a colored sensation V, that if these different sensa- 
tions are transmitted to me by the same retinal fiber, a is iden- 
tical with b, but that in general the colored sensations a' and V 
produced by different bodies are different ? In that case it would 
be the identity of the sensation a which accompanies a' with the 
sensation b which accompanies &', which would teU that all these 
sensations are transmitted to me by the same fiber. 

However it may be with this hypothesis and although I am 
led to prefer to it others considerably more complicated, it is 
certain that we are told in some way that there is something in 
common between these sensations a -{-a' and b + 5', without 
which we should have no means of recognizing that the object B 
has taken the place of the object A. 

Therefore I do not further insist and I recall the hypothesis I 
have just made: I suppose that I have ascertained that the im- 
pressions which I attribute to B are transmitted to me at the 
instant p by the same fibers, optic as well as tactile, which, at the 
instant a, had transmitted to me the impressions that I attributed 
to A. If it is so, we shall not hesitate to declare that the point 
occupied by B at the instant p is identical with the point occu- 
pied by A at the instant a. 

I have just enunciated two conditions for these points being 
identical ; one is relative to sight, the other to touch. Let us con- 
sider them separately. The first is necessary, but is not suffi- 
cient. The second is at once necessary and sufficient. A person 
knowing geometry could easily explain this in the following 
manner : Let be the point of the retina where is formed at the 
instant a the image of the body A ; let ilf be the point of space 
occupied at the instant a by this body A ; let M' be the point of 

space occupied at the instant p by the body B. For this body B 
to form its image in O, it is not necessary that tlie points U and 
M' coincide; since vision acts at a distance, it suffices for the 
three points M M' to be in a straight line. This condition that 
the two objects form their image on is therefore necessary, but 
not sufBcient for the points .¥ and M' to coincide. Let now P be 
the point occupied by my finger and where it remains, sinee it 
does not budge. As touch docs not act at a distance, if the 
body A touches my finger at the instant a, it is because M and 
P coincide; if B touches my finger at the instant p, it is because 
Jtf' and P coincide. Therefore M and M' coincide. Thus this 
condition that if A touches my finger at the instant a, B touches 
it at the instant fi, is at once necessary and sufficient for M and 
M' to coincide. 

But we who, as yet, do not know geometry can not reason 
thus; all that we can do is to ascertain experimentally that tlie 
first condition relative to sight may be fulfilled without the 
second, which is relative to touch, but that the second can not 
be fulfilled without the first. 

Suppose experience had taught us the contrary, as might well 
be ; this hypothesis contains nothing absurd. Suppose, therefore, 
that we had ascertained experimentally that the condition rela- 
tive to touch may be fulfilled without that of sight being fulfilled 
and that, on the contrary, that of sight can not be fulfilled i\-ith- 
ont that of touch being also. It is clear that if this were so we 
should conclude that it is touch which may be esereised at a dis- 
tance, and that sight does not operate at a distance. 

But this is not all; up to this time 1 have supposed that to 
determine the place of an object I have made use only of my 
eye and a single finger; hut I could just as well have employed 
other means, for example, all my other fiogera. 

I suppose that my first finger receives at the instant a a tactile 
impression which I attribute to the object A. I make a scries of 
movements, corresponding to a series S of muscular sensations. 
After these movements, at the instant a', my second finger re- 
ceives a tactile impression that I attribute likewise to A. After- 
ward, at the instant p, without my having budged, as my mus- 
cular sense tells me, this same second finger transmits to me 


anew a tactile impression which I attribute this time to the 
object B ; I then make a series of movements, corresponding to 
a series 8' of muscular sensations. I know that this series 8' is 
the inverse of the series 8 and corresponds to contrary move- 
ments. I know this because many previous experiences have 
shown me that if I made successively the two series of movements 
corresponding to 8 and to 8', the primitive impressions would be 
reestablished, in other words, that the two series mutually com- 
pensate. That settled, should I expect that at the instant ^9^, 
when the second series of movements is ended, my first finger 
would feel a tactile impression attributable to the object B f 

To answer this question, those already knowing geometry 
would reason as follows : There are chances that the object A has 
not budged, between the instants a and a', nor the object B 
between the instants p and p'; assume this. At the instant a, 
the object A occupied a certain point M of space. Now at this 
instant it touched my first finger, and as touch does not operate 
at a distance, my first finger was likewise at the point If. I 
afterward made the series 8 of movements and at the end of 
this series, at the instant a', I ascertained that the object A 
touched my second finger. I thence conclude that this second 
finger was then at M, that is, that the movements 8 had the result 
of bringing the second finger to the place of the first. At the 
instant p the object B has come in contact with my second finger : 
as I have not budged, this second finger has remained at M\ 
therefore the object B has come to ilf ; by hypothesis it does not 
budge up to the instant p^. But between the instants p and p! 
I have made the movements 8'\ as these movements are the in- 
verse of the movements 8, they must have for effect bringing the 
first finger in the place of the second. At the instant p^ this 
first finger will, therefore, be at M ; and as the object B is like- 
wise at M, this object B will touch my first finger. To the ques- 
tion put, the answer should therefore be yes. 

We who do not yet know geometry can not reason thus ; but 
we ascertain that this anticipation is ordinarily realized ; and we 
can always explain the exceptions by saying that the object A 
has moved between the instants a and a', or the object B between 
the instants p and ^. 

But could not experience have given a contrary results Would 
this contrary result have been abaurd in itself! Evidently not. 
What should we have done then if experience had given this 
contrary resultf Would all geometry thus have become impos- 
sible t Not the least in the world. "We should have contented 
ourselves with concluding that touch can operate at a distance. 

When I say, touch does not operate at a distance, but sight 
operates at a distance, this assertion has only one meaning, 
which is as follows: To recognize whether B occupies at the 
instant j? the point occupied by A at the instant a, I can use 
& moltitude of different criteria. In one my eye intervenes, 
in another my first finger, in another my second finger, etc. 
Well, it is sufficient for the criterion relative to one of my fin- 
gers to be satisfied in order that all the others should be satisfied, 
but it is not sufBcient that the criterion relative to the eye should 
be. This b the sense of my assertion, I content myself with 
affirming an experimental fact which is ordinarily verified. 

At the end of the preceding chapter we analyzed visual space; 
we saw that to engender this space it is necessary to bring in the 
retinal sensations, the sensation of convergence and the sensa- 
tion of accommodation ; that if these last two were not always 
in accord, visual space would have four dimensions in place of 
three ; we also saw that if we brought in only the retinal sensa- 
tioDS, we should obtain 'simple visual space,' of only two dimen- 
sions. On the other hand, consider tactile space, limiting our- 
selves to the sensations of a single finger, that is in sum to the 
assemblage of positions this finger can occupy. This tactile 
space that we shall analyze in the following section and which 
consequently I ask permission not to consider further for the 
moment, this tactile space, I say, has three dimensions. Why 
has space properly so called as many dimensions as tactile space 
and more than simple visual space T It is because touch does not 
operate at a distance, while vision does operate at a distance. 
These two assertions have the same meaning and we have just 
seen what this is. 

Now I return to a point over which I passed rapidly in order 
not to interrupt the discussion. How do we know that the im- 
pre^ioDS made on our retina by A at the instant a and B at the 


instant p are transmitted by the same retinal fiber, although 
these impressions are qualitatively different? I have suggested 
a simple hypothesis, while adding that other hypotheses, decid- 
edly more complex, would seem to me more probably true. Here 
then are these hypotheses, of which I have already said a word. 
How do we know that the impressions produced by the red object 
A at the instant a, and by the blue object B at the instant ^, if 
these two objects have been imaged on the same point of the 
retina, have something in common? The simple hypothesis 
above made may be rejected and we may suppose that these two 
impressions, qualitatively different, are transmitted by two dif- 
ferent though contiguous nervous fibers. What means have I 
then of knowing that these fibers are contiguous ? It is probable 
that we should have none, if the eye were immovable. It is the 
movements of the eye which have told us that there is the same 
relation between the sensation of blue at the point A and the sen- 
sation of blue at the point B of the retina as between the sensation 
of red at the point A and the sensation of red at the point B. 
They have shown us, in fact, that the same movements, corre- 
sponding to the same muscular sensations, carry us from the 
first to the second, or from the third to the fourth. I do not 
emphasize these considerations, which belong, as one sees, to the 
question of local signs raised by Lotze. 

3. Tactile Space 

Thus I know how to recognize the identity of two points, the 
point occupied by A at the instant a and the point occupied by 
B at the instant p, but only on one condition, namely, that I have 
not budged between the instants a and p. That does not suiBSce 
for our object. Suppose, therefore, that I have moved in any 
manner in the interval between these two instants, how shall I 
know whether the point occupied by A at the instant a is identi- 
cal with the point occupied by B at the instant pi I suppose 
that at the instant a, the object A was in contact with my first 
finger and that in the same way, at the instant p, the object B 
touches this first finger ; but at the same time, my muscular sense 
has told me that in the interval my body has moved. I have 
considered above two series of muscular sensations S and 8\ and 



I have said it sometimes liappetts that we are led to coDsider two 
sach series S and S' as inverse one of the other, because we have 
often observed that when these two series succeed one another 
our primitive impressions are reestablished. 

If then my muscular sense tells me that I have moved between 
the two instants a and ff, but so as to feel successively the two 
series of muscular sensations S and S' that I consider inverses, 
I shall still conclude, just as if I had not budged, that the points 
occupied by A at the instant a and by B at the instant p are 
identical, if I ascertain that my lirst finger touches A at the 
instant a, and B at the instant y8. 

This solution is not yet completely satisfactory, as one will see. 
Let us see, in fact, how many dimensions it would make us at- 
tribute to space. I wish to compare the two points occupied by A 
and B at the instants a and j3, or (what amounts to the same 
thing since I suppose that my finger touches A at the instant a 
and B at the instant 0] I wish to compare the two points occu- 
pied by my finger at the two instants a and /3. The sole means 

1 use for this comparison is the series S of muscular sensations 
which have accompanied the movements of my body between 
these two instants. The different imaginable series 2 form evi- 
dently a physical continuum of which the number of dimensions 
is very great. Let us agree, as I have done, not to consider as 
distinct the two series 2 and 2 + S + S\ when S and S' are in- 
verses one of the other in the sense above given to this word; 
in spite of this agreement, the aggregate of distinct series 2 will 
still form a physical continuum and the number of dimensions 
will be less but still very great. 

To each of these series 2 corresponds a point of space ; to two 
series 5 and S' thus correspond two points If and M". The means 
we have hitherto used enable us to recognize that M and M' are 
not distinct in two cases: (1) if 2 is identical with 2'; (2) if 2' = 

2 -|- S -f S', S and S' being inverses one of the otlier. If in all 
the other cases we should regard M and M' as distinct, the mani- 
fold of points would have as many dimensions as the aggregate 
of distinct series 2, that is, much more than three. 

For those who already know geometry, the following esplana- 
lion would be easily comprehensible. Among the imaginable 


series of muscular sensations, there are those which correspond 
to series of movements where the finger does not budge. I say 
that if one does not consider as distinct the series S and S -|- <r, 
where the series <r corresponds to movements where the fingar 
does not budge, the aggregate of series will constitute a con- 
tinuum of three dimensions, but that if one regards as distinct 
two series S and S' unless S' = S + /S + /S', 8 and 8' being in- 
verses, the aggregate of series will constitute a continuum of 
more than three dimensions. 

In fact, let there be in space a surface A, on this surface a 
line By on this line a point M. Let Co be the aggregate of all 
series S. Let C^ be the aggregate of all the series S, such that 
at the end of corresponding movements the finger is found upon 
the surface A, and C, or C^ the aggregate of series 2 such that 
at the end the finger is found on B, or at M. It is clear, first that 
Ci will constitute a cut which will divide Co, that C, will be a cut 
which will divide Ci, and Cj a cut which will divide C,. Thence 
it results, in accordance with our definitions, that if C^ is a con- 
tinuum of n dimensions, Co will be a physical continuum of 
n -j- 3 dimensions. 

Therefore, let 5 and S' = 5 + o- be two series forming part 
of Cj ; for both, at the end of the movements, the finger is found 
at M ; thence results that at the beginning and at the end of the 
series o-, the finger is at the same point M. This series a is there- 
fore one of those which correspond to movements where the 
finger does not budge. If 5 and 5 + o- are not regarded as dis- 
tinct, all the series of Cj blend into one ; therefore Cj will have 
dimension, and Cq will have 3, as I washed to prove. If, on 
the contrary, I do not regard 5 and 5 + o- as blending (unless 
(r=8 -\-8', 8 and 8' being inverses), it is clear that C^ will con- 
tain a great number of series of distinct sensations; because, 
without the finger budging, the body may take a multitude of 
different attitudes. Then Cg will form a continuum and Co will 
have more than three dimensions, and this also I wished to prove. 

We who do not yet know geometry can not reason in this way; 
we can only verify. But then a question arises; how, before 
knowing geometry, have we been led to distinguish from the 
others these series o- where the finger does not budget It is, in 



Jact, only after having made thia distinction that we could be led 
to regard S and S + (r as identical, and it is on this condition 
alone, as we have just seen, that we can arrive at apace of three 

We are led to distinguish the series <r, because it often happens 
that when we have executed the movements which correspond to 
these series a of muscular sensations, the tactile sensations which 
are transmitted to us by the nerve of the finger that we have 
called the first finger, persist and are not altered by these move- 
ments. Experience alone tells us that and it alone could tell us. 

If we have distinguished the series of muscular sensations 
5 + S' formed by the union of two inverse series, it is because 
they preserve the totality of our impressions; if now we distin- 
guish the series a, it is because they preserve certain of our im- 
pressions. (When I say that a series of muscular sensations S 
'preserves' one of our impressions A, I mean that we ascertain 
that if we feel the impression A, then the muscular sensations 8, 
we still feel the impression A after these sensations .S.) 

I have said above it often happens that the series a do not 
alter the tactile impressions felt by our Brst finger ; I said ofien, 
I did not say always. This it is that we express in our ordinary 
language by saying that the tactile impressions would not be 
altered if the finger has not moved, on the condition that neither 
has the object -d, which was in contact with this finger, moved. 
Before knowing geometry, we could not give this explanation; 
all we could do is to ascertain that the impression often per^ 
gists, but not always. 

But that the impression often continues is enough to make the 
series o appear remarkable to us, to lead us to put in the sai 
class the series 2 and 2 + 0-, and hence not regard them as dis- 
tinct. Under these conditions we have seen that they will en- 
gender a physical continuum of three dimensions. 

Behold then a space of three dimensions engendered by my 
first finger. Each of my fingers will create one like it. It re- 
mains to consider how we are led to regard them as identical 
with visual space, as identical with geometric space. 

Bnt one reflection before going further ; according to the fore- 
going, we know the points of space, or more generally the final 


situation of our body, only by the series of muscular sensations 
revealing to us the movements which have carried us from a 
certain initial situation to this final situation. But it is clear 
that this final situation will depend, on the one hand, upon 
these movements and, on the other hand, upon the initial situa- 
tion from which we set out. Now these movements are re- 
vealed to us by our muscular sensations ; but nothing tells us the 
initial situation; nothing can distinguish it for us from all the 
other possible situations. This puts well in evidence the essential 
relativity of space. 

4. Identity of the Different Spaces 

We are therefore led to compare the two continua C and C 
engendered, for instance, one by my first finger D, the other by 
my second finger D\ These two physical continua both have 
three dimensions. To each element of the continuum C, or, if 
you prefer, to each point of the first tactile space, corresponds a 
series of muscular sensations S, which carry me from a certain 
initial situation to a certain final situation.^ Moreover, the same 
point of this first space will correspond to S and to S + cr, if cr 
is a series of which we know that it does not make the finger D 

Similarly to each element of the continuum C\ or to each point 
of the second tactile space, corresponds a series of sensations S', 
and the same point will correspond to 2' and to 2' + <t', if a' is a 
series which does not make the finger D' move. 

What makes us distinguish the various series designated o- from 
those called o-' is that the first do not alter the tactile impressions 
felt by the finger D and the second preserve those the finger ly 

Now see what we ascertain : in the beginning my finger D' feels 
a sensation A' ; I make movements which produce muscular sen- 
sations 8; my finger D feels the impression A; I make move- 
ments which produce a series of sensations a; my finger D con- 
tinues to feel the impression A, since this is the characteristic 

1 In place of saying that we refer space to axes rigidly bound to our 
body, perhaps it would be better to say, in conformity to what precedes, 
that we refer it to axes rigidly bound to the initial situation of our body. 


property of the series a ; I then make movements which produce 
the series S' of muscular sensations, inverse to S io the sense 
above given to this word. I ascertain then that my finger D' 
feels anew the impression A'. (It is of course understood that 
8 has been suitably chosen.) 

This means that the series 8 -\-a-\-S', preserving the tactile 
impressions of the finger D', is one of the series I have called t/. 
Inversely, if one takes any series a', S' + o' + S will be one of 
the series that we call a. 

Thas if S is suitably chosen, S -j- o' + S' will be a series a', and 
by making a vary in all possible ways, we shall obtain all the 
possible series </. 

Not yet knowing geometry, we limit ourselves to verifying all 
that, but here is how those who know geometrj- would explain the 
fact. In the beginning my finger ly is at the point M, in contact 
with the object a, which makes it feel the impression A'. I make 
the movements corresponding to the series S; I have said that 
this series should be suitably chosen, I should so make this 
choice tliat these movements carry the finger D to the point 
originally occupied by the finger D', that is, to the point M ; this 
finger D will thus be in contact with the object a, which will 
make it feel the impression A. 

I then make the movements corresponding to the series a; in 
these movements, by hypothesis, the position of the finger D does 
not change, this finger therefore remains in contact with the ob- 
ject a and continues to feel the impression A. Finally I make 
the movements corresponding to the series S'. As S' is inverse 
Io S, these movements carry the finger D' to the point previously 
occupied by the finger D, that is, to the point M, If, as may be 
supposed, tJie object a has not budged, this finger D' will be 
in contact with this object and will feel anew the impression 
A'.. . . Q. E. D. 

Let us see the consequences. I consider a series of muscular 
sensations 2. To this series will correspond a point M of the 
first tactile space. Now take again the two series S and 8', in- 
verses of one another, of which we have just spoken. To the 
series S -}- 1-^-8' will correspond a point N of the second tac- 
I tilfi space, since to any series of muscular sensations corresponds, 


as we have said, a point, whether in the first space or in the 

I am going to consider the two points N and M, thus defined, 
as corresponding. What authorizes me so to do? For this 
correspondence to be admissible, it is necessary that if two points 
M and M\ corresponding in the first space to two series S and S', 
are identical, so also are the two corresponding points of the 
second space N and N', that is the two points which correspond 
to the two series /S + S + S' and S + S' + 8\ Now we shall see 
that this condition is fulfilled. 

First a remark. As S and S' are inverses of one another, we 
•shall have /S + /S' = 0, and consequently flf + flf' + S = S + 5 + 
8' =t S, or again S + flf + /S' + 5' = 5 + S'; but it does not fol- 
low that we have 8 -{-1, -{- S' = i; because, though we have used 
the addition sign to represent the succession of our sensations, 
it is clear that the order of this succession is not indifferent: 
we can not, therefore, as in ordinary addition, invert the order 
of the terms ; to use abridged language, our operations are asso- 
ciative, but not commutative. 

That fixed, in order that S and S' should correspond to the 
same point M=^M' of the first space, it is necessary and suffi- 
cient for us to have 2' = 5 + o-. We shall then have : S -{- S' + 

flf' = fif + 5+cr + iS' = /Sf + 5 + iS' + flf + cr + flf'. 

But we have just ascertained that S -\- g -\- S' was one of the 
series a'. We shall therefore have :/S + S' + iS' = /S-fS + 
8' + or', which means that the series /S + 2' + ^' aiid iSf + 2 + 
8' correspond to the same point N=N' of the second space. 
Q. E. D. 

Our two spaces therefore correspond point for point ; they can 
be 'transformed' one into the other; they are isomorphic. How 
are we led to conclude thence that they are identical ? 

Consider the two series o- and 8 -\- <t -\- 8' ==^ <t\ I have said 
that often, but not always, the series a preserves the tactile impres- 
sion A felt by the finger D ; and similarly it often happens, but 
not always, that the series </ preserves the tactile impression A' 
felt by the finger D\ Now I ascertain that it happens very often 
(that is, much more often than what I have just called 'often') 
that when the series o- has preserved the impression A of the 


finger D, the series </ preserves at the same time the impression 
A^ of the finger D' ; and, inversely, that if the first impression is 
altered, the second is likewise. That happens very often, but not 

We interpret this experimental fact by saying that the un- 
known object a which gives the impression A to the finger D is 
identical with the unknown object a' which gives the impression 
A' to the finger Z>'. And in fact when the first object moves, 
which the disappearance of the impression A tells us, the second 
likewise moves, since the impression A' disappears likewise. 
When the first object remains motionless, the second remains 
motionless. If these two objects are identical, as the first is at 
the point M of the first space and the second at the point N 
of the second space, these two points are identical. This is how 
we are led to regard these two spaces as identical ; or better, this 
is what we mean when we say that they are identical. 

What we have just said of the identity of the two tactile 
spaces makes unnecessary our discussing the question of the 
identity of tactile space and visual space, which could be treated 
in the same way. 

5. Space and Empiricism 

It seems that I am about to be led to conclusions in conformity 
with empiristic ideas. I have, in fact, sought to put in evidence 
the role of experience and to analyze the experimental facts 
which intervene in the genesis of space of three dimensions. But 
whatever may be the importance of these facts, there is one thing 
we must not forget and to which besides I have more than once 
called attention. These experimental facts are often verified 
but not always. That evidently does not mean that space has 
often three dimensions, but not always. 

I know well that it is easy to save oneself and that, if the 
facts do not verify, it will be easily explained by saying that 
the exterior objects have moved. If experience succeeds, we say 
that it teaches us about space; if it does not succeed, we hie to 
exterior objects which we accuse of having moved; in other 
words, if it does not succeed, it is given a fillip. 

These fillips are legitimate ; I do not refuse to admit them ; but 


they sofSce to tell us that the properties of space are not experi- 
mental truths, properly so called. If we had wished to verify 
other laws,, we could have succeeded also, by giving other analo- 
gous fillips. Should we not always have been able to justify 
these fillips by the same reasons ? One could at most have said to 
us: 'Your fillips are doubtless legitimate, but you abuse them; 
why move the exterior objects so often t* 

To sum up, experience does not prove to us that space has 
three dimensions ; it only proves to us that it is convenient to at- 
tribute three to it, because thus the number of fillips is reduced 
to a minimum. 

I will add that experience brings us into contact only with 
representative space, which is a physical continuum, never with 
geometric space, which is a mathematical continuum. At the 
very most it would appear to tell us that it is convenient to give 
to geometric space three dimensions, so that it may have as 
many as representative space. 

The empiric question may be put under another form. Is it 
impossible to conceive physical phenomena, the mechanical phe- 
nomena, for example, otherwise than in space of three dimen- 
sions? We should thus have an objective experimental proof, 
so to speak, independent of our physiology, of our modes of 

But it is not so; I shall not here discuss the question com- 
pletely, I shall confine myself to recalling the striking example 
given us by the mechanics of Hertz. You know that the great 
physicist did not believe in the existence of forces, properly so 
called ; he supposed that visible material points are subjected to 
certain invisible bonds which join them to other invisible points 
and that it is the effect of these invisible bonds that we attribute 
to forces. 

But that is only a part of his ideas. Suppose a system formed 
of n material points, visible or not ; that will give in all 3n coor- 
dinates ; let us regard them as the coordinates of a single point 
in space of 3n dimensions. This single point would be con- 
strained to remain upon a surface (of any number of dimensions 
< 3n) in virtue of the bonds of which we have just spoken; to 
go on this surface from one point to another, it would always 



tabe the shortest way ; this would be the single principle which 
would sum up all mechanics. 

Whatever should be thought of this hypothesis, whether we be 
allured by its simplicity, or repelled by its artificial character, 
the simple fact that Hertz was able to conceive it, and to regard 
it as more convenient than our habitual hypotheses, suEBces to 
prove that our ordinary ideas, and, in particular, the three di- 
mensions of space, are in no wise imposed upon mechanics with 
an invincible force. 

6. Mind and Space 

Experience, therefore, has played only a single role, it has 
served as occasion. But this role was none the less very impor- 
tant ; and I have thought it necessary to give it prominence. 
This role would have been useless if there existed an a priori 
form imposing itself upon our sensitivity, and which was space 
of three dimensions. 

Does this form exist, or, if you choose, can we represent to oar- 
selves space of more than three dimensioos ! And first what does 
this question mean! In the true sense of the word, it is clear 
that we can not represent to ourselves space of four, nor space 
of three, dimensions ; we can not first represent them to ourselves 
empty, and no more can we represent to ourselves an object 
either in space of four, or in space of three, dimensions: (1) 
Because these spaces are both infinite and we can not represent 
to ourselves a figure in space, that is, the part in the whole, with- 
out representing the whole, and that is impossible, because it is 
infinite; (2) because these spaces are both mathematical con- 
tinna, and we can represent to ourselves only the physical con- 
tinnnm; (3) because these spaces are both homogeneous, and 
the frames in which we enclose oar sensations, being limited, can 
not be homogeneous. 

Thus the question put can only be nnderstood in one way; 
is it possible to imagine that, the results of the experiences 
related above having been different, we might have been led to 
attribute to space more than three dimensions; to imagine, for 
instance, that the sensation of accommodation might not be con- 
stantly in accord with the sensation of convergence of the eyes; 


or indeed that the exi)eriences of which we have spoken in § 2, 
and of which we express the result by saying ' that touch does 
not operate at a distance/ might have led us to an inverse con- 

And then yes evidently that is possible ; from the moment one 
imagines an experience, one imagines just there by the two con- 
trary results it may give. That is possible, but that is diffi- 
cult, because we have to overcome a multitude of associations of 
ideas, which are the fruit of a long personal experience and of 
the still longer experience of the race. Is it these associations 
(or at least those of them that we have inherited from our an- 
cestors), which constitute this a priori form of which it is said 
that we have pure intuition? Then I do not see why one should 
declare it refractory to analysis and should deny me the right 
of investigating its origin. 

When it is said that our sensations are 'extended' only one 
thing can be meant, that is that they are always associated with 
the idea of certain muscular sensations, corresponding to the 
movements which enable us to reach the object which causes 
them, which enable us, in other words, to defend ourselves against 
it. And it is just because this association is useful for the de- 
fense of the organism, that it is so old in the history of the species 
and that it seems to us indestructible. Nevertheless, it is only 
an association and we can conceive that it may be broken; so 
that we may not say that sensation can not enter consciousness 
without entering in space, but that in fact it does not enter con- 
sciousness without entering in space, which means, without being 
entangled in this association. 

No more can I understand one's saying that the idea of time 
is logically subsequent to space, since we can represent it to our- 
selves only under the form of a straight line; as well say that 
time is logically subsequent to the cultivation of the prairies, 
since it is usually represented armed with a scythe. That one 
can not represent to himself simultaneously the different parts of 
time, goes without saying, since the essential character of these 
parts is precisely not to be simultaneous. That does not mean 
that we have not the intuition of time. So far as that goes, no 
more should we have that of space, because neither can we rep- 


resent it, in the proper sense of the word, for the reasons I have 
mentioned. What we represent to ourselves under the name of 
straight is a crude image which as ill resembles the geometric 
straight as it does time itself. 

Why has it been said that every attempt to ^ve a fourth dimen- 
sion to space always carries this one back to one of the other 
three! It is easy to understand. Consider our muscular sen- 
sations and the 'series' they may form. In consequence of nu- 
merous experiences, the ideas of these series are associated to- 
gether in a very complex woof, our series are classed. Allow 
me, for convenience of language, to express my thought in a 
way altogether crude and even inexact by saying that our scries 
of muscular sensations are classed in three classes correspond- 
ing to the three dimensions of space. Of course this classiBca- 
tion is much more complicated than that, but that will suffice 
to make my reasoning understood. If I wish to imagine a fourth 
dimensiou, I shall suppose another series of muscular sensations, 
making part of a fourth class. But as all my muscular sensa- 
tions have already been classed in one of the three preexistent 
classes, I can only represent to myself a series belonging to one 
of these three classes, so that my fourth dimension is carried 
back to one of the other three. 

What does that prove! This; that it woixld have been neces- 
sary first to destroy the old classification and replace it by a new 
one in which the series of muscular sensations should have been 
distributed into four classes. The difBculty would have dis- 

It is presented sometimes under a more striking form. Sup- 
pose I am enclosed in a chamber between the six impassable 
boundaries formed by the four walls, the Soor and the ceiling; 
it will be impossible for me to get out and to imagine my getting 
out. Pardon, can you not imagine that the door opens, or that 
two of these walla separate! But of course, you answer, one 
must suppose that these walls remain immovable. Yea, but it is 
evident that I have the right to move ; and then the walls that we 
suppose absolutely at rest will be in motion with regard to me. 
Yes, but such a relative motion can not be arbitrary; when ob- 
jects are at rest, their relative motion with regard to any axea 


is that of a rigid solid; now, the apparent motions that yon 
imagine are not in conformity with the laws of motion of a rigid 
solid. Yes, but it is experience which has taught us the laws 
of motion of a rigid solid ; nothing would prevent our imagimng 
them different. To sum up, for me to imagine that I get out of 
my prison, I have only to imagine that the walls seem to open, 
when I move. 

I believe, therefore, that if by space is understood a mathemat- 
ical continuum of three dimensions, were it otherwise amorphous, 
it is the mind which constructs it, but it does not construct it out 
of nothing; it needs materials and models. These materials, 
like these models, preexist within it. But there is not a single 
model which is imposed upon it; it has choice; it may choose, 
for instance, between space of four and space of three dimen- 
sions. What then is the role of experience? It gives the indi- 
cations following which the choice is made. 

Another thing: whence does space get its quantitative char- 
acter? It comes from the role which the series of muscular sen- 
sations play in its genesis. These are series which may repeat 
themselves, and it is from their repetition that number comes ; it 
is because they can repeat themselves indefinitely that space is 
infinite. And finally we have seen, at the end of section 3, that 
it is also because of this that space is relative. So it is repeti- 
tion which has given to space its essential characteristics; now, 
repetition supposes time; this is enough to tell that time is 
logically anterior to space. 

7. Role of the Semicircular Canals 

I have not hitherto spoken of the role of certain organs to 
which the physiologists attribute with reason a capital impor- 
tance, I mean the semicircular canals. Numerous experiments 
have suflSciently shown that these canals are necessary to our 
sense of orientation; but the physiologists are not entirely in 
accord ; two opposing theories have been proposed, that of Mach- 
Delage and that of M. de Cyon. 

M. de Cyon is a physiologist who has made his name illustrious 
by important discoveries on the innervation of the heart; I can 
not, however, agree with his ideas on the question before us. Not 


being a physiologist, I hesitate to criticize the experiments he has 
directed against the adverse theory of Mach-Delage; it seems 
to me, however, that they are not convincing, because in many 
of them the total pressure was made to vary in one of the canals, 
while, physiologically, what varies is the difference between the 
pressures on the two extremities of the canal; in others the 
organs were subjected to profound lesions, which must alter their 

Besides, this is not important; the experiments, if they were 
irreproachable, might be convincing against the old theory. They 
would not be convincing for the new theory. In fact, if I have 
rightly understood the theory, my explaining it wiU be enough 
for one to understand that it is impossible to conceive of an experi- 
ment confirming it. 

The three pairs of canals would have as sole function to tell us 
that space has three dimensions. Japanese mice have only two 
pairs of canals ; they believe, it would seem, that space has only 
two dimensions, and they manifest this opinion in the strangest 
way ; they put themselves in a circle, and, so ordered, they spin 
rapidly around. The lampreys, having only one pair of canals, 
believe that space has only one dimension, but their manifesta- 
tions are less turbulent. 

It is evident that such a theory is inadmissible. The sense- 
organs are designed to tell us of changes which happen in the 
exterior world. We could not understand why the Creator should 
have given us organs destined to cry without cease : Remember 
that space has three dimensions, since the number of these three 
dimensions is not subject to change. 

We must, therefore, come back to the theory of Mach-Delage. 
What the nerves of the canals can tell us is the difference of pres- 
sure on the two extremities of the same canal, and thereby: (1) 
the direction of the vertical with regard to three axes rigidly 
bound to the head; (2) the three components of the acceleration 
of translation of the center of gravity of the head; (3) the cen- 
trifugal forces developed by the rotation of the head; (4) the 
acceleration of the motion of rotation of the head. 

It follows from the experiments of M. Delage that it is this 
last indication which is much the most important; doubtless be- 


eaiue the nenres are leas sensible to the difference of pressoie 
itself than to the brusque variations of this difference. The first 
three indications may thus be neglected. 

Knowing the acceleration of the motion of rotation of the head 
at each instant, we deduce from it, by an unconscious integrar 
tion, the final orientation of the head, referred to a certain initial 
orientation taken as origin. The circular canals contribute, there- 
fore, to inform us of the movements that we have executed, and 
that on the same ground as the muscular sensations. When, 
therefore, above we speak of the series S or of the series S, we 
should say, not that these were series of muscular sensations 
alone, but that they were series at the same time of muscular 
sensations and of sensations due to the semicircular canals. 
Apart from this additioii, we should have nothing to change in 
what precedes. 

In the series S and S, these sensations of the semicircular canals 
evidently hold a very important place. Yet alone they would 
not suffice, because they can tell us only of the movements of the 
head ; they tell us nothing of the relative movements of the body 
or of the members in regard to the head. And more, it seems thai 
they tell us only of the rotations of the head and not of the trans- 
lations it may undergo. 




Analysis and Physics 


You have doubtless often been asked of what good is mathe- 
matics and whether these delicate constmctions entirely mind- 
made are not artificial and bom of our caprice. 

Among those who put this question I should make a distinc- 
tion ; practical people ask of us only the means of money-making. 
These merit no reply j rather would it be proper to ask of them 
what is the good of accumulating so much wealth and whether, 
to get time to acquire it, we are to neglect art and science, which 
alone give us souls capable of enjoying it, 'and for life's sake to 
sacrifice all reasons for living.' 

Besides, a science made solely in view of applications is impos- 
sible; truths are fecund only if bound together. If we devote 
ourselves solely to those truths whence we expect an immediate 
result, the intermediary links are wanting and there will no 
longer be a chain. 

The men most disdainful of theory get from it, without sus- 
pecting it, their daily bread; deprived of this food, progress 
would quickly cease, and we should soon congeal into the im- 
mobility of old China. 

But enough of uncompromising practicians! Besides these, 
there are those who are only interested in nature and who ask us 
if we can enable them to know it better. 

To answer these, we have only to show them the two monu- 
ments already rough-hewn. Celestial Mechanics and Mathematical 



They would doubtless concede that these structures are wdl 
worth the trouble they have cost us. But this is not enough. 
Mathematics has a triple aim. It must furnish an instrument 
for the study of nature. But that is not all : it has a philosophic 
aim and, I dare maintain, an esthetic aim. It must aid the 
philosopher to fathom the notions of number, of space, of time. 
And above all, its adepts find therein delights analogous to those 
given by painting and music. They admire the delicate harmony 
of numbers and forms ; they marvel when a new discovery opens 
to them an unexpected perspective ; and has not the joy they thus 
feel the esthetic character, even though the senses take no part 
therein t Only a privileged few are called to enjoy it fully, it is 
true, but is not this the case for all the noblest arts ? 

This is why I do not hesitate to say that mathematics deserves 
to be cultivated for its own sake, and the theories inapplicable 
to physics as well as the others. Even if the physical aim and 
the esthetic aim were not united, we ought not to sacrifice either. 

But more : these two aims are inseparable and the best means 
of attaining one is to aim at the other, or at least never to lose 
sight of it. This is what I am about to try to demonstrate in 
setting forth the nature of the relations between the pure sci- 
ence and its applications. 

The mathematician should not be for the physicist a mere pur- 
veyor of formulas ; there should be between them a more intimate 
collaboration. Mathematical physics and pure analysis are not 
merely adjacent powers, maintaining good neighborly relations; 
they mutually interpenetrate and their spirit is the same. This 
will be better understood when I have shown what physics gets 
from mathematics and what mathematics, in return, borrows 
from physics. 


The physicist can not ask of the analyst to reveal to him a new 
truth ; the latter could at most only aid him to foresee it. It is a 
long time since one still dreamt of forestalling experiment, or of 
constructing the entire world on certain premature hypotheses. 
Since all those constructions in which one yet took a naive de- 
light it is an age, to-day only their ruins remain. 


AU laws are therefore deduced from experiment; but to eniin- 
ciate tlieni, a special language is needful ; ordinary language is 
too poor, it is besides too vague, to express relations so delicate, 
BO rich, and ao precise. 

This therefore is one reason why the physicist can not do with- 
out mathematics ; it furnishes him the only language he can speak. 
And a well-made language is no indifferent thing; not to go 
beyond physics, the unknown man who invented the word heat 
devoted many generations to error. Heat has been treated as a 
sabstance, simply because it was designated by a substantive, and 
it has been thought indestructible. 

On the other hand, he who invented the word electricity had 
the unmerited good fortune to implicitly endow physics with a 
new law, that of the conservation of electricity, which, by a pure 
chance, has been found exact, at least until now. 

Well, to continue the simile, the writers who embellish s lan- 
guage, who treat it as an object of art, make of it at the same time 
a more supple instrument, more apt for rendering shades of 

We understand, then, how the analyst, who pursues a purely 
esthetic aim, helps create, just by that, a language more fit to 
satisfy the physicist. 

But this is not all : law springs from experiment, but not im- 
mediately. Experiment is individual, the law deduced from it is 
general ; experiment is only approximate, the law is precise, or at 
least pretends to be. Experiment is made under conditions 
always complex, the enunciation of the luw eliminates these com- 
plications. This is what is called ' correcting the systematic errors. * 

In a word, to get the law from esperiment, it is necessary to 
generalize; this is a necessity imposed upon the most circum- 
spect observer. But how generalize T Every particular truth 
may evidently be extended in an infinity o£ ways. Among these 
thousand routes opening before us, it is necessary to make a 
choice, at least provisional ; in this choice, what shall guide us? 

It can only be analogy. But how vague is this word 1 Primi- 
tive man knew only crude analogies, those which strike the senses, 
those of colors or of sounds. He never would have dreamt of 
likening light to radiant heat. 


What has taught us to know the true, profound analogies, those 
the eyes do not see but reason divines? 

It is the mathematical spirit, which disdains matter to ding 
only to pure form. This it is which has taught us to give the same 
name to things differing only in material, to call by the same 
name, for instance, the multiplication of quaternions and that of 
whole numbers. 

If quaternions, of which I have just spoken, had not been so 
promptly utilized by the English physicists, many persons would 
doubtless see in them only a useless fancy, and yet, in teaching us 
to liken what appearances separate, they would have already 
rendered us more apt to penetrate the secrets of nature. 

Such are the services the physicist should expect of analysis; 
but for this science to be able to render them, it must be culti- 
vated in the broadest fashion without immediate expectation of 
utility — the mathematician must have worked as artist. 

What we ask of him is to help us to see, to discern our way in 
the labyrinth which opens before us. Now, he sees best who 
stands highest. Examples abound, and I limit myself to the most 

The first will show us how to change the language suffices to 
reveal generalizations not before suspected. 

A\Tien Newton's law has been substituted for Kepler's we still 
know only elliptic motion. Now, in so far as concerns this motion, 
the two laws differ only in form ; we pass from one to the other 
by a simple differentiation. And yet from Newton's law may be 
deduced by an immediate generalization all the effects of pertur- 
bations and the whole of celestial mechanics. If, on the other 
hand, Kepler's enunciation had been retained, no one would ever 
have regarded the orbits of the perturbed plants, those compli- 
cated curves of which no one has ever written the equation, as 
the natural generalizations of the ellipse. The progress of obser- 
vations would only have served to create belief in chaos. 

The second example is equally deserving of consideration. 

When Maxwell began his work, the laws of electro-dynamics 
admitted up to his time accounted for all the known facts. It was 
not a new experiment which came to invalidate them. But in 
looking at them under a new bias. Maxwell saw that the equa- 


lions became more symmetrical when a term was added, and 
besides, this term was too small to produce effects appreciable 
with the old methods. 

You know that Maxwell's a priori views awaited for twenty 
years an e^cperimental confirmation ; or, if you prefer, Maxwell 
was twenty years ahead of experiment. How was this triomph 

It was because Maxwell was profoundly steeped in the sense of 
mathematical symmetry ; would he have been so, if others before 
him had not studied this symmetry for its own beauty t 

It was because Maxwell was accustomed to 'think in vectors,' 
and yet it was through the theory of imaginaries (neomonics) 
that vectors were introduced into analysis. And those who in- 
vented imaginaries hardly suspected the advantage which would 
be obtained from them for the study of the real world, of this the 
name given them is proof sufScient. 

In a word, Maxwell was perhaps not an able analyst, but this 
ability would have been for him only a useless and bothersome 
baggage. On the other hand, he had iu the highest degree the 
intimate sense of mathematical analogies. Therefore it is that he 
made good mathematical physics. 

Maxwell's example teaches us still another thing. 

How should the equations of mathematical physics be treated ! 
Should we simply deduce all the consequences, and regard them 
as intangible realities t Par from it; what they should teach us 
above all is what can and what should be changed. It is thus 
that we get from them something useful. 

The third example goes to show us bow we may perceive mathe- 
matical analogies between phenomena which have physically no 
relation either apparent or real, so that the laws of one of these 
phenomena aid us to divine those of the other. 

The very same equation, that of Laplace, is met in the theory i 
of Newtonian attraction, in that of the motion of liquids, in that 
of the electric potential, in that of magnetism, in that of the 
propagation of heat and in still many others. What is the result* 
These theories seem images copied one from the other; they are 
mutually illuminating, borrowing their language from each other ; 
ask electricians if they do not felicitate themselves on having in- 



vented the phrase flow of force, suggested by hydrodynamics and 
the theory of heat. 

Thus mathematical analogies not only may make us foresee 
physical analogies, but besides do not not cease to be useful when 
these latter fail. 

To sum up, the aim of mathematical physics is not only to 
facilitate for the physicist the numerical calculation of certain 
constants or the int^ration of certain differential equations. It 
is besides, it is above all, to reveal to him the hidden harmony of 
things in making him see them in a new way. 

Of all the parts of analysis, the most elevated, the purest, so 
to speak, will be the most fruitful in the hands of those who know 
how to use them. 


Let us now see what analysis owes to physics. 

It would be necessary to have completely forgotten the history 
of science not to remember that the desire to understand nature 
has had on the development of mathematics the most constant and 
happiest influence. 

In the flrst place the physicist sets us problems whose solution 
he expects of us. But in proposing them to us, he has largely 
paid us in advance for the service we shall render him, if we 
solve them. 

If I may be allowed to continue my comparison with the fine 
arts, the pure mathematician who should forget the existence of 
the exterior world would be like a painter who knew how to har- 
moniously combine colors and forms, but who lacked models. His 
creative power would soon be exhausted. 

The combinations which numbers and symbols may form are an 
infinite multitude. In this multitude how shall we choose those 
which are worthy to fix our attention t Shall we let ourselves be 
guided solely by our caprice f This caprice, which itself would 
besides soon tire, would doubtless carry us very far apart and we 
should quickly cease to understand each other. 

