Science and method

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Science and method.

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CIENCE AND
METHOD

BY

HENRI POINCARE. TRANSLATED
BY FRANCIS MAITLAND. WITH
A PREFACE BY THE HON.
BERTRAND RUSSELL, F.R.S.

THOMAS NELSON AND SONS
LONDON, EDINBURGH, DUBLIN, & NEW YORK

tEV.

'75-

CONTENTS

Preface

II. MATHEMATICAL REASONING.
[IJThe Relativity of Space

II. Mathematical Definitions and Education
III. Mathematics and Logic
IV". The New Logics

V. The Last Efforts of the Logisticians

5

Introduction ........ 9

I. THE SCIENTIST AND SCIENCE.
I. The Selection of Facts . . . .15
II. The Future of Mathematics . . .25

III. Mathematical Discovery .... 46

IV. Chance 64

93
117

143
160

177

IIL THE NEW MECHANICS.

I. Mechanics and Radium 199

II. Mechanics and Optics 213

IIL The New Mechanics and Astronomy . -235

IV. ASTRONOMICAL SCIENCE.

I. The Milky Way and the Theory of Gases . 253

II. French Geodesy 270

General Conclusions 284

PREFACE.

Henri Poincar^ was, by general agreement, the
most eminent scientific man of his generation — more
eminent, one is tempted to think, than any man of
science now living. From the mere variety of the
subjects which he illuminated, there is certainly no
one who can appreciate critically the whole of his
work. Some conception of his amazing comprehen-
siveness may be derived from the obituary number of
the Revue de Mitaphysique et de Morale (September
1913), where, in the course of 130 pages, four eminent
men — a philosopher, a mathematician, an astronomer,
and a physicist — tell in outline the contributions which
he made to their several subjects. In all we find the
same characteristics — swiftness, comprehensiveness,
unexampled lucidity, and the perception of recondite
but fertile analogies.

Poincare's philosophical writings, of which the pres-
ent volume is a good example, are not those of a
professional philosopher : they are the untrammelled
reflections of a broad and cultivated mind upon the
procedure and the postulates of scientific discovery.
The writing of professional philosophers on such sub-
jects has too often the deadness of merely external
description ; Poincar6's writing, on the contrary, as
the reader of this book may see in his account of
mathematical invention, has the freshness of actual
experience, of vivid, intimate contact with what he is

6 PREFACE.

describing. There results a certain richness and
resonance in his words : the sound emitted is not
hollow, but comes from a great mass of which only
the polished surface appears. His wit, his easy mas-
tery, and his artistic love of concealing the labour of
thought, may hide from the non-mathematical reader
the background of solid knowledge from which his
apparent paradoxes emerge : often, behind what may
seem a light remark, there lies a whole region of
mathematics which he himself has helped to explore.

A philosophy of science is growing increasingly
necessary at the present time, for a variety of reasons.
Owing to increasing specialization, and to the con-
stantly accelerated accumulation of new facts, the
general bearings of scientific systems become more
and more lost to view, and the synthesis that depends
on coexistence of multifarious knowledge in a single
mind becomes increasingly difficult. In order to over-
come this difficulty, it is necessary that, from time to
time, a specialist capable of detachment from details
should set forth the main lines and essential structure
of his science as it exists at the moment. But it is
not results, which are what mainly interests the man
in the street, that are what is essential in a science :
what is essential is its method, and it is with method
that Poincar^'s philosophical writings are concerned.

Another reason which makes a philosophy of science
specially useful at the present time is the revolutionary
progress, the sweeping away of what had seemed fixed
landmarks, which has so far characterized this century,
especially in physics. The conception of the " working
hypothesis,'' provisional, approximate, and merely use-
ful, has more and more pushed aside the comfortable

PREFACE. 7

eighteenth century conception of "laws of nature."
Even the Newtonian dynamics, which for over two
hundred years had seemed to embody a definite con-
quest, must now be regarded as doubtful, and as
probably only a first rough sketch of the ways of
matter. And thus, in virtue of the very rapidity of
our progress, a new theory of knowledge has to be
sought, more tentative and more modest than that of
more confident but less successful generations. Of
this necessity Poincar6 was acutely conscious, and it
gave to his writings a tone of doubt which was hailed
with joy by sceptics and pragmatists. But he was in
truth no sceptic : however conscious of the difficulty
of attaining knowledge, he never admitted its impos-
sibilit)'. " It is a mistake to believe," he said, " that the
love of truth is indistinguishable from the love of cer-
tainty ;" and again: "To doubt everything or to believe
everything are two equally convenient solutions ; both
dispense with the necessity of reflection." His was the
active, eager doubt that inspires a new scrutiny, not
the idle doubt that acquiesces contentedly in nescience-
'Two opposite and conflicting qualities are required
for the successful practice of philosophy — comprehen-
siveness of outlook, and minute, patient analysis. Both
exist in the highest degree in Descartes and Leibniz ;
but in their day comprehensiveness was less difficult
than it is now. Since Leibniz, I do not know of any
philosopher who has possessed both : broadly speaking,
British philosophers have excelled in analysis, while
those of the Continent have excelled in breadth and
scope. In this respect, Poincare is no exception : in
philosophy, his mind was intuitive and synthetic;
wonderfully skilful, it is true, in analysing a science

8 PREFACE.

until he had extracted its philosophical essence, and
in combining this essence with those of other sciences,
but not very apt in those further stages of analysis
which fall within the domain of philosophy itself. He
built wonderful edifices with the philosophic materials
that he found ready to hand, but he lacked the patience
and the minuteness of attention required for the crea-
tion of new materials. For this reason, his philosophy,
though brilliant, stimulating, and instructive, is not
among those that revolutionize fundamentals, or com-
pel us to remould our imaginative conception of the
nature of things. In fundamentals, broadly speaking,
he remained faithful to the authority of Kant.

Readers of the following pages will not be surprised
to learn that his criticisms of mathematical logic do
not appear to me to be among the best parts of his
work. He was already an old man when he became
aware of the existence of this subject, and he was led, by
certain indiscreet advocates, to suppose it in some way
opposed to those quick flashes of insight in mathe-
matical discovery which he has so admirably described-
No such opposition in fact exists ; but the misconcep-
tion, however regrettable, was in no way surprising.

To be always right is not possible in philosophy ;
but Poincar^'s opinions, right or wrong, are always the
expression of a powerful and original mind with a
quite unrivalled scientific equipment ; a masterly style,
great wit, and a profound devotion to the advance-
ment of knowledge. Through these merits, his books
supply, better than any others known to me, the
growing need for a generally intelligible account of
the philosophic outcome of modern science.

Bertrand Russell.

INTRODUCTION.

In this work I have collected various studies which are
more or less directly concerned with scientific metho-
dology. [The scientific method consists in observation
and experiment. If the scientist had an infinity of
time at his disposal, it would be sufficient to say to
him, " Look, and look carefully." But, since he has
not time to look at everything, and above all to look
carefully, and since it is better not to look at all than
to look carelessly, he is forced to make a selection.
The first question, then, is to know how to make this
selection. This question confronts the physicist as
well as the historian ; it also confronts the mathema-
tician, and the principles which should guide them all
are not very dissimilar. The scientist conforms to
them instinctively, and by reflecting on these principles
one can foresee the possible future of mathematics.

We shall understand this still better if we observe
the scientist at work ; and, to begin with, we must have
some acquaintance with the psychological mechanism
of discovery, more especially that of mathematical dis-
covery. Observation of the mathematician's method
of working is specially instructive for the psychologist.

In all sciences depending on observation, we must

TO INTRODUCTION.

reckon with errors due to imperfections of our senses
and of our instruments. Happily we may admit that,
under certain conditions, there is a partial compensa-
tion of these errors, so that they disappear in averages.
This compensation is due to chance. But what is
chance? It is a notion which is difficult of justilica-
tion, and even of definition ; and yet what I have just
said with regard to errors of observation, shows that
the scientist cannot get on without it. It is necessary,
therefore, to give as accurate a definition as possible
of this notion, at once so indispensable and so elusive.

These are generalities which apply in the main to
all sciences. For instance, there is no appreciable
difference between the mechanism of mathematical
discovery and the mechanism of discovery in general.
Further on I approach questions more particularly
concerned with certain special sciences, beginning with
pure mathematics.

In the chapters devoted to them, I am obliged to
treat of somewhat more abstract subjects, and, to begin
with, I have to speak of the notion of space. Every one
knows that space is relative, or rather every one says
so, but how many people think still as if they con-
sidered it absolute. Nevertheless, a little reflection
will show to what contradictions they are exposed.

Questions concerning methods of instruction are of
importance, firstly, on their own account, and secondly,
because one cannot reflect on the best method of
imbuing virgin brains with new notions without, at
the same time, reflecting on the manner in which
these notions have been acquired by our ancestors,
and consequently on their true origin — that is in
reality, on their true nature. Why is it that, in most

INTRODUCTION. ii

cases, the definitions which satisfy scientists mean
nothing at all to children? Why is it necessary to
give them other definitions ? This is the question I
have set myself in the chapter which follows, and its
solution might, I think, suggest useful reflections to
philosophers interested in the logic of sciences.

On the other hand, there are many geometricians
who believe that mathematics can be reduced to the
rules of formal logic. Untold efforts have been made
in this direction. To attain their object they have not
hesitated, for instance, to reverse the historical order of
the genesis of our conceptions, and have endeavoured
to explain the finite by the infinite. I think I have suc-
ceeded in showing, for all who approach the problem
with an open mind, that there is in this a deceptive
illusion. I trust the reader will understand the im-
portance of the question, and will pardon the aridity
of the pages I have been constrained to devote to it.

The last chapters, relating to mechanics and astron-
omy, will be found easier reading.

Mechanics seem to be on the point of undergoing a
complete revolution. The ideas which seemed most
firmly established are being shattered by daring
innovators. It would certainly be premature to
decide in their favour from the start, solely because
they are innovators ; but it is interesting to state
their views, and this is what I have tried to do. As
far as possible I have followed the historical order,
for the new ideas would appear too surprising if we
did not see the manner in which they had come into
existence.

Astronomy offers us magnificent spectacles, and
raises tremendous problems. We cannot dream of

12 INTRODUCTION.

applying the experimental method to them directly ;
our laboratories are too small. But analogy with the
phenomena which these laboratories enable us to reach
may nevertheless serve as a guide to the astronomer.
The Milky Way, for instance, is an assemblage of suns
whose motions appear at first sight capricious. But
may not this assemblage be compared with that of
the molecules of a gas whose properties we have
learnt from the kinetic theory of gases ? Thus the
method of the physicist may come to the aid of the
astronomer by a side-track.

Lastly, I have attempted to sketch in a few lines the
history of the development of French geodesy. I have
■shown at what cost, and by what persevering efforts
and often dangers, geodesists have secured for us the
few notions we possess about the shape of the earth.
Is this really a question of method ? Yes, for this
history certainly teaches us what precautions must
surround any serious scientific operation, and what
time and trouble are involved in the conquest of a
single new decimal.

BOOK I.
THE SCIENTIST AND SCIENCE.

I.

THE SELECTION OF FACTS.

Tolstoi explains somewhere in his writings why, in
his opinion, " Science for Science's sake " is an absurd
conception. We cannot know all the facts, since they
are practically infinite in number. We must make a
selection ; and that being so, can this selection be
-governed by the mere caprice of our curiosity? Is
it not better to be guided by utility, by our practical,
and more especially our moral, necessities ? Have we
not some better occupation than counting the number
of lady-birds in existence on this planet ?

It is clear that for him the word utility has not the
meaning assigned to it by business men, and, after
them, by the greater number of our contemporaries.
He cares but little for the industrial applications of
science, for the marvels of electricity or of auto-
mobilism, which he regards rather as hindrances to
moral progress. For him the useful is exclusively
what is capable of making men better.

It is hardly necessary for me to state that, for my
part, I could not be satisfied with either of these
ideals. I have no liking either for a greedy and
narrow plutocracy, or for a virtuous unaspiring
democracy, solely occupied in turning the other

i6 SCIENCE AND METHOD.

cheek, in which we should find good people devoid of
curiosity, who, avoiding all excesses, would not die
of any disease — save boredom. But it is all a matter
of taste, and that is not the point I wish to discuss.

None the less the question remains, and it claims
our attention. If our selection is only determined by
caprice or by immediate necessity, there can be no
science for science's sake, and consequently no science.
Is this true? There is no disputing the fact that a
selection must be made : however great our activity,
facts outstrip us, and we can never overtake them ;
while the scientist is discovering one fact, millions
and millions are produced in every cubic inch of his
body. Trying to make science contain nature is like
trying to make the part contain the whole.

But scientists believe that there is a hierarchy
of facts, and that a judicious selection can be made.
They are right, for otherwise there would be no science,
and science does exist. One has only to open one's
eyes to see that the triumphs of industry, which have
enriched so many practical men, would never have
seen the light if only these practical men had existed,
and if they had not been preceded by disinterested
fools who died poor, who never thought of the useful,
and yet had a guide that was not their own caprice.

What these fools did, as Mach has said, was to save
their successors the trouble of thinking. If they had
worked solely in view of an immediate application,
they would have left nothing behind them, and in face
of a new requirement, all would have had to be done
again. Now the majority of men do not like thinking,
and this is perhaps a good thing, since instinct guides
them, and very often better than reason would o-uide
(1.777) '^

THE SELECTION OF FACTS. 17

a pure intelligence, at least whenever they are pursuing
an end that is immediate and always the same. But
instinct is routine, and if it were not fertilized by
thought, it would advance no further with man than
with the bee or the ant. It is necessary, therefore, to
think for those who do not like thinking, and as they
are many, each one of our thoughts must be useful
in as many circumstances as possible. For this
reason, the more general a law is, the greater is its
value.

This shows us how our selection should be made.
The most interesting facts are those which can be
used several times, those which have a chance of
recurring. We have been fortunate enough to be born
in a world where there are such facts. Suppose that
instead of eighty chemical elements we had eighty
millions, and that they were not some common and
others rare, but uniformly distributed. Then each
time we picked up a new pebble there would be a
strong probability that it was composed of some un-
known substance. Nothing that we knew of other
pebbles would tell us anything about it. Before each
new object we should be like a new-born child ; like
him we could but obey our caprices or our necessities.
In such a world there would be no science, perhaps
thought and even life would be impossible, since
evolution could not have developed the instincts of
self-preservation. Providentially it is not so ; but this
blessing, like all those to which we are accustomed, is
not appreciated at its true value. The biologist would
be equall}'- embarrassed if there were only individuals
and no species, and if heredity did not make children
resemble their parents.

(1,777J 2

i8 SCIENCE AND METHOD.

Which, then, are the facts that have a chance of
recurring? In the first place, simple facts. It is
evident that in a complex fact many circumstances
are united by chance, and that only a still more
improbable chance could ever so unite them again.
But are there such things as simple facts ? and if there
are, how are we to recognize them ? Who can tell
that what we believe to be simple does not conceal
an alarming complexity? All that we can say is
that we must prefer facts which appear simple, to
those in which our rude vision detects dissimilar
elements. Then only two alternatives are possible ;
either this simplicity is real, or else the elements
are so intimately mingled that they do not admit of
being distinguished. In the first case we have a
chance of meeting the same simple fact again, either
in all its purity, or itself entering as an element into
some complex whole. In the second case the intimate
mixture has similarly a greater chance of being re-
produced than a heterogeneous assemblage. Chance
can mingle, but it cannot unmingle, and a combination
of various elements in a well-ordered edifice in which
something can be distinguished, can only be made
deliberately. There is, therefore, but little chance that
an assemblage in which different things can be dis-
tinguished should ever be reproduced. On the other
hand, there is great probability that a mixture which
appears homogeneous at first sight will be reproduced
several times. Accordingly facts which appear simple,
even if they are not so in reality, will be more easily
brought about again by chance.

It is this that justifies the method instinctively
adopted by scientists, and what perhaps justifies it

THE SELECTION OF FACTS. 19

still better is that facts which occur frequently appear
to us simple just because we are accustomed to
them.

But where is the simple fact ? Scientists have tried
to find 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 only as a point and
qualitative differences disappear, and because a point
is simpler than a body which has shape and qualities.
The physicist, on the other hand, has sought the
elementary phenomenon in an imaginary division of
bodies into infinitely small atoms, because the con-
ditions of the problem, which undergo slow and con-
tinuous variations as we pass from one point of the
body to another, may be regarded as constant within
each of these little atoms. Similarly the biologist has
been led instinctively to regard the cell as more interest-
ing than the whole animal, and the event has proved
him right, since cells belonging to the most diverse
organisms have greater resemblances, for those who can
recognize them, than the organisms themselves. The
sociologist is in a more embarrassing position. The
elements, which for him are men, are too dissimilar, too
variable, too capricious, in a word, too complex them-
selves. Furthermore, history does not repeat itself;
how, then, is he to select the interesting fact, the fact
which is repeated ? Method is precisely the selection
of facts, and accordingly our first care must be to
devise a method. Many have been devised because
none holds the field undisputed. Nearly every socio-
logical thesis proposes a new method, which, however,
its author is very careful not to apply, so that sociology

20 SCIENCE AND METHOD.

is the science with the greatest number of methods
and the least results.

It is with regular facts, therefore, that we ought to
begin ; but as soon as the rule is well established, as
soon as it is no longer in doubt, the facts which are in
complete conformity with it lose their interest, since
they can teach us nothing new. Then it is the excep-
tion which becomes important. We cease to look for
resemblances, and apply ourselves before all else to
differences, and of these differences we select first
those that are most accentuated, not only because
they are the most striking, but because they will be
the most instructive. This will be best explained by a
simple example. Suppose we are seeking to determine
a curve by observing some of the points on it. The
practical man who looked only to immediate utility
would merely observe the points he required for some
special object ; these points would be badly distributed
on the curve, they would be crowded together in cer-
tain parts and scarce in others, so that it would be
impossible to connect them by a continuous line, and
they would be useless for any other application. The
scientist would proceed in a different manner. Since
he wishes to study the curve for itself, he will distribute
the points to be observed regularly, and as soon as he
knows some of them, he will join them by a regular
line, and he will then have the complete curve. But
how is he to accomplish this ? If he has determined
one extreme point on the curve, he will not remain
close to this extremity, but will move to the other end.
After the two extremities, the central point is the most
instructive, and so on.

Thus when a rule has been established, we have first

THE SELECTION OF FACTS. 21

to look for the cases in which the rule stands the best
chance of being found in fault. This is one of many
reasons for the interest of astronomical facts and of
geological ages. By making long excursions in
space or in time, we may find our ordinary rules
completely upset, and these great upsettings will give
us a clearer view and better comprehension of such
small changes as may occur nearer us, in the small
corner of the world in which we are called to live and
move. We shall know this corner better for the
journey we have taken into distant lands where we
had no concern.

1"" But what we must aim at is not so much to ascertain
resemblances and differences, as to discover similarities

Lhidden under apparent discrepancies. The individual
rules appear at first discordant, but on looking closer
we can generally detect a resemblance ; though differ-
ing in matter, they approximate in form and in the
order of their parts. When we examine them from
this point of view, we shall see them widen and tend
to embrace everything. This is what gives a value to
certain facts that come to complete a whole, and
show that it is the faithful image of other known
wholes.

I cannot dwell further on this point, but these few
words will suffice to show that the scientist does not
make a random selection of the facts to be observed.
He does not count lady-birds, as Tolstoi says, because
the number of these insects, interesting as they are, is
subject to capricious variations. He tries to condense
a great deal of experience and a great deal of thought
into a small volume, and that is why a little book on
physics contains so many past experiments, and a

22 SCIENCE AND METHOD.

thousand times as many possible ones, whose results
are known in advance.

But so far we have only considered one side of the
question. The scientist does not study nature because
it is useful to do so. He studies it because he takes
pleasure in it, and he takes pleasure in it because it is
beautiful. If nature were not beautiful it would not be
worth knowing, and life would not be worth living. I
am not speaking, of course, of that beauty which
strikes the senses, of the beauty of qualities and ap-
pearances. I am far from despising this, but it has
nothing to do with science. What I mean is that
more intimate beauty which comes from the harmo-
nious order of its parts, and which a pure intelligence
can grasp. It is this that gives a body a skeleton,
so to speak, to the shimmering visions that flatter
our senses, and without this support the beauty
of these fleeting dreams would be imperfect, because
it would be indefinite and ever elusive. Intellectual
beauty, on the contrary, is self-sufficing, and it is for
it, more perhaps than for the future good of humanity,
that the scientist condemns himself to long and painful
labours.

It is, then, the search for this special beauty, the
sense of the harmony of the world, that makes us
select the facts best suited to contribute to this har-
mony ; just as the artist selects those features of his
sitter which complete the portrait and give it character
and life. And there is no fear that this instinctive
and unacknowledged preoccupation will divert the
scientist from the search for truth. We may dream
of a harmonious world, but how far it will fall short
of the real world ! The Greeks, the greatest artists

THE SELECTION OF FACTS. 23

that ever were, constructed a heaven for themselves ;
how poor a thing it is beside the heaven as we know it !

It is because simpHcity and vastness are both beau-
tiful that we seek by preference simple facts and vast
facts ; that we take delight, now in following the giant
courses of the stars, now in scrutinizing with a micro-
scope that prodigious smallness which is also a vastness,
and now in seeking in geological ages the traces of a
past that attracts us because of its remoteness.

Thus we see that care for the beautiful leads us to
the same selection as care for the useful. Similarly
economy of thought, that economy of effort which,
according to Mach, is the constant tendency of science,
is a source of beauty as well as a practical advantage.
The buildings we admire are those in which the archi-
tect has succeeded in proportioning the means to the
end, in which the columns seem to carry the burdens
imposed on them lightly and without effort, like the
graceful caryatids of the Erechtheum.

Whence comes this concordance? Is it merely
that things which seem to us beautiful are those
which are best adapted to our intelligence, and that
consequently they are at the same time the tools that
intelligence knows best how to handle ? Or is it due
rather to evolution and natural selection ? Have the
peoples whose ideal conformed best to their own in-
terests, properly understood, exterminated the others
and taken their place? One and all pursued their
ideal without considering the consequences, but while
this pursuit led some to their destruction, it gave
empire to others. We are tempted to believe this,
for if the Greeks triumphed over the barbarians, and
if Europe, heir of the thought of the Greeks, dominates

24 SCIENCE AND METHOD.

the world, it is due to the fact that the savages loved
garish colours and the blatant noise of the drum, which
appealed to their senses, while the Greeks loved the
intellectual beauty hidden behind sensible beauty, and
that it is this beauty which gives certainty and strength
to the intelligence.

No doubt Tolstoi would be horrified at such a
triumph, and he would refuse to admit that it could
be truly useful. But this disinterested pursuit of truth
for its own beauty is also wholesome, and can make
men better. I know very well there are disappoint-
ments, that the thinker does not always find the
serenity he should, and even that some scientists have
thoroughly bad tempers.

Must we therefore say that science should be
abandoned, and morality alone be studied ? Does
any one suppose that moralists themselves are entirely
above reproach when they have come down from the
pulpit?

II.

THE FUTURE OF MATHEMATICS.

If we wish to foresee the future of mathematics, our
proper course is to study the history and present
condition of the science.

For us mathematicians, is not this procedure to
some extent professional ? We are accustomed to
extrapolation, which is a method of deducing the
future from the past and the present ; and since we
are well aware of its limitations, we run no risk of
deluding ourselves as to the scope of the results it
gives us.

In the past there have been prophets of ill. They
took pleasure in repeating that all problems suscep-
tible of being solved had already been solved, and that
after them there would be nothing left but gleanings.
Happily we are reassured by the example of the
past. Many times already men have thought that
they had solved all the problems, or at least that
they had made an inventory of all that admit of
solution. And then the meaning of the word solution
has been extended ; the insoluble problems have
become the most interesting of all, and other problems
hitherto undreamed of have presented themselves.
For the Greeks a good solution was one that em-

26 SCIENCE AND METHOD.

ployed only rule and compass ; later it became one
obtained by the extraction of radicals, then one in
which algebraical functions and radicals alone figured.
Thus the pessimists found themselves continually
passed over, continually forced to retreat, so that at
present I verily believe there are none left.

My intention, therefore, is not to refute them, since
they are dead. We know very well that mathematics
will continue to develop, but we have to find out in
what direction. I shall be told "in all directions,"
and that is partly true ; but if it were altogether true,
it would become somewhat alarming. Our riches
would soon become embarrassing, and their accumula-
tion would soon produce a mass just as impenetrable
as the unknown truth was to the ignorant.

The historian and the physicist himself must make
a selection of facts. The scientist's brain, which is
only a corner of the universe, will never be able to
contain the whole universe ; whence it follows that,
of the innumerable facts offered by nature, we shall
leave some aside and retain others. The same is
true, a fortiori, in mathematics. The mathematician
similarly cannot retain pell-mell all the facts that are
presented to him, the more so that it is himself— I was
almost going to say his own caprice — that creates these
facts. It is he who assembles the elements and con-
structs a new combination from top to bottom ; it is
generally not brought to him ready-made by nature.

No doubt it is sometimes the case that a mathe-
matician attacks a problem to satisfy some require-
ment of physics, that the physicist or the engineer
asks him to make a calculation in view of some par-
ticular application. Will it be said that we geometri-

THE FUTURE OF MATHEMATICS. 27

clans are to confine ourselves to waiting for orders,
and, instead of cultivating our science for our own
pleasure, to have no other care but that of accom-
modating ourselves to our clients' tastes? If the only
object of mathematics is to come to the help of those
who make a study of nature, it is to them we must
look for the word of command. Is this the correct
view of the matter ? Certainly not ; for if we had not
cultivated the exact sciences for themselves, we should
never have created the mathematical instrument, and
when the word of command came from the physicist
we should have been found without arms.

Similarly, physicists do not wait to study a phenom-
enon until some pressing need of material life makes
it an absolute necessity, and they are quite right. If
the scientists of the eighteenth century had dis-
regarded electricity, because it appeared to them
merely a curiosity having no practical interest, we
should not have, in the twentieth century, either
telegraphy or electro-chemistry or electro -traction.
Physicists forced to select are not guided in their
selection solely by utility. What method, then, do
they pursue in making a selection between the dif-
ferent natural facts ? I have explained this in the
preceding chapter. The facts that interest them are
those that may lead to the discovery of a law, those
• that have an analogy with many other facts and do
not appear to us as isolated, but as closely grouped
with others. The isolated fact attracts the attention
of all, of the layman as well as the scientist. But
what the true scientist alone can see is the link that
unites several facts which have a deep but hidden
analogy. The anecdote of Newton's apple is probably

28 SCIENCE AND METHOD.

not true, but it is symbolical, so we will treat it as if
it were true. Well, we must suppose that before
Newton's day many men had seen apples fall, but
none had been able to draw any conclusion. Facts
would be barren if there were not minds capable of
selecting between them and distinguishing those which
have something hidden behind them and recognizing
what is hidden — minds which, behind the bare fact,
can detect the soul of the fact

In mathematics we do exactly the same thing. Of
the various elements at our disposal we can form
millions of different combinations, but any one of
these combinations, so long as it is isolated, is ab-
solutely without value ; often we have taken great
trouble to construct it, but it is of absolutely no use,
unless it be, perhaps, to supply a subject for an exer-
cise in secondary schools. It will be quite different
as soon as this combination takes its place in a class
of analogous combinations whose analogy we have
recognized ; we shall then be no longer in presence of
a fact, but of a law. And then the true discoverer
will not be the workman who has patiently built up
some of these combinations, but the man who has
brought out their relation. The former has only seen
the bare fact, the latter alone has detected the soul of
the fact. The invention of a new word will often
be sufficient to bring out the relation, and the word
will be creative. The history of science furnishes us
with a host of examples that are familiar to all.

The celebrated Viennese philosopher Mach has said
that the part of science is to effect economy of thought,
just as a machine effects economy of effort, and this is
very true. The savage calculates on his fingers, or

THE FUTURE OF MATHEMATICS. 29

by putting togethei' pebbles. By teaching children the
multiplication table we save them later on countless
operations with pebbles. Some one once recognized,
whether by pebbles or otherwise, that 6 times 7
are 42, and had the idea of recording the result, and
that is the reason why we do not need to repeat the
operation. His time was not wasted even if he was
only calculating for his own amusement. His opera-
tion only took him two minutes, but it would have
taken two million, if a million people had had to
repeat it after him. ^

Thus the importance of a fact is measured by the
return it gives — that is, by the amount of thought it
enables us to economize.

In physics, the facts which give a large return are
those which take their place in a very general law,
because they enable us to foresee a very large number
of others, and it is exactly the same in mathematics.
Suppose I apply myself to a complicated calculation
and with much difficulty arrive at a result, I shall
have gained nothing by my trouble if it has not
enabled me to foresee the results of other analogous
calculations, and to direct them with certainty, avoid-
ing the blind groping with which I had to be con-
tented the first time. On the contrary, my time will
not have been lost if this very groping has succeeded
in revealing to me the profound analogy between the
problem just dealt with and a much more extensive
class of other problems ; if it has shown me at once
their resemblances and their differences ; if, in a word,
it has enabled me to perceive the possibility of a
generalization. Then it will not be merely a new
result that I have acquired, but a new force.

30 SCIENCE AND METHOD.

An algebraical formula which gives us the solution
of a type of numerical problems, if we finally replace
the letters by numbers, is the simple example which
occurs to one's mind at once. Thanks to the formula,
a single algebraical calculation saves us the trouble of
a constant repetition of numerical calculations. But
this is only a rough example : every one feels that
there are analogies which cannot be expressed by a
formula, and that they are the most valuable.

If a new result is to have any value, it must unite
elements long since known, but till then scattered
and seemingly foreign to each other, and suddenly
introduce order where the appearance of disorder
reigned. Then it enables us to see at a glance each
of these elements in the place it occupies in the whole.
Not only is the new fact valuable on its own account,
but it alone gives a value to the old facts it unites.
Our mind is frail as our senses are ; it would lose
itself in the complexity of the world if that complexity
were not harmonious ; like the short-sighted, it would
only see the details, and would be obliged to forget
each of these details before examining the next,
because it would be incapable of taking in the whole.
The only facts worthy of our attention are those
which introduce order into this complexity and so
make it accessible to us.

Mathematicians attach a great importance to the
elegance of their methods and of their results, and
this is not mere dilettantism. What is it that gives
us the feeling of elegance in a solution or a demonstra-
tion ? It is the harmony of the different parts, their
symmetry, and their happy adjustment ; it is, in a
word, all that introduces order, all that gives them

THE FUTURE OF MATHEMATICS. 31

unity, that enables us to obtain a clear comprehension
of the whole as well as of the parts. But that is
also precisely what causes it to give a large return ;
and in fact the more we see this whole clearly and
at a single glance, the better we shall perceive the
analogies with other neighbouring objects, and con-
sequently the better chance we shall have of guessing
the possible generalizations. Elegance may result
from the feeling of surprise caused by the un-
looked-for occurrence together of objects not habitu-
ally associated. In this, again, it is fruitful, since it
thus discloses relations till then unrecognized. It is
also fruitful even when it only results from the con-
trast between the simplicity of the means and the
complexity of the problem presented, for it then causes
us to reflect on the reason for this contrast, and gener-
ally shows us that this reason is not chance, but is to
be found in some unsuspected law. Briefly stated, the
sentiment of mathematical elegance is nothing but the
satisfaction due to some conformity between the solu-
tion we wish to discover and the necessities of our
mind, and it is on account of this very conformity
that the solution can be an instrument for us. This
aesthetic satisfaction is consequently connected with
the economy of thought. Again the comparison with
the Erechtheum occurs to me, but I do not wish to
serve it up too often.

It is for the same reason that, when a somewhat
lengthy calculation has conducted us to some simple
and striking result, we are not satisfied until we have
shown that we might have foreseen, if not the whole
result, at least its most characteristic features. Why
is this ? What is it that prevents our being contented

32 SCIENCE AND METHOD.

with a calculation which has taught us apparently all
that we wished to know? The reason is that, in
analogous cases, the lengthy calculation might not be
able to be used again, while this is not true of the
reasoning, often semi-intuitive, which might have
enabled us to foresee the result. This reasoning
being short, we can see all the parts at a single glance,
so that we perceive immediately what must be changed
to adapt it to all the problems of a similar nature
that may be presented. And since it enables us to
foresee whether the solution of these problems will
be simple, it shows us at least whether the calculation
is worth undertaking.

What I have just said is sufficient to show how vain
it would be to attempt to replace the mathematician's
free initiative by a mechanical process of any kind.
In order to obtain a result having any real value, it
is not enough to grind out calculations, or to have
a machine for putting things in order : it is not order
only, but unexpected order, that has a value. A
machine can take hold of the bare fact, but the soul
of the fact will always escape it.

Since the middle of last century, mathematicians
have become more and more anxious to attain to
absolute exactness. They are quite right, and this
tendency will become more and more marked. In
mathematics, exactness is not everything, but without
it there is nothing : a demonstration which lacks
exactness is nothing at all. This is a truth that I
think no one will dispute, but if it is taken too
literally it leads us to the conclusion that before 1820,
for instance, there was no such thing as mathematics,
and this is clearly an exaggeration. The geometri-

THE FUTURE OF MATHEMATICS. 33

cians of that day were willing to assume what we
explain by prolix dissertations. This does not mean
that they did not see it at all, but they passed it
over too hastily, and, in order to see it clearly, they
would have had to take the trouble to state it.

Only, is it always necessary to state it so many
times ? Those who were the first to pay special
attention to exactness have given us reasonings that
we may attempt to imitate ; but if the demonstrations
of the future are to be constructed on this model,
mathematical works will become exceedingly long,
and if I dread length, it is not only because I am
afraid of the congestion of our libraries, but because
I fear that as they grow in length our demonstrations
will lose that appearance of harmony which plays such
a useful part, as I have just explained.

It is economy of thought that we should aim at,
and therefore it is not sufficient to give models to
be copied. We must enable those that come after
us to do without the models, and not to repeat a
previous reasoning, but summarize it in a few lines.
And this has already been done successfully in certain
cases. For instance, there was a whole class of reason-
ings that resembled each other, and were found every-
where ; they were perfectly exact, but they were long.
One day some one thought of the term " uniformity of
convergence," and this term alone made them useless ;
it was no longer necessary to repeat them, since they
could now be assumed. Thus the hair-splitters can
render us a double service, first by teaching us to
do as they do if necessary, but more especially by
enabling us as often as possible not to do as they

do, and yet make no sacrifice of exactness.
(1.777) 3

34 SCIENCE AND METHOD.

One example has just shown us the importance
of terms in mathematics ; but I could quote many
others. It is hardly possible to believe what economy
of thought, as Mach used to say, can be effected by
a well-chosen term. I think I have already said
somewhere that mathematics is the art of giving the
same name to different things. It is enough that
these things, though differing in matter, should be
similar in form, to permit of their being, so to speak,
run in the same mould. When language has been
well chosen, one is astonished to find that all demon-
strations made for a known object apply immediately
to many new objects : nothing requires to be changed,
not even the terms, since the names have become the
same.

A well-chosen term is very often sufficient to remove
the exceptions permitted by the rules as stated in the
old phraseology. This accounts for the invention of
negative quantities, imaginary quantities, decimals to
infinity, and I know not what else. And we must
never forget that exceptions are pernicious, because
they conceal laws.

This is one of the characteristics by which we re-
cognize facts which give a great return : they are the
facts which permit of these happy innovations of
language. The bare fact, then, has sometimes no great
interest : it may have been noted many times without
rendering any great service to science ; it only acquires
a value when some more careful thinker perceives the
connexion it brings out, and symbolizes it by a term.

The physicists also proceed in exactly the same
way. They have invented the term " energy," and the
term has been enormously fruitful, because it also

THE FUTURE OF MATHEMATICS. 35

creates a law by eliminating exceptions ; because it
gives the same name to things which differ in matter,
but are similar in form.

Among the terms which have exercised the most
happy influence I would note "group" and "invariable."
They have enabled us to perceive the essence of many
mathematical reasonings, and have shown us in how
many cases the old mathematicians were dealing with
groups without knowing it, and how, believing them-
selves far removed from each other, they suddenly
found themselves close together without understanding
why.