But this is only the smaller side of the question. Physics will 
doubtless prevent our straying, but it will also preserve us from 
a danger much more formidable ; it will prevent our ceaselessly 
going around in the same circle. 

History proves that physics has not only forced us to choose 
amoDg problems which came in a crowd ; it has imposed upon as 
such as we should without it never have dreamed of. However 
varied may be the imagination of man, nature is still a thousand 
times richer. To follow her we must take ways we have 
neglected, and these patlis lead us often to summits whence we 
discover new countries. "What could be more u.%eful ! 

It is with mathematical symbola as with physical realities; it is 
in comparing the different aspects of things that we are able to 
comprehend their inner harmony, which alone is beautiful and 
consequently worthy of our efforts. 

The first example I shall cite is so old we are tempted to foi^et 
it; it is nevertheless the most important of all. 

The sole natural object of mathematical thought is the whole 
number. It is the external world which has imposed the con- 
tinuum upon U9, which we doubtless have invented, but which it 
has forced us to invent. Without it there would be no infini- 
tesimal analysis ; all mathematical science would reduce itself to 
antbmetic or to the theory of substitutions. 

On the contrary, we have devoted to the study of the con- 
tinuum almost all our time and all our strength. Who will regret 
it; who will think that this time and this strength have been 
wasted! Analysis unfolds before us infinite perspectives that 
arithmetic never suspects; it shows us at a glance a majestic 
assemblage whose array is simple and symmetric; on the con- 
trary, in the theory of numbers, where reigns the unforeseen, the 
view is, so to speak, arrested at every step. 

Doubtless it will be said that outside of the whole number there 
is no rigor, and consequently no mathematical truth; that the 
whole number hides everywhere, and that we must strive to render 
transparent the screens which cloak it, even if to do so we must 
resign ourselves to interminable repetitions. Let us not be such 
pnrists and let us be grateful to the continuum, which, if oU 
springs from the whole number, was alone capable of making 
to much proceed therefrom. 

Need I also recall that M. Hermite obtained a surprising ad- 
vantage from the introduction of continuous variables into the 
theory of numbersi Thus the whole number's own domain is 


itself invaded, and this invasion has established order where dis- 
order reigned. 

See what we owe to the continuum and consequently to phys- 
ical nature. 

Fourier's series is a precious instrument of which analysis 
makes continual use, it is by this means that it has been able to 
represent discontinuous functions ; Fourier invented it to solve a 
problem of physics relative to the propagation of heat. If this 
problem had not come up naturally, we should never have dared 
to give discontinuity its rights ; we should still long have regarded 
continuous functions as the only true functions. 

The notion of function has been thereby considerably extended 
and has received from some logician-analysts an unforeseen de- 
velopment. These analysts have thus adventured into regions 
where reigns the purest abstraction and have gone as far away 
as possible from the real world. Yet it is a problem of physics 
which has furnished them the occasion. 

After Fourier's series, other analogous series have entered the 
domain of analysis; they have entered by the same door; they 
have been imagined in view of applications. 

The theory of partial differential equations of the second 
order has an analogous history. It has been developed chiefly 
by and for physics. But it may take many forms, because such 
an equation does not suflBce to determine the unknown function, 
it is necessary to adjoin to it complementary conditions which 
are called conditions at the limits; whence many different 

If the analysts had abandoned themselves to their natural tend- 
encies, they would never have known but one, that which 
Madame Kovalevski has treated in her celebrated memoir. But 
there are a multitude of others which they would have ignored. 
Each of the theories of physics, that of electricity, that of heat, 
presents us these equations under a new aspect. It may, there- 
fore, be said that without these theories we should not know 
partial differential equations. 

It is needless to multiply examples. I have given enough to 
be able to conclude : when physicists ask of us the solution of a 
problem, it is not a duty-service they impose upon us, it is on 
the contrary we who owe them thanks. 



But this is not all; physics not only gives ns the occasion to 
solve problems ; it aids us to find the means thereto, and that in 
two ways. It makes us foresee the solution; it suggests argu- 
ments to us. 

I have spoken above of Laplace's equation which is met in a 
multitude of diverse physical theories. It is found again in 
geometry, in the theory of conf ormal representation and in pure 
analysis, in that of imaginaries. 

In this way, in the study of functions of complex variables, the 
analyst, alongside of the geometric image, which is his usual in- 
strument, finds many physical images which he may make 
use of with the same success. Thanks to these images, he can 
see at a glance what pure deduction would show him only suc- 
cessively. He masses thus the separate elements of the solu- 
tion, and by a sort of intuition divines before being able to 

To divine before demonstrating ! Need I recall that thus have 
been made all the important discoveries? How many are the 
truths that physical analogies permit us to present and that we 
are not in condition to establish by rigorous reasoning! 

For example, mathematical physics introduces a great number 
of developments in series. No one doubts that these develop- 
ments converge ; but the mathematical certitude is lacking. These 
are so many conquests assured for the investigators who shall 
come after us. 

On the other hand, physics furnishes us not alone solutions; 
it furnishes us besides, in a certain measure, arguments. It will 
sufSce to recall how Felix BUein, in a question relative to Rie- 
mann surfaces, has had recourse to the properties of electric 

It is true, the arguments of this species are not rigorous, in 
the sense the analyst attaches to this word. And here a question 
arises: How can a demonstration not suflBciently rigorous for 
the analyst sufSce for the physicist! It seems there can not be 
two rigors, that rigor is or is not, and that, where it is not there 
can not be deduction. 

This apparent paradox will be better understood by recalling 


under what conditions number is applied to natural phenomena. 
Whence come in general the difficulties encountered in seeking 
rigorf We strike them almost always in seeking to estaUiah 
that some quantity tends to some limit, or that some function is 
continuous, or that it has a derivative. 

Now the numbers the physicist measures by experiment are 
never known except approximately; and besides, any function 
always differs as littie as you choose from a discontinuous func- 
tion, and at the same time it differs as littie as you choose from 
a continuous function. The physicist may, therefore, at will 
suppose that the function studied is continuous, or that it is dis- 
continuous; that it has or has not a derivative; and may do so 
without fear of ever being contradicted, either by present ex- 
perience or by any future experiment We see that with such 
liberty he makes sport of difficulties which stop the analyst. He 
may always reason as if all the functions which occur in his 
calculations were entire polynomials. 

Thus the sketch which suffices for physics is not the deduc- 
tion which analysis requires. It does not follow thence that one 
can not aid in finding the other. So many physical sketches have 
already been transformed into rigorous demonstrations that 
to-day this transformation is easy. There would be plenty of 
examples did I not fear in citing them to tire the reader. 

I hope I have said enough to show that pure analysis and 
mathematical physics may serve one another without making any 
sacrifice one to the other, and that each of these two sciences 
should rejoice in all which elevates its associate. 


Governments and parliaments must find that astronomy is one 
of the sciences which cost most dear: the least instrument costs 
hundreds of thousands of dollars, the least observatory costs 
millions; each eclipse carries with it supplementary appropria- 
tions. And all that for stars which are so far away, which are 
complete strangers to our electoral contests, and in all probability 
will never take any part in them. It must be that our politi- 
cians have retained a remnant of idealism, a vague instinct for 
what is grand; truly, I think they have been calumniated; they 
should be encouraged and shown that this instinct does not de- 
ceive them, that they are not dupes of that idealism. 

We might indeed speak to them of navigation, of which no 
one can underestimate the importance, and which has need of 
astronomy. But this would be to take the question by its 
smaller side. 

Astronomy is useful because it raises us above ourselves ; it is 
useful because it is grand ; that is what we should say. It shows 
us how small is man's body, how great his mind, since his intel- 
ligence can embrace the whole of this dazzling immensity, where 
his body is only an obscure point, and enjoy its silent harmony. 
Thus we attain the consciousness of our power, and this is some- 
thing which can not cost too dear, since this consciousness makes 
us mightier. 

But what I should wish before all to show is, to what point 
astronomy has facilitated the work of the other sciences, more 
directly useful, since it has given us a soul capable of compre- 
hending nature. 

Think how diminished humanity would be if, under heavens 
constantly overclouded, as Jupiter's must be, it had forever 
remained ignorant of the stars. Do you think that in such a 
world we should be what we are! I know well that under this 
somber vault we should have been deprived of the light of the 
20 289 


son, necessary to organisms like those which inhabit the eartL 
But if you please, we shall assume that these clouds are phos- 
phorescent and emit a soft and constant light. Since we are 
making hypotheses, another will cost no more. Well ! I repeat 
my question: Do you think that in such a world we should be 
what we are f 

The stars send us not only that visible and gross light which 
strikes our bodily eyes, but from them also comes to us a light far 
more subtle, which illuminates our minds and whose effects I 
shall try to show you. You know what man was on the earth 
some thousands of years ago, and what he is to-day. Isolated 
amidst a nature where everything was a mystery to him, terrified 
at each unexpected manifestation of incomprehensible forces, he 
was incapable of seeing in the conduct of the universe anything 
but caprice ; he attributed all phenomena to the action of a mul- 
titude of little genii, fantastic and exacting, and to act on the 
world he sought to conciliate them by means analogous to those 
employed to gain the good graces of a minister or a deputy. 
Even his failures did not enlighten him, any more than to-day 
a beggar refused is discouraged to the point of ceasing to beg. 

To-day we no longer beg of nature ; we command her, because 
we have discovered certain of her secrets and shall discover 
others each day. We command her in the name of laws she can 
not challenge, because they are hers ; these laws we do not madly 
ask her to change, we are the first to submit to them. Nature 
can only be governed by obeying her. 

What a change must our souls have undergone to pass from the 
one state to the other! Does any one believe that, without the 
lessons of the stars, under the heavens perpetually overclouded 
that I have just supposed, they would have changed so quickly? 
Would the metamorphosis have been possible, or at least would it 
not have been much slower? 

And first of all, astronomy it is which taught that there are 
laws. The Chaldeans, who were the first to observe the heavens 
with some attention, saw that this multitude of luminous points 
is not a confused crowd wandering at random, but rather a disci- 
plined army. Doubtless the rules of this discipline escaped them, 
but the harmonious spectacle of the starry night suflSced to give 


them the impression of regularity, and that was in itself already 
a great thing. Besides, these rules were discerned by Hippar- 
chus, Ptolemy, Copernicus, Kepler, one after another, and finally, 
it is needless to recall that Newton it was who enunciated the 
oldest, the most precise, the most simple, the most general of all 
natural laws. 

And then, taught by this example, we have seen our little ter- 
restrial world better and, under the apparent disorder, there also 
we have found again the harmony that the study of the heavens 
had revealed to us. It also is regular, it also obeys immutable 
laws, but they are more complicated, in apparent conflict one with 
another, and an eye untrained by other sights would have seen 
there only chaos and the reign of chance or caprice. If we had 
not known the stars, some bold spirits might perhaps have 
sought to foresee physical phenomena; but their failures would 
have been frequent, and they would have escited only the deri- 
sion of the vulgar; do we not see, that even in our day the 
meteorologists sometimes deceive themselves, and that certain 
persons are inclined to laugh at them. 

How often would the physicists, disheartened by so many 
(^ecks, have fallen into discouragement, if they had not had, to 
soBtain their confidence, the brilliant example of the success of 
the astronomers! This success showed them that nature obejTB 
laws; it only remained to know what laws; for that they only 
needed patience, and they had the right to demand that tbo 
sceptics should give them credit. 

This is not all : astronomy has not only taught us that there are 
laws, but that from these laws there is no escape, that with them 
there is no possible compromise. How much time should we have 
n«eded to comprehend that fact, if we had known only the ter- 
restrial world, where each elemental force would always seem to 
OS in conflict with other forces T Astronomy has taught us that 
the laws are infinitely precise, and that if those we enunciate 
are approximative, it is because we do not know them well. Aris- 
totle, the most scientific mind of antiquity, still accorded a part 
to accident, to chance, and seemed to think that the laws of na- 
ture, at least here below, determine only the large features of 
phenomena. How much has the ever-increasing precision of 


astronomical predictions contributed to correct such an error, 
which would have rendered nature unintelligible! 

But are these laws not local, varying in different places, like 
those which men make ; does not that which is truth in one comer 
of the universe, on our globe, for instance, or in our little solar 
system, become error a little farther away ? And then could it 
not be asked whether laws depending on space do not also depend 
upon time, whether they are not simple habitudes, transitory, 
therefore, and ephemeral f Again it is astronomy that answers 
this question. Consider the double stars; all describe conies; 
thus, as far as the telescope carries, it does not reach the limits 
of the domain which obeys Newton's law. 

Even the simplicity of this law is a lesson for us; how many 
complicated phenomena are contained in the two lines of its 
enunciation ; persons who do not understand celestial mechanics 
may form some idea of it at least from the size of the treatises 
devoted to this science ; and then it may be hoped that the com- 
plication of physical phenomena likevdse hides from us some 
simple cause still unknown. 

It is therefore astronomy which has shown us what are the 
general characteristics of natural laws ; but among these charac- 
teristics there is one, the most subtle and the most important of 
all, which I shall ask leave to stress. 

How was the order of the universe understood by the 
ancients; for instance, by Pythagoras, Plato or Aristotle? It 
was either an immutable type fixed once for all, or an ideal to 
which the world sought to approach. Kepler himself still 
thought thus when, for instance, he sought whether the distances 
of the planets from the sun had not some relation to the five reg- 
ular polyhedrons. This idea contained nothing absurd, but it 
was sterile, since nature is not so made. Newton has shown us 
that a law is only a necessary relation between the present state 
of the world and its immediately subsequent state. All the 
other laws since discovered are nothing else; they are in sum, 
differential equations; but it is astronomy which furnished the 
first model for them, without which we should doubtless long 
have erred. 

Astronomy has also taught us to set at naught appearances. 



The day Copemicna proved that what was thought the most stable 
was in motion, that what was thought raoviug was fixed, he 
showed us how deceptive could be the infantile reasonings which 
spring directly from the immediate data of our senses. True, 
bis ideas did not easily triumph, but since this triumph there is 
no longer a prejudice so inveterate that we can not shake it off. 
How can we estimate the value of the new weapon thus wonf 

The ancients thought everything was made for man, and this 
illusion must be very tenacious, since it must ever be combated. 
Yet it is necessary to divest oneself of it ; or else one will be only 
an eternal myope, incapable of seeing the truth. To comprehend 
nature one must be able to get out of self, so to speak, and to 
contemplate her from many different points of view; other^vise 
we never shall know more than one side. Now, to get out of 
self is what he who refers everj'thing to himself can not do. Who 
deUvered us from this illusion J It was those who showed us that 
t&e earth is only one of the smallest planets of the solar system, 
and that the solar system itself is only an imperceptible point 
in the infinite spaces of the stellar universe. 

At the same time astronomy taught us not to be afraid of big 
numbers. This was needful, not only for knowing the heavens, 
but to know the earth itself; and was not so easj- as it seems to 
us to-day. Let us try to go back and picture to ourselves what a 
Greek would have thought if told that red light vibrates four 
hundred millions of millions of times per second. Without any 
doubt, such an assertion would have appeared to him pure mad- 
ness, and he never would have lowered himself to test it. To- 
day a hypothesis will no longer appear absurd to us because it 
obligea us to imagine objects much larger or smaller than those 
onr senses are capable of showing us, and we no longer com- 
prehend those scruples which arrested our predecessors and pre- 
vented them from discovering certain truths simply because they 
were afraid of them. But whyT It is because we have seen 
the heavens enlarging and enlarging without cease; because we 
know that the sun is 150 millions of kilometers from the earth 
and that the distatiees of the nearest stars are hundreds of 
thoosands of times greater yet. Habituated to the contempla- 
tioD of the infinitely great, we have become apt to comprehend 


the infinitely small. Thanks to the education it has reoeived, 
our imagination, like the eagle 's eye that the sun does not dazzle, 
can look truth in the face. 

Was I wrong in saying that it is astronomy which has made 
us a soul capable of comprehending nature ; that under heavens 
always overcast and starless, the earth itself would have been for 
us eternally unintelligible; that we should there have seen only 
caprice and disorder; and that, not knowing the world, we should 
never have been able to subdue itf What science could have 
been more useful t And in thus speaking I put myself at the 
point of view of those who only value practical applications. 
Certainly, this point of view is not mine ; as for me, on the con- 
trary, if I admire the conquests of industry, it is above all be- 
cause if they free us from material cares, they will one day give 
to all the leisure to contemplate nature. I do not say: Science 
is useful, because it teaches us to construct machines. I say: 
Machines are useful, because in working for us, they will some 
day leave us more time to make science. But finally it is worth 
remarking that between the two points of view there is no antag- 
onism, and that man having pursued a disinterested aim, all else 
has been added unto him. 

Auguste Comte has said somewhere, that it would be idle to 
seek to know the composition of the sun, since this knowledge 
would be of no use to sociology. How could he be so short- 
sighted ? Have we not just seen that it is by astronomy that, to 
speak his language, humanity has passed from the theological to 
the positive state? He found an explanation for that because 
it had happened. But how has he not understood that what 
remained to do was not less considerable and would be not less 
profitable? Physical astronomy, which he seems to condemn, 
has already begun to bear fruit, and it will give us much more, 
for it only dates from yesterday. 

First was discovered the nature of the sun, what the founder of 
positivism wished to deny us, and there bodies were found which 
exist on the earth, but had here remained undiscovered ; for ex- 
ample, helium, that gas almost as light as hydrogen. That al- 
ready contradicted Comte. But to the spectroscope we owe a 
lesson precious in a quite different way ; in the most distant stars. 

A8TB0N0UI 295 

it shows us the same substances. It might have been asked 
whether the terrestrial elements were not due to some chance 
which had brought together more tenuous atoms to construct of 
them the more complex edifice that the chemists call atom; 
whether, in other regions of the universe, other fortuitous meet- 
ings had not engendered edifices entirely different. Now we know 
that this is not so, that the laws of our chemistry are the gen- 
eral laws of nature, and that they owe nothing to the chance 
which caused us to be bom on the earth. 

But, it will be said, astronomy has given to the other sciences 
all it can give them, and now that the heavens have procured for 
us the instruments which enable us to study terrestrial nature, 
they could without danger veil themselves forever. After what 
we have just said, is there still need to answer this objection t 
One could have reasoned the same in Ptolemy's time; then also 
men thought they knew everything, and they still had almost 
everything to learn. 

The stars are majestic laboratories, gigantic crucibles, such as 
no chemist could dream. There reign temperatures impossible 
for us to realize. Their only defect is being a little far away; 
but the telescope will soon bring them near to us, and then we 
shall see how matter acts there. What good fortune for the 
physicist and the chemist ! 

Matter will there exhibit itself to us under a thousand different 
states, from those rarefied gases which seem to form the nebul® 
and which are luminous with I know not what glimmering of 
mysterious origin, even to the incandescent stars and to the 
planets so near and yet so different. 

Perchance even, the stars will some day teach us something 
about life ; that seems an insensate dream and I do not at all see 
how it can be realized ; but, a hundred years ago, would not the 
chemistry of the stars have also appeared a mad dream f 

But limiting our views to horizons less distant, there still will 
remain to us promises less contingent and yet sufBciently seduc- 
tive. If the past has given us much, we may rest assured that 
the future will give us still more. 

In sum, it is incredible how useful belief in astrology has 
been to humanity. If Kepler and Tycho Brahe made a living. 


it was because they sold to naive kings predictions founded cm 
the conjunctions of the stars. If these princes had not been so 
credulous, we should perhaps still believe that nature obeys 
caprice, and we should still wallow in ignorance. 

The History of Mathematical Physics 

The Past and the Future of Physics. — What is the present 
state of mathematical physics t What are the problems it is led 
to set itself t What is its future t Is its orientation about to be 
modified t 

Ten years hence will the aim and the methods of this science 
appear to our immediate successors in the same light as to our- 
selves; or, on the contrary, are we about to witness a profound 
transformation t Such are the questions we are forced to raise 
in entering to-day upon our investigation. 

If it is easy to propound them: to answer is difScult. If we 
felt tempted to risk a prediction, we should easily resist this 
temptation, by thinking of all the stupidities the most eminent 
savants of a hundred years ago would have uttered, if some one 
had asked them what the science of the nineteenth century 
would be. They would have thought themselves bold in their 
predictions, and after the event, how very timid we should have 
found them. Do not, therefore, expect of me any prophecy. 

But if, like all prudent physicians, I shun giving a prognosis, 
yet I can not dispense with a little diagnostic ; well, yes, there are 
indications of a serious crisis, as if we might expect an approach- 
ing transformation. Still, be not too anxious: we are sure the 
patient will not die of it, and we may even hope that this crisis 
will be salutary, for the history of the past seems to guarantee us 
this. This crisis, in fact, is not the first, and to understand it, 
it is important to recall those which have preceded. Pardon then 
a brief historical sketch. 

The Physics of Central Forces. — ^Mathematical physics, as we 
know, was bom of celestial mechanics, which gave birth to it at 
the end of the eighteenth century, at the moment when it itself 
attained its complete development. During its first years espe- 
cially, the infant strikingly resembled its mother. 



The astronomic universe is formed of masses, very great, no 
doubt, but separated by intervals so immense that they appear 
to us only as material points. These points attract each other 
inversely as the square of the distance, and this attraction is the 
sole force which influences their movements. But if our senses 
were sufSciently keen to show us all the details of the bodies 
which the physicist studies, the spectacle thus disclosed would 
scarcely differ from the one the astronomer contemplates. There 
also we should see material points, separated from one another 
by intervals, enormous in comparison with their dimensions, and 
describing orbits according to regular laws. These infinitesimal 
stars are the atoms. Like the stars proper, they attract or repel 
each other, and this attraction or this repulsion, following the 
straight line which joins them, depends only on the distance. 
The law according to which this force varies as function of the 
distance is perhaps not the law of Newton, but it is an analogous 
law; in place of the exponent — ^2, we have probably a different 
exponent, and it is from this change of exponent that arises all 
the diversity of physical phenomena, the variety of qualities and 
of sensations, all the world, colored and sonorous, which sur- 
rounds us; in a word, all nature. 

Such is the primitive conception in all its purity. It only 
remains to seek in the different cases what value should be given 
to this exponent in order to explain all the facts. It is on this 
model that Laplace, for example, constructed his beautiful theory 
of capillarity; he regards it only as a particular case of attrac- 
tion, or, as he says, of universal gravitation, and no one is as- 
tonished to find it in the middle of one of the five volumes of the 
*Mecanique celeste.' More recently Briot believes he penetrated 
the final secret of optics in demonstrating that the atoms of ether 
attract each other in the inverse ratio of the sixth power of the 
distance; and Maxwell himself, does he not say somewhere that 
the atoms of gases repel each other in the inverse ratio of the 
fifth power of the distance? We have the exponent — 6, or — 5, 
in place of the exponent — 2, but it is always an exponent. 

Among the theories of this epoch, one alone is an exception, 
that of Fourier ; in it are indeed atoms acting at a distance one 
upon the other; they mutually transmit heat, but they do not 


attract, they never budge. From this point of view, Fourier's 
theory must have appeared to the eyes of his contemporaries, to 
those of Fourier himself, as imperfect and provisional. 

This conception was not without grandeur; it was seductive, 
and many among us have not finally renounced it ; they know that 
one will attain the ultimate elements of things only by patiently 
disentangling the complicated skein that our senses give us ; that 
it is necessary to advance step by step, neglecting no interme- 
diary ; that our fathers were wrong in wishing to skip stations ; 
but they believe that when one shall have arrived at these ulti- 
mate elements, there again will be found the majestic simplicity 
of celestial mechanics. 

Neither has this conception been useless ; it has rendered us an 
inestimable service, since it has contributed to make precise the 
fundamental notion of the physical law. 

I will explain myself; how did the ancients understand lawt 
It was for them an internal harmony, static, so to say, and im- 
mutable ; or else it was like a model that nature tried to imitate. 
For us a law is something quite different; it is a constant rela- 
tion between the phenomenon of to-day and that of to-morrow; 
in a word, it is a differential equation. 

Behold the ideal form of physical law; well, it is Newton's law 
which first clothed it forth. If then one has acclimated this form 
in physics, it is precisely by copying as far as possible this law of 
Newton, that is by imitating celestial mechanics. This is, more- 
over, the idea I have tried to bring out in Chapter VI. 

The Physics of the Principles. — ^Nevertheless, a day arrived 
when the conception of central forces no longer appeared suffi- 
cient, and this is the first of those crises of which I just now 

What was done then? The attempt to penetrate into the 
detail of the structure of the universe, to isolate the pieces of this 
vast mechanism, to analyze one by one the forces which put them 
in motion, was abandoned, and we were content to take as guides 
certain general principles the express object of which is to spare 
us this minute study. How so ? Suppose we have before us any 
machine; the initial wheel work and the final wheel work alone 


are visible, but the transmission, the intermediary machinery 1^ 
which the movement is communicated from one to the other, is 
hidden in the interior and escapes our view; we do not know 
whether the communication is made by gearing or by belts^ by 
connecting-rods or by other contrivances. Do we say that it 
is impossible for us to understand anything about this machine 
so long as we are not permitted to take it to pieces t You know 
well we do not, and that the principle of the conservation of 
energy sufSces to determine for us the most interesting point 
We easily ascertain that the final wheel turns ten times less 
quickly than the initial wheel, since these two wheels are visible; 
we are able thence to conclude that a couple applied to the one 
will be balanced by a couple ten times greater applied to the 
other. For that there is no need to penetrate the mechanism 
of this equilibrium and to know how the forces compensate each 
other in the interior of the machine; it sufSces to be assured 
that this compensation can not fail to occur. 

Well, in regard to the universe, the principle of the conserva- 
tion of energy is able to render us the same service. The uni- 
verse is also a machine, much more complicated than all those of 
industry, of which almost all the parts are profoundly hidden 
from us; but in observing the motion of those that we can see, 
we are able, by the aid of this principle, to draw conclusions 
which remain true whatever may be the details of the invisible 
mechanism which animates them. 

The principle of the conservation of energy, or Mayer's prin- 
ciple, is certainly the most important, but it is not the only one ; 
there are others from which we can derive the same advantage. 
These are: 

Camot's principle, or the principle of the degradation of 

Newton's principle, or the principle of the equality of action 
and reaction. 

The principle of relativity, according to which the laws of 
physical phenomena must be the same for a stationary observer 
as for an observer carried along in a uniform motion of trans- 
lation ; so that we have not and can not have any means of dis- 
cerning whether or not we are carried along in such a motion. 


The principle of the conservation of mass, or Lavoisier's 

I will add the principle of least action. 

The application of these five or six general principles to the 
different physical phenomena is sufficient for our learning of 
them all that we could reasonably hope to know of them. The 
most remarkable example of this new mathematical physics is, 
beyond question, Maxwell's electromagnetic theory of light. 

"SVe know nothing as to what the ether is, how its molecules are 
disposed, whether they attract or repel each other ; but we know 
that this medium transmits at the same time the optical pertur- 
bations and the electrical perturbations ; we know that this trans- 
mission must take place in conformity with the general princi- 
ples of mechanics, and that suffices us for the establishment of 
the equations of the electromagnetic field. 

These principles are results of experiments boldly generalized ; 
but they seem to derive from their very generality a high degree 
of certainty. la fact, the more general they are, the more fre- 
quent are the opportunities to check them, and the verifications 
multiplying, taking the most varied, the most unexpected forms, 
end by no longer leaving place for doubt. 

Uiility of the Old Physics. — Such is the second phase of the 
history of mathematical physics and we have not yet emerged 
from it. Shall we say that the first has been uselessT that dur- 
ing fifty years science went the wrong way, and that there is 
nothing left but to forget so many accumulated efforts that a 
vicious conception condemned in advance to failure! Not the 
least in the world. Do you think the second phase could have 
come into existence without the first T The hypothesis of central 
forces contained all the principles ; it involved them as necessary 
consequences; it involved both the conservation of energy and 
that of masses, and the equality of action and reaction, and the 
law of least action, which appeared, it is true, not as experimental 
truths, but as theorems; the enunciation of which had at the 
same time something more precise and less general than under 
their present form. 

It is the mathematical physics of our fathers which has famil- 
iarized us little by little with these various principles; which has 


habituated us to recognize them under the different vestments in 
which they disguise themselves. They have been compared with 
the data of experience, it has been seen how it was necessary to 
modify their enunciation to adapt them to these data; thereby 
they have been extended and consolidated. Thus th^ came 
to be regarded as experimental truths ; the conception of central 
forces became then a useless support, or rather an embarrasi- 
ment, since it made the principles partake of its hypothetical 

The frames then have not broken, because they are elastic ; but 
they have enlarged; our fathers, who established them, did not 
labor in vain, and we recognize in the science of to-day the gen- 
eral traits of the sketch which they traced. 

The Present Crisis of Mathematical Physics 

The New Crisis. — ^Are we now about to enter upon a third 
period t Are we on the eve of a second crisis t These principles 
on which we have built all, are they about to crumble away in 
their tumt This has been for some time a pertinent question. 

When I speak thus, you no doubt think of radium, that grand 
revolutionist of the present time, and in fact I shall come back 
to it presently; but there is something else. It is not alone the 
conservation of energy which is in question ; all the other princi- 
ples are equally in danger, as we shall see in passing them succes- 
sively in review. 

Camoi's Principle. — ^Let us commence with the principle of 
Carnot. This is the only one which does not present itself as an 
immediate consequence of the hypothesis of central forces ; more 
than that, it seems, if not to directly contradict that hypothesis, 
at least not to be reconciled with it without a certain effort. If 
physical phenomena were due exclusively to the movements of 
atoms whose mutual attraction depended only on the distance, it 
seems that all these phenomena should be reversible ; if all the in- 
itial velocities were reversed, these atoms, always subjected to 
the same forces, ought to go over their trajectories in the contrary 
sense, just as the earth would describe in the retrograde sense 
this same elliptic orbit which it describes in the direct sense, if 
the initial conditions of its motion had been reversed. On this 
account, if a physical phenomenon is possible, the inverse phe- 
nomenon should be equally so, and one should be able to reascend 
the course of time. Now, it is not so in nature, and this is pre- 
cisely what the principle of Carnot teaches us; heat can pass 
from the warm body to the cold body ; it is impossible afterward 
to make it take the inverse route and to reestablish differences 
of temperature which have been effaced. Motion can be wholly 
dissipated and transformed into heat by friction; the contrary 
transformation can never be made except partially. 



We have striven to reconcile this apparent contradiction. If 
the world tends toward uniformity, this is not because its ulti- 
mate parts, at first unlike, tend to become less and less different; 
it is because, shifting at random, they end by blending. For an 
eye which should distinguish all the elements, the variety would 
remain always as great; each grain of this dust preserves its 
originality and does not model itself on its neighbors; but as the 
blend becomes more and more intimate, our gross senses perceive 
only the uniformity. This is why for example, temperatures 
tend to a level, without the possibility of going backwards. 

A drop of wine falls into a glass of water; whatever may be 
the law of the internal motion of the liquid, we shall soon see it 
colored of a uniform rosy tint, and however much from this 
moment one may shake it afterwards, the wine and the water 
do not seem capable of again separating. Here we have the 
type of the irreversible physical phenomenon : to hide a grain of 
barley in a heap of wheat, this is easy; afterwards to find it 
again and get it out, this is practically impossible. All this 
Maxwell and Boltzmann have explained; but the one who has 
seen it most clearly, in a book too little read because it is a little 
diflBeult to read, is Gibbs, in his ' Elementary Principles of Statis- 
tical Mechanics.' 

For those who take this point of view, Camot's principle is 
only an imperfect principle, a sort of concession to the infirmity 
of our senses ; it is because our eyes are too gross that we do not 
distinguish the elements of the blend ; it is because our hands are 
too gross that we can not force them to separate ; the imaginary 
demon of Maxwell, who is able to sort the molecules one by one, 
could well constrain the world to return backward. Can it re- 
turn of itself? That is not impossible; that is only infinitely 
improbable. The chances are that we should wait a long time 
for the concourse of circumstances which would permit a retro- 
gradation ; but sooner or later they will occur, after years whose 
number it would take millions of figures to write. These reser- 
vations, however, all remained theoretic ; they were not very dis- 
quieting, and Carnot's principle retained all its practical value. 
But here the scene changes. The biologist, armed with his micro- 
scope, long ago noticed in his preparations irregular movements 


of little particles la suspension ; this is the Brownian movement. 
He first thought this was a vital phenomenon, but soon he saw 
that the inanimate bodies danced with no less ardor than the 
others; then he turned the matter over to the physicists. Un- 
happily, the physicists remained long uninterested in this ques- 
tion; one concentrates the tight to illuminate the microscopic 
preparation, thought they; with light goes heat; thence inequal- 
ities of temperature and in the liquid interior currents which 
produce the movements referred to. It occurred to M. Oouy to 
look more closely, and he saw, or thought he saw, that this ex- 
planation is untenable, that the movements become brisker as the 
particles are smaller, but that they are not influenced by the 
mode of illumination. If then these movements never cease, or 
rather are reborn without cease, without borrowii^ anything 
from an external source of energy, what ought we to believe t 
To be sure, we should not on this account renounce our belief 
in the conservation of energy, but we see under our eyes now 
motion transformed into heat by friction, now inversely heat 
changed into motion, and that without loss since the movement 
lasts forever. This is the contrary of Carnot's principle. If 
this be so, to see the world return backward, we no longer have 
need of the infinitely keen eye of Maxwell's demon; our micro- 
scope sufBees, Bodies too large, those, for example, which are 
a tenth of a millimeter, are hit from all sides by moving atoma, 
but they do not budge, because these shocks are very numerous 
and the law of chance makes them compensate each other; but 
the smaller particles receive too few shocks for this compensation 
to take place with certainty and are incessantly knocked about. 
And behold already one of our principles in peril. 

The Principle of Relafivity. — Let us pass to the principle of 
relativity: this not only is confirmed by daily experience, not 
only is it a necessary consequence of the hypothesis of central 
forces, hut it is irresistibly imposed upon our good sense, and 
yet it also is assailed. Consider two electrified bodies; though 
they seem to us at rest, they are both carried along by the mo- 
tion of the earth; an electric eharge in motion, Rowland has 
taught us, ia equivalent to a current; these two charged bodies 
I ate, therefore, equivalent to two parallel currents of the same 


sense and these two currents should attract each other. In meas- 
uring this attraction, we shall measure the velocity of the earth; 
not its velocity in relation to the sun or the fixed stars, but its 
absolute velocity. 

I well know what will be said: It is not its absolute velocity 
that is measured, it is its velocity in relation to the ether. How 
unsatisfactory that is I Is it not evident that from the principle 
so understood we could no longer infer anything f It could no 
longer tell us anything just because it would no longer fear any 
contradiction. If we succeed in measuring anything, we shall 
always be free to say that this is not the absolute velocity, and if 
it is not the velocity in relation to the ether, it might always be 
the velocity in relation to some new unknown fluid with which 
we might fill space. 

Indeed, experiment has taken upon itself to ruin this interpre- 
tation of the principle of relativity; all attempts to measure the 
velocity of the earth in relation to the ether have led to nega- 
tive results. This time experimental physics has been more 
faithful to the principle than mathematical ph3rsics; the theorists, 
to put in accord their other general views, would not have spared 
it; but experiment has been stubborn in confirming it. The 
means have been varied; finally Michelson pushed precision to 
its last limits; nothing came of it. It is precisely to explain 
this obstinacy that the mathematicians are forced to-day to em- 
ploy all their ingenuity. 

Their task was not easy, and if Lorentz has got through it, it is 
only by accumulating hypotheses. 

The most ingenious idea was that of local time. Imagine twp 
observers who wish to adjust their timepieces by optical signals; 
they exchange signals, but as they know that the transmission 
of light is not instantaneous, they are careful to cross them. 
When station B perceives the signal from station A, its clock 
should not mark the same hour as that of station A at the 
moment of sending the signal, but this hour augmented by a 
constant representing the duration of the transmission. Sup- 
pose, for example, that station A sends its signal when its clock 
marks the hour 0, and that station B perceives it when its clock 
marks the hour t. The clocks are adjusted if the slowness equal 



to 1 represeDts the duration of the transmission, and to verify 
it, station B sends in its turn a signal when its clock marks O; 
then station A should perceive it when its clock marks t. The 
timepieces are then adjusted. 

And in fact they mark the same hour at the same pbTBical 
instant, but on the one condition, that the two stations are fixed. 
Otherwise the doration of the transmission will not be the same 
in the two senses, since the station A, for example, moves for^ 
ward to meet the optical perturbation emanating from B, whereas 
the station B Sees before the perturbation emanating from A. 
The watches adjusted in that way will not mark, therefore, the 
true time ; they will mark what may be called the local lime, so 
that one of them will be alow of the other. It matters little, since 
we have do means of perceiving it. All the phenomena which 
happen at A, for example, will be late, but all will be equally 
80, and the observer will not perceive it, since his watch is slow; 
so, as the principle of relativity requires, he will have no means 
of knowing whether he is at rest or in absolute motion. 

Unhappily, that does not sufSce, and complementary hypoth- 
eses are necessary; it is necessary to admit that bodies in mo- 
tion undergo a uniform contraction in the sense of the motion. 
One of the diameters of the earth, for example, is shrunk by 
one two-hundred-millionth in consequence of our planet's motion, 
while the other diameter retains its normal length. Thus the last 
little differences are compensated. And then, there is still the 
hypothesis about forces. Forces, whatever be their origin, grav- 
ity as well as elasticity, would be reduced in a certain propor- 
tion in a world animated by a uniform translation; or, rather, 
this would happen for the components perpendicular to the 
translation ; the components parallel would not change. Re- 
sume, then, our example of two eleetri6ed bodies; these bodies 
repel each other, but at the same time if all is carried along in a 
uniform translation, they are equivalent to two parallel currents 
of the same sense which attract each other. This eleetrodynamio 
attraction diminishes, therefore, the electrostatic repulsion, and 
the total repulsion is feebler than if the two bodies were at rest 
Bat since to measure this repulsion we must balance it by another 

ree, and all these other forces are reduced in the same pro- 


portion, we perceive nothing. Thus all seems arranged, but are 
all the doubts dissipated? What would happen if one could 
communicate by non-luminous signals whose velocity of propa- 
gation differed from that of light? If, after having adjusted 
the watches by the optical procedure, we wished to verify the 
adjustment by the aid of these new signals, we should observe 
discrepancies which would render evident the common transla- 
tion of the two stations. And are such signals inconceivable, if 
we admit with Laplace that universal gravitation is transmitted 
a million times more rapidly than light? 

Thus, the principle of relativity has been valiantly defended 
in these latter times, but the very energy of the defense proves 
how serious was the attack. 

Newton* 8 Principle. — ^Let us speak now of the principle of 
Newton, on the equality of action and reaction. This is inti- 
mately bound up with the preceding, and it seems indeed that the 
fall of the one would involve that of the other. Thus we must 
not be astonished to find here the same difficulties. 

Electrical phenomena, according to the theory of Lorentz, are 
due to the displacements of little charged particles, called elec- 
trons, immersed in the medium we call ether. The movements 
of these electrons produce perturbations in the neighboring ether; 
these perturbations propagate themselves in every direction with 
the velocity of light, and in turn other electrons, originally at 
rest, are made to vibrate when the perturbation reaches the parts 
of the ether which touch them. The electrons, therefore, act on 
one another, but this action is not direct, it is accomplished 
through the ether as intermediary. Under these conditions can 
there be compensation between action and reaction, at least for 
an observer who should take account only of the movements 
of matter, that is, of the electrons, and who should be ignorant 
of those of the ether that he could not see? Evidently not. 
Even if the compensation should be exact, it could not be simul- 
taneous. The perturbation is propagated with a finite velocity; 
it, therefore, reaches the second electron only when the first has 
long ago entered upon its rest. This second electron, therefore, 
will undergo, after a delay, the action of the first, but will cer- 
tainly not at that moment react upon it, since around this first 
electron nothing any longer budges. 

The analysis of the facta penoits us to be still more precise. 
Imagine, for example, a Hertzian oscillator, like those used in 
wireless telegraphy ; it sends out energy in every direction ; but 
we can provide it with a parabolic mirror, as Hertz did with bis 
smallest oscillators, so as to send all the energy produced in a 
single direction. "What happens then according to the theoryt 
The apparatus recoils, as if it were a cannon and the projected 
energy a ball; and that is contrary to the principle of Newton, 
since our projectile here has no mass, it is not matter, it is energy. 
The ease is still the same, moreover, with a beacon light provided 
with a reflector, since light is nothing but a perturbation of the 
eleotromagnetic field. This beacon light should recoil as if the 
light it sends out were a projectile. What is the force that 
should produce this recoil T It is what is called the Alaxwell- 
Bartholi pressure. It is very minute, and it has been difficult 
to put it in evidence even with the most sensitive radiometers; 
but it suffices that it exists. 

If all the energy issuing from our oscillator falls on a receiver, 
this will act as if it had received a mechanical shock, which will 
represent in a sense the compensation of the oscillator's recoil; 
the reaction will be equal to the action, but it will not be simul- 
taneous; the receiver will move on, but not at the moment when 
the oscillator recoils. If tlie energy propagates itself indefihitely 
without encountering a receiver, the compensation will never 

SbfiU we say that the space which separates the oscillator from 
the receiver and which the perturbation must pass over in going 
from the one to the other is not void, that it is full not only of 
ether, but of air, or even in the interplanetary spaces of some 
fluid subtile but still ponderable ; that this matter undergoes the 
shock like the receiver at the moment when the energy reaches 
it, and recoils in its turn when the perturbation quits itt That 
would save NchIou's principle, but that is not true. If energy 
in its diffusion remained always attached to some material sub- 
slratam, then matter in motion would carry along light with it, 
and Fizeau has demonstrated that it does nothing of the sort, 
at least for air. Michelson and Moriey have since confirmed 
this. It might be supposed also that the movements of matter 


proper are exactly compensated by those of the ether; but that 
would lead us to the same reflections as before now. The prin- 
ciple so understood will explain everything, since, whatever 
might be the visible movements, we always could imagine hypo- 
thetical movements which compensate them. But if it is able 
to explain everything, this is because it does not enable us to 
foresee anything; it does not enable us to decide between the 
different possible hypotheses, since it explains everything be- 
forehand. It therefore becomes useless. 

And then the suppositions that it would be necessary to make 
on the movements of the ether are not very satisfactory. If the 
electric charges double, it would be natural to imagine that the 
velocities of the diverse atoms of ether double also ; but, for the 
compensation, it would be necessary that the mean velocity of 
the ether quadruple. 

This is why I have long thought that these consequences of 
theory, contrary to Newton's principle, would end some day by 
being abandoned, and yet the recent experiments on the move- 
ments of the electrons issuing from radium seem rather to con- 
firm them. 

Lavoisier's Principle, — I arrive at the principle of Lavoisier on 
the conservation of mass. Certainly, this is one not to be 
touched without unsettling all mechanics. And now certain per- 
sons think that it seems true to us only because in mechanics 
merely moderate velocities are considered, but that it would cease 
to be true for bodies animated by velocities comparable to that 
of light. Now these velocities are believed at present to have 
been realized; the cathode rays and those of radium may be 
formed of very minute particles or of electrons which are dis- 
placed with velocities smaller no doubt than that of light, but 
which might be its one tenth or one third. 

These rays can be deflected, whether by an electric fleld, or 
by a magnetic fleld, and we are able, by comparing these deflec- 
tions, to measure at the same time the velocity of the electrons 
and their mass (or rather the relation of their mass to their 
charge). But when it was seen that these velocities approached 
that of light, it was decided that a correction was necessary. 
These molecules, being electrified, can not be displaced without 



agitating the ether ; to put them in motion it is necessary to over- 
come a double inertia, that of the molecule itself and that of the 
ether. The total or apparent mass that one measures is com- 
posed, therefore, of two parts: the real or mechanical mass of 
the molecule and the electrodynamie mass representing the 
inertia of the ether. 

The calculations of Abraham and the experiments of Kanf- 
mann have then shown that the mechanical mass, properly so 
called, is null, and that the mass of the electrons, or, at least, of 
the negative electrons, is of exclusively electrodynamic origin. 
This is what forces us to change the definition of mass; we can 
not any longer distinguish mechanical mass and electrodynamic 
mass, since then the first would vanish; there is no mass other 
than electrodynamic inertia. But in this case the mass can no 
longer be constant; it augments with the velocity, and it even 
depends on the direction, and a body animated by a notable 
velocity will not oppose the same inertia to the forces which tend 
to deflect it from its route, as to those which tend to accelerate 
or to retard its progress. 

There is still a resource; the ultimate elements of bodies are 
electrons, some charged negatively, the others charged positively. 
The negative electrons have no mass, this is understood; but the 
positive electrons, from the little we know of them, seem much 
greater. Perhaps they have, besides their electrodynamic mass, 
a true mechanical mass. The real mass of a body would, then, 
be the sum of the mechanical masses of its positive electrons, the 
negative electrons not counting; mass so defined might still be 

Alas! this resource also evades us. Recall what we have said 
of the principle of relativity and of the efforts made to save it. 
And it is not merely a principle which it is a question of saving, 
it is the indubitable results of the experiments of Michelson. 

Well, as was above seen, Lorentz, to account for these results, 
was obliged to suppose that all forces, whatever their origin, 
were reduced in the same proportion in a medium animated by a 
uniform translation ; this is not suflScient ; it is not enough that 
this take place for the real forces, it must also be the same for 
the forces of inertia ; it is therefore necessary, he says, that the 


masses of all the particles he influenced by a translation to the 
same degree as the electromagnetic masses of the electrons. 

So the mechanical masses must vary in accordance with the 
same laws as the electrodynamic masses ; th^ can not, therefore, 
be constant. 

Need I point out that the fall of Lavoisier's principle involves 
that of Newton's ? This latter signifies that the center of gravity 
of an isolated system moves in a straight line ; but if there is no 
longer a constant mass, there is no longer a center of gravity, 
we no longer know even what this is. This is why I said above 
that the experiments on the cathode rays appeared to justify 
the doubts of Lorentz concerning Newton's principle. 

From all these results, if they were confirmed, would arise an 
entirely new mechanics, which would be, above all, characterized 
by this fact, that no velocity could surpass that of light,^ any 
more than any temperature can fall below absolute zero. 

No more for an observer, carried along himself in a transla- 
tion he does not suspect, could any apparent velocity surpass 
that of light ; and this would be then a contradiction, if we did 
not recall that this observer would not use the same clocks as a 
fixed observer, but, indeed, clocks marking * local time.' 

Here we are then facing a question I content myself with stat- 
ing. If there is no longer any mass, what becomes of Newton's 
law? Mass has two aspects : it is at the same time a coefiScient of 
inertia and an attracting mass entering as factor into Newtonian 
attraction. If the coefficient of inertia is not constant, can the 
attracting mass be ? That is the question. 

Mayer's Principle, — At least, the principle of the conservation 
of energy yet remained to us, and this seemed more solid. Shall 
I recall to you how it was in its turn thrown into discredit? 
This event has made more noise than the preceding, and it is in 
all the memoirs. From the first works of Becquerel, and, above 
all, when the Curies had discovered radium, it was seen that 
every radioactive body was an inexhaustible source of radiation. 
Its activity seemed to subsist without alteration throughout the 
months and the years. This was in itself a strain on the prin- 

1 Because bodies would oppose an increasing inertia to the causes which 
would tend to accelerate their motion ; and this inertia would become infinite 
when one approached the velocity of light. 


ciples; these radiations were in fact energy, and from the same 
morsel of radium this issued and forever issued. But these 
quantities of energy were too slight to be measured ; at least that 
was the belief and we were not much disquieted. 

The scene changed when Curie bethought himself to put ra- 
dium in a calorimeter ; it was then seen that the quantity of heat 
incessantly created was very notable. 

The explanations proposed were numerous; but in such case 
we can not say, the more the better. In so far as no one of them 
has prevailed over the others, we can not be sure there is a good 
one among them. Since some time, however, one of these ex- 
planations seems to be getting the upper hand and we may rea- 
sonably hope that we hold the key to the mystery. 

Sir W. Bamsay has striven to show that radium is in process 
of transformation, that it contains a store of energy enormous 
but not inexhaustible. The transformation of radium then 
would produce a million times more heat than all known trans- 
formations ; radium would wear itself out in 1,250 years ; this is 
quite short, and you see that we are at least certain to have this 
point settled some hundreds of years from now. While wait- 
ing, our doubts remain. 

The Future of Mathematical Physics 

The Principles and Experiment. — ^In the midst of so much 
ruin, what remains standing? The principle of least action is 
hitherto intact, and Larmor appears to believe that it will long 
survive the others; in reality, it is still more vague and more 

In presence of this general collapse of the principles, what at- 
titude will mathematical physics take? And first, before too 
much excitement, it is proper to ask if all that is really true. 
All these derogations to the principles are encountered only 
among infinitesimals; the microscope is necessary to see the 
Brownian movement; electrons are very light; radium is very 
rare, and one never has more than some milligrams of it at a 
time. And, then, it may be asked whether, besides the infinites- 
imal seen, there was not another infinitesimal unseen counterpoise 
to the first. 

So there is an interlocutory question, and, as it seems, only 
experiment can solve it. We shall, therefore, only have to hand 
over the matter to the experimenters, and, while waiting for them 
to finally decide the debate, not to preoccupy ourselves with these 
disquieting problems, and to tranquilly continue our work as if 
the principles were still uncontested. Certes, we have much to 
do without leaving the domain where they may be applied in all 
security; we have enough to employ our activity during this 
period of doubts. 

The Bole of the Analy[st. — ^And as to these doubts, is it indeed 
true that we can do nothing to disembarrass science of themf 
It must indeed be said, it is not alone experimental physics that 
has given birth to them ; mathematical physics has well contrib- 
uted. It is the experimenters who have seen radium throw out 
energy, but it is the theorists who have put in evidence all the 
diflSculties raised by the propagation of light across a medium in 
motion; but for these it is probable we should not have become 




conscious of tliem, "Well, then, if they have done their best to 
put us into this embarrassment, it is proper also that they help us 
to get out of it. 

They must subject to critical examination all these new viewa 
I have just outlined before you, and abandon the principles only 
after Laving made a loyal effort to save them. What can they 
do in this sense I That is what I will try to explain. 

It is a question before all of endeavoring to obtain a more 
satisfactory- theory of the electrodynamics of bodies in motion. 
It is there especially, as I have sufficiently shown above, that 
difficulties accumulate. It is useless to heap up hypotheses, 
we can not satisfy all the principles at once; so far, one has 
succeeded in safeguarding some only on condition of sacrificing 
the others; hut all hope of obtaining better results ia not yet 
lost. Let us take, then, the theory of Lorentz, turn it in all 
senses, modify it little by little, and perhaps everything will 
arrange itself. 

Thus in place of supposing that bodies in motion undergo a 
contraction in the sense of the motion, and that this contraction 
is the same whatever be the nature of these bodies and the forces 
to which they are otherwise subjected, could we not make a more 
simple and natural hypothesis f We might imagine, for example, 
that it ia the ether which is modified when it is in relative motion 
in reference to the material medium which penetrates it, that, 
when it is thus modified, it no longer transmits perturbations 
with the same velocity in every direction. It might tranamit 
more rapidly those which are propagated parallel to the motion 
of the medium, whether in the same sense or in the opposite sense, 
and less rapidly those which are propagated perpendicularly. 
The wave surfaces would no longer be spheres, but ellipsoids, 
and we could dispense with that extraordinary contraction of all 

I cite this only as an example, since the modifications that 
might be essayed would be evidently sxisceptible of infinite varia- 

Aberralion and Astronomy. — It ia possible also that astronomy 
may some day furnish us data on this point; she it was in the 
main who raised the question in making us acquainted with the 


phenomenon of the aberration of light. If we make craddy the 
theory of aberration, we reach a very curious result. The ap- 
parent positions of the stars differ from their real positions be- 
cause of the earth's motion, and as this motion is variable, these 
apparent positions vary. The real position we can not ascertain, 
but we can observe the variations of the apparent position. The 
observations of the aberration show us, therefore, not the earth's 
motion, but the variations of this motion; they can not, there- 
fore, give us information about the absolute motion of the earth. 

At least this is true in first approximation, but the case would 
be no longer the same if we could appreciate the thousandths of 
a second. Then it would be seen that the amplitude of the oscil- 
lation depends not alone on the variation of the motion, a varia- 
tion which is well known, since it is the motion of our globe on 
its elliptic orbit, but on the mean value of this motion, so that 
the constant of aberration would not be quite the same for all the 
stars, and the differences would tell us the absolute motion of the 
earth in space. 

This, then, would be, under another form, the ruin of the prin- 
ciple of relativity. We are far, it is true, from appreciating the 
thousandth of a second, but, after all, say some, the earth's total 
absolute velocity is perhaps much greater than its relative veloc- 
ity with respect to the sun. If, for example, it were 300 kilo- 
meters per second in place of 30, this would suffice to make the 
phenomenon observable. 

I believe that in reasoning thus one admits a too simple theory 
of aberration. Michelson has shown us, I have told you, that the 
physical procedures are powerless to put in evidence absolute 
motion ; I am persuaded that the same will be true of the astro- 
nomic procedures, however far precision be carried. 

However that may be, the data astronomy will furnish us in 
this regard will some day be precious to the physicist. Mean- 
while, I believe that the theorists, recalling the experience of 
Michelson, may anticipate a negative result, and that they would 
accomplish a useful work in constructing a theory of aberration 
which would explain this in advance. 

Electrons and Spectra, — This dynamics of electrons can be ap- 
proached from many sides, but among the ways leading thither is 



one which has been somewhKt neglected, and yet this is one of 
those which promise as the most surprises. It is movements of 
electrons which produce the lines of the emiGsion spectra ; this is 
proved by the Zeeman effect ; in an incandescent body what vi- 
brates is sensitive to the magnet, therefore electrified. This is a 
very important first point, but no one has gone farther. Why 
are the lines of the spectrum distributed in accordance with a 
regular law T These laws have been studied hy the experimenters 
in their least details; they are very precise and comparatively 
simple. A first study of these distributions recalls the harmon- 
ics encountered in acoustics; but the difference is great. Not 
only are the numbers of vibrations not the successive multiples 
of a single number, but we do not even find anything analogous 
to the roots of those transcendental equations to which we are 
led by 80 many problems of matliematical physics: that of the 
vibrations of an elastic body of any form, that of the Hertzian 
oscillations in a generator of any form, the problem of Fourier 
for the cooling of a solid body. 

The laws are simpler, but they are of wholly other nature, and 
to cite only one of these differences, for the harmonics of high 
order, the number of vibrations tends toward a finite limit, 
instead of increasing indefinitely. 

That has not yet been accounted for, and I believe that there 
we have one of the most important secrets of nature. A Japa- 
nese physicist, M. Nagaoka, has recently proposed an explana- 
tion; according to him, atoms are composed of a large podtive 
electron surrounded by a ring formed of a great number of very 
small negative electrons. Such is the planet Saturn with ita 
rings. This is a very interesting attempt, but not yet wholly 
satisfactory; this attempt should be renewed. We will pene- 
trate, 80 to speak, into the inmost recess of matter. And from 
the particular point of view which we to-day occupy, when we 
know why the vibrations of incandescent bodies differ thus from 
ordinary elastic vibrations, why the electrons do not behave like 
the matter which is familiar to us, we shall better comprehend the 
dynamics of electrons nnd it will be perhaps more easy for us 
to reconcile it with the principles. 

Conventions Preceding Experiment. — Suppose, now, that all 


these efforts fail, and, after all, I do not believe they will, what 
most be done? Will it be necessary to seek to mend the broken 
principles by giving what we French call a coup de paucef That 
evidently is always possible, and I retract nothing of what I have 
said above. 

Have yon not written, you might say if you wished to seek a 
quarrel with me — ^have you not written that the principles, 
though of experimental origin, are now unassailable by experi- 
ment because they have become conventions f And now you 
have just told us that the most recent conquests of experiment 
put these principles in danger. 

Well, formerly I was right and to-day I am not wrong. For- 
merly I was right, and what is now happening is a new proof of 
it. Take, for example, the calorimetric experiment of Curie on 
radium. Is it possible to reconcile it with the principle of the 
conservation of energy ? This has been attempted in many ways. 
But there is among them one I should like you to notice; this is 
not the explanation which tends to-day to prevail, but it is one 
of those which have been proposed. It has been conjectured 
that radium was only an intermediary, that it only stored radia- 
tions of unknown nature which flashed through space in every 
direction, traversing all bodies, save radium, without being al- 
tered by this passage and without exercising any action upon 
them. Radium alone took from them a little of their energy and 
afterward gave it out to us in various forms. 

What an advantageous explanation, and how convenient! 
First, it is unverifiable and thus irrefutable. Then again it will 
serve to account for any derogation whatever to Mayer's prin- 
ciple ; it answers in advance not only the objection of Curie, but 
all the objections that future experimenters might accumulate. 
This new and unknown energy would serve for everything. 

This is just what I said, and therewith we are shown that our 
principle is unassailable by experiment. 

But then, what have we gained by this stroke? The principle 
is intact, but thenceforth of what use is it ? It enabled us to fore- 
see that in such or such circumstance we could count on such a 
total quantity of energy ; it limited us ; but now that this indefi- 
nite provision of new energy is placed at our disposal, we are no 


longer limited by anything; and, as I have written in 'Science 
and Hypothesis,' if a principle ceases to be fecund, experiment 
without contradicting it directly will nevertheless have con- 
demned it. 

Future Mathematical Physics. — This, therefore, is not what 
would have to be done; it would be necessary to rebuild anew. 
If we were reduced to this necessity, we could moreover console 
ourselves. It would not be necessary thence to conclude that 
science can weave only a Penelope's web, that it can raise only 
ephemeral structures, which it is soon forced to demolish from 
top to bottom with its own hands. 

As I have said, we have already passed through a like crisis. 
I have shown you that in the second mathematical physics, that 
of the principles, we find traces of the first, that of central 
forces ; it will be just the same if we must know a third. Just so 
with the animal that exuviates, that breaks its too narrow cara- 
pace and makes itself a fresh one; under the new envelope one 
will recognize the essential traits of the organism which have 

We can not foresee in what way we are about to expand ; per- 
haps it is the kinetic theory of gases which is about to undergo 
development and serve as model to the others. Then the facts 
which first appeared to us as simple thereafter would be merely 
resultants of a very great number of elementary facts which only 
the laws of chance would make cooperate for a common end. 
Physical law would then assume an entirely new aspect ; it would 
no longer be solely a differential equation, it would take the char- 
acter of a statistical law. 

Perhaps, too, we shall have to construct an entirely new me- 
chanics that we only succeed in catching a glimpse of, where, 
inertia increasing with the velocity, the velocity of light would 
become an impassable limit. The ordinary mechanics, more 
simple, would remain a first approximation, since it would be 
true for velocities not too great, so that the old dynamics would 
still be found under the new. We should not have to regret hav- 
ing believed in the principles, and even, since velocities too great 
for the old formulas would always be only exceptional, the sur- 
est way in practise would be still to act as if we continued to 


believe in them. They are so useful, it would be neceflsary to 
keep a place for them. To determine to exclude them altogether 
would be to deprive oneself of a precious weapon. I hasten to 
say in conclusion that we are not yet there, and as yet nothing 
proves that the principles will not come forth from out the fray 
victorious and intact.^ 

1 These considerations on mathematical physics are borrowed from wj 
St Louis address. 

PART in 


Is Science AsTiFicuLf 

1. The Philosophy of M. LeBoy 

There are many reasons for being sceptics; should we push 
this scepticism to the very end or stop on the way? To go to the 
end is the most tempting solution, the easiest, and that which 
many have adopted, despairing of saving anything from the 

Among the writings inspired by this tendency it is proper to 
place in the first rank those of M. LeBoy. This thinker is not 
only a philosopher and a writer of the greatest merit, but he has 
acquired a deep knowledge of the exact and physical sciences, 
and even has shown rare powers of mathematical invention. Let 
us recapitulate in a few words his doctrine, which has given rise 
to numerous discussions. 

Science consists only of conventions, and to this circumstance 
solely does it owe its apparent certitude ; the facts of science and, 
a fortiori, its laws are the artificial work of the scientist; science 
therefore can teach us nothing of the truth; it can only serve 
us as rule of action. 

Here we recognize the philosophic theory known under the 
name of nominalism ; all is not false in this theory ; its legitimate 
domain must be left it, but out of this it should not be allowed 
to go. 

This is not all; M. LeRoy's doctrine is not only nominalistic ; 
it has besides another characteristic which it doubtless owes to M. 
Bergson, it is anti-intellectualistic. According to M. LeRoy, the 
22 321 


intellect deforms all its touches, and that is still more true of its 
necessary instrument 'discourse/ There is reality only in our 
fugitive and changing impressions, and even this reality, when 
touched, vanishes. 

And yet M. LeBoy is not a sceptic ; if he regards the intellect 
as incurably powerless, it is only to give more scope to other 
sources of knowledge, to the heart, for instance, to sentiment, to 
instinct or to faith. 

However great my esteem for M. LeBoy 's talent, whatever the 
ingenuity of this thesis, I can not wholly accept it. Certes, I 
am in accord on many points with M. LeBoy, and he has even 
cited, in support of his view, various passages of my writings 
which I am by no means disposed to reject. I think myself only 
the more bound to explain why I can not go with him all the way. 

M. LeBoy often complains of being accused of scepticism. 
He could not help being, though this accusation is probably un- 
just. Are not appearances against himf Nominalist in doc- 
trine, but realist at heart, he seems to escape absolute nominaUsm 
only by a desperate act of faith. 

The fact is that anti-intellectualistic philosophy in rejecting 
analysis and 'discourse,' just by that condemns itself to being 
intransmissible ; it is a philosophy essentially internal, or, at the 
very least, only its negations can be transmitted; what wonder 
then that for an external observer it takes the shape of scepticism t 

Therein lies the weak point of this philosophy ; if it strives to 
remain faithful to itself, its energy is spent in a negation and a 
cry of enthusiasm. Each author may repeat this negation and 
this cry, may vary their form, but without adding anything. 

And yet, would it not be more logical in remaining silent t 
See, you have written long articles; for that, it was necessary 
to use words. And therein have you not been much more 'dis- 
cursive' and consequently much farther from life and truth than 
the animal who simply lives without philosophizing t Would 
not this animal be the true philosopher? 

However, because no painter has made a perfect portrait, 
should we conclude that the best painting is not to paint ? When 
a zoologist dissects an animal, certainly he 'alters it.' Yes, in 
dissecting it, he condemns himself to never know all of it ; but in 


not dissecting it, he would condemn himself to never know any- 
thing of it and consequently to never see anything of it. 

Certes^ in man are other forces besides his intellect; no one 
has ever been mad enough to deny that. The first comer makes 
these blind forces act or lets them act; the philosopher must 
speak of them ; to speak of them, he must know of them the little 
that can be known, he should therefore see them act. Howf 
With what eyes, if not with his intellect? Heart, instinct, may 
guide it, but not render it useless ; they may direct the look, but 
not replace the eye. It may be granted that the heart is the 
workman, and the intellect only the instrument. Yet is it an 
instrument not to be done without, if not for action, at least for 
philosophizing f Therefore a philosopher really anti-intellectual« 
istic is impossible. Perhaps we shall have to declare for the 
supremacy of action; always it is our intellect which will thus 
conclude ; in allowing precedence to action it will thus retain the 
superiority of the thinking reed. This also is a supremacy not 
to be disdained. 

Pardon these brief reflections and pardon also their brevity, 
scarcely skimming the question. The process of intellectualism 
is not the subject I wish to treat : I wish to speak of science, and 
about it there is no doubt ; by definition, so to speak, it will be 
intellectualistic or it will not be at all. Precisely the question is, 
whether it will be. 

2. Science, Rule of Action 

For M. LeRoy, science is only a rule of action. We are pow- 
erless to know anything and yet we are launched, we must act, 
and at all hazards we have established rules. It is the aggregate 
of these rules that is called science. 

It is thus that men, desirous of diversion, have instituted rules 
of play, like those of tric-trac for instance, which, better than 
science itself, could rely upon the proof by universal consent. 
It is thus likewise that, unable to choose, but forced to choose, we 
toss up a coin, head or tail to win. 

The rule of tric-trac is indeed a rule of action like science, 
but does any one think the comparison just and not see 
the difference? The rules of the game are arbitrary conven- 


tions and the contrary convention might have been adopted, 
wJUch would have been none the less good. On the contrary, 
science is a rule of action which is successful, generally at least, 
and I add, while the contrary rule would not have succeeded. 

If I say, to make hydrogen cause an acid to act on zinc, I for- 
mulate a rule which succeeds; I could have said, make distilled 
water act on gold ; that also would have been a rule, only it would 
not have succeeded. If, therefore, scientific ^recipes' have a 
value, as rule of action, it is because we know they succeed, gener- 
ally at least. But to know this is to know something and th^ 
why tell us we can know nothing f 

Science foresees, and it is because it foresees that it can be 
useful and serve as rule of action. I well know that its pre- 
visions are often contradicted by the event; that shows that 
science is imperfect, and if I add that it will always remain so, 
I am certain that this is a prevision which, at least, will nev^ 
be contradicted. Always the scientist is less often mistaken 
than a prophet who should predict at random. Besides the 
progress though slow is continuous, so that scientists, though 
more and more bold, are less and less misled. This is little, but 
it is enough. 

I well know that M. LeRoy has somewhere said that science 
was mistaken of tener than one thought, that comets sometimes 
played tricks on astronomers, that scientists, who apparently are 
men, did not willingly speak of their failures, and that, if they 
should speak of them, they would have to count more defeats 
than victories. 

That day, M. LeRoy evidently overreached himself. If science 
did not succeed, it could not serve as rule of action; whence 
would it get its value? Because it is * lived,' that is, because we 
love it and believe in it? The alchemists had recipes for making 
gold, they loved them and had faith in them, and yet our recipes 
are the good ones, although our faith be less lively, because they 

There is no escape from this dilemma ; either science does not 
enable us to foresee, and then it is valueless as rule of action ; or 
else it enables us to foresee, in a fashion more or less imperfect, 
and then it is not without value as means of knowledge. 

It should not even be said that action is the goal of science ; 
shoald we condemn studies of the star Sinus, under pretext that 
we shall probably never exercise any influence on tliat starl To 
my eyes, on the contrary, it is the knowledge which is the end, 
and the action which is the means. If I felicitate myself on the 
industrial development, it is not alone because it furnishes a 
facile argument to the advocates of science ; it is above all because 
it gives to the scientist faith in himself and also because it offers 
him an immense field of experience where he clashes againat 
forces too colossal to be tampered with. Without this ballast, 
who knows whether he would not quit solid ground, seduced by 
the mirage of some scholastic novelty, or whether he would not 
despair, believing he had fashioned only a dream! 

3. The Crude Fact arid the Scientific Fact 

What was most paradoxical in M. LeRoy's thesis was that 
affirmation that the scientist creates th-e fact; this was at the 
same time its essential point and it is one of those which have 
been most discussed. 

Perhaps, says he (I well believe that this was a concession), 
it is not the scientist that creates the fact in the rough ; it is at 
least he who creates the scientific fact. 

This distinction between the fact in the rough and the scien- 
tific fact does not by itself appear to me illegitimate. But I 
complain first that the boundary has not been traced either 
exactly or precisely ; and then that the author has seemed to sup- 
pose that the crude fact, not being scientific, is outside of science. 

Pinally, I can not admit that the scientist creates without re- 
straint the scientific fact, since it is the crude fact which impoaea 
it upon him. 

The examples given by M. LeRoy have greatly astonished me. 
The first is taken from the notion of atom. The atom chosen as 
example of fact! I avow that this choice has so disconcerted 
me that I prefer to say nothing about it. I have evidently mia- 
underatood the author's thought and I could not fruitfully dis- 
cuss it. 

The second case taken as example is that of an eclipse where 
tlie erode phenomenon is a play of light and shadow, but where 


the astronomer can not intervene without introducing two foreign 
elements, to wit, a clock and Newton's law. 

Finally, M. heRoy cites the rotation of the earth; it has been 
answered : but this is not a fact, and he has replied : it was one 
for Galileo, who afSrmed it, as for the inquisitor, who denied it 
It always remains that this is not a fact in the same sense as 
those just spoken of and that to give them the same name is to 
expose one's self to many confusions. 

Here then are four degrees : 

1°. It grows dark, says the clown. 

2°. The eclipse happened at nine o'clock, says the astronomer. 

3°. The eclipse happened at the time deducible from the tables 
constructed according to Newton's law, says he again. 

4^. That results from the earth's turning around the sun, says 
Qalileo finally. 

Where then is the boundary between the fact in the rough 
and the scientific factf To read M. LeBoy one would believe 
that it is between the first and the second stage, but who does not 
see that there is a greater distance from the second to the third, 
and still more from the third to the fourth. 

Allow me to cite two examples which perhaps will enlighten us 
a little. 

I observe the deviation of a galvanometer by the aid of a mov- 
able mirror which projects a luminous image or spot on a divided 
scale. The crude fact is this : I see the spot displace itself on the 
scale, and the scientific fact is this : a current passes in the circuit. 

Or again: when I make an experiment I should subject the 
result to certain corrections, because I know I must have made 
errors. These errors are of two kinds, some are accidental and 
these I shall correct by taking the mean ; the others are systematic 
and I shall be able to correct those only by a thorough study of 
their causes. The first result obtained is then the fact in the 
rough, while the scientific fact is the final result after the 
finished corrections. 

Reflecting on this latter example, we are led to subdivide our 
second stage, and in place of saying : 

2. The eclipse happened at nine o'clock, we shall say: 

2a. The eclipse happened when my clock pointed to nine, and 


2b. Mjr clock being ten minutes slow, the eclipse happened at 
ten minutes past nine. 

And this is not all : the first stage also should be subdivided, 
and not between these two subdivisions will be the least distance; 
it is necessary to distinguish between the impression of obscur- 
ity felt by one witnessing an eclipse, and the affirmation : It grows 
dark, which this impression extorts from him. In a sense it is 
the first which is the only true fact in the rough, aud Ihe second 
is already a sort of scientific fact. 

Now then our scale has six stages, and even though there is no 
reason for halting at this figure, there we shall stop. 

What strikes me at the start is this. At the first of our six 
stages, the fact, still completely in the rough, is, so to speak, in- 
dividual, it is completely distinct from all other possible facts. 
From the second stage, already it is no longer the same. The 
entmciation of the fact would suit an infinity of other facts. 
So soon as language intervenes, I have at my command only a 
finite number of terms to express the shades, in number infinite, 
that my impressions might cover. When I say: It grows dark, 
that well expresses the impressions I feel in being present at an 
eclipse; but even in obscurity a multitude of shades could be 
imagined, and if, instead of that actually realized, had happened 
a slightly different shade, yet I should still have enunciated this 
other fact by saying: It grows dark. 

Second remark: even at the second stage, the enunciation of 
a fact can only be (rue or false. This is not so of any proposi- 
tion; if this proposition is the enunciation of a convention, it 
can not be said that this enunciation is true, in the proper sense 
of the word, since it could not be true apart from me and is tme 
only because T wish it to be. 

When, for instance, I say the unit for length is the meter, this 
is a decree that I promulgate, it is not something ascertained 
which forces itself upon me. It is the same, as I think I have 
elsewhere shown, when it is a question, for example, of Euclid's 

When I am asked: Is it growing darki I always know whether 
I ought to reply yes or no. Although an infinity of possible 
facts may be susceptible of this same enunciation, it grows dark, 


I shall always know whether the fact realized belongs or does not 
belong among those which answer to this enunciation. Facts are 
classed in categories, and if I am asked whether the fact that I 
ascertain belongs or does not belong in such a category, I shall 
not hesitate. 

Doubtless this classification is su£Sciently arbitrary to leave a 
large part to man's freedom or caprice. In a word, this classifi- 
cation is a convention. This convention being given, if I am 
asked : Is such a fact true ? I shall always know what to answer, 
and my reply will be imposed upon me by the witness of my 

If therefore, during an eclipse, it is asked : Is it growing darkf 
all the world will answer yes. Doubtless those speaking a lan- 
guage where bright was called dark, and dark bright, would 
answer no. But of what importance is thatf 

In the same way, in mathematics, when I have laid doum the 
definitions, and the postulates which are conventions, a theorem 
henceforth can only be true or false. But to answer the ques- 
tion : Is this theorem true ? it is no longer to the witness of my 
senses that I shall have recourse, but to reasoning. 

A statement of fact is always verifiable, and for the verifica- 
tion we have recourse either to the witness of our senses, or to 
the memory of this witness. This is properly what characterizes 
a fact. If you put the question to me : Is such a fact true 1 I 
shall begin by asking you, if there is occasion, to state precisely 
the conventions, by asking you, in other words, what language you 
have spoken; then once settled on this point, I shall interrogate 
my senses and shall answer yes or no. But it will be my senses 
that will have made answer, it will not be you when you say to 
me: I have spoken to you in English or in French. 

Is there something to change in all that when we pass to the 
following stages! When I observe a galvanometer, as I have 
just said, if I ask an ignorant visitor: Is the current passing? 
he looks at the wire to try to see something pass ; but if I put the 
same question to my assistant who understands my language, he 
will know I mean: Does the spot move? and he will look at the 

What difference is there then between the statement of a fact 


in the rough and the statement of a scientific fact ! The same 
difference as between the statement of the same crude fact in 
French and in German, The scientific statement is the transla- 
tion of the crude statement into a language which is distinguished 
above all from the common German or French, because it is 
spoken by a veiy much smaller number of people. 

Yet let us not go too fast. To measure a current I may use 
a very great number of types of galvaBomet^rs or besides an 
eleetrodynamometer. And then when I shall say there is r unni ng 
in tbia circuit a current of so many amperes, that will mean: 
if I adapt to this circuit such a galvanometer I shall see the 
spot come to the division a; but that will mean equally: if I 
adapt to this circuit such an eleetrodynamometer, I shall see the 
spot go to the division b. And that will mean still many other 
things, because the current can manifest itself not only by me- 
chanical effects, but by effects chemical, thermal, luminous, etc. 

Here then is one same statement which siuts a very great num- 
ber of facts absolutely different. WhyT It is because I assuma | 
a law according to which, whenever such a mechanical effect shall 
happen, such a chemical effect will happen also. Previous experi- 
ments, very numerous, have never shown this law to fail, and 
then I have understood that I could express by the same state- 
ment two facta BO invariably bound one to the other. 

When I am asked: Is the current passing 1 I can understand 
that that means ; Will such a mechanical effect happen I But I 
can understand also: Will such a chemical effect happen? I 
shall then verify either the existence of the mechanical effect, or 
that of the chemical effect ; that will be indifferent, since in both 
caaes the answer must be Uie same. 

And if the law should one day be found false T If it was per- 
ceived that the concordance of the two effects, mechanical and 
chemical, is not constant! That day it would be necessary to 
change the scientific language to free it from a grave ambiguity. 

And after that 1 Is it thought that ordinary language by aid 
of which are expressed the facts of daily life is exempt from 
ambiguity 1 

Shalt we thence conclude that the fads of daily life are Ih^ 
work of the grammariansT 


You ask me : Is there a current f I try whether the mechanical 
effect exists, I ascertain it and I answer: Yes, there is a current 
You understand at once that that means that the mechanical 
effect exists, and that the chemical effect, that I have not investi- 
gated, exists likewise. Imagine now, supposing an impossibility, 
the law we believe true, not to be, and the chemical effect not to 
exist. Under this hypothesis there will be two distinct facts, the 
one directly observed and which is true, the other inferred and 
which is false. It may strictly be said that we have created the 
second. So that error is the part of man's personal collabora- 
tion in the creation of the scientific fact. 

But if we can say that the fact in question is false, is this not 
just because it is not a free and arbitrary creation of our mind, a 
disguised convention, in which case it would be neither true nor 
false. And in fact it was verifiable ; I had not made the verifica- 
tion, but I could have made it If I answered amiss, it was be- 
cause I chose to reply too quickly, without having asked nature, 
who alone knew the secret. 

When, after an experiment, I correct the accidental and sys- 
tematic errors to bring out the scientific fact, the case is the same; 
the scientific fact will never be anything but the crude fact trans- 
lated into another language. When I shall say: It is such an 
hour, that will be a short way of saying : There is such a relation 
between the hour indicated by my clock, and the hour it marked 
at the moment of the passing of such a star and such another 
star across the meridian. And this convention of language once 
adopted, when I shall be asked: Is it such an hour? it will not 
depend upon me to answer yes or no. 

Let us pass to the stage before the last : the eclipse happened at 
the hour given by the tables deduced from Newton's laws. This 
is still a convention of language which is perfectly clear for those 
who know celestial mechanics or simply for those who have the 
tables calculated by the astronomers. I am asked: Did the 
eclipse happen at the hour predicted? I look in the nautical 
almanac, I see that the eclipse was announced for nine o'clock 
and I understand that the question means: Did the eclipse 
happen at nine o'clock? There still we have nothing to change 
in our conclusions. The scientific fact is only the crude fact 
translated into a convenient language. 

It is true that at the last stage things change. Does the 
earth rotate t Is this a veriiiaMe fact? Could Galileo and the 
Grand Inquisitor, to settle the matter, appeal to the witness of 
their senses 1 On the contrary, they were in accord about the 
appearances, and whatever had been the accumulated expe- 
riences, they would have remained in accord with regard to the 
appearances without ever agreeing on their interpretation. It 
is just on that account that they were obliged to have recourse 
to procedures of discussion so unscientific. 

This is why I think they did not disagree about a fact: we 
have not the right to give the same name to the rotation of the 
earth, which was the object of their discussion, and to the facta 
crude or scientific we have hitherto passed in review. 

After what precedes, it seems superfluous to investigate 
whether the fact in the rough is outside of science, because there 
can neither be science without scientific fact, nor scientific fact 
without fact in the rough, since the first is only the translation 
of the second. 

And then, has one the right to say that the scientist creates the 
scientific factf First of all, he does not create it from nothing, 
since he makes it with the fact in the rough. Consequently he 
does not make it freely and as he chooses. However able the 
worker may be, his freedom is always limited by the properties of 
the raw material on which he works. 

After all, what do you mean when you speak of this free 
creation of the scientific fact and when you take as example the 
astronomer who intervenes actively in the phenomenon of the 
eclipse by bringing his clock 1 Do you mean: The eclipse hap- 
pened at nine o'clock; but if the astronomer had wished it to 
happen at ten, that depended only on him, he had only to 
advance his clock an hour! 

But the astronomer, in perpetrating that bad joke, would 
evidently have been guilty of an equivocation. When he tells 
me : The eclipse happened at nine, I understand that nine is the 
hour deduced from the crude indication of the pendulum by the 
usual series of corrections. If he has given me solely that crude 
indication, or if he has made corrections contrary to the babitnal 
roles, he has changed the language agreed upon without fore- 


warning me. If, on the contrary, he took care to forewarn me, 
I have nothing to complain of, but then it is always the same 
fact expressed in another language. 

In sum, aU the scientist creates in a fact is tliie language ivC 
which he enunciates it. If he predicts a fact, he will employ this 
language, and for all those who can speak and understand it, his 
prediction is free from ambiguity. Moreover, this prediction 
once made, it evidently does not depend upon him whether it is 
fulfilled or not. 

What then remains of M. LeRoy's thesis f This remains: the 
scientist intervenes actively in choosing the facts worth observ- 
ing. An isolated fact has by itself no interest ; it becomes inter- 
esting if one has reason to think that it may aid in the prediction 
of other facts; or better, if, having been predicted, its verifies^ 
tion is the confirmation of a law. Who shall choose the facts 
which, corresponding to these conditions, are worthy the freedom 
of the city in science ? This is the free activity of the scientist 

And that is not all. I have said that the scientific fact is the 
translation of a crude fact into a certain language ; I should add 
that every scientific fact is formed of many crude facts. This is 
suflBciently shown by the examples cited above. For instance, 
for the hour of the eclipse my clock marked the hour a at the 
instant of the eclipse ; it marked the hour p at the moment of the 
last transit of the meridian of a certain star that we take as 
origin of right ascensions ; it marked the hour y at the moment 
of the preceding transit of this same star. There are three dis- 
tinct facts (still it will be noticed that each of them results itself 
from two simultaneous facts in the rough; but let us pass this 
over). In place of that I say: The eclipse happened at the hour 
24 {oL-p) / ip-^) , and the three facts are combined in a single 
scientific fact. I have concluded that the three readings a, j8, y 
made on my clock at three different moments lacked interest and 
that the only thing interesting was the combination (oL-p)/(p^) 
of the three. In this conclusion is found the free activity of my 

But I have thus used up my power ; I can not make this com- 
bination {cL-p)/{p-^) have such a value and not such another, 
since I can not influence either the value of a, or that of j8, of 
that of y, which are imposed upon me as crude facts. 


In sum, facts are facts, and if it happens that they satisfy a 
prediction, this is not an effect of our free activity. There is no 
precise frontier between the fact in the rough and the scientific 
fact ; it can only be said that such an enunciation of fact is more 
crude or, on the contrary, more scientific than such another. 

4. ^Nominalism* and *the Universal Invariant* 

If from facts we pass to laws, it is clear that the part of the 
free activity of the scientist will become much greater. But 
did not M. LeBoy make it still too great f This is what we are 
about to examine. 

Recall first the examples he has given. When I say: Phos- 
phorus melts at 44^, I think I am enunciating a law; in reality 
it is just the definition of phosphorus; if one should discover a 
body which, possessing otherwise all the properties of phosphorus, 
did not melt at 44^, we should give it another name, that is all, 
and the law would remain true. 

Just so when I say: Heavy bodies falling freely pass over 
spaces proportional to the squares of the times, I only give the 
definition of free fall. Whenever the condition shall not be 
fulfilled, I shall say that the fall is not free, so that the law 
wiU never be wrong. It is clear that if laws were reduced to that, 
they could not serve in prediction ; then they would be good for 
nothing, either as means of knowledge or as principle of action. 

When I say: Phosphorus melts at 44°, I mean by that: All 
bodies possessing such or such a property (to wit, all the prop- 
erties of phosphorus, save fusing-point) fuse at 44**. So under- 
stood, my proposition is indeed a law, and this law may be use- 
ful to me, because if I meet a body possessing these properties 
I shall be able to predict that it will fuse at 44°. 

Doubtless the law may be found to be false. Then we shall 
read in the treatises on chemistry: ** There are two bodies which 
chemists long confounded under the name of phosphorus; these 
two bodies differ only by their points of fusion." That would 
evidently not be the first time for chemists to attain to the separa- 
tion of two bodies they were at first not able to distinguish ; such, 
for example, are neodymium and praseodymium, long confounded 
under the name of didymium. 


I do not think the chemists much fear that a like mischance 
will ever happen to phosphorus. And if, to suppose the impos- 
sible, it should happen, the two bodies would probably not have 
identically the same density, identicaUy the same specific heat» 
etc., so that after having determined with care the density, for 
instance, one could still foresee the fusion point. 

It is, moreover, unimportant; it su£Sces to remark that there 
is a law, and that this law, true or false, does not reduce to a 

Will it be said that if we do not know on the earth a body 
which does not fuse at 44° while having all the other properties 
of phosphorus, we can not know whether it does not exist on other 
planets f Doubtless that may be maintained, and it would then 
be inferred that the law in question, which may serve as a rule 
of action to us who inhabit the earth, has yet no general value 
from the point of view of knowledge, and owes its interest only 
to the chance which has placed us on this globe. This is possible, 
but, if it were so, the law would be valueless, not because it re- 
duced to a convention, but because it would be false. 

The same is true in what concerns the fall of bodies. It would 
do me no good to have given the name of free fall to falls which 
happen in conformity with Galileo 's law, if I did not know that 
elsewhere, in such circumstances, the fall will be probably free or 
approximately free. That then is a law which may be true or 
false, but which does not reduce to a convention. 

Suppose the astronomers discover that the stars do not exactly 
obey Newton's law. They will have the choice between two 
attitudes; they may say that gravitation does not vary exactly 
as the inverse of the square of the distance, or else they may say 
that gravitation is not the only force which acts on the stars and 
that there is in addition a different sort of force. 

In the second case, Newton's law will be considered as the 
definition of gravitation. This will be the nominalist attitude. 
The choice between the two attitudes is free, and is made from 
considerations of convenience, though these considerations are 
most often so strong that there remains practically little of this 

We can break up this proposition : (1) The stars obey Newton's 


law, into two others; (2) gravitation obeys Newton's law; (3) 
gravitation is the only force acting on the stars. In this case 
proposition (2) is no longer anything but a definition and is 
beyond the test of experiment ; but then it will be on proposition 
(3) that this check can be exercised. This is indeed necessary, 
since the resulting proposition (1) predicts verifiable facts in the 

It is thanks to these artifices that by an unconscious nomi- 
nalism the scientists have elevated above the laws what they call 
principles. When a law has received a sufficient confirmation 
from experiment, we may adopt two attitudes: either we may 
leave this law in the fray; it will then remain subjected to an 
incessant revision, which without any doubt will end by demon- 
strating that it is only approximative. Or else we may elevate 
it into a principle by adopting conventions such that the propo- 
sition may be certainly true. For that the procedure is always 
the same. The primitive law enunciated a relation between two 
facts in the rough, A and B; between these two crude facts is 
introduced an abstract intermediary C, more or less fictitious 
(such was in the preceding example the impalpable entity, gravi- 
tation). And then we have a relation between A and C that we 
may suppose rigorous and which is the principle; and another 
between C and B which remains a law subject to revision. 

The principle, henceforth crystallized, so to speak, is no longer 
subject to the test of experiment. It is not true or false, it is 

Great advantages have often been found in proceeding in that 
way, but it is clear that if all the laws had been transformed 
into principles nothing would be left of science. Every law may 
be broken up into a principle and a law, but thereby it is very 
clear that, however far this partition be pushed, there will always 
remain laws. 

Nominalism has therefore limits, and this is what one might 
fail to recognize if one took to the very letter M. LeRoy's 

A rapid review of the sciences will make us comprehend better 
what are these limits. The nominalist attitude is justified only 
when it is convenient ; when is it so ? 