To-day we should say that they had been examining
isomorphic groups. We now know that, in a group, the
matter is of little interest, that the form only is of
importance, and that when we are well acquainted
with one group, we know by that very fact all the
isomorphic groups. Thanks to the terms " group " and
"isomorphism," which sum up this subtle rule in a
few syllables, and make it readily familiar to all minds,
the passage is immediate, and can be made without
expending any effort of thinking. The idea of group
is, moreover, connected with that of transformation.
Why do we attach so much value to the discovery
of a new transformation ? It is because, from a single
theorem, it enables us to draw ten or twenty others.
It has the same value as a zero added to the right
of a whole number.

This is what has determined the direction of the
movement of mathematical science up to the present,
and it is also most certainly what will determine it
in the future. But the nature of the problems which
present themselves contributes to it in an equal degree.

36 SCIENCE AND METHOD.

We cannot forget what our aim should be, and in my
opinion this aim is a double one. Our science borders
on both philosophy and physics, and it is for these
two neighboMrs that we must work. And so we have
always seen, and we shall still see, mathematicians
advancing in two opposite directions.

On the one side, mathematical science must reflect
upon itself, and this is useful because reflecting upon
itself is reflecting upon the human mind which has
created it ; the more so because, of all its creations,
mathematics is the one for which it has borrowed
least from outside. This is the reason for the utility
of certain mathematical speculations, such as those
which have in view the study of postulates, of un-
usual geometries, of functions with strange behaviour.
The more these speculations depart from the most
ordinary conceptions, and, consequently, from nature
and applications to natural problems, the better will
they show us what the human mind can do when it
is more and more withdrawn from the tyranny of
the exterior world ; the better, consequently, will they
make us know this mind itself

But it is to the opposite side, to the side of nature,
that we must direct our main forces.

There we meet the physicist or the engineer, who
says, " Will you integrate this differential equation for
me ; I shall need it within a week for a piece of
construction work that has to be completed by a
certain date ? " " This equation," we answer, " is not
included in one of the types that can be integrated,
of which you know there are not very many." " Yes,
I know ; but, then, what good are you ? " More often
than not a mutual understanding is sufficient. The

THE FUTURE OF MATHEMATICS. 37

engineer does not really require the integral in finite
terms, he only requires to know the general behaviour
of the integral function, or he merely wants a certain
figure which would be easily deduced from this in-
tegral if we knew it. Ordinarily we do not know
it, but we could calculate the figure without it, if we
knew just what figure and what degree of exactness
the engineer required.

Formerly an equation was not considered to have
been solved until the solution had been expressed
by means of a finite number of known functions.
But this is impossible in about ninety-nine cases
out of a hundred. What we can always do, or rather
what we should always try to do, is to solve the
problem qualitatively, so to speak — that is, to try to
know approximately the general form of the curve
which represents the unknown function.

It then remains to find the exact solution of the
problem. But if the unknown cannot be determined
by a finite calculation, we can always represent it
by an infinite converging series which enables us to
calculate it. Can this be regarded as a true solu-
tion ? The story goes that Newton once communi-
cated to Leibnitz an anagram somewhat like the
following : aaaaabbbeeeeii, etc. Naturally, Leibnitz
did not understand it at all, but we who have the
key know that the anagram, translated into modern
phraseology, means, " I know how to integrate all
differential equations," and we are tempted to make
the comment that Newton was either exceedingly
fortunate or that he had very singular illusions.
What he meant to say was simply that he could
form (by means of indeterminate coefficients) a

38 SCIENCE AND METHOD.

series of powers formally satisfying the equation
presented.

To-day a similar solution would no longer satisfy
us, for two reasons — because the convergence is too
slow, and because the terms succeed one another
without obeying any law. On the other hand the
series 9 appears to us to leave nothing to be desired,
first, because it converges very rapidly (this is for
the practical man who wants his number as quickly
as possible), and secondly, because we perceive at a
glance the law of the terms, which satisfies the
esthetic requirements of the theorist.

There are, therefore, no longer some problems
solved and others unsolved, there are only problems
more or less solved, according as this is accomplished
by a series of more or less rapid convergence or
regulated by a more or less harmonious law. Never-
theless an imperfect solution may happen to lead
us towards a better one.

Sometimes the series is of such slow convergence
that the calculation is impracticable, and we have
only succeeded in demonstrating the possibility of
the problem. The engineer considers this absurd,
and he is right, since it will not help him to com-
plete his construction within the time allowed. He
doesn't trouble himself with the question whether it
will be of use to the engineers of the twenty-second
century. We think differently, and we are sometimes
more pleased at having economized a day's work
for our grandchildren than an hour for our contem-
poraries.

Sometimes by groping, so to speak, empirically,
we arrive at a formula that is sufficiently convergent.

THE FUTURE OF MATHEMATICS. 39

What more would you have? says the engineer; and
yet, in spite of everything, we are not satisfied, for
we should have liked to be able to predict the con-
vergence. And why? Because if we had known
how to predict it in the one case, we should know
how to predict it in another. We have been success-
ful, it is true, but that is little in our eyes if we have
no real hope of repeating our success.

In proportion as the science develops, it becomes
more difficult to take it in in its entirety. Then an
attempt is made to cut it in pieces and to be satisfied
with one of these pieces — in a word, to specialize. Too
great a movement in this direction would constitute
a serious obstacle to the progress of the science. As
I have said, it is by unexpected concurrences between
its different parts that it can make progress. Too
much specializing would prohibit these concurrences.
Let us hope that congresses, such as those of Heidel-
berg and Rome, by putting us in touch with each
other, will open up a view of our neighbours' territory,
and force us to compare it with our own, and so
escape in a measure from our own little village. In
this way they will be the best remedy against the
danger I have just noted.

But I have delayed too long over generalities ; it
is time to enter into details.

Let us review the different particular sciences which
go to make up mathematics ; let us see what each of
them has done, in what direction it is tending, and
what we may expect of it. If the preceding views
are correct, we should see that the great progress of
the past has been made when two of these sciences
have been brought into conjunction, when men have

40 SCIENCE AND METHOD.

become aware of the similarity of their form in spite
of the dissimilarity of their matter, when they have
modelled themselves upon each other in such a way
that each could profit by the triumphs of the other.
At the same time we should look to concurrences of
a similar nature for progress in the future.

Arithmetic.

The progress of arithmetic has been much slower
than that of algebra and analysis, and it is easy to
understand the reason. The feeling of continuity is
a precious guide which fails the arithmetician.
Every whole number is separated from the rest, and
has, so to speak, its own individuality ; each of them
is a sort of exception, and that is the reason why
general theorems will always be less common in
the theory of numbers, and also why those that do
exist will be more hidden and will longer escape
detection.

If arithmetic is backward as compared with algebra
and analysis, the best thing for it to do is to try to
model itself on these sciences, in order to profit by
their advance. The arithmetician then should be
guided by the analogies with algebra. These analo-
gies are numerous, and if in many cases they have
not yet been studied sufficiently closely to become
serviceable, they have at least been long foreshadowed,
and the very language of the two sciences shows
that they have been perceived. Thus we speak of
transcendental numbers, and so become aware of
the fact that the future classification of these numbers
has already a model in the classification of transcen-
dental functions. However, it is not yet very clear

THE FUTURE OF MATHEMATICS. 41

how we are to pass from one classification to the
other ; but if it were clear it would be already done,
and would no longer be the work of the future.

The first example that comes to my mind is the
theory of congruents, in which we find a perfect
parallelism with that of algebraic equations. We
shall certainly succeed in completing this parallelism,
which must exist, for instance, between the theory of
algebraic curves and that of congruents with two
variables. When the problems relating to congruents
with several variables have been solved, we shall have
made the first step towards the solution of many ques-
tions of indeterminate analysis.

Algebra.

The theory of algebraic equations will long continue
to attract the attention of geometricians, the sides by
which it may be approached being so numerous and
so different

It must not be supposed that algebra is finished
because it furnishes rules for forming all possible
combinations ; it still remains to find interesting com-
binations, those that satisfy such and such conditions.
Thus there will be built up a kind of indeterminate
analysis, in which the unknown quantities will no
longer be whole numbers but polynomials. So this
time it is algebra that will model itself on arithmetic,
being guided by the analogy of the whole number,
either with the whole polynomial with indefinite
coefificients, or with the whole polynomial with whole
coefficients.

42 SCIENCE AND METHOD.

Geometry.

It would seem that geometry can contain nothing
that is not already contained in algebra or analysis, and
that geometric facts are nothing but the facts of algebra
or analysis expressed in another language. It might
be supposed, then, that after the review that has just
been made, there would be nothing left to say having
any special bearing on geometry. But this would
imply a failure to recognize the great importance of a
well-formed language, or to understand what is added
to things themselves by the method of expressing, and
consequently of grouping, those things.

To begin with, geometric considerations lead us to
set ourselves new problems. These are certainly, if
you will, analytical problems, but they are problems
we should never have set ourselves on the score of
analysis. Analysis, however, profits by them, as it
profits by those it is obliged to solve in order to
satisfy the requirements of physics.

One great advantage of geometry lies precisely in
the fact that the senses can come to the assistance of
the intellect, and help to determine the road to be
followed, and many minds prefer to reduce the
problems of analysis to geometric form. Unfortu-
nately our senses cannot carry us very far, and they
leave us in the lurch as soon as we wish to pass
outside the three classical dimensions. Does this
mean that when we have left this restricted domain
in which they would seem to wish to imprison us, we
must no longer count on anything but pure analysis,
and that all geometry of more than three dimensions
is vain and without object ? In the generation which

THE FUTURE OF MATHEMATICS. 43

preceded ours, the greatest masters would have an-
swered "Yes." To-day we are so familiar with this
notion that we can speak of it, even in a university
course, without exciting too much astonishment.

But of what use can it be ? This is easy to see. In
the first place it gives us a very convenient language,
which expresses in very concise terms what the ordi-
nary language of analysis would state in long-winded
phrases. More than that, this language causes us to
give the same name to things which resemble one
another, and states analogies which it does not allow
us to forget. It thus enables us still to find our way
in that space which is too great for us, by calling to
our mind continually the visible space, which is only
an imperfect image of it, no doubt, but still an image.
Here again, as in all the preceding examples, it is
the analogy with what is simple that enables us to
understand what is complex.

This geometry of more than three dimensions is
not a simple analytical geometry, it is not purely
quantitative, but also qualitative, and it is principally
on this ground that it becomes interesting. There is a
science called Geometry of Position, which has for its
object the study of the relations of position of the
different elements of a figure, after eliminating their
magnitudes. This geometry is purely qualitative ; its
theorems would remain true if the figures, instead of
being exact, were rudely imitated by a child. We can
also construct a Geometry of Position of more than
three dimensions. The importance of Geometry of
Position is immense, and I cannot insist upon it too
much ; what Riemann, one of its principal creators,
has gained from it would be sufficient to demonstrate

44 SCIENCE AND METHOD.

this. We must succeed in constructing it completely
in the higher spaces, and we shall then have an instru-
ment which will enable us really to see into hyperspace
and to supplement our senses.

The problems of Geometry of Position would perhaps
not have presented themselves if only the language of
analysis had been used. Or rather I am wrong, for
they would certainly have presented themselves, since
their solution is necessary for a host of questions of
analysis, but they would have presented themselves
isolated, one after the other, and without our being
able to perceive their common link.

Cantorism.

I have spoken above of the need we have of
returning continually to the first principles of our
science, and of the advantage of this process to the
study of the human mind. It is this need which has
inspired two attempts which have held a very great
place in the most recent history of mathematics. The
first is Cantorism, and the services it has rendered to
the science are well known. Cantor introduced into
the science a new method of considering mathematical
infinity, and I shall have occasion to speak of it again
in Book tl., chapter iii. One of the characteristic
features of Cantorism is that, instead of rising to the
general by erecting more and more complicated con-
structions, and defining by construction, it starts with
the genus supreinum and only defines, as the scholastics
would have said, per genus proximum et differe?ttiam
specificam. Hence the horror he has sometimes in-
spired in certain minds, such as Hermitte's, whose
favourite idea was to compare the mathematical with

THE FUTURE OF MATHEMATICS. 45

the natural sciences. For the greater number of us
these prejudices had been dissipated, but it has come
about that we have run against certain paradoxes and
apparent contradictions, which would have rejoiced
the heart of Zeno of Elea and the school of Megara.
Then began the business of searching for a remedy,
each man his own way. For my part I think, and I
am not alone in so thinking, that the important thing
is never to introduce any entities but such as can be
completely defined in a finite number of words. What-
ever be the remedy adopted, we can promise ourselves
the joy of the doctor called in to follow a fine patho-
logical case.

The Search for Postulates.

Attempts have been made, from another point of
view, to enumerate the axioms and postulates more
or less concealed which form the foundation of the
different mathematical theories, and in this direction
Mr. Hilbert has obtained the most brilliant results.
It seems at first that this domain must be strictly
limited, and that there will be nothing more to do
when the inventory has been completed, which cannot
be long. But when everything has been enumerated,
there will be many ways of classifying it all. A good
librarian always finds work to do, and each new classi-
fication will be instructive for the philosopher.

I here close this review, which I cannot dream of
making complete. I think that these examples will
have been sufficient to show the mechanism by which
the mathematical sciences have progressed in the past,
and the direction in which they must advance in the
future.

III.

MATHEMATICAL DISCOVERY.

The genesis of mathematical discovery is a problem
which must inspire the psychologist with the keenest
interest. For this is the process in which the human
mind seems to borrow least from the exterior world,
in which it acts, or appears to act, only by itself and
on itself, so that by studying the process of geometric
thought we may hope to arrive at what is most
essential in the human mind.

This has long been understood, and a few months
ago a review called l' Enseignement Mathematique,
edited by MM. Laisant and Fehr, instituted an en-
quiry into the habits of mind and methods of work
of different mathematicians. I had outlined the
principal features of this article when the results of
the enquiry were published, so that I have hardly been
able to make any use of them, and I will content
myself with saying that the majority of the evidence
confirms my conclusions. I do not say there is
unanimity, for on an appeal to universal suffrage we
cannot hope to obtain unanimity.

One first fact must astonish us, or rather would
astonish us if we were not too much accustomed to
it. How does it happen that there are people who

MATHEMATICAL DISCOVERY. 47

do not understand mathematics? If the science
invokes only the rules of logic, those accepted by
all well-formed minds, if its evidence is founded on
principles that are common to all men, and that none
but a madman would attempt to deny, how does it
happen that there are so many people who are
entirely impervious to it ?

There is nothing mysterious in the fact that every
one is not capable of discovery. That every one
should not be able to retain a demonstration he has
once learnt is still comprehensible. But what does
seem most surprising, when we consider it, is that
any one should be unable to understand a mathe-
matical argument at the very moment it is stated to
him. And yet those who can only follow the argu-
ment with difficulty are in a majority ; this is incon-
testable, and the experience of teachers of secondary
education will certainly not contradict me.

And still further, how is error possible in mathe-
matics ? A healthy intellect should not be guilty
of any error in logic, and yet there are very keen
minds which will not make a false step in a short
argument such as those we have to make in the
ordinary actions of life, which yet are incapable of
following or repeating without error the demonstra-
tions of mathematics which are longer, but which
are, after all, only accumulations of short arguments
exactly analogous to those they make so easily. Is it
necessary to add that mathematicians themselves are
not infallible?

The answer appears to me obvious. Imagine a
long series of syllogisms in which the conclusions of
those that precede form the premises of those that

48 SCIENCE AND METHOD.

follow. We shall be capable of grasping each of the
syllogisms, and it is not in the passage from premises
to conclusion that we are in danger of going astray.
But between the moment when we meet a proposition
for the first time as the conclusion of one syllogism,
and the moment when we find it once more as the
premise of another syllogism, much time will some-
times have elapsed, and we shall have unfolded many
links of the chain ; accordingly it may well happen
that we shall have forgotten it, or, what is more serious,
forgotten its meaning. So we may chance to replace
it by a somewhat different proposition, or to preserve
the same statement but give it a slightly different
meaning, and thus we are in danger of falling into
error.

A mathematician must often use a rule, and, natur-
ally, he begins by demonstrating the rule. At the
moment the demonstration is quite fresh in his
memory he understands perfectly its meaning and
significance, and he is in no danger of changing it.
But later on he commits it to memory, and only
applies it in a mechanical way, and then, if his
memory fails him, he may apply it wrongly. It is
thus, to take a simple and almost vulgar example,
that we sometimes make mistakes in calculation,
because we have forgotten our multiplication table.

On this view special aptitude for mathematics
would be due to nothing but a very certain memory
or a tremendous power of attention. It would be a
quality analogous to that of the whist player who
can remember the cards played, or, to rise a step
higher, to that of the chess player who can picture
a very great number of combinations and retain them

MATHEMATICAL DISCOVERY. 49

in his memory. Every good mathematician should
also be a good chess player and vice versa, and
similarly he should be a good numerical calculator.
Certainly this sometimes happens, and thus Gauss
was at once a geometrician of genius and a very
precocious and very certain calculator.

But there are exceptions, or rather I am wrong,
for I cannot call them exceptions, otherwise the excep-
tions would be more numerous than the cases of con-
formity with the rule. On the contrary, it was Gauss
who was an exception. As for myself, I must confess
I am absolutely incapable of doing an addition sum
without a mistake. Similarly I should be a very bad
chess player. I could easily calculate that by playing
in a certain way I should be exposed to such and
such a danger ; I should then review many other
moves, which I should reject for other reasons, and
I should end by making the move I first examined,
having forgotten in the interval the danger I had
foreseen.

In a word, my memory is not bad, but it would be
insufficient to make me a good chess player. Why,
then, does it not fail me in a difficult mathematical
argument in which the majority of chess players
would be lost ? Clearly because it is guided by the
general trend of the argument. A mathematical
demonstration is not a simple juxtaposition of syl-
logisms ; it consists of syllogisms placed in a certain
order, and the order in which these elements are
placed is much more important than the elements
themselves. If I have the feeling, so to speak the
intuition, of this order, so that I can perceive the

whole of the argument at a glance, I need no longer
(1,777) 4

50 SCIENCE AND METHOD.

be afraid of forgetting one of the elements ; each of
them will place itself naturally in the position pre-
pared for it, without my having to make any effort
of memory.

It seems to me, then, as I repeat an argument I
have learnt, that I could have discovered it. This
is often only an illusion ; but even then, even if I am
not clever enough to create for myself, I rediscover
it myself as I repeat it.

We can understand that this feeling, this intuition
of mathematical order, which enables us to guess
hidden harmonies and relations, cannot belong to
every one. Some have neither this delicate feeling
that is difficult to define, nor a power of memory and
attention above the common, and so they are abso-
lutely incapable of understanding even the first steps
of higher mathematics. This applies to the majority
of people. Others have the feeling only in a slight
degree, but they are gifted with an uncommon
memory and a great capacity for attention. They
learn the details one after the other by heart, they
can understand mathemathics and sometimes apply
them, but they are not in a condition to create.
Lastly, others possess the special intuition I have
spoken of more or less highly developed, and they
can not only understand mathematics, even though
their memory is in no way extraordinary, but they
can become creators, and seek to make discovery
with more or less chance of success, according as their
intuition is more or less developed.

What, in fact, is mathematical discovery? It does
not consist in making new combinations with mathe-
matical entities that are already known. That can

MATHEMATICAL DISCOVERY. 51

be done by any one, and the combinations that could
be so formed would be infinite in number, and the
greater part of them would be absolutely devoid of
interest. Discovery consists precisely in not con-
structing useless combinations, but in constructing
those that are useful, which are an infinitely small
minority. Discovery is discernment, selection.

How this selection is to be made I have explained
above. Mathematical facts worthy of being studied
are those which, by their analogy with other facts,
are capable of conducting us to the knowledge of a
mathematical law, in the same way that experimental
facts conduct us to the knowledge of a physical law.
They are those which reveal unsuspected relations
between other facts, long since known, but wrongly
believed to be unrelated to each other.

Among the combinations we choose, the most fruit-
ful are often those which are formed of elements
borrowed from widely separated domains. I do not
mean to say that for discovery it is sufficient to bring
together objects that are as incongruous as possible.
The greater part of the combinations so formed would
be entirely fruitless, but some among them, though
very rare, are the most fruitful of all.

Discovery, as I have said, is selection. But this is
perhaps not quite the right word. It suggests a pur-
chaser who has been shown a large number of samples,
and examines them one after the other in order to
make his selection. In our case the samples would be
so numerous that a whole life would not give sufficient
time to examine them. Things do not happen in this
way. Unfruitful combinations do not so much as
present themselves to the mind of the discoverer. In

52 SCIENCE AND METHOD.

the field of his consciousness there never appear any
but really useful combinations, and some that he
rejects, which, however, partake to some extent of
the character of useful combinations. Everything
happens as if the discoverer were a secondary examiner
who had only to interrogate candidates declared eli-
gible after passing a preliminary test.

But what I have said up to now is only what can
be observed or inferred by reading the works of
geometricians, provided they are read with some
reflection.

It is time to penetrate further, and to see what
happens in the very soul of the mathematician. For
this purpose I think I cannot do better than recount
my personal recollections. Only I am going to confine
myself to relating how I wrote my first treatise on
Fuchsian functions. I must apologize, for I am going
to introduce some technical expressions, but they need
not alarm the reader, for he has no need to under-
stand them. I shall say, for instance, that I found the
demonstration of such and such a theorem under such
and such circumstances ; the theorem will have a
barbarous name that many will not know, but that
is of no importance. What is interesting for the
psychologist is not the theorem but the circumstances.

For a fortnight I had been attempting to prove
that there could not be any function analogous to
what I have since called Fuchsian functions. I was at
that time very ignorant. Every day I sat down at my
table and spent an hour or two trying a great number
of combinations, and I arrived at no result. One
night I took some black coffee, contrary to my custom,
and was unable to sleep. A host of ideas kept surging

MATHEMATICAL DISCOVERY. 53

in my head ; I could almost feel then jostling one
another, until two of them coalesced, so to speak, to
form a stable combination. When morning came, I
had established the existence of one class of Fuchsian
functions, those that are derived from the hyper-
geometric series. I had only to verify the results,
which only took a few hours.

Then I wished to represent these functions by the
quotient of two series. This idea was perfectly con-
scious and deliberate ; I was guided by the analogy
with elliptical functions. I asked myself what must
be the properties of these series, if they existed, and
I succeeded without difficulty in forming the series
that I have called Theta-Fuchsian.

At this moment I left Caen, where I was then living,
to take part in a geological conference arranged by
the School of Mines. The incidents of the journey
made me forget my mathematical work. When we
arrived at Coutances, we got into a break to go
for a drive, and, just as I put my foot on the
step, the idea came to me, though nothing in my
former thoughts seemed to have prepared me for it,
that the transformations I had used to define Fuchsian
functions were identical with those of non-Euclidian
geometry. I made no verification, and had no time to
do so, since I took up the conversation again as soon
as I had sat down in the break, but I felt absolute
certainty at once. When I got back to Caen I verified
the result at my leisure to satisfy my conscience.

I then began to study arithmetical questions without
any great apparent result, and without suspecting that
they could have the least connexion with my previous
researches. Disgusted at my want of success, I went

54 SCIENCE AND METHOD.

away to spend a few days at the seaside, and
thought of entirely different things. One day, as I
was walking on the cliff, the idea came to me, again
with the same characteristics of conciseness, sudden-
ness, and immediate certainty, that arithmetical trans-
formations of indefinite ternary quadratic forms are
identical with those of non-Euclidian geometry.

Returning to Caen, I reflected on this result and
deduced its consequences. The example of quadratic
forms showed me that there are Fuchsian groups
other than those which correspond with the hyper-
geometric series ; I saw that I could apply to them
the theory of the Theta-Fuchsian series, and that,
consequently, there are Fuchsian functions other than
those which are derived from the hypergeometric series,
the only ones I knew up to that time. Naturally, I
proposed to form all these functions. I laid siege
to them systematically and captured all the outworks
one after the other. There was one, however, which
still held out, whose fall would carry with it that of the
central fortress. But all my efforts were of no avail at
first, except to make me better understand the difficulty,
which was already something. All this work was per-
fectly conscious.

Thereupon I left for Mont-Valerien, where I had
to serve my time in the army, and so my mind was
preoccupied with very different matters. One day, as
I was crossing the street, the solution of the difficulty
which had brought me to a standstill came to me
all at once. I did not try to fathom it immediately,
and it was only after my service was finished that
I returned to the question. I had all the elements,
and had only to assemble and arrange them. Accord-

MATHEMATICAL DISCOVERY. 55

ingly I composed my definitive treatise at a sitting
and without any difficulty.

It is useless to multiply examples, and I will con-
tent myself with this one alone. As regards my other
researches, the accounts I should give would be exactly
similar, and the observations related by other mathe-
maticians in the enquiry of V Enseigtiement Math'e-
niatique would only confirm them.

One is at once struck by these appearances of
sudden illumination, obvious indications of a long
course of previous unconscious work. The part played
by this unconscious work in mathematical discovery
seems to me indisputable, and we shall find traces
of it in other cases where it is less evident. Often
when a man is working at a difficult question, he
accomplishes nothing the first time he sets to work.
Then he takes more or less of a rest, and sits down
again at his table. During the first half-hour he still
finds nothing, and then all at once the decisive idea
presents itself to his mind. We might say that the
conscious work proved more fruitful because it was
interrupted and the rest restored force and freshness
to the mind. But it is more probable that the rest
was occupied with unconscious work, and that the
result of this work was afterwards revealed to the
geometrician exactly as in the cases 1 have quoted,
except that the revelation, instead of coming to light
during a walk or a journey, came during a period
of conscious work, but independently of that work,
which at most only performs the unlocking process,
as if it were the spur that excited into conscious form
the results already acquired during the rest, which till
then remained unconscious.

56 SCIENCE AND METHOD.

There is another remark to be made regarding
the conditions of this unconscious work, which is, that
it is not possible, or in any case not fruitful, unless
it is first preceded and then followed by a period
of conscious work. These sudden inspirations are
never produced (and this is sufficiently proved already
by the examples I have quoted) except after some
days of voluntary efforts which appeared absolutely
fruitless, in which one thought one had accomplished
nothing, and seemed to be on a totally wrong track.
These efforts, however, were not as barren as one
thought ; they set the unconscious machine in motion,
and without them it would not have worked at all,
and would not have produced anything.

The necessity for the second period of conscious
work can be even more readily understood. It is
necessary to work out the results of the inspiration,
to deduce the immediate consequences and put them
in order and to set out the demonstrations ; but, above
all, it is necessary to verify them. I have spoken
of the feeling of absolute certainty which accompanies
the inspiration ; in the cases quoted this feeling was
not deceptive, and more often than not this will be
the case. But we must beware of thinking that this
is a rule without exceptions. Often the feeling de-
ceives us without being any less distinct on that
account, and we only detect it when we attempt to
establish the demonstration. I have observed this
fact most notably with regard to ideas that have come
to me in the morning or at night when I have been
in bed in a semi-somnolent condition.

Such are the facts of the case, and they suggest the
following reflections. The result of all that precedes

MATHEMATICAL DISCOVERY. 57

is to show that the unconscious ego, or, as it is called,
the subliminal ego, plays a most important part
in mathematical discovery. But the subliminal ego
is generally thought of as purely automatic. Now we
have seen that mathematical work is not a simple
mechanical work, and that it could not be entrusted
to any machine, whatever the degree of perfection we
suppose it to have been brought to. It is not merely
a question of applying certain rules, of manufacturing
as many combinations as possible according to certain
fixed laws. The combinations so obtained would
be extremely numerous, useless, and encumbering. '
The real work of the discoverer consists in choosing
between these combinations with a view to eliminating
those that are useless, or rather not giving himself 1
the trouble of making them at all. The rules which ;
must guide this choice are extremely subtle and
delicate, and it is practically impossible to state them
in precise language ; they must be felt rather than for- <
mulated. Under these conditions, how can we imagine ;
a sieve capable of applying them mechanically ?

The following, then, presents itself as a first hypoth-
esis. The subliminal ego is in no way inferior to the
conscious ego ; it is not purely automatic ; it is capable
of discernment ; it has tact and lightness of touch ;
it can select, and it can divine. More than that,
it can divine better than the conscious ego, since
it succeeds where the latter fails. In a word, is not
the subliminal ego superior to the conscious ego?
The importance of this question will be readily
understood. In a recent lecture, M. Boutroux showed
how it had arisen on entirely different occasions, and
what consequences would be involved by an answer

58 SCIENCE AND METHOD.

in the affirmative. (See also the same author's
Science et Religion, pp. 313^/" seq})

Are we forced to give this affirmative answer by
the facts I have just stated ? I confess that, for my
part, I should be loth to accept it. Let us, then,
return to the facts, and see if they do not admit of
some other explanation.

It is certain that the combinations which present
themselves to the mind in a kind of sudden illumina-
tion after a somewhat prolonged period of unconscious
work are generally useful and fruitful combinations,
which appear to be the result of a preliminary sifting.
Does it follow from this that the subliminal ego,
having divined by a delicate intuition that these
combinations could be useful, has formed none but
these, or has it formed a great many others which
were devoid of interest, and remained unconscious ?

Under this second aspect, all the combinations are
formed as a result of the automatic action of the
subliminal ego, but those only which are interesting
find their way into the field of consciousness. This, too,
is most mysterious. How can we explain the fact that,
of the thousand products of our unconscious activity,
some are invited to cross the threshold, while others
remain outside? Is it mere chance that gives them
this privilege? Evidently not. For instance, of
all the excitements of our senses, it is only the most
intense that retain our attention, unless it has been
directed upon them by other causes. More commonly
the privileged unconscious phenomena, those that are
capable of becoming conscious, are those which,
directly or indirectly, most deeply affect our sen-
sibility.

MATHEMATICAL DISCOVERY. 59

It may appear surprising that sensibility should
be introduced in connexion with mathematical de-
monstrations, which, it would seem, can only interest
the intellect. But not if we bear in mind the feeling
of mathematical beauty, of the harmony of numbers
and forms and of geometric elegance. It is a real
jesthetic feeling that all true mathematicians recognize,
and this is truly sensibility.

Now, what are the mathematical entities to which
we attribute this character of beauty and elegance,
which are capable of developing in us a kind of
aesthetic emotion ? Those whose elements are har-
moniously arranged so that the mind can, without
effort, take in the whole without neglecting the details.
This harmony is at once a satisfaction to our aesthetic
requirements, and an assistance to the mind which
it supports and guides. At the same time, by setting
before our eyes a well-ordered whole, it gives us
a presentiment of a mathematical law. Now, as I
have said above, the only mathematical facts worthy
of retaining our attention and capable of being useful
are those which can make us acquainted with a
mathematical law. Accordingly we arrive at the
following conclusion. The useful combinations are
precisely the most beautiful, I mean those that can
most charm that special sensibility that all mathe-
maticians know, but of which laymen are so ignorant
that they are often tempted to smile at it.

What follows, then ? Of the very large number of
combinations which the subliminal ego blindly forms,
almost all are without interest and without utility.
But, for that very reason, they are without action on
the aesthetic sensibility ; the consciousness will never

6o SCIENCE AND METHOD.

know them. A few only are harmonious, and con-
sequently at once useful and beautiful, and they
will be capable of affecting the geometrician's special
sensibility 1 have been speaking of; which, once
aroused, will direct our attention upon them, and will
thus give them the opportunity of becoming conscious.

This is only a hypothesis, and yet there is an
observation which tends to confirm it. When a
sudden illumination invades the mathematician's mind,
it most frequently happens that it does not mislead
him. But it also happens sometimes, as I have said,
that it will not stand the test of verification. Well,
it is to be observed almost always that this false idea,
if it had been correct, would have flattered our natural
instinct for mathematical elegance.

Thus it is this special assthetic sensibility that plays
the part of the delicate sieve of which I spoke above,
and this makes it sufficiently clear why the man who
has it not will never be a real discoverer.

All the difficulties, however, have not disappeared.
The conscious ego is strictly limited, but as regards
the subliminal ego, we do not know its limitations,
and that is why we are not too loth to suppose
that in a brief space of time it can form more
different combinations than could be comprised in
the whole life of a conscient being. These limitations
do exist, however. Is it conceivable that it can form
all the possible combinations, whose number staggers
the imagination ? Nevertheless this would seem to be
necessary, for if it produces only a small portion of the
combinations, and that by chance, there vv^ill be very
small likelihood of the right one, the one that must be
selected, being found among them.

MATHEMATICAL DISCOVERY. 6i

Perhaps we must look for the explanation in that
period of preliminary conscious work which always
precedes all fruitful unconscious work. If I may
be permitted a crude comparison, let us represent the
future elements of our combinations as something
resembling Epicurus's hooked atoms. When the mind
is in complete repose these atoms are immovable ;
they are, so to speak, attached to the wall. This com-
plete repose may continue indefinitely without the
atoms meeting, and, consequently, without the pos-
sibility of the formation of any combination.

On the other hand, during a period of apparent
repose, but of unconscious work, some of them are
detached from the wall and set in motion. They
plough through space in all directions, like a swarm
of gnats, for instance, or, if we prefer a more learned
comparison, like the gaseous molecules in the kinetic
theory of gases. Their mutual collisions may then
produce new combinations.

What is the part to be played by the preliminary
conscious work ? Clearly it is to liberate some of
these atoms, to detach them from the wall and set
them in motion. We think we have accomplished
nothing, when we have stirred up the elements in a
thousand different ways to try to arrange them, and
have not succeeded in finding a satisfactory arrange-
ment. But after this agitation imparted to them by
our will, they do not return to their original repose,
but continue to circulate freely.

Now our will did not select them at random, but
in pursuit of a perfectly definite aim. Those it has
liberated are not, therefore, chance atoms ; they are
those from which we may reasonably expect the

62 SCIENCE AND METHOD.

desired solution. The liberated atoms will then
experience collisions, either with each other, or with
the atoms that have remained stationary, which
they will run against in their course. I apologize
once more. My comparison is very crude, but I
cannot well see how I could explain my thought
in any other way.

However it be, the only combinations that have
any chance of being formed are those in which one
at least of the elements is one of the atoms deliber-
ately selected by our will. Now it is evidently
among these that what I called just now the right
combination is to be found. Perhaps there is here
a means of modifying what was paradoxical in the
original hypothesis.

Yet another observation. It never happens that
unconscious work supplies ready-made the result of
a lengthy calculation in which we have only to apply
fixed rules. It might be supposed that the sub-
liminal ego, purely automatic as it is, was peculiarly
fitted for this kind of work, which is, in a sense, ex-
clusively mechanical. It would seem that, by think-
ing overnight of the factors of a multiplication sum,
we might hope to find the product ready-made for
us on waking ; or, again, that an algebraical calcula-
tion, for instance, or a verification could be made
unconsciously. Observation proves that such is by no
means the case. All that we can hope from these
inspirations, which are the fruits of unconscious
work, is to obtain points of departure for such
calculations. As for the calculations themselves,
they must be made in the second period of conscious
work which follows the inspiration, and in which

MATHEMATICAL DISCOVERY. 63

the results of the inspiration are verified and the
consequences deduced. The rules of these calcula-
tions are strict and complicated ; they demand disci-
pline, attention, will, and consequently consciousness.
In the subliminal ego, on the contrary, there reigns
what I would call liberty, if one could give this
name to the mere absence of discipline and to dis-
order born of chance. Only, this very disorder permits
of unexpected couplings.

I will make one last remark. When I related
above some personal observations, I spoke of a night
of excitement, on which I worked as though in spite
of myself The cases of this are frequent, and it is
not necessary that the abnormal cerebral activity
should be caused by a physical stimulant, as in the
case quoted. Well, it appears that, in these cases,
we are ourselves assisting at our own unconscious
work, which becomes partly perceptible to the over-
excited consciousness, but does not on that account
change its nature. We then become vaguely aware
of what distinguishes the two mechanisms, or, if you
will, of the methods of working of the two egos.
The psychological observations I have thus suc-
ceeded in making appear to me, in their general
characteristics, to confirm the views I have been
enunciating.