Experiment teaches us relations between bodies ; this is the fact 
in the rough ; these relations are extremely complicated. Instead 
of envisaging directly the relation of the body A and the body B^ 
we introduce between them an intermediary, which is space, and 
we envisage three distinct relations : that of the body A with the 
figure A' of space, that of the body B with the figure B' of space, 
that of the two figures A' and B' to each other. Why is this 
detour advantageous f Because the relation of A and B was com- 
plicated, but differed little from that of A' and B'y which is 
simple ; so that this complicated relation may be replaced by the 
simple relation between A' and £' and by two other relations 
which tell us that the differences between A and A', on the one 
hand, between B and B\ on the other hand, are very small. For 
example, if A and B are two natural solid bodies which are dis- 
placed with slight deformation, we envisage two movable rigid 
figures A' and B\ The laws of the relative displacement of these 
figures A' and B' will be very simple ; they will be those of geom- 
etry. And we shall afterward add that the body A^ which always 
differs very little from A', dilates from the effect of heat and 
bends from the effect of elasticity. These dilatations and flexions, 
just because they are very small, will be for our mind relatively 
easy to study. Just imagine to what complexities of language 
it would have been necessary to be resigned if we had wished to 
comprehend in the same enunciation the displacement of the 
solid, its dilatation and its flexure ? 

The relation between A and B was a rough law, and was broken 
up ; we now have two laws which express the relations of A and A\ 
of B and B\ and a principle which expresses that of A' with B'. 
It is the aggregate of these principles that is called geometry. 

Two other remarks. We have a relation between two bodies A 
and B, which we have replaced by a relation between two figures 
A' and B' \ but this same relation between the same two figures 
A' and B' could just as well have replaced advantageously a 
relation between two other bodies A" and B", entirely different 
from A and B. And that in many ways. If the principles and 
geometry had not been invented, after having studied the rela- 
tion of A and -B, it would be necessary to begin again ab ovo the 
study of the relation of A" and B'\ That is why geometry is so 


preoiouB. A geometrical relation can advantageously replace a 
relation which, considered in the rough state, should be regarded 
as mechanical, it can replace another which should be regarded 
as optical, etc. 

Yet let no one sa; : Bat that proves geometry an experimental 
science ; in separating its principles from laws whence they have 
been drawn, you artificially separate it itself from the sciences 
which have given birth to it. The other sciences have likewise 
principles, but that does not preclude our having to call them 

It must be recognized that it would have been difficult not to 
make this separation that is pretended to be artificial. "We know 
the role that the kinematics of solid bodies has played in the 
genesis of geometry ; should it then he said that geometry is only 
a branch of experimental kinematics? But the laws of the recti- 
linear propagation of light have also contributed to the forma- 
tion of its principles. Must geometry be regarded both as a 
branch of kinematics and as a branch of optics t I recall besides 
that our Euclidean space which is the proper object of geometry 
has been chosen, for reasons of convenience, from among a cer- 
tain number of types which preexist in our mind and which are 
called groups. 

If we pass to mechanics, we still see great principles whose 
origin is analogous, and, as their 'radius of action,' so to speak, 
is smaller, there is no longer reason to separate them from 
mechanics proper and to regard this science as deductive. 

In physics, finally, the role of the principles is still more dimin- 
ished. And in fact they are only introduced when it is of ad- 
vantage. Now they are advantageous precisely because they are 
few, since each of them very nearly replaces a great numher of 
laws. Therefore it is not of interest to multiply them. Besides 
an outcome is necessary, and for that it is needful to end by leav- 
ing abstraction to take hold of reality. 

Such are the limits of nominalism, and they are narrow. 

M. LeRoy has insisted, however, and he has put the question 
under another form. 

Since the enunciation of our laws may vary with the conven- 
tions that we adopt, since these conventions may modify even the 


natural relations of these laws, is there in the manifold of these 
laws something independent of these conventions and which may, 
so to speak, play the role of universal invariant t For instance, 
the fiction has been introduced of beings who, having been edu- 
cated in a world different from ours, would have been led to 
create a non-Euclidean geometry. If these beings were after- 
ward suddenly transported into our world, they would observe 
the same laws as we, but they would enunciate them in an 
entirely different way. In truth there would still be something 
in common between the two enunciations, but this is because these 
beings do not yet differ enough from us. Beings still more strange 
may be imagined, and the part common to the two systems of 
enunciations will shrink more and more. Will it thus shrink 
in convergence toward zero, or will there remain an irreducible 
residue which will then be the universal invariant sought f 

The question calls for precise statement. Is it desired that 
this common part of the enunciations be expressible in words f 
It is clear, then, that there are not words common to all languages, 
and we can not pretend to construct I know not what universal 
invariant which should be understood both by us and by the 
fictitious non-Euclidean geometers of whom I have just spoken; 
no more than we can construct a phrase which can be understood 
both by Germans who do not understand French and by French 
who do not understand German. But we have fixed rules which 
permit us to translate the French enunciations into German, 
and inversely. It is for that that grammars and dictionaries 
have been made. There are also fixed rules for translating the 
Euclidean language into the non-Euclidean language, or, if there 
are not, they could be made. 

And even if there were neither interpreter nor dictionary, if 
the Germans and the French, after having lived centuries in 
separate worlds, found themselves all at once in contact, do you 
think there would be nothing in common between the science 
of the German books and that of the French books 1 The French 
and the Germans would certainly end by understanding each 
other, as the American Indians ended by understanding the 
language of their conquerors after the arrival of the Spanish. 

But, it will be said, doubtless the French would be capable of 


nnderstanding the Qermans even without having learned Qer- 
man, but this is because there remains between the French and 
the Germans something in common, since both are men. We 
should still attain to an understanding with our hypothetical non- 
EuclideanSy though they be not men, because they would still 
retain something human. But in any case a minimum of humanity 
is necessary. 

This is possible, but I shall observe first that this little human- 
ness which would remain in the non-Euclideans would suffice not 
only to make possible the translation of a Utile of their language, 
but to make possible the translation of all their language. 

Now, that there must be a minimum is what I concede ; suppose 
there exists I know not what fiuid which penetrates between the 
molecules of our matter, without having any action on it and 
without being subject to any action coming from it. Suppose 
beings sensible to the influence of this fluid and insensible to 
that of our matter. It is clear that the science of these beings 
would differ absolutely from ours and that it would be idle to 
seek an 'invariant' common to these two sciences. Or again, if 
these beings rejected our logic and did not admit, for instance, 
the principle of contradiction. 

But truly I think it without interest to examine such 

And then, if we do not push whimsicality so far, if we intro- 
duce only fictitious beings having senses analogous to ours and 
sensible to the same impressions, and moreover admitting the 
principles of our logic, we shall then be able to conclude that 
their language, however different from ours it may be, would 
always be capable of translation. Now the possibility of trans- 
lation implies the existence of an invariant. To translate is 
precisely to disengage this invariant. Thus, to decipher a crypto- 
gram is to seek what in this document remains invariant, when 
the letters are permuted. 

What now is the nature of this invariant it is easy to under- 
stand, and a word will suffice us. The invariant laws are the 
relations between the crude facts, while the relations between the 
'scientific facts' remain always dependent on certain conventions. 

Science and Beauty 

5. Contingence and Detefminism 

I DO not intend to treat here the question of the eontingenoe of 
the laws of nature, which is evidently insoluble, and on which so 
much has already been written. I only wish to call attention to 
what different meanings have been given to this word, contm- 
gence, and how advantageous it would be to distinguish them. 

If we look at any particular law, we may be certain in advance 
that it can only be approximate. It is, in fact, deduced from 
experimental verifications, and these verifications were and could 
be only approximate. We should always expect that more precise 
measurements will oblige us to add new terms to our formulas; 
this is what has happened, for instance, in the case of Mariotte's 

Moreover the statement of any law is necessarily incomplete. 
This enunciation should comprise the enumeration of (M the 
antecedents in virtue of which a given consequent can happen. 
I should first describe all the conditions of the experiment to be 
made and the law would then be stated : If all the conditions are 
fulfilled, the phenomenon will happen. 

But we shall be sure of not having forgotten any of these con- 
ditions only when we shall have described the state of the entire 
universe at the instant t ; all the parts of this universe may, in 
fact, exercise an influence more or less great on the phenomenon 
which must happen at the instant t -j- dt. 

Now it is clear that such a description could not be found in 
the enunciation of the law; besides, if it were made, the law 
would become incapable of application ; if one required so many 
conditions, there would be very little chance of their ever being 
all realized at any moment. 

Then as one can never be certain of not having forgotten some 
essential condition, it can not be said: If such and such condi- 


tions are realized, each a pbenomenoD uill occur; it can only be 
said; H such and such conditions are realized, it is probable that 
aueh a phenomenon will occur, very nearly. 

Take the law of gravitation, which ia the least imperfect of all 
known laws. It enables us to foresee the motions of the planets. 
When I use it, for instance, to calculate the orbit of Saturn, I 
neglect the action of the staro, and iu doing so I am certain of 
not deceiving myself, because I know that these stars are too fan 
away for their action to be sensible, 

I announce, then, with a quasi-certitude that the coordinates 
of Saturn at such an hour will be comprised between such and 
such limits. Yet is that certitude absolute t Could there not 
exist in the universe some gigantic mass, much greater than th&t 
of all the known stars and whose action could make itself felt 
at great distances! That mass might be animated by a colossal 
velocity, and after having circulated from all time at such dis- 
tances that its influence had remained hitherto insensible to na, 
it might come all at once to pass near us. Surely it would pro- 
duce in our solar system enormous perturbations that we could 
not have foreseen. All that can be said is that such an event ia 
wholly improbable, and then, instead of saying: Saturn will be 
near sueh a point of the heavens, we must limit ourselves to say- 
ing : Saturn will probably be near such a point of the heavens. 
Although this probability may be practically equivalent to cer- 
tainty, it is only a probability. 

For all these reasons, no particular law will ever he more than 
approximate and probable. Scientists have never failed to recog- 
nize this truth ; only they believe, right or wrong, that every law 
may be replaced by another closer and more probable, that this 
new law wiU itself be only provisional, but that the same move- 
ment can continue indefinitely, so that science in progressing will 
possess laws more and more probable, that the approximation will 
end by differing as little as you choose from exactitude and the 
probability from certitude. 

If the scientists who thinfe thus are right, still could it be said 
that the laws of nature are contingent, even though each law, 
taken in particular, may be quali6ed as contingentt Or must one 1 
Teqnire, before concluding the contingence of the natural lawi^J 


that this process have an end, that the scientist finish some day 
by being arrested in his search for a closer and closer approxi- 
mation, and that, beyond a cettain limit, he thereafter meet in 
nature only caprice f 

In the conception of which I have just spoken (and which I 
shall call the scientific conception), every law is only a statement 
imperfect and provisional, but it must one day be replaced by 
another, a superior law, of which it is only a crude image. No 
place therefore remains for the intervention of a free will. 

It seems to me that the kinetic theory of gases will furnish 
us a striking example. 

You know that in this theory all the properties of gases are 
explained by a simple hypothesis; it is supposed that all the 
gaseous molecules move in every direction with great velocities 
and that they follow rectilineal paths which are disturbed only 
when one molecule passes very near the sides of the vessel or 
another molecule. The effects our crude senses enable us to 
observe are the mean effects, and in these means, the great devia- 
tions compensate, or at least it is very improbable that they do 
not compensate ; so that the observable phenomena follow simple 
laws such as that of Mariotte or of Gay-Lussac. But this com- 
pensation of deviations is only probable. The molecules inces- 
santly change place and in these continual displacements the 
figures they form pass successively through all possible combina- 
tions. Singly these combinations are very numerous ; almost all 
are in conformity with Mariotte 's law, only a few deviate from 
it. These also will happen, only it would be necessary to wait 
a long time for them. If a gas were observed during a sufB- 
ciently long time, it would certainly be finally seen to deviate, 
for a very short time, from Mariotte 's law. How long would it 
be necessary to wait? If it were desired to calculate the prob- 
able number of years, it would be found that this number is so 
great that to write only the number of places of figures employed 
would still require half a score places of figures. No matter; 
enough that it may be done. 

I do not care to discuss here the value of this theory. It is 
evident that if it be adopted, Mariotte 's law will thereafter 
appear only as contingent, since a day will come when it will not 


be true. And yet, think you the partisans of the kinetic theory 
are adversaries of determinism f Far from it; they are the 
most ultra of mechanists. Their molecules follow rigid paths, 
from which they depart only under the influence of forces which 
vary with the distance, following a perfectly determinate law. 
There remains in their system not the smallest place either for 
freedom, or for an evolutionary factor, properly so-called, or for 
anything whatever that could be called contingence. I add, to 
avoid mistake, that neither is there any evolution of Mariotte's 
law itself ; it ceases to be true after I know not how many cen- 
turies ; but at the end of a fraction of a second it again becomes 
true and that for an incalculable number of centuries. 

And since I have pronounced the word evolution, let us clear 
away another mistake. It is often said: Who knows whether 
the laws do not evolve and whether we shall not one day discover 
that they were not at the Carboniferous epoch what they are 
to-day t What are we to understand by thatt What we think 
we know about the past state of our globe, we deduce from its 
present state. And how is this deduction madet It is by means 
of laws supposed known. The law, being a relation between the 
antecedent and the consequent, enables us equally well to deduce 
the consequent from the antecedent, that is, to foresee the future, 
and to deduce the antecedent from the consequent, that is, to 
conclude from the present to the past. The astronomer who 
knows the present situation of the stars can from it deduce their 
future situation by Newton's law, and this is what he does when 
he constructs ephemerides; and he can equally deduce from it 
their past situation. The calculations he thus can make can not 
teach him that Newton's law will cease to be true in the future, 
since this law is precisely his point of departure; not more can 
they tell him it was not true in the past. Still, in what concerns 
the future, his ephemerides can one day be tested and our de- 
scendants will perhaps recognize that they were false. But in 
what concerns the past, the geologic past which had no witnesses, 
the results of his calculation, like those of all speculations where 
we seek to deduce the past from the present, escape by their 
very nature every species of test. So that if the laws of nature 
were not the same in the Carboniferous age as at the present 


epoch, we shall never be able to know it, since we can know 
nothing of this age, only what we deduce from the hyx>otheBi8 of 
the permanence of these laws. 

Perhaps it will be said that this hypothesis might lead to con- 
tradictory results and that we shall be obliged to abandon it 
Thus, in what concerns the origin of life, we may conclude that 
there have always been living beings, since the present world 
shows us always life springing from life ; and we may also con- 
clude that there have not always been, since the application of 
the existent laws of physics to the present state of our globe 
teaches us that there was a time when this globe was so warm that 
life on it was impossible. But contradictions of this sort can 
always be removed in two ways; it may be supposed that the 
actual laws of nature are not exactly what we have assumed; 
or else it may be supposed that the laws of nature actually are 
what we have assumed, but that it has not always been so. 

It is evident that the actual laws will never be sufSciently well 
known for us not to be able to adopt the first of these two solu- 
tions and for us to be constrained to infer the evolution of 
natural laws. 

On the other hand, suppose such an evolution ; assume, if you 
wish, that humanity lasts sufficiently long for this evolution to 
have witnesses. The same antecedent shall produce, for instance, 
different consequents at the Carboniferous epoch and at the 
Quaternary. That evidently means that the antecedents are 
closely alike ; if all the circumstances were identical, the Carbon- 
iferous epoch would be indistinguishable from the Quaternary. 
Evidently this is not what is supposed. What remains is that 
such antecedent, accompanied by such accessory circumstance, 
produces such consequent ; and that the same antecedent, accom- 
panied by such other accessory circumstance, produces such 
other consequent. Time does not enter into the affair. 

The law, such as ill-informed science would have stated it, and 
which would have affirmed that this antecedent always produces 
this consequent, without taking account of the accessory circum- 
stances, this law, which was only approximate and probable, 
must be replaced by another law more approximate and more 
probable, which brings in these accessory circumstances. We 


always come back, therefore, to that same process which we have 
analyzed above, and if humanity should discover something o£ 
this sort, it would not say tliat it is the laws which have evolated, 
but the circumstances which have changed. 

Here, therefore, are several different senses of the word eon- 
tingence. M. LeRoy retains them all and he does not sufficiently 
distinguish them, but he introduces a new one. Experimental 
laws are only approximate, and if some appear to us as exact, it 
is because we have artificially transformed them into what I have 
above called a principle. We have made this transformation 
freely, and as the caprice which has determined ua to make it 
is something eminently contingent, we have communicated this 
contingence to the law itself. It is in this sense that we have the 
right to say that determinism supposes freedom, since it is freely 
that we become determinists. Perhaps it will be found that this 
is to give large scope to nominalism and that the introduction 
of this new sense of the word contingence will not help much to 
solve all those questions which naturally arise and of which we 
have just been speaking. 

I do not at all wish to investigate here the foundations of the 
principle of induction; I know very well that I should not suc- 
ceed ; it is as difficult to justify this principle as to get on with- 
out it. I only wish to show how scientists apply it and are 
forced to apply it. 

"When the same antecedent recurs, the same consequent must 
likewise recur; such is the ordinary statement. But reduced 
to these terms this principle could be of no use. For one to be 
able to say that the same antecedent recurred, it would be neces- 
sary for the circumstances ail to be reproduced, since no one 
is absolutely indifferent, and for them to be exacfly reproduced. 
And, as that will never happen, the principle can have no 

We should therefore modify the enunciation and say: If an 
antecedent A has once produced a consequent B, an antecedent 
A', slightly different from A, will produce a consequent B', 
slightly different from B. But how shall we recognize that the 
antecedents A and A' are 'slightly different'I If some one of thu 
etremuBtances can be expressed by a number, and this number 


has in the two cases values very near together, the sense of the 
phrase ' slightly different ' is relatively clear; the principle then 
signifies that the consequent is a continuous function of the ante- 
cedent. And as a practical rule, we reach this conclusion that 
we have the right to interpolate. This is in fact what scientists 
do every day, and without interpolation all science would be 

Yet observe one thing. The law sought may be represented by 
a curve. Experiment has taught us certain points of this curve. 
In virtue of the principle we have just stated, we believe these 
points may be connected by a continuous graph. We trace this 
graph with the eye. New experiments will furnish us new points 
of the curve. If these points are outside of the graph traced in 
advance, we shall have to modify our curve, but not to abandon 
our principle. Through any points, however numerous they may 
be, a continuous curve may always be passed. Doubtless, if this 
curve is too capricious, we shall be shocked (and we shall even 
suspect errors of experiment), but the principle will not be 
directly put at fault. 

Furthermore, among the circumstances of a phenomenon, there 
are some that we regard as negligible, and we shall consider A 
and A' as slightly different if they differ only by these accessory 
circumstances. For instance, I have ascertained that hydrogen 
unites with oxygen under the influence of the electric spark, and 
I am certain that these two gases will unite anew, although the 
longitude of Jupiter may have changed considerably in the 
interval. We assume, for instance, that the state of distant 
bodies can have no sensible influence on terrestrial phenomena, 
and that seems in fact requisite, but there are cases where the 
choice of these practically indifferent circumstances admits of 
more arbitrariness or, if you choose, requires more tact. 

One more remark: The principle of induction would be inap- 
plicable if there did not exist in nature a great quantity of 
bodies like one another, or almost alike, and if we could not 
infer, for instance, from one bit of phosphorus to another bit of 

If we reflect on these considerations, the problem of deter- 
minism and of contingence will appear to us in a new light. 


Suppose we were able to embrace the series of all phenomena 
of the universe in the whole sequence of time. We could envis- 
age what might be called the sequences; I mean relations between 
antecedent and consequent. I do not wish to speak of constant 
relations or laws, I envisage separately (individually, so to 
speak) the different sequences realized. 

We should then recognize that among these sequences there 

are no two altogether alike. But, if the principle of induction, 

as we have just stated it, is true, there will be those almost alike 

and that can be classed alongside one another. In other words, 

^it is possible to make a classification of sequences. 

^^■At is to the possibility and the legitimacy of such a classifica- 

^^fti that determinism, in the end, reduces. This is all that the 

^^^eceding analysis leaves of it. Perhaps under this modest form 

it will seem less appalling to the moralist. 

It will doubtless be said that this is to come back by a detonr 
to M. LeRoy's conclusion which a moment ago we seemed to 
reject: we are determinists voluntarily. And in fact all classi- 
fication supposes the active intervention of the classifier. I a^tree 
that this may be maintained, but it seems to me that this detour 
will not have been useless and will have contributed to enlighten 
^JU a little. 

^^H 6. Objectivity of Science 

^^Bk arrive at the question set by the title of this article : What is 
^^Be objective value of science T And first what should we under- 
stand by objectivity 1 

What guarantees the objectivity of the world in which we live 
is that this world is common to us with other thinking beings. 
Through the communications that we have with other men, we 
receive from them ready-raade reasonings; we know that these 
reasonings do not come from us and at the same time we recog- 
nize in them the work of reasonable beings like ourselves. And 
as these reasonings appear to fit the world of our sensations, we 
think we may infer that these reasonable beings have seen the 
same thing as we; thus it is we know we have not been dreaming. 
Such, therefore, is the first condition of objectivity; what is 
objective must be common to many minds and consequently trans- 
miBgible from one to the other, and as this transmisaioo can only 


come about by that 'discourse' which inspires so much distrust 
in M. LeRoy, we are even forced to conclude : no discourse, no 

The sensations of others will be for us a world eternally dosed. 
We have no means of verifying that the sensation I call red is 
the same as that which my neighbor calls red. 

Suppose that a cherry and a red poppy produce on me the 
sensation A and on him the sensation B and that, on the con- 
trary, a leaf produces on me the sensation B and on him the 
sensation A. It is clear we shall never know anything about it; 
since I shall call red the sensation A and green the sensation B^ 
while he will call the first green and the second red. In com- 
pensation, what we shall be able to ascertain is that, for him as 
for me, the cherry and the red poppy produce the same sensa< 
tion, since he gives the same name to the sensations he feels and 
I do the same. 

Sensations are therefore intransmissible, or rather all that is 
pure quality in them is intransmissible and forever impenetrable. 
But it is not the same with relations between these sensations. 

From this point of view, all that is objective is devoid of all 
quality and is only pure relation. Certes, I shall not go so far 
as to say that objectivity is only pure quantity (this would be 
to particularize too far the nature of the relations in question), 
but we understand how some one could have been carried away 
into saying that the world is only a differential equation. 

With due reserve regarding this paradoxical proposition, we 
must nevertheless admit that nothing is objective which is not 
transmissible, and consequently that the relations between the 
sensations can alone have an objective value. 

Perhaps it will be said that the esthetic emotion, which is 
common to all mankind, is proof that the qualities of our sensa- 
tions are also the same for all men and hence are objective. But 
if we think about this, we shall see that the proof is not com- 
plete ; what is proved is that this emotion is aroused in John as 
in James by the sensations to which James and John give the 
same name or by the corresponding combinations of these sensa- 
tions; either because this emotion is associated in John with 
the sensation A, which John calls red, while parallelly it is asso- 


elated in Jamea with the sensation B, which James calls red; 
or better because this emotion is aroused, not by the qualities 
themselves of the sensations, but by the harmonious combina- 
tion of their relations of which we undergo the uneonscioos 

Such a sensation is beautiful, not because it possesses such a 
quality, but because it occupies such a place in the woof of our 
associations of ideas, so that it can not be excited without putting 
in motion the 'receiver' which is at the other end of the thread 
and which corresponds to the artistic emotion. 

Whether we talie the moral, the esthetic or the scientific point 
of view, it is always the same thing. Nothing is objective except 
what is identical for all ; now we can only speak of such an 
identity if a comparison is possible, and can be translated into a 
'money of exchange' capable of transmission from one mind to 
another. Nothing, therefore, will have objective value except 
what is transmissible by 'discourse,' that is, intelligible. 

But this is only one side of the question. An absolutely dis- 
ordered aggregate could not have objective value since it would 
be unintelligible, but no more can a well-ordered assemblage 
have it. if it does not correspond to sensations really experienced. 
It seems to me superfluous to recall this condition, and I should 
not have dreamed of it, if it had not lately been maintained that 
physics is not an experimental science. Although this opinion 
has no chance of being adopted either by physicists or by phi- 
losophers, it is well to be warned so as not to let oneself slip over 
the declivity which would lead thither. Two conditions are 
therefore to be fulfilled, and if the first separates reality' from 
the dream, the second distinguishes it from the romance. 

Now what is sciencet I have explained in the preceding 
article, it is before all a classification, a manner of bringing 
together facts which appearances separate, though they were 
bound together by some natural and hidden kinship. Science, 
in other words, is a system of relations. Now we have just said, 
it is in the relations alone that objectivity must be sought; it 

1 1 lt«re OM the nord real as ■ ajmoDj^m. of objective; I tlius eonfonn (o 
eomaon nMge; perhaps I am wtong, our dreams are real, but the^r are not 


would be vain to seek it in beings considered as isolated from one 

To say that science can not have objective value since it teaches 
us only relations, this is to reason backward, since, precisely, it 
is relations alone which can be regarded as objective. 

External objects, for instance, for which the word abject was 
invented, are really objects and not fleeting and fugitive appear- 
ances, because they are not only groups of sensations, but groups 
cemented by a constant bond. It is this bond, and this bond 
alone, which is the object in itself, and this bond is a relation. 

Therefore, when we ask what is the objective value of science, 
that does not mean: Does science teach us the true nature of 
things? but it means: Does it teach us the true relations of 
things T 

To the first question, no one would hesitate to reply, no ; but I 
think we may go farther; not. only science can not teach us the 
nature of things ; but nothing is capable of teaching it to us, and 
if any god knew it, he could not find words to express it. Not 
only can we not divine the response, but if it were given to us 
we could understand nothing of it; I ask myself even whether 
we really understand the question. 

When, therefore, a scientific theory pretends to teach us what 
heat is, or what is electricity, or life, it is condemned beforehand; 
all it can give us is only a crude image. It is, therefore, pro- 
visional and crumbling. 

The first question being out of reason, the second remains. 
Can science teach us the true relations of things? What it joins 
together should that be put asunder, what it puts asunder should 
that be joined together? 

To understand the meaning of this new question, it is needful 
to refer to what was said above on the conditions of objectivity. 
Have these relations an objective value? That means: Are 
these relations the same for all ? Will they still be the same for 
those who shall come after us ? 

It is clear that they are not the same for the scientist and the 
ignorant person. But that is unimportant, because if the ignorant 
person does not see them all at once, the scientist may succeed in 
making him see them by a series of experiments and reasonings. 


The thing essential ie that there are points on which all those 
acquainted with the experiments made can reach accord. 

The question is to know whether this accord will be durable and 
whether it will persist for our successors. It may be asked 
whether tlie unions that the science of to-day makes will be con- 
Srmed by the science of to-morrow. To ftflarm that it will be so 
we can not invoke any a priori reason ; but this is a question of 
fact, and science has already lived long enough for us to he able 
to find out by asking its history whether the edifices it builds 
stand the test of time, or whether they are only ephemeral con- 

Now what do we see I At the first blush it seems to us that the 
theories last only a day and that ruins upon ruins accumulate. 
To-day the theories are bom, to-morrow they are the fashion, the 
day after to-morrow they are classic, the fourth day they are 
superannuated, and the fifth they are forgotten- But if we look 
more closely, we see that what thus succumb are the theories 
properly so called, those which pretend to teach us what things 
are. But there is in them something which usually survives. 
If one of them taught us a true relation, this relation is defini- 
tively acquired, and it will be found again under a new disguise 
in the other theories which will successively come to reign in 
place of the old. 

Take only a single example: The theory of the undulations of 
the ether taught ns that light is a motion ; to-day fashion favors 
the electromagnetic theory which teaches us that light is a cur- 
rent. We do not consider whether we could reconcile them and 
say that light is a current, and that this current is a motion. As 
it is probable in any ease that this motion would not be identical 
with that which the partisans of the old theory presume, we might 
think ourselves justified in saying that this old theory is de- 
throned. And yet something of it remains, since between the 
hypothetical currents which Maxwell supposes there arc the same 
relations as between the hypothetical motions that Presnel sup- 
posed. There is, therefore, something which remains over and 
this something is the essential. This it is which explains how 
we see the present physicists pass without any embarrassment 
trom the language of Fresnel to that of Maxwell. Doubtless 


many connections that were believed well established have been 
abandoned, but the greatest number remain and it would seem 
must remain. 

And for these, then, what is the measure of their objectivity f 
Well, it is precisely the same as for our belief in external objects. 
These latter are real in this, that the sensations they make us feel 
appear to us as united to each other by I know not what inde- 
structible cement and not by the hazard of a day. In the same 
way science reveals to us between phenomena other bonds finer 
but not less solid; these are threads so slender that they long 
remained unperceived, but once noticed there remains no way of 
not seeing them ; they are therefore not less real than those which 
give their reality to external objects ; small matter that they are 
more recently known, since neither can perish before the other. 

It may be said, for instance, that the ether is no less real than 
any external body; to say this body exists is to say there is 
between the color of this body, its taste, its smell, an intimate 
bond, solid and persistent; to say the ether exists is to say there 
is a natural kinship between all the optical phenomena, and 
neither of the two propositions has less value than the other. 

And the scientific syntheses have in a sense even more reality 
than those of the ordinary senses, since they embrace more terms 
and tend to absorb in them the partial syntheses. 

It will be said that science is only a classification and that a 
classification can not be true, but convenient. But it is true that 
it is convenient, it is true that it is so not only for me, but for 
all men ; it is true that it will remain convenient for our descend- 
ants; it is true finally that this can not be by chance. 

In sum, the sole objective reality consists in the relations of 
things whence results the universial harmony. Doubtless these 
relations, this harmony, could not be conceived outside of a mind 
which conceives them. But they are nevertheless objective be- 
cause they are, will become, or will remain, conunon to all think- 
ing beings. 

This will permit us to revert to the question of the rotation of 
the earth which will give us at the same time a chance to make 
clear what precedes by an example. 


7. The Rotation of the Earth 

**. . . Therefore," have I said in Science and Hypothesis, 
''this affirmation, the earth turns round, has no meaning ... or 
rather these two propositions, the earth turns round, and, it is 
more convenient to suppose that the earth turns round, have one 
and the same meaning." 

These words have given rise to the strangest interpretations. 
Some have thought they saw in them the rehabilitation of 
Ptolemy's system, and perhaps the justification of Galileo's 

Those who had read attentively the whole volume could not, 
however, delude themselves. This truth, the earth turns round, 
was put on the same footing as Euclid's postulate, for example. 
Was that to reject itT But better; in the same language it may 
very well be said: These two propositions, the external world 
exists, or, it is more convenient to suppose that it exists, have one 
and the same meaning. So the hypothesis of the rotation of the 
earth would have the same degree of certitude as the very exist- 
ence of external objects. 

But after what we have just explained in the fourth part, we 
may go farther. A physical theory, we have said, is by so much 
the more true as it puts in evidence more true relations. In the 
light of this new principle, let us examine the question which 
occupies us. 

No, there is no absolute space ; these two contradictory propo- 
sitions: *The earth turns round' and 'The earth does not turn 
round' are, therefore, neither of them more true than the other. 
To affirm one while denying the other, in the kinematic sense, 
would be to admit the existence of absolute space. 

But if the one reveals true relations that the other hides from 
us, we can nevertheless regard it as physically more true than the 
other, since it has a richer content. Now in this regard no doubt 
is possible. 

Behold the apparent diurnal motion of the stars, and the 

diurnal motion of the other heavenly bodies, and besides, the 

flattening of the earth, the rotation of Foucault's pendulum, the 

gyration of cyclones, the trade- winds, what not else? For the 



Ptolemaist all these phenomena have no bond between them; for 
the Copemican they are produced by the one same cause. In 
saying, the earth turns round, I afSrm that all these phenomena 
have an intimate relation, and that is true, and that remains true, 
although there is not and can not be absolute space. 

So much for the rotation of the earth upon itself ; what shall we 
say of its revolution around the sun T Here again, we have three 
phenomena which for the Ptolemaist are absolutely independent 
and which for the Copemican are referred back to the same 
origin; they are the apparent displacements of the planets on 
the celestial sphere, the aberration of the fixed stars, the parallax 
of these same stars. Is it by chance that all the planets admit an 
inequality whose period is a year, and that this period is precisely 
equal to that of aberration, precisely equal besides to that of 
parallax? To adopt Ptolemy's system is to answer, yes ; to adopt 
that of Copernicus is to answer, no ; this is to affirm that there is 
a bond between the three phenomena, and that also is true, 
although there is no absolute space. 

In Ptolemy's system, the motions of the heavenly bodies can 
not be explained by the action of central forces, celestial 
mechanics is impossible. The intimate relations that celestial 
mechanics reveals to us between all the celestial phenomena are 
true relations; to affirm the immobility of the earth would be to 
deny these relations, that would be to fool ourselves. 

The truth for which Galileo suffered remains, therefore, the 
truth, although it has not altogether the same meaning as for 
the vulgar, and its true meaning is much more subtle, more pro- 
found and more rich. 

8. Science for Its Own Sake 

Not against M. LeRoy do I wish to defend science for its own 
sake ; maybe this is what he condemns, but this is what he culti- 
vates, since he loves and seeks truth and could not live without it. 
But I have some thoughts to express. 

We can not know all facts and it is necessary to choose those 
which are worthy of being known. According to Tolstoi, scien- 
tists make this choice at random, instead of making it, which 
would be reasonable, with a view to practical applications. On 


the contrary, scientists think that certain facts are more interest- 
ing than others, because they complete an unfinished harmony, 
or because they make one foresee a great number of other facts. 
If they are wrong, if this hierarchy of facts that they implicitly 
postulate is only an idle illusion, there could be no science for its 
own sake, and consequently there could be no science. As for 
me, I believe they are right, and, for example, I have shown above 
what is the high value of astronomical facts, not because they 
are capable of practical applications, but because they are the 
most instructive of all. 

It is only through science and art that civilization is of value. 
Some have wondered at the formula: science for its own sake; 
and yet it is as good as life for its own sake, if life is only misery ; 
and even as happiness for its own sake, if we do not believe that 
all pleasures are of the same quality, if we do not wish to admit 
that the goal of civilization is to furnish alcohol to people who 
love to drink. 

Every act should have an aim. We must suffer, we must work, 
we must pay for our place at the game, but this is for seeing 's 
sake ; or at the very least that others may one day see. 

All that is not thought is pure nothingness ; since we can think 
only thoughts and all the words we use to speak of things can 
express only thoughts, to say there is something other than 
thought, is therefore an affirmation which can have no meaning. 

And yet — strange contradiction for those who believe in time — 
geologic history shows us that life is only a short episode between 
two eternities of death, and that, even in this episode, conscious 
thought has lasted and will last only a moment. Thought is only 
a gleam in the midst of a long night. 

But it is this gleam which is everything. 



- w 



I BRiNa together here different studies relating more or less 
directly to questions of scientific methodology. The scientific 
method consists in observing and experimenting ; if the scientist 
had at his disposal infinite time, it would only be necessary to 
say to him : ' Look and notice well ' ; but, as there is not time to 
see everything, and as it is better not to see than to see wrongly, 
it is necessary for him to make choice. The first question, there- 
forC; is how he should make this choice. This question presents 
itself as well to the physicist as to the historian; it presents 
itself equally to the mathematician, and the principles which 
should guide each are not without analogy. The scientist con- 
forms to them instinctively, and one can, reflecting on these prin- 
ciples, foretell the future of mathematics. 

We shall understand them better yet if we observe the scien- 
tist at work, and first of all it is necessary to know the xxsycho- 
logic mechanism of invention and, in particular, that of mathe- 
matical creation. Observation of the processes of the work of 
the mathematician is particularly instructive for the psychologist. 

In all the sciences of observation account must be taken of the 
errors due to the imperfections of our senses and our instru- 
ments. Luckily, we may assume that, under certain conditions, 
these errors are in part self-compensating, so as to disappear in 
the average; this compensation is due to chance. But what is 
chance? This idea is difficult to justify or even to define; and 
yet what I have just said about the errors of observation, shows 
that the scientist can not neglect it. It therefore is necessary to 
give a definition as precise as possible of this concept, so indis- 
pensable yet so illusive. 

These are generalities applicable in sum to all the sciences; 
and for example the mechanism of mathematical invention does 
not differ sensibly from the mechanism of invention in general. 
Later I attack questions relating more particularly to certain 
special sciences and first to pure mathematics. 



In the chapters devoted to these, I have to treat sabjects 
a little more abstract I have first to speak of the notion of 
space ; every one knows space is relative, or rather every one says 
so, but many think still as if they believed it absolute ; it soffices 
to reflect a little however to perceive to what contradictions they 
are exposed. 

The questions of teaching have their importance, first in them- 
selves, then because reflecting on the best way to make new 
ideas penetrate virgin minds is at the same time reflecting on 
how these notions were acquired by our ancestors, and conse- 
quently on their true origin, that is to say, in reality on their 
true nature. Why do children usually understand nothing of 
the definitions which satisfy scientists T Why is it necessary to 
give them others f This is the question I set myself in the suc- 
ceeding chapter and whose solution should, I think, suggest use- 
ful reflections to the philosophers occupied with the logic of 
the sciences. 

On the other hand, many geometers believe we can reduce 
mathematics to the rules of formal logic. Unheard-of efforts 
have been made to do this; to accomplish it, some have not 
hesitated, for example, to reverse the historic order of the genesis 
of our conceptions and to try to explain the finite by the infinite. 
I believe I have succeeded in showing, for all those who attack 
the problem unprejudiced, that here there is a fallacious illusion. 
I hope the reader will understand the importance of the question 
and pardon me the aridity of the pages devoted to it. 

The concluding chapters relative to mechanics and astronomy 
will be easier to read. 

Mechanics seems on the point of undergoing a complete revo- 
lution. Ideas which appeared best established are assailed by 
bold innovators. Certainly it would be premature to decide in 
their favor at once simply because they are innovators. 

But it is of interest to make known their doctrines, and this 
is what I have tried to do. As far as possible I have followed 
the historic order; for the new ideas would seem too astonish- 
ing unless we saw how they arose. 

Astronomy offers us majestic spectacles and raises gigantic 
problems. We can not dream of applying to them directly the 


experimental method; our laboratories are too small. But anid- 
ogy with phenomena these laboratories permit us to attain may 
nevertheless guide the astronomer. The Milky Way, for ex- 
ample, is an assemblage of suns whose movements seem at first 
capricious. But may not this assemblage be compared to that of 
the molecules of a gas, whose properties the kinetic theory of 
gases has made known to usf It is thus by a roundabout way 
that the method of the physicist may come to the aid of the 

Finally I have endeavored to give in a few lines the history 
of the development of French geodesy; I have shown through 
what persevering efforts, and often what dangers, the geodesists 
have procured for us the knowledge we have of the figure of the 
earth. Is this then a question of method f Yes, without doubt, 
this history teaches us in fact by what precautions it is necessary 
to surround a serious scientific operation and how much time and 
pains it costs to conquer one new decimal. 




The Choice op Pacts 

Tolstoi somewhere explains why 'science for its own sake' is 
in his eyes an absurd conception. We can not know all facts, 
since their number is practically infinite. It is necessary to 
choose; then we may let this choice depend on the pure caprice 
of our curiosity ; would it not be better to let ourselves be guided 
by utility, by our practical and above all by our moral needs ; 
have we nothing better to do than to count the number of lady- 
bugs on our planet T 

It is clear the word utility has not for him the sense men of 
affairs give it, and following them most of our contemporaries. 
Little cares he for industrial applications, for the marvels of 
electricity or of automobilism, which he regards rather as ob- 
stacles to moral progress ; utility for him is solely what can make 
man better. 

For my part, it need scarce be said, I could never be content 
with either the one or the other ideal ; I want neither that plutoc- 
racy grasping and mean, nor that democracy goody and mediocre, 
occupied solely in turning the other cheek, where would dwell 
sages without curiosity, who, shunning excess, would not die of 
disease, but would surely die of ennui. But that is a matter of 
taste and is not what I wish to discuss. 

The question nevertheless remains and should fix our attention ; 
if our choice can only be determined by caprice or by immediate 
utility, there can be no science for its own sake, and consequently 
no science. But is that true? That a choice must be made is 
incontestable ; whatever be our activity, facts go quicker than we, 
and we can not catch them ; while the scientist discovers one fact, 



there happen milliards of milliards in a cubic millimeter of his 
body. To wish to comprise nature in science would be to want 
to put the whole into the part. 

But scientists believe there is a hierarchy of facts and that 
among them may be made a judicious choice. They are right, 
since otherwise there would be no science, yet science exists. One 
need only open the eyes to see that the conquests of industry which 
have enriched so many practical men would never have seen the 
light, if these practical men alone had existed and if they had not 
been preceded by unselfish devotees who died poor, who never 
thought of utility, and yet had a guide far other than caprice. 

As Mach says, these devotees have spared their successors the 
trouble of thinking. Those who might have worked solely in 
view of an immediate application would have left nothing behind 
them, and, in face of a new need, all must have been begun over 
again. Now most men do not love to think, and this is perhaps 
fortunate when instinct guides them, for most often, when they 
pursue an aim which is immediate and ever the same, instinct 
guides them better than reason would guide a pure intelligence. 
But instinct is routine, and if thought did not fecundate it, it 
would no more progress in man than in the bee or ant. It is 
needful then to think for those who love not thinking, and, as 
they are numerous, it is needful that each of our thoughts be as 
often useful as possible, and this is why a law will be the more 
precious the more general it is. 

This shows us how we should choose : the most interesting facts 
are those which may serve many times ; these are the facts which 
have a chance of coming up again. We have been so fortunate as 
to be bom in a world where there are such. Suppose that in- 
stead of 60 chemical elements there were 60 milliards of them, 
that they were not some common, the others rare, but that they 
were uniformly distributed. Then, every time we picked up a 
new pebble there would be great probability of its being formed 
of some unknown substance; all that we knew of other pebbles 
would be worthless for it ; before each new object we should be 
as the new-bom babe ; like it we could only obey our caprices or 
our needs. Biologists would be just as much at a loss if there 
were only individuals and no species and if heredity did not 
make sons like their fathers. 


In such a world there would be no science; perhaps thought 
and even life would be impossible, since evolution could not there 
develop the preservational instincts. Happily it is not so; like 
all good fortune to which we are accustomed, this is not appre- 
ciated at its true worth. 

Which then are the facts likely to reappear T They are first 
the simple facts. It is clear that in a complex fact a thousand 
circumstances are united by chance, and that only a chance still 
much less probable could reunite them anew. But are there any 
simple facts T And if there are, how recognize themT What 
assurance is there that a thing we think simple does not hide a 
dreadful complexity? All we can say is that we ought to prefer 
the facts which seem simple to those where our crude eye discerns 
unlike elements. And then one of two things : either this simplic- 
ity is real, or else the elements are so intimately mingled as not 
to be distinguishable. In the first case there is chance of our 
meeting anew this same simple fact, either in all its purity or 
entering itself as element in a complex manifold. In the second 
case this intimate mixture has likewise more chances of recurring 
than a heterogeneous assemblage; chance knows how to mix, it 
knows not how to disentangle, and to make with multiple elements 
a well-ordered edifice in which something is distinguishable, it 
must be made expressly. The facts which appear simple, even 
if they are not so, will therefore be more easily revived by chance. 
This it is which justifies the method instinctively adopted by the 
scientist, and what justifies it still better, perhaps, is that oft- 
recurring facts appear to us simple, precisely because we are 
used to them. 

But where is the simple fact? Scientists have been seeking 
it in the two extremes, in the infinitely great and in the infinitely 
small. The astronomer has found it because the distances of 
the stars are immense, so great that each of them appears but 
as a point, so great that the qualitative differences are effaced, 
and because a point is simpler than a body which has form and 
qualities. The physicist on the other hand has sought the ele- 
mentary phenomenon in fictively cutting up bodies into infinites- 
imal cubes, because the conditions of the problem, which undergo 
slow and continuous variation in passing from one point of the 


body to another, may be regarded as constant in the interior of 
each of these little cubes. In the same way the biologist has 
been instinctively led to regard the cell as more interesting than 
the whole animal, and the outcome has shown his wisdom, since 
cells belonging to organisms the most different are more alike, 
for the one who can recognize their resemblances, than are these 
organisms themselves. The sociologist is more embarrassed ; the 
elements, which for him are men, are too unlike, too variable, too 
capricious, in a word, too complex ; besides, history never begins 
over again. How then choose the interesting fact, which is that 
which begins again T Method is precisely the choice of facts; it 
is needful then to be occupied first with creating a method, and 
many have been imagined, since none imposes itself, so that so- 
ciology is the science which has the most methods and the fewest 

Therefore it is by the regular facts that it is proper to begin ; 
but after the rule is well established, after it is beyond all doubt, 
the facts in fidl conformity with it are erelong without interest 
since they no longer teach us anything new. It is then the ex- 
ception which becomes important. We cease to seek resem- 
blances; we devote ourselves above all to the differences, and 
among the differences are chosen first the most accentuated, not 
only because they are the most striking, but because they will 
be the most instructive. A simple example will make my thought 
plainer : Suppose one wishes to determine a curve by observing 
some of its points. The practician who concerns himself only 
with immediate utility would observe only the points he might 
need for some special object. These points would be badly dis- 
tributed on the curve ; they would be crowded in certain regions, 
rare in others, so that it would be impossible to join them by a 
continuous line, and they would be unavailable for other applica- 
tions. The scientist will proceed differently; as he wishes to 
study the curve for itself, he will distribute regularly the points 
to be observed, and when enough are known he will join them 
by a regular line and then he will have the entire curve. But 
for that how does he proceed ? If he has determined an extreme 
point of the curve, he does not stay near this extremity, but goes 
first to the other end ; after the two extremities the most instruc- 
tive point will be the mid-point, and so on. 


So when a rule is established we should first seek the cases 
where this rule has the greatest chance of failing. Thence, 
among other reasons, come the interest of astronomic facts, and 
the interest of the geologic past ; by going very far away in space 
or very far away in time, we may find our usual rules entirely 
overturned, and these grand overtumings aid us the better to see 
or the better to understand the little changes which may happen 
nearer to us, in the little comer of the world where we are called 
to Uve and act. We shall better know this comer for having 
traveled in distant countries with which we have nothing to do. 

But what we ought to aim at is less the ascertainment of resem- 
blances and differences than the recognition of likenesses hidden 
under apparent divergences. Particular rules seem at first dis- 
cordant, but looking more closely we see in general that they 
resemble each other; different as to matter, they are alike as to 
form, as to the order of their parts. When we look at them with 
this bias, we shall see them enlarge and tend to embrace every- 
thing. And this it is which makes the value of certain facts 
which come to complete an assemblage and to show that it is the 
faithful image of other known assemblages. 

I will not further insist, but these few words suffice to show 
that the scientist does not choose at random the facte he observes. 
He does not, as Tolstoi says, count the lady-bugs, because, how- 
ever interesting lady-bugs may be, their number is subject to 
capricious variations. He seeks to condense much experience 
and much thought into a slender volume ; and that is why a little 
book on physics contains so many past experiences and a thou- 
sand times as many possible experiences whose result is known 

But we have as yet looked at only one side of the question. 
The scientist does not study nature because it is useful ; he studies 
it because he delights in it, and he delights in it because it is 
beautiful. If nature were not beautiful, it would not be worth 
knowing, and if nature were not worth knowing, life would not 
be worth living. Of course I do not here speak of that beauty 
which strikes the senses, the beauty of qualities and of appear- 
ances ; not that I undervalue such beauty, far from it, but it has 
nothing to do with science ; I mean that prof ounder beauty which 


comes from the harmonious order of the parts and which a pure 
intelligence can grasp. This it is which gives body, a structure 
so to speak, to the iridescent appearances which flatter our senses, 
and without this support the beauty of these fugitive dreams 
would be only imperfect, because it would be vague and always 
fleeting. On the contrary, intellectual beauty is sufficient unto 
itself, and it is for its sake, more perhaps than for the future 
good of humanity, that the scientist devotes himself to long and 
difficult labors. 

It is, therefore, the quest of this especial beauty, the sense of 
the harmony of the cosmos, which makes us choose the facts 
most fitting to contribute to this harmony, just as the artist 
chooses from among the features of his model those which perfect 
the picture and give it character and life. And we need not 
fear that this instinctive and unavowed prepossession will turn 
the scientist aside from the search for the true. One may dream 
a harmonious world, but how far the real world will leave it 
behind! The greatest artists that ever lived, the Greeks, made 
their heavens ; how shabby it is beside the true heavens, ours ! 

And it is because simplicity, because grandeur, is beautiful, 
that we preferably seek simple facts, sublime facts, that we de- 
light now to follow the majestic course of the stars, now to ex- 
amine with the microscope that prodigious littleness which is 
also a grandeur, now to seek in geologic time the traces of a past 
which attracts because it is far away. 

We see too that the longing for the beautiful leads us to the 
same choice as the longing for the useful. And so it is that this 
economy of thought, this economy of effort, which is, according 
to Mach, the constant tendency of science, is at the same time 
a source of beauty and a practical advantage. The edifices that 
we admire are those where the architect has known how to pro- 
portion the means to the end, where the columns seem to carry 
gaily, without effort, the weight placed upon them, like the 
gracious caryatids of the Erechtheum. 

Whence comes this concordance f Is it simply that the things 
which seem to us beautiful are those which best adapt themselves 
to our intelligence, and that consequently they are at the same 
time the implement this intelligence knows best how to usef 


Or 18 there here a play of evolutioii and natural adeetionf Have 
the peoples whoee ideal most conformed to their highest interest 
exterminated the others and taken their place f All puraaed 
their ideals without reference to consequences, but while this 
quest led some to destruction, to others it gave empire. One is 
tempted to belieye it If the Oreeks triumphed over the bar- 
barians and if Europe, heir of Greek thought, dominates the 
world, it is because the savages loved loud colors and the clamor- 
ous tones of the drum which occupied only their senses, while the 
Greeks loved the intellectual beauty which hides beneath sen- 
suous beauty, and this intellectual beauty it is which makes in- 
telligence sure and strong. 

Doubtless such a triumph would horrify Tolstoi, and he would 
not like to acknowledge that it might be tmly usef uL But this 
disinterested quest of the true for its own beauty is sane also and 
able to make man better. I well know that there are mistakes, 
that the thinker does not always draw thence the serenity he 
should find therein, and even that there are scientists of bad 
character. Must we, therefore, abandon science and study only 
morals f What ! Do you think the moralists themselves are irre- 
proachable when they come down from their pedestal f 

The Futube of Mathematics 

To foresee the future of mathematics, the true method is to 
study its history and its present state. 

Is this not for us mathematicians in a way a professional pro- 
cedure f We are accustomed to exirapoUrie, which is a means 
of deducing the future from the past and present, and as we well 
know what this amounts to, we run no risk of deceiving ourselves 
about the range of the results it gives us. 

We have had hitherto prophets of evil. They blithely reiterate 
that all problems capable of solution have already been solved, 
and that nothing is left but gleaning. Happily the case of the 
past reassures us. Often it was thought all problems were solved 
or at least an inventory was made of all admitting solution. 
And then the sense of the word solution enlarged, the insoluble 
problems became the most interesting of all, and others unfore- 
seen presented themselves. For the Greeks a good solution was 
one employing only ruler and compasses; then it became one 
obtained by the extraction of roots, then one using only algebraic 
or logarithmic functions. The pessimists thus found themselves 
always outflanked, always forced to retreat, so that at present I 
think there are no more. 

My intention, therefore, is not to combat them, as they are 
dead ; we well know that mathematics will continue to develop, 
but the question is how, in what direction? You will answer, 
'in every direction,' and that is partly true; but if it were 
wholly true it would be a little appalling. Our riches would 
soon become encumbering and their accumulation would produce 
a medley as impenetrable as the unknown true was for the 

The historian, the physicist, even, must make a choice among 
facts; the head of the scientist, which is only a comer of the 
universe, could never contain the universe entire ; so that among 
the innumerable facts nature offers, some will be passed by, 
others retained. 

25 369 


Jujst so, a fortiori, in mathenuitics ; no more can the geometer 
hold fast pell-mell all the facts presenting themselyes to him; 
all the more becanae he it is, almost I had said his caprice, that 
creates these facts. He constructs a wholly new combination by 
patting together its elements; nature does not in general give it 
to him ready made. 

Doubtless it sometimes happens that the mathematician under- 
takes a problem to satisfy a need in physics; that the physicist 
or engineer asks him to calculate a number for a certain applica- 
tion. Shall it be said that we geometers should limit ourselyes 
to awaiting orders, and, in place of cultivating our science for 
our own delectation, try only to accommodate ourselves to the 
wants of our patrons? If mathematics has no other object be- 
sides aiding those who study nature, it is from these we should 
await orders. Is this way of looking at it legitimate f Certainly 
not; if we had not cultivated the exact sciences for themselves, 
we should not have created mathematics the instrument^ and the 
day the call came from the physicist we should have been 

Nor do the physicists wait to study a phenomenon until some 
urgent need of material life has made it a necessity for them; 
and they are right. If the scientists of the eighteenth century 
had neglected electricity as being in their eyes only a curiosity 
without practical interest, we should have had in the twentieth 
century neither telegraphy, nor electro-chemistry, nor electro- 
technics. The physicists, compelled to choose, are therefore not 
guided in their choice solely by utility. How then do they choose 
between the facts of nature ? We have explained it in the pre- 
ceding chapter: the facts which interest them are those capable 
of leading to the discovery of a law, and so they are analogous 
to many other facts which do not seem to us isolated, but closely 
grouped with others. The isolated fact attracts all eyes, those of 
the layman as well as of the scientist. But what the genuine 
physicist alone knows how to see, is the bond which unites many 
facts whose analogy is profound but hidden. The story of New- 
ton's apple is probably not true, but it is symbolic; let us speak 
of it then as if it were true. Well then, we must believe that 
before Newton plenty of men had seen apples fall ; not one knew 


how to coDclude anything therefrom. Facts would be fiterile 
were there not minds capable of choosing among them, discern- 
mg those behind which something was hidden, and of recognizing 
what is hiding, minds which under the cmde fact perceive the 
eool of the fact. 

We find just the same thing in mathematics. From the varied 
elements at our disposal we can get millions of different com- 
binations ; but one of these combinations, in so far as it is isolated, 
is absolutely void of value. Often we have taken great pains to 
construct it, but it serves no purpose, if not perhaps to furnish a 
task in secondary education. Quite otherwise will it be when 
thi>t combination shall find place in a class of analogous combina- 
tions and we shall have noticed this analogy. We are no longer 
in the presence of a fact, but of a law. And upon that day the 
real discoverer will not be the workman who shall have patiently 
built up certain of these combinations; it will be he who brings 
to light their kinship. The first will have seen merely the crude 
fact, only the other will have perceived the soul of the fact. 
Often to fix this kinship it suffices him to make a new word, and 
this word is creative. The history of science furnishes us a 
crowd of examples familiar to all. 

The celebrated Vienna philosopher Maeh has said that the r61e 
of science is to produce economy of thought, just as machines 
produce economy of effort. And that is very true. The savage 
reckons on his fingers or by heaping pebbles. In teaching chil- 
dren the multiplication table we spare them later innumerable 
pebble bunchings. Some one has already found out, with pebbles 
or otherwise, that 6 times 7 is 42 and lias had the idea of noting 
the result, and so we need not do it over again. He did not 
waste his time even if he reckoned for pleasure: his operation 
took him only two minutes ; it would have taken in all two mil- 
liards if a milliard men had had to do it over after him. 

The importance of a fact then is measured by its yield, that is 
to say, by the amount of thought it permits ua to spare. 

In physics the facts of great yield are those entering into a 
very general law, since from it they enable us to foresee a great 
number of others, and just so it is in mathematics. Suppose I 
bare undertaken a complicated calculation and laborioualjp 


reached a result: I shall not be compensated for my trouble if 
thereby I have not become capable of foreseeing the results of 
other analogous calculations and guiding them with a certainty 
that avoids the gropings to which one must be resigned in a 
fiist attempt. On the other hand, I shall not have wasted my 
time if these gropings themselves have ended by revealing to me 
the profound analogy of the problem just treated with a much 
more extended class of other problems; if they have shown me 
at once the resemblances and differences of these, if in a word 
they have made me perceive the possibility of a generalization. 
Then it is not a new result I have won, it is a new power. 

The simple example that comes first to mind is that of an alge- 
braic formula which gives us the solution of a type of numeric 
problems when finally we replace the letters by numbers. Thanks 
to it, a single algebraic calculation saves us the pains of cease- 
lessly beginning over again new numeric calculations. But this 
is only a crude example; we all know there are analogies inex- 
pressible by a formula and all the more precious. 

A new result is of value, if at all, when in unifying elements 
long known but hitherto separate and seeming strangers one to 
another it suddenly introduces order where apparently disorder 
reigned. It then permits us to see at a glance each of these 
elements and its place in the assemblage. This new fact is not 
merely precious by itself, but it alone gives value to all the old 
facts it combines. Our mind is weak as are the senses ; it would 
lose itself in the world's complexity were this complexity not har- 
monious ; like a near-sighted person, it would see only the details 
and would be forced to forget each of these details before exam- 
ining the following, since it would be incapable of embracing all. 
The only facts worthy our attention are those which introduce 
order into this complexity and so make it accessible. 

Mathematicians attach great importance to the elegance of 
their methods and their results. This is not pure dilettantism. 
What is it indeed that gives us the feeling of elegance in a solu- 
tion, in a demonstration 1 It is the harmony of the diverse parts, 
their symmetry, their happy balance; in a word it is all that 
introduces order, all that gives unity, that permits us to see 
clearly and to comprehend at once both the ensemble and the 

details. But this is exactly what yielda great results; in fact the 
more we see this aggregate clearly and at a single glance, the 
better we perceive its analogies with other neighboring objects, 
consequently the more chances we have of divining the possible 
generalizations. Elegance may produce the feeling of the unfore- 
seen by the unexpected meeting of objects we are not accustomed 
to bring together; there again it is fruitful, since it thus unveils 
for us kinships before unrecognized. It is fruitful even when it 
results only from the contrast between the simplicity of the 
means and the complexity of the problem set ; it makes us then 
think of the reason for this contrast and very often makes t]s 
see that chance ia not the reason ; that it is to be found in some 
unexpected law. In a word, the feeling of mathematical ele- 
gance is only the satisfaction due to any adaptation of the solu- 
tion to the needs of our mind, and it is because of this very 
adaptation that this solution can be for us an instrument. Con- 
sequently this esthetic satisfaction is bound up with the econ- 
omy of thought. Again the comparison of the Erechthcum 
comes to my mind, but I must not use it too often. 

It is for the same reason that, when a rather long calculation 
has led to some simple and striking result, we are not satisfied 
until we have shown that we should have been able to foresee, 
if not this entire result, at least its most characteristic traits. 
Why I "What prevents our beii^ content with a calculation 
which has told us, it seems, all we wished to knowT It is be- 
cause, in analogous eases, the long calculation might not again 
avail, and that this is not so about the reasoning often half in- 
tuitive which would have enabled us to foresee. This reasoning 
being short, we see at a single glance all its parts, so that we im- 
mediately perceive what must be changed to adapt it to all the 
problems of the same nature which can occur. And then it 
enables as to foresee if the solution of these problems will be 
simple, it shows us at least if the calculation is worth under- 

What we have just said suffices to show how vain it would be 
to seek to replace by any mechanical procedure the free initiative 
of the mathematician. To obtain a result of real value, it is not 
enough to grind out calculations, or to have a machine to put 


things in order; it is not order alone, it is unexpected order, 
which is worth while. The machine may gnaw on the crude fact, 
the soul of the fact will always escape it. 

Since the middle of the last century, mathematicians are more 
and more desirous of attaining absolute rigor; they are right, 
and this tendency will be more and more accentuated. In math- 
ematics rigor is not everything, but without it there is nothing. 
A demonstration which is not rigorous is nothingness. I think 
no one will contest this truth. But if it were taken too literally, 
we should be led to conclude that before 1820, for example, there 
was no mathematics; this would be manifestly excessive; the 
geometers of that time understood voluntarily what we explain 
by prolix discourse. This does not mean that they did not see it 
at all ; but they passed over it too rapidly, and to see it well would 
have necessitated taking the pains to say it. 

But is it always needful to say it so many times; those who 
were the first to emphasize exactness before all else have given 
us arguments that we may try to imitate ; but if the demonstra- 
tions of the future are to be built on this model, mathematical 
treatises will be very long; and if I fear the lengthenings, it is 
not solely because I deprecate encumbering libraries, but because 
I fear that in being lengthened out, our demonstrations may lose 
that appearance of harmony whose usefulness I have just 

The economy of thought is what we should aim at, so it is not 
enough to supply models for imitation. It is needful for those 
after us to be able to dispense with these models and, in place of 
repeating an argument already made, summarize it in a few 
words. And this has already been attained at times. For in- 
stance, there was a type of reasoning found everywhere, and 
everywhere alike. They were perfectly exact but long. Then 
all at once the phrase * uniformity of convergence ' was hit upon 
and this phrase made those arguments needless; we were no 
longer called upon to repeat them, since they could be under- 
stood. Those who conquer difficulties then do us a double service : 
first they teach us to do as they at need, but above all they 
enable us as often as possible to avoid doing as they, yet without 
sacrifice of exactness. 


We have just seen by one example the importance of words in 
mathematics, but many others could be cited. It is hard to be- 
lieve how much a well-chosen word can economize thought, as 
Mach says. Perhaps I have already said somewhere that mathe- 
matics is the art of giving the same name to different things. It 
is proper that these things, differing in matter, be alike in 
form, that they may, so to speak, run in the same mold. When 
the language has been well chosen, we are astonished to see that 
all the proofs made for a certain object apply immediately to 
many new objects ; there is nothing to change, not even the words, 
since the names have become the same. 

A well-chosen word usually suffices to do away witii the ex- 
ceptions from which the rules stated in the old way suffer; this 
is why we have created negative quantities, imaginaries, points 
at infinity, and what not. And exceptions, we must not forget, 
are pernicious because they hide the laws. 

Well, this is one of the characteristics by which we recognize 
the facts which yield great results. They are those which allow 
of these happy innovations of language. The crude fact then 
is often of no great interest; we may point it out many times 
without having rendered great service to science. It takes value 
only when a wiser thinker perceives the relation for which it 
stands, and symbolizes it by a word. 

Moreover the physicists do just the same. They have in- 
vented the word 'energy,* and this word has been prodigiously 
fruitful, because it also made the law by eliminating the excep- 
tions, since it gave the same name to things differing in matter 
and like in form. 

Among words that have had the most fortunate influence I 
would select * group' and * invariant.' They have made us see 
the essence of many mathematical reasonings; they have shown 
us in how many cases the old mathematicians considered groups 
without knowing it, and how, believing themselves far from one 
another, they suddenly found themselves near without knowing 

To-day we should say that they had dealt with isomorphic 
groups. We now know that in a group the matter is of little 
interest, the form alone counts, and that when we know a group 


we thus know all the isomorphic groups; and thanks to these 
words 'group' and isomorphism,' which condense in a few syl- 
lables this subtile rule and quickly make it familiar to all minds, 
the transition is immediate and can be done with every econ- 
omy of thought effort. The idea of group besides attaches to that 
of transformation. Why do we put such a value on the in- 
vention of a new transformation f Because from a single the- 
orem it enables us to get ten or twenty; it has the same value as 
a zero adjoined to the right of a whole number. 

This then it is which has hitherto determined the direction of 
mathematical advance, and just as certainly will determine it in 
the future. But to this end the nature of the problems which 
come up contributes equally. We can not forget what must be 
our aim. In my opinion this aim is double. Our science borders 
upon both philosophy and physics, and we work for our two 
neighbors; so we have always seen and shall still see mathema- 
ticians advancing in two opposite directions. 

On the one hand, mathematical science must reflect upon itself, 
and that is useful since reflecting on itself is reflecting on the 
human mind which has created it, all the more because it is the 
very one of its creations for which it has borrowed least from 
without. This is why certain mathematical speculations are 
useful, such as those devoted to the study of the postulates, of 
unusual geometries, of peculiar functions. The more these spec- 
ulations diverge from ordinary conceptions, and consequently 
from nature and applications, the better they show us what the 
human mind can create when it frees itself more and more from 
the tyranny of the external world, the better therefore they let 
us know it in itself. 

But it is toward the other side, the side of nature, that we must 
direct the bulk of our army. There we meet the physicist or 
the engineer, who says to us: ** Please integrate this differential 
equation for me ; I might need it in a week in view of a construc- 
tion which should be finished by that time." **This equation," 
we answer, **does not come under one of the integrable types; 
you know there are not many." **Yes, I know; but then what 
good are you?" Usually to understand each other is enough; 
the engineer in reality does not need the integral in finite terms; 


he needs to know the general look of the integral function, or he 
simply wants a certain nmnher which could readily be deduced 
from this integral if it were known. Usually it is not known, 
but the number can be calculated without it if we know exactly 
what number the engraeer needs and with what approximation. 

Formerly an equation was considered solved only when its 
solution had been expressed by aid of a finite number of known 
functions ; but that is possible scarcely once in a hundred times. 
What we always can do, or rather what we should always seek 
to do, is to solve the problem qtuilitatively so to speak; that is to 
say, seek to know the general form of the curve which represents 
the unknown function. 

It remains to find the quantitaiive solution of the problem; 
but if the unknown can not be determined by a finite calculation, 
it may always be represented by a convergent infinite series 
vhich enables us to calculate it. Can that be regarded as a true 
solution? We are told that Newton sent Leibnitz an anagram 
almost like this: aaaaabbbeeeeii, etc. Leibnitz naturally under- 
stood nothing at all of it ; but we, who have the key, know that 
this anagram meant, translated into modem terms: '^I can inte- 
grate all differential equations ' ' ; and we are tempted to say that 
Newton had either great luck or strange delusions. He merely 
wished to say he could form (by the method of indeterminate 
coefiScients) a series of powers formally satisfying the proposed 

Such a solution would not satisfy us to-day, and for two 
reasons: because the convergence is too slow and because the 
terms follow each other without obeying any law. On the con- 
trary, the series seems to us to leave nothing to be desired, first 
because it converges very quickly (this is for the practical man 
who wishes to get at a number as quickly as possible) and next 
because we see at a glance the law of the terms (this is to satisfy 
the esthetic need of the theorist). 

But then there are no longer solved problems and others 
which are not; there are only problems more or less solved, 
according as they are solved by a series converging more or less 
rapidly, or ruled by a law more or less harmonious. It often 
happens however that an imperfect solution guides us toward a 


better one. Sometimes the series converges so slowly that the 
computation is impracticable and we have only succeeded in 
proving the possibility of the problem. 

And then the engineer finds this a mockery, and justly, since 
it will not aid him to complete his construction by the date fixed. 
He little cares to know if it will benefit engineers of the twenty- 
second century. But as for us, we think differently and we arc 
sometimes happier to have spared our grandchildren a day's 
work than to have saved our contemporaries an hour. 

Sometimes by groping, empirically, so to speak, we reach a 
formula sufficiently convergent. **What more do you wantt" 
says the engineer. And yet, in spite of all, we are not satisfied; 
we should have liked to foresee that convergence. Why? Be- 
cause if we had known how to foresee it once, we woidd know how 
to foresee it another time. We have succeeded ; that is a smaU 
matter in our eyes if we can not validly expect to do so again. 

In proportion as science develops, its total comprehension 
becomes more difficult; then we seek to cut it in pieces and to 
be satisfied with one of these pieces: in a word, to specialize. 
If we went on in this way, it would be a grievous obstacle to the 
progress of science. As we have said, it is by unexpected union 
between its diverse parts that it progresses. To specialize too 
much would be to forbid these drawings together. It is to be 
hoped that congresses like those of Heidelberg and Rome, by 
putting us in touch with one another, will open for us vistas over 
neighboring domains and oblige us to compare them with our 
own, to range somewhat abroad from our own little village ; thus 
they will be the best remedy for the danger just mentioned. 

But I have lingered too long over generalities; it is time to 
enter into detail. 

Let us pass in review the various special sciences which com- 
bined make mathematics ; let us see what each has accomplished, 
whither it tends and what we may hope from it. If the pre- 
ceding views are correct, we should see that the greatest advances 
in the past have happened when two of these sciences have united, 
when we have become conscious of the similarity of their form, 
despite the difference of their matter, when they have so modeled 
themselves upon each other that each could profit by the other's 


conquests. We should at the same time foresee in combinations 
of the same sort the progress of the future. 


Progress in arithmetic has been much slower than in algebra 
and analysis, and it is easy to see why. The feeling of continuity 
is a precious guide which the arithmetician lacks; each whole 
number is separated from the others, — ^it has, so to speak, its own 
individuality. Each of them is a sort of exception and this is 
why general theorems are rarer in the theory of numbers; this 
is also why those which exist are more hidden and longer elude 
the searchers. 

If arithmetic is behind algebra and analysis, the best thing for 
it to do is to seek to model itself upon these sciences so as to 
profit by their advance. The arithmetician ought therefore to 
take as guide the analogies with algebra. These analogies are 
numerous and if, in many cases, they haVe not yet been studied 
sufficiently closely to become utilizable, they at least have long 
been foreseen, and even the language of the two sciences shows 
they have been recognized. Thus we speak of transcendent 
numbers and thus we account for the future classification of 
these numbers already having as model the classification of tran- 
scendent functions, and still we do not as yet very well see how 
to pass from one classification to the other; but had it been seen, 
it would already have been accomplished and would no longer 
be the work of the future. 

The first example that comes to my mind is the theory of con- 
gruences, where is found a perfect parallelism to the theory of 
algebraic equations. Surely we shall succeed in completing this 
parallelism, which must hold for instance between the theory of 
algebraic curves and that of congruences with two variables. 
And when the problems relative to congruences with several 
variables shall be solved, this will be a first step toward the solu- 
tion of many questions of indeterminate analysis. 


The theory of algebraic equations will still long hold the atten- 
tion of geometers; numerous and very different are the sides 
whence it may be attacked. 


We need not think algebra is ended because it gives ns roles 
to form all possible combinations ; it remains to find the interest- 
ing combinations, those which satisfy such and such a condition. 
Thus will be formed a sort of indeterminate analysis where the 
unknowns will no longer be whole numbers, but polynomials. 
This time it is algebra which will model itself upon arithmetic, 
following the analogy of the whole number to the integ^ral poly- 
nomial with any coefiScients or to the integral polynomial with 
integral coefiScients. 


It looks as if geometry could contain nothing which is not 
already included in algebra or analysis; that geometric facts are 
only algebraic or analytic facts expressed in another language. 
It might then be thought that after our review there would 
remain nothing more for us to say relating specially to geometry. 
This would be to fail to recognize the importance of well-oon- 
strueted language, not to comprehend what is added to the things 
themselves by the method of expressing these things and conse- 
quently of grouping them. 

First the geometric considerations lead us to set ourselves new 
problems; these may be, if you choose, analytic problems, but 
such as we never would have set ourselves in connection with 
analysis. Analysis profits by them however, as it profits by those 
it has to solve to satisfy the needs of physics. 

A great advantage of geometry lies in the fact that in it the 
senses can come to the aid of thought, and help find the path to 
follow, and many minds prefer to put the problems of analysis 
into geometric form. Unhappily our senses can not carry us very 
far, and they desert us when we wish to soar beyond the classical 
three dimensions. Does this mean that, beyond the restricted 
domain wherein they seem to wish to imprison us, we should 
rely only on pure analysis and that all geometry of more than 
three dimensions is vain and objectless? The greatest masters 
of a preceding generation would have answered *yes* ; to-day we 
are so familiarized with this notion that we can speak of it, even 
in a university course, without arousing too much astonishment 

But what good is it T That is easy to see : First it gives us a 


very convenient terminology, which expresses concisely what the 
ordinary analytic language would say in prolix phrases. More- 
over, this language makes us call like things by the same name 
and emphasize analogies it will never again let us forget. It 
enables us therefore still to find our way in this space which is 
too big for us and which we can not see, always recalling visible 
space, which is only an imperfect image of it doubtless, but which 
is nevertheless an image. Here again, as in all the preceding 
examples, it is analogy with the simple which enables us to com- 
prehend the complex. 

This geometry of more than three dimensions is not a simple 
analytic geometry; it is not purely quantitative, but qualitative 
also, and it is in this respect above all that it becomes interesting. 
There is a science called analysis situs and which has for its 
object the study of the positional relations of the different ele- 
ments of a figure, apart from their sizes. This geometry is purely 
qualitative ; its theorems would remain true if the figures, instead 
of being exact, were roughly imitated by a child. We may also 
make an analysis situs of more than three dimensions. The 
importance of analysis situs is enormous and can not be too much 
emphasized ; the advantage obtained from it by Riemann, one of 
its chief creators, would suffice to prove this. We must achieve 
its complete construction in the higher spaces ; then we shall have 
an instrument which will enable us really to see in hyperspace 
and supplement our senses. 

The problems of analysis situs would perhaps not have sug- 
gested themselves if the analytic language alone had been spoken ; 
or rather, I am mistaken, they would have occurred surely, since 
their solution is essential to a crowd of questions in analysis, but 
they would have come singly, one after another, and without our 
being able to perceive their common bond. 


I have spoken above of our need to go back continually to the 
first principles of our science, and of the advantage of this for 
the study of the human mind. This need has inspired two en- 
deavors which have taken a very prominent place in the most 
recent annals of mathematics. The first is Cantorism, which has 


rendered our science such conspicuous service. Cantor intro- 
duced into science a new way of considering mathematical in- 
finity. One of the characteristic traits of Cantorism is that in 
place of going up to the general by building up constructions 
more and more complicated and defining by construction, it starts 
from the genus supremum and defines only, as the scholastics 
would have said, per genus proximum et differentiam spedficam. 
Thence comes the horror it has sometimes inspired in certain 
minds, for instance in Hermite, whose favorite idea was to com- 
pare the mathematical to the natural sciences. With most of 
us these prejudices have been dissipated, but it has come to 
pass that we have encountered certain paradoxes, certain appar- 
ent contradictions that would have delighted Zeno the Eleatic 
and the school of Megara. And then each must seek the remedy. 
For my part, I think, and I am not the only one, that the impor- 
tant thing is never to introduce entities not completely definable 
in a finite number of words. Whatever be the cure adopted, we 
may promise ourselves the joy of the doctor called in to follow 
a beautiful pathologic case. 

The Investigation of the Postulates 

On the other hand, efforts have been made to enumerate the 
axioms and postulates, more or less hidden, which serve as foun- 
dation to the different theories of mathematics. Professor Hilbert 
has obtained the most brilliant results. It seems at first that this 
domain would be very restricted and there would be nothing 
more to do when the inventory should be ended, which could not 
take long. But when we shall have enumerated all, there will be 
many ways of classifying all ; a good librarian always finds some- 
thing to do, and each new classification will be instructive for 
the philosopher. 

Here I end this review which I could not dream of making 
complete. I think these examples will suffice to show by what 
mechanism the mathematical sciences have made their progress 
in the past and in what direction they must advance in the future. 

Mathematioal Cbeation 

The genesis of mathematical creation is a problem which 
should intensely interest the psychologist. It is the activity in 
which the human mind seems to take least from the outside 
world, in which it acts or seems to act only of itself and on itself, 
so that in studying the procedure of geometric thought we may 
hope to reach what is most essential in man's mind. 

This has long been appreciated, and some time back the journal 
called L^enseignement mathematique, edited by Laisant and 
Fehr, began an investigation of the mental habits and methods 
of work of different mathematicians. I had finished the main 
outlines of this article when the results of that inquiry were 
published, so I have hardly been able to utilize them and shall 
confine myself to saying that the majority of witnesses confirm 
my conclusions; I do not say all, for when the appeal is to uni- 
versal suffrage unanimity is not to be hoped. 

A first fact should surprise us, or rather would surprise us if 
we were not so used to it. How does it happen there are people 
who do not understand mathematics? If mathematics invokes 
only the rules of logic, such as are accepted by all normal minds; 
if its evidence is based on principles common to all men, and that 
none could deny without being mad, how does it come about that 
so many persons are here refractory t 

That not every one can invent is nowise mysterious. That 
not every one can retain a demonstration once learned may also 
pass. But that not every one can understand mathematical 
reasoning when explained appears very surprising when we think 
of it. And yet those who can follow this reasoning only with 
difficulty are in the majority : that is undeniable, and will surely 
not be gainsaid by the experience of secondary-school teachers. 

And further: how is error possible in mathematics t A sane 
mind should not be guilty of a logical fallacy, and yet there are 



Yery fine minds who do not trip in brief reasoning such as oocors 
in the ordinary doings of life, and who are incapable of follow- 
ing or repeating without error the mathematical demonstrations 
which are longer, but which after all are only an acoumnlation 
of brief reasonings wholly analogous to those they make so easily. 
Need we add that mathematicians themselves are not infallible t 

The answer seems to me evident. Imagine a long series of 
syllogisms, and that the conclusions of the first serve as premises 
of the following: we shall be able to catch each of these syllo- 
gisms, and it is not in passing from premises to conclusion that 
we are in danger of deceiving ourselves. But between the 
moment in which we first meet a proposition as conclusion of one 
syllogism, and that in which we reencounter it as premise of 
another syllogism occasionally some time will elapse, several links 
of the chain will have unrolled ; so it may happen that we have 
forgotten it, or worse, that we have forgotten its meaning. So 
it may happen that we replace it by a slightly different propo- 
sition, or that, while retaining the same enunciation, we attribute 
to it a slightly different meaning, and thus it is that we are 
exposed to error. 

Often the mathematician uses a rule. Naturally he begins by 
demonstrating this rule ; and at the time when this proof is fresh 
in his memory he understands perfectly its meaning and its bear- 
ing, and he is in no danger of changing it. But subsequently he 
trusts his memory and afterward only applies it in a mechanical 
way; and then if his memory fails him, he may apply it all 
wrong. Thus it is, to take a simple example, that we sometimes 
make slips in calculation because we have forgotten our multi- 
plication table. 

According to this, the special aptitude for mathematics would 
be due only to a very sure memory or to a prodigious force of 
attention. It would be a power like that of the whist-player who 
remembers the cards played ; or, to go up a step, like that of the 
chess-player who can visualize a great number of combinations 
and hold them in his memory. Every good mathematician ought 
to be a good chess-player, and inversely ; likewise he should be a 
good computer. Of course that sometimes happens; thus Gauss 


was at the same time a geometer of genius and a very precocious 
and accurate computer. 

But there are exceptions; or rather I err; I can not call them 
exceptions without the exceptions being more than the rule. 
Gauss it is, on the contrary, who was an exception. As for my- 
self, I must confess, I am absolutely incapable even of adding 
without mistakes. In the same way I should be but a poor chess- 
player; I would perceive that by a certain play I should expose 
myself to a certain danger; I would pass in review several other 
plays, rejecting them for other reasons, and then finally I should 
make the move first examined, having meantime forgotten the 
danger I had foreseen. 

In a word, my memory is not bad, but it would be insufiScient 
to make me a good chess-player. Why then does it not fail me in 
a difiScult piece of mathematical reasoning where most chess- 
players would lose themselves t Evidently because it is guided 
by the general march of the reasoning. A mathematical demon- 
stration is not a simple juxtaposition of syllogisms, it is syllo- 
gisms placed in a certain order, and the order in which these 
elements are placed is much more important than the elements 
themselves. If I have the feeling, the intuition, so to speak, of 
this order, so as to perceive at a glance the reasoning as a whole, 
I need no longer fear lest I forget one of the elements, for each 
of them will take its allotted place in the array, and that with- 
out any effort of memory on my part. 

It seems to me then, in repeating a reasoning learned, that I 
could have invented it. This is often only an illusion ; but even 
then, even if I am not so gifted as to create it by myself, I my- 
self re-invent it in so far as I repeat it. 

We know that this feeling, this intuition of mathematical 
order, that makes us divine hidden harmonies and relations, can 
not be possessed by every one. Some will not have either this 
delicate feeling so difficult to define, or a strength of memory 
and attention beyond the ordinary, and then they will be abso- 
lutely incapable of understanding higher mathematics. Such are 
the majority. Others will have this feeling only in a slight 
degree, but they will be gifted with an uncommon memory and 
a great power of attention. They will learn by heart the details 


one after another; they can understand mathematics and some- 
times make applications, but they cannot create. Others, finally, 
will possess in a less or greater degree the special intuition 
referred to, and then not only can they understand mathematics 
even if their memory is nothing extraordinary, but they may 
become creators and try to invent with more or less success 
according as this intuition is more or less developed in them. 

In fact, what is mathematical creation t It does not consist 
in making new combinations with mathematical entities already 
known. Any one could do that, but the combinations so made 
would be infinite in number and most of them absolutely with- 
out interest. To create consists precisely in not making useless 
combinations and in making those which are useful and which 
are only a small minority. Invention is discernment, choice. 

How to make this choice I have before explained; the mathe- 
matical facts worthy of being studied are those which, by their 
analogy with other facts, are capable of leading us to the knowl- 
edge of a mathematical law just as experimental facts lead us to 
the knowledge of a physical law. They are those which reveal 
to us unsuspected kinship between other facts, long known, but 
wrongly believed to be strangers to one another. 

Among chosen combinations the most fertile will often be those 
formed of elements drawn from domains which are far apart. 
Not that I mean as suflBcing for invention the bringing together 
of objects as disparate as possible ; most combinations so formed 
would be entirely sterile. But certain among them, very rare, 
are the most fruitful of all. 

To invent, I have said, is to choose ; but the word is perhaps 
not wholly exact. It makes one think of a purchaser before whom 
are displayed a large number of samples, and who examines 
them, one after the other, to make a choice. Here the samples 
would be so numerous that a whole lifetime would not suffice to 
examine them. This is not the actual state of things. The sterile 
combinations do not even present themselves to the mind of the 
inventor. Never in the field of his consciousness do combina- 
tions appear that are not really useful, except some that he rejects 
but which have to some extent the characteristics of useful com- 
binations. All goes on as if the inventor were an examiner for 


the second degree who would only have to question the candi- 
dates who had passed a previous examination. 

But what I have hitherto said is what may be observed or 
inferred in reading the writings of the geometers, reading 

It is time to penetrate deeper and to see what goes on in the 
very soul of the mathematician. For this, I believe, I can do best 
by recalling memories of my own. But I shall limit myself to 
telling how I wrote my first memoir on Fuchsian functions. I 
beg the reader's pardon ; I am about to use some technical expres- 
sions, but they need not frighten him, for he is not obliged to 
understand them. I shall say, for example, that I have found 
the demonstration of such a theorem under such circumstances. 
This theorem will have a barbarous name, unfamiliar to many, 
but that is unimportant ; what is of interest for the psychologist 
is not the theorem but the circumstances. 

For fifteen days I strove to prove that there could not be any 
functions like those I have since called Fuchsian functions. I 
was then very ignorant; every day I seated myself at my work 
table, stayed an hour or two, tried a great number of combina- 
tions and reached no results. One evening, contrary to my 
custom, I drank black coffee and could not sleep. Ideas rose in 
crowds; I felt them collide until pairs interlocked, so to speak, 
making a stable combination. By the next morning I had estab- 
lished the existence of a class of Fuchsian functions, those which 
come from the hypergeometric series; I had only to write out 
the results, which took but a few hours. 

Then I wanted to represent these functions by the quotient of 
two series; this idea was perfectly conscious and deliberate, the 
analogy with elliptic functions guided me. I asked myself what 
properties these series must have if they existed, and I succeeded 
without diflBculty in forming the series I have called theta- 

Just at this time I left Caen, where I was then living, to go on 
a geologic excursion under the auspices of the school of mines. 
The changes of travel made me forget my mathematical work. 
^Having reached Coutances, we entered an omnibus to go some 
place or other. At the moment when I put my foot on the step 


the idea came to me, without anything in my former thoughts 
seeming to have paved the way for it, that the transformations 
I had used to define the Fuchsian functions were identical with 
those of non-Euclidean geometry. I did not verify the idea; I 
should not have had time, as, upon taking my seat in the omni- 
bus, I went on with a conversation already commenced^ but I 
felt a perfect certainty. On my return to Caen, for conscience' 
sake I verified the result at my leisure. 

Then I turned my attention to the study of some arithmetical 
questions apparently without much success and without a sus- 
picion of any connection with my preceding researches. Dis- 
gusted with my failure, I went to spend a few days at the sea- 
side, and thought of something else. One morning, walking on 
the bluff, the idea came to me, with just the same characteristics 
of brevity, suddenness and immediate certainty, that the arith- 
metic transformations of indeterminate ternary quadratic forms 
were identical with those of non-Euclidean geometry. 

Returned to Caen, I meditated on this result and deduced the 
consequences. The example of quadratic forms showed me that 
there were Fuchsian groups other than those corresponding to 
the hypergeometric series ; I saw that I could apply to them the 
theory of theta-Fuehsian series and that consequently there 
existed Fuchsian functions other than those from the hyper- 
geometric series, the ones I then knew. Naturally I set my- 
self to form all these functions. I made a systematic attack upon 
them and carried all the outworks, one after another. There was 
one however that still held out, whose fall would involve that of 
the whole place. But all my efforts only served at first the better 
to show me the diflSculty, which indeed was something. All this 
work was perfectly conscious. 

Thereupon I left for Mont-Val6rien, where I was to go through 
my military service; so I was very differently occupied. One 
day, going along the street, the solution of the diflSculty which 
had stopped me suddenly appeared to me. I did not try to go 
deep into it immediately, and only after my service did I again 
take up the question. I had all the elements and had only to 
arrange them and put them together. So I wrote out my final 
memoir at a single stroke and without diflSculty. 



I shall limit myself to this single example; it is useless to 
multiply them. In regard to my other researches I would have 
to say analogous things, and the observations of other mathe- 
maticians given in L'enseignement matkimaHqxie would od17 
conUrm them. 

Most striking at first is this appearance of sudden illumina- 
tion, a manifest sign of long, unconscious prior work. The role 
of this unconscious work in mathematical invention appears to 
me incontestable, and traces of it would be found in other cases 
where it is less evident. Often when one works at a hard ques- 
tion, nothing good is accomplished at the first attack. Then 
one takes a rest, longer or shorter, and sita down anew to the 
work, Durinfj the first half-hour, as before, nothing is found, 
and then all of a sudden the decisive idea presents itself to the 
mind. It might be said that the conscious work has been more 
fruitful because it has been interrupted and the rest has given 
back to the mind its force and freshness. But it is more prob- 
able that this rest has been filled out with unconscious work and 
that the result of this work has afterward revealed itself to the 
geometer just as in the eases I have cited ; only the revelation, 
instead of coming during a walk or a journey, has happened 
during a period of conscious work, but independently of this 
work which plays at most a role of excitant, as if it were the goad 
stimulating the results already reached during rest, but remain- 
ing unconscious, to assume tbe conscious form. 