Truly there is great need of this, for in spite of
everything they are and remain largely hypothetical.
The interest of the question is so great that I do
not regret having submitted them to the reader.

IV.

CHANCE.

I.

" How can we venture to speak of the laws of chance ?
Is not chance the antithesis of all law ? " It is thus
that Bertrand expresses himself at the beginning of
his "Calculus of Probabilities." Probability is the
opposite of certainty ; it is thus what we are ignorant
of, and consequently it would seem to be what we
cannot calculate. There is here at least an apparent
contradiction, and one on which much has already
been written.

To begin with, what is chance ? The ancients
distinguished between the phenomena which seemed
to obey harmonious laws, established once for all,
and those that they attributed to chance, which were
those that could not be predicted because they were
not subject to any law. In each domain the precise
laws did not decide everything, they only marked
the limits within which chance was allowed to move.
In this conception, the word chance had a precise,
objective meaning ; what was chance for one was
also chance for the other and even for the gods.

But this conception is not ours. We have become
complete determinists, and even those who wish to

CHANCE. 65

reserve the right of human free will at least allow
determinism to reign undisputed in the inorganic
world. Every phenomenon, however trifling it be,
has a cause, and a mind infinitely powerful and
infinitely well-informed concerning the laws of nature
could have foreseen it from the beginning of the ages.
If a being with such a mind existed, we could play
no game of chance with him ; we should always
lose.

For him, in fact, the word chance would have no
meaning, or rather there would be no such thing as
chance. That there is for us is only on account of
our frailty and our ignorance. And even without
going beyond our frail humanity, what is chance
for the ignorant is no longer chance for the learned.
Chance is only the measure of our ignorance. For-
tuitous phenomena are, by definition, +hose whose
laws we are ignorant of

But is this definition very satisfactory? \Vhen the
first Chaldean shepherds followed with their eyes
the movements of the stars, they did not yet know
the laws of astronomy, but would they have dreamed
of saying that the stars move by chance? If a
modern physicist is studying a new phenomenon,
and if he discovers its law on Tuesday, would he
have said on Monday that the phenomenon was
fortuitous ? But more than this, do we not often
invoke what Bertrand calls the laws of chance in
order to predict a phenomenon ? For instance, in
the kinetic theory of gases, we find the well-known
laws of Mariotte and of Gay-Lussac, thanks to the
hypothesis that the velocities of the gaseous mole-
cules vary irregularly, that is to say, by chance.

(1.777) 5

66 SCIENCE AND METHOD.

The observable laws would be much less simple,
say all the physicists, if the velocities were regulated
by some simple elementary law, if the molecules
were, as they say, organized, if they were subject to
some discipline. It is thanks to chance — that is to
say, thanks to our ignorance, that we can arrive at con-
clusions. Then if the word chance is merely synony-
mous with ignorance, vthat does this mean ? Must
we translate as follows ? —

"You ask me to predict the phenomena that will
be produced. If I had the misfortune to know the
laws of these phenomena, I could not succeed except
by inextricable calculations, and I should have to
give up the attempt to answer you ; but since I am
fortunate enough to be ignorant of them, I will
give you an answer at once. And, what is more
extraordinary still, my answer will be right."

Chance, then, must be something more than the
name we' give to our ignorance. Among the phe-
nomena whose causes we are ignorant of, we must
distinguish between fortuitous phenomena, about
which the calculation of probabilities will give us
provisional information, and those that are not for-
tuitous, about which we can say nothing, so long
as we have not determined the laws that govern
them. And as regards the fortuitous phenomena
themselves, it is clear that the information that the
calculation of probabilities supplies will not cease to
be true when the phenomena are better known.

The manager of a life insurance company does
not know when each of the assured will die, but he
relies upon the calculation of probabilities and on
the law of large numbers, and he does not make a

CHANCE. 67

mistake, since he is able to pay dividends to his
shareholders. These dividends would not vanish if
a very far-sighted and very indiscreet doctor came,
when once the policies were signed, and gave the
manager information on the chances of life of the
as.sured. The doctor would dissipate the ignorance
of the manager, but he would have no effect upon
the dividends, which are evidently not a result of
that ignorance.

11.

In order to find the best definition of chance, we
must examine some of the facts which it is agreed
to regard as fortuitous, to which the calculation of
probabilities seems to apply. We will then try to
find their common characteristics.

We will select unstable equilibrium as our first
example. If a cone is balanced on its point, we know
very well that it will fall, but we do not know to
which side ; it seems that chance alone will decide.
If the cone were perfectly symmetrical, if its axis
were perfectly vertical, if it were subject to no other
force but gravity, it would not fall at all. But the
slightest defect of symmetry will make it lean slightly
to one side or other, and as soon as it leans, be it
ever so little, it will fall altogether to that side.
Even if the symmetry is perfect, a very slight trepida-
tion, or a breath of air, may make it incline a few
seconds of arc, and that will be enough to determine
its fall and even the direction of its fall, which will be
that of the original inclination.

A very small cause which escapes our notice
determines a considerable effect that we cannot fail
to see, and then we say that that effect is due to

68 SCIENCE AND METHOD.

chance. If we knew exactly the laws of nature and
the situation of the universe at the initial moment,
we could predict exactly the situation of that same
universe at a succeeding moment. But, even if it
were the case that the natural laws had no longer
any secret tor us, we could still only know the initial
situation approximately. If that enabled us to predict
the succeeding situation with the same approximation,
that is all we require, and we should say that the
phenomenon had been predicted, that it is governed
by laws. But it is not always so ; it may happen that
small differences in the initial conditions produce very
great ones in the final phenomena. A small error in
the former will produce an enormous error in the
latter. Prediction becomes impossible, and we have
the fortuitous phenomenon.

Our second example will be very much like our
first, and we will borrow it from meteorology. Why
have meteorologists such difficulty in predicting the
weather with any certainty ? Why is it that showers
and even storms seem to come by chance, so that
many people think it quite natural to pray for rain
or fine weather, though they would consider it
ridiculous to ask for an eclipse by prayer ? We see
that great disturbances are generally produced in
regions where the atmosphere is in unstable equilib-
rium. The meteorologists see very well that the
equilibrium is unstable, that a cyclone will be formed
somewhere, but exactly where they are not in a
position to say ; a tenth of a degree more or less at
any given point, and the cyclone will burst here and
not there, and extend its ravages over districts it
would otherwise have spared. If they had been aware

CHANCE. 69

of this tenth of a degree, they could have known
it beforehand, but the observations were neither
sufficiently comprehensive nor sufficiently precise, and
that is the reason why it all seems due to the
intervention of chance. Here, again, we find the
same contrast between a very trifling cause that
is inappreciable to the observer, and considerable
effects, that are sometimes terrible disasters.

Let us pass to another example, the distribution of
the minor planets on the Zodiac. Their initial
longitudes may have had some definite order, but
their mean motions were different and they have been
revolving for so long that we may say that practically
they are distributed by chance throughout the Zodiac.
Very small initial differences in their distances from
the sun, or, what amounts to the same thing, in their
mean motions, have resulted in enormous differences
in their actual longitudes. A difference of a thousandth
part of a second in the mean daily motion will have
the effect of a second in three years, a degree in ten
thousand years, a whole circumference in three or
four millions of years, and what is that beside the
time that has elapsed since the minor planets became
detached from Laplace's nebula ? Here, again, we
have a small cause and a great effect, or better, small
differences in the cause and great differences in the
effect.

The eame of roulette does not take us so far as it
might appear from the preceding example. Imagine
a needle that can be turned about a pivot on a dial
divided into a hundred alternate red and black
sections. If the needle stops at a red section we win ;
if not, we lose. Clearly, all depends on the initial

70 SCIENCE AND METHOD.

impulse we give to the needle. I assume that the
needle will make ten or twenty revolutions, but it
will stop earlier or later according to the strength
of the spin I have given it. Only a variation of a
thousandth or a two-thousandth in the impulse is
sufficient to determine whether my needle will stop
at a black section or at the following section, which
is red. These are differences that the muscular sense
cannot appreciate, which would escape even more
delicate instruments. It is, accordingly, impossible for
me to predict what the needle I have just spun will
do, and that is why my heart beats and I hope for
everything from chance. The difference in the cause
is imperceptible, and the difference in the effect is
for me of the highest importance, since it affects my
whole stake.

III.

In this connexion I wish to make a reflection that
is somewhat foreign to my subject. Some years
ago a certain philosopher said that the future was
determined by the past, but not the past by the
future ; or, in other words, that from the knowledge
of the present we could deduce that of the future
but not that of the past ; because, he said, one cause
can produce only one effect, while tne same effect can
be produced by several different causes. It is obvious
that no scientist can accept this conclusion. The laws
of nature link the antecedent to the consequent in
such a way that the antecedent is determined by the
consequent just as much as the consequent is by the
antecedent. But what can have been the origin of
the philosopher's error? We know that, in virtue
of Carnot's principle, physical phenomena are irrevers-

CHANCE. 71

ible and that the world is tending towards uniformity.
When two bodies of different temperatures are in
conjunction, the warmer gives up heat to the colder,
and accordingly we can predict that the temperatures
will become equal. But once the temperatures have
become equal, if we are asked about the previous state,
what can we answer ? We can certainly say that one
of the bodies was hot and the other cold, but we
cannot guess which of the two was formerly the
warmer.

And yet in reality the temperatures never arrive
at perfect equality. The difference between the
temperatures only tends towards zero asymptotically.
Accordingly there comes a moment when our
thermometers are powerless to disclose it. But if
we had thermometers a thousand or a hundred
thousand times more sensitive, we should recognize
that there is still a small difference, and that one of
the bodies has remained a little warmer than the
other, and then we should be able to state that this
is the one which was formerly very much hotter than
the other.

So we have, then, the reverse of what we found in
the preceding examples, great differences in the cause
and small differences in the effect. Flammarion once
imagined an observer moving away from the earth
at a velocity greater than that of light. For him
time would have its sign changed, history would be
reversed, and Waterloo would come before Austerlitz.
Well, for this observer effects and causes would be
inverted, unstable equilibrium would no longer be the
exception ; on account of the universal irreversibility,
everything would seem to him to come out of a kind

72 SCIENCE AND METHOD.

of chaos in unstable equilibrium, and the whole of
nature would appear to him to be given up to chance.

IV.

We come now to other arguments, in which we
shall see somewhat different characteristics appearing,
and first let us take the kinetic theory of gases. How
are we to picture a receptacle full of gas ? Innumer-
able molecules, animated with great velocities, course
through the receptacle in all directions ; every moment
they collide with the sides or else with one another,
and these collisions take place under the most varied
conditions. What strikes us most in this case is not
the smallness of the causes, but their complexity.
And yet the former element is still found here, and
plays an important part. If a molecule deviated
from its trajectory to left or right in a very small
degree as compared with the radius of action of the
gaseous molecules, it would avoid a collision, or would
suffer it under different conditions, and that would
alter the direction of its velocity after the collision
perhaps by 90 or 180 degrees.

That is not all. It is enough, as we have just seen,
that the molecule should deviate before the collision
in an infinitely small degree, to make it deviate after
the collision in a finite degree. Then, if the molecule
suffers two successive collisions, it is enough that it
should deviate before the first collision in a degree of
infinite smallness of the second order, to make it deviate
after the first collision in a degree of infinite small-
ness of the first order, and after the second collision
in a finite degree. And the molecule will not suffer
two collisions only, but a great number each second.

CHANCE. 73

So that if the first collision multiplied the deviation
by a very large number, A, after n collisions it will be
multiplied by A". It vi^ill, therefore, have become very
great, not only because A is large — that is to say,
because small causes produce great effects — but be-
cause the exponent n is large, that is to say, because
the collisions are very numerous and the causes very
complex.

Let us pass to a second example. Why is it that
in a shower the drops of rain appear to us to be
distributed by chance ? It is again because of the
complexity of the causes which determine their
formation. Ions have been distributed through the
atmosphere ; for a long time they have been sub-
jected to constantly changing air currents, they have
been involved in whirlwinds of very small dimensions,
so that their final distribution has no longer any
relation to their original distribution. Suddenly the
temperature falls, the vapour condenses, and each of
these ions becomes the centre of a raindrop. In
order to know how these drops will be distributed
and how many will fall on each stone of the pave-
ment, it is not enough to know the original position
of the ions, but we must calculate the effect of a
thousand minute and capricious air currents.

It is the same thing again if we take grains of dust
in suspension in water. The vessel is permeated by
currents whose law we know nothing of except that
it is very complicated. After a certain length of
time the grains will be distributed by chance, that
is to say uniformly, throughout the vessel, and this
is entirely due to the complication of the currents
If they obeyed some simple law — if, for instance

74 SCIENCE AND METHOD.

the vessel were revolving and the currents revolved
in circles about its axis — the case would be altered,
for each grain would retain its original height and
its original distance from the axis.

We should arrive at the same result by picturing
the mixing of two liquids or of two fine powders.
To take a rougher example, it is also what
happens when a pack of cards is shuffled. At
each shuffle the cards undergo a permutation similar
to that studied in the theory of substitutions.
What will be the resulting permutation? The prob-
ability that it will be any particular permutation (for
instance, that which brings the card occupying the
position <^ {n) before the permutation into the position
n), this probability, I say, depends on the habits of
the player. But if the player shuffles the cards long
enough, there will be a great number of successive
permutations, and the final order which results will
no longer be governed by anj'thing but chance ; I
mean that all the possible orders will be equally
probable. This result is due to the great number
of successive permutations, that is to say, to the
complexity of the phenomenon.

A final word on the theory of errors. It is a case
in which the causes have complexity and multiplicity.
How numerous are the traps to which the observer
is exposed, even with the best instrument. He must
take pains to look out for and avoid the most flagrant,
those which give birth to systematic errors. But
when he has eliminated these, admitting that he
succeeds in so doing, there still remain many which,
though small, may become dangerous by the ac-
cumulation of their effects. It is from these that

CHANCE. 75

accidental errors arise, and we attribute them to
chance, because their causes are too complicated and
too numerous. Here again we have only small causes,
but each of them would only produce a small effect ;
it is by their union and their number that their effects
become formidable.

V.

There is yet a third point of view, which is less im-
portant than the two former, on which I will not lay so
much stress. When we are attempting to predict a
fact and making an examination of the antecedents,
we endeavour to enquire into the anterior situation.
But we cannot do this for every part of the universe,
and we are content with knowing what is going
on in the neighbourhood of the place where the fact
will occur, or what appears to have some connexion
with the fact. Our enquiry cannot be complete, and
we must know how to select. But we may happen
to overlook circumstances which, at first sight, seemed
completely foreign to the anticipated fact, to which
we should never have dreamed of attributing any
influence, which nevertheless, contrary to all anticipa-
tion, come to play an important part.

A man passes in the street on the way to his
business. Some one familiar with his business could
say what reason he had for starting at such an hour
and why he went by such a street. On the roof a
slater is at work. The contractor who employs him
could, to a certain extent, predict what he will do.
But the man has no thought for the slater, nor the
slater for him ; they seem to belong to two worlds
completely foreign to one another. Nevertheless
the slater drops a tile v/hich kills the man, and we

76 SCIENCE AND METHOD.

should have no hesitation in saying that this was
chance.

Our frailty does not permit us to take in the whole
universe, but forces us to cut it up in slices. We
attempt to make this as little artificial as possible,
and yet it happens, from time to time, that two of
these slices react upon each other, and then the effects
of this mutual action appear to us to be due to chance.

Is this a third way of conceiving of chance ? Not
always ; in fact, in the majority of cases, we come
back to the first or second. Each time that two
worlds, generally foreign to one another, thus come
to act upon each other, the laws of this reaction
cannot fail to be very complex, and moreover a very
small change in the initial conditions of the two
worlds would have been enough to prevent the
reaction from taking place. How very little it would
have taken to make the man pass a moment later,
or the slater drop his tile a moment earlier !

VI.

Nothing that has been said so far explains why
chance is obedient to laws. Is the fact that the
causes are small, or that they are complex, sufficient
to enable us to predict, if not what the effects will
be m each case, at least what they will be on the
average ? In order to answer this question, it will
be best to return to some of the examples quoted
above.

I will begin with that of roulette. I said that the
point where the needle stops will depend on the
initial impulse given it. What is the probability that
this impulse will be of any particular strength .? I

CHANCE. yy

do not know, but it is difficult not to admit that
this probability is represented by a continuous
analytical function. The probability that the impulse
will be comprised between a and a + e will, then,
clearly be equal to the probability that it will be
comprised between a + e and a + ze, pi-ovided that € is
very small. This is a property common to all
analytical functions. Small variations of the function
are proportional to small variations of the variable.

But we have assumed that a very small variation in
the impulse is sufficient to change the colour of the
section opposite which the needle finally stops.
From u to a + « is red, from a + e to a + 2e is black.
The probability of each red section is accordingly the
same as that of the succeeding black section, and
consequently the total probability of red is equal
to the total probability of black.

The datum in the case is the analytical function
which represents the probability of a particular
initial impulse. But the theorem remains true, what-
ever this datum may be, because it depends on a
property common to all analytical functions. From
this it results finally that we have no longer any need
of the datum.

What has just been said of the case of roulette
applies also to the example of the minor planets.
The Zodiac may be regarded as an immense roulette
board on which the Creator has thrown a very great
number of small balls, to which he has imparted
different initial impulses, varying, however, according
to some sort of law. Their actual distribution is
uniform and independent of that law, for the same
reason as in the preceding case. Thus we see why

78 SCIENCE AND METHOD.

phenomena obey the laws of chance when small
differences in the causes are sufficient to produce
great differences in the effects. The probabilities of
these small differences can then be regarded as
proportional to the differences themselves, just be-
cause these differences are small, and small increases
of a continuous function are proportional to those
of the variable.

Let us pass to a totally different example, in which
the complexity of the causes is the principal factor.
I imagine a card-player shuffling a pack of cards.
At each shuffle he changes the order of the cards,
and he may change it in various ways. Let us take
three cards only in order to simplify the explanation.
The cards which, before the shuffle, occupied the
positions 123 respectively may, after the shuffle,
occupy the positions

123, 231, 312, 321, 132, 213.

Each of these six hypotheses is possible, and their
probabilities are respectively

/i. /a. /s, A. /s. A-
The sum of these six numbers is equal to i, but that
is all we know about them. The six probabilities
natural!}' depend upon the player's habits, which we
do not know.

At the second shuffle the process is repeated, and
under the same conditions. I mean, for instance,
that p^ always represents the probability that the
three cards which occupied the positions 123 after
the n"' shuffle and before the w+i'", will occupy the
positions 321 after the n+\"' shuffle. And this re-
mains true, whatever the number n may be, since the

CHANCE. 79

player's habits and his method of shuffling remain
the same.

But if the number of shuffles is very large, the cards
which occupied the positions 123 before the first shuffle
may, after the last shuffle, occupy the positions

123, 231, 312, 321, 132, 213,
and the probability of each of these six hypotheses is
clearly the same and equal to I- ; and this is true what-
ever be the numbers A • • • A. which we do not know.
The great number of shuffles, that is to say, the com-
plexity of the causes, has produced uniformity.

This would apply without change if there were more
than three cards, but even with three the demonstra-
tion would be complicated, so I will content myself
with giving it for two cards only. We have now only
two hypotheses

12, 21,

with the probabilities A and A = I -A- Assume that
there are n shuffles, and that I win a shilling if the
cards are finally in the initial order, and that I lose one
if they are finally reversed. Then my mathematical
expectation will be

(A -A)"

The difference A ~A is certainly smaller than i, so
that if n is very large, the value of my expectation
will be nothing, and we do not require to know A
and A to know that the game is fair.

Nevertheless there would be an exception if one of
the numbers A and A was equal to i and the other to
nothing. // would then hold good no longer, because
our original hypotheses would be too simple.

What we have just seen applies not only to the

8o SCIENCE AND METHOD.

mixing of cards, but to all mixing, to that of powders
and liquids, and even to that of the gaseous molecules
in the kinetic theory of gases. To return to this theory,
let us imagine for a moment a gas whose molecules
cannot collide mutually, but can be deviated by col-
lisions with the sides of the vessel in which the gas
is enclosed. If the form of the vessel is sufficiently
i.omplicated, it will not be long before the distribution
of the molecules and that of their velocities become
uniform. This will not happen if the vessel is spherical,
or if it has the form of a rectangular parallelepiped.
And why not? Because in the former case the dis-
tance of any particular trajectory from the centre
remains constant, and in the latter case we have
the absolute value of the angle of each trajectory
with the sides of the parallelepiped.

Thus we see what we must understand by conditions
that are too simple. They are conditions which pre-
serve something of the original state as an invariable.
Are the differential equations of the problem too
simple to enable us to apply the laws of chance?
This question appears at first sight devoid of any pre-
cise meaning, but we know now what it means. They
are too simple if something is preserved, if they
admit a uniform integral. If something of the initial
conditions remains unchanged, it is clear that the
final situation can no longer be independent of the
initial situation.

We come, lastl}^ to the theory of errors. We are
ignorant of what accidental errors are due to, and it is
just because of this ignorance that we know they will
obey Gauss's law. Such is the paradox. It is ex-
plained in somewhat the same way as the preceding

CHANCE. 8i

cases. We only need to know one thing — that the
errors are very numerous, that they are very small,
and that each of them can be equally well negative
or positive. What is the curve of probability of each
of them ? We do not know, but only assume that it
is symmetrical. We can then show that the resultant
error will follow Gauss's law, and this resultant law is
independent of the particular laws which we do not
know. Here again the simplicity of the result actually
owes its existence to the complication of the data.

VII.

But we have not come to the end of paradoxes. I
recalled just above Flammarion's fiction of the man
who travels faster than light, for whom time has its
sign changed. I said that for him all phenomena
would seem to be due to chance. This is true from
a certain point of view, and yet, at any given moment,
all these phenomena would not be distributed in con-
formity with the laws of chance, since they would be
just as they are for us, who, seeing them unfolded
harmoniously and not emerging from a primitive
chaos, do not look upon them as governed by chance.

What does this mean ? For Flammarion's imagi-
nary Lumen, small causes seem to produce great
effects ; why, then, do things not happen as they do
for us when we think we see great effects due to small
causes ? Is not the same reasoning applicable to
his case?

Let us return to this reasoning. When small dif-
ferences in the causes produce great differences in
the effects, why are the effects distributed according
to the laws of chance ? Suppose a difference of an

(1,777) 6

82 SCIENCE AND METHOD.

inch in the cause produces a difference of a mile in
the effect. If I am to win in case the effect corre-
sponds with a mile bearing an even number, my
probability of winning will be -|. Why is this ?
Because, in order that it should be so, the cause must
correspond with an inch bearing an even number.
Now, according to all appearance, the probability
that the cause will vary between certain limits is
proportional to the distance of those limits, provided
that distance is very small. If this hypothesis be not
admitted, there would no longer be any means of
representing the probability by a continuous function.
Now what will happen when great causes produce
small effects ? This is the case in which we shall not
attribute the phenomenon to chance, and in which
Lumen, on the contrary, would attribute it to chance.
A difference of a mile in the cause corresponds to
a difference of an inch in the effect. Will the
probability that the cause will be comprised between
two limits n miles apart still be proportional to «?
We have no reason to suppose it, since this dis-
tance of n miles is great. But the probability that
the effect will be comprised between two limits n
inches apart will be precisely the same, and ac-
cordingly it will not be proportional to n, and that
notwithstanding the fact that this distance of n
inches is small. There is, then, no means of repre-
senting the law of probability of the effects by a
continuous curve. I do not mean to say that the
curve may not remain continuous in the mtalytical
sense of the word. To infinitely small variations
of the abscissa there will correspond infinitely small
variations of the ordinate. But practically it would

CHANCE. 83

not be continuous, since to very small variations of
the abscissa there would not correspond very small
variations of the ordinate. It would become impos-
sible to trace the curve with an ordinary pencil : that
is what I mean.

What conclusion are we then to draw ? Lumen has
no right to say that the probability of the cause (that
of his cause, which is our effect) must necessarily be
represented by a continuous function. But if that be
so, why have we the right ? It is because that state of
unstable equilibrium that I spoke of just now as initial,
is itself only the termination of a long anterior history.
In the course of this history complex causes have been
at work, and they have been at work for a long time.
They have contributed to bring about the mixture oi"
the elements, and they have tended to make everything
uniform, at least in a small space. They have rounded
off the corners, levelled the mountains, and filled up
the valleys. However capricious and irregular the
original curve they have been given, they have worked
so much to regularize it that they will finally give us
a continuous curve, and that is why we can quite con-
fidently admit its continuity.

Lumen would not have the same reasons for drawing
this conclusion. For him complex causes would not
appear as agents of regularity and of levelling ; on the
contrary, they would only create differentiation and
inequality. He would see a more and more varied
world emerge from a sort of primitive chaos. The
changes he would observe would be for him unfore-
seen and impossible to foresee. They would seem
to him due to some caprice, but that caprice would
not be at all the same as our chance, since it would

84 SCIENCE AND METHOD.

not be amenable to any law, while our chance has its
own laws. All these points would require a much
longer development, which would help us perhaps to
a better comprehension of the irreversibility of the
universe.

VIII.

We have attempted to define chance, and it would
be well now to ask ourselves a question. Has chance,
thus defined so far as it can be, an objective character?

We may well ask it. I have spoken of very small
or very complex causes, but may not what is very
small for one be great for another, and may not what
seems very complex to one appear simple to another ?
I have already given a partial answer, since I stated
above most precisely the case in which differential
equations become too simple for the laws of chance
to remain applicable. But it would be well to exam-
ine the thing somewhat more closely, for there are
still other points of view we may take.

What is the meaning of the word small ? To
understand it, we have only to refer to what has
been said above. A difference is very small, an
interval is small, when within the limits of that in-
terval the probability remains appreciably constant.
Why can that probability be regarded as constant
in a small interval? It is because we admit that the
law of probability is represented by a continuous
curve, not only continuous in the analytical sense of
the word, but practically continuous, as I explained
above. This means not only that it will present no
absolute hiatus, but also that it will have no projections
or depressions too acute or too much accentuated.

What gives us the right to make this hypothesis?

CHANCE. 85

As I said above, it is because, from the begihning of
the ages, there are complex causes that never cease
to operate in the same direction, which cause the
world to tend constantly towards uniformity without
the possibility of ever going back. It is these causes
which, little by little, have levelled the projections and
filled up the depressions, and it is for this reason that
our curves of probability present none but gentle undu-
lations. In millions and millions of centuries we shall
have progressed another step towards uniformity, and
these undulations will be ten times more gentle still.
The radius of mean curvature of our curve will have
become ten times longer. And then a length that
to-day does not seem to us very small, because an
arc of such a length cannot be regarded as rectilineal,
will at that period be properly qualified as very small,
since the curvature will have become ten times less,
and an arc of such a length will not differ appreciably
from a straight line.

Thus the word very small remains relative, but it
is not relative to this man or that, it is relative to
the actual state of the world. It will change its
meaning when the world becomes more uniform and
all things are still more mixed. But then, no doubt,
men will no longer be able to live, but will have to
make way for other beings, shall I say much smaller
or much larger? So that our criterion, remaining
true for all men, retains an objective meaning.

And, further, what is the meaning of the word very
complex ? I have already given one solution, that
which I referred to again at the beginning of this
section ; but there are others. Complex causes, I have
said, produce a more and more intimate mixture, but

86 SCIENCE AND METHOD.

how long will it be before this mixture satisfies us ?
When shall we have accumulated enough complica-
tions ? When will the cards be sufficiently shuffled ?
If we mix two powders, one blue and the other white,
there comes a time when the colour of the mixture
appears uniform. This is on account of the infirmity
of our senses ; it would be uniform for the long-
sighted, obliged to look at it from a distance, when
it would not yet be so for the short-sighted. Even
when it had become uniform for all sights, we could
still set back the limit by employing instruments.
There is no possibility that any man will ever dis-
tinguish the infinite variety that is hidden under the
uniform appearance of a gas, if the kinetic theory is
true. Nevertheless, if we adopt Gouy's ideas on the
Brownian movement, does not the microscope seem to
be on the point of showing us something analogous ?

This new criterion is thus relative like the first, and
if it preserves an objective character, it is because all
men have about the same senses, the power of their
instruments is limited, and, moreover, they only make
use of them occasionally.

IX.

It is the same in the moral sciences, and particularly
in history. The historian is obliged to make a selec-
tion of the events in the period he is studying, and he
only recounts those that seem to him the most im-
portant. Thus he contents himself with relating the
most considerable events of the i6th century, for
instance, and similarly the most remarkable facts of
the 17th century. If the former are sufficient to
explain the latter, we say that these latter conform

CHANCE. 87

to the laws of history. But if a great event of the
17th century owes its cause to a small fact of the
l6th century that no history reports and that every
one has neglected, then we say that this event is due
to chance, and so the word has the same sense as in
the physical sciences ; it means that small causes
have produced great effects.

The greatest chance is the birth of a great man.
It is only by chance that the meeting occurs of two
genital cells of different sex that contain precisely,
each on its side, the mysterious elements whose mutual
reaction is destined to produce genius. It will be
readily admitted that these elements must be rare^
and that their meeting is still rarer. How little it
would have taken to make the spermatozoid which
carried them deviate from its course. It would have
been enough to deflect it a hundredth part of a inch,
and Napoleon would not have been born and the
destinies of a continent would have been changed.
No example can give a better comprehension of the
true character of chance.

One word more about the paradoxes to which the
application of the calculation of probabilities to the
moral sciences has given rise. It has been demon-
strated that no parliament would ever contain a
single member of the opposition, or at least that such
an event would be so improbable that it would be
quite safe to bet against it, and to bet a million to
one. Condorcet attempted to calculate how many
jurymen it would require to make a miscarriage of
justice practically impossible. If we used the results
of this calculation, we should certainly be exposed
to the same disillusionment as by betting on the

88 SCIENCE AND METHOD.

strength of the calculation that the opposition would
never have a single representative.

The laws of chance do not apply to these questions.
If justice does not always decide on good grounds,
it does not make so much use as is generally supposed
of Bridoye's method. This is perhaps unfortunate,
since, if it did, Condorcet's method would protect us
against miscarriages.

What does this mean ? 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 complex, since
they preserve something, and we have seen that this
is the distinguishing mark of "too simple" causes.
When men are brought together, they no longer
decide by chance and independently of each other,
but react upon one another. Many causes come into
action, they trouble the men and draw them this way
and that, but there is one thing they cannot destroy,
the habits they have of Panurge's sheep. And it is this
that is preserved.

X.

The application of the calculation of probabilities
to the exact sciences also involves many difficulties.
Why are the decimals of a table of logarithms or of
the number x distributed in accordance with the laws
of chance ? I have elsewhere studied the question
in regard to logarithms, and there it is eas}'. It is
clear that a small difference in the argument will give
a small difference in the logarithm, but a great differ-
ence in the sixth decimal of the logarithm. We still
find the same criterion.

CHANCE. 89

But as regards the number tt the question presents
more difficulties, and for the moment I have no
satisfactory explanation to give.

There are many other questions that might be
raised, if I wished to attack them before answering
the one I have more especially set myself When we
arrive at a simple result, when, for instance, we find
a round number, we say that such a result cannot be
due to chance, and we seek for a non-fortuitous cause
to explain it. And in fact there is only a very slight
likelihood that, out of 10,000 numbers, chance will
give us a round number, the number 10,000 for in-
stance ; there is only one chance in 10,000. But
neither is there more than one chance in 10,000 that
it will give us any other particular number, and yet
this result does not astonish us, and we feel no hesita-
tion about attributing it to chance, and that merely
because it is less striking.

Is this a simple illusion on our part, or are there
cases in which this view is legitimate ? We must
hope so, for otherwise all science would be impossible.
When we wish to check a hypothesis, what do we
do? We cannot verify all its consequences, since
they are infinite in number. We content ourselves
with verifying a few, and, if we succeed, we declare
that the hypothesis is confirmed, for so much success
could not be due to chance. It is always at bottom
the same reasoning.

I cannot justify it here completely, it would take
me too long, but I can say at least this. We find
ourselves faced by two hypotheses, either a simple
cause or else that assemblage of complex causes we
call chance. We find it natural to admit that the

go SCIENCE AND METHOD.

former must produce a simple result, and then, if we
arrive at this simple result, the round number for
instance, it appears to us more reasonable to attribute
it to the simple cause, which was almost certain to
give it us, than to chance, which could only give it
us once in 10,000 times. It will not be the same
if we arrive at a result that is not simple. It is true
that chance also will not give it more than once in
10,000 times, but the simple cause has no greater
chance of producing it.

BOOK II.
MATHEMATICAL REASONING.

I.

THE RELATIVITY OF SPACE.

I.

It is impossible to picture empty space. All our
efforts to imagine pure space from which the changing
images of material objects are excluded can only
result in a representation in which highly-coloured
surfaces, for instance, are replaced by lines of slight
colouration, and if we continued in this direction to the
end, everything would disappear and end in nothing.
-Hence arises the irreducible relativity of space.

Whoever speaks of absolute space uses a word de-
void of meaning. This is a truth that has been long
proclaimed by all who have reflected on the question,
but one which we are too often inclined to forget.

If I am at a definite point in Paris, at the Place
du Pantheon, for instance, and I say, " I will come
back here to-morrow ; " if I am asked, " Do you mean
that you will come back to the same point in space?"
I should be tempted to answer yes. Yet I should
be wrong, since between now and to-morrow the earth
will have moved, carrying with it the Place du Pan-
theon, which will have travelled more than a million
miles. And if I wished to speak more accurately, I
should gain nothing, since this million of miles has

94 SCIENCE AND METHOD.

been covered by our globe in its motion in relation
to the sun, and the sun in its turn moves in relation
to the Milky Way, and the Milky Way itself is no
doubt in motion without our being able to recognize
its velocity. So that we are, and shall always be,
completely ignorant how far the Place du Pantheon
moves in a day. In fact, what I meant to say was,
" To-morrow I shall see once more the dome and
pediment of the Pantheon," and if there was no
Pantheon my sentence would have no meaning" and
space would disappear.

This is one of the most commonplace forms of the
principle of the relativity of space, but there is another
on which Delbeuf has laid particular stress. Suppose
that in one night all the dimensions of the universe
became a thousand times larger. The world will*
remain similar to itself, if we give the word similitude
the meaning it has in the third book of Euclid.
Only, what was formerly a metre long will now measure
a kilometre, and what was a millimetre long will
become a metre. The bed in which I went to sleep
and my body itself will have grown in the same
proportion. When I wake in the morning what will
be my feeling in face of such an astonishing trans-
formation ? Well, I shall not notice anything at all.
The most exact measures will be incapable of revealing
anything of this tremendous change, since the yard-
measures I shall use will have varied in exactly the
same proportions as the objects I shall attempt to
measure. In reality the change only exists for those
who argue as if space were absolute. If I have argued
for a moment as they do, it was only in order to make
it clearer that their view implies a contradiction. In

THE RELATIVITY OF SPACE. 95

reality it would be better to say that as space is
relative, nothing at all has happened, and that it is
for that reason that we have noticed nothing.

Have we any right, therefore, to say that we know
the distance between two points? No, since that
distance could undergo enormous variations without
our being able to perceive it, provided other distances
varied in the same proportions. We saw just now
that when I say I shall be here to-morrow, that does
not mean that to-morrow I shall be at the point in
space where I am to-day, but that to-morrow I shall
be at the same distance from the Panthdon as I am
to-day. And already this statement is not sufficient,
and I ought to say that to-morrow and to-day my
distance from the Pantheon will be equal to the same
number of times the length of my body.