There is another remark to be made about the conditions of 
this unconscious work: it is possible, and of a certainty it is only 
fruitful, if it is on the one hand preceded and on the other hand 
followed by a period of conscious work. These sudden inspira- 
tions (and the examples already cited sufficiently prove this) 
never happen except after some days of voluntary effort which 
has appeared absolutely fruitless and whence nothing good scemn 
to have come, where the way taken seems totally astray. These 
efforts then have not been as sterile as one thinks; they have set 
agoing the unconscious machine and without them it would not 
have moved and would have produced nothing. 

The need for tlie second period of conscious work, after the 
inspiration, is atiU easier to nnderstand. It is neceBBary to put 


in shape the results of this inspiration, to deduce from them the 
immediate consequences, to arrange them, to word the demonstra- 
tions, but above all is verification necessary. I have spoken of 
the feeling of absolute certitude accompanying the inspiration; 
in the cases cited this feeling was no deceiver, nor is it usually. 
But do not think this a rule without exception ; often this feeling 
deceives us without being any the less vivid, and we only find it 
out when we seek to put on foot the demonstration. I have 
especially noticed this fact in regard to ideas coming to me in the 
morning or evening in bed while in a semi-hypnagogic state. 

Such are the realities; now for the thoughts they force upon 
us. The unconscious, or, as we say, the subliminal self plays an 
important role in mathematical creation ; this follows from what 
we have said. But usually the subliminal self is considered as 
purely automatic. Now we have seen that mathematical work is 
not simply mechanical, that it could not be done by a machine, 
however perfect. It is not merely a question of applying rules, 
of making the most combinations possible according to certain 
fixed laws. The combinations so obtained would be exceedingly 
numerous, useless and cumbersome. The true work of the in- 
ventor consists in choosing among these combinations so as to 
eliminate the useless ones or rather to avoid the trouble of mak- 
ing them, and the rules which must guide this choice are extremely 
fine and delicate. It is almost impossible to state them precisely; 
they are felt rather than formulated. Under these conditions, 
how imagine a sieve capable of applying them mechanically! 

A first hypothesis now presents itself : the subliminal self is in 
no way inferior to the conscious self; it is not purely automatic; 
it is capable of discernment ; it has tact, delicacy ; it knows how 
to choose, to divine. What do I say? It knows better how to 
divine than the conscious self, since it succeeds where that has 
failed. In a word, is not the subliminal self superior to the 
conscious self? You recognize the full importance of this ques- 
tion. Boutroux in a recent lecture has shown how it came up 
on a very different occasion, and what consequences would follow 
an affirmative answer. (See also, by the same author. Science 
et Religion, pp. 313 ff.) 

Is this affirmative answer forced upon us by the facts I have 


just given t I confess that, for my part, I should hate to accept 
it Reexamine the facts then and see if they are not compatible 
with another explanation. 

It is certain that the combinations which present themselves to 
the mind in a sort of sadden illumination, after an unconscious 
working somewhat prolonged, are generally useful and fertile 
combinations, which seem the result of a first impression. Does 
it follow that the subliminal self, having divined by a delicate 
intuition that these combinations would be useful, has formed 
only these, or has it rather formed many others which were 
lacking in interest and have remained unconscious t 

In this second way of looking at it, all the combinations would 
be formed in consequence of the automatism of the subliminal 
self, but only the interesting ones would break into the domain 
of consciousness. And this is still very mysterious. What is the 
cause that, among the thousand products of our unconscious 
activity, some are called to pass the threshold, while others remain 
below t Is it a simple chance which confers this privilege t Evi- 
dently not ; among all the stimuli of our senses, for example, only 
the most intense fix our attention, unless it has been drawn to 
them by other causes. More generally the privileged uncon- 
scious phenomena, those susceptible of becoming conscious, are 
those which, directly or indirectly, affect most profoundly our 
emotional sensibility. 

It may be surprising to see emotional sensibility invoked 
d propos of mathematical demonstrations which, it would seem, 
can interest only the intellect. This would be to forget the feel- 
ing of mathematical beauty, of the harmony of numbers and 
forms, of geometric elegance. This is a true esthetic feeling that 
all real mathematicians know, and surely it belongs to emo- 
tional sensibility. 

Now, what are the mathematic entities to which we attribute 
this character of beauty and elegance, and which are capable of 
developing in us a sort of esthetic emotion t They are thoae 
whose elements are harmoniously disposed so that the mind with- 
out effort can embrace their totality while realizing the details. 
This harmony is at once a satisfaction of our esthetic needs and 
an aid to the mind, sustaining and guiding. And at the same 


time, in putting under our eyes a well-ordered whole, it makes 
us foresee a mathematical law. Now, as we have said above, the 
only mathematical facts worthy of fixing our attention and 
capable of being useful are those which can teach us a mathe- 
matical law. So that we reach the following conclusion: The 
useful combinations are precisely the most 'beautiful, I mean 
those best able to charm this special sensibility that all mathe- 
maticians know, but of which the profane are so ignorant as 
often to be tempted to smile at it. 

What happens then t Among the great numbers of combina- 
tions blindly formed by the subliminal self, almost all are without 
interest and without utility; but just for that reason they are 
also without effect upon the esthetic sensibility. Consciousness 
will never know them; only certain ones are harmonious, and, 
consequently, at once useful and beautiful. They will be capable 
of touching this special sensibility of the geometer of which I 
have just spoken, and which, once aroused, will call our atten- 
tion to them, and thus give them occasion to become conscious. 

This is only a hypothesis, and yet here is an observation which 
may confirm it: when a sudden illumination seizes upon the 
mind of the mathematician, it usually happens that it does not 
deceive him, but it also sometimes happens, as I have said, that 
it does not stand the test of verification ; well, we almost always 
notice that this false idea, had it been true, would have gratified 
our natural feeling for mathematical elegance. 

Thus it is this special esthetic sensibility which plays the role 
of the delicate sieve of which I spoke, and that suflSciently ex- 
plains why the one lacking it will never be a real creator. 

Yet all the diflBculties have not disappeared. The conscious 
self is narrowly limited, and as for the subliminal self we know 
not its limitations, and this is why we are not too reluctant in 
supposing that it has been able in a short time to make more 
different combinations than the whole life of a conscious being 
could encompass. Yet these limitations exist. Is it likely that 
it is able to form all the possible combinations, whose number 
would frighten the imagination t Nevertheless that would seem 
necessary, because if it produces only a small part of these com- 
binations, and if it makes them at random, there would be small { 



chance that the good, the one we shotQd choose, would be found 
among them. 

Perhaps we ought to seek the explanation in that preliminary 
period of conscious work which always precedes all fruitful 
unconscious labor. Permit me a rough comparison. Figure 
the future elements of our combinations as something like the 
hooked atoms of Epicurus. During the complete repose of the 
mind, these atoms are motionless, they are, so to speak, hooked 
to the wall ; so this complete rest may be indefinitely prolonged 
without the atoms meeting, and consequently without any com- 
bination between them. 

On the other hand, during a period of apparent rest and 
unconscious work, certain of them are detached from the wall and 
put in motion. They flash in every direction through the space 
(I was about to say the room) where they are enclosed, as would, 
for example, a swarm of gnats or, if you prefer a more learned 
comparison, like the molecules of gas in the kinematic theory of 
gases. Then their mutual impacts may produce new combinations. 

What is the role of the preliminary conscious workt It is 
evidently to mobilize certain of these atoms, to unhook them from 
the wall and put them in swing. We think we have done no 
good, because we have moved these elements a thousand different 
ways in seeking to assemble them, and have found no satisfactory 
aggregate. But, after this shaking up imposed upon them by our 
will, these atoms do not return to their primitive rest. They 
freely continue their dance. 

Now, our will did not choose them at random; it pursued a 
perfectly determined aim. The mobilized atoms are therefore 
not any atoms whatsoever; they are those from which we might 
reasonably expect the desired solution. Then the mobilized atoms 
undergo impacts which make them enter into combinations among 
themselves or with other atoms at rest which they struck against 
in their course. Again I beg pardon, my comparison is very 
rough, but I scarcely know how otherwise to make my thought 

However it may be, the only combinations that have a chance 
of forming are those where at least one of the elements is one 
of those atoms freely chosen by our will. Now, it is evidently 


among these that is found what I called the good combination. 
Perhaps this is a way of lessening the paradoxical in the original 

Another observation. It never happens that the unconscious 
work gives us the result of a somewhat long calculation all made, 
where we have only to apply fixed rules. We might think the 
wholly automatic subliminal self particularly apt for this sort of 
work, which is in a way exclusively mechanical. It seems that 
thinking in the evening upon the factors of a multiplication we 
might hope to find the product ready made upon our awaken- 
ing, or again that an algebraic calculation, for example a veri- 
fication, would be made unconsciously. Nothing of the sort, as 
observation proves. All one may hope from these inspirations, 
fruits of unconscious work, is a point of departure for such calcu- 
lations. As for the calculations themselves, they must be made 
in the second period of conscious work, that which follows the 
inspiration, that in which one verifies the results of this inspira- 
tion and deduces their consequences. The rules of these calcu- 
lations are strict and complicated. They require discipline, atten- 
tion, will, and therefore consciousness. In the subliminal self, on 
the contrary, reigns what I should call liberty, if we might give 
this name to the simple absence of discipline and to the disorder 
bom of chance. Only, this disorder itself permits unexpected 

I shall make a last remark : when above I made certain personal 
observations, I spoke of a night of excitement when I worked in 
spite of myself. Such cases are frequent, and it is not necessary 
that the abnormal cerebral activity be caused by a physical exci- 
tant as in that I mentioned. It seems, in such cases, that one is 
present at his own unconscious work, made partially perceptible 
to the over-excited consciousness, yet without having changed its 
nature. Then we vaguely comprehend what distinguishes the 
two mechanisms or, if you wish, the working methods of the two 
egos. And the psychologic observations I have been able thus 
to make seem to me to confirm in their general outlines the views 
I have given. 

Surely they have need of it, for they are and remain in spite 
of all very hypothetical : the interest of the questions is so great 
that I do not repent of having submitted them to the reader. 



''How dare we speak of the laws of chance t Is not chance 
the antithesis of all lawt" So says Bertrand at the beginning of 
his Calcul des probabilitSs. Probability is opposed to certitude ; 
so it is what we do not know and consequently it seems what we 
could not calculate. Here is at least apparently a contradiction, 
and about it much has already been written. 

And first, what is chance t The ancients distinguished between 
phenomena seemingly obeying harmonious laws, established once 
for all, and those which they attributed to chance; these were 
the ones unpredictable because rebellious to all law. In each 
domain the precise laws did not decide everything, they only 
drew limits between which chance might act. In this conception 
the word chance had a precise and objective meaning : what was 
chance for one was also chance for another and even for the gods. 

But this conception is not ours to-day. We have become abso- 
lute determinists, and even those who want to reserve the rights 
of human free will let determinism reign undividedly in the inor- 
ganic world at least. Every phenomenon, however minute, has 
a cause ; and a mind infinitely powerful, infinitely well-informed 
about the laws of nature, could have foreseen it from the begin- 
ning of the centuries. If such a mind existed, we could not play 
with it at any game of chance ; we should always lose. 

In fact for it the word chance would not have any meaning, 
or rather there would be no chance. It is because of our weak- 
ness and our ignorance that the word has a meaning for us. And, 
even without going beyond our feeble humanity, what is chance 
for the ignorant is not chance for the scientist. Chance is only 
the measure of our ignorance. Fortuitous phenomena are, by 
definition, those whose laws we do not know. 

But is this definition altogether satisfactory t When the first 



Chaldean shepherds followed with their eyes the movements of 
the stars, they knew not as yet the laws of astronomy ; would ih^ 
have dreamed of saying that the stars move at random t If a 
modem physicist studies a new phenomenon, and if he discovers 
its law Tuesday, would he have said Monday that this phenom- 
enon was fortuitous t Moreover, do we not often invoke what 
Bertrand calls the laws of chance, to predict a phenomenon t 
For example, in the kinetic theory of gases we obtain the known 
laws of Mariotte and of Gay-Lussac by means of the hypothesis 
that the velocities of the molecules of gas vary irregularly, that 
is to say at random. All physicists will agree that the observable 
laws would be much less simple if the velocities were ruled by 
any simple elementary law whatsoever, if the molecules were, 
as we say, organized, if they were subject to some discipline. It 
is due to chance, that is to say, to our ignorance, that we can draw 
our conclusions ; and then if the word chance is simply synony- 
mous with ignorance what does that meant Must we therefore 
translate as follows! 

"You ask me to predict for you the phenomena about to 
happen. If, unluckily, I knew the laws of these phenomena I 
could make the prediction only by inextricable calculations and 
would have to renounce attempting to answer you ; but as I have 
the good fortune not to know them, I will answer you at once. 
And what is most surprising, my answer will be right.'' 

So it must well be that chance is something other than the 
name we give our ignorance, that among phenomena whose 
causes are unknown to us we must distinguish fortuitous phe- 
nomena about which the calculus of probabilities will provision- 
ally give information, from those which are not fortuitous and of 
which we can say nothing so long as we shall not have determined 
the laws governing them. For the fortuitous phenomena them- 
selves, it is clear that the information given us by the calculus 
of probabilities will not cease to be true upon the day when these 
phenomena shall be better known. 

The director of a life insurance company does not know when 
each of the insured will die, but he relies upon the calculus of 
probabilities and on the law of great numbers, and he is not 
deceived, since he distributes dividends to his stockholders. These 


dividends would not vanish if a very penetrating and very indis- 
crete physician should, after the policies were signed, reveal to 
the director the life chances of the insured. This doctor would 
dissipate the ignorance of the director, but he would have no 
influence on the dividends, which evidently are not an outcome 
of this ignorance. 


To find a better definition of chance we must examine some of 
the facts which we agree to regard as fortuitous, and to which 
the calculus of probabilities seems to apply ; we then shall investi- 
gate what are their common characteristics. 

The first example we select is that of unstable equilibrium ; if 
a cone rests upon its apex, we know well that it will fall, but we 
do not know toward what side ; it seems to us chance alone will 
decide. If the cone were perfectly symmetric, if its axis were 
perfectly vertical, if it were acted upon by no force other than 
gravity, it would not fall at all. But the least defect in symmetry 
will make it lean slightly toward one side or the other, and if it 
leans, however little, it will fall altogether toward that side. 
Even if the symmetry were perfect, a very slight tremor, a breath 
of air could make it incline some seconds of arc; this will be 
enough to determine its fall and even the sense of its fall which 
will be that of the initial inclination. 

A very slight cause, which escapes us, determines a consider- 
able effect which we can not help seeing, and then we say this 
effect is due to chance. If we could know exactly the laws of 
nature and the situation of the universe at the initial instant, 
we should be able to predict exactly the situation of this same 
universe at a subsequent instant. But even when the natural 
laws should have no further secret for us, we could know the 
initial situation only approxinuUely. If that permits us to fore- 
see the subsequent situation tviih the same degree of approxima- 
tion, this is all we require, we say the phenomenon has been 
predicted, that it is ruled by laws. But this is not always the 
case ; it may happen that slight differences in the initial condi- 
tions produce very great differences in the final phenomena; a 
slight error in the former would make an enormous error in the 


latter. Prediction becomes impossible and we have the fortuitous 

Our second example will be very analogous to the first and we 
shall take it from meteorology. Why have the meteorologists such 
difficulty in predicting the weather with any certainty! Why 
do the rains, the tempests themselves seem to us to come by 
chance, so that many persons find it quite natural to pray for 
rain or shine, when they would think it ridiculous to pray for 
an eclipse t We see that great perturbations generally happen in 
regions where the atmosphere is in unstable equilibrium. The 
meteorologists are aware that this equilibrium is unstable, that a 
cyclone is arising somewhere; but where they can not tell; one- 
tenth of a degree more or less at any point, and the cyclone 
bursts here and not there, and spreads its ravages over countries 
it would have spared. This we could have foreseen if we had 
known that tenth of a degree, but the observations were neither 
sufficiently close nor sufficiently precise, and for this reason all 
seems due to the agency of chance. Here again we find the same 
contrast between a very slight cause, unappreciable to the ob- 
server, and important effects, which are sometimes tremendous 

Let us pass to another example, the distribution of the minor 
planets on the zodiac. Their initial longitudes may have been 
any longitudes whatever ; but their mean motions were different 
and they have revolved for so long a time that we may say they 
are now distributed at random along the zodiac. Very slight 
initial differences between their distances from the sun, or, what 
comes to the same thing, between their mean motions, have 
ended by giving enormous differences between their present 
longitudes. An excess of the thousandth of a second in the daily 
mean motion will give in fact a second in three years, a degree 
in ten thousand years, an entire circumference in three or four 
million years, and what is that to the time which has passed since 
the minor planets detached themselves from the nebula of 
Laplace? Again therefore we see a slight cause and a great 
effect ; or better, slight differences in the cause and great differ- 
ences in the effect. 

The game of roulette does not take us as far as might seem 


from the preceding example. Assume a needle to be turned on a 
pivot over a dial divided into a hundred sectors alternately red 
and black. If it stops on a red sector I win ; if not, I lose. Evi-: 
dently all depends upon the initial impulse I give the needle. 
The needle will make, suppose, ten or twenty turns, but it will 
stop sooner or not so soon, according as I shall have pushed it 
more or less strongly. It suffices that the impulse vary only by 
a thousandth or a two thousandth to make the needle stop over a 
black sector or over the following red one. These are differences 
the muscular sense can not distinguish and which elude even the 
most delicate instruments. So it is impossible for me to foresee 
what the needle I have started will do, and this is why my heart 
throbs and I hope ever3rthing from luck. The difference in the 
cause is imperceptible, and the difference in the effect is for me 
of the highest importance, since it means my whole stake. 


Permit me, in this connection, a thought somewhat foreign to 
my subject. Some years ago a philosopher said that the future 
is determined by the past, but not the past by the future ; or, in 
other words, from knowledge of the present we could deduce the 
future, but not the past ; because, said he, a cause can have only 
one effect, while the same effect might be produced by several 
different causes. It is clear no scientist can subscribe to this 
conclusion. The laws of nature bind the antecedent to the conse- 
quent in such a way that the antecedent is as well determined by 
the consequent as the consequent by the antecedent. But whence 
came the error of this philosopher t We know that in virtue of 
Carnot's principle physical phenomena are irreversible and the 
world tends toward uniformity. When two bodies of different 
temperature come in contact, the warmer gives up heat to the 
colder; so we may foresee that the temperature will equalize. 
But once equal, if asked about the anterior state, what can we 
answer? We might say that one was warm and the other cold, 
but not be able to divine which formerly was the warmer. 

And yet in reality the temperatures will never reach perfect 
equality. The difference of the temperatures only tends asymp- 
totically toward zero. There comes a moment when our ther- 


mometers are powerless to make it known. But if we had ther- 
mometers a thousand times, a hundred thousand times as sensi- 
tive, we should recognize that there still is a slight difference, and 
that one of the bodies remains a little warmer than the other, and 
so we could say this it is which formerly was much the warmer. 

So then there are, contrary to what we found in the former 
examples, great differences in cause and slight differences in 
effect. Flammarion once imagined an observer going away from 
the earth with a velocity greater than that of light; for him time 
would have changed sign. History would be turned about, and 
Waterloo would precede Austerlitz. Well, for this observer, 
effects and causes would be inverted ; unstable equilibrium would 
no longer be the exception. Because of the universal irreversi- 
bility, all would seem to him to come out of a sort of chaos in 
unstable equilibrium. All nature would appear to him delivered 
over to chance. 


Now for other examples where we shall see somewhat different 
characteristics. Take first the kinetic theory of gases. How 
should we picture a receptacle filled with gast Innumerable 
molecules, moving at high speeds, flash through this receptacle 
in every direction. At every instant they strike against its walls 
or each other, and these collisions happen under the most diverse 
conditions. What above all impresses us here is not the little- 
ness of the causes, but their complexity, and yet the former ele- 
ment is still found here and plays an important role. If a mole- 
cule deviated right or left from its trajectory, by a very small 
quantity, comparable to the radius of action of the gaseous mole- 
cules, it would avoid a collision or sustain it under different con- 
ditions, and that would vary the direction of its velocity after 
the impact, perhaps by ninety degrees or by a hundred and 
eighty degrees. 

And this is not all; we have just seen that it is necessary to 
deflect the molecule before the clash by only an infinitesimal, to 
produce its deviation after the collision by a finite quantity. If 
then the molecule undergoes two successive shocks, it will suflSce 
to deflect it before the first by an infinitesimal of the second 
order, for it to deviate after the first encounter by an infinites- 



imal of the first order, and after the second hit, by a finite quan- 
tity. And the molecule will not undergo merely two shocks; it 
will undergo a very great number per second. So that if the 
first shock has multiplied the deviation by a very large number 
A, after n shocks it will be multiplied by A^. It will therefore 
become very great not merely because A is large, that is to say 
because little causes produce big effects, but because the exponent 
n is large, that is to say because the shocks are very numerous 
and the causes very complex. 

Take a second example. Why do the drops of rain in a 
shower seem to be distributed at random t This is again because 
of the complexity of the causes which determine their formation. 
Ions are distributed in the atmosphere. For a long while they 
have been subjected to air-currents constantly changing, they 
have been caught in very small whirlwinds, so that their final 
distribution has no longer any relation to their initial distribu- 
tion. Suddenly the temperature falls, vapor condenses, and each 
of these ions becomes the center of a drop of rain. To know 
what will be the distribution of these drops and how many will 
fall on each paving-stone, it would not be sufBcient to know the 
initial situation of the ions, it would be necessary to compute 
the effect of a thousand little capricious air-currents. 

And again it is the same if we put grains of powder in sus- 
pension in water. The vase is ploughed by currents whose law 
we know not, we only know it is very complicated. At the 
end of a certain time the grains will be distributed at random, 
that is to say uniformly, in the vase ; and this is due precisely to 
the complexity of these currents. If they obeyed some simple 
law, if, for example the vase revolved and the currents circulated 
around the axis of the vase, describing circles, it would no 
longer be the same, since each grain ^ ould retain its initial alti- 
tude and its initial distance from the axis. 

We should reach the same result in considering the mixing of 
two liquids or of two fine-grained powders. And to take a 
grosser example, this is also what happens when we shuflSe play- 
ing-cards. At each stroke the cards undergo a permutation 
(analogous to that studied in the theory of substitutions). What 
will happen? The probability of a particular permutation (for 


example, that bringing to the nth place the eard occupying the 
^(n}th place before the permutation} dependa upon the player's 
habits. But if this player shuflSes the cards long enough, there 
will be a great number of successive permutations, and the re- 
sulting final order will no longer be governed by aught but 
chance; I mean to say that all possible orders will be equally 
probable. It is to the great number of successive permutations, 
that is to say to the complexity of the phenomenon, that this 
result is due. 

A final word about the theory of errors. Here it is that the 
causes are complex and multiple. To how many snares is not 
the observer exposed, even with the best instrument 1 He should 
apply himself to finding out the largest and avoiding them. 
These are the ones giving birth to gystematic errors. But when 
he has eliminated those, admitting that he succeeds, there remain 
many small ones which, their effects accumulating, may be- 
come dangerous. Thence come the accidental errors; and we at- 
tribute them to diance because their causes are too complicated 
and too numerous. Here again we have only little causes, but 
each of them would produce only a slight effect; it is by their 
union and their number that their effects become formidable. 


We may take still a third point of view, less important than 
the first two and upon which I shall lay less stress. When we 
seek to foresee an event and examine its antecedents, we strive 
to search into the anterior situation. This could not be done for 
all parts of the universe and we are content to know what is 
passing in the neighborhood of the point where the event should 
occur, or what would appear to have some relation to it. An 
examination can not be complete and we must know how to choose. 
But it may happen that we have passed by circumstances which 
at first sight seemed completely foreign to the foreseen happen- 
ing, to which one would never have dreamed of attributing any 
influence and which nevertheless, contrary to all anticipation, 
come to play an important role. 

A man passes in the street going to his business; some one 
knowing the business could have told why he started at such a 


time and went by such a street. On the roof works a tiler. 
The contractor employing him could in a certain measure fore- 
see what he would do. But the passer-by scarcely thinks of the 
tiler, nor the tiler of him; they seem to belong to two worlds 
completely foreign to one another. And yet the tiler drops a 
tile which kills the man, and we do not hesitate to say this is 

Our weakness forbids our considering the entire universe 
and makes us cut it up into slices. We try to do this as little 
artificially as possible. And yet it happens from time to time 
that two of these slices react upon each other. The effects 
of this mutual action then seem to us to be due to chance. 

Is this a third way of conceiving chance t Not always; in 
fact most often we are carried back to the first or the second. 
Whenever two worlds usually foreign to one another come thus 
to react upon each other, the laws of this reaction must be very 
complex. On the other hand, a very slight change in the initial 
conditions of these two worlds would have been su£Bcient for the 
reaction not to have happened. How little was needed for the 
man to pass a second later or the tiler to drop his tile a second 


All we have said still does not explain why chance obeys laws. 
Does the fact that the causes are slight or complex suffice for 
our foreseeing, if not their effects in each case, at least what their 
effects will be, on the average? To answer this question we had 
better take up again some of the examples already cited. 

I shall begin with that of the roulette. I have said that the 
point where the needle will stop depends upon the initial push 
given it. What is the probability of this push having this or 
that value? I know nothing about it, but it is difficult for me 
not to suppose that this probability is represented by a continuous 
analytic function. The probability that the push is comprised 
between a and a + « will then be sensibly equal to the probability 
of its being comprised between a + e and a + 2€, provided € be 
very S7nall. This is a property common to all analytic functions. 
Minute variations of the function are proportional to minute 
variations of the variable. 


But we have assumed that an exceedingly slight variation of 
the push suffices to change the color of the sector over which the 
needle finally stops. From a to a + e it is red, from a + e to 
a -|- 2€ it is black ; the probability of each red sector is therefore 
the same as of the following black, and consequently the total 
probability of red equals the total probability of black. 

The datum of the question is the analytic function representing 
the probability of a particular initial push. But the theorem 
remains true whatever be this datum, since it depends upon a 
property common to all analytic functions. From this it follows 
finally that we no longer need the datum. 

What we have just said for the case of the roulette applies 
also to the example of the minor planets. The zodiac may be 
regarded as an immense roulette on which have been tossed many 
little balls with different initial impulses varying according to 
some law. Their present distribution is uniform and independ- 
ent of this law, for the same reason as in the preceding case. 
Thus we see why phenomena obey the laws of chance when 
slight differences in the causes suffice to bring on great differences 
in the effects. The probabilities of these slight differences may 
then be regarded as proportional to these differences themselves, 
just because these differences are minute, and the infinitesimal 
increments of a continuous function are proportional to those of 
the variable. 

Take an entirely different example, where intervenes especially 
the complexity of the causes. Suppose a player shuffles a pack 
of cards. At each shuffle he changes the order of the cards, and 
he may change them in many ways. To simplify the exposition, 
consider only three cards. The cards which before the shuffle 
occupied respectively the places 123, may after the shuffle occupy 
the places 

123, 231, 312, 321, 132, 213. 

Each of these six hypotheses is possible and they have respec- 
tively for probabilities : 

Vu Vzt Ps, P<y Pb, Pa- 

The sum of these six numbers equals 1 ; but this is all we know 
of them ; these six probabilities depend naturally upon the habits 
of the player which we do not know. 


At the second shuffle and the following, this will recommence, 
and under the same conditions ; I mean that p^ for example rep- 
resents always the probability that the three cards which occu- 
pied after the nth shuffle and before the n -{- 1th the places 123, 
occupy the places 321 after the n-|-lth shuffle. And this re- 
mains true whatever be the number n, since the habits of the 
player and his way of shuffling remain the same. 

But if the number of shuffles is very great, the cards which 
before the first shuffle occupied the places 123 may, after the 
last shuffle, occupy the places 

123, 231, 312, 321, 132, 213 

and the probability of these six hypotheses will be sensibly the 
same and equal to 1/6; and this will be true whatever be the 
numbers Pi • • • Pa which we do not know. The gn>^at num- 
ber of shuffles, that is to say the complexity of the causes, has 
produced uniformity. 

This would apply without change if there were more than 
three cards, but even with three cards the demonstration would 
be complicated ; let it suffice to give it for only two cards. Then 
we have only two possibilities 12, 21 with the probabilities p^ and 

V2 = 1 — Pi- 

Suppose n shuffles and suppose I win one franc if the cards 

are finally in the initial order and lose one if they are finally 

inverted. Then, my mathematical expectation will be (Pi — P2)*- 

The difference pj — pj is certainly less than 1; so that if n 
is very great my expectation will be zero; we need not learn p^ 
and P2 to be aware that the game is equitable. 

There would always be an exception if one of the numbers 
Pi and P2 was equal to 1 and the other naught. Then it would 
not apply because our initial hypotheses would he too simple. 

What we have just seen applies not only to the mixing of 
cards, but to all mixings, to those of powders and of liquids; 
and even to those of the molecules of gases in the kinetic theory 
of gases. 

To return to this theory, suppose for a moment a gas whose 
molecules can not mutually clash, but may be deviated by hitting 
the insides of the vase wherein the gas is confined. If the form 


of the vase is sufficiently complex the distribution of the mole- 
cules and that of the velocities will not be long in becoming uni- 
form. But this will not be so if the vase is spherical or if it 
has the shape of a cuboid. Whyf Because in the first case the 
distance from the center to any trajectory will remain constant; 
in the second case this will be the absolute value of the angle of 
each trajectory with the faces of the cuboid. 

So we see what should be understood by conditions too simple; 
they are those which conserve something, which leave an invariant 
remaining. Are the differential equations of the problem too 
simple for us to apply the laws of chance 1 This question would 
seem at first view to lack precise meaning ; now we know what it 
means. They are too simple if they conserve something, if they 
admit a uniform integral. If something in the initial conditions 
remains unchanged, it is clear the final situation can no longer 
be independent of the initial situation. 

We come finally to the theory of errors. We know not to 
what are due the accidental errors, and precisely because we do 
not know, we are aware they obey the law of Gauss. Such is the 
paradox. The explanation is nearly the same as in the preceding 
cases. We need know only one thing: that the errors are very 
numerous, that they are very slight, that each may be as well 
negative as positive. What is the curve of probability of each 
of them? We do not know; we only suppose it is symmetric. 
We prove then that the resultant error will follow Gauss's law, 
and this resulting law is independent of the particular laws 
which we do not know. Here again the simplicity of the result 
is born of the very complexity of the data. 


But we are not through with paradoxes. I have just recalled 
the figment of Flammarion, that of the man going quicker than 
light, for whom time changes sign. I said that for him all phe- 
nomena would seem due to chance. That is true from a certain 
point of view, and yet all these phenomena at a given moment 
would not be distributed in conformity with the laws of chance, 
since the distribution would be the same as for us, who, seeing 
them unfold harmoniously and without coming out of a primal 
chaos, do not regard them as ruled by chance. 


What does that meant For Lumen, Flammarion's man, slight 
causes seem to produce great effects ; why do not things go on as 
for us when we think we see grand effects due to little causes t 
Would not the same reasoning be applicable in his caset 

Let us return to the argument. When slight differences in the 
causes produce vast differences in the effects, why are these effects 
distributed according to the laws of chance f Suppose a differ- 
ence of a millimeter in the cause produces a difference of a kilo- 
meter in the effect. If I win in case the effect corresponds to a 
kilometer bearing an even number, my probability of winning 
will be 1/2. Why f Because to make that, the cause must corre- 
spond to a millimeter with an even number. Now, according to 
all appearance, the probability of the cause varying between 
certain limits will be proportional to the distance apart of these 
limits, provided this distance be very small. If this hypothesis 
were not admitted there would no longer be any way of repre- 
senting the probability by a continuous function. 

What now will happen when great causes produce small 
effects ? This is the case where we should not attribute the phe- 
nomenon to chance and where on the contrary Lumen would 
attribute it to chance. To a difference of a kilometer in the 
cause would correspond a difference of a millimeter in the effect. 
Would the probability of the cause being comprised between two 
limits n kilometers apart still be proportional to n? We have 
no reason to suppose so, since this distance, n kilometers, is 
great. But the probability that the effect lies between two 
limits n millimeters apart will be precisely the same, so it will not 
be proportional to n, even though this distance, n millimeters, 
be small. There is no way therefore of representing the law of 
probability of effects by a continuous curve. This curve, un- 
derstand, may remain continuous in the analytic sense of the 
word; to infinitesimal variations of the abscissa will correspond 
infinitesimal variations of the ordinate. But practically it will 
not be continuous, since very small variations of the ordinate 
would not correspond to very small variations of the abscissa. It 
would become impossible to trace the curve with an ordinary 
pencil ; that is what I mean. 

So what must we conclude? Lumen has no right to say that 


the probability of the cause {his cause, our effect) ahould be 
represented necessarily by a continuous function. But then why 
have we this right t It is because this state of unstable equilib- 
rium which we have been calling initial is itself only the final 
outcome of a long previous history. In the course of this history 
complex causes have worked a great while : they have contributed 
to produce the mixture of elements and they have tended to make 
everything uniform at least within a small region; they have 
rounded off the comers, smoothed down the hills and filled up 
the valleys. However capricious and irregular may have been the 
primitive curve given over to them, they have worked so much 
toward making it reg^ular that finally they deliver over to us a 
continuous curve. And this is why we may in all confidence 
assume its continuity. 

Lumen would not have the same reasons for such a conclusion. 
For him complex causes would not seem agents of equalization 
and regularity, but on the contrary would create only inequality 
and differentiation. He would see a world more and more varied 
come forth from a sort of primitive chaos. The changes he 
could observe would be for him unforeseen and impossible to 
foresee. They would seem to him due to some caprice or another; 
but this caprice would be quite different from our chance, since 
it would be opposed to all law, while our chance still has its laws. 
All these points call for lengthy explications, which perhaps 
would aid in the better comprehension of the irreversibility of 
the universe. 


We have sought to define chance, and now it is proper to put a 
question. Has chance thus defined, in so far as this is possible, 

It may be questioned. I have spoken of very slight or very 
complex causes. But what is very little for one may be very 
big for another, and what seems very complex to one may seem 
simple to another. In part I have already answered by saying 
precisely in what cases differential equations become too simple 
for the laws of chance to remain applicable. But it is fitting to 
examine the matter a little more closely, because we may take 
still other points of view. 


What means the phrase Wery slight'? To understand it we 
need only go back to what has already been said. A difference 
is very slight, an interval is very small, when within the limits 
of this interval the probability remains sensibly constant. And 
why may this probability be regarded as constant within a 
small interval? It is because we assume that the law of proba- 
bility is represented by a continuous curve, continuous not only 
in the analytic sense, but practically continuous, as already ex- 
plained. This means that it not only presents no absolute hiatus, 
but that it has neither salients nor reentrants too acute or too 

And what gives us the right to make this hypothesis? We 
have already said it is because, since the beginning of the ages, 
there have always been complex causes ceaselessly acting in the 
same way and making the world tend toward uniformity without 
ever being able to turn back. These are the causes which little 
by little have flattened the salients and filled up the reentrants, 
and this is why our probability curves now show only gentle un- 
dulations. In milliards of milliards of ages another step will 
have been made toward uniformity, and these undulations will be 
ten times as gentle; the radius of mean curvature of our curve 
will have become ten times as great. And then such a length as 
seems to us to-day not very small, since on our curve an arc of 
this length can not be regarded as rectilineal, should on the con- 
trary at that epoch be called very little, since the curvature will 
have become ten times less and an arc of this length may be 
sensibly identified with a sect. 

Thus the phrase *very slight' remains relative; but it is not 
relative to such or such a man, it is relative to the actual state of 
the world. It will change its meaning when the world shall have 
become more uniform, when all things shall have blended still 
more. But then doubtless men can no longer live and must give 
place to other beings — should I say far smaller or far larger? 
So that our criterion, remaining true for all men, retains an 
objective sense. 

And on the other hand what means the phrase 'very complex'? 
I have already given one solution, but there are others. Com- 
plex causes we have said produce a blend more and more inti- 


mate, but after how long a time will this blend satisfy ns t When 
will it have accumulated sufScient complexity t When shall we 
have sufSciently shufBed the cards t If we mix two powders, one 
blue, the other white, there comes a moment when the tint of the 
mixture seems to us uniform because of the feebleness of our 
senses; it will be uniform for the presbyte, forced to gaze &<»n 
afar, before it will be so for the myope. And when it has become 
uniform for all eyes, we still could push back the limit by the use 
of instruments. There is no chance for any man ever to discern 
the infinite variety which, if the kinetic theory is true, hides 
under the uniform appearance of a gas. And yet if we accept 
Gk)uy 's ideas on the Brownian movement, does not the microscope 
seem on the point of showing us something analogous? 

This new criterion is therefore relative like the first ; and if it 
retains an objective character, it is because all men have ap- 
proximately the same senses, the power of their instruments is 
limited, and besides they use them only exceptionally. 


It is just the same in the moral sciences and particularly in 
history. The historian is obliged to make a choice among the 
events of the epoch he studies; he recounts only those which 
seem to him the most important. He therefore contents himself 
with relating the most momentous events of the sixteenth cen- 
tury, for example, as likewise the most remarkable facts of the 
seventeenth century. If the first suflBce to explain the second, 
we say these conform to the laws of history. But if a great event 
of the seventeenth century should have for cause a small fact of 
the sixteenth century which no history reports, which all the 
world has neglected, then we say this event is due to chance. 
This word has therefore the same sense as in the physical sci- 
ences ; it means that slight causes have produced great effects. 

The greatest bit of chance is the birth of a great man. It is 
only by chance that meeting of two germinal cells, of different 
sex, containing precisely, each on its side, the mysterious ele- 
ments whose mutual reaction must produce the genius. One will 
agree that these elements must be rare and that their meeting is 
still more rare. How slight a thing it would have required to de- 
flect from its route the carrying spermatozoon. It would have 


sufSced to deflect it a tenth of a millimeter and Napoleon would 
not have been bom and the destinies of a continent would have 
been changed. No example can better make us understand the 
veritable characteristics of chance. 

One more word about the paradoxes brought out by the appli- 
cation of the calculus of probabilities to the moral sciences. It 
has been proved that no Chamber of Deputies will ever fail to 
contain a member of the opposition, or at least such an event 
would be so improbable that we might without fear wager the 
contrary, and bet a million against a sou. 

Condorcet has striven to calculate how many jurors it would 
require to make a judicial error practically impossible. If we 
had used the results of this calculation, we should certainly have 
been exposed to the same disappointments as in betting, on the 
faith of the calculus, that the opposition would never be without 
a representative. 

The laws of chance do not apply to these questions. If justice 
be not always meted out to accord with the best reasons, it uses 
less than we think the method of Bridoye. This is perhaps to 
be regretted, for then the system of Condorcet would shield us 
from judicial errors. 

What is the meaning of thisf We are tempted to attribute 
facts of this nature to chance because their causes are obscure; 
but this is not true chance. The causes are unknown to us, it is 
true, and they are even complex ; but they are not sufficiently so, 
since they conserve something. We have seen that this it is which 
distinguishes causes 4oo simple.' When men are brought to- 
gether they no longer decide at random and independently one 
of another; they influence one another. Multiplex causes come 
into action. They worry men, dragging them to right or left, 
but one thing there is they can not destroy, this is their Panurge 
flock-of -sheep habits. And this is an invariant. 


Difficulties are indeed involved in the application of the 
calculus of probabilities to the exact sciences. Why are the 
decimals of a table of logarithms, why are those of the number 
IT distributed in accordance with the laws of chance t Elsewhere 
I have already studied the question in so far as it concerns log- 


arithms, and there it is easy. It is clear that a slight difference 
of argument will give a slight difference of logarithm, but a great 
difference in the sixth decimal of the logarithm. Always we 
find again the same criterion. 

But as for the number v, that presents more difficulties, and I 
have at the moment nothing worth while to say. 

There would be many other questions to resolve, had I wished 
to attack them before solving that which I more specially set 
myself. When we reach a simple result, when we find for ex- 
ample a round number, we say that such a result can not be due 
to chance, and we seek, for its explanation, a non-fortuitous 
cause. And in fact there is only a very slight probability that 
among 10,000 numbers chance will give a round number; for 
example, the number 10,000. This has only one chance in 10,000. 
But there is only one chance in 10,000 for the occurrence of any 
other one number; and yet this result will not astonish us, nor 
will it be hard for us to attribute it to chance ; and that simply 
because it will be less striking. 

Is this a simple illusion of ours, or are there cases where this 
way of thinking is legitimate t We must hope so, else were all 
science impossible. When we wish to check a hypothesis, what 
do we do? We can not verify all its consequences, since they 
would be infinite in number ; we content ourselves with verifying 
certain ones and if we succeed we declare the hypothesis con- 
firmed, because so much success could not be due to chance. 
And this is always at bottom the same reasoning. 

I can not completely justify it here, since it would take too 
much time; but I may at least say that we find ourselves con- 
fronted by two hypotheses, either a simple cause or that aggre- 
gate of complex causes we call chance. We find it natural to 
suppose that the first should produce a simple result, and then, 
if we find that simple result, the round number for example, it 
seems more likely to us to be attributable to the simple cause 
which must give it almost certainly, than to chance which could 
only give it once in 10,000 times. It will not be the same if we 
find a result which is not simple ; chance, it is true, will not give 
this more than once in 10,000 times ; but neither has the simple 
cause any more chance of producing it. 



Thb Relativity op Space 


It is impossible to represent to oneself empty space; all our 
efforts to imagine a pure space, whence should be excluded the 
changing images of material objects, can result only in a repre- 
sentation where vividly colored surfaces, for example, are re- 
placed by lines of faint coloration, and we can not go to the very 
end in his way without all vanishing and terminating in nothing- 
ness. Thence comes the irreducible relativity of space. 

Whoever speaks of absolute space uses a meaningless phrase. 
This is a truth long proclaimed by all who have reflected upon 
the matter, but which we are too often led to forget. 

I am at a determinate point in Paris, place du Pantheon for in- 
stance, and I say: I shall come back here to-morrow. If I be 
asked : Do you mean you will return to the same point of space, 
I shall be tempted to answer: yes; and yet I shall be wrong, 
since by to-morrow the earth will have journeyed hence, carrying 
with it the place du Pantheon, which will have traveled over 
more than two million kilometers. And if I tried to speak more 
precisely, I should gain nothing, since our globe has run over 
these two million kilometers in its motion with relation to the sun, 
while the sun in its turn is displaced with reference to the Milky 
Way, while the Milky Way itself is doubtless in motion without 
our being able to perceive its velocity. So that we are completely 
ignorant, and always shall be, of how much the place du Pan- 
theon is displaced in a day. 

In sum, I meant to say : To-morrow I shall see again the dome 



and the pediment of the Panth6on, and if there were no Pan- 
theon my phrase would be meaningless and space would vanish. 

This is one of the most commonplace forms of the principle 
of the relativity of space; but there is another, upon which 
Delbeuf has particularly insisted. Suppose that in the night 
all the dimensions of the universe become a thousand times 
greater : the world will have remained similar to itself, giving to 
the word similitude the same meaning as in Euclid, Book YL 
Only what was a meter long will measure thenceforth a kilometer, 
what was a millimeter long will become a meter. The bed where- 
on I lie and my body itself will be enlarged in the same pro- 

When I awake to-morrow morning, what sensation shall I feel 
in presence of such an astounding transformation t Well, I shall 
perceive nothing at all. The most precise measurements will be 
incapable of revealing to me anything of this immense convul- 
sion, since the measures I use will have varied precisely in the 
same proportion as the objects I seek to measure. In reality, 
this convulsion exists only for those who reason as if space were 
absolute. If I for a moment have reasoned as they do, it is the 
better to bring out that their way of seeing implies contradic- 
tion. In fact it would be better to say that, space being relative, 
nothing at all has happened, which is why we have perceived 

Has one the right, therefore, to say he knows the distance be- 
tween two points? No, since this distance could undergo enor- 
mous variations without our being able to perceive them, pro- 
vided the other distances have varied in the same proportion. 
We have just seen that when I say: I shall be here to-morrow, 
this does not mean : To-morrow I shall be at the same point of 
space where I am to-day, but rather : To-morrow I shall be at the 
same distance from the Pantheon as to-day. And we see that 
this statement is no longer suflScient and that I should say : To- 
morrow and to-day my distance from the Pantheon will be equal 
to the same number of times the height of my body. 

But this is not all ; I have supposed the dimensions of the world 
to vary, but that at least the world remained always similar to 
itself. We might go much further, and one of the most aston- 
ishing theories of modern physics furnishes us the occasion. 


According to Lorentz and Fitzgerald, all the bodies borne along 
in the motion of the earth undergo a deformation. 

This deformation is, in reality, very slight, since all dimensions 
parallel to the movement of the earth diminish by a hundred 
millionth, while the dimensions perpendicular to this movement 
are unchanged. But it matters little that it is slight, that it 
exists sufiBces for the conclusion I am about to draw. And be- 
sides, I have said it was slight, but in reality I know nothing 
about it; I have myself been victim of the tenacious illusion 
which makes us believe we conceive an absolute space; I have 
thought of the motion of the earth in its elliptic orbit around 
the sun, and I have allowed thirty kilometers as its velocity. 
But its real velocity (I mean, this time, not its absolute velocity, 
which is meaningless, but its velocity with relation to the ether) , 
I do not know that, and have no means of knowing it : it is per- 
haps 10, 100 times greater, and then the deformation will be 100, 
10,000 times more. 

Can we show this deformation t Evidently not ; here is a cube 
with edge one meter; in consequence of the earth's displacement 
it is deformed, one of its edges, that parallel to the motion, 
becomes smaller, the others do not change. If I wish to assure 
myself of it by aid of a meter measure, I shall measure first 
one of the edges perpendicular to the motion and shall find that 
my standard meter fits this edge exactly ; and in fact neither of 
these two lengths is changed, since both are perpendicular to 
the motion. Then I wish to measure the other edge, that parallel 
to the motion ; to do this I displace my meter and turn it so aJs to 
apply it to the edge. But the meter, having changed orienta- 
tion and become parallel to the motion, has undergone, in its 
turn, the deformation, so that though the edge be not a meter 
long, it will fit exactly, I shall find out nothing. 

You ask then of what use is the hypothesis of Lorentz and 
of Fitzgerald if no experiment can permit of its verification t 
It is my exposition that has been incomplete ; I have spoken only 
of measurements that can be made with a meter; but we can 
also measure a length by the time it takes light to traverse it, on 
condition we suppose the velocity of light constant and inde- 
pendent of direction. Lorentz could have accounted for the 


facts by supposing the velocity of light greater in the direction 
of the earth's motion than in the perpendicular direction. 
He preferred to suppose that the velocity is the same in these 
different directions, but that the bodies are smaller in the one 
than in the other. If the wave surfaces of light had undergone 
the same deformations as the material bodies we should never 
have perceived the Lorentz-Fitzgerald deformation. 

In either case, it is not a question of absolute magnitude, but 
of the measure of this magnitude by means of some instrument; 
this instrument may be a meter, or the path traversed by light; 
it is only the relation of the magnitude to the instrument that 
we measure; and if this relation is altered, we have no way of 
knowing whether it is the magnitude or the instrument which 
has changed. 

But what I wish to bring out is, that in this deformation the 
world has not remained similar to itself; squares have become 
rectangles, circles ellipses, spheres ellipsoids. And yet we have 
no way of knowing whether this deformation be real. 

Evidently one could go much further : in place of the Lorents- 
Fitzgerald deformation, whose laws are particularly simple, we 
could imagine any deformation whatsoever. Bodies could be 
deformed according to any laws, as complicated as we might wish, 
we never should notice it provided all bodies without exception 
were deformed according to the same laws. In saying, all bodies 
without exception, I include of course our own body and the 
light rays emanating from different objects. 

If we look at the world in one of those mirrors of complicated 
shape which deform objects in a bizarre way, the mutual relations 
of the different parts of this world would not be altered; if, in 
fact two real objects touch, their images likewise seem to touch. 
Of course when we look in such a mirror we see indeed the 
deformation, but this is because the real world subsists along- 
side of its deformed image ; and then even were this real world 
hidden from us, something there is could not be hidden, ourself ; 
we could not cease to see, or at least to feel, our body and our 
limbs which have not been deformed and which continue to serve 
us as instruments of measure. 

But if we imagine our body itself deformed in the same way 


as if seen in the mirror, these instruments of measure in their 
turn will fail us and the deformation will no longer be ascer- 

Consider in the same way two worlds images of one another; 
to each object P of the world A corresponds in the world B an 
object P', its image ; the coordinates of this image P* are deter- 
minate functions of those of the object P; moreover these func- 
tions may be any whatsoever; I only suppose them chosen once 
for all. Between the position of P and that of P* there is a 
constant relation ; what this relation is, matters not ; enough that 
it be constant. 

Well, these two worlds will be indistinguishable one from the 
other. I mean the first will be for its inhabitants what the 
second is for its. And so it will be as long as the two worlds 
remain strangers to each other. Suppose we live in world A, we 
shall have constructed our science and in particular our geom- 
etry ; during this time the inhabitants of world B will have con- 
structed a science, and as their world is the image of ours, their 
geometry will also be the image of ours or, better, it will be the 
same. But if for us some day a window is opened upon world 
B, how we shall pity them: **Poor things," we shall say, **they 
think they have made a geometry, but what they call so is only 
a grotesque image of ours; their straights are all twisted, their 
circles are humped, their spheres have capricious inequalities." 
And we shall never suspect they say the same of us, and one 
never will know who is right. 

We see in how broad a sense should be understood the rela- 
tivity of space; space is in reality amorphous and the things 
which are therein alone give it a form. What then should be 
thought of that direct intuition we should have of the straight 
or of distance t So little have we intuition of distance in itself 
that in the night, as we have said, a distance might become a 
thousand times greater without our being able to perceive it, if 
all other distances had undergone the same alteration. And even 
in a night the world B might be substituted for the world A 
without our having any way of knowing it, and then the straight 
lines of yesterday would have ceased to be straight and we 
should never notice. 


One part of space is not by itself and in the absolute sense of 
the word equal to another part of space ; because if so it is for 
us, it would not be for the dwellers in world B ; and these have 
just as much right to reject our opinion as we to condemn theirs. 

I have elsewhere shown what are the consequences of these 
facts from the viewpoint of the idea we should form of non- 
Euclidean geometry and other analogous geometries; to that I 
do not care to return ; and to-day I shall take a somewhat dif- 
ferent point of view. 


If this intuition of distance, of direction, of the straight line, 
if this direct intuition of space in a word does not exist, whence 
comes our belief that we have itt If this is only an illusion, 
why is this illusion so tenacious t It is proper to examine into 
this. We have said there is no direct intuition of size and we 
can only arrive at the relation of this magnitude to our instru- 
ments of measure. We should therefore not have been able to 
construct space if we had not had an instrument to measure it; 
well, this instrument to which we relate everything, which we 
use instinctively, it is our own body. It is in relation to our 
body that we place exterior objects, and the only spatial rela- 
tions of these objects that we can represent are their relations 
to our body. It is our body which serves us, so to speak, as 
system of axes of coordinates. 

For example, at an instant a, the presence of the object A is 
revealed to me by the sense of sight; at another instant, fi, the 
presence of another object, B, is revealed to me by another sense, 
that of hearing or of touch, for instance. I judge that this 
object B occupies the same place as the object A. What does 
that mean? First that does not signify that these two objects 
occupy, at two different moments, the same point of an absolute 
space, which even if it existed would escape our cognition, since, 
between the instants a and p, the solar system has moved and 
we can not know its displacement. That means these two objects 
occupy the same relative position with reference to our body. 

But even this, what does it mean ? The impressions that have 
come to us from these objects have followed paths absolutely 


different, the optic nerve for the object A, the acoustic nerve for 
the object jB. They have nothing in common from the qualita- 
tive point of view. The representations we are able to make of 
these two objects are absolutely heterogeneous, irreducible one to 
the other. Only I know that to reach the object A I have just 
to extend the right arm in a certain way ; even when I abstain 
from doing it, I represent to myself the muscular sensations and 
other analogous sensations which would accompany this exten- 
sion, and this representation is associated with that of the 
object A. 

Now, I likewise know I can reach the object B by extending my 
right arm in the same manner, an extension accompanied by the 
same train of muscular sensations. And when I say these two 
objects occupy the same place, I mean nothing more. 

I also know I could have reached the object A by another 
appropriate motion of the left arm and I represent to myself the 
muscular sensations which would have accompanied this move- 
ment ; and by this same motion of the left arm, accompanied by 
the same sensations, I likewise could have reached the object B. 

And that is very important, since thus I can defend myself 
against dangers menacing me from the object A or the object B. 
With each of the blows we can be hit, nature has associated 
one or more parries which permit of our guarding ourselves. 
The same parry may respond to several strokes ; and so it is, for 
instance, that the same motion of the right arm would have 
allowed us to guard at the instant a against the object A and at 
the instant p against the object B. Just so, the same stroke can 
be parried in several ways, and we have said, for instance, the 
object A could be reached indifferently either by a certain move- 
ment of the right arm or by a certain movement of the left arm. 

All these parries have nothing in common except warding off 
the same blow, and this it is, and nothing else, which is meant 
when we say they are movements terminating at the same point 
of space. Just so, these objects, of which we say they occupy 
the same point of space, have nothing in common, except that the 
same parry guards against them. 

Or, if you choose, imagine innumerable telegraph wires, some 
centripetal, others centrifugal. The centripetal wires warn us of 


accidents happening without; the centrifugal wires carry the 
reparation. Connections are so established that when a cen- 
tripetal wire is traversed by a current this acts on a relay and so 
starts a current in one of the centrifugal wires, and things are 
so arranged that several centripetal wires may act on the same 
centrifugal wire if the same remedy suits several ills, and that a 
centripetal wire may agitate different centrifugal wires, either 
simultaneously or in lieu one of the other when the same ill may 
be cured by several remedies. 

It is this complex system of associations, it is this table of distri- 
bution, so to speak, which is all our geometry or, if you wish, 
all in our geometry that is instinctive. What we call our intui- 
tion of the straight line or of distance is the consciousness we 
have of these associations and of their imperious character. 

And it is easy to understand whence comes this imperious 
character itself. An association will seem to us by so much the 
more indestructible as it is more ancient. But these associations 
are not, for the most part, conquests of the individual, since their 
trace is seen in the new-bom babe : they are conquests of the race. 
Natural selection had to bring about these conquests by so much 
the more quickly as they were the more necessary. 

On this account, those of which we speak must have been of 
the earliest in date, since without them the defense of the organ- 
ism would have been impossible. From the time when the cell- 
ules were no longer merely juxtaposed, but were called upon to 
give mutual aid, it was needful that a mechanism organize anal- 
ogous to what we have described, so that this aid miss not its 
way, but forestall the peril. 

When a frog is decapitated, and a drop of acid is placed on a 
point of its skin, it seeks to wipe off the acid with the nearest foot, 
and, if this foot be amputated, it sweeps it off with the foot of 
the opposite side. There we have the double parry of which I 
have just spoken, allowing the combating of an ill by a second 
remedy, if the first fails. And it is this multiplicity of parries, 
and the resulting coordination, which is space. 

We see to what depths of the unconscious we must descend 
to find the first traces of these spatial associations, since only 
the inferior parts of the nervous system are involved. Why be 


astonished then at the resistance we oppose to every attempt 
made to dissociate what so long has been associated t Now, it is 
just this resistance that we call the evidence for the geometric 
truths ; this evidence is nothing but the repugnance we feel toward 
breaking with very old habits which have always proved good. 


The space so created is only a little space extending no farther 
than my arm can reach ; the intervention of the memory is neces- 
sary to push back its limits. There are points which will remain 
out of my reach, whatever effort I make to stretch forth my hand ; 
if I were fastened to the ground like a hydra polyp, for instancei 
which can only extend its tentacles, all these points would be 
outside of space, since the sensations we could experience from 
the action of bodies there situated, would be associated with the 
idea of no movement allowing us to reach them, of no appro- 
priate parry. These sensations would not seem to us to have 
any spatial character and we should not seek to localize them. 

But we are not fixed to the ground like the lower animals ; we 
can, if the enemy be too far away, advance toward him first and 
extend the hand when we are sufiSciently near. This is still a 
parry, but a parry at long range. On the other hand, it is a 
complex parry, and into the representation we make of it enter 
the representation of the muscular sensations caused by the 
movements of the legs, that of the muscular sensations caused 
by the final movement of the arm, that of the sensations of the 
semicircular canals, etc. We must, besides, represent to our- 
selves, not a complex of simultaneous sensations, but a complex 
of successive sensations, following each other in a determinate 
order, and this is why I have just said the intervention of memory 
was necessary. Notice moreover that, to reach the same point, 
I may approach nearer the mark to be attained, so as to have to 
stretch my arm less. What more t It is not one, it is a thousand 
parries I can oppose to the same danger. All these parries are 
made of sensations which may have nothing in common and yet 
we regard them as defining the same point of space, since they 
may respond to the same danger and are all associated with the 
notion of this danger. It is the potentiality of warding off the 


same stroke which makes the unity of these different parries, as 
it is the possibility of being parried in the same way which makes 
the unity of the strokes so different in kind, which may menace 
us from the same point of space. It is this double unity which 
makes the individuality of each point of space, and, in the 
notion of point, there is nothing else. 

The space before considered, which might be called restricted 
space, was referred to coordinate axes bound to my body; these 
axes were fixed, since my body did not move and only my mem- 
bers were displaced. What are the axes to which we naturally 
refer the extended space? that is to say the new space just 
defined. We define a point by the sequence of movements to be 
made to reach it, starting from a certain initial position of the 
body. The axes are therefore fixed to this initial position of the 

But the position I call initial may be arbitrarily chosen among 
all the positions my body has successively occupied ; if the memory 
more or less unconscious of these successive positions is necessary 
for the genesis of the notion of space, this memory may go back 
more or less far into the past. Thence results in the definition 
itself of space a certain indetermination, and it is precisely this 
indetermination which constitutes its relativity. 

There is no absolute space, there is only space relative to a 
certain initial position of the body. For a conscious being fixed 
to the ground like the lower animals, and consequently knowing 
only restricted space, space would still be relative (since it would 
have reference to his body), but this being would not be conscious 
of this relativity, because the axes of reference for this restricted 
space would be unchanging! Doubtless the rock to which this 
being would be fettered would not be motionless, since it would 
be carried along in the movement of our planet; for us conse- 
quently these axes would change at each instant ; but for him they 
would be changeless. We have the faculty of referring our 
extended space now to the position A of our body, considered as 
initial, again to the position B^ which it had some moments 
afterward, and which we are free to regard in its turn as initial ; 
we make therefore at each instant unconscious transformations 
of coordinates. This faculty would be lacking in our imaginary 


being, and from not having traveled, he would think space abso- 
lute. At every instant, his system of axes would be imposed 
upon him ; this system would have to change greatly in reality, 
but for him it would be always the same, since it would be 
always the only system. Quite otherwise is it with us, who at 
each instant have many systems between which we may choose at 
will, on condition of going back by memory more or less far into 
the past. 

This is not all; restricted space would not be homogeneous; 
the different points of this space could not be regarded as equiva- 
lent, since some could be reached only at the cost of the greatest 
efforts, while others could be easily attained. On the contrary, 
our extended space seems to us homogeneous, and we say all its 
points are equivalent. What does that meant 

If we start from a certain place A, we can, from this position, 
make certain movements, M, characterized by a certain complex 
of muscular sensations. But, starting from another position, B, 
we make movements M' characterized by the same muscular sen- 
sations. Let a, then, be the situation of a certain point of the 
body, the end of the index finger of the right hand for example, 
in the initial position A, and b the situation of this same index 
when, starting from this position A, we have made the motions M. 
Afterwards, let a' be the situation of this index in the position B, 
and b' its situation when, starting from the position B, we have 
made the motions 3f^ 

Well, I am accustomed to say that the points of space a and b 
are related to each other just as the points a' and b\ and this 
simply means that the two series of movements M and ilf' are 
accompanied by the same muscular sensations. And as I am 
conscious that, in passing from the position A to the position £, 
my body has remained capable of the same movements, I know 
there is a point of space related to the point of just as any point 
b is to the point a, so that the two points a and a' are equivalent. 
This is what is called the homogeneity of space. And, at the same 
time, this is why space is relative, since its properties remain the 
same whether it be referred to the axes A or to the axes B. So 
that the relativity of space and its homogeneity are one sole and 
same thing. 


r, if I wiih to pav to Ae great wp&ot^ wliidi no longer 
aenres onl j for me, bat where I maj lodge the muretae, I get 
there bj an act of imaginatioiL. I imagine how a giant would 
feel who could reach the planets in a few steps; or, if yoa ehooae, 
what I mjself should feel in presence of a miniature world whore 
these planets were replaced hj little balls, while on one of these 
little balls moved a liliputian I should call mjrself. Bat this aet 
of imagination woald be impossible for me had I not previoosij 
eonstracted my restricted space and my eztoided space for my 
own use. 


Why now have all these spaces three dimensions ? Go back 
to the "table of distribution" of which we hare spoken. We 
have on the one side the list of the different possible dangers; 
designate them by Al^ A2^ etc. ; and, on the other side, the list 
of the different remedies which I shall call in the same way 
Bly jB2, etc. We have then connections between the contact studs 
or push buttons of the first list and those of the second, so that 
when, for instance, the announcer of danger AZ functions, it 
will put or may put in action the relay corresponding to the 
parry 54. 

As I have spoken above of centripetal or centrifugal wires, I 
fear lest one see in all this, not a simple comparison, but a descrip- 
tion of the nervous system. Such is not my thought, and that 
for several reasons : first I should not permit myself to put forth 
an opinion on the structure of the nervous system which I do 
not know, while those who have studied it speak only circum- 
spectly; again because, despite my incompetence, I well know 
this scheme would be too simplistic; and finally because on my 
list of parries, some would figure very complex, which might even, 
in the case of extended space, as we have seen above, consist of 
many steps followed by a movement of the arm. It is not a ques- 
tion then of physical connection between two real conductors, 
but of psychologic association between two series of sensations. 

If Al and A2 for instance are both associated with the parry 
51, and if Al is likewise associated with the parry 52, it will 
generally happen that A2 and 52 will also themselves be asso- 
ciated. If this fundamental law were not generally true, there 


would exist only an immense confusion and there would be 
nothing resembling a conception of space or a geometry. How 
in fact have we defined a point of space. We have done it in two 
ways: it is on the one hand the aggregate of announcers A in 
connection with the same parry JB; it is on the other hand the 
aggregate of parries B in connection with the same announcer A. 
If our law was not true, we should say ill and A2 correspond 
to the same point since they are both in connection with Bl ; but 
we should likewise say they do not correspond to the same point, 
since Al would be in connection with B2 and the same would 
not be true of A2. This would be a contradiction. 

But, from another side, if the law were rigorously and always 
true, space would be very diiSEerent from what it is. We should 
have categories strongly contrasted between which would be 
portioned out on the one hand the announcers A, on the other 
hand the parries B; these categories would be excessively nu- 
merous, but they would be entirely separated one from another. 
Space would be composed of points very numerous, but discrete; 
it would be discontinuous. There would be no reason for rang- 
ing these points in one order rather than another, nor conse- 
quently for attributing to space three dimensions. 

But it is not so ; permit me to resume for a moment the lan- 
guage of those who already know geometry ; this is quite proper 
since this is the language best understood by those I wish to make 
understand me. 

When I desire to parry the stroke, I seek to attain the point 
whence comes this blow, but it suffices that I approach quite near. 
Then the parry Bl may answer for Al and for A2, if the point 
which corresponds to Bl is sufficiently near both to that corre- 
sponding to Al and to that corresponding to A2. But it may 
happen that the point corresponding to another parry B2 may be 
sufficiently near the point corresponding to ^1 and not suffi- 
ciently near the point corresponding to -42 ; so that the parry B2 
may answer for Al without answering for A2. For one who 
does not yet know geometry, this translates itself simply by a 
derogation of the law stated above. And then things will happen 

Two parries JBl and B2 will be associated with the same warn- 


ing ill and with a large number of warnings which we ahall 
range in the same category as Al and which we shall make corre- 
spond to the same point of space. But we may find warnings 
A2 which will be associated with B2 without being associated 
with Bly and which in compensation will be associated with £3, 
which jB3 was not associated with Al, and so forth, so that we 
may write the series 

Bl, ^1, B2, A2, BZ, A3, B4, A4, 

where each term is associated with the following and the preced- 
ing, but not with the terms several places away. 

Needless to add that each of the terms of these series is not 
isolated, but forms part of a very numerous category of other 
warnings or of other parries which have the same connections as 
it, and which may be regarded as belonging to the same point of 

The fundamental law, though admitting of exceptions, remains 
therefore almost always true. Only, in consequence of these 
exceptions, these categories, in place of being entirely separated, 
encroach partially one upon another and mutually penetrate in 
a certain measure, so that space becomes continuous. 

On the other hand, the order in which these categories are to 
be ranged is no longer arbitrary, and if we refer to the preceding 
series, we see it is necessary to put 52 between Al and A2 and 
consequently between Bl and B3 and that we could not for 
instance put it between BS and B4. 

There is therefore an order in which are naturally arranged 
our categories which correspond to the points of space, and experi- 
ence teaches us that this order presents itself under the form 
of a table of triple entry, and this is why space has three 


So the characteristic property of space, that of having three 
dimensions, is only a property of our table of distribution, an 
internal property of the human intelligence, so to speak. It 
would suffice to destroy certain of these connections, that is to 
say of the associations of ideas to give a different table of dis- 
tribution, and that might be enough for space to acquire a fourth 


Some persons will be astonished at such a result. The external 
world, they will think, should count for something. If the num- 
ber of dimensions comes from the way we are made, there might 
be thinking beings living in our world, but who might be made 
differently from us and who would believe space has more or less 
than three dimensions. Has not M. de Cyon said that the Jap- 
anese mice, having only two pair of semicircular canals, believe 
that space is two-dimensional T And then this thinking being, if 
he is capable of constructing a physics, would he not make a phys- 
ics of two or of four dimensions, and which in a sense would 
still be the same as ours, since it would be the description of the 
same world in another language! 

It seems in fact that it would be possible to translate our phys- 
ics into the language of geometry of four dimensions ; to attempt 
this translation would be to take great pains for little profit, and 
I shall confine myself to citing the mechanics of Hertz where we 
have something analogous. However, it seems that the transla- 
tion would always be less simple than the text, and that it would 
always have the air of a translation, that the language of three 
dimensions seems the better fitted to the description of our world, 
although this description can be rigorously made in another 
idiom. Besides, our table of distribution was not made at ran- 
dom. There is connection between the warning Al and the 
parry jBI, this is an internal property of our intelligence; but 
why this connection? It is because the parry jBI affords means 
effectively to guard against the danger Al; and this is a fact 
exterior to us, this is a property of the exterior world. Our 
table of distribution is therefore only the translation of an ag- 
gregate of exterior facts; if it has three dimensions, this is be- 
cause it has adapted itself to a world having certain properties ; 
and the chief of these properties is that there exist natural solids 
whose displacements follow sensibly the laws we call laws of 
motion of rigid solids. If therefore the language of three di- 
mensions is that which permits us most easily to describe our 
world, we should not be astonished ; this language is copied from 
our table of distribution ; and it is in order to be able to live in 
this world that this table has been established. 

I have said we could conceive, living in our world, thinking 


beings whose table of distribution would be four-dimensional 
and who consequently would think in hyperspace. It is not 
certain however that such beings, admitting they were bom there, 
could live there and defend themselves against the thousand 
dangers by which they would there be assailed. 


A few remarks to end with. There is a striking contrast be- 
tween the roughness of this primitive geometry, reducible to 
what I call a table of distribution, and the infinite precision of 
the geometers' geometry. And yet this is bom of that; but not 
of that alone ; it must be made fecund by the faculty we have of 
constructing mathematical concepts, such as that of group, for 
instance; it was needful to seek among the pure concepts that 
which best adapts itself to this rough space whose genesis I have 
sought to explain and which is common to us and the higher 

The evidence for certain geometric postulates, we have said, is 
only our repugnance to renouncing very old habits. But these 
postulates are infinitely precise, while these habits have some- 
thing about them essentially pliant. When we wish to think, we 
need postulates infinitely precise, since this is the only way to 
avoid contradiction ; but among all the possible systems of postu- 
lates, there are some we dislike to choose because they are not 
suflBciently in accord with our habits; however pliant, however 
elastic they may be, these have a limit of elasticity. 

We see that if geometry is not an experimental science, it is a 
science bom apropos of experience; that we have created the 
space it studies, but adapting it to the world wherein we live. 
We have selected the most convenient space, but experience has 
guided our choice; as this choice has been unconscious, we think 
it has been imposed upon us; some say experience imposes it, 
others that we are bom with our space ready made ; we see from 
the preceding considerations, what in these two opinions is the 
part of truth, what of error. 

In this progressive education whose outcome has been the con- 
struction of space, it is very diflScult to determine what is the 


part of the individaal, what the part of the race. How far could 
one of us, transported from birth to an entirely different world, 
where were dominant, for instance, bodies moving in conformity 
to the laws of motion of non-Euclidean solids, renounce the an- 
cestral space to build a space completely newt 

The part of the race seems indeed preponderant ; yet if to it we 
owe rough space, the soft space I have spoken of, the space of 
the higher animals, is it not to the unconscious experience of the 
individual we owe the infinitely precise space of the geometer! 
This is a question not easy to solve. Yet we cite a fact showing 
that the space our ancestors have bequeathed us still retains a 
certain plasticity. Some hunters learn to shoot fish under water, 
though the image of these fish be turned up by refraction. Be- 
sides they do it instinctively: they therefore have learned to 
modify their old instinct of direction ; or, if you choose, to sub- 
stitute for the association Al, j51, another association Al, j52, 
because experience showed them the first would not work. 

Mathematical Definitions and Teachinq 

1. I SHOULD speak here of general definitions in mathematicB; 
at least that is the title, but it will be impossible to confine my- 
self to the subject as strictly as the rule of unity of action would 
require ; I shall not be able to treat it without touching upon a 
few other related questions, and if thus I am forced from time 
to time to walk on the bordering flower-beds on the right or left^ 
I pray you bear with me. 

What is a good definition T For the philosopher or the scientist 
it is a definition which applies to all the objects defined, and only 
those ; it is the one satisfying the rules of logic. But in teach- 
ing it is not that; a good definition is one understood by the 

How does it happen that so many refuse to understand math- 
ematics T Is that not something of a paradox! Lo and behold I 
a science appealing only to the fundamental principles of logic, 
to the principle of contradiction, for instance, to that which is 
the skeleton, so to speak, of our intelligence, to that of which we 
can not divest ourselves without ceasing to think, and there are 
people who find it obscure! and they are even in the majority! 
That they are incapable of inventing may pass, but that they do 
not understand the demonstrations shown them, that they re- 
main blind when we show them a light which seems to us flash- 
ing pure flame, this it is which is altogether prodigious. 

And yet there is no need of a wide experience with examina- 
tions to know that these blind men are in no wise exceptional 
beings. This is a problem not easy to solve, but which should 
engage the attention of all those wishing to devote themselves to 

What is it, to understand? Has this word the same meaning 
for all the world ? To understand the demonstration of a theorem, 
is that to examine successively each of the syllogisms composing 
it and to ascertain its correctness, its conformity to the rules of 



the garnet Likewise, to understand a definition, is this merely 
to recognize that one already knows the meaning of all the terms 
employed and to ascertain that it implies no contradiction T 

For some, yes ; when they have done this, they will say : I un- 

For the majority, no. Almost all are much more exacting; 
they wish to know not merely whether all the syllogisms of a 
demonstration are correct, but why they link together in this 
order rather than another. In so far as to them they seem en- 
gendered by caprice and not by an intelligence always conscious 
of the end to be attained, they do not believe they understand. 

Doubtless they are not themselves just conscious of what they 
crave and they could not formulate their desire, but if they do 
not get satisfaction, they vaguely feel that something is lacking. 
Then what happens t In the beginning they still perceive the 
proofs one puts under their eyes; but as these are connected 
only by too slender a thread to those which precede and those 
which follow, they pass without leaving any trace in their head ; 
they are soon forgotten ; a moment bright, they quickly vanish in 
night eternal. When they are farther on, they will no longer see 
even this ephemeral light, since the theorems lean one upon 
another and those they would need are forgotten ; thus it is they 
become incapable of understanding mathematics. 

This is not always the fault of their teacher ; often their mind, 
which needs to perceive the guiding thread, is too lazy to seek 
and find it. But to come to their aid, we first must know just 
what hinders them. 

Others will always ask of what use is it; they will not have 
understood if they do not find about them, in practise or in 
nature, the justification of such and such a mathematical concept. 
Under each word they wish to put a sensible image ; the definition 
must evoke this image, so that at each stage of the demonstration 
they may see it transform and evolve. Only upon this condition 
do they comprehend and retain. Often these deceive themselves ; 
they do not listen to the reasoning, they look at the figures ; they 
think they have understood and they have only seen. 

2. How many different tendencies! Must we combat themt 
Must we use them ? And if we wish to combat them, which should 


be favored t Must we show those content with the pure logic that 
they have seen only one side of the matter t Or need we say to 
those not so cheaply satisfied that what they demand is not 

In other words, should we constrain the young people to change 
the nature of their minds? Such an attempt would be vain; we 
do not possess the philosopher's stone which would enable us to 
transmute one into another the metals confided to us; all we 
can do is to work with them, adapting ourselves to their 

Many children are incapable of becoming mathematicians, to 
whom however it is necessary to teach mathematics; and the 
mathematicians themselves are not all cast in the same mold. 
To read their works suffices to distinguish among them two 
sorts of minds, the logicians like Weierstrass for example, the 
intuitives like Biemann. There is the same difference among 
our students. The one sort prefer to treat their problems 'by 
analysis' as they say, the others *by geometry.' 

It is useless to seek to change anything of that, and besides 
would it be desirable! It is well that there are logicians and 
that there are intuitives; who would dare say whether he pre- 
ferred that Weierstrass had never written or that there never 
had been a Riemann. We must therefore resign ourselves to the 
diversity of minds, or better we must rejoice in it. 

3. Since the word understand has many meanings, the defi- 
nitions which will be best understood by some will not be best 
suited to others. We have those which seek to produce an image, 
and those where we confine ourselves to combining empty forms, 
perfectly intelligible, but purely intelligible, which abstraction 
has deprived of all matter. 

I know not whether it be necessary to cite examples. Let us 
cite them, anyhow, and first the definition of fractions will furnish 
us an extreme case. In the primary schools, to define a frac- 
tion, one cuts up an apple or a pie; it is cut up mentally of 
course and not in reality, because I do not suppose the budget 
of the primary instruction allows of such prodigality. At the 
Normal School, on the other hand, or at the college, it is said: 
a fraction is the combination of two whole numbers separated by 


a horizontal bar; vre define by conventions the operations to 
which these symbols may be submitted ; it is proved that the rules 
ot these operations are the same as in calculating with whole 
numbers, sjid we ascertain finally that multiplying the fraction, 
according to these rules, by the denominator gives the numerator. 
This is all very well because we are addressing young people 
long familiarized with the notion of fractions through having cut 
up apples or other objects, and whose mind, matured by a hard 
mathematical education, has come little by little to desire a purely 
logical definition. But the debutant to whom one should try to 
give it, how dumf ounded ! 

Such also are the definitions found in a book justly admired 
and greatly honored, the Foundations of Geometry by Hilbert. 
See in fact how he begins: We think three systems of thinqs 
which we shail call points, straights and planes. What are these 
'things' T 

We know not, nor need we know ; it would even be a pity to 
seek to know ; all we have the right to know of them is what the 
assumptions tell us ; this for example : Two distinct points olwoyr 
determine a straight, which is followed by this remark: in place 
of determine, we may say the two points are on the straight, or 
the straight goes through these two points or joins the ttvo points. 

Thus 'to be on a straight' is simply defined as synonymouu 
with 'determine a straight.' Behold a book of which I think 
much good, but which I should not recommend to a school boy. 
Yet I could do so without fear, he would not read much of it, 
I have taken extreme examples and no teacher would dream ot 
going that far. But even stopping short of such models, doea 
he not already expose himself to the same dangerl 

Suppose we are in a class; the professor dictates; the circle is 
the locns of points of the plane equidistant from an interior point 
called the center. The good scholar writes this phrase in his 
note-booki the bad scholar draws faces; hut neither understands; 
then the professor takes the chalk and draws a circle ou the board. 
"Ahl" think the scholars, "why did he not say at once: a circle 
is a ring, we should have uuderstood." Doubtless the professor 
is right. The scholars' definition would have been of no avail, 
since it could serve for no demonstration, since besides it would 


not give them the salutary habit of analyzing their conceptions. 
But one should show them that they do not comprehend what 
they think they know, lead them to be conscious of the roughness 
of their primitive conception, and of themselves to wish it puri- 
fied and made precise. 

4. I shall return to these examples ; I only wished to show you 
the two opposed conceptions ; they are in violent contrast. This 
contrast the history of science explains. If we read a book 
written fifty years ago, most of the reasoning we find there seems 
lacking in rigor. Then it was assumed a continuous function 
can change sign only by vanishing ; to-day we prove it. It was 
assumed the ordinary rules of calculation are applicable to 
inconmiensurable numbers; to-day we prove it. Many other 
things were assumed which sometimes were false. 

We trusted to intuition ; but intuition can not give rigor, nor 
even certainty ; we see this more and more. It tells us for instance 
that every curve has a tangent, that is to say that every con- 
tinuous function has a derivative, and that is false. And as we 
sought certainty, we had to make less and less the part of 

What has made necessary this evolution? We have not been 
slow to perceive that rigor could not be established in the reason- 
ings, if it were not first put into the definitions. 

The objects occupying mathematicians were long ill defined; 
we thought we knew them because we represented them with the 
senses or the imagination; but we had of them only a rough 
image and not a precise concept upon which reasoning could take 
hold. It is there that the logicians would have done well to direct 
their efforts. 

So for the incommensurable number, the vague idea of con- 
tinuity, which we owe to intuition, has resolved itself into a com- 
plicated system of inequalities bearing on whole numbers. Thus 
have finally vanished all those difiiculties which frightened our 
fathers when they reflected upon the foundations of the infini- 
tesimal calculus. To-day only whole numbers are left in analysis, 
or systems finite or infinite of whole numbers, bound by a 
plexus of equalities and inequalities. Mathematics we say is 


5. But do you tliink mathematics has attained absolute rigor 
without making any sacrifice t Not at all; what it has gained in 
rigor it has lost in objectivity. It is by separating itself from 
reality that it has acquired this perfect purity. We may freely 
run over its whole domain, formerly bristling with obstacles, but 
these obstacles have not disappeared. They have only been 
moved to the frontier, and it would be necessary to vanquish 
them anew if we wished to break over this frontier to enter the 
realm of the practical. 

We had a vague notion, formed of incongruous elements, some 
a priori, others coming from experiences more or less digested; 
we thought we knew, by intuition, its principal properties. To- 
day we reject the empiric elements, retaining only the a priori; 
one of the properties serves as definition and all the others are 
deduced from it by rigorous reasoning. This is all very well, 
but it remains to be proved that this properly, which has become 
a definition, pertains to the real objects which experience had 
made known to us and whence we drew our vague intuitive 
notion. To prove that, it would be necessary to appeal to experi- 
ence, or to make an effort of intuition, and if we could not prove 
it, our theorems would be perfectly rigorous, but perfectly 

Logic sometimes makes monsters. Since half a century we 
have seen arise a crowd of bizarre functions which seem to try 
to resemble as little as possible the honest functions which serve 
some purpose. No longer continuity, or perhaps continuity, but 
no derivatives, etc. Nay more, from the logical point of view, 
it is these strange functions which are the most general, those 
one meets without seeking no longer appear except as particular 
case. There remains for them only a very small comer. 

Heretofore when a new function was invented, it was for some 
practical end ; to-day they are invented expressly to put at fault 
the reasonings of our fathers, and one never will get from them 
anything more than that. 

If logic were the sole guide of the teacher, it would be neces- 
sary to begin with the most general functions, that is to say with 
the most bizarre. It is the beginner that would have to be set 


grappling with this teratologic museum. If you do not do it, 
the logicians might say, you will achieve rigor only by stages. 

6. Yes, perhaps, but we can not make so cheap of reality, and 
I mean not only the reality of the sensible world, which however 
has its worth, since it is to combat against it that nine tenths of 
your students ask of you weapons. There is a reality more 
subtile, which makes the very life of the mathematical beings, 
and which is quite other than logic. 

Our body is formed of cells, and the cells of atoms ; are these 
cells and these atoms then all the reality of the human bodyt 
The way these cells are arranged, whence results the unity of the 
individual, is it not also a reality and much more interesting t 

A naturalist who never had studied the elephant except in 
the microscope, would he think he knew the animal adequately! 
It is the same in mathematics. When the logician shall have 
broken up each demonstration into a multitude of elementary 
operations, all correct, he still will not possess the whole reality ; 
this I know not what which makes the unity of the demonstration 
will completely escape him. 

In the edifices built up by our masters, of what use to admire 
the work of the mason if we can not comprehend the plan of 
the architect? Now pure logic can not give us this appreciation 
of the total effect ; this we must ask of intuition. 

Take for instance the idea of continuous function. This is at 
first only a sensible image, a mark traced by the chalk on the 
blackboard. Little by little it is refined ; we use it to construct 
a complicated system of inequalities, which reproduces all the 
features of the primitive image; when all is done, we have 
removed the centering, as after the construction of an arch; 
this rough representation, support thenceforth useless, has dis- 
appeared and there remains only the edifice itself, irreproachable 
in the eyes of the logician. And yet, if the professor did not 
recall the primitive image, if he did not restore momentarily the 
centering, how could the student divine by what caprice all these 
inequalities have been scaffolded in this fashion one upon another? 
The definition would be logically correct, but it would not show 
him the veritable reality. 

7. So back we must return ; doubtless it is hard for a master 



to teach what does not entirely satisfy him; but the satisfaction 
of the master is not the unique object of teaching; we should first 
give attention to what the mind of the pupil is and to what we 
wish it to become. 

Zoologists maintain that the embryonic development of an 
animal recapitulates in brief the whole history of its ancestors 
throughout geologic time. It seems it is the same in the develop- 
ment of minds. The teacher should make the child go over the 
path his fathers trod; more rapidly, but without skipping sta- 
tions. For this reason, the history of science should be our first 

Our fathers thought they knew what a fraction was, or con- 
tinuity, or the area of a curved surface ; we have found they did 
not know it. Just so our scholars think they know it when they 
begin the serious study of mathematics. If without warning I 
tell them : **No, you do not know it; what you think you under- 
stand, you do not understand ; I must prove to you what seems 
to you evident," and if in the demonstration I support myself 
upon premises which to them seem less evident than the con- 
clusion, what shall the unfortunates think? They will think that 
the science of mathematics is only an arbitrary mass of useless 
subtilities ; either they will be disgusted with it, or they will play 
it as a game and will reach a state of mind like that of the Greek 

Later, on the contrary, when the mind of the scholar, familiar- 
ized with mathematical reasoning, has been matured by this long 
frequentation, the doubts will arise of themselves and then your 
demonstration will be welcome. It will awaken new doubts, and 
the questions will arise successively to the child, as they arose suc- 
cessively to our fathers, until perfect rigor alone can satisfy him. 
To doubt everything does not suflSce, one must know why he 

8. The principal aim of mathematical teaching is to develop 
certain faculties of the mind, and among them intuition is not the 
least precious. It is through it that the mathematical world 
remains in contact with the real world, and if pure mathematics 
could do without it, it would always be necessary to have recoone 
to it to fill up the chasm which separates the symbol from reality. 


The practician will always have need of it, and for one pure 
geometer there should be a hundred practicians. 

The engineer should receive a complete mathematical educa- 
tion, but for what should it serve him ? 

To see the different aspects of things and see them quickly; 
he has no time to hunt mice. It is necessary that, in the com- 
plex physical objects presented to him, he should promptly recog- 
nize the point where the mathematical tools we have put in his 
hands can take hold. How could he do it if we should leave 
between instruments and objects the deep chasm hollowed out 
by the logicians! 

9. Besides the engineers, other scholars, less numerous, are in 
their turn to become teachers; they therefore must go to the 
very bottom; a knowledge deep and rigorous of the firist prin- 
ciples is for them before all indispensable. But this is no reason 
not to cultivate in them intuition ; for they would get a false idea 
of the science if they never looked at it except from a single side, 
and besides they could not develop in their students a quality 
they did not themselves possess. 

For the pure geometer himself, this faculty is necessary; it 
is by logic one demonstrates, by intuition one invents. To know 
how to criticize is good, to know how to create is better. You 
know how to recognize if a combination is correct; what a pre- 
dicament if you have not the art of choosing among all the pos- 
sible combinations. Logic tells us that on such and such a way 
we are sure not to meet any obstacle ; it does not say which way 
leads to the end. For that it is necessary to see the end from 
afar, and the faculty which teaches us to see is intuition. With- 
out it the geometer would be like a writer who should be versed 
in grammar but had no ideas. Now how could this faculty 
develop if, as soon as it showed itself, we chase it away and pro- 
scribe it, if we learn to set it at naught before knowing the 
good of it. 

And here permit a parenthesis to insist upon the importance of 
written exercises. Written compositions are perhaps not suflS- 
ciently emphasized in certain examinations, at the polytechnic 
school, for instance. I am told they would close the door 


against very good scholars who have mastered the course, thor- 
oughly understanding it, and who nevertheless are incapable of 
making the slightest application. I have just said the word 
understand has several meanings: such students only understand 
in the first way, and we have seen that suffices neither to make an 
engineer nor a geometer. Well, since choice must be made, I pre- 
fer those who understand completely. 

10. But is the art of sound reasoning not also a precious 
thing, which the professor of mathematics ought before all to 
cultivate! I take good care not to forget that. It should oc- 
cupy our attention and from the very beginning. I should be 
distressed to see geometry degenerate into I know not what tach- 
ymetry of low grade and I by no means subscribe to the extreme 
doctrines of certain German Oberlehrer. But there are occa- 
sions enough to exercise the scholars in correct reasoning in the 
parts of mathematics where the inconveniences I have pointed 
out do not present themselves. There are long chains of the- 
orems where absolute logic has reigned from the very first and, 
so to speak, quite naturally, where the first geometers have given 
us models we should constantly imitate and admire. 