But that is not all. I imagined the dimensions of
the world changing, but at least the world remaining
always similar to itself We can go much further than
that, and one of the most surprising theories of modern
physicists will furnish the occasion. According to
a hypothesis of Lorentz and Fitzgerald,* all bodies
carried forward in the earth's motion undergo a de-
formation. This deformation is, in truth, very slight,
since all dimensions parallel with the earth's motion
are diminished by a hundred-millionth, while dimen-
sions perpendicular to this motion are not altered.
But it matters little that it is slight ; it is enough
that it should exist for the conclusion I am soon
going to draw from it. Besides, though I said that
it is slight, I really know nothing about it. I have
myself fallen a victim to the tenacious illusion that

* Vide infra. Book III. Chap. ii.

96 SCIENCE AND METHOD.

makes us believe that we think of an absolute space.
I was thinking of the earth's motion on its elliptical
orbit round the sun, and I allowed 1 8 miles a second
for its velocity. But its true velocity (I mean this
time, not its absolute velocity, which has no sense,
but its velocity in relation to the ether), this I do not
know and have no means of knowing. It is, perhaps,
lo or lOO times as high, and then the deformation
will be 100 or io,000 times as great.

It is evident that we cannot demonstrate this de-
formation. Take a cube with sides a yard long. It
is deformed on account of the earth's velocity ; one
of its sides, that parallel with the motion, becomes
smaller, the others do not vary. If I wish to assure
myself of this with the help of a yard-measure, I shall
measure first one of the sides perpendicular to the
motion, and satisfy myself that my measure fits this
side exactly ; and indeed neither one nor other of
these lengths is altered, since they are both perpendic-
ular to the motion. I then wish to measure the other
side, that parallel with the motion ; for this purpose
I change the position of my measure, and turn it so
as to apply it to this side. But the yard-measure,
having changed its direction and having become paral-
lel with the motion, has in its turn undergone the
deformation, so that, though the side is no longer a
yard long, it will still fit it exactly, and I shall be
aware of nothing.

What, then, I shall be asked, is the use of the
hypothesis of Lorentz and Fitzgerald if no experiment
can enable us to verify it ? The fact is that my state-
ment has been incomplete. I have only spoken of
measurements that can be made with a yard-measure.

THE RELATIVITY OF SPACE. 97

but we can also measure a distance by the time that
light takes to traverse it, on condition that we admit
that the velocity of light is constant, and independent
of its direction. Lorentz could have accounted for the
facts by supposing that the velocity of light is greater
in the direction of the earth's motion than in the
perpendicular direction. He preferred to admit that
the velocity is the same in the two directions, but that
bodies are smaller in the former than in the latter. If
the surfaces of the waves of light had undergone the
same deformations as material bodies, we should never
have perceived the Lorentz-Fitzgerald deformation.

In the one case as in the other, there can be no
question of absolute magnitude, but of the meas-
urement of that magnitude by means of some instru-
ment. This instrument may be a yard-measure 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 means of
knowing whether it is the magnitude or the instrument
that has changed.

But what I wish to make clear is, that in this
deformation the world has not remained similar to
itself. Squares have become rectangles or parallel-
ograms, circles ellipses, and spheres ellipsoids. And
yet we have no means of knowing whether this de-
formation is real.

It is clear that we might go much further. Instead
of the Lorentz-Fitzgerald deformation, with its ex-
tremely simple laws, we might imagine a deformation
of any kind whatever ; bodies might be deformed in
accordance with any laws, as complicated as we liked,
and we should not perceive it, provided all bodies

(1,777) 7

98 SCIENCE AND METHOD.

without exception were deformed in accordance with
the same laws. When I say all bodies without excep-
tion, I include, of course, our own bodies and the rays
of light emanating from the different objects.

If we look at the world in one of those mirrors
of complicated form which deform objects in an odd
way, the mutual relations of the different parts of the
world are not altered ; if, in fact, two real objects
touch, their images likewise appear to touch. In truth,
when we look in such a mirror we readily perceive the
deformation, but it is because the real world exists
beside its deformed image. And even if this real
world were hidden from us, there is something which
cannot be hidden, and that is ourselves. We cannot
help seeing, or at least feeling, our body and our
members which have not been deformed, and continue
to act as measuring instruments. But if we imagine
our body itself deformed, and in the same way as if
it were seen in the mirror, these measuring instruments
will fail us in their turn, and the deformation will
no longer be able to be ascertained.

Imagine, in the same way, two universes which are
the image one of the other. With each object P in
the universe A, there corresponds, in the universe B,
an object P^ which is its image. The co-ordinates
of this image P^ are determinate functions of those
of the object P ; moreover, these functions may be
of any kind whatever — I assume only that they are
chosen once for all. Between the position of P and
that of P^ there is a constant relation ; it matters little
what that relation may be, it is enough that it should
be constant.

Well, these two universes will be indistinguishable.

THE RELATIVITY OF SPACE.

99

I mean to say that the former will be for its inhab-
itants what the second is for its own. This would
be true so long as the two universes remained foreign
to one another. Suppose we are inhabitants of the
universe A ; we have constructed our science and
particularly our geometry. During this time the in-
habitants of the universe B have constructed a science,
and as their world is the image of ours, their geometry
will also be the image of ours, or, more accurately,
it will be the same. But if one day a window were to
open for us upon the universe B, we should feel
contempt for them, and we should say, "These
wretched people imagine that they have made a
geometry, but what they so name is only a grotesque
image of ours ; their straight lines are all twisted,
their circles are hunchbacked, and their spheres have
capricious inequalities." We should have no suspicion
that they were saying the same of us, and that no
one will ever know which is right.

We see in how large a sense we must understand
the relativity of space. Space is in reality amorphous,
and it is only the things that are in it that give it
a form. What are we to think, then, of that direct
intuition we have of a straight line or of distance ?
We have so little the intuition of distance in itself
that, in a single night, as we have said, a distance
could become a thousand times greater without our
being able to perceive it, if all other distances had
undergone the same alteration. And in a night the
universe B might even be substituted for the universe
A without our having any means of knowing it, and
then the straight lines of yesterday would have ceased
to be straight, and we should not be aware of anything.

100 SCIENCE AND METHOD.

One part of space is not by itself and in the absolute
sense of the word equal to another part of space, for
if it is so for us, it will not be so for the inhabitants of
the universe B, and they have precisely as much right
to reject our opinion as we have to condemn theirs.

I have shown elsewhere what are the consequences
of these facts from the point of view of the idea that
we should construct non-Euclidian and other analogous
geometries. I do not wish to return to this, and
I will take a somewhat different point of view.

II.

If this intuition of distance, of direction, of the
straight line, if, in a word, this direct intuition of space
does not exist, whence comes it that we imagine
we have it? If this is only an illusion, whence comes
it that the illusion is so tenacious ? This is what
we must examine. There is no direct intuition of
magnitude, as we have said, and we can only arrive
at the relation of the magnitude to our measuring
instruments. Accordingly we could not have con-
structed space if we had not had an instrument
for measuring it. Well, that instrument to which we
refer everything, which we use instinctively, is our
own body. It is in reference to our own body that we
locate exterior objects, and the only special relations
of these objects that we can picture to ourselves are
their relations with our body. It is our body that
serves us, so to speak, as a system of axes of
co-ordinates.

For instance, at a moment a the presence of an
object A is revealed to me by the sense of sight ; at
another moment /i the presence of another object

THE RELATIVITY OF SPACE. loi

B is revealed by another sense, that, for instance,
of hearing or of touch. I judge that this object B
occupies the same place as the object A. What does
this mean? To begin with, it does not imply that
these two objects occupy, at two different moments,
the same point in an absolute space, which, even
if it existed, would escape our knowledge, since
between the moments a and fi the solar system has
been displaced and we cannot know what this dis-
placement is. It means that these two objects occupy
the same relative position in reference to our body.

But what is meant even by this ? The impressions
that have come to us from these objects have followed
absolutely different paths — the optic nerve for the
object A, and the acoustic nerve for the object B ;
they have nothing in common from the qualitative
point of view. The representations we can form of
these two objects are absolutely heterogeneous and
irreducible one to the other. Only I know that,
in order to reach the object A, I have only to extend
my right arm in a certain way ; even though I refrain
from doing it, I represent to myself the muscular and
other analogous sensations which accompany that
extension, and that representation is associated with
that of the object A.

Now I know equally that I can reach the object B
by extending my right arm in the same way, an
extension accompanied by the same train of muscular
sensations. And I mean nothing else but this when
I say that these two objects occupy the same
position.

I know also that I could have reached the object A
by another appropriate movement of the left arm,

102 SCIENCE AND METHOD.

and I represent to myself the muscular sensations that
would have accompanied the movement. And by
the same movement of the left arm, accompanied
by the same sensations, I could equally have reached
the object B.

And this is very important, since it is in this
way that I could defend myself against the dangers
with which the object A or the object B might threaten
me. With each of the blows that may strike us,
nature has associated one or several parries which
enable us to protect ourselves against them. The
same parry may answer to several blows. It is
thus, for instance, that the same movement of the
right arm would have enabled us to defend our-
selves at the moment a against the object A, and
at the moment /3 against the object B. Similarly, the
same blow may be parried in several ways, and we
have said, for instance, that we could reach the object
A equally well either by a certain movement of the
right arm, or by a certain movement of the left.

All these parries have nothing in common with one
another, except that they enable us to avoid the same
blow, and it is that, and nothing but that, we
mean when we say that they are movements ending
in the same point in space. Similarly, these objects,
of which we say that they occupy the same point in
space, have nothing in common, except that the same
parry can enable us to defend ourselves against them.

Or, if we prefer it, let us imagine innumerable
telegraph wires, some centripetal and others centri-
fugal. The centripetal wires warn us of accidents
that occur outside, the centrifugal wires have to
provide the remedy. Connexions are established

THE RELATIVITY OF SPACE. 103

in such a way that when one of the centripetal wires
is traversed by a current, this current acts on a central
exchange, and so excites a current in one of the
centrifugal wires, and matters are so arranged that
several centripetal wires can act on the same centri-
fugal wire, if the same remedy is applicable to several
evils, and that one centripetal wire can disturb several
centrifugal wires, either simultaneously or one in
default of the other, every time that the same evil
can be cured by several remedies.

It is this complex system of associations, it is this
distribution board, so to speak, that is our whole
geometry, or, if you will, all that is distinctive in our
geometry. What we call our intuition of a straight
line or of distance is the consciousness we have of
these associations and of their imperious character.

Whence this imperious character itself comes, it
is easy to understand. The older an association is,
the more indestructible it will appear to us. But
these associations are not, for the most part, conquests
made by the individual, since we see traces of them
in the newly-born infant; they are conquests made
by the race. The more necessary these conquests
were, the more quickly they must have been brought
about by natural selection.

On this account those we have been speaking
of must have been among the earliest, since without
them the defence of the organism would have been
impossible. As soon as the cells were no longer
merely in juxtaposition, as soon as they were called
upon to give mutual assistance to each other, some
such mechanism as we have been describing must
necessarily have been organized in order that the

104 SCIENCE AND METHOD.

assistance should meet the danger without mis-
carrying.

When a frog's head has been cut off, and a drop of
acid is placed at some point on its skin, it tries
to rub off the acid with the nearest foot ; and if that
foot is cut off, it removes it with the other foot. Here
we have, clearly, that double parry I spoke of just now,
making it possible to oppose an evil by a second
remedy if the first fails. It is this multiplicity of
parries, and the resulting co-ordination, that is space.

We see to what depths of unconsciousness we have
to descend to find the first traces of these spacial
associations, since the lowest parts of the nervous
system alone come into play. Once we have rea-
lized this, how can we be astonished at the resistance
we oppose to any attempt to dissociate what has been
so long associated ? Now, it is this very resistance
that we call the evidence of the truths of geometry.
This evidence is nothing else than the repugnance we
feel at breaking with very old habits with which we
have always got on very well.

III.

The space thus created is only a small space that
does not extend beyond what my arm can reach,
and the intervention of memory is necessary to set
back its limits. There are points that will always
remain out of my reach, whatever effort I may make
to stretch out my hand to them. If I were attached
to the ground, like a sea-polype, for instance, which
can only extend its tentacles, all these points would
be outside space, since the sensations we might
experience from the action of bodies placed there

THE RELATIVITY OF SPACE. 105

would not be associated with the idea of any move-
ment enabling us to reach them, or with any appro-
priate parry. These sensations would not seem to us
to have any spacial character, and we should not
attempt to locate them.

But we are not fixed to the ground like the inferior
animals. If the enemy is too far off, we can advance
upon him first and extend our hand when we are near
enough. This is still a parry, but a long-distance
parry. Moreover, it is a complex parry, and into the
representation we make of it there enter the repre-
sentation of the muscular sensations caused by the
movement of the legs, that of the muscular sensations
caused by the final movement of the arm, that of the
sensations of the semi-circular canals, etc. Besides, we
have to make a representation, not of a complexus
of simultaneous sensations, but of a complexus of
successive sensations, following one another in a deter-
mined order, and it is for this reason that I said just
now that the intervention of memory is necessary.

We must further observe that, to reach the same
point, I can approach nearer the object to be attained,
in order not to have to extend my hand so far. And
how much more might be said ? It is not one only, but
a thousand parries I can oppose to the same danger.
All these parries are formed of sensations that may
have nothing in common, and yet we regard them
as defining the same point in space, because they can
answer to the same danger and are one and all
of them associated with the notion of that danger. It
is the possibility of parrying the same blow which
makes the unity of these different parries, just as
it is the possibility of being parried in the same way

io6 SCIENCE AND METHOD.

which makes the unity of the blows of such different
kinds that can threaten us from the same point
in space. It is this double unity that makes the
individuality of each point in space, and in the notion
of such a point there is nothing else but this.

The space I pictured in the preceding section,
which I might call restricted space, was referred to
axes of co-ordinates attached to my body. These axes
were fixed, since my body did not move, and it
was only my limbs that changed their position. What
are the axes to which the extended space is naturally
referred — that is to say, the new space I have just
defined ? We define a point by the succession of
movements we require to make to reach it, starting
from a certain initial position of the body. The axes
are accordingly attached to this initial position of the
body.

But the position I call initial may be arbitrarily
chosen from among all the positions my body has
successively occupied. If a more or less unconscious
memory of these successive positions is necessary for
the genesis of the notion of space, this memory can go
back more or less into the past. Hence results a
certain indeterminateness in the very definition of
space, and it is precisely this indeterminateness which
constitutes its relativity.

Absolute space exists no longer ; there is only space
relative to a certain initial position of the body. For
a conscious being, fixed to the ground like the inferior
animals, who would consequently only know restricted
space, space would still be relative, since it would be
referred to his body, but this being would not be
conscious of the relativity, because the axes to which

THE RELATIVITY OF SPACE. 107

he referred this restricted space would not change.
No doubt the rock to which he was chained would
not be motionless, since it would be involved in the
motion of our planet ; for us, consequently, these axes
would change every moment, but for him they would
not change. We have the faculty of referring our
extended space at one time to the position A of our
body considered as initial, at another to the position
B which it occupied some moments later, which we
are free to consider in its turn as initial, and, accord-
ingly, we make unconscious changes in the co-ordinates
every moment. This faculty would fail our imaginary
being, and, through not having travelled, he would
think space absolute. Every moment his system of
axes would be imposed on him ; this system might
change to any extent in reality, for him it would be
always the same, since it would always be the unique
system. It is not the same for us who possess, each
moment, several systems between which we can choose
at will, and on condition of going back by memory
more or less into the past.

That is not all, for the restricted space would not
be homogeneous. The different points of this space
could not be regarded as equivalent, since some could
only be reached at the cost of the greatest efforts,
while others could be reached with ease. On the
contrary, our extended space appears to us homoge-
neous, and we say that all its points are equivalent.
What does this mean ?

If we start from a certain position A, we can,
starting from that position, effect certain movements
M, characterized by a certain complexus of muscular
sensations. But, starting from another position B,

io8 SCIENCE AND METHOD.

we can execute movements M^ which will be char-
acterized by the same muscular sensations. Then let
a be the situation of a certain point in the body, the
tip of the forefinger of the right hand, for instance,
in the initial position A, and let b be the position of
this same forefinger when, starting from that position

A, we have executed the movements M. Then let d^
be the situation of the forefinger in the position B,
and b^ its situation when, starting from the position

B, we have executed the movements M^

Well, I am in the habit of saying that the points a
and b are, in relation to each other, as the points a)-
and b^, and that means simply that the two series of
movements M and M^ are accompanied by the same
muscular sensations. And as I am conscious that,
in passing from the position A to the position B, my
body has remained capable of the same movements,
I know that there is a point in space which is to the
point a'- what some point b is to the point a, so that
the two points a and a)- are equivalent. It is this that
is called the homogeneity of space, and at the same
time it is for this reason that space is relative, since
its properties remain the same whether they are
referred to the axes A or to the axes B. So that the
relativity of space and its homogeneity are one and
the same thing.

Now, if I wish to pass to the great space, which is
no longer to serve for my individual use only, but in
which I can lodge the universe, I shall arrive at it by
an act of imagination. I shall imagine what a giant
would experience who could reach the planets in a
few steps, or, if we prefer, what I should feel myself
in presence of a world in miniature, in which these

THE RELATIVITY OF SPACE. 109

planets would be replaced by little balls, while on
one of these little balls there would move a Lilliputian
that I should call myself. But this act of imagination
would be impossible for me if I had not previously
constructed my restricted space and my extended
space for my personal use.

IV.

Now we come to the question why all these spaces
have three dimensions. Let us refer to the " distribu-
tion board" spoken of above. We have, on the one
side, a list of the different possible dangers — let us
designate them as Ai, A 2, etc. — and, on the other side,
the list of the different remedies, which I will call in
the same way Bi, B2, etc. Then we have connexions
between the contact studs of the first list and those of
the second in such a way that when, for instance, the
alarm for danger A3 works, it sets in motion or
may set in motion the relay corresponding to the
parry B4.

As I spoke above of centripetal or centrifugal wires,
I am afraid that all I have said may be taken, not as
a simple comparison, but as a description of the
nervous system. Such is not my thought, and that
for several reasons. Firstly, I should not presume to
pronounce an opinion on the structure of the nervous
system which I do not know, while those who have
studied it only do so with circumspection. Secondly,
because, in spite of my incompetence, I fully realize
that this scheme would be far too simple. And lastly,
because, on my list of parries, there appear some that
are very complex, which may even, in the case of
extended space, as we have seen above, consist of

no SCIENCE AND METHOD.

several steps followed by a movement of the arm. It
is not a question, then, of physical connexion between
two real conductors, but of psychological association
between two series of sensations.

If A I and A2, for instance, are both of them
associated with the parry Bl, and if Al is similarly
associated with B2, it will generally be the case that
A2 and B2 will also be associated. If this fundamental
law were not generally true, there would only be an
immense confusion, and there would be nothing that
could bear any resemblance to a conception of space
or to a geometry. How, indeed, have we defined a
point in space ? We defined it in two ways : on the
one hand, it is the whole of the alarms A which are
in connexion with the same parry B ; on the other,
it is the whole of the parries B which are in connexion
with the same alarm A. If our law were not true, we
should be obliged to say that A I and A2 correspond
with the same point, since they are both in con-
nexion with Bl ; but we should be equally obliged
to say that they do not correspond with the same
point, since Ai would be in connexion with B2, and
this would not be true of A2 — which would be a
contradiction.

But from another aspect, if the law were rigorously
and invariably true, space would be quite different
from what it is. We should have well-defined cate-
gories, among which would be apportioned the alarms
A on the one side and the parries B on the other.
These categories would be exceedingly numerous, but
they would be entirely separated one from the other.
Space would be formed of points, very numerous but
discrete ; it would be discontinuous. There would be

THE RELATIVITY OF SPACE. iir

no reason for arranging these points in one order
rather than another, nor, consequently, for attributing
three dimensions to space.

But this is not the case. May I be permitted for
a moment to use the language of those who know
geometry already? It is necessary that I should do
so, since it is the language best understood by those
to whom I wish to make myself clear. When I wish
to parry the blow, I try to reach the point whence
the blow comes, but it is enough if I come fairly near
it. Then the parry Bi may answer to Ai, and to
A2 if the point which corresponds with Bi is sufficiently
close both to that which corresponds with Ai and to
that which corresponds with A2. But it may happen
that the point which corresponds with another parry
B2 is near enough to the point corresponding with
A I, and not near enough to the point corresponding
with A2. And so the parry B2 may answer to Ai
and not be able to answer to A 2.

For those who do not yet know geometry, this may
be translated simply by a modification of the law
enunciated above. Then what happens is as follows.
Two parries, Bi and B2, are associated with one alarm
A I, and with a very great number of alarms that we
will place in the same category as A I, and make to
correspond with the same point in space. But we
may find alarms A2 which are associated with B2 and
not with Bi, but on the other hand are associated with
B3, which are not with Ai, and so on in succession,
so that we may write the sequence

Bi, Ai, B2, A2, B3, A3, B4, A4,
in which each term is associated with the succeeding

112 SCIENCE AND METHOD.

and preceding terms, but not with those that are
several places removed.

It is unnecessary to add that each of the terms of
these sequences is not isolated, but forms part of a
very numerous category of other alarms or other
parries which has the same connexions as it, and
may be regarded as belonging to the same point in
space. Thus the fundamental law, though admitting
of exceptions, remains almost always true. Only, in
consequence of these exceptions, these categories,
instead of being entirely separate, partially encroach
upon each other and mutually overlap to a certain
extent, so that space becomes continuous.

Furthermore, the order in which these categories
must be arranged is no longer arbitrary, and a
reference to the preceding sequence will make it
clear that B2 must be placed between Ai and A2,
and, consequently, between Br and B3, and that it
could not be placed, for instance, between B3
and B4.

Accordingly there is an order in which our cate-
gories range themselves naturally which corresponds
with the points in space, and experience teaches us
that this order presents itself in the form of a three-
circuit distribution board, and it is for this reason
that space has three dimensions.

V.

Thus the characteristic property of space, that of
having three dimensions, is only a property of our
distribution board, a property residing, so to speak,
in the human intelligence. The destruction of some
of these connexions, that is to say, of these associa-

THE RELATIVITY OF SPACE. 113

tions of ideas, would be sufficient to give us a dif-
ferent distribution board, and that might be enough
to endow space with a fourth dimension.

Some people will be astonished at such a result.
The exterior world, they think, must surely count
for something. If the number of dimensions comes
from the way in which we are made, there might
be thinking beings living in our world, but made
differently from us, who would think that space has
more or less than three dimensions. Has not M.
de Cyon said that Japanese mice, having only two
pairs of semicircular canals, think that space has
two dimensions ? Then will not this thinking being,
if he is capable of constructing a physical system,
make a system of two or four dimensions, which
yet, in a sense, will be the same as ours, since it will
be the description of the same world in another
language ?

It quite seems, indeed, that it would be possible to
translate our physics into the language of geometry
of four dimensions. Attempting such a translation
would be giving oneself a great deal of trouble for
little profit, and I will content myself with men-
tioning Hertz's mechanics, in which something of
the kind may be seen. Yet it seems that the
translation would always be less simple than the
text, and that it would never lose the appearance of
a translation, for the language of three dimensions
seems the best suited to the description of our
world, even though that description may be made,
in case of necessity, in another idiom.

Besides, it is not by chance that our distribution
board has been formed. There is a connexion
(1.777) 8

114 SCIENCE AND METHOD.

between the alarm A I and the parry Bl, that is, a
property residing in our intelh'gence. But why is
there this connexion? It is because the parry Bi
enables us effectively to defend ourselves against the
danger Ai, and that is a fact exterior to us, a
property of the exterior world. Our distribution
board, then, is only the translation of an assemblage
of exterior facts ; if it has three dimensions, it is
because it has adapted itself to. a world having
certain properties, and the most important of these
properties is that there exist natural solids which
are clearly displaced in accordance with the laws
we call laws of motion of unvarying solids. If, then,
the language of three dimensions is that which
enables us most easily to describe our world, we
must not be surprised. This language is founded
on our distribution board, and it is in order to
enable us to live in this world that this board has
been established.

I have said that we could conceive of thinking
beings, living in our world, whose distribution board
would have four dimensions, who would, consequently,
think in hyperspace. It is not certain, however, that
such beings, admitting that they were born, would
be able to live and defend themselves against the
thousand dangers by which they would be assailed.

VI.

A few remarks in conclusion. There is a striking
contrast between the roughness of this primitive
geometry which is reduced to what I call a distribu-
tion board, and the infinite precision of the geometry
of geometricians. And yet the latter is the child ot

THE RELATIVITY OF SPACE. 115

the former, but not of it alone ; it required to be
fertilized by the faculty we have of constructing
mathematical concepts, such, for instance, as that of
the group. It was necessary to find among these
pure concepts the one that was best adapted to
this rough space, whose genesis I have tried to
explain in the preceding pages, the space which is
common to us and the higher animals.

The evidence of certain geometrical postulates is
only, as I have said, our unwillingness to give up
very old habits. But these postulates are infinitely
precise, while the habits have about them some-
thing essentially fluid. As soon as we wish to think,
we are bound to have infinitely precise postulates,
since this is the only means of avoiding contradic-
tion. But among all the possible systems of postu-
lates, there are some that we shall be unwilling to
choose, because they do not accord sufficiently with
our habits. However fluid and elastic these may be,
they have a limit of elasticity.

It will be seen that though geometry is not an
experimental science, it is a science born in con-
nexion with experience ; that we have created the
space it studies, but adapting it to the world in
which we live. We have chosen the most con-
venient space, but experience guided our choice.
As the choice was unconscious, it appears to be
imposed upon us. Some say that it is imposed by
experience, and others that we are born with our
space ready-made. After the preceding considera-
tions, it will be seen what proportion of truth and
of error there is in these two opinions.

In this progressive education which has resulted

ii6 SCIENCE AND METHOD.

in the construction of space, it is very difficult to
determine what is the share of the individual and
what of the race. To what extent could one of us,
transported from his birth into an entirely different
world, where, for instance, there existed bodies dis-
placed in accordance with the laws of motion of
non-Euclidian solids — to what extent, I say, would
he be able to give up the ancestral space in order
to build up an entirely new space ?

The share of the race seems to preponderate largely,
and yet if it is to it that we owe the rough space,
the fluid space of which I spoke just now, the space
of the higher animals, is it not to the unconscious
experience of the individual that we owe the in-
finitely precise space of the geometrician? This is
a question that is not easy of solution. I would
mention, however, a fact which shows that the space
bequeathed to us by our ancestors still preserves a
certain plasticity. Certain hunters learn to shoot
fish under the water, although the image of these
fish is raised by refraction ; and, moreover, they do
it instinctively. Accordingly they have learnt to
modify their ancient instinct of direction, or, if you
will, to substitute for the association Ai, Bi, another
association Ai, B2, because experience has shown
them that the former does not succeed.

II.

MATHEMATICAL DEFINITIONS AND
EDUCATION.

I. I have to speak here of general definitions in
mathematics. At least that is what the title of the
chapter says, but it will be impossible for me to
confine myself to the subject as strictly as the rule
of unity of action demands. I shall not be able to
treat it without speaking to some extent of other
allied questions, and I must ask your kind forgiveness
if I am thus obliged from time to time to walk among
the flower-beds to right or left.

What is a good definition ? For the philosopher
or the scientist, it is a definition which applies to
all the objects to be defined, and applies only to
them ; it is that which satisfies the rules of logic.
But in education it is not that ; it is one that can be
understood by the pupils.

How is it that there are so many minds that are
incapable of understanding mathematics? Is there
not something paradoxical in this ? Here is a
science which appeals only to the fundamental
principles of logic, to the principle of contradiction,
for instance, to what forms, so to speak, the skeleton
of our understanding, to what we could not be de-
prived of without ceasing to think, and yet there are

ii8 SCIENCE AND METHOD.

people who find it obscure, and actually they are the
majority. That they should be incapable of discovery
we can understand, but that they should fail to under-
stand the demonstrations expounded to them, that
they should remain blind when they are shown a
light that seems to us to shine with a pure brilliance,
it is this that is altogether miraculous.

And yet one need have no great experience of
examinations to know that these blind people are
by no means exceptional beings. We have here a
problem that is not easy of solution, but yet must
engage the attention of all who wish to devote them-
selves to education.

What is understanding ? Has the word the same
meaning for everybody? Does understanding the
demonstration of a theorem consist in examining each
of the syllogisms of which it is composed in succession,
and being convinced that it is correct and conforms
to the rules of the game? In the same way, does
understanding a definition consist simply in recog-
nizing that the meaning of all the terms employed
is already known, and being convinced that it in-
volves no contradiction ?

Yes, for some it is ; when they have arrived at the
conviction, they will say, I understand. But not
for the majority. Almost all are more exacting ;
they want to know not only whether all the syllo-
gisms of a demonstration are correct, but why they
are linked together in one order rather than in
another. As long as they appear to them engendered
by caprice, and not by an intelligence constantly
conscious of the end to be attained, they do not think
they have understood.

DEFINITIONS AND EDUCATION. 119

No doubt they are not themselves fully aware of
what they require and could not formulate their
desire, but if they do not obtain satisfaction, they
feel vaguely that something is wanting. Then what
happens? At first they still perceive the evidences
that are placed before their eyes, but, as they are
connected by too attenuated a thread with those that
precede and those that follow, they pass without
leaving a trace in their brains, and are immediately
forgotten ; illuminated for a moment, they relapse
at once into an eternal night. As they advance
further, they will no longer see even this ephemeral
light, because the theorems depend one upon another,
and those they require have been forgotten. Thus
it is that they become incapable of understanding
mathematics.

It is not always the fault of their instructor. Often
their intellect, which requires to perceive the connect-
ing thread, is too sluggish to seek it and find it. But
in order to come to their assistance, we must first of
all thoroughly understand what it is that stops them.

Others will always ask themselves what use it is.
They will not have understood, unless they find
around them, in practice or in nature, the object of
such and such a mathematical notion. Under each
word they wish to put a sensible image ; the definition
must call up this image, and at each stage of the
demonstration they must see it being transformed
and evolved. On this condition only will they under-
stand and retain what they have understood. These
often deceive themselves : they do not listen to the
reasoning, they look at the figures ; they imagine that
they have understood when they have only seen.

120 SCIENCE AND METHOD.

2. What different tendencies we have here ! Are
we to oppose them, or are we to make use of them ?
And if we wish to oppose them, which are we to
favour? Are we to show those who content them-
selves with the pure logic that they have only seen
one side of the matter, or must we tell those who are
not so easily satisfied that what they demand is not
necessary ?

In other words, should we constrain young people
to change the nature of their minds ? Such an
attempt would be useless ; we do not possess the
philosopher's stone that would enable us to transmute
the metals entrusted to us one into the other. All
that we can do is to work them, accommodating our-
selves to their properties.

Many children are incapable of becoming mathe-
maticians who must none the less be taught
mathematics ; and mathematicians themselves are
not all cast in the same mould. We have only to
read their works to distinguish among them two kinds
of minds — logicians like Weierstrass, for instance, and
intuitionists like Riemann. There is the same
difference among our students. Some prefer to treat
their problems " by analysis," as they say, others " by
geometry."

It is quite useless to seek to change anything in
this, and besides, it would not be desirable. It is
well that there should be logicians and that there
should be intuitionists. Who would venture to
say whether he would prefer that Weierstrass had
never written or that there had never been a Rie-
mann ? And so we must resign ourselves to the
diversity of minds, or rather we must be glad of it.

DEFINITIONS AND EDUCATION. 121

3. Since the word understand has several meanings,
the definitions that will be best understood by some
are not those that will be best suited to others. We
have those who seek to create an image, and those
who restrict themselves to combining empty forms,
perfectly intelligible, but purely intelligible, and de-
prived by abstraction of all matter.

I do not know whether it is necessary to quote
any examples, but I will quote some nevertheless,
and, first, the definition of fractions will furnish us with
an extreme example. In the primary schools, when
they want to define a fraction, they cut up an apple or
a pie. Of course this is done only in imagination and
not in reality, for I do not suppose the budget of primary
education would allow such an extravagance. In the
higher normal school, on the contrary, or in the
universities, they say : a fraction is the combination
of two whole numbers separated by a horizontal line.
By conventions they define the operations that these
symbols can undergo ; they demonstrate that the rules
of these operations are the same as in the calculation
of whole numbers ; and, lastly, they establish that
multiplication of the fraction by the denominator,
in accordance with these rules, gives the numerator.
This is very well, because it is addressed to young
people long since familiarized with the notion of
fractions by dint of cutting up apples and other
objects, so that their mind, refined by a considerable
mathematical education, has, little by little, come to
desire a purely logical definition. But what would
be the consternation of the beginner to whom we
attempted to offer it ?

Such, also, are the definitions to be found in a

122 SCIENCE AND METHOD.

book that has been justly admired and has received
several awards of merit — Hilbert's " Grundlagen der
Geometric." Let us see how he begins. " Imagine
three systems of THINGS, which we will call points,
straight lines, and planes." What these " things " are
we do not know, and we do not need to know — it
would even be unfortunate that we should seek to
know ; all that we have the right to know about them
is that we should learn their axioms, this one, for
instance : " Two different points always determine
a straight line," which is followed by this comment-
ary : " Instead of determine we may say that the
straight line passes through these two points, or that
it joins these two points, or that the two points are
situated on the straight line." Thus " being situated
on a straight line" is simply defined as synonymous
with " determining a straight line." Here is a book
of which I think very highly, but which I should not
recommend to a schoolboy. For the matter of that
I might do it without fear ; he would not carry his
reading very lar.

I have taken extreme examples, and no instructor
would dream of going so far. But, even though he
comes nowhere near such models, is he not still
exposed to the same danger?

We are in a class of the fourth grade. The teacher
is dictating : " A circle is the position of the points
in a plane which are the same distance from an in-
terior point called the centre." The good pupil writes
this phrase in his copy-book and the bad pupil draws
faces, but neither of them understands. Then the
teacher takes the chalk and draws a circle on the
board. "Ah," think the pupils, "why didn't he say

DEFINITIONS AND EDUCATION. 123

at once, a circle is a round, and we should have
understood." No doubt it is the teacher who is
right. The pupils' definition would have been of no
value, because it could not have been used for any
demonstration, and chiefly because it could not have
given them the salutary habit of analyzing their con-
ceptions. But they should be made to see that they
do not understand what they think they understand,
and brought to realize the roughness of their primitive
concept, and to be anxious themselves that it should
be purified and refined.

4. I shall return to these examples ; I only wished
to show the two opposite conceptions. There is a
violent contrast between them, and this contrast is
explained by the history of the science. If we read
a book written fifty years ago, the greater part of the
arguments appear to us devoid of exactness.

At that period they assumed that a continuous func-
tion cannot change its sign without passing through
zero, but to-day we prove it. They assumed that the
ordinary rules of calculus are applicable to incommen-
surable numbers ; to-day we prove it. They assumed
many other things that were sometimes untrue.

They trusted to intuition, but intuition cannot give
us exactness, nor even certainty, and this has been
recognized more and more. It teaches us, for instance,
that every curve has a tangent — that is to say, that
every continuous function has a derivative — and that
is untrue. As certainty was required, it has been
necessary to give less and less place to intuition.

How has this necessary evolution come about ? It
was not long before it was recognized that exactness

124 SCIENCE AND METHOD.

cannot be established in the arguments unless it is
first introduced into the definitions.

For a long time the objects that occupied the atten-
tion of mathematicians were badly defined. They
thought they knew them because they represented
them by their senses or their imagination, but they
had only a rough image, and not a precise idea such
as reasoning can take hold of.

It is to this that the logicians have had to apply their
efforts, and similarly for incommensurable numbers.

The vague idea of continuity which we owe to
intuition has resolved itself into a complicated system
of inequalities bearing on whole numbers. Thus it
is that all those difficulties which terrified our ances-
tors when they reflected upon the foundations of the
infinitesimal calculus have finally vanished.

In analysis to-day there is no longer anything but
whole numbers, or finite or infinite systems of whole
numbers, bound together by a network of equalities
and inequalities. Mathematics, as it has been said,
has been arithmetized.

5. But we must not imagine that the science of
mathematics has attained to absolute exactness with-
out making any sacrifice. What it has gained in
exactness it has lost in objectivity. It is by with-
drawing from reality that it has acquired this perfect
purity. We can now move freely over its whole
domain, which formerly bristled with obstacles. But
these obstacles have not disappeared ; they have only
been removed to the frontier, and will have to be
conquered again if we wish to cross the frontier and
penetrate into the realms of practice.

DEFINITIONS AND EDUCATION. 125

We used to possess a vague notion, formed of in-
congruous elements, some a priori and others derived
from more or less digested experiences, and we im-
agined we knew its principal properties by intuition.
To-day we reject the empirical element and preserve
only the a priori ones. One of the properties
serves as definition, and all the others are de-
duced from it by exact reasoning. This is very well,
but it still remains to prove that this property, which
has become a definition, belongs to the real objects
taught us by experience, from which we had drawn
our vague intuitive notion. In order to prove it we
shall certainly have to appeal to experience or make
an effort of intuition ; and if we cannot prove it, our
theorems will be perfectly exact but perfectly useless.

Logic sometimes breeds monsters. For half a
century there has been springing up a host of weird
functions, which seem to strive to have as little resem-
blance as possible to honest functions that are of some
use. No more continuity, or else continuity but no
derivatives, etc. More than this, from the point of
view of logic, it is these strange functions that are
the most general ; those that are met without being
looked for no longer appear as more than a particular
case, and they have only quite a little corner left them.

Formerly, when a new function was invented, it
was in view of some practical end. To-day they are
invented on purpose to show our ancestors' reasonings
at fault, and we shall never get anything more than
that out of them.

If logic were the teacher's only guide, he would
have to begin with the most general, that is to say,
with the most weird, functions. He would have to

126 SCIENCE AND METHOD.

set the beginner to wrestle with this collection of
monstrosities. If you don't do so, the logicians might
say, you will only reach exactness by stages.

6. Possibly this may be true, but we cannot take
such poor account of reality, and I do not mean
merely the reality of the sensible world, which has
its value nevertheless, since it is for battling with
it that nine-tenths of our pupils are asking for arms.
There is a more subtle reality which constitutes the
life of mathematical entities, and is something more
than logic.

Our body is composed of cells, and the cells of
atoms, but are these cells and atoms the whole reality
of the human body? Is not the manner in which
these cells are adjusted, from which results the unity
of the individual, also a reality, and of much greater
interest ?

Would a naturalist imagine that he had an adequate
knowledge of the elephant if he had never studied the
animal except through a microscope?

It is the same in mathematics. When the logician
has resolved each demonstration into a host of ele-
mentary operations, all of them correct, he will not yet
be in possession of the whole reality ; that indefinable
something that constitutes the unity of the demonstra-
tion will still escape him completely.

What good is it to admire the mason's work in the
edifices erected by great architects, if we cannot under-
stand the general plan of the master ? Now pure logic
cannot give us this view of the whole ; it is to intuition
we must look for it.

Take, for instance, the idea of the continuous func-

DEFINITIONS AND EDUCATION. 127

tion. To begin with, it is only a perceptible image,
a line drawn with chalk on a blackboard. Little by
little it is purified ; it is used for constructing a com-
plicated system of inequalities which reproduces all
the lines of the original image ; when the work is
quite finished, the centering is removed, as it is after
the construction of an arch ; this crude representation
is henceforth a useless support, and disappears, and
there remains only the edifice itself, irreproachable in
the eyes of the logician. And yet, if the instructor
did not recall the original image, if he did not replace
the centering for a moment, how would the pupil
guess by what caprice all these inequalities had been
scaffolded in this way one upon another? The defini-
tion would be logically correct, but it would not show
him the true reality.

7. And so we are obliged to make a step back-
wards. No doubt it is hard for a master to teach
what does not satisfy him entirely, but the satisfaction
of the master is not the sole object of education. We
have first to concern ourselves with the pupil's state
of mind, and what we want it to become.

Zoologists declare that the embryonic development
of an animal repeats in a very short period of time
the whole history of its ancestors of the geological
ages. It seems to be the same with the development
of minds. The educator must make the child pass
through all that his fathers have passed through, more
rapidly, but without missing a stage. On this account,
the history of any science must be our first guide.

Our fathers imagined they knew what a fraction
was, or continuity, or the area of a curved surface ; it

128 SCIENCE AND METHOD.

is we who have reahzed that they did not. In the
same way our pupils imagine that they know it when
they begin to study mathematics seriously. If, with-
out any other preparation, I come and say to them :
" No, you do not know it ; you do not understand
what you imagine you understand ; I must demon-
strate to you what appears to you evident ; " and if,
in the demonstration, I rely on premises that seem
to them less evident than the conclusion, what will
the wretched pupils think ? They will think that the
science of mathematics is nothing but an arbitrary
aggregation of useless subtleties ; or they will lose
their taste for it ; or else they will look upon it as
an amusing game, and arrive at a state of mind
analogous to that of the Greek sophists.

Later on, on the contrary, when the pupil's mind
has been familiarized with mathematical reasoning
and ripened by this long intimacy, doubts will spring
up of their own accord, and then your demonstration
will be welcome. It will arouse new doubts, and
questions will present themselves successively to the
child, as they presented themselves successively to
our fathers, until they reach a point when only perfect
exactness will satisfy them. It is not enough to feel
doubts about everything; we must know why we doubt.

8. The principal aim of mathematical education is
to develop certain faculties of the mind, and among
these intuition is not the least precious. It is through
it that the mathematical world remains in touch with
the real world, and even if pure mathematics could
do without it, we should still have to have recourse
to it to fill up the gulf that separates the symbol

DEFINITIONS AND EDUCATION. 129

from reality. The practitioner will always need it,
and for every pure geometrician there must be a
hundred practitioners.

The engineer must receive a complete mathematical
training, but of what use is it to be to him, except to
enable him to see the different aspects of things and
to see them quickly ? He has no time to split hairs.
In the complex physical objects that present them-
selves to him, he must promptly recognize the point
where he can apply the mathematical instruments we
have put in his hands. How could he do this if we left
between the former and the latter that deep gulf dug
by the logicians ?

9. Beside the future engineers are other less numerous
pupils, destined in their turn to become teachers, and
so they must go to the very root of the matter ; a
profound and exact knowledge of first principles is
above all indispensable for them. But that is no
reason for not cultivating their intuition, for they
would form a wrong idea of the science if they never
looked at it on more than one side, and, besides, they
could not develop in their pupils a quality they did
not possess themselves.

For the pure geometrician hintself this faculty is
necessary : it is by logic that we prove, but by intui-
tion that we discover. To know how to criticize is
good, but to know how to create is better. You
know how to recognize whether a combination is
correct, but much use this will be if you do not
possess the art of selecting among all the possible
combinations. Logic teaches us that on such and
such a road we are sure of not meeting an obstacle ;

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130 SCIENCE AND METHOD.

it does not tell us which is the road that leads to the
desired end. For this it is necessary to see the end
from afar, and the faculty which teaches us to see is
intuition. Without it, the geometrician would be like
a writer well up in grammar but destitute of ideas.
Now how is this faculty to develop, if, as soon as it
shows itself, it is hounded out and proscribed, if we
learn to distrust it before we know what good can be
got from it ?

And here let me insert a parenthesis to insist on
the importance of written exercises. Compositions
in writing are perhaps not given sufficient prominence
in certain examinations. In the 'kcole Poly technique, for
instance, I am told that insistence on such compositions
would close the door to very good pupils who know
their subject and understand it very well, and yet are
incapable of applying it in the smallest degree. I
said just above that the word understand has several
meanings. Such pupils only understand in the first
sense of the word, and we have just seen that this
is not sufficient to make either an engineer or a
geometrician. Well, since we have to make a choice,
I prefer to choose those who understand thoroughly.

lo. But is not the art of exact reasoning also a
precious quality that the teacher of mathematics
should cultivate above all else? I am in no danger
of forgetting it : we must give it attention, and that
from the beginning. I should be distressed to see
geometry degenerate into some sort of low - grade
tachymetrics, and I do not by any means subscribe
to the extreme doctrines of certain German professors.
But we have sufficient opportunity of training pupils

DEFINITIONS AND EDUCATION. 131

in correct reasoning in those parts of mathematics in
which the disadvantages I have mentioned do not
occur. We have long series of theorems in which
absolute logic has ruled from the very start and, so to
speak, naturally, in which the first geometricians have
given us models that we must continually imitate and
admire.

It is in expounding the first principles that we must
avoid too much subtlety, for there it would be too
disheartening, and useless besides. We cannot prove
everything, we cannot define everything, and it will
always be necessary to draw upon intuition. What
does it matter whether we do this a little sooner or a
little later, and even whether we ask for a little more
or a little less, provided that, making a correct use
of the premises it gives us, we learn to reason
accurately ?

ir. Is it possible to satisfy so many opposite
conditions? Is it possible especially when it is a
question of giving a definition ? How are we to find
a statement that will at the same time satisfy the
inexorable laws of logic and our desire to understand
the new notion's place in the general scheme of the
science, our need of thinking in images ? More often
than not we shall not find it, and that is why the
statement of a definition is not enough ; it must be
prepared and it must be justified.

What do I mean by this ? You know that it has
often been said that every definition implies an axiom,
since it asserts the existence of the object defined.
The definition, then, will not be justified, from the
purely logical point of view, until we have proved that

132 SCIENCE AND METHOD.

it involves no contradiction either in its terms or with
the truths previously admitted.

But that is not enough. A definition is stated as
a convention, but the majority of minds will revolt
if you try to impose it upon them as an arbitrary
convention. They will have no rest until you have
answered a great number of questions.

Mathematical definitions are most frequently, as
M. Liard has shown, actual constructions built up
throughout of simpler notions. But why should these
elements have been assembled in this manner, when
a thousand other assemblages were possible ? Is it
simply caprice? If not, why had this combination
more right to existence than any of the others ? What
need does it fill ? How was it foreseen that it would
play an important part in the development of the
science, that it would shorten our reasoning and our
calculations? Is there any familiar object in nature
that is, so to speak, its indistinct and rough image?

That is not all. If you give a satisfactory answer
to all these questions, we shall realize that the new-
comer had the right to be baptized. But the choice of
a name is not arbitrary either ; we must explain what
analogies have guided us, and that if we have given
analogous names to different things, these things at
least differ only in matter, and have some resemblance
in form, that their properties are analogous and, so to
speak, parallel.

It is on these terms that we shall satisfy all propen-
sities. If the statement is sufficiently exact to please
the logician, the justification will satisfy the intui-
tionist. But we can do better still. Whenever it is
possible, the justification will precede the statement

DEFINITIONS AND EDUCATION. 133

and prepare it. The general statement will be led up
to by the study of some particular examples.

One word more. The aim of each part of the
statement of a definition is to distinguish the object
to be defined from a class of other neighbouring
objects. The definition will not be understood until
you have shown not only the object defined, but the
neighbouring objects from which it has to be dis-
tinguished, until you have made it possible to grasp
the difference, and have added explicitly your reason
for saying this or that in stating the definition.

But it is time to leave generalities and to enquire
how the somewhat abstract principles I have been
expounding can be applied in arithmetic, in geometry,
in analysis, and in mechanics.

Arithmetic.

12. We do not have to define the whole number.
On the other hand, operations on whole numbers are
generally defined, and I think the pupils learn these
definitions by heart and attach no meaning to them.
For this there are two reasons : first, they are taught
them too early, while their mind still feels no need of
them ; and then these definitions are not satisfactory
from the logical point of view. For addition, we
cannot find a good one, simply because we must
stop somewhere, and cannot define everything. The
definition of addition is to say that it consists in adding.
All that we can do is to start with a certain number
of concrete examples and say, the operation that has
just been performed is called addition.

For subtraction it is another matter. It can be
defined logically as the inverse operation of addition.

134 SCIENCE AND METHOD.

But is that how we should begin ? Here, again, we
should start with examples, and show by these
examples the relation of the two operations. Thus
the definition will be prepared and justified.

In the same way for multiplication. We shall take
a particular problem ; we shall show that it can be
solved by adding several equal numbers together ;
we shall then point out that we arrive at the result
quicker by multiplication, the operation the pupils
perform already by rote, and the logical definition will
spring from this quite naturally.

We shall define division as the inverse operation
of multiplication ; but we shall begin with an example
drawn from the familiar notion of sharing, and we
shall show by this example that multiplication
reproduces the dividend.

There remain the operations on fractions. There is
no difficulty except in the case of multiplication. The
best way is first to expound the theory of proportions,
as it is from it alone that the logical definition can
spring. But, in order to gain acceptance for the
definitions that are met with at the start in this theory,
we must prepare them by numerous examples drawn
from classical problems of the rule of three, and we
shall be careful to introduce fractional data. We shall
not hesitate, either, to familiarize the pupils with the
notion of proportion by geometrical figures ; either
appealing to their recollection if they have already
done any geometry, or having recourse to direct
intuition if they have not, which, moreover, will prepare
them to do it. I would add, in conclusion, that after
having defined the multiplication of fractions, we must
justify this definition by demonstration that it is

DEFINITIONS AND EDUCATION. 135

commutative, associative, and distributive, making it
quite clear to the listeners that the verification has
been made in order to justify the definition.

We see what part is played in all this by geometrical
figures, and this part is justified by the philosophy and
the history of the science. If arithmetic had remained
free from all intermixture with geometry, it would
never have known anything but the whole number.
It was in order to adapt itself to the requirements of
geometry that it discovered something else.

Geometry,

In geometry we meet at once the notion of the
straight line. Is it possible to define the straight
line ? The common definition, the shortest path from
one point to another, does not satisfy me at all. I
should start simply with the ruler, and I should first
show the pupil how we can verify a ruler by revolving
it. This verification is the true definition of a straight
line, for a straight line is an axis of rotation. We
should then show him how to verify the ruler by
sliding it, and we should have one of the most im-
portant properties of a straight line. As for that
other property, that of being the shortest path from
one point to another, it is a theorem that can be
demonstrated apodeictically, but the demonstration is
too advanced to find a place in secondary education.
It will be better to show that a ruler previously veri-
fied can be applied to a taut thread. We must not
hesitate, in the presence of difficulties of this kind,
to multiply the axioms, justifying them by rough
examples.

Some axioms we must admit ; and if we admit a

136 SCIENCE AND METHOD.

few more than is strictly necessary, the harm is not
great. The essential thing is to learn to reason
exactly with the axioms once admitted. Uncle
Sarcey, who loved to repeat himself, often said that
the audience at a theatre willingly accepts all the
postulates imposed at the start, but that once the
curtain has gone up it becomes inexorable on the
score of logic. Well, it is just the same in mathe-
matics.

For the circle we can start with the compass. The
pupils will readily recognize the curve drawn. We
shall then point out to them that the distance of the
two points of the instrument remains constant, that
one of these points is fixed and the other movable,
and we shall thus be led naturally to the logical
definition.

The definition of a plane implies an axiom, and
we must not attempt to conceal the fact. Take a
drawing-board and point out how a movable ruler
can be applied constantly to the board, and that
while still retaining three degrees of freedom. We
should compare this with the cylinder and the cone,
surfaces to which a straight line cannot be applied
unless we allow it only two degrees of freedom.
Then we should take three drawing-boards, and we
should show first that they can slide while still re-
maining in contact with one another, and that with
three degrees of freedom. And lastly, in order to
distinguish the plane from the sphere, that two of
these boards that can be applied to a third can also
be applied to one another.

Perhaps you will be surprised at this constant use
of movable instruments. It is not a rough artifice.

DEFINITIONS AND EDUCATION. 137

and it is much more philosophical than it would
appear at first sight. What is geometry for the
philosopher ? It is the study of a group. And what
group ? That of the movements of solid bodies. How
are we to define this group, then, without making some
solid bodies move?

Are we to preserve the classical definition of par-
allels, and say that we give this name to two straight
lines, situated in the same plane, which, being pro-
duced ever so far, never meet? No, because this
definition is negative, because it cannot be verified
by experience, and cannot consequently be regarded
as an immediate datum of intuition, but chiefly because
it is totally foreign to the notion of group and to the
consideration of the motion of solid bodies, which is,
as I have said, the true source of geometry. Would
it not be better to define first the rectilineal trans-
position of an invariable figure as a motion in which
all the points of this figure have rectilineal trajectories,
and to show that such a transposition is possible,
making a square slide on a ruler ? From this experi-
mental verification, raised to the form of an axiom,
it would be easy to educe the notion of parallel and
Euclid's postulate itself.

Mechanics.

I need not go back to the definition of velocity or
of acceleration or of the other kinematic notions :
they will be more properly connected with ideas of
space and time, which alone they involve.

On the contrary, I will dwell on the dynamic
notions of force and mass.

There is one thing that strikes me, and that is, how

138 SCIENCE AND METHOD.

far young people who have received a secondary
education are from applying the mechanical laws
they have been taught to the real world. It is not
only that they are incapable of doing so, but they
do not even think of it. For them the world of
science and that of reality are shut off in water-tight
compartments. It is not uncommon to see a well-
dressed man, probably a university man, sitting in
a carriage and imagining that he is helping it on by
pushing on the dash-board, and that in disregard of
the principle of action and reaction.

If we try to analyze the state of mind of our pupils,
this will surprise us less. What is for them the true
definition of force ? Not the one they repeat, but the
one that is hidden away in a corner of their intellect,
and from thence directs it all. This is their definition:
Forces are arrows that parallelograms are made of;
these arrows are imaginary things that have nothing
to do with anything that exists in nature. This would
not happen if they were shown forces in reality before
having them represented by arrows.

How are we to define force? If we want a logical
definition, there is no good one, as I think I have
shown satisfactorily elsewhere. There is the anthro-
pomorphic definition, the sensation of muscular effort ;
but this is really too crude, and we cannot extract
anything useful from it.

This is the course we ought to pursue. First, in
order to impart a knowledge of the genus force, we
must show, one after the other, all the species of this
genus. They are very numerous and of great variety.
There is the pressure of liquids on the sides of the
vessels in which they are contained, the tension of

DEFINITIONS AND EDUCATION. 139

cords, the elasticity of a spring, gravity that acts 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 ; we have to
learn to measure a force. For this purpose we shall
show first that we can replace one force by another
without disturbing the equilibrium, and we shall find
the first example of this substitution in the balance
and Borda's double scales. Then we shall show that
we can replace a weight not only by another weight,
but by forces of different nature ; for example, Prony's
dynamometer break enables us to replace a weight
by friction.

From all this arises the notion of the equivalence
of two forces.

We must also define the direction of a force. If
a force F is equivalent to another force F^ that is
applied to the body we are dealing with through the
medium of a taut cord, in such a way that F can be
replaced by F-^ without disturbing the equilibrium,
then the point of attachment of the cord will be, by
definition, the point of application of the force F^ and
that of the equivalent force F, and the direction of the
cord will be the direction of the force F-^ and also that
of the equivalent force F.

From this we shall pass to the comparison of the
magnitude of forces. If one force can replace two
others of the same direction, it must be equal to their
sum, and we shall show, for instance, that a weight of
20 ounces can replace two weights of 10 ounces.

But this is not all. We know now how to compare
the intensity of two forces which have the same direc-
tion and the same point of application, but we have

140 SCIENCE AND METHOD.

to learn to do this when the directions are different.
For this purpose we imagine a cord stretched by a
weight and passing over a pulley ; we say that the
tension of the two portions of the cord is the same,
and equal to the weight

Here is our definition. It enables us to compare
the tensions of our two portions, and, by using the
preceding definitions, to compare two forces of any
kind having the same direction as these two portions.
We have to justify it by showing that the tension of
the last portion remains the same for the same weight,
whatever be the number and the disposition of the
pulleys. We must then complete it by showing that
this is not true unless the pulleys are without friction.

Once we have mastered these definitions we must
show that the point of application, the direction, and
the intensity are sufficient to determine a force ; that
two forces for which these three elements are the same
are always equivalent, and can always be replaced one
by the other, either in equilibrium or in motion, and
that whatever be the other forces coming into play.

We must show that two concurrent forces can
always be replaced by a single resultant force, and
that this resultant remains the same whether the body
is in repose or in motion, and whatever be the other
forces applied to it.

Lastly, we must show that forces defined as we have
defined them satisfy the principle of the equality of
action and reaction.

All this we learn by experiment, and by experiment
alone.

It will be sufficient to quote some common experi-
ments that the pupils make every day without being

DEFINITIONS AND EDUCATION. 141

aware of it, and to perform before them a small
number of simple and well-selected experiments.

It is not until we have passed through all these
roundabout ways that we can represent forces by
arrows, and even then I think it would be well, from
time to time, as the argument develops, to come back
from the symbol to the reality. It would not be
difficult, for instance, to illustrate the parallelogram
of forces with the help of an apparatus composed of
three cords passing over pulleys, stretched by weights,
and producing equilibrium by pulling on the same
point

Once we know force, it is easy to define mass.
This time the definition must be borrowed from
dynamics. We cannot do otherwise, since the end
in view is to make clear the distinction between mass
and weight. Here, again, the definition must be pre-
pared by experiments. There is, indeed, a machine
that seems to be made on purpose to show what
mass is, and that is Atwood's machine. Besides this
we shall recall the laws of falling bodies, and how
acceleration of gravity is the same for heavy as for
light bodies, and varies according to latitude, etc.

Now if you tell me that all the methods I advocate
have long since been applied in schools, I shall be
more pleased than surprised to hear it. I know that
on the whole our mathematical education is good ; I
do not wish to upset it, and should even be distressed
at this result ; I only desire gradual, progressive im-
provements. This education must not undergo sudden
variations at the capricious breath of ephemeral fashions.
In such storms its high educative value would soon
founder. A good and sound logic must continue to

142 SCIENCE AND METHOD.

form its toundation. Definition by example is always
necessary, but it must prepare the logical definition
and not take its place ; it must at least make its want
felt in cases where the true logical definition cannot be
given to any purpose except in higher education.

You will understand that what I have said here in
no sense implies the abandonment of what I have
written elsewhere. I have often had occasion to
criticize certain definitions which I advocate to-day.
These criticisms hold good in their entirety ; the
definitions can only be provisional, but it is through
them that we must advance.

III.

MATHEMATICS AND LOGIC.

Introduction.

Can mathematics be reduced to logic without having
to appeal to principles peculiar to itself? There is a
whole school full of ardour and faith who make it
their business to establish the possibility. They have
their own special language, in which words are used
no longer, but only signs. This language can be
understood only by the few initiated, so that the
vulgar are inclined to bow before the decisive affirma-
tions of the adepts. It will, perhaps, be useful to
examine these affirmations somewhat more closely, in
order to see whether they justify the peremptory tone
in which they are made.

But in order that the nature of the question should
be properly understood, it is necessary to enter into
some historical details, and more particularly to review
the character of Cantor's work.

The notion of infinity had long since been introduced
into mathematics, but this infinity was what philoso-
phers call a becoming. Mathematical infinity was only
a quantity susceptible of growing beyond all limit ; it
was a variable quantity of which it could not be said
that it had passed, but only that it would pass, all limits.

144 SCIENCE AND METHOD.

Cantor undertook to introduce into mathematics an
actual infinity — that is to say, a quantity which is not
only susceptible of passing all limits, but which is
regarded as having already done so. He set himself
such questions as these : Are there more points in
space than there are whole numbers ? Are there more
points in space than there are points in a plane ? etc.

Then the number of whole numbers, that of points
in space, etc., constitutes what he terms a transfinite
cardinal number — that is to say, a cardinal number
greater than all the ordinary cardinal numbers. And
he amused himself by comparing these transfinite car-
dinal numbers, by arranging in suitable order the
elements of a whole which contains an infinite number
of elements ; and he also imagined what he terms
transfinite ordinal numbers, on which I will not dwell
further.

Many mathematicians have followed in his tracks,
and have set themselves a series of questions of the
same kind. They have become so familiar with trans-
finite numbers that they have reached the point of
making the theory of finite numbers depend on that
of Cantor's cardinal numbers. In their opinion, if we
wish to teach arithmetic in a truly logical way, we
ought to begin by establishing the general properties
of the transfinite cardinal numbers, and then distin-
guish from among them quite a small class, that of the
ordinary whole numbers. Thanks to this roundabout
proceeding, we might succeed in proving all the propo-
sitions relating to this small class (that is to say, our
whole arithmetic and algebra) without making use of
a single principle foreign to logic.

This method is evidently contrary to all healthy

MATHEMATICS AND LOGIC. 145

psychology. It is certainly not in this manner that
the human mind proceeded to construct mathematics,
and I imagine, too, its authors do not dream of intro-
ducing it into secondary education. But is it at least
logical, or, more properly speaking, is it accurate?
We may well doubt it.

Nevertheless, the geometricians who have employed
it are very numerous. They have accumulated formulas
and imagined that they rid themselves of all that is not
pure logic by writing treatises in which the formulas
are no longer interspersed with explanatory text, as in
the ordinary works on mathematics, but in which the
text has disappeared entirely.

Unfortunately, they have arrived at contradictory
results, at what are called the Cantorian antinomies,
to which we shall have occasion to return. These
contradictions have not discouraged them, and they
have attempted to modify their rules, in order to
dispose of those that had already appeared, but with-
out gaining any assurance by so doing that no new
ones would appear.

It is time that these exaggerations were treated as
they deserve. I have no hope of convincing these
logicians, for they have lived too long in this atmo-
sphere. Besides, when we have refuted one of their
demonstrations, we are quite sure to find it cropping
up again with insignificant changes, and some of them
have already risen several times from their ashes.
Such in old times was the Lernaan hydra, with its
famous heads that always grew again. Hercules was
successful because his hydra had only nine heads
(unless, indeed, it was eleven), but in this case there are
too many, they are in England, in Germany, in Italy,

(1.777) 10

146 SCIENCE AND METHOD.

and in France, and he would be forced to abandon the
task. And so I appeal only to unprejudiced people of
common sense.

I.

In these latter years a large number of works have
been published on pure mathematics and the philosophy
of mathematics, with a view to disengaging and isolat-
ing the logical elements of mathematical reasoning.
These works have been analyzed and expounded
very lucidly by M. Couturat in a work entitled
" Les Principes des Math^matiques."

In M. Couturat's opinion the new works, and more
particularly those of Mr. Russell and Signor Peano,
have definitely settled the controversy so long in
dispute between Leibnitz and Kant. They have
shown that there is no such thing as an a priori
synthetic judgment (the term employed by Kant to
designate the judgments that can neither be demon-
strated analytically, nor reduced to identity, nor
established experimentally); they have shown that
mathematics is entirely reducible to logic, and that
intuition plays no part in it whatever.

This is what M. Couturat sets forth in the work I
have just quoted. He also stated the same opinions
even more explicitly in his speech at Kant's jubilee;
so much so that I overheard my neighbour whisper :
" It's quite evident that this is the centenary of Kant's
death'.'

Can we subscribe to this decisive condemnation ?
I do not think so, and I will try to show why.

MATHEMATICS AND LOGIC. 147

II.

What strikes us first of all in the new mathennatics
is its purely formal character. " Imagine," says Hilbert,
"three kinds of things, which we will call points,
straight lines, and planes ; let us agree that a straight
line shall be determined by two points, and that, in-
stead of saying that this straight line is determined by
these two points, we may say that it passes through
these two points, or that these two points are situated
on the straight line." What these things are, not only
do we not know, but we must not seek to know. It is
unnecessary, and any one who had never seen either a
point or a straight line or a plane could do geometry
just as well as we can. In order that the words /a.yj
through or the words be situated on should not call up
any image in our minds, the former is merely regarded
as the synonym of be determined^ and the latter of
determine.

Thus it will be readily understood that, in order to
demonstrate a theorem, it is not necessary or even
useful to know what it means. We might replace
geometry by the reasoning piano imagined by Stanley
Jevons ; or, if we prefer, we might imagine a machine
where we should put in axioms at one end and take
out theorems at the other, like that legendary machine
in Chicago where pigs go in alive and come out trans-
formed into hams and sausages. It is no more neces-
sary for the mathematician than it is for these machines
to know what he is doing.

I do not blame Hilbert for this formal character of
his geometry. He was bound to tend in this direction,
given the problem he set himself. He wished to reduce

148 SCIENCE AND METHOD.

to a minimum the number of the fundamental axioms
of geometry, and to make a complete enumeration of
them. Now, in the arguments in which our mind
remains active, in those in which intuition still plays
a part, in the living arguments, so to speak, it is
difficult not to introduce an axiom or a postulate that
passes unnoticed. Accordingly, it was not till he had
reduced all geometrical arguments to a purely me-
chanical form that he could be certain of having
succeeded in his design and accomplished his work.

What Hilbert had done for geometry, others have
tried to do for arithmetic and analysis. Even if they
had been entirely successful, would the Kantians be
finally condemned to silence ? Perhaps not, for it is
certain that we cannot reduce mathematical thought
to an empty form without mutilating it. Even admit-
ting that it has been established that all theorems can
be deduced by purely analytical processes, by simple
logical combinations of a finite number of axioms, and
that these axioms are nothing but conventions, the
philosopher would still retain the right to seek the
origin of these conventions, and to ask why they were
iudged preferable to the contrary conventions.

And, further, the logical correctness of the argu-
ments that lead from axioms to theorems is not the
only thing we have to attend to. Do the rules of
perfect logic constitute the whole of mathematics?
As well say that the art of the chess-player reduces
itself to the rules for the movement of the pieces.
A selection must be made out of all the construc-
tions that can be combined with the materials
furnished by logic. The true geometrician makes
this selection judiciously, because he is guided by

MATHEMATICS AND LOGIC. 149

a sure instinct, or by some vague consciousness of
I know not what profounder and more hidden geom-
etry, which alone gives a value to the constructed
edifice.

To seek the origin of this instinct, and to study
the laws of this profound geometry which can be
felt but not expressed, would be a noble task for
the philosophers who will not allow that logic is
all. But this is not the point of view I wish to
take, and this is not the way I wish to state
the question. This instinct I have been speaking
of is necessary to the discoverer, but it seems at
first as if we could do without it for the study of
the science once created. Well, what I want to find
out is, whether it is true that once the principles of
logic are admitted we can, I will not say discover,
but demonstrate all mathematical truths without
making a fresh appeal to intuition.

III.

To this question I formerly gave a negative answer.
(See " Science et Hypothese," Chapter I.) Must our
answer be modified by recent works ? I said no,
because " the principle of complete induction " ap-
peared to me at once necessary to the mathematician,
and irreducible to logic. We know the statement of
the principle : " If a property is true of the number
I, and if it is established that it is true of n+i pro-
vided it is true of n, it will be true of all whole
numbers." I recognized in this the typical mathe-
matical argument. I did not mean to say, as has
been supposed, that all mathematical arguments can
be reduced to an application of this principle.

150 SCIENCE AND METHOD.

Examining these arguments somewhat closely, we
should discover the application of many other similar
principles, offering the same essential characteristics.
In this category of principles, that of complete induc-
tion is only the simplest of all, and it is for that
reason that I selected it as a type.

The term principle of complete induction which
has been adopted is not justifiable. This method
of reasoning is none the less a true mathematical
induction itself, which only differs from the ordinary
induction by its certainty.

IV.

Definitions and Axioms.

The existence of such principles is a difficulty for
the inexorable logicians. How do they attempt to
escape it? The principle of complete induction, they
say, is not an axiom properly so called, or an a
prion synthetic judgment ; it is simply the defini-
tion of the whole number. Accordingly it is a mere
convention. In order to discuss this view, it will be
necessary to make a close examination of the rela-
tions between definitions and axioms.

We will first refer to an article by M. Couturat
on mathematical definitions which appeared in
r Enseignement MatMmatique, a review published by
Gauthier-Villars and by Georg in Geneva. We find
a distinction between direct definition and definition
by postulates.

" Definition by postulates," says M. Couturat,
" applies not to a single notion, but to a system of
notions ; it consists in enumerating the fundamental

MATHEMATICS AND LOGIC. 151

relations that unite them, which make it possible to
demonstrate all their other properties : these relations
are postulates . . ."

If we have previously defined all these notions
with one exception, then this last will be by defini-
tion the object which verifies these postulates.

Thus certain indemonstrable axioms of mathe-
matics would be nothing but disguised definitions.
This point of view is often legitimate, and I have
myself admitted it, for instance, in regard to Euclid's
postulate.

The other axioms of geometry are not sufficient to
define distance completely. Distance, then, will be
by definition, the one among all the magnitudes
which satisfy the other axioms, that is of such a
nature as to make Euclid's postulate true.

Well, the logicians admit for the principle of com-
plete induction what I admit for Euclid's postulate,
and they see nothing in it but a disguised definition.

But to give us this right, there are two conditions
that must be fulfilled. John Stuart Mill used to say
that every definition implies an axiom, that in which
we affirm the existence of the object defined. On
this score, it would no longer be the axiom that
might be a disguised definition, but, on the contrary,
the definition that would be a disguised axiom.
Mill understood the word existence in a material
and empirical sense ; he meant that in defining a
circle we assert that there are round things in
nature.

In this form his opinion is inadmissible. Mathe-
matics is independent of the existence of material
objects. In mathematics the word exist can only

152 SCIENCE AND METHOD.

have one meaning ; it signifies exemption from
contradiction. Thus rectified, Mill's thought becomes
accurate. In defining an object, we assert that the
definition involves no contradiction.

If, then, we have a system of postulates, and if we
can demonstrate that these postulates involve no
contradiction, we shall have the right to consider
them as representing the definition of one of the
notions found among them. If we cannot demon-
strate this, we must admit it without demonstration,
and then it will be an axiom. So that if we wished
to find the definition behind the postulate, we should
discover the axiom behind the definition.

Generally, for the purpose of showing that a
definition does not involve any contradiction, we
proceed by example, and try to form an example of
an object satisfying the definition. Take the case
of a definition by postulates. We wish to define a
notion A, and we say that, by definition, an A is
any object for which certain postulates are true. If
we can demonstrate directly that all these postulates
are true of a certain object B, the definition will be
justified, and the object B will be an example of A.
We shall be certain thc^t the postulates are not
contradictory, since there are cases in which they
are all true at once.

But such a direct demonstration by example is
not always possible. Then, in order to establish
that the postulates do not involve contradiction, we
must picture all the propositions that can be de-
duced from these postulates considered as premises,
and show that among these propositions there are
no two of which one is the contradiction of the

MATHEMATICS AND LOGIC. 153

other. If the number of these propositions is finite,
a direct verification is possible ; but this is a case
that is not frequent, and, moreover, of little interest.

If the number of the propositions is infinite, we
can no longer make this direct verification. We
must then have recourse to processes of demonstra-
tion, in which we shall generally be forced to invoke
that very principle of complete induction that we are
attempting to verify.

I have just explained one of the conditions which
the logicians were bound to satisfy, and we shall see
further on that tliey have not done so.

V.

There is a second condition. When we give a
definition, it is for the purpose of using it.

Accordingly, we shall find the word defined in the
text that follows. Have we the right to assert, of
the object represented by this word, the postulate
that served as definition ? Evidently we have, if the
word has preserved its meaning, if we have not
assigned it a different meaning by implication. Now
this is what sometimes happens, and it is generally
difficult to detect it. We must see how the word
was introduced into our text, and whether the door
through which it came does not really imply a
different definition from the one enunciated.

This difficulty is encountered in all applications of
mathematics. The mathematical notion has received
a highly purified and exact definition, and for the
pure mathematician all hesitation has disappeared.
But when we come to apply it, to the physical
sciences, for instance, we are no longer dealing with

154 SCIENCE AND METHOD.

this pure notion, but with a concrete object which is
often only a rough image of it. To say that this
object satisfies the definition, even approximately, is
to enunciate a new truth, which has no longer the
character of a conventional postulate, and that expe-
rience alone can establish beyond a doubt.

But, without departing from pure mathematics, we
still meet with the same difficulty. You give a
subtle definition of number, and then, once the
definition has been given, you think no more about
it, because in reality it is not your definition that
has taught you what a number is, you knew it long
before, and when you come to write the word
number farther on, you give it the same meaning
as anybody else. In order to know what this
meaning is, and if it is indeed the same in this
phrase and in that, we must see how you have been
led to speak of number and to introduce the word
into the two phrases. I will not explain my point
any further for the moment, for we shall have occa-
sion to return to it.

Thus we have a word to which we have explicitly
given a definition A. We then proceed to make use
of it in our text in a way which implicitly supposes
another definition B. It is possible that these two
definitions may designate the same object, but that
such is the case is a new truth that must either be
demonstrated or else admitted as an independent
axiom.

JVe shall see further on that the logicians have not
fulfilled this second condition any better than the first.

MATHEMATICS AND LOGIC. 155

VI.

The definitions of number are very numerous and
of great variety, and I will not attempt to enumerate
even their names and their authors. We must not be
surprised that there are so many. If any one of them
was satisfactory we should not get any new ones. If
each new philosopher who has applied himself to the
question has thought it necessary to invent another,
it is because he was not satisfied with those of his
predecessors ; and if he was not satisfied, it was because
he thought he detected 2. petitio principii.

I have always experienced a profound sentiment
of uneasiness in reading the works devoted to this
problem. I constantly expect to run against a petitio
principii, and when I do not detect it at once I am
afraid that I have not looked sufficiently carefully.

The fact is that it is impossible to give a definition
without enunciating a phrase, and difficult to enun-
ciate a phrase without putting in a name of number,
or at least the word several, or at least a word in the
plural. Then the slope becomes slippery, and every
moment we are in danger of falling into the petitio
principii.

I will concern myself in what follows with those
only of these definitions in which the petitio principii
is most skilfully concealed.

VII.

Pasigraphy.

The symbolical language created by Signor Peano
plays a very large part in these new researches. It is

156 SCIENCE AND METHOD.

capable of rendering some service, but it appears to
me that M. Couturat attaches to it an exaggerated
importance that must have astonished Peano himself.

The essential element of this language consists in
certain algebraical signs vi^hich represent the con-
junctions : if, and, or, therefore. That these signs may
be convenient is very possible, but that they should be
destined to change the face of the whole philosophy is
quite another matter. It is difficult to admit that
the word if acquires, when written o, a virtue it did
■iiot possess when written if.

This invention of Peano was first called pasigraphy,
that is to say, the art of writing a treatise on mathe-
matics without using a single word of the ordinary
language. This name defined its scope most exactly.
Since then it has been elevated to a more exalted
dignity, by having conferred upon it the title of
logistic. The same word is used, it appears, in the Ecole
de Guerre to designate the art of the quartermaster,
the art of moving and quartering troops.* But no
confusion need be feared, and we see at once that the
new name implies the design of revolutionizing logic.

We may see the new method at work in a mathe-
matical treatise by Signor Burali-Forti entitled " Una
Questione sui Numeri transfiniti" (An Enquiry concern-
ing transfinite Numbers), included in Volume XI. of the
" Rendiconti del circolo vtatematico di Palermo " (Reports
of the mathematical club of Palermo).

I will begin by saying that this treatise is very
interesting, and, if I take it here as an example, it

* In the French the confusion is with " lop'slique" the art of the
"marechal des logis^'' or quartermaster. In English the possibility of
confusion does not arise.

MATHEMATICS AND LOGIC. 157

is precisely because it is the most important of all
that have been written in the new language. Besides,
the uninitiated can read it, thanks to an interlined
Italian translation.

What gives importance to this treatise is the fact that
it presented the first example of those antinomies met
with in the study of transfinite numbers, which have
become, during the last few years, the despair of
mathematicians. The object of this note, says Signor
Burali-Forti, is to show that there can be two trans-
finite (ordinal) numbers, a and b, such that a is neither
equal to, greater than, nor smaller than, b.

The reader may set his mind at rest. In order to
understand the considerations that will follow, he does
not require to know what a transfinite ordinal number is.

Now Cantor had definitely proved that between
two transfinite numbers, as between two finite num-
bers, there can be no relation other than equality or
inequality in one direction or the other. But it is
not of the matter of this treatise that I desire to speak
here ; this would take me much too far from my
subject. I only wish to concern myself with the form,
and I ask definitely whether this form makes it gain
much in the way of exactness, and whether it thereby
compensates for the efforts it imposes upon the
writer and the reader.

To begin with, we find that Signor Burali-Forti
defines the number i in the following manner : —

I = t T' {Ko^(«,/^) e (ui One},

a definition eminently fitted to give an idea of the
number i to people who had never heard it before.
I do not understand Peanian well enough to ven-

158 SCIENCE AND METHOD.

ture to risk a criticism, but I am very much afraid
that this definition contains a petitio principii, seeing
that I notice the figure i in the first half and the
word One in the second.

However that may be, Signer Burah-Forti starts
with this definition, and, after a short calculation,
arrives at the equation

(27) I e No,

which teaches us that One is a number.

And since I am on the subject of these definitions
of the first numbers, I may mention that M. Couturat
has also defined both o and i.

What is zero ? It is the number of elements in the
class nil. And what is the class nil ? It is the class
which contains none.

To define zero as nil and nil as none is really an
abuse of the wealth of language, and so M. Couturat
has introduced an improvement into his definition by
writing

= 1 a: ^^ = A- 3- A = ixe<f>x),

which means in English : zero is the number of the
objects that satisfy a condition that is never fulfilled.
But as never means zn no case, I do not see that any
very great progress has been made.

I hasten to add that the definition M. Couturat
gives of the number i is more satisfactory.

One, he says in substance, is the number of the
elements of a class in which any two elements are
identical.

It is more satisfactory, as I said, in this sense,
that in order to define i, he does not use the word
one ; on the other hand, he does use the word two.

MATHEMATICS AND LOGIC. 159

But I am afraid that if we asked M. Couturat what
two is, he would be obliged to use the word one.

VIII.

But let us return to the treatise of Signor Burali-
Forti. I said that his conclusions are in direct
opposition to those of Cantor. Well, one day I
received a visit from M. Hadamard, and the conversa-
tion turned upon this antinomy.

" Does not Burali-Forti's reasoning," I said, " seem
to you irreproachable ? "

" No," he answered ; " and, on the contrary, I have
no fault to find with Cantor's. Besides, Burali-Forti
had no right to speak of the whole of all the ordinal
numbers."

" Excuse me, he had that right, since he could
always make the supposition that

ft = T' (No, i >).

I should like to know who could prevent him. And
can we say that an object does not exist when we
have called it 12 ? "

It was quite useless ; I could not convince him
(besides, ft would have been unfortunate if I had, since
he was right). Was it only because I did not speak
Feanian with sufficient eloquence ? Possibly, but,
between ourselves, I do not think so.

Thus, in spite of all this pasigraphical apparatus,
the question is not solved, What does this prove?
So long as it is merely a question of demonstrating
that one is a number, pasigraphy is equal to the task ;
but if a difficulty presents itself, if there is an anti-
nomy to be resolved, pasigraphy becomes powerless.

IV.

THE NEW LOGICS.

I.

Russell's Logic.

In order to justify its pretensions, logic has had to
transform itself We have seen new logics spring
up, and the most interesting of these is Mr. Bertrand
Russell's. It seems as if there could be nothing new-
written about formal logic, and as if Aristotle had gone
to the very bottom of the subject. But the field
that Mr. Russell assigns to logic is infinitely more
extensive than that of the classical logic, and he
has succeeded in expressing views on this subject that
are original and sometimes true.

To begin with, while Aristotle's logic was, above all,
the logic of classes, and took as its starting-point
the relation of subject and predicate, Mr. Russell
subordinates the logic of classes to that of propositions.
The classical 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."
This is, in my opinion, one of the happiest of ideas,
for the classical syllogism is easily reduced to the
hypothetical syllogism, while the inverse transforma-
tion cannot be made without considerable difficulty.

THE NEW LOGICS. i6i

But this is not all. Mr. Russell's logic of propo-
sitions is the study of the laws in accordance with
which combinations are formed with the conjunctions
if, and, or, and the negative not. This is a consider-
able extension of the ancient logic. The properties of
the classical syllogism can be extended without any
difficulty to the hypothetical syllogism, and in the
forms of this latter we can easily recognize the
scholastic forms ; we recover what is essential in the
classical logic. But the theory of the syllogism is still
only the syntax of the conjunction if and, perhaps,
of the negative.

By adding two other conjunctions, and and or,
Mr. Russell opens up a new domain to logic. The
signs and and or follow the same laws as the two
signs X and +, that is to say, the commutative,
associative, and distributive laws. Thus and repre-
sents logical multiplication, while or represents logical
addition. This, again, is most interesting.

Mr. Russell arrives at the conclusion that a false
proposition of any kind involves all the other pro-
positions, whether true or false. M. Couturat says
that this conclusion will appear paradoxical at first
sight. However, one has only to correct a bad
mathematical paper to recognize how true Mr.
Russell's view is. The candidate often takes an
immense amount of trouble to find the first false
equation ; but as soon as he has obtained it, it is
no more than child's play for him to accumulate
the most surprising results, some of which may
actually be correct.

(1.777) XI

i62 SCIENCE AND METHOD.

II.

We see how much richer this new logic is than
the classical logic. The symbols have been multiplied
and admit of varied combinations, which are no longer
of limited number. Have we any right to give this
extension of meaning to the word logic} It would be
idle to examine this question, and to quarrel with
Mr. Russell merely on the score of words. We will
grant him what he asks ; but we must not be sur-
prised if we find that certain truths which had been
declared to be irreducible to logic, in the old sense
of the word, have become reducible to logic, in its
new sense, which is quite different.

We have introduced a large number of new notions,
and they are not mere combinations of the old.
Moreover, Mr. Russell is not deceived on this point,
and not only at the beginning of his first chapter — that
is to say, his logic of propositions — but at the beginning
of his second and third chapters also — that is to say,
his logic of classes and relations — he introduces new
words which he declares to be undefinable.

And that is not all. He similarly introduces prin-
ciples which he declares to be undemonstrable. But
these undemonstrable principles are appeals to in-
tuition, a priori synthetic judgments. We regarded
them as intuitive when we met them more or less
explicitly enunciated in treatises on mathematics.
Have they altered in character because the meaning
of the word logic has been extended, and we find
them now in a book entitled "Treatise on Logic"?
They have not changed in nature, but only in position.

THE NEW LOGICS. 163

III.

Could these principles be considered as disguised
definitions ? That they should be so, we should
require to be able to demonstrate that they involve
no contradiction. We should have to establish that,
however far we pursue the series of deductions, we
shall never be in danger of contradicting ourselves.

We might attempt to argue as follows. We can
verify the fact that the operations of the new logic,
applied to premises free from contradiction, can only
give consequences equally free from contradiction. If
then, after n operations, we have not met with contra-
diction, we shall not meet it any more after n+i.
Accordingly, it is impossible that there can be a
moment when contradiction will begin, which shows
that we shall never meet it. Have we the right
to argue in this way? No, for it would be making
complete induction, and we must not forget that
we do not yet know the principle of complete induction.

Therefore we have no right to regard these axioms
as disguised definitions, and we have only one course
left. Each one of them, we admit, is a new act of
intuition. This is, moreover, as I believe, the thought
of Mr. Russell and M. Couturat.

Thus each of the nine undefinable notions and
twenty undemonstrable propositions (I feel sure that,
if I had made the count, I should have found one
or two more) which form the groundwork of the
new logic — of the logic in the broad sense — pre-
supposes a new and independent act of our intuition,
and why should we not term it a true a /r^Wz synthetic
judgment.? On this point everybody seems to be

i64 SCIENCE AND METHOD.

agreed ; but what Mr. Russell claims, and what appears
to me doubtful, is that after these appeals to intuition
we shall have finished : we shall have no more to make,
and we shall be able to construct the whole of mathe-
matics without bringing in a single new element.

IV.

M. Couturat is fond of repeating that this new logic
is quite independent of the idea of number. I will
not amuse myself by counting how many instances
his statement contains of adjectives of number,
cardinal as well as ordinal, or of indefinite adjectives
such as several. However, I will quote a few
examples : —

" The logical product of two or of several propo-
sitions is "

" All propositions are susceptible of two values only,
truth or falsehood."

" The relative product of two relations is a relation."
" A relation is established between two terms."
Sometimes this difficulty would not be impossible
to avoid, but sometimes it is essential. A relation is
incomprehensible without two terms. It is impossible
to have the intuition of a relation, without having
at the same time the intuition of its two terms, and
without remarking that they are two, since, for a
relation to be conceivable, they must be two and
two only.

V.

Arithmetic.

I come now to what M. Couturat calls the ordinal
theory, which is the groundwork of arithmetic properly

THE NEW LOGICS. 165

so called. M. Couturat begins by enunciating Peano's
five axioms, which are independent, as Signor Peano
and Signor Padoa have demonstrated.

1. Zero is a whole number.

2. Zero is not the sequent of any whole number.

3. The sequent of a whole number is a whole
number. To which it would be well to add : every
whole number has a sequent.

4. Two whole numbers are equal if their sequents
are equal.

The Sth axiom is the principle ot complete induction.

M. Couturat considers these axioms as disguised
definitions ; they constitute the definition by postulates
of zero, of the " sequent," and of the whole number.

But we have seen that, in order to allow of a
definition by postulates being accepted, we must be
able to establish that it implies no contradiction.

Is this the case here ? Not in the very least.

The demonstration cannot be made by example.
We cannot select a portion of whole numbers — for
instance, the three first — and demonstrate that they
satisfy the definition.

If I take the series o, r, 2, I can readily see that
it satisfies axioms i, 2, 4, and 5 ; but in order that
it should satisfy axiom 3, it is further necessary that
3 should be a whole number, and consequently that
the series o, I, 2, 3 should satisfy the axioms. We
could verify that it satisfies axioms I, 2, 4, and 5,
but axiom 3 requires besides that 4 should be a
whole number, and that the series o, i, 2, 3, 4 should
satisfy the axioms, and so on indefinitely.

It is, therefore, impossible to demonstrate the
axioms for some whole numbers without demonstrat-

i66 SCIENCE AND METHOD.

ing them for all, and so we must give up the
demonstration by example.

It is necessary, then, to take all the consequences
of our axioms and see whether they contain any
contradiction. If the number of these consequences
were finite, this would be easy ; but their number
is infinite — they are the whole of mathematics, or at
least the whole of arithmetic.

What are we to do, then ? Perhaps, if driven to
it, we might repeat the reasoning of Section III.
But, as I have said, this reasoning is complete induction,
and it is precisely the principle of complete induction
that we are engaged in justifying.

VI.

Hilbert's Logic.

I come now to Mr. Hilbert's important work,
addressed to the Mathematical Congress at Heidelberg,
a French translation of which, by M. Pierre Boutroux,
appeared in I' Enseignement Mathematique, while an
English translation by Mr. Halsted appeared in The
Monist. In this work, in which we find the most
profound thought, the author pursues an aim similar
to Mr. Russell's, but he diverges on many points from
his predecessor.

" However,'' he says, " if we look closely, we recog-
nize that in logical principles, as they are com-
monly presented, certain arithmetical notions are
found already implied ; for instance, the notion of
whole, and, to a certain extent, the notion of number.
Thus we find ourselves caught in a circle, and that
is why it seems to me necessary, if we wish to avoid

THE NEW LOGICS. 167

all paradox, to develop the principles of logic and of
arithmetic simultaneously."

We have seen above that what Mr. Hilbert says
of the principles of logic, as they are commonly pre-
sented, applies equally to Mr. Russell's logic. For
Mr. Russell logic is anterior to arithmetic, and for
Mr. Hilbert they are " simultaneous." Further on we
shall find other and yet deeper differences ; but we
will note them as they occur. I prefer to follow the
development of Hilbert's thought step by step, quoting
the more important passages verbatim.

" Let us first take into consideration the object i."
We notice that in acting thus we do not in any way
imply the notion of number, for it is clearly understood
that I here is nothing but a symbol, and that we do
not in any way concern ourselves with knowing its
signification. " The groups formed with this object,
two, three, or several times repeated . . ." This
time the case is quite altered, for if we introduce the
words two, three, and, above all, several, we introduce
the notion of number ; and then the definition of the
finite whole number that we find later on comes a
trifle late. The author was much too wary not to
perceive this petitio principii. And so, at the end of
his work, he seeks to effect a real patching-up.

Hilbert then introduces two simple objects, i and
=, and pictures all the combinations of these two
objects, all the combinations of their combinations,
and so on. It goes without saying that we must
forget the ordinary signification of these two signs,
and not attribute any to them. He then divides these
combinations into two classes, that of entities and that
of nonentities, and, until further orders, this partition

i68 SCIENCE AND METHOD.

is entirely arbitrary. Every affirmative proposition
teaches us that a combination belongs to the class of
entities, and every negative proposition teaches us
that a certain combination belongs to the class ot
nonentities.

VII.

We must now note a difference that is of the
highest importance. For Mr. Russell a chance object,
which he designates by ;r, is an absolutely indeterminate
object, about which he assumes nothing- For Hilbert
it is one of those combinations formed with the symbols
I and = ; he will not allow the introduction of any-
thing but combinations of objects already defined.
Moreover, Hilbert formulates his thought in the most
concise manner, and I think I ought to reproduce
his statement in extenso : " The indeterminates which
figure in the axioms (in place of the 'some' or the
' all ' of ordinary logic) represent exclusively the whole
of the objects and combinations that we have already
acquired in the actual state of the theory, or that we
are in course of introducing. Therefore, when we
deduce propositions from the axioms under considera-
tion, it is these objects and these combinations alone
that we have the right to substitute for the indeter-
minates. Neither must we forget that when we
increase the number of the fundamental objects, the
axioms at the same time acquire a new extension, and
must, in consequence, be put to the proof afresh and,
if necessary, modified."

The contrast with Mr. Russell's point of view is
complete. According to this latter philosopher, we
may substitute in place of x not only objects already

THE NEW LOGICS. 169

known, but anything whatsoever. Russell is faithful
to his point of view, which is that of comprehension.
He starts with the general idea of entity, and enriches
it more and more, even while he restricts it, by adding
to it new qualities. Hilbert, on the contrary, only
recognizes as possible entities combinations of objects
already known ; so that (looking only at one side of
his thought) we might say that he takes the point
of view of extension.

vni.

Let us proceed with the exposition of Hilbert's
ideas. He introduces two axioms which he enunciates
in his symbolical language, but which signify, in the
language of the uninitiated like us, that every quantity
is equal to itself, and that every operation upon two
identical quantities gives identical results. So stated
they are evident, but such a presentation of them
does not faithfully represent Hilbert's thought. For
him mathematics has to combine only pure symbols,
and a true mathematician must base his reasoning
upon them without concerning himself with their
meaning. Accordingly, his axioms are not for him
what they are for the ordinary man.

He considers them as representing the definition by
postulates of the symbol :=, up to this time devoid
of all signification. But in order to justify this defini-
tion, it is necessary to show that these two axioms do
not lead to any contradiction.

For this purpose Hilbert makes use of the reasoning
of Section HL, without apparently perceiving that he
is making complete induction.

170 SCIENCE AND METHOD.

IX.

The end of Mr. Hilbert's treatise is altogether
enigmatical, and I will not dwell upon it. It is full
of contradictions, and one feels that the author is
vaguely conscious of the petitio principii he has been
guilty of, and that he is vainly trying to plaster up
the cracks in his reasoning.

What does this mean ? It means that when he
co77ies to demonstrate that the definition of the whole
number by the axiom of complete induction does not
involve contradiction, Mr. Hilbert breaks down, just as
Mr. Russell and M. Couturat broke down, because the
difficulty is too great.

X.

Geometry.

Geometry, M. Couturat says, is a vast body of
doctrine upon which complete induction does not
intrude. This is true to a certain extent : we cannot
say that it does not intrude at all, but that it intrudes
very little. If we refer to Mr. Halsted's " Rational
Geometry " (New York : John Wiley and Sons,
1904), founded on Hilbert's principles, we find the
principle of induction intruding for the first time
at page 114 (unless, indeed, 1 have not searched care-
fully enough, which is quite possible).

Thus geometry, which seemed, only a few years
ago, the domain in which intuition held undisputed
sway, is to-day the field in which the logisticians
appear to triumph. Nothing could give a better
measure of the importance of Hilbert's geometrical
works, and of the profound impression they have left
upon our conceptions.

THE NEW LOGICS. 171

But we must not deceive ourselves. What is, in
fad, the fundamental theorem of geometry ? It is that
the axioms of geometry do not involve contradiction, and
this cannot be demonstrated without the principle of
induction.

How does Hilbert demonstrate this essential point ?
He does it by relying upon analysis, and, through it,
upon arithmetic, and, through it, upon the principle
of induction.

If another demonstration is ever discovered, it will
still be necessary to rely on this principle, since the
number of the possible consequences of the axioms
which we have to show are not contradictory is
infinite.

XI.

Conclusion.

Our conclusion is, first of all, that the principle of
induction cannot be regarded as the disguised definition
of the whole number.

Here are three truths : —

The principle of complete induction ;
Euclid's postulate ;

The physical law by which phosphorus melts
at 44° centigrade (quoted by M. Le Roy).

We say : these are three disguised definitions — the
first that of the whole number, the second that of the
straight line, and the third that of phosphorus.

I admit it for the second, but I do not admit it
for the two others, and I must explain the reason of
this apparent inconsistency.

In the first place, we have seen that a definition

172 SCIENCE AND METHOD.

is only acceptable if it is established that it does not
involve contradiction. We have also shown that, in
the case of the first definition, this demonstration is
impossible ; while in the case of the second, on the
contrary, we have just recalled the fact that Hilbert
has given a complete demonstration.

So far as the third is concerned, it is clear that it
does not involve contradiction. But does this mean
that this definition guarantees, as it should, the
existence of the object defined ? We are here no
longer concerned with the mathematical sciences, but
with the physical sciences, and the word existence has
no longer the same meaning ; it no longer signifies
absence of contradiction, but objective existence.

This is one reason already for the distinction I make
between the three cases, but there is a second. In
the applications we have to make of these three
notions, do they present themselves as defined by
these three postulates ?

The possible applications of the principle of induc-
tion are innumerable. Take, for instance, one of those
we have expounded above, in which it is sought to
establish that a collection of axioms cannot lead to
a contradiction. For this purpose we consider one of
the series of syllogisms that can be followed out, start-
ing with these axioms as premises.

When we have completed the n*'^ syllogism, we see
that we can form still another, which will be the
(«+i)''^: thus the number n serves for counting a
series of successive operations ; it is a number that
can be obtained by successive additions. Accordingly,
it is a number from which we can return to unity by
successive subtractions. It is evident that we could

THE NEW LOGICS. 173

not do so if we had n = n-i, for then subtraction
would always give us the same number. Thus, then,
the way in which we have been brought to consider
this number n involves a definition of the finite whole
number, and this definition is as follows: a finite
whole number is that which can be obtained by suc-
cessive additions, and which is such that n is not equal
to n-l.

This being established, what do we proceed to do ?
We show that if no contradiction has occurred up to
the w'* syllogism, it will not occur any the more at
the {n+iy^, and we conclude that it will never occur.
You say I have the right to conclude thus, because
whole numbers are, by definition, those for which such
reasoning is legitimate. But that involves another
definition of the whole number, which is as follows :
a whole numbej- is that about which we can reason by
recurrence. In the species it is that of which we can
state that, if absence of contradiction at the moment
of occurrence of a syllogism whose number is a whole
number carries with it the absence of contradiction
at the moment of occurrence of the syllogism whose
number is the following whole number, then we need
not fear any contradiction for any of the syllogisms
whose numbers are whole numbers.

The two definitions are not identical. They are
equivalent, no doubt, but they are so by virtue of an
a priori synthetic judgment; we cannot pass from
one to the other by purely logical processes. Con-
sequently, we have no right to adopt the second after
having introduced the whole number by a road which
presupposes the first.

On the contrary, what happens in the case of the

174 SCIENCE AND METHOD.

straight line ? I have already explained this so often
that I feel some hesitation about repeating myself
once more. I will content myself with a brief sum-
mary of my thought.

We have not, as in the previous case, two equivalent
definitions logically irreducible one to the other. We
have only one expressible in words. It may be said
that there is another that we feel without being able
to enunciate it, because we have the intuition of a
straight line, or because we can picture a straight
line. But, in the first place, we cannot picture it in
geometric space, but only in representative space ;
and then we can equally well picture objects which
possess the other properties of a straight line, and
not that of satisfying Euclid's postulate. These
objects are "non- Euclidian straight lines," which,
from a certain point of view, are not entities
destitute of meaning, but circles (true circles of true
space) orthogonal to a certain sphere. If, among
these objects equally susceptible of being pictured,
it is the former (the Euclidian straight lines) that
we call straight lines, and not the latter (the non-
Euclidian straight lines), it is certainly so by definition.

And if we come at last to the third example, the
definition of phosphorus, we see that the true defini-
tion would be: phosphorus is this piece of matter
that I see before me in this bottle.

XII.

Since I am on the subject, let me say one word
more. Concerning the example of phosphorus, I
said : " This proposition is a true physical law that
can be verified, for it means : all bodies which possess

THE NEW LOGICS. 175

all the properties of phosphorus except its melting-
point, melt, as it does, at 44° centigrade." It has been
objected that this law is not verifiable, for if we came
to verify that two bodies resembling phosphorus melt
one at 44° and the other at 50° centigrade, we could
always say that there is, no doubt, besides the melting-
point, some other property in which they differ.

This was not exactly what I meant to say, and I
should have written : " all bodies which possess such
and such properties in finite number (namely, the
properties of phosphorus given in chemistry books,
with the exception of its melting-point) melt at 44°
centigrade.''

In order to make still clearer the difference between
the case of the straight line and that of phosphorus,
I will make one more remark. The straight line has
several more or less imperfect images in nature, the
chief of which are rays of light and the axis of
rotation of a solid body. Assuming that we ascertain
that the ray of light does not satisfy Euclid's postulate
(by showing, for instance, that a star has a negative
parallax), what shall we do ? Shall we conclude that,
as a straight line is by definition the trajectory of
light, it does not satisfy the definition, or, on the
contrary, that, as a straight line by definition satisfies
the postulate, the ray of light is not rectilineal ?

Certainly we are free to adopt either definition,
and, consequently, either conclusion. But it would be
foolish to adopt the former, because the ray of light
probably satisfies in a most imperfect way not only
Euclid's postulate but the other properties of the
straight line ; because, while it deviates from the
Euclidian straight, it deviates none the less from the

176 SCIENCE AND METHOD.

axis of rotation of solid bodies, which is another
imperfect image of the, straight line ; and lastly,
because it is, no doubt, subject to change, so that
such and such a line which was straight yesterday-
will no longer be so to-morrow if some physical cir-
cumstance has altered.

Assume, now, that we succeed in discovering that
phosphorus melts not at 44° but at 43'9° centigrade.
Shall we conclude that, as phosphorus is by definition
that which melts at 44°, this substance that we called
phosphorus is not true phosphorus, or, on the contrary,
that phosphorus melts at 43 '9°? Here, again, we are
free to adopt either definition, and, consequently, either
conclusion ; but it would be foolish to adopt the
former, because we cannot change the name of a
substance every time we add a fresh decimal to its
melting-point.

xni.

To sum up, Mr. Russell and Mr. Hilbert have both
made a great effort, and have both of them written
a book full of views that are original, profound, and
often very true. These two books furnish us with
subject for much thought, and there is much that we
can learn from them. Not a few of their results are
substantial and destined to survive.

But to say that they have definitely settled the
controversy between Kant and Leibnitz and destroyed
the Kantian theory of mathematics is evidently un-
true. I do not know whether they actually imagined
they had done it, but if they did they were mistaken.

V.

THE LAST EFFORTS OF THE LOGISTICIANS.

I.

The logisticians have attempted to answer the fore-
going considerations. For this purpose they have
been obHged to transform logistic, and Mr. Russell
in particular has modified his original views on certain
points. Without entering into the details of the con-
troversy, I should like to return to what are, in my
opinion, the two most important questions. Have the
rules of logistic given any proof of fruitfulness and of
infallibility? Is it true that they make it possible to
demonstrate the principle of complete induction with-
out any appeal to intuition ?

II.

The Infallibility of Logistic.

As regards fruitfulness, it seems that M. Couturat
has most childish illusions. Logistic, according to
him, lends " stilts and wings " to discovery, and on the
following page he says, " // is ten years since Signor
Peano published the first edition of his " Formulaire."

What ! you have had wings for ten years, and you
haven't flown yet !

I have the greatest esteem for Signor Peano, who

(1.777) 12

178 SCIENCE AND METHOD.

has done some very fine things (for instance, his curve
which fills a whole area) ; but, after all, he has not
gone any farther, or higher, or faster than the majority
of wingless mathematicians, and he could have done
everything just as well on his feet.

On the contrary, I find nothing in logistic for the
discoverer but shackles. It does not help us at all
in the direction of conciseness, far from it ; and if it
requires 27 equations to establish that I is a num-
ber, how many will it require to demonstrate a real
theorem ? If we distinguish, as Mr. Whitehead does,
the individual x, the class whose only member is x,
which we call ix, then the class whose only member
is the class whose only member is x, which we call
ux, do we imagine that these distinctions, however
useful they may be, will greatly expedite our progress ?

Logistic forces us to say all that we commonly
assume, it forces us to advance step by step ; it is
perhaps surer, but it is not more expeditious.

It is not wings you have given us, but leading-
strings. But we have the right to demand that these
leading-strings should keep us from falling ; this is
their only excuse. When an investment does not pay
a high rate of interest, it must at least be a gilt-edged
security.

Must we follow your rules blindly? Certainly, for
otherwise it would be intuition alone that would enable
us to distinguish between them. But in that case they
must be infallible, for it is only in an infallible author-
ity that we can have blind confidence. Accordingly,
this is a necessity for you : you must be infallible or
cease to exist.

You have no right to say to us: " We make mistakes,

LAST EFFORTS OF LOGISTICIANS. 179

it is true, but you make mistakes too.'' For us, making
mistakes is a misfortune, a very great misfortune, but
for you it is death.

Neither must you say, " Does the infallibility of arith-
metic prevent errors of addition ? " The rules of calcula-
tion are infallible, and yet we find people making
mistakes through not applying these rules. But a
revision of their calculation will show at once just
where they went astray. Here the case is quite dif-
ferent. The logisticians have applied their rules, and
yet they have fallen into contradiction. So true is
this, that they are preparing to alter these rules and
•'sacrifice the notion of class." Why alter them if
they were infallible ?

" We are not obliged," you say, " to solve hie et nunc
all possible problems." Oh, we do not ask as much as
that. If, in face of a problem, you gave no solution,
we should have nothing to say ; but, on the contrary,
you give two, and these two are contradictory, and
consequently one at least of them is false, and it is
this that constitutes a failure.

Mr. Russell attempts to reconcile these contradic-
tions, which can only be done, according to him, " by
restricting or even sacrificing the notion of class."
And M. Couturat, discounting the success of this
attempt, adds: "If logisticians succeed where others
have failed, M. Poincare will surely recollect this sen-
tence, and give logistic the credit of the solution."

Certainly not Logistic exists ; it has its code, which
has already gone through four editions ; or, rather, it
is this code which is logistic itself Is Mr. Russell
preparing to show that one at least of the two contra-
dictory arguments has transgressed the code ? Not in

i8o SCIENCE AND METHOD.

the very least ; he is preparing to alter these laws and
to revoke a certain number of them. If he succeeds,
I shall give credit to Mr. Russell's intuition, and not to
Peanian Logistic, which he will have destroyed.

III.
Liberty of Contradiction.

I offered two principal objections to the definition
of the whole number adopted by the logisticians.
What is M. Couturat's answer to the first of these
objections ?

What is the meaning in mathematics of the word
to exist? It means, I said, to be free from contradic-
tion. This is what M. Couturat disputes. " Logical
existence," he says, " is quite a different thing from
absence of contradiction. It consists in the fact that
a class is not empty. To say that some «'s exist is,
by definition, to assert that the class a is not void."
And, no doubt, to assert that the class a is not void
is, by definition, to assert that some ds exist. But
one of these assertions is just as destitute of meaning
as the other if they do not both signify either that
we can see or touch a, which is the meaning given
them by physicists or naturalists, or else that we can
conceive of an a without being involved in contradic-
tions, which is the meaning given them by logicians
and mathematicians.

In M. Couturat's opinion it is not non-contradiction
that proves existence, but existence that proves non-
contradiction. In order to establish the existence of a
class, we must accordingly establish, by an example,
that there is an individual belonging to that class.

LAST EFFORTS OF LOGISTICIANS. i8i

"But it will be said, How do we demonstrate the
existence of this individual ? Is it not necessary that
this existence should be established, to enable us to
deduce the existence of the class of which it forms
part? It is not so. Paradoxical as the assertion
may appear, we never demonstrate the existence of
an individual. Individuals, from the very fact that
they are individuals, are always considered as existing.
We have never to declare that an individual exists,
absolutely speaking, but only that it exists in a class."
M. Couturat finds his own assertion paradoxical, and
he will certainly not be alone in so finding it. Never-
theless it must have some sense, and it means, no
doubt, that the existence of an individual alone in
the world, of which nothing is asserted, cannot involve
contradiction. As long as it is quite alone, it is
evident that it cannot interfere with any one. Well,
be it so ; we will admit the existence of the individual,
" absolutely speaking," but with it we have nothing to
do. It still remains to demonstrate the existence
of the individual " in a class," and, in order to do
this, you will still have to prove that the assertion that
such an individual belongs to such a class is neither
contradictory in itself nor with the other postulates
adopted.

" Accordingly," M. Couturat continues, " to assert
that a definition is not valid unless it is first proved
that it is not contradictory, is to impose an arbitrary
and improper condition." The claim for the liberty
of contradiction could not be stated in more emphatic
or haughtier terms. " In any case, the onus probandi
rests with those who think these principles are contra-
dictory." Postulates are presumed to be compatible.

i82 SCIENCE AND METHCD.

just as a prisoner is presumed to be innocent, until the
contrary is proved.

It is unnecessary to add that I do not acquiesce
in this claim. But, you say, the de-honstration you
demand of us is impossible, and yoa cannot require
us to " aim at the moon." Excuse me; it is impossible
for you, but not for us who admit the principle of
induction as an a priori synthetic judgment. This
would be necessary for you as it is for us.

In order to demonstrate that a system of postulates
does not involve contradiction, it is necessary to apply
the principle of complete induction. Not only is there
nothing " extraordinary " in this method of reasoning,
but it is the only correct one. It is not " incon-
ceivable " that any one should ever have used it, and
it is not difficult to find "examples and precedents."
In my article I have quoted two, and they were
borrowed from Hilbert's pamphlet. He is not alone
in having made use of it, and those who have
not done so have been wrong. What I reproach
Hilbert with, is not that he has had recourse to it
(a born mathematician such as he could not but see
that a demonstration is required, and that this is the
only possible one), but that he has had recourse to it
without recognizing the reasoning by recurrence.

IV.

The Second Objection.

1 had noted a second error of the logisticians in
Hilbert's article. To-day Hilbert is excommuni-
cated, and M. Couturat no longer considers him as
a logistician. He will therefore, ask me if I have

LAST EFFORTS OF LOGISTICIANS. 183

found the same mistake in the orthodox logfis-
ticians. I have not seen it in the pages I have read,
but I do not know vi^hether I should find it in the
three hundred pages they have written that I have no
wish to read.

Only, they will have to commit the error as soon
as they attempt to make any sort of an application
of mathematical science. The eternal contemplation
of its own navel is not the sole object of this science.
It touches nature, and one day or other it will come
into contact with it. Then it will be necessary to
shake off purely verbal definitions and no longer to
content ourselves with words.

Let us return to Mr. Hilbert's example. It is still
a question of reasoning by recurrence and of knowing
whether a system of postulates is not contradictory.
M. Couturat will no doubt tell me that in that case
it does not concern him, but it may perhaps interest
those who do not claim, as he does, the liberty of
contradiction.

We wish to establish, as above, that we shall not
meet with contradiction after some particular number
of arguments, a number which may be as large as you
please, provided it is finite. For this purpose we
must apply the principle of induction. Are we to
understand here by finite number every number to
which the principle of induction applies ? Evidently
not, for otherwise we should be involved in the most
awkward consequences.

To have the right to lay down a system of postu-
lates, we must be assured that they are not contra-
dictory. This is a truth that is admitted by the
majority of scientists ; I should have said all before

i84 SCIENCE AND METHOD.

reading M. Couturat's last article. But what does it
signify ? Does it mean that we must be sure of not
meeting with contradiction after a finite number of
propositions, the finite number being, by definition,
that which possesses all the properties of a recurrent
nature in such a way that if one of these properties
were found wanting — if, for instance, we came upon a
contradiction — we should agree to say that the number
in question was not finite ?

In other words, do we mean that we must be sure
of not meeting a contradiction, with this condition,
that we agree to stop just at the moment when we are
on the point of meeting one ? The mere statement
of such a proposition is its sufficient condemnation.

Thus not only does Mr. Hilbert's reasoning assume
the principle of induction, but he assumes that this
principle is given us, not as a simple definition, but
as an a priori synthetic judgment.

I would sum up as follows : —

A demonstration is necessary.

The only possible demonstration is the demonstra-
tion by recurrence.

This demonstration is legitimate only if the prin-
ciple of induction is admitted, and if it is regarded
not as a definition but as a synthetic judgment.

V.

The Cantorian Antinomies.

I will now take up the examination of Mr. Russell's
new treatise. This treatise was written with the object
ofovercoming the difficulties raised by those Cantorian

LAST EFFORTS OF LOGISTICIANS. 185

antinomies to which I have already made frequent
allusion. Cantor thought it possible to construct a
Science of the Infinite. Others have advanced further
along the path he had opened, but they very soon ran
against strange contradictions. These antinomies are
already numerous, but the most celebrated are : —

1. Burali-Forti's antinomy.

2. The Zermelo-Konig antinomy.

3. Richard's antinomy.

Cantor had demonstrated that ordinal numbers (it
is a question of transfinite ordinal numbers, a new
notion introduced by him) can be arranged in a lineal
series ; that is to say, that of two unequal ordinal
numbers, there is always one that is smaller than the
other. Burali-Forti demonstrates the contrary ; and
indeed, as he says in substance, if we could arrange all
the ordinal numbers in a lineal series, this series
would define an ordinal number that would be
greater than all the others, to which we could then
add I and so obtain yet another ordinal number
which would be still greater. And this is contra-
dictory.

We will return later to the Zermelo-Konig anti-
nomy, which is of a somewhat different nature.
Richard's antinomy is as follows {Revue ginerale des
Sciences, June 30, 1905). Let us consider all the
decimal numbers that can be defined with the help of
a finite number of words. These decimal numbers form
an aggregate E, and it is easy to see that this aggregate
is denumerable — that is to say, that it is possible to
number the decimal numbers of this aggregate from one
to infinity. Suppose the numeration effected, and let

i86 SCIENCE AND METHOD.

us define a number N in the following manner. If
the «'" decimal of the «'" number of the aggregate E is

o, I, 2, 3, 4, 5, 6, 7, 8, or 9,
the n'^ decimal of N will be

r, 2, 3, 4, 5, 6, 7, 8, I, or i.

As we see, N is not equal to the n'" number of E,
and since n is any chance number, N does not belong
to E, and yet N should belong to this aggregate, since
we have defined it in a finite number of words.

We shall see further on that M. Richard himself
has, with much acuteness, given the explanation of his
paradox, and that his explanation can be extended,
mutatis mutandis, to the other paradoxes of like
nature. Mr. Russell quotes another rather amusing
antinomy :

What is the smallest whole number that cannot be
defined in a sentence formed of less than a hundred
English words ?

This number exists, and, indeed, the number of
numbers capable of being defined by such a sentence
is evidently finite, since the number of words in the
English language is not infinite. Therefore among
them there will be one that is smaller than all the
others.

On the other hand the number does not exist, for
its definition involves contradiction. The number, in
fact, is found to be defined by the sentence in italics,
which is formed of less than a hundred English words,
and, by definition, the number must not be capable
of being defined by such a sentence.

LAST EFFORTS OF LOGISTICIANS. 187

VI.
Zigzag Theory and No Classes Theory.

What is Mr. Rus.sell's attitude in face of these con-
tradictions ? After analysing those I have just spoken
of, and quoting others, after putting them in a form
that recalls Epimenides, he does not hesitate to con-
clude as follows : —

" A propositional function of one variable does not
always determine a class." * A " propositional func-
tion " (that is to say, a definition) or " norm " can be
"non-predicative." And this does not mean that these
non-predicative propositions determine a class that is
empty or void ; it does not mean that there is no
value oi X that satisfies the definition and can be one of
the elements of the class. The elements exist, but they
have no right to be grouped together to form a class.

But this is only the beginning, and we must know
how to recognize whether a definition is or is not
predicative. For the purpose of solving this problem,
Mr. Russell hesitates between three theories, which he
calls —

A. The zigzag theory.

B. The theory of limitation of size.

C. The no classes theory.

According to the zigzag theory, "definitions (pro-
positional functions) determine a class when they are
fairly simple, and only fail to do so when they are
complicated and recondite." Now who is to decide

* This and the following quotations are from Mr. Russell's paper,
" On some difficulties in the theory of transPnite numbers and order
types, ^'Proceedings of the London Mathematical Society. Ser. 2, Vol. 4,
Part I.

i88 SCIENCE AND METHOD.

whether a definition can be regarded as sufficiently
simple to be acceptable ? To this question we get no
answer except a candid confession of powerlessness.
" The axionns as to what functions are predicative
have to be exceedingly complicated, and cannot be
recommended by any intrinsic plausibility. This is a
defect which might be remedied by greater ingenuity,
or by the help of some hitherto unnoticed distinction.
But hitherto, in attempting to set up axioms for this
theory, I have found no guiding principle except the
avoidance of contradictions."

This theory therefore remains very obscure. In the
darkness there is a single glimmer, and that is the
word zigzag. What Mr. Russell calls zigzagginess is
no doubt this special character which distinguishes the
argument of Epimenides.

According to the theory of limitation of size, a
class must not be too extensive. It may, perhaps,
be infinite, but it must not be too infinite.

But we still come to the same difficulty. At what
precise moment will it begin to be too extensive ? Of
course this difficulty is not solved, and Mr. Russell
passes to the third theory.

In the no classes theory all mention of the word
class is prohibited, and the word has to be replaced by
various periphrases. What a change for the logis-
ticians who speak of nothing but class and classes of
classes ! The whole of Logistic will have to be re-
fashioned. Can we imagine the appearance of a page
of Logistic when all propositions dealing with class
have been suppressed ? There will be nothing left
but a few scattered survivors in the midst of a blank
page. Apparent rari nantes in gurgite vasto.

LAST EFFORTS OF LOGISTICIANS. 189

However that may be, we understand Mr. Russell's
hesitation at the modifications to which he is about
to submit the fundamental principles he has hitherto
adopted. Criteria will be necessary to decide whether
a definition is too complicated or too extensive, and
these criteria cannot be justified except by an appeal
to intuition.

It is towards the no classes theory that Mr. Russell
eventually inclines.

However it be, Logistic must be refashioned, and it
is not yet known how much of it can be saved. It is
unnecessary to add that it is Cantorism and Logistic
alone that are in question. The true mathematics, the
mathematics that is of some use, may continue to
develop according to its own principles, taking no
heed of the tempests that rage without, and step
by step it will pursue its wonted conquests, which are
decisive and have never to be abandoned.

VIL
The True Solution.

How are we to choose between these different
theories? It seems to me that the solution is con-
tained in M. Richard's letter mentioned above, which
will be found in the Revue G'enerale des Sciences of June
30, 1905. After stating the antinomy that I have called
Richard's antinomy, he gives the explanation.

Let us refer to what was said of this antinomy in
Section V. E is the aggregate oi all the numbers that
can be defined by a finite number of words, without
introducing the notion of the aggregate E itself, otherwise

igo SCIENCE AND METHOD.

the definition of E would contain a vicious circle, for
we cannot define E by the aggregate E itself.

Now we have defined N by a finite number of
words, it is true, but only with the help of the notion
of the aggregate E, and that is the reason why N does
not form a part of E.

In the example chosen by M. Richard, the con-
clusion is presented with complete evidence, and the
evidence becomes the more apparent on a reference to
the actual text of the letter. But the same explana-
tion serves for the other antinomies, as may be easily
verified.

Thus the definitions that must be regarded as non-
predicative are those which contain a vicious circle.
The above examples show sufficiently clearly what
I mean by this. Is this what Mr. Russell calls
" zigzagginess " ? I merely ask the question without
answering it.

VIII.

The Demonstrations of the Principle
OF Induction.

We will now examine the so-called demonstrations
of the principle of induction, and more particularly
those of Mr. Whitehead and Signor Burali-Forti.

And first we will speak of Whitehead's, availing our-
selves of some new denominations happily introduced
by Mr. Russell in his recent treatise.

We will call recurrent class every class of numbers
that includes zero, and also includes « + 1 if it
includes ;/.

We will call inductive number every number which
forms a part of all recurrent classes.

LAST EFFORTS OF LOGISTICIANS. 191

Upon what condition will this latter definition,
which plays an essential part in Whitehead's demon-
stration, be " predicative" and consequently acceptable?

Following upon what has been said above, we must
understand by all recurrent classes all those whose
definition does not contain the notion of inductive
number ; otherwise we shall be involved in the vicious
circle which engendered the antinomies.

Now, Whitehead has not taken this precaution.

Whitehead's argument is therefore vicious ; it is the
same that led to the antinomies. It was illegitimate
when it gave untrue results, and it remains illegitimate
when it leads by chance to a true result.

A definition which contains a vicious circle defines
nothing. It is of no use to say we are sure, whatever
be the meaning given to our definition, that there is
at least zero which belongs to the class of inductive
numbers. It is not a question of knowing whether
this class is empty, but whether it can be rigidly
delimited. A " non-predicative class " is not an empty
class, but a class with uncertain boundaries.

It is unnecessary to add that this particular objection
does not invalidate the general objections that apply
to all the demonstrations.

IX.

Signor Burali-Forti has given another demonstration
in his article " Le Classi finite" {Atti di Torino,
Vol. xxxii). But he is obliged to admit two postulates :

The first is that there exists always at least one
infinite class.

The second is stated thus : —

« e K (K - t /\). 3. 2< < v u.

192 SCIENCE AND METHOD.

The first postulate is no more evident than the
principle to be demonstrated. The second is not
only not evident, but it is untrue, as Mr. Whitehead
has shown, as, moreover, the veriest schoolboy could
have seen at the first glance if the axiom had been
stated in intelligible language, since it means : the
number of combinations that can be formed with
several objects is smaller than the number of those
objects.

X.

Zermelo's Axiom.

In a celebrated demonstration. Signer Zermelo
relies on the following axiom :

In an aggregate of any kind (or even in each of
the aggregates of an aggregate of aggregates) we
can always select one element at random (even if
the aggregate of aggregates contains an infinity
of aggregates).

This axiom had been applied a thousand times with-
out being stated, but as soon as it was stated, it raised
doubts. Some mathematicians, like M. Borel, rejected
it resolutely, while others admitted it. Let us see what
Mr. Russell thinks of it according to his last article.

He pronounces no opinion, but the considerations
which he gives are most suggestive.

To begin with a picturesque example, suppose that
we have as many pairs of boots as there are whole
numbers, so that we can number the pairs from i to
infinity, how many boots shall we have? Will the
number of boots be equal to the number of pairs ?
It will be so if, in each pair, the right boot is dis-

LAST EFFORTS OF LOGISTICIANS. 193

tinguishable from the left ; it will be sufficient in fact to
give the number 2« - i to the right boot of the n'"
pair, and the number 2« to the left boot of the n'"
pair. But it will not be so if the right boot is similar
to the left, because such an operation then becomes
impossible ; unless we admit Zermelo's axiom, since
in that case we can select at random from each pair
the boot we regard as the right.

XI.

Conclusions.

A demonstration really based upon the principles of
Analytical Logic will be composed of a succession of
propositions ; some, which will serve as premises, will
be identities or definitions ; others will be deduced
from the former step by step ; but although the con-
nexion between each proposition and the succeeding
proposition can be grasped immediately, it is not
obvious at a glance how it has been possible to pass
from the first to the last, which we may be tempted
to look upon as a new truth. But if we replace
successively the various expressions that are used by
their definitions, and if we pursue this operation to the
furthest possible limit, there will be nothing left at the
end but identities, so that all will be reduced to one
immense tautology. Logic therefore remains barren,
unless it is fertilized by intuition.

This is what I wrote formerly. The logisticians
assert the contrary, and imagine that they have proved
it by effectively demonstrating new truths. But what
mechanism have they used ?

(1,777) 13

194 SCIENCE AND METHOD.

Why is it that by applying to their arguments the
procedure I have just described, that is, by replacing
the terms defined by their definitions, we do not see
them melt into identities like the ordinary arguments ?
It is because the procedure is not applicable to them.
And why is this? Because their definitions are non-
predicative and present that kind of hidden vicious
circle I have pointed out above, and non-predicative
definitions cannot be substituted for the term defined.
Under these conditions. Logistic is no longer barren, it
engenders antinomies.

It is the belief in the existence of actual infinity that
has given birth to these non-predicative definitions. I
must explain myself. In these definitions we find the
word all, as we saw in the examples quoted above.
The word all has a very precise meaning when it is a
question of a finite * number of objects ; but for it still
to have a precise meaning when the number of the
objects is infinite, it is necessary that there should
exist an actual infinity. Otherwise all these objects
cannot be conceived as existing prior to their definition,
and then, if the definition of a notion N depends on
all the objects A, it may be tainted with the vicious
circle, if among the objects A there is one that cannot
be defined without bringing in the notion N itself.

The rules of formal logic simply express the pro-
perties of all the possible classifications. But in order
that they should be applicable, it is necessary that
these classifications should be immutable and not
require to be modified in the course of the argument.
If we have only to classify a finite number of objects,
it is easy to preserve these classifications without

* The original has " infinite,'' obviously a slip.

LAST EFFORTS OF LOGISTICIANS. 195

change. If the number of the objects is indefinite,
that is to say if we are constantly liable to find new
and unforeseen objects springing up, it may happen
that the appearance of a new object will oblige us to
modify the classification, and it is thus that we are
exposed to the antinomies.

There is no actual infinity. The Cantorians forgot
this, and so fell into contradiction. It is true that
Cantorism has been useful, but that was when it was
applied to a real problem, whose terms were clearly
defined, and then it was possible to advance without
danger.

Like the Cantorians, the logisticians have forgotten
the fact, and they have met with the same difficulties.
But it is a question whether they took this path by
accident or whether it was a necessity for them,

In my view, there is no doubt about the matter ;
belief in an actual infinity is essential in the Russellian
logistic, and this is exactly what distinguishes it from
the Hilbertian logistic. Hilbert takes the point of
view of extension precisely in order to avoid the
Cantorian antinomies. Russell takes the point of
view of comprehension, and consequently for him the
genus is prior to the species, and the summuni genus
prior to all. This would involve no difficulty if the
sunimum genus were finite ; but if it is infinite, it is
necessary to place the infinite before the finite — that is
to say, to regard the infinite as actual.

And we have not only infinite classes ; when we
pass from the genus to the species by restricting the
concept by new conditions, the number of these
conditions is still infinite, for they generally express
that the object under consideration is in such and

196 SCIENCE AND METHOD.

such a relation with all the objects of an infinite
class.

But all this is ancient history. Mr. Russell has
realized the danger and is going to reconsider the
matter. He is going to change everything, and we
must understand clearly that he is preparing not only
to introduce new principles which permit of operations
formerly prohibited, but also to prohibit opera-
tions which he formerly considered legitimate. He
is not content with adoring what he once burnt, but
he is going to burn what he once adored, which is
more serious. He is not adding a new wing to the
building, but sapping its foundations.

The old Logistic is dead, and so true is this, that
the zigzag theory and the no classes theory are
already disputing the succession. We will wait until
the new exists before we attempt to judge it.

BOOK III.
THE NEW MECHANICS.

I.

MECHANICS AND RADIUM.

I.

Introduction.

Are the general principles of Dynamics, which have
served since Newton's day as the foundation of Physi-
cal Science, and appear immutable, on the point of
being abandoned, or, at the very least, profoundly
modified ? This is the question many people have
been asking for the last few years. According to
them the discovery of radium has upset what were
considered the most firmly rooted scientific doctrines,
the impossibility of the transmutation of metals on the
one hand, and, on the other, the fundamental postu-
lates of Mechanics. Perhaps they have been in too
great haste to consider these novelties as definitely
established, and to shatter our idols of yesterday ;
perhaps it would be well to await more numerous
and more convincing experiments. It is none the less
necessary that we should at once acquire a knowledge
of the new doctrines and of the arguments, already
most weighty, upon which they rely.

I will first recall in a few words what these prin-
ciples are.

200 SCIENCE AND METHOD.

A. The motion of a material point, isolated and un-
affected by any exterior force, is rectilineal and
uniform. This is the principle of inertia ; no accelera-
tion without force.

B. The acceleration of a moving point has the same
direction as the resultant of all the forces to which the
point is subjected ; it is equal to the quotient of this
resultant by a coefficient called the mass of the moving
point.

The mass of a moving point, thus defined, is con-
stant; it does not depend upon the velocity acquired by
the point, it is the same whether the force is parallel
to this velocity and only tends to accelerate or retard
the motion of the point, or whether it is, on the con-
trary, perpendicular to that velocity and tends to
cause the motion to deviate to right or left, that is to
say to airve the trajectory.

C. All the forces to which a material point is sub-
jected arise from the action of other material points ;
they depend only upon the relative positions and
velocities of these different material points.

By combining the two principles B and C we
arrive at the principle of relative motion, by virtue of
which the laws of motion of a system are the same
whether we refer the system to fixed axes, or whether
we refer it to moving axes animated with a rectilineal
and uniform forward motion, so that it is impossible
to distinguish absolute motion from a relative motion
referred to such moving axes.

D. If a material point A acts upon another material
point B, the body B reacts upon A, and these two
actions are two forces that are equal and directly
opposite to one another. This is the principle of the

MECHANICS AND RADIUM. 201

equality of action and reaction, or more briefly, the
principle of reaction.

Astronomical observations, and the commonest
physical phenomena, seem to have afforded the most
complete, unvarying, and precise confirmation of these
principles. That is true, they tell us now, but only
because we have never dealt with any but low velo-
cities. Mercury, for instance, which moves faster than
any of the other planets, scarcely travels sixty miles a
second — Would it behave in the same way if it travelled
a thousand times as fast? It is clear that we have still
no cause for anxiety ; whatever may be the progress
of automobilism, it will be some time yet before we
have to give up applying the classical principles of
Dynamics to our machines.

How is it then that we have succeeded in realizing
velocities a thousand times greater than that of
Mercury, equal, for instance, to a tenth or a third of
the velocity of light, or coming nearer to it even than
that? It is by the help of the cathode rays and
the rays of radium.

We know that radium emits three kinds of rays,
which are designated by the three Greek letters «, ^, y.
In what follows, unless I specifically state the contrary,
I shall always speak of the fi rays, which are analogous
to the cathode rays.

After the discovery of the cathode rays, two opposite
theories were propounded. Crookes attributed the
phenomena to an actual molecular bombardment,
Hertz to peculiar undulations of the ether. It was a
repetition of the controversy that had divided physi-
cists a century before with regard to light. Crookes
returned to the emission theory, abandoned in the case

202 SCIENCE AND METHOD.

of light, while Hertz held to the undulatory theory.
The facts seemed to be in favour of Crookes.

It was recognized in the first place that the cathode
rays carry with them a negative electric charge : they
are deviated by a magnetic and by an electric field,
and these deviations are precisely what would be pro-
duced by these same fields upon projectiles animated
with a very great velocity, and highly charged with
negative electricity. These two deviations depend
upon two quantities ; the velocity on the one hand,
and the proportion of the projectile's electric charge to
its mass on the other. We cannot know the absolute
value of this mass, nor that of the charge, but only
their proportion. It is clear in fact, that if we double
both the charge and the mass, without changing the
velocity, we shall double the force that tends to deviate
the projectile ; but as its mass is similarly doubled,
the observable acceleration and deviation will not be
changed. Observation of the two deviations will
accordingly furnish us with two equations for deter-
mining these two unknown quantities. We find a
velocity of 6,000 to 20,000 miles a second. As for
the proportion of the charge to the mass, it is very
great ; it may be compared with the corresponding
proportion in the case of a hydrogen ion in electro-
lysis, and we find then that a cathode projectile
carries with it about a thousand times as much
electricity as an equal mass of hydrogen in an
electrolyte.

In order to confirm these views, we should require a
direct measure of this velocity, that could then be
compared with the velocity so calculated. Some old
experiments of Sir J. J. Thomson's had given results

MECHANICS AND RADIUM. 203

more than a hundred times too low, but they were
subject to certain causes of error. The question has
been taken up again by Wiechert, with the help of an
arrangement by which he makes use of the Hertzian
oscillations, and this has given results in accordance
with the theory, at least in the matter of magnitude,
and it would be most interesting to take up these
experiments again. However it be, the theory of
undulations seems to be incapable of accounting for
this body of facts.

The same calculations made upon the /3 rays of
radium have yielded still higher velocities — 60,000,
120,000 miles a second, and even more. These
velocities greatly surpass any that we know. It is
true that light, as we have long known, travels 186,000
miles a second, but it is not a transportation of matter,
while, if we adopt the emission theory for the cathode
rays, we have material molecules actually animated
with the velocities in question, and we have to enquire
whether the ordinary laws of Mechanics are still
applicable to them.

II.

Longitudinal and Transversal Mass.

We know that electric currents give rise to pheno-
mena of induction, in particular to self-induction.
When a current increases it develops an electro-motive
force of self-induction which tends to oppose the
current. On the contrary, when the current decreases,
the electro-motive force of self-induction tends to
maintain the current. Self-induction then opposes
all variation in the intensity of a current, just as in

204 SCIENCE AND METHOD.

Mechanics, the inertia of a body opposes all variation
in its velocity. Self-induction is an actual inertia.
Everything takes place as if the current could not be
set up without setting the surrounding ether in motion,
and as if the inertia of this ether consequently tended
to keep the intensity of the current constant. The
inertia must be overcome to set up the current, and it
must be overcome again to make it cease.

A cathode ray, which is a rain of projectiles charged
with negative electricity, can be likened to a current.
No doubt this current differs, at first sight at any rate,
from the ordinary conduction currents, where the
matter is motionless and the electricity circulates
through the matter. It is a convection current, where
the electricity is attached to a material vehicle and
carried by the movement of that vehicle. But Rowland
has proved that convection currents produce the same
magnetic effects as conduction currents. They must
also produce the same effects of induction. Firstly, if
it were not so, the principle of the conservation of
energy would be violated ; and secondly, Cremien and
Pender have employed a method in which these effects
of induction are directly demonstrated.

If the velocity of a cathode corpuscle happens to
vary, the intensity of the corresponding current will
vary equally, and there will be developed effects of
self-induction which tend to oppose this variation.
These corpuscles must therefore possess a double
inertia, first their actual inertia, and then an apparent
inertia due to self-induction, which produces the same
effects. They will therefore have a total apparent
mass, composed of their real mass and of a fictitious
mass of electro-magnetic origin. Calculation shows

MECHANICS AND RADIUM. 205

that this fictitious mass varies with the velocity (when
this is comparable with the velocity of light), and that
the force of the inertia of self-induction is not the
same when the velocity of the projectile is increased
or diminished, as when its direction is changed, and
accordingly the same holds good of the apparent total
force of inertia.

The total apparent mass is therefore not the same
when the actual force applied to the corpuscle is
parallel with its velocity and tends to accelerate its
movement, as when it is perpendicular to the velocity
and tends to alter its direction. Accordingly we must
distinguish between the total longitudinal mass and the
total transversal mass, and, moreover, these two total
masses depend upon the velocity. Such are the
results of Abraham's theoretical work.

In the measurements spoken of in the last section,
what was it that was determined by measuring
the two deviations ? The velocity on the one hand,
and on the other the proportion of the charge to the
total transversal mass. Under these conditions, how
are we to determine what are the proportions, in this
total mass, of the actual mass and of the fictitious
electro-magnetic mass? If we had only the cathode
rays properly so called, we could not dream of doing
so, but fortunately we have the rays of radium, whose
velocity, as we have seen, is considerably higher.
These rays are not all identical, and do not behave
in the same way under the action of an electric and a
magnetic field. We find that the electric deviation
is a function of the magnetic deviation, and by re-
ceiving upon a sensitive plate rays of radium that
have been subjected to the action of the two fields,

2o6 SCIENCE AND METHOD.

we can photograph the curve which represents the
relation between these two deviations. This is what
Kaufmann has done, and he has deduced the rela-
tion between the velocity and the proportion of the
charge to the total apparent mass, a proportion that
we call e.

We might suppose that there exist several kinds
of rays, each characterized by a particular velocity,
by a particular charge, and by a particular mass ;
but this hypothesis is most improbable. What reason
indeed could there be why all the corpuscles of the
same mass should always have the same velocity ? It
is more natural to suppose that the charge and
the actual mass are the same for all the projectiles,
and that they differ only in velocity. If the propor-
tion « is a function of the velocity, it is not because
the actual mass varies with the velocity, but, as the
fictitious electro-magnetic mass depends upon that
velocity, the total apparent mass, which is alone
observable, must depend upon it also, even though
the actual mass does not depend upon it but is
constant.

Abraham's calculations make us acquainted with
the law in accordance with which the fictitious mass
varies as a function of the velocity, and Kaufmann's
experiment makes us acquainted with the law of
variation of the total mass. A comparison of these
two laws will therefore enable us to determine the
proportion of the actual mass to the total mass.

Such is the method employed by Kaufmann to
determine this proportion. The result is most sur-
prising : the actual mass is nil.

We have thus been led to quite unexpected con-

MECHANICS AND RADIUM. 207

ceptions. What had been proved only in the case
of the cathode corpuscles has been extended to all
bodies. What we call mass would seem to be nothing
but an appearance, and all inertia to be of electro-
magnetic origin. But if this be true, mass is no
longer constant; it increases with the velocity: while
apparently constant for velocities up to as much as
600 miles a second, it grows thenceforward and be-
comes infinite for the velocity of light. Transversal
mass is no longer equal to longitudinal mass, but only
about equal if the velocity is not too great. Principle
B of mechanics is no longer true.

III.

Canal-Rays.

At the point we have reached, this conclusion may
seem premature. Can we apply to the whole of
matter what has only been established for these
very light corpuscles which are only an emanation
of matter and perhaps not true matter? But before
broaching this question, we must say a word about
another kind of rays — I mean the canal-rays, Gold-
stein's Kanaktrahlen. Simultaneously with the cathode
rays charged with negative electricity, the cathode
emits canal-rays charged with positive electricity. In
general these canal-rays, not being repelled by the cath-
ode, remain confined in the immediate neighbourhood
of that cathode, where they form the " buff stratum "
that is not very easy to detect. But if the cathode is
pierced with holes and blocks the tube almost com-
pletely, the canal-rays will be generated behind the

2o8 SCIENCE AND METHOD.

cathode, in the opposite direction from that of the
cathode rays, and it will become possible to study
them. It is thus that we have been enabled to
demon.strate their positive charge and to show that
the magnetic and electric deviations still exist, as
in the case of the cathode rays, though they are much
weaker.

Radium likewise emits rays similar to the canal-
rays, and relatively very absorbable, which are called
a rays.

As in the case of the cathode rays, we can measure
the two deviations and deduce the velocity and the
proportion e. The results are less constant than in
the case of the cathode rays, but the velocity is lower,
as is also the proportion «. The positive corpuscles
are less highly charged than the negative corpuscles ;
or if, as is more natural, we suppose that the charges
are equal and of opposite sign, the positive corpuscles
are much larger. These corpuscles, charged some
positively and others negatively, have been given the
name of electrons*

IV.

LoRENTz's Theory.

But the electrons do not only give evidence of
their existence in these rays in which they appear

* The name is now applied only to the negative corpuscles, which
seem to possess no actual mass and only a fictitious electro-magnetic
mass, and not to the canal-rays, which appear to consist of ordinary
chemical atoms positively charged, owing to the fact that they have
lost one or more of the electrons they possess in their ordinary neutral
state.

MECHANICS AND RADIUM. 209

to us animated with enormous velocities. We shall
see them in very different parts, and it is they that
explain for us the principal phenomena of optics and
of electricity. The brilliant synthesis about which I
am going to say a few words is due to Lorentz.

Matter is entirely formed of electrons bearing enor-
mous charges, and if it appears to us neutral, it is
because the electrons' charges of opposite sign balance.
For instance, we can picture a kind of solar system
consisting of one great positive electron, about which
gravitate numerous small planets which are negative
electrons, attracted by the electricity of opposite sign
with which the central electron is charged. The
negative charges of these planets balance the positive
charge of the sun, so that the algebraic sum of all
these charges is nil.

All these electrons are immersed in ether. The
ether is everywhere identical with itself, and perturba-
tions are produced in it, following the same laws as
light or the Hertzian oscillations in empty space.
Beyond the electrons and the ether there is nothing.
When a luminous wave penetrates a part of the ether
where the electrons are numerous, these electrons are
set in motion under the influence of the perturbation
of the ether, and then react upon the ether. This
accounts for refraction, dispersion, double refraction,
and absorption. In the same way, if an electron was
set in motion for any reason, it would disturb the
ether about it and give birth to luminous waves, and
this explains the emission of light by incandescent
bodies.

In certain bodies — metals, for instance — we have
motionless electrons, about which circulate movable

(1,777) 14

210 SCIENCE AND METHOD.

electrons, enjoying complete liberty, except of leaving
the metallic body and crossing the surface that sepa-
rates it from exterior space, or from the air, or from
any other non-metallic body. These movable elec-
trons behave then inside the metallic body as do the
molecules of a gas, according to the kinetic theory of
gases, inside the vessel in which the gas is contained.
But under the influence of a difference of potential
the negative movable electrons would all tend to go
to one side and the positive movable electrons to the
other. This is what produces electric currents, and it
is for this reason that such bodies act as conductors.
Moreover, the velocities of our electrons will become
greater as the temperature rises, if we accept the
analogy of the kinetic theory of gases. When one
of these movable electrons meets the surface of the
metallic body, a surface it cannot cross, it is deflected
like a billiard ball that has touched the cushion, and
its velocity undergoes a sudden change of direction.
But when an electron changes its direction, as we
shall see further on, it becomes the source of a lumin-
ous wave, and it is for this reason that hot metals are
incandescent.

In other bodies, such as dielectric and transparent
bodies, the movable electrons enjoy much less liberty.
They remain, as it were, attached to fixed electrons
which attract them. The further they stray, the
greater becomes the attraction that tends to bring
them back. Accordingly they can only suffer slight
displacements ; they cannot circulate throughout the
body, but only oscillate about their mean position.
It is for this reason that these bodies are non-
conductors ; they are, moreover, generally trans-

MECHANICS AND RADIUM. 211

parent, and they are refractive because the luminous
vibrations are communicated to the movable electrons
which are susceptible of oscillation, and a refraction
of the original beam of light results.

I cannot here give the details of the calculations.
I will content myself with saying that this theory
accounts for all the known facts, and has enabled us
to foresee new ones, such as Zeeman's phenomenon.

V.

MECHANICAL CONSEQUENCES.

Now we can form two hypotheses in explanation of
the above facts.

1. The positive electrons possess an actual mass,
much greater than their fictitious electro-magnetic
mass, and the negative electrons alone are devoid of
actual mass. We may even suppose that, besides the
electrons of both signs, there are neutral atoms which
have no other mass than their actual mass. In this
case Mechanics is not affected, we have no need to
touch its laws, actual mass is constant, only the move-
ments are disturbed by the effects of self-induction, as
has always been known. These perturbations are,
moreover, almost negligible, except in the case of the
negative electrons which, having no actual mass, are
not true matter.

2. But there is another point of view. We may sup-
pose that the neutral atom does not exist, and that the
positive electrons are devoid of actual mass just as
much as the negative electrons. But if this be so,
actual mass disappears, and either the word mass will

212 SCIENCE AND METHOD.

have no further meaning, or else it must designate the
fictitious electro-magnetic mass ; in that case mass will
no longer be constant, transversal mass will no longer
be equal to longitudinal mass, and the principles of
Mechanics will be upset.

And first a word by way of explanation. I said
that, for the same charge, the total mass of a positive
electron is much greater than that of a negative electron.
Then it is natural to suppose that this difference is
explained by the fact that the positive electron has,
in addition to its fictitious mass, a considerable actual
mass, which would bring us back to the first hypothesis.
But we may equally well admit that the actual mass
is nil for the one as for the other, but that the fictitious
mass of the positive electron is much greater, because
this electron is much smaller. I say advisedly, much
smaller. And indeed, in this hypothesis, inertia is of
exclusively electro-magnetic origin, and is reduced to
the inertia of the ether ; the electrons are no longer
anything in themselves, they are only holes in the
ether, around which the ether is agitated ; the smaller
these holes are, the more ether there will be, and the
greater, consequently, will be its inertia.

How are we to decide between these two hypotheses ?
By working upon the canal-rays, as Kaufmann has
done upon the /? rays? This is impossible, for the
velocity of these rays is much too low. So each must
decide according to his temperament, the conservatives
taking one side and the lovers of novelty the other.
But perhaps, to gain a complete understanding of
the innovators' arguments, we must turn to other
considerations.

II.

MECHANICS AND OPTICS.
I.

ABERRATION.

We know the nature of the phenomenon of aberration
discovered by Bradley. The light emanating from a
star takes a certain time to traverse the telescope.
During this time the telescope is displaced by the
Earth's motion. If, therefore, the telescope were
pointed in the tme direction of the star, the image
would be formed at the point occupied by the crossed
threads of the reticule when the light reached the
object-glass. When the light reached the plane of the
reticule the crossed threads would no longer be in the
same spot, owing to the Earth's motion. We are there-
fore obliged to alter the direction of the telescope to
bring the image back to the crossed threads. It
follows that the astronomer will not point his telescope
exactly in the direction of the absolute velocity of the
light from the star — that is to say, upon the true position
of the star — but in the direction of the relative velocity
of the light in relation to the Earth — that is to say, upon
what is called the apparent position of the star.
The velocity of light is known, and accordingly we

214 SCIENCE AND METHOD.

might imagine that we have the means of calculating
the absolute velocity of the Earth. (I shall explain the
meaning of this word "absolute" later.) But it is not
so at all. We certainly know the apparent position of
the star we are observing, but we do not know its true
position. We know the velocity of light only in terms
of magnitude and not of direction.

If, therefore, the Earth's velocity were rectilineal and
uniform, we should never have suspected the pheno-
menon of aberration. But it is variable : it is composed
of two parts — the velocity of the Solar System, which
is, as far as we know, rectilineal and uniform ; and the
velocity of the Earth in relation to the Sun, which is
variable. If the velocity of the Solar System — that is
to say the constant part — alone existed, the observed
direction would be invariable. The position we should
thus observe is called the mean apparent position of
the star.

Now if we take into account at once both parts of
the Earth's velocity, we shall get the actual apparent
position, which describes a small ellipse about the
mean apparent position, and it is this ellipse that is
observed.

Neglecting very small quantities, we shall see that
the dimensions of this ellipse depend only upon the
relation between the Earth's velocity in relation to the
Sun and the velocity of light, so that the relative
velocity of the Earth in relation to the Sun is alone
in question.

We must pause, however. This result is not exact,
but only approximate. Let us push the approxima-
tion a step further. The dimensions of the ellipse will
then depend upon the absolute velocity of the Earth.

MECHANICS AND OPTICS. 215

If we compare the great axes of ellipse for the different
stars, we shall have, theoretically at least, the means
determining this absolute velocity.

This is perhaps less startling than it seems at first.
It is not a question, indeed, of the velocity in relation
to absolute space, but of the velocity in relation to the
ethics, which is regarded, by definition, as being in
absolute repose.

Moreover, this method is purely theoretical. In fact
the aberration is very small, and the possible variations
of the ellipse of aberration are much smaller still, and,
acccordingly, if we regard the aberration as of the first
order, the variations must be regarded as of the second
order, about a thousandth of a second of arc, and
absolutely inappreciable by our instruments. Lastly,
we shall see further on why the foregoing theory must
be rejected, and why we could not determine this
absolute velocity even though our instruments were
ten thousand times as accurate.

Another method may be devised, and, indeed, has
been devised. The velocity of light is not the same in
the water as in the air : could we not compare the two
apparent positions of a star seen through a telescope
filled first with air and then with water ? The results
have been negative ; the apparent laws of reflection
and of refraction are not altered by the Earth's motion.
This phenomenon admits of two explanations.

I. We may suppose that the ether is not in repose,
but that it is displaced by bodies in motion. It would
not then be astonishing that the phenomenon of re-
fraction should not be altered by the Earth's motion,
since everything — lenses, telescopes, and ether — would
be carried along together by the same motion. As for

2i6 SCIENCE AND METHOD.

aberration itself, it would be explained by a kind of
refraction produced at the surface of separation of the
ether in repose in the interstellar spaces and the ether
carried along by the Earth's movement. It is upon
this hypothesis (the total translation of the ether) that
Hertz's theory of the Electro-dynamics of bodies in
motion is founded.

2. Fresnel, on the contrary, supposes that the ether
is in absolute repose in space, and almost in absolute
repose in the air, whatever be the velocity of that air,
and that it is partially displaced by refringent mediums.
Lorentz has given this theory a more satisfactory form.
In his view the ether is in repose and the electrons
alone are in motion. In space, where the ether alone
comes into play, and in the air, where it comes almost
alone into play, the displacement is nil or almost nil.
In refringent mediums, where the perturbation is pro-
duced both by the vibrations of the ether and by
those of the electrons set in motion by the agitation of
the ether, the undulations 2X& partially carried along.

To help us to decide between these two hypotheses,
we have the experiment of Fizeau, who compared, by
measurements of fringes of interference, the velocity of
light in the air in repose and in motion as well as in
water in repose and in motion. These experiments
have confirmed Fresnel's hypothesis of partial dis-
placement, and they have been repeated with the
same result by Michelson. Hertz's theory, tlierefore,
must be rejected.

MECHANICS AND OPTICS. 217

11.
The Principle of Relativity.

But if the ether is not displaced by the Earth's
motion, is it possible by means of optical phenomena
to demonstrate the absolute velocity of the Earth, or
rather its velocity in relation to the motionless ether ?
Experience has given a negative reply, and yet the
experimental processes have been varied in every
possible way. Whatever be the method employed,
we shall never succeed in disclosing any but relative
velocities ; I mean the velocities of certain material
bodies in relation to other material bodies. Indeed,
when the source of the light and the apparatus for
observation are both on the Earth and participate in
its motion, the experimental results have always been
the same, whatever be the direction of the apparatus
in relation to the direction of the Earth's orbital motion.
That astronomical aberration takes place is due to the
fact that the source, which is a star, is in motion in
relation to the observer.

The hypotheses formed up to now account perfectly
for this general result, if we neglect very small quanti-
ties on the order of the square of aberration. The
explanation relies on the notion o{ local time introduced
by Lorentz, which I will try to make clear. Imagine
two observers placed, one at a point A and the other
at a point B, wishing to set their watches by means of
optical signals. They agree that B shall send a signal
to A at a given hour by his watch, and A sets his
watch to that hour as soon as he sees the signal. If
the operation were performed in this way only, there

2i8 SCIENCE AND METHOD.

would be a systematic error ; for, since light takes a
certain time, t, to travel from B to A, A's watch would
always be slower than B's to the extent of /. This
error is easily corrected, for it is sufficient to inter-
change the signals. A in his turn must send signals
to B, and after this new setting it will be B's watch
that will be slower than A's to the extent of t. Then
it will only be necessary to take the arithmetic mean
between the two settings.

But this method of operating assumes that light
takes the same time to travel from A to B and to
return from B to A. This is true if the observers are
motionless, but it is no longer true if they are involved
in a common transposition, because in that case A, for
instance, will be meeting the light that comes from B,
while B is retreating from the light that comes from
A. Accordingly, if the observers are involved in a
common transposition without suspecting it, their set-
ting will be defective ; their watches will not show the
same time, but each of them will mark the local time
proper to the place where it is.

The two observers will have no means of detecting
this, if the motionless ether can only transmit luminous
signals all travelling at the same velocity, and if the
other signals they can send are transmitted to them
by mediums involved with them in their transposition.
The phenomenon each of them observes will be either
early or late — it will not occur at the moment it would
have if there were no transposition ; but since their
observations are made with a watch defectively set,
they will not detect it, and the appearances will not
be altered.

It follows from this that the compensation is easy to

MECHANICS AND OPTICS. 219

explain so long as we neglect the square of aberration,
and for a long time experiments were not sufficiently
accurate to make it necessary to take this into account.
But one day Michelson thought out a much more
delicate process. He introduced rays that had
traversed different distances after being reflected by
mirrors. Each of the distances being about a yard,
and the fringes of interference making it possible to
detect differences of a fraction of a millionth of a
millimeter (2-riinnnnTTrth of an inch), the square of
aberration could no longer be neglected, and yet the
results were still negative. Accordingly, the theory
required to be completed, and this has been done by
the hypothesis of Lorentz and fitz-Gerald.

These two physicists assume that all bodies in-
volved in a transposition undergo a contraction in the
direction of this transposition, while their dimensions
perpendicular to the transposition remain invariable.
This cont7'action is the same for all bodies. It is, more-
over, very slight, about one part in two hundred million
for a velocity such as that of the Earth. Moreover,
our measuring instruments could not disclose it, even
though they were very much more accurate, since
indeed the yard-measures with which we measure
undergo the same contraction as the objects to be
measured. If a body fits exactly to a measure when
the body, and consequently the measure, are turned in
the direction of the Earth's motion, it will not cease to
fit exactly to the measure when turned in another
direction, in spite of the fact that the body and the
measure have changed their length in changing their
direction, precisely because the change is the same for
both. But it is not so if we measure a distance, no

220 SCIENCE AND METHOD.

longer with a yard-measure, but by the time light
takes to traverse it, and this is exactly what
Michelson has done.

A body that is spherical when in repose will thus
assume the form of a flattened ellipsoid of revolution
when it is in motion. But the observer will always
believe it to be spherical, because he has himself under-
gone an analogous deformation, as well as all the
objects that serve him as points of reference. On the
contrary, the surfaces of the waves of light, which have
remained exactly spherical, will appear to him as
elongated ellipsoids.

What will happen then ? Imagine an observer and
a source involved together in the transposition. The
wave surfaces emanating from the source will be
spheres, having as centre the successive positions of
the source. The distance of this centre from the actual
position of the source will be proportional to the time
elapsed since the emission — that is to say, to the radius
of the sphere. All these spheres are accordingly
homothetic one to the other, in relation to the actual
position S of the source. But for our observer, on
account of the contraction, all these spheres will
appear as elongated ellipsoids, and all these ellip-
soids will still be homothetic in relation to the point
S ; the excentricity of all the ellipsoids is the
same, and depends solely upon the Earth's velocity.
We shall select our law of contraction in such a way
that S zvill be the focus of the meridian section of the
ellipsoid.

This time the compensation is exact, and this is
explained by Michelson's experiments.

I said above that, according to the ordinary theories,

MECHANICS AND OPTICS. 221

observations of astronomical aberration could make us
acquainted with the absolute velocity of the Earth, if
our instruments were a thousand times as accurate,
but this conclusion must be modified. It is true that
the angles observed would be modified by the effect of
this absolute velocity, but the graduated circles we use
for measuring the angles would be deformed by the
motion ; they would become ellipses, the result would
be an error in the angle measured, and this second
error would exactly compensate the forjner.

This hypothesis of Lorentz and Fitz-Gerald will
appear most extraordinary at first sight. All that can
be said in its favour for the moment is that it is merely
the immediate interpretation of Michelson's experi-
mental result, if we define distances by the time taken
by light to traverse them.

However that be, it is impossible to escape the
impression that the Principle of Relativity is a general
law of Nature, and that we shall never succeed, by any
imaginable method, in demonstrating any but relative
velocities ; and by this I mean not merely the velocities
of bodies in relation to the ether, but the velocities ot
bodies in relation to each other. So many different
experiments have given similar results that we cannot
but feel tempted to attribute to this Principle of
Relativity a value comparable, for instance, to that of
the Principle of Equivalence. It is well in any case to
see what are the consequences to which this point of
view would lead, and then to submit these consequences
to the test of experiment.

232 SCIENCE AND METHOD.

III.
The Principle of Reaction.

Let us see what becomes, under Lorentz's theory,
of the principle of the equality of action and reaction.
Take an electron. A, which ie set in motion by some
means. It produces a disturbance in the ether, and
after a certain time this disturbance reaches another
electron, B, which will be thrown out of its posi-
tion of equilibrium. Under these conditions there
can be no equality between the action and the re-
action, at least if we do not consider the ether, but
only the electrons which are alone observable, since
our matter is composed of electrons.

It is indeed the electron A that has disturbed the
electron B ; but even if the electron B reacts upon A,
this reaction, though possibly equal to the action,
cannot in any case be simultaneous, since the electron
B cannot be set in motion until after a certain length
of time necessary for the effect to travel through the
ether. If we submit the problem to a more precise
calculation, we arrive at the following result. Imagine
a Hertz excitator placed at the focus of a parabolic
mirror to which it is attached mechanically ; this
excitator emits electro-magnetic waves, and the mirror
drives all these waves in the same direction : the
excitator will accordingly radiate energy in a particular
direction. Well, calculations show that the excitator
will recoil like a cannon that has fired a projectile.
In the case of the cannon, the recoil is the natural
result of the equality of action and reaction. The

MECHANICS AND OPTICS. 223

cannon recoils because the projectile on which it has
acted reacts upon it.

But here the case is not the same. What we have
fired away is no longer a material projectile ; it is
energy, and energy has no mass — there is no counter-
part. Instead of an excitator, we might have con-
sidered simply a lamp with a reflector concentrating
its rays in a single direction.

It is true that if the energy emanating from the
excitator or the lamp happens to reach a material
object, this object will experience a mechanical thrust
as if it had been struck by an actual projectile, and
this thrust will be equal to the recoil of the excitator
or the lamp, if no energy has been lost on the way,
and if the object absorbs the energy in its entirety.
We should then be tempted to say that there is still
compensation between the action and the reaction.
But this compensation, even though it is complete, is
always late. It never occurs at all if the light, after
leaving the source, strays in the interstellar spaces
without ever meeting a material body, and it is
incomplete if the body it strikes is not perfectly
absorbent.

Are these mechanical actions too small to be
measured, or are they appreciable by experiment ?
They are none other than the actions due to the
Maxwell-Bartholi pressures. Maxwell had predicted
these pressures by calculations relating to Electro-
statics and Magnetism, and Bartholi had arrived at
the same results on Thermodynamic grounds.

It is in this way that tails of comets are explained.
Small particles are detached from the head of the
comet, they are struck by the light of the Sun, which

224 SCIENCE AND METHOD.

repels them just as would a shower of projectiles
coming from the Sun. The mass of these particles is
so small that this repulsion overcomes the Newtonian
gravitation, and accordingly they form the tail as they
retreat from the Sun.

Direct experimental verification of this pressure ot
radiation was not easy to obtain. The first attempt
led to the construction of the radiometer. But this
apparatus turns the wrong way, the reverse of the
theoretical direction, and the explanation of its rota-
tion, which has since been discovered, is entirely
different. Success has been attained at last by creat-
ing a more perfect vacuum on the one hand ; and
on the other, by not blackening one of the faces of
the plates, and by directing a luminous beam upon
one of these faces. The radiometric effects and other
disturbing causes are eliminated by a series of minute
precautions, and a deviation is obtained which is
extremely small, but is, it appears, in conformity with
the theory.

The same effects of the Maxwell-Bartholi pressure
are similarly predicted by Hertz's theory, of which I
spoke above, and by that of Lorentz, but there is a
difference. Suppose the energy, in the form of light,
for instance, travels from a luminous source to any
body through a transparent medium. The Maxwell-
Bartholi pressure will act not only upon the source at
its start and upon the body lighted at its arrival, but
also upon the matter of the transparent medium it
traverses. At the moment the luminous wave reaches
a new portion of this medium, the pressure will drive
forward the matter there distributed, and will drive it
back again when the wave leaves that portion. So

MECHANICS AND OPTICS. 225

that the recoil of the source has for its counterpart the
forward motion of the transparent matter that is in
contact with the source ; a Httle later the recoil of
this same matter has for its counterpart the forward
motion of the transparent matter a little further off,
and so on.

Only, is the compensation perfect ? Is the action of
the Maxwell-Bartholi pressure upon the matter of the
transparent medium equal to its reaction upon the
source, and that, whatever that matter may be ? Or
rather, is the action less in proportion as the medium
is less refringent and more rarefied, becoming nil in a
vacuum ? If we admit Hertz's theory, which regards
the ether as mechanically attached to matter, so that
the ether is completely carried along by matter, we
must answer the first and not the second question in
the affirmative.

There would then be perfect compensation, such as
the principle of the equality of action and reaction
demands, even in the least refringent media, even in
the air, even in the interplanetary space, where it
would be sufficient to imagine a bare remnant of
matter, however attenuated. If we admit Lorentz's
theory, on the contrary, the compensation, always
imperfect, is inappreciable in the air, and becomes nil
in space.

But we have seen above that Fizeau's experiment
does not permit of our retaining Hertz's theory. We
must accordingly adopt Lorentz's theory, and conse-
quently give up the principle of reaction.

(1,777) 15

226 SCIENCE AND METHOD.

IV.

Consequences of the Principle of
Relativity.

We have seen above the reasons that incline us to
regard the Principle of Relativity as a general law of
Nature. Let us see what consequences the principle
will lead us to if we regard it as definitely proved.

First of all, it compels us to generalize the hypo-
thesis of Lorentz and Fitz-Gerald on the contraction
of all bodies in the direction of their transposition.
More particularly, we must extend the hypothesis to
the electrons themselves. Abraham considered these
electrons as spherical and undeformable, but we shall
have to admit that the electrons, while spherical when
in repose, undergo Lorentz's contraction when they
are in motion, and then take the form of flattened
ellipsoids.

This deformation of the electrons will have an
influence upon their mechanical properties. In fact,
I have said that the displacement of these charged
electrons is an actual convection current, and that
their apparent inertia is due to the self-induction of
this current, exclusively so in the case of the negative
electrons, but whether exclusively or not in the case of
the positive electrons we do not yet know.

On these terms the compensation will be perfect,
and in conformity with the requirements of the
Principle of Relativity, but only upon two con-
ditions : —

I. That the positive electrons have no real mass,
but only a fictitious electro-magnetic mass ; or at least

MECHANICS AND OPTICS. 227

that their real mass, if it exists, is not constant, but
varies with the velocity, following the same laws as
their fictitious mass.

2. That all forces are of electro-magnetic origin, or
at least that they vary with the velocity, following the
same laws as forces of electro-magnetic origin.

It is Lorentz again who has made this remarkable
synthesis. Let us pause a moment to consider what
results from it. In the first place, there is no more
matter, since the positive electrons have no longer
any real mass, or at least no constant real mass. The
actual principles of our Mechanics, based upon the
constancy of mass, must accordingly be modified.

Secondly, we must seek an electro-magnetic ex-
planation of all known forces, and especially of gravi-
tation, or at least modify the law of gravitation in the
sense that this force must be altered by velocity in
the same way as electro-magnetic forces. We shall
return to this point.

All this appears somewhat artificial at first sight,
and more particularly the deformation of the electrons
seems extremely hypothetical. But the matter can
be presented differently, so as to avoid taking this
hypothesis of deformation as the basis of the argu-
ment. Let us imagine the electrons as material points,
and enquire how their mass ought to vary as a function
of the velocity so as not to violate the Principle of
Relativity. Or rather let us further enquire what should
be their acceleration under the influence of an electric
or magnetic field, so that the principle should not be
violated and that we should return to the ordinary
laws when we imagine the velocity very low. We
shall find that the variations of this mass or of these

228 SCIENCE AND METHOD.

accelerations must occur as if the electron underwent
Lorentz's deformation.

V.

Kaufmann's Experiment.

Two theories are thus presented to us : one in
which the electrons are undeformable, which is Abra-
ham's ; the other, in which they undergo Lorentz's
deformation. In either case their mass grows with
their velocity, becoming infinite when that velocity
becomes equal to that of light ; but the law of the
variation is not the same. The method employed by
Kaufmann to demonstrate the law of variation of the
mass would accordingly seem to give us the means of
deciding experimentally between the two theories.

Unfortunately his first experiments were not suffi-
ciently accurate for this purpose, so much so that he
has thought it necessary to repeat them with more
precautions, and measuring the intensity of the fields
with greater care. In their new form they have shown
Abraham's theory to be right. Accordingly, it would seem
that the Principle of Relativity has not the exact value
we have been tempted to give it, and that we have no
longer any reason for supposing that the positive elec-
trons are devoid of real mass like the negative electrons.

Nevertheless, before adopting this conclusion some
reflexion is necessary. The question is one of such
importance that one would wish to see Kaufmann's
experiment repeated by another experimenter.*

* At the moment of going to press we learn that M. Bucherer has
repeated the experiment, surrounding it with new precautions, and that,
unlike Kaufmann, he has obtained results confirming Lorentz's views.

MECHANICS AND OPTICS. 229

Unfortunately, the experiment is a very delicate
one, and cannot be performed successfully, except by
a physicist as skilful as Kaufmann. All suitable pre-
cautions have been taken, and one cannot well see
what objection can be brought.

There is, nevertheless, one point to which I should
wish to call attention, and that is the measurement of
the electrostatic field, the measurement upon which
everything depends. This field was produced between
the two armatures of a condenser, and between these
two armatures an extremely perfect vacuum had to
be created in order to obtain complete isolation. The
difference in the potential of the two armatures was
then measured, and the field was obtained by dividing
this difference by the distance between the armatures.
This assumes that the field is uniform ; but is this
certain ? May it not be that there is a sudden drop
in the potential in the neighbourhood of one of the
armatures, of the negative armature, for instance?
There may be a difference in potential at the point
of contact between the metal and the vacuum, and it
may be that this difference is not the same on the
positive as on the negative side. What leads me to
think this is the electric valve effect between mercury
and vacuum. It would seem that we must at least
take into account the possibility of this occurring,
however slight the probability may be.

VI.

The Principle of Inertia.

In the new Dynamics the Principle of Inertia is still
true — that is to say, that an isolated electron will have

230 SCIENCE AND METHOD.

a rectilineal and uniform motion. At least it is gener-
ally agreed to admit it, though Lindemann has raised
objections to the assumption. I do not wish to take
sides in the discussion, which I cannot set out here
on account of its extremely difficult nature. In any
case, the theory would only require slight modifications
to escape Lindemann's objections.

We know that a body immersed in a fluid meets
with considerable resistance when it is in motion ; but
that is because our fluids are viscous. In an ideal
fluid, absolutely devoid of viscidity, the body would
excite behind it a liquid stern-wave, a kind of wake.
At the start, it would require a great effort to set it
in motion, since it would be necessary to disturb not
only the body itself but the liquid of its wake. But
once the motion was acquired, it would continue
without resistance, since the body, as it advanced,
would simply carry with it the disturbance of the
liquid, without any increase in the total vis viva of
the liquid. Everything would take place, therefore,
as if its inertia had been increased. An electron
advancing through the ether will behave in the same
way. About it the ether will be disturbed, but this
disturbance will accompany the body in its motion, so
that, to an observer moving with the electron, the
electric and magnetic fields which accompany the
electron would appear invariable, and could only
change if the velocity of the electron happened to
vary. An effort is therefore required to set the
electron in motion, since it is necessary to create the
energy of these fields. On the other hand, once the
motion is acquired, no effort is necessary to maintain
it, since the energy created has only to follow the

MECHANICS AND OPTICS. 231

electron like a wake. This energy, therefore, can only
increase the inertia of the electron, as the agitation
of the liquid increases that of the body immersed in
a perfect fluid. And actually the electrons, at any
rate the negative electrons, have no other inertia but
this.

In Lorentz's hypothesis, the vis viva, which is
nothing but the energy of the ether, is not propor-
tional to v^. No doubt if v is very small, the vis
viva is apparently proportional to v"^, the amount of
momentum apparently proportional to v, and the two
masses apparently constant and equal to one another.
But when the velocity approaches the velocity of light,
the vis viva, the amount of momentum, and the two
masses increase beyond all limit.

In Abraham's hypothesis the expressions are some-
what more complicated, but what has just been said
holds good in its essential features.

Thus the mass, the amount of momentum, and the
vis viva become infinite when the velocity is equal to
that of light. Hence it follows that no body can, by
any possibility, attain a velocity higher than that of
light. And, indeed, as its velocity increases its mass
increases, so that its inertia opposes a more and more
serious obstacle to any fresh increase in its velocity.

A question then presents itself. Admitting the
Principle of Relativity, an observer in motion can have
no means of perceiving his own motion. If, therefore,
no body in its actual motion can exceed the velocity
of light, but can come as near it as we like, it must be the
same with regard to its relative motion in relation to
our observer. Then we might be tempted to reason
as follows : — The observer can attain a velocity of

232 SCIENCE AND METHOD.

120,000 miles a second, the body in its relative motion
in relation to the observer can attain the same velocity ;
its absolute velocity will then be 240,000 miles, which
is impossible, since this is a figure higher than that of
the velocity of light. But this is only an appearance
which vanishes when we take into account Lorentz's
method of valuing local times.

VII.
The Wave of Acceleratlon.

When an electron is in motion it produces a dis-
turbance in the ether which surrounds it. If its
motion is rectilineal and uniform, this disturbance is
reduced to the wake I spoke of in the last section.
But it is not so if the motion is in a curve or not
uniform. The disturbance may then be regarded as
the superposition of two others, to which Langevin
has given the names of wave of velocity and wave of
acceleration.

The wave of velocity is nothing else than the wake
produced by the uniform motion.

As for the wave of acceleration, it is a disturbance
absolutely similar to light waves, which starts from
the electron the moment it undergoes an acceleration,
and is then transmitted in successive spherical waves
with the velocity of light.

Hence it follows that in a rectilineal and uniform
motion there is complete conservation of energy, but
as soon as there is acceleration there is loss of energy,
which is dissipated in the form of light waves and
disappears into infinite space through the ether.

MECHANICS AND OPTICS. 233

Nevertheless, the effects of this wave of acceleration,
and more particularly the corresponding loss of energy,
are negligible in the majority of cases — that is to say,
not only in the ordinary Mechanics and in the motions
of the celestial bodies, but even in the case of the radium
rays, where the velocity, but not the acceleration, is
very great. We may then content ourselves with the
application of the laws of Mechanics, stating that the
force is equal to the product of the acceleration and
the mass, this mass, however, varying with the velocity
according to the laws set forth above. The motion is
then said to be quasi-stationary.

It is not so in all the cases where the acceleration is
great, the chief of which are as follows, (i.) In incan-
descent gases certain electrons take on an oscillatory
motion of very high frequency ; the displacements are
very small, the velocities finite, and the accelerations
very great ; the energy is then communicated to the
ether, and it is for this reason that these gases radiate
light of the same periodicity as the oscillations of the
electron. (2.) Inversely, when a gas receives light,
these same electrons are set in motion with violent
accelerations, and they absorb light. (3.) In Hertz's
excitator, the electrons which circulate in the metallic
mass undergo a sudden acceleration at the moment of
the discharge, and then take on an oscillatory motion
of high frequency. It follows that a part of the energy
is radiated in the form of Hertzian waves. (4.) In an
incandescent metal, the electrons enclosed in the metal
are animated with great velocities. On arriving at the
surface of the metal, which they cannot cross, they are
deflected, and so undergo a considerable acceleration,
and it is for this reason that the metal emits light.

234 SCIENCE AND METHOD.

This I have already explained in Book III., Chap. I.,
Sec. 4. The details of the laws of the emission of
light by dark bodies are perfectly explained by this
hypothesis. (5.) Lastly, when the cathode rays strike
the anticathode, the negative electrons constituting
these rays, which are animated with very great velo-
cities, are suddenly stopped. In consequence of the
acceleration they thus undergo, they produce undula-
tions in the ether. This, according to certain
physicists, is the origin of the Rontgen rays, which are
nothing else than light rays of very short wave length.

III.

THE NEW MECHANICS AND ASTRONOMY.

I.

Gravitation.

Mass may be defined in two ways — firstly, as the
quotient of the force by the acceleration, the true
definition of mass, which is the measure of the body's
inertia ; and secondly, as the attraction exercised by
the body upon a foreign body, by virtue of Newton's
law. We have therefore to distinguish between mass,
the coefficient of inertia, and mass, the coefficient of
attraction. According to Newton's law, there is a
rigorous proportion between these two coefficients, but
this is only demonstrated in the case of velocities to
which the general principles of Dynamics are appli-
cable. Now we have seen that the mass coefficient of
inertia increases with the velocity ; must we conclude
that the mass coefficient of attraction increases
similarly with the velocity, and remains proportional
to the coefficient of inertia, or rather that the
coefficient of attraction remains constant ? This is a
question that we have no means of deciding.

On the other hand, if the coefficient of attraction
depends upon the velocity, as the velocities of bodies

236 SCIENCE AND METHOD.

mutually attracting each other are generally not the
same, how can this coefficient depend upon these two
velocities ?

Upon this subject wc can but form hypotheses, but
we are naturally led to enquire which of these hypo-
theses will be compatible with the Principle of
Relativity. There are a great number, but the only
one I will mention here is Lorentz's hypothesis, which
I will state briefly.

Imagine first of all electrons in repose. Two
electrons of similar sign repel one another, and two
electrons of opposite sign attract one another. Accord-
ing to the ordinary theory, their mutual actions are
proportional to their electric charges. If, therefore,
we have four electrons, two positive, A and A', and
two negative, B and B', and the charges of these four
electrons are the same in absolute value, the repulsion
of A upon A' will be, at the same distance, equal to
the repulsion of B upon B', and also equal to the
attraction of A upon B' or of A' upon B. Then if A and
B are very close to each other, as also A' and B', and
we examine the action of the system A -t- B upon
the system A'-i-B', we shall have two repulsions and
two attractions that are exactly compensated, and the
resultant action will be nil.

Now material molecules must precisely be regarded
as kinds of solar systems in which the electrons circulate,
some positive and others negative, in such a way that
the algebraic sum of all the charges is nil. A material
molecule is thus in all points comparable to the system
A + B I have just spoken of, so that the total
electric action of two molecules upon each other
should be nil.

NEW MECHANICS AND ASTRONOMY. 237

But experience shows us that these molecules attract
one another in accordance with Newtonian gravitation,
and that being so we can form two hypotheses. We
may suppose that gravitation has no connexion with
electrostatic attraction, that it is due to an entirely
different cause, and that it is merely superimposed
upon it ; or else we may admit that there is no pro-
portion between the attractions and the charges, and
that the attraction exercised by a charge + 1 upon a
charge - I is greater than the mutual repulsion of two
charges + i or of two charges - i.

In other words, the electric field produced by the
positive electrons and that produced by the negative
electrons are superimposed and remain distinct. The
positive electrons are more sensitive to the field pro-
duced by the negative electrons than to the field pro-
duced by the positive electrons, and contrariwise for
the negative electrons. It is clear that this hypothesis
somewhat complicates electrostatics, but makes it
include gravitation. It was, in the main, Franklin's
hypothesis.

jNow, what happens if the electrons are in motion ?
The positive electrons will create a disturbance in the
ether, and will give rise in it to an electric field and a
magnetic field. The same will be true of the negative
electrons. The electrons, whether positive or negative,
then receive a mechanical impulse by the action of
these different fields. In the ordinary theory, the
electro-magnetic field due to the motion of the positive
electrons exercises, upon two electrons of opposite
sign and of the same absolute charge, actions that are
equal and of opposite sign. We may, then, without
impropriety make no distinction between the field due

238 SCIENCE AND METHOD.

to the motion of the positive electrons and the field
due to the motion of the negative electrons, and
consider only the algebraic sum of these two fields^
that is to say, the resultant field.

In the new theory, on the contrary, the action upon
the positive electrons of the electro-magnetic field due
to the positive electrons takes place in accordance
with the ordinary laws, and the same is true of the
action upon the negative electrons of the field due
to the negative electrons. Let us now consider the
action of the field due to the positive electrons upon
the negative electrons, or vice versa. It will still
follow the same laws, but with a different coefficient.
Each electron is more sensitive to the field created
by the electrons of opposite denomination than to
the field created by the electrons of the same de-
nomination.

Such is Lorentz's hypothesis, which is reduced to
Franklin's hypothesis for low velocities. It agrees
with Newton's law in the case of these low velocities.
More than that, as gravitation is brought down to
forces of electro-dynamic origin, Lorentz's general
theory will be applicable to it, and consequently the
Principle of Relativity will not be violated.

We see that Newton's law is no longer applicable to
great velocities, and that it must be modified, for
bodies in motion, precisely in the same way as the
laws of Electrostatics have to be for electricity in
motion.

We know that electro-magnetic disturbances are
transmitted with the velocity of light. We shall
therefore be tempted to reject the foregoing theory,
remembering that gravitation is transmitted, according

NEW MECHANICS AND ASTRONOMY. 239

to Laplace's calculations, at least ten million times as
quickly as light, and that consequently it cannot be of
electro-magnetic origin. Laplace's result is well known,
but its significance is generally lost sight of. Laplace
assumed that, if the transmission of gravitation is not
in.stantaneous, its velocity of transmission combines
with that of the attracted body, as happens in the case
of light in the phenomenon of astronomical aberration,
in such a way that the effective force is not directed
along the straight line joining the two bodies, but
makes a small angle with that straight line. This is
quite an individual hypothesis, not very well sub-
stantiated, and in any case entirely different from that
of Lorentz. Laplace's result proves nothing against
Lorentz's theory.
(

n.

Comparison with Astronomical
Observations.

Are the foregoing theories reconcilable with astro-
nomical observations ? To begin with, if we adopt
them, the energy of the planetary motions will be
constantly dissipated by the effect of the wave of
acceleration. It would follow from this that there would
be a constant acceleration of the mean motions of the
planets, as if these planets were moving in a resisting
medium. But this effect is exceedingly slight, much
too slight to be disclosed by the most minute obser-
vation.s. The acceleration of the celestial bodies is
relatively small, so that the effects of the wave of
acceleration are negligible, and the motion may be
regarded as quasi-stationary. It is true that the

240 SCIENCE AND METHOD.

effects of the wave of acceleration are constantly
accumulating, but this accumulation itself is so slow
that it would certainly require thousands of years of
observation before it became perceptible.

Let us therefore make the calculation, taking the
motion as quasi-stationary, and that under the three
following hypotheses : —

A. Admitting Abraham's hypothesis (undeformable
electrons), and retaining Newton's law in its ordinary
form.

B. Admitting Lorentz's hypothesis concerning the
deformation of the electrons, and retaining Newton's
ordinary law.

C. Admitting Lorentz's hypothesis concerning the
electrons, and modifying Newton's law, as in the fore-
going section, so as to make it compatible with the
Principle of Relativity.

It is in the motion of Mercury that the effect will
be most perceptible, because it is the planet that has
the highest velocity. Tisserand formerly made a
similar calculation, admitting Weber's law. I would
remind the reader that Weber attempted to explain
both the electrostatic and the electro-dynamic phe-
nomena, assuming that the electrons (whose name had
not yet been invented) exercise upon each other attrac-
tions and repulsions in the direction of the straight
line joining them, and depending not only upon their
distances, but also upon the first and second deriva-
tives of these distances, that is consequently upon
their velocities and their accelerations. This law of
Weber's, different as it is from those that tend to gain
acceptance to-day, presents none the less a certain
analogy with them.

NEW MECHANICS AND ASTRONOMY. 241

Tisserand found that if the Newtonian attraction
took place in conformity with Weber's law, there would
result, in the perihelion of Mercury, a secular variation
of 14", in the same direction as that which has been
observed and not explained, but smaller, since the
latter is 38".

Let us return to the hypotheses A, B, and C, and study
first the motion of a planet attracted by a fixed centre.
In this case there will be no distinction between
hypotheses B and C, since, if the attracting point is
fixed, the field it produces is a purely electrostatic
field, in which the attraction varies in the inverse
ratio of the square of the distance, in conformity with
Coulomb's electrostatic law, which is identical with
Newton's.

The vis viva equation holds good if we accept the
new definition of vis viva. In the same way the
equation of the areas is replaced by another equivalent.
The moment of the quantity of motion is a constant,
but the quantity of motion must be defined in the
new way.

The only observable effect will be a secular motion
of the perihelion. For this motion we shall get, with
Lorentz's theory, a half, and with Abraham's theory
two-fifths, of what was given by Weber's law.

If we now imagine two moving bodies gravitating
about their common centre of gravity, the effects are
but very slightly different, although the calculations
are somewhat more complicated. The motion of
Mercury's perihelion will then be 7" in Lorentz's
theory, and 5.6" in Abraham's.

The effect is, moreover, proportional to «V, n being
the mean motion of the planet, and a the radius of its
(1.777) 16

242 SCIENCE AND METHOD.

orbit. Accordingly for the planets, by virtue of
Kepler's law, the effect varies in the inverse ratio of
sja^, and it is therefore imperceptible except in the
case of Mercury.

It is equally imperceptible in the case of the Moon,
because, though n is large, a is extremely small.
In short, it is five times as small for Venus, and six
hundred times as small for the Moon, as it is for
Mercury. I would add that as regards Venus and
the Earth, the motion of the perihelion (for the same
angular velocity of this motion) would be much more
difficult to detect by astronomical observations, because
the excentricity of their orbits is much slighter than in
the case of Mercury.

To sum up, the only appreciable effect upon astronom-
ical observations would be a motion of Mercury's peri-
helion, in the same direction as that which has been
observed without being explained, but considerably
smaller.

This cannot be regarded as an argument in favour
of the new Dynamics, since we still have to seek
another explanation of the greater part of the anomaly
connected with Mercury ; but still less can it be
regarded as an argument against it.

III.

Lesage's Theory,

It would be well to set these considerations beside
a theory put forward long ago to explain universal
gravitation. Imagine the interplanetary spaces full of
very tiny corpuscles, travelling in all directions at very

NEW MECHANICS AND ASTRONOMY. 243

high velocities. An isolated body in space will not
be affected apparently by the collisions with these
corpuscles, since the collisions are distributed equally
in all directions. But if two bodies, A and B, are in
proximity, the body B will act as a screen, and inter-
cept a portion of the corpuscles, which, but for it,
would have struck A. Then the collisions received
by A from the side away from B will have no counter-
part, or will be only imperfectly compensated, and will
drive A towards B.

Such is Lesage's theory, and we will discuss it first
from the point of view of ordinary mechanics. To begin
with, how must the collisions required by this theory
occur? Must it be in accordance with the laws of
perfectly elastic bodies, or of bodies devoid of elasticity,
or in accordance with some intermediate law ? Lesage's
corpuscles cannot behave like perfectly elastic bodies,
for in that case the effect would be nil, because the
corpuscles intercepted by the body B would be replaced
by others which would have rebounded from B, and
calculation proves that the compensation would be
perfect.

The collision must therefore cause a loss of energy
to the corpuscles, and this energy should reappear in
the form of heat. But what would be the amount of
heat so produced? We notice that the attraction
passes through the body, and we must accordingly
picture the Earth, for instance, not as a complete
screen, but as composed of a very large number of
extremely small spherical molecules, acting individually
as little screens, but allowing Lesage's corpuscles to
travel freely between them. Thus, not only is the
Earth not a complete screen, but it is not even a

244 SCIENCE AND METHOD.

strainer, since the unoccupied spaces are much larger
than the occupied. To reahze this, we must remem-
ber that Laplace demonstrated that the attraction, in
passing through the Earth, suffers a loss, at the very-
most, of