It is in the exposition of first principles that it is necessary 
to avoid too much subtility ; there it would be most discouraging 
and moreover useless. We can not prove everything and we can 
not define everything ; and it will always be necessary to borrow 
from intuition; what does it matter whether it be done a little 
sooner or a little later, provided that in using correctly prem- 
ises it has furnished us, we learn to reason soundly. 

11. Is it possible to fulfill so many opposing conditions t Is 
this possible in particular when it is a question of giving a defi- 
nition T How find a concise statement satisfying at once the un- 
compromising rules of logic, our desire to grasp the place of the 
new notion in the totality of the science, our need of thinking 
with images! Usually it will not be found, and this is why it is 
not enough to state a definition; it must be prepared for and 

What does that meant You know it has often been said: 
every definition implies an assumption, since it affirms the exist- 
ence of the object defined. The definition then will not be ju»- 


tified, from the purely logical i>oint of view, until one shall have 
proved that it involves no contradiction, neither in the terms, 
nor with the verities previously admitted. 

But this is not enough ; the definition is stated to us as a con- 
vention ; but most minds will revolt if we wish to impose it ux>on 
them as an arbitrary convention* They will be satisfied only 
when you have answered numerous questions. 

Usually mathematical definitions, as M. Liard has shown, are 
veritable constructions built up wholly of more simple notions. 
But why assemble these elements in this way when a thousand 
other combinations were possible! 

Is it by caprice T If not, why had this combination more right 
to exist than all the others! To what need does it respond! 
How was it foreseen that it would play an important role in the 
development of the science, that it would abridge our reason- 
ings and our calculations! Is there in nature some familiar 
object which is so to speak the rough and vague image of it! 

This is not all; if you answer all these questions in a satis- 
factory manner, we shall see indeed that the new-bom had the 
right to be baptized; but neither is the choice of a name arbi- 
trary; it is needful to explain by what analogies one has been 
guided and that if analogous names have been given to different 
things, these things at least differ only in material and are allied 
in form; that their properties are analogous and so to say 

At this cost we may satisfy all inclinations. If the statement 
is correct enough to please the logician, the justification will 
satisfy the intuitive. But there is still a better procedure; 
wherever possible, the justification should precede the statement 
and prepare for it; one should be led on to the general state- 
ment by the study of some particular examples. 

Still another thing: each of the parts of the statement of a 
definition has as aim to distinguish the thing to be defined from 
a class of other neighboring objects. The definition will be un- 
derstood only when you have shown, not merely the object de- 
fined, but the neighboring objects from which it is proper to dis- 
tinguish it, when you have given a grasp of the difference and 
when you have added explicitly : this is why in stating the defini- 
tion I have said this or that. 


But it is time to leave generalities and examine how the some- 
what abstract principles I have expounded may be applied in 
arithmetic, geometry, analysis and mechanics. 


12. The whole number is not to be defined ; in return, one or- 
dinarily defines the operations upon whole numbers; I believe 
the scholars learn these definitions by heart and attach no mean- 
ing to them. For that there are two reasons : first they are made 
to learn them too soon, when their mind as yet feels no need of 
them ; then these definitions are not satisfactory from the logical 
point of view. A good definition for addition is not to be found 
just simply because we must stop and can not define everything. 
It is not defining addition to say it consists in adding. All that 
can be done is to start from a certain number of concrete exam- 
ples and say : the operation we have performed is called addition. 

For subtraction it is quite otherwise; it may be logically de- 
fined as the operation inverse to addition; but should we begin 
in that wayt Here also start with examples, show on these ex- 
amples the reciprocity of the two operations ; thus the definition 
will be prepared for and justified. 

Just so again for multiplication; take a particular problem; 
show that it may be solved by adding several equal numbers; 
then show that we reach the result more quickly by a multiplica- 
tion, an operation the scholars already know how to do by routine 
and out of that the logical definition will issue naturally. 

Division is defined as the operation inverse to multiplication; 
but begin by an example taken from the familiar notion of par- 
tition and show on this example that multiplication reproduces 
the dividend. 

There still remain the operations on fractions. The only 
difficulty is for multiplication. It is best to expound first the 
theory of proportion ; from it alone can come a logical definition ; 
but to make acceptable the definitions met at the beginning of 
this theory, it is necessary to prepare for them by numerous ex- 
amples taken from classic problems of the rule of three, taking 
pains to introduce fractional data. 

Neither should we fear to familiarize the scholars with the 


notion of proportion by geometric images, either by appealing to 
what they remember if they have already studied geometry, or 
in having recourse to direct intuition, if they have not studied 
it, which besides will prepare them to study it. Finally I shall 
add that after defining multiplication of fractions, it is needful 
to justify this definition by showing that it is commutative, as- 
sociative and distributive, and calling to the attention of the 
auditors that this is established to justify the definition. 

One sees what a role geometric images play in all this; and 
this role is justified by the philosophy and the history of the 
science. If arithmetic had remained free from all admixture 
of geometry, it would have known only the whole number ; it is 
to adapt itself to the needs of geometry that it invented any- 
thing else. 


In geometry we meet forthwith the notion of the straight line. 
Can the straight line be defined? The well-known definition, 
the shortest path from one point to another, scarcely satisfies 
me. I should start simply with the ruler and show at first to 
the scholar how one may verify a ruler by turning; this verifi- 
cation is the true definition of the straight line; the straight 
line is an axis of rotation. Next he should be shown how to 
verify the ruler by sliding and he would have one of the most 
important properties of the straight line. 

As to this other property of being the shortest path from one 
point to another, it is a theorem which can be demonstrated 
apodictically, but the demonstration is too delicate to find a place 
in secondary teaching. It will be worth more to show that a 
ruler previously verified fits on a stretched thread. In presence 
of difficulties like these one need not dread to multiply assump- 
tions, justifying them by rough experiments. 

It is needful to grant these assumptions, and if one admits a 
few more of them than is strictly necessary, the evil is not very 
great; the essential thing is to learn to reason soundly on the 
assumptions admitted. Uncle Sarcey, who loved to repeat, often 
said that at the theater the spectator accepts willingly all the 
postulates imposed upon him at the beginning, but the curtain 


once raised, he becomes uncompromising on the logic. Well, it 
is just the same in mathematics. 

For the circle, we may start with the compasses; the scholars 
will recognize at the first glance the curve traced; then make 
them observe that the distance of the two points of the instru- 
ment remains constant, that one of these points is fixed and the 
other movable, and so we shall be led naturally to the logical 

The definition of the plane implies an axiom and this need not 
be hidden. Take a drawing board and show that a moving ruler 
may be kept constantly in complete contact with this plane and 
yet retain three degrees of freedomu Compare with the cylin- 
der and the cone, surfaces on which an applied straight retains 
only two degrees of freedom; next take three drawing boards; 
show first that they will glide while remaining applied to one an- 
other and this with three degrees of freedom ; and finally to dis- 
tinguish the plane from the sphere, show that two of these boards 
which fit a third will fit each other. 

Perhaps you are surprised at this incessant employment of 
moving things; this is not a rough artifice; it is much more 
philosophic than one would at first think. What is geometry 
for the philosopher? It is the study of a group. And what 
group? That of the motions of solid bodies. How define this 
group then without moving some solids f 

Should we retain the classic definition of parallels and say 
parallels are two coplanar straights which do not meet, however 
far they be prolonged? No, since this definition is negative, 
since it is unverifiable by experiment, and consequently can not 
be regarded as an immediate datum of intuition. No, above all 
because it is wholly strange to the notion of group, to the consid- 
eration of the motion of solid bodies which is, as I have said, the 
true source of geometry. Would it not be better to define first 
the rectilinear translation of an invariable figure, as a motion 
wherein all the points of this figure have rectilinear trajectories; 
to show that such a translation is possible by making a square 
glide on a ruler? 

From this experimental ascertainment, set up as an assump- 
tion, it would be easy to derive the notion of parallel and 
Euclid's postulate itself. 



I need not return to the definition of velocity, or acceleration, 
or other kinematic notions; they may be advantageously con- 
nected with that of the derivative. 

I shall insist, on the other hand, upon the dynamic notions of 
force and mass. 

I am struck by one thing : how very far the young people who 
have received a high-school education are from applying to the 
real world the mechanical laws they have been taught. It is not 
only that they are incapable of it ; they do not even think of it 
For them the world of science and the world of reality are sepa- 
rated by an impervious partition wall. 

If we try to analyze the state of mind of our scholars, this will 
astonish us less. What is for them the real definition of force f 
Not that which they recite, but that which, crouching in a nook 
of their mind, from there directs it wholly. Here is the definition : 
forces are arrows with which one makes parallelograms. These 
arrows are imaginary things which have nothing to do with any- 
thing existing in nature. This would not happen if they had been 
shown forces in reality before representing them by arrows. 

How shall we define force 1 

I think I have elsewhere suflSciently shown there is no good 
logical definition. There is the anthropomorphic definition, the 
sensation of muscular eflfort ; this is really too rough and nothing 
useful can be drawn from it. 

Here is how we should go: first, to make known the genus 
force, we must show one after the other all the species of this 
genus; they are very numerous and very different; there is the 
pressure of fluids on the insides of the vases wherein they are 
contained ; the tension of threads ; the elasticity of a spring ; the 
gravity working on all the molecules of a body; friction; the 
normal mutual action and reaction of two solids in contact. 

This is only a qualitative definition; it is necessary to learn 
to measure force. For that begin by showing that one force may 
be replaced by another without destroying equilibrium ; we may 
find the first example of this substitution in the balance and 
Borda's double weighing. 

Then show that a weight may be replaced, not only by another 


weight, but by force of a different nature: for instance, Prony's 
brake permits replacing weight by friction. 

From all this arises the notion of the equivalence of two forces. 

The direction of a force must be defined. If a force F is equiv- 
alent to another force P' applied to the body considered by means 
of a stretched string, so that F may be replaced by P' without 
affecting the equilibrium, then the point of attachment of the 
string will be by definition the point of application of the force 
F', and that of the equivalent force F; the direction of the string 
will be the direction of the force F' and that of the equivalent 
force F. 

From that, pass to the comparison of the magnitude of forces. 
If a force can replace two others with the same direction, it 
equals their sum; show for example that a weight of 20 grams 
may replace two 10-gram weights. 

Is this enough f Not yet. We now know how to compare the 
intensity of two forces which have the same direction and same 
point of application ; we must learn to do it when the directions 
are different. For that, imagine a string stretched by a weight 
and passing over a pulley; we shall say that the tensor of the 
two legs of the string is the same and equal to the tension weight. 

This definition of ours enables us to compare the tensions of 
the two pieces of our string, and, using the preceding defini- 
tions, to compare any two forces having the same direction as 
these two pieces. It should be justified by showing that the 
tension of the last piece of the string remains the same for the 
same tensor weight, whatever be the number and the disposition 
of the reflecting pulleys. It has still to be completed by showing 
this is only true if the pulleys are frictionless. 

Once master of these definitions, it is to be shown that the 
point of application, the direction and the intensity suffice to 
determine a force ; that two forces for which these three elements 
are the same are always equivalent and may always be replaced 
by one another, whether in equilibrium or in movement, and this 
whatever be the other forces acting. 

It must be shown that two concurrent forces may always be 
replaced by a unique resultant ; and that this resultant remains 


the same, whether the body be at rest or in motion and whatever 
be the other forces applied to it. 

Finally it must be shown that forces thus defined satisfy the 
principle of the equality of action and reaction. 

Experiment it is, and experiment alone, which can teach us 
all that. It will suffice to cite certain common experiments, 
which the scholars make daily without suspecting it, and to per- 
form before them a few experiments, simple and well chosen. 

It is after having passed through all these meanders that one 
may represent forces by arrows, and I should even wish that in 
the development of the reasonings return were made from time 
to time from the symbol to the reality. For instance it would 
not be difficult to illustrate the parallelogram of forces by aid 
of an apparatus formed of three strings, passing over puUeys, 
stretched by weights and in equilibrium while pulling on the 
same point. 

Knowing force, it is easy to define mass ; this time the defini- 
tion should be borrowed from dynamics ; there is no way of doing 
otherwise, since the end to be attained is to give understanding 
of the distinction between mass and weight. Here again, the 
definition should be led up to by experiments ; there is in fact a 
machine which seems made expressly to show what mass is, 
Atwood's machine; recall also the laws of the fall of bodies, that 
the acceleration of gravity is the same for heavy as for light 
bodies, and that it varies with the latitude, etc. 

Now, if you tell me that all the methods I extol have long been 
applied in the schools, I shall rejoice over it more than be sur- 
prised at it. I know that on the whole our mathematical teach- 
ing is good. I do not wish it overturned ; that would even dis- 
tress me. I only desire betterments slowly progressive. This 
teaching should not be subjected to brusque oscillations under 
the capricious blast of ephemeral fads. In such tempests its 
high educative value would soon founder. A good and sound 
logic should continue to be its basis. The definition by example 
is always necessary, but it should prepare the way for the logical 
definition, it should not replace it; it should at least make this 
wished for, in the cases where the true logical definition can be 
advantageously given only in advanced teaching. 


Understand that what I have here said does not imply giving 
up what I have written elsewhere. I have often had occasion to 
criticize certain definitions I extol to-day. These criticisms hold 
good completely. These definitions can only be provisory. But 
it is by way of them that we must pass. 

Mathematics and Logic 


Can mathematics be reduced to logic without having to appeal 
to principles peculiar to mathematics? There is a whole school, 
abounding in ardor and full of faith, striving to prove it. They 
have their own special language, which is without words, using 
only signs. This language is understood only by the initiates, 
so that commoners are disposed to bow to the trenchant affirma- 
tions of the adepts. It is perhaps not unprofitable to examine 
these affirmations somewhat closely, to see if they justify the 
peremptory tone with which they are presented. 

But to make clear the nature of the question it is necessary to 
enter upon certain historical details and in particular to recall 
the character of the works of Cantor. 

Since long ago the notion of infinity had been introduced 
into mathematics; but this infinite was what philosophers call 
a becoming. The mathematical infinite was only a quantity 
capable of increasing beyond all limit: it was a variable quan- 
tity of which it could not be said that it had passed all limits, 
but only that it could pass them. 

Cantor has undertaken to introduce into mathematics an 
actual infinite, that is to say a quantity which not only is capable 
of passing all limits, but which is regarded as having already 
passed them. He has set himself questions like these : Are there 
more points in space than whole numbers? Are there more 
points in space than points in a plane? etc. 

And then the number of whole numbers, that of the points of 
space, etc., constitutes what he calls a transfinite cardinal number, 
that is to say a cardinal number greater than all the ordinary 
cardinal numbers. And he has occupied himself in comparing 
these transfinite cardinal numbers. In arranging in a proper 
order the elements of an aggregate containing an infinity of 



them, he has also imagined what he calls transfinite ordinal 
numbers upon which I shall not dwell. 

Many mathematicians followed his lead and set a series of 
questions of the sort. They so familiarized themselves with 
transfinite numbers that they have come to make the theory of 
finite numbers depend upon that of Cantor's cardinal numbers. 
In their eyes, to teach arithmetic in a way truly logical, one 
should begin by establishing the general properties of trans- 
finite cardinal numbers, then distinguish among them a very 
small class, that of the ordinary whole numbers. Thanks to this 
d6tour, one might succeed in proving all the propositions relative 
to this little class (that is to say all our arithmetic and our 
algebra) without using any principle foreign to logic. This 
method is evidently contrary to all sane psychology; it is cer- 
tainly not in this way that the human mind proceeded in con- 
structing mathematics ; so its authors do not dream, I think, of 
introducing it into secondary teaching. But is it at least logic, 
or, better, is it correct 1 It may be doubted. 

The geometers who have employed it are however very numer- 
ous. They have accumulated formulas and they have thought 
to free themselves from what was not pure logic by writing 
memoirs where the formulas no longer alternate with explana- 
tory discourse as in the books of ordinary mathematics, but 
where this discourse has completely disappeared. 

Unfortunately they have reached contradictory results, what 
are called the caniorian aniinamies, to which we shall have 
occasion to return. These contradictions have not discouraged 
them and they have tried to modify their rules so as to make 
those disappear which had already shown themselves, without 
being sure, for all that, that new ones would not manifest 

It is time to administer justice on these exaggerations. I do 
not hope to convince them ; for they have lived too long in this 
atmosphere. Besides, when one of their demonstrations has 
been refuted, we are sure to see it resurrected with insignificant 
alterations, and some of them have already risen several times 
from their ashes. Such long ago was the Lernaean hydra with its 
famous heads which alwa]^s grew again. Hercules got through, 


since his hydra had only nine heads, or eleven ; bat here there are 
too many, some in England, some in Germany, in Italy, in 
France, and he would have to give up the straggle. So I appeal 
only to men of good judgment unprejudiced. 

In these latter years numerous works have been published on 
pure mathematics and the philosophy of mathematics, trying to 
separate and isolate the logical elements of mathematical reason- 
ing. These works have been analyzed and expounded veiy 
clearly by M. Couturat in a book entitled: The Principles of 

For M. Couturat, the new works, and in particular those of 
Russell and Peano, have finally settled the controversy, so long 
pending between Leibnitz and Kant. They have shown that 
there are no synthetic judgments a priori (Kant's phrase to 
designate judgments which can neither be demonstrated analyti- 
cally, nor reduced to identities, nor established experimentally), 
they have shown that mathematics is entirely reducible to logic 
and that intuition here plays no role. 

This is what M. Couturat has set forth in the work just cited ; 
this he says still more explicitly in his Kant jubilee discourse, 
so that I heard my neighbor whisper: **I well see this is the 
centenary of Kant's death,** 

Can we subscribe to this conclusive condemnation! I think 
not, and I shall try to show why. 


What strikes us first in the new mathematics is its purely 
formal character: **We think," says Hilbert, '* three sorts of 
things, which we shall call points, straights and planes. We 
convene that a straight shall be determined by two points, and 
that in place of saying this straight is determined by these two 
points, we may say it passes through these two points, or that 
these two points are situated on this straight." What these 
things are, not only we do not know, but we should not seek to 
know. We have no need to, and one who never had seen either 
point or straight or plane could geometrize as well as we. That 


the phrase to pctss through, or the phrase to be sitiiated upon 
may arouse in us no image, the first is simply a synonym of to 
be determined and the second of to determine. 

Thus, be it understood, to demonstrate a theorem, it is neither 
necessary nor even advantageous to know what it means. The 
geometer might be replaced by the logic piano imagined by 
Stanley Jevons ; or, if you choose, a machine might be imagined 
where the assumptions were put in at one end, while the theorems 
came out at the other, like the legendary Chicago machine where 
the pigs go in alive and come out transformed into hams and 
sausages. No more than these machines need the mathematician 
know what he does. 

I do not make this formal character of his geometry a reproach 
to Hilbert. This is the way he should go, given the problem he 
set himself. He wished to reduce to a minimum the number of 
the fundamental assumptions of geometry and completely enu- 
merate them ; now, in reasonings where our mind remains active, 
in those where intuition still plays a part, in living reasonings, 
so to speak, it is difficult not to introduce an assumption or a 
postulate which passes unperceived. It is therefore only after 
having carried back all the geometric reasonings to a form purely 
mechanical that he could be sure of having accomplished his 
design and finished his work. 

What Hilbert did for geometry, others have tried to do for 
arithmetic and analysis. Even if they had entirely succeeded, 
would the Kantians be finally condemned to silence f Perhaps 
not, for in reducing mathematical thought to an empty form, 
it is certainly mutilated. 

Even admitting it were established that all the theorems could 
be deduced by procedures purely analytic, by simple logical 
combinations of a finite number of assumptions, and that these 
assumptions are only conventions; the philosopher would still 
have the right to investigate the origins of these conventions, 
to see why they have been judged preferable to the contrary 

And then the logical correctness of the reasonings leading 
from the assumptions to the theorems is not the only thing 
which should occupy us. The rules of perfect logic, are th^ 


the whole of mathematics T As well say the whole art of play- 
ing chess reduces to the rules of the moves of the pieces. Among 
all the constructs which can be built up of the materials fur- 
nished by logic, choice must be made; the true geometer makes 
this choice judiciously because he is guided by a sure instinct, 
or by some vague consciousness of I know not what more pro- 
found and more hidden geometry, which alone gives value to the 
edifice constructed. 

To seek the origin of this instinct, to study the laws of this 
deep geometry, felt, not stated, would also be a fine employment 
for the philosophers who do not want logic to be all. But it is 
not at this point of view I wish to put myself, it is not thus I 
wish to consider the question. The instinct mentioned is neces- 
sary for the inventor, but it would seem at first we might do 
without it in studying the science once created. Well, what I 
wish to investigate is if it be true that, the principles of logic 
once admitted, one can, I do not say discover, but demonstrate, 
all the mathematical verities without making a new appeal to 


I once said no to this question :^ should our reply be modified 
by the recent works? My saying no was because '*the principle 
of complete induction*' seemed to me at once necessary to the 
mathematician and irreducible to logic. The statement of this 
principle is: **If a property be true of the number 1, and if we 
establish that it is true oi n-\-l provided it be of n, it will be 
true of all the whole numbers." Therein I see the mathematical 
reasoning par excellence. I did not mean to say, as has been 
supposed, that all mathematical reasonings can be reduced to 
an application of this principle. Examining these reasonings 
<;losely, we there should see applied many other analogous princi- 
ples, presenting the same essential characteristics. In this cate- 
gory of principles, that of complete induction is only the simplest 
of all and this is why I have chosen it as type. 

The current name, principle of complete induction, is not 
justified. This mode of reasoning is none the less a true mathe- 

1 See Science and Hypothesis, chapter I. 


matical induction which differs from ordinary induction only by 
its certitude. 


Definitions and Assumptions 

The existence of such principles is a difficulty for the uncom- 
promising logicians; how do they pretend to get out of itf The 
principle of complete induction, they say, is not an assumption 
properly so called or a i^nthetic judgment a priori; it is just 
simply the definition of whole number. It is therefore a simple 
convention. To discuss this way of looking at it, we must ex- 
amine a little closely the relations between definitions and 

Let us go back first to an article by M. Couturat on mathe- 
matical definitions which appeared in VEnseignement mathe- 
matique, a magazine published by Qauthier-Villars and by Georg 
at Geneva. We shall see there a distinction between the direct 
definition and the definition by postulates. 

**The definition by postulates," says M. Couturat, "applies, 
not to a single notion, but to a system of notions ; it consists in 
enumerating the fundamental relations which unite them and 
which enable us to demonstrate all their other properties; these 
relations are postulates.'' 

If previously have been defined all these notions but one, then 
this last will be by definition the thing which verifies these pos- 
tulates. Thus certain indemonstrable assumptions of mathe- 
matics would be only disguised definitions. This point of view 
is often legitimate ; and I have myself admitted it in regard for 
instance to Euclid's postulate. 

The other assumptions of geometry do not suffice to completely 
define distance ; the distance then will be, by definition, among all 
the magnitudes which satisfy these other assumptions, that 
which is such as to make Euclid's postulate true. 

Well the logicians suppose true for the principle of complete 
induction what I admit for Euclid's postulate; they want to 
see in it only a disguised definition. 

But to give them this right, two conditions must be fulfilled. 
Stuart Mill says every definition implies an assumption, that by 
which the existence of the defined object is affirmed. According 


to that, it would no longer be the assumption which might be a 
disguised definition, it would on the contrary be the definition 
which would be a disguised assumption. Stuart Mill meant the 
word existence in a material and empirical sense; he meant to 
say that in defining the circle we afSrm there are round things in 

Under this form, his opinion is inadmissible. Mathematics is 
independent of the existence of material objects ; in mathematics 
the word exist can have only one meaning, it means free from 
contradiction. Thus rectified, Stuart Mill's thought becomes 
exact ; in defining a thing, we affirm that the definition implies no 

If therefore we have a system of postulates, and if we can 
demonstrate that these postulates imply no contradiction, we 
shall have the right to consider them as representing the defini- 
tion of one of the notions entering therein. If we can not demon- 
strate that, it must be admitted without proof, and that then 
will be an assumption; so that, seeking the definition under the 
postulate, we should find the assumption under the definition. 

Usually, to show that a definition implies no contradiction, we 
proceed 61/ example, we try to make an example of a thing satis- 
fying the definition. Take the case of a definition by postulates ; 
we wish to define a notion A, and we say that, by definition, an 
A is anything for which certain postulates are true. If we can 
prove directly that all these postulates are true of a certain object 
B, the definition will be justified ; the object B will be an example 
of an A. We shall be certain that the postulates are not contra- 
dictory, since there are cases where they are all true at the same 

But such a direct demonstration by example is not always 

To establish that the postulates imply no contradiction, it is 
then necessary to consider all the propositions deducible from 
these postulates considered as premises, and to show that, among 
these propositions, no two are contradictory. If these proposi- 
tions are finite in number, a direct verification is possible. This 
case is infrequent and uninteresting. If these propositions are 
infinite in number, this direct verification can no longer be made; 


recourse must be had to procedures where in general it is neces- 
sary to invoke just this principle of complete induction which is 
precisely the thing to be proved. 

This is an explanation of one of the conditions the logicians 
should satisfy, and further on we shdU see they have not done it. 


There is a second. When we give a definition, it is to use it. 

We therefore shall find in the sequel of the exposition the 
word defined; have we the right to affirm, of the thing repre- 
sented by this word, the postulate which has served for definition f 
Yes, evidently, if the word has retained its meaning, if we do 
not attribute to it implicitly a different meaning. Now this is 
what sometimes happens and it is usually difficult to perceive it; 
it is needful to see how this word comes into our discourse, and 
if the gate by which it has entered does not imply in reality a 
definition other than that stated. 

This difficulty presents itself in all the applications of math- 
ematics. The mathematical notion has been given a definition 
very refined and very rigorous ; and for the pure mathematician 
all doubt has disappeared; but if one wishes to apply it to the 
physical sciences for instance, it is no longer a question of this 
pure notion, but of a concrete object which is often only a rough 
image of it. To say that this object satisfies, at least approx- 
imately, the definition, is to state a new truth, which experience 
alone can put beyond doubt, and which no longer has the char- 
acter of a conventional postulate. 

But without going beyond pure mathematics, we also meet the 
same difficulty. 

You give a subtile definition of numbers ; then, once this defini- 
tion given, you think no more of it ; because, in reality, it is not 
it which has taught you what number is; you long ago knew 
that, and when the word number further on is found under your 
pen, you give it the same sense as the first comer. To know what 
is this meaning and whether it is the same in this phrase or that, 
it is needful to see how you have been led to speak of number and 
to introduce this word into these two phrases. I shall not for 
the moment dilate upon this point, because we shall have occasion 
to return to it. 


Thus consider a word of which we have given explicitly a defi- 
nition A ; afterwards in the discourse we make a use of it which 
implicitly supposes another definition B, It is possible that 
these two definitions designate the same thing. But that this is 
so is a new truth which must either be demonstrated or admitted 
as an independent assumption. 

We shall see farther on that the logicians have not fulfilled the 
second condition any better than the first, 


The definitions of number are very numerous and very differ- 
ent ; I forego the enumeration even of the names of their authors. 
We should not be astonished that there are so many. If one 
among them was satisfactory, no new one would be given. If 
each new philosopher occupying himself with this question has 
thought he must invent another one, this was because he was not 
satisfied with those of his predecessors, and he was not satisfied 
with them because he thought he saw a petitio principii. 

I have always felt, in reading the writings devoted to this prob- 
lem, a profound feeling of discomfort ; I was always expecting to 
run against a petitio principii, and when I did not immediately 
perceive it, I feared I had overlooked it. 

This is because it is impossible to give a definition without 
using a sentence, and diflBcult to make a sentence without using 
a number word, or at least the word several, or at least a word 
in the plural. And then the declivity is slippery and at each 
instant there is risk of a fall into petitio principii. 

I shall devote my attention in what follows only to those of 
these definitions where the petitio principii is most ably con- 



The symbolic language created by Peano plays a very grand 
role in these new researches. It is capable of rendering some 
service, but I think M. Couturat attaches to it an exaggerated 
importance which must astonish Peano himself. 

The essential element of this language is certain algebraic 


signs which represent the different conjunctions: if, and, or, 
therefore. That these signs may be convenient is possible; but 
that they are destined to revolutionize all philosophy is a differ- 
ent matter. It is difficult to admit that the word if acquires, 
when written 'q, a virtue it had not when written if. This in- 
vention of Peano was first called pasigraphy, that is to say the 
art of writing a treatise on mathematics without using a single 
word of ordinary language. This name defined its range very 
exactly. Later, it was raised to a more eminent dignity by con- 
ferring on it the title of logistic. This word is, it appears, em- 
ployed at the Military Academy, to designate the art of the 
quartermaster of cavalry, the art of marching and cantoning 
troops; but here no confusion need be feared, and it is at once 
seen that this new name implies the design of revolutionizing 

We may see the new method at work in a mathematical memoir 
by Burali-Forti, intitled: Una Questione sui numeri transfiniti, 
inserted in Volume XI of the Bendiconti del circolo matematico 
di Palermo. 

I begin by saying this memoir is very interesting, and my tak- 
ing it here as example is precisely because it is the most im- 
portant of all those written in the new language. Besides, the un- 
initiated may read it, thanks to an Italian interlinear translation. 

What constitutes the importance of this memoir is that it has 
given the first example of those antinomies met in the study of 
transfinite numbers and making since some years the despair of 
mathematicians. The aim, says Burali-Forti, of this note is to 
show there may be two transfinite numbers (ordinals), a and b, 
such that a is neither equal to, greater than, nor less than b. 

To reassure the reader, to comprehend the considerations which 
follow, he has no need of knowing what a transfinite ordinal 
number is. 

Now, Cantor had precisely proved that between two transfinite 
numbers as between two finite, there can be no other relation 
than equality, or inequality in one sense or the other. But it is 
not of the substance of this memoir that I wish to speak here; 
that would carry me much too far from my subject ; I only wish 
to consider the form, and just to ask if this form makes it gain 


much in rigor and whether it thus compensates for the efforts it 
imposes upon the writer and the reader. 
First we see Burali-Forti define the number 1 as follows: 

a definition eminently fitted to give an idea of the number 1 to 
persons who had never heard speak of it. 

I understand Peanian too ill to dare risk a critique, but still I 
fear this definition contains a petitio principii, considering that 
I see the figure 1 in the first member and Tin in letters in the 

However that may be, Burali-Forti starts from this definition 
and, after a short calculation, reaches the equation : 

(27) leNo, 

which tells us that One is a number. 

And since we are on these definitions of the first numbers, we 
recall that M. Couturat has also defined and 1. 

What is zero T It is the number of elements of the null dass. 
And what is the null class t It is that containing no element. 

To define zero by null, and null by no, is really to abuse the 
wealth of language ; so M. Couturat has introduced an improve- 
ment in his definition, by writing : 

which means : zero is the number of things satisfying a condition 
never satisfied. 

But as never means in no case I do not see that the progress 
is great. 

I hasten to add that the definition M. Couturat gives of the 
number 1 is more satisfactory. 

One, says he in substance, is the number of elements in a class 
in which any two elements are identical. 

It is more satisfactory, I have said, in this sense that to define 
1, he does not use the word one; in compensation, he uses the 
word two. But I fear, if asked what is two, M. Couturat would 
have to use the word one. 



But to return to the memoir of Burali-Forti ; I have said his 
conclusions are in direct opposition to those of Cantor. Now, one 
day M. Hadamard came to see me and the talk fell upon this 

"Burali-Forti's reasoning," I said, **does it not seem to you 
irreproachable!" *'No, and on the contrary I find nothing to 
object to in that of Cantor. Besides, Burali-Forti had no right 
to speak of the aggregate of all the ordinal numbers." 

'' Pardon, he had the right, since he could always put 


I should like to know who was to prevent him, and can it be 
said a thing does not exist, when we have called it OT" 

It was in vain, I could not convince him (which besides would 
have been sad, since he was right). Was it merely because I do 
not speak the Peanian with enough eloquence! Perhaps; but 
between ourselves I do not think so. 

Thus, despite all this pasigraphic apparatus, the question was 
not solved. What does that prove t In so far as it is a question 
only of proving one a number, pasigraphy sufSces, but if a diffi- 
culty presents itself, if there is an antinomy to solve, pasigraphy 
becomes impotent. 

The New Logics 

The SusseU Logic 

To justify its pretensions, logic had to change. We hxve seen 
new logics arise of which the most interesting is that of BnasdL 
It seems he has nothing new to write about formal logic, as if 
Aristotle there had touched bottom. But the domain BusseD 
attributes to logic is infinitely more extended than that of the 
classic logic, and he has put forth on the subject views which are 
original and at times well warranted. 

First, Bussell subordinates the logic of classes to that of prop- 
ositions, while the logic of Aristotle was above all the logic of 
classes and took as its point of departure the relation of subject 
to predicate. The classic syllogism, ''Socrates is a man," etc., 
gives place to the hypothetical syllogism: **If A is true, B is 
true; now if B is true, C is true," etc. And this is, I think, a 
most happy idea, because the classic syllogism is easy to carry 
back to the hypothetical syllogism, while the inverse transfor- 
mation is not without diflSculty. 

And then this is not all. Russeirs logic of propositions is the 
study of the laws of combination of the conjunctions if, and, or, 
and the negation not. 

In adding here two other conjunctions and and or, Russell 
opens to logic a new field. The symbols and, or follow the same 
laws as the two signs X and +, that is to say the commutative 
associative and distributive laws. Thus and represents logical 
multiplication, while or represents logical addition. This also 
is very interesting. 

Russell reaches the conclusion that any false proposition im- 
plies all other propositions true or false. M. Couturat says this 
conclusion will at first seem paradoxical. It is sufficient how- 
ever to have corrected a bad thesis in mathematics to recognize 



how right Bussell is. The candidate often is at great pains to 
get the first false equation; but that once obtained, it is only 
sport then for him to accumulate the most surprising results, 
some of which even may be true. 


We see how much richer the new logic is than the classic logic ; 
the symbols are multiplied and allow of varied combinations 
which are no longer limited in number. Has one the right to 
give this extension to the meaning of the word logic t It would 
be useless to examine this question and to seek with Bussell a 
mere quarrel about words. Grant him what he demands; but be 
not astonished if certain verities declared irreducible to logic 
in the old sense of the word find themselves now reducible to 
logic in the new sense — something very different. 

A great number of new notions have been introduced, and 
these are not simply combinations of the old. Bussell knows 
this, and not only at the beginning of the first chapter, 'The 
Logic of Propositions,' but at the beginning of the second and 
third, 'The Logic of Classes' and 'The Logic of Belations,' he 
introduces new words that he declares indefinable. 

And this is not all; he likewise introduces principles he de- 
clares indemonstrable. But these indemonstrable principles are 
appeals to intuition, synthetic judgments a priori. We regard 
them as intuitive when we meet them more or less explicitly 
enunciated in mathematical treatises; have they changed char- 
acter because the meaning of the word logic has been enlarged 
and we now find them in a book entitled Treatise on Logic? 
They have not changed nature; they have only changed place. 


Could these principles be considered as disguised definitions! 
It would then be necessary to have some way of proving that 
they imply no contradiction. It would be necessary to establish 
that, however far one followed the series of deductions, he would 
never be exposed to contradicting himself. 

We might attempt to reason as follows: We can verify that 


the operations of the new logic applied to premises exempt from 
contradiction can only give consequences equally exempt from 
contradiction* If therefore after n operations we have not met 
contradiction, we shall not encounter it after n -{-1. Thus it is 
impossible that there should be a moment when contradiction 
begins, which shows we shall never meet it. Have we the right to 
reason in this way t No, for this would be to make use of com- 
plete induction ; and remember, we do not yet know the principU 
of complete induction. 

We therefore have not the right to regard these assumptions 
as disguised definitions and only one resource remains for us, to 
admit a new act of intuition for each of them. Moreover I be- 
lieve this is indeed the thought of Bussell and M. Coutorat. 

Thus each of the nine indefinable notions and of the twenty 
indemonstrable propositions (I believe if it were I that did the 
counting, I should have found some more) which are the founda- 
tion of the new logic, logic in the broad sense, presupposes a new 
and independent act of our intuition and (why not say itt) a 
veritable synthetic judgment a priori. On this point all seem 
agreed, but what Bussell claims, and what seems to me doubtful, 
is that after these appeals to intuition, that will be the end of it; 
we need make no otlicrs and can build all mathematics without 
the intervention of any new element. 


M. Couturat often repeats that this new logic is altogether in- 
dependent of the idea of number. I shall not amuse myself by 
counting how many numeral adjectives his exposition contains, 
both cardinal and ordinal, or indefinite adjectives such as several. 
We may cite, however, some examples: 

**The logical product of two or more propositions is . . ."; 

**A11 propositions are capable only of two values, true and 

The relative product of two relations is a relation"; 
A relation exists between two terms,'' etc., etc. 

Sometimes this inconvenience would not be unavoidable, but 
sometimes also it is essential. A relation is incomprehensible 

( ( 
( { 


without two terms; it is impossible to have the intuition of the 
relation, without having at the same time that of its two terms, 
and without noticing they are two, because, if the relation is to 
be conceivable, it is necessary that there be two and only two. 


I reach what M. Couturat calls the ordinal theory which is 
the foundation of arithmetic properly so called. M. Couturat 
begins by stating Peano's five assumptions, which are independ- 
ent, as has been proved by Peano and Padoa. 

1. Zero is an integer. 

2. Zero is not the successor of any integer. 

3. The successor of an integer is an integer. 
To this it would be proper to add, 

Every integer has a successor. 

4. Two integers are equal if their successors are. 

The fifth assumption is the principle of complete induction. 

M. Couturat considers these assumptions as disguised defini- 
tions; they constitute the definition by postulates of zero, of 
successor, and of integer. 

But we have seen that for a definition by postulates to be 
acceptable we must be able to prove that it implies no contra- 

Is this the case heret Not at all. 

The demonstration can not be made by example. We can not 
take a part of the integers, for instance the first three, and 
prove they satisfy the definition. 

If I take the series 0, 1, 2, I see it fulfils the assumptions 1, 
2, 4 and 5 ; but to satisfy assumption 3 it still is necessary that 
3 be an integer, and consequently that the series 0, 1, 2, 3, fulfil 
the assumptions; we might prove that it satisfies assumptions 
1, 2, 4, 5, but assumption 3 requires besides that 4 be an integer 
and that the series 0, 1, 2, 3, 4 fulfil the assumptions, and so on. 

It is therefore impossible to demonstrate the assumptions for 
certain integers without proving them for all; we must give up 
proof by example. 


It is necessary then to take all the consequences of our aasomp- 
tions and see if they contain no contradiction* 

If these consequences were finite in number, this would be 
easy; but they are infinite in number; they are the whole of 
mathematics, or at least all arithmetic 

What then is to be done! Perhaps strictly we could repeat 
the reasoning of number III. 

But as we have said, this reasoning is complete induction, and 
it is precisely the principle of complete induction whose justifi- 
cation would be the point in question. 


The Logic of Hilbert 

I come now to the capital work of Hilbert which he com- 
municated to the Congress of Mathematicians at Heidelberg, and 
of which a French translation by M. Pierre Boutroux appeared 
in VEnseignement mathematique, while an English translation 
due to Halsted appeared in The Monist.^ In this work, which 
contains profound thoughts, the author's aim is analogous to 
that of Russell, but on many points he diverges from his 

**But,'' he says {Monist, p. 340), **on attentive consideration 
we become aware that in the usual exposition of the laws of logic 
certain fundamental concepts of arithmetic are already employed ; 
for example, the concept of the aggregate, in part also the concept 
of number. 

'* We fall thus into a vicious circle and therefore to avoid para- 
doxes a partly simultaneous development of the laws of logic and 
arithmetic is requisite." 

We have seen above that what Hilbert says of the principles 
of logic in the usual exposition applies likewise to the logic of 
Russell. So for Russell logic is prior to arithmetic ; for Hilbert 
they are * simultaneous.' We shall find further on other differ- 
ences still greater, but we shall point them out as we come 
to them. I prefer to follow step by step the development 
of Hilbert 's thought, quoting textually the most important 

i*The Foundations of Logic and Arithmetic,' Monistf XV., 338-352. 


"Let us take as the basis of our consideration first of all a 
thought-thing 1 (one) " (p. 341). Notice that in so doing we in 
no wise imply the notion of number, because it is understood that 
1 is here only a symbol and that we do not at all seek to know 
its meaning. ''The taking of this thing together with itself 
respectively two, three or more times. ..." Ah ! this time it is 
no longer the same; if we introduce the words 'two,' 'three,' and 
above all 'more,' 'several,' we introduce the notion of number; 
and then the definition of finite whole number which we shall 
presently find, will come too late. Our author was too circum- 
spect not to perceive this begging of the question. So at the end 
of his work he tries to proceed to a truly patching-up process. 

Hilbert then introduces two simple objects 1 and =, and con- 
siders all the combinations of these two objects, all the combina- 
tions of their combinations, etc. It goes without saying that we 
must forget the ordinary meaning of these two signs and not 
attribute any to them. 

Afterwards he separates these combinations into two classes, 
the class of the existent and the class of the non-existent, and 
till further orders this separation is entirely arbitrary. Every 
affirmative statement tells us that a certain combination belongs 
to the class of the existent; every negative statement tells us that 
a certain combination belongs to the class of the non-existent. 


Note now a difference of the highest importance. For Russell 
any object whatsoever, which he designates by a;, is an object 
absolutely undetermined and about which he supposes nothing; 
for Hilbert it is one of the combinations formed with the symbols 
1 and = ; he could not conceive of the introduction of anything 
other than combinations of objects already defined. Moreover 
Hilbert formulates his thought in the neatest way, and I think 
I must reproduce in extenso his statement (p. 348) : 

"In the assumptions the arbitraries (as equivalent for the 
concept 'every' and 'all' in the customary logic) represent only 
those thought-things and their combinations with one another, 
which at this stage are laid down as fundamental or are to be 



newly defined. Therefore in the dedaetion of inferences from 
the amimptiong, the arbitraries, which oceor in the assump- 
tions, can be replaced onlj by sach thonght-things and their 

"Also we most duly remember, that through the saper-addi- 
tion and making fundamental of a new thought-thing the pre- 
ceding assumptions undergo an enlargement of their validity, 
and where necessary, are to be subjected to a change in con- 
formity with the sense." 

The contrast with Russell's ^ew-point is complete. For this 
philosopher we may substitute for x not only objects already 
known, but anything. 

Russell is faithful to his point of view, which is that of com- 
prehension* He starts from the general idea of being, and 
enriches it more and more while restricting it, by adding new 
qualities. Hilbert on the contrary recognizes as possible beings 
only combinations of objects already known ; so that (looking at 
only one side of his thought) we might say he takes the view- 
point of extension. 


Let us continue with the exposition of Hilbert 's ideas. He 
introduces two assumptions which he states in his symbolic 
language but which signify, in the language of the uninitiated, 
that every quality is equal to itself and that every operation per- 
formed upon two identical quantities gives identical results. 

So stated, they are evident, but thus to present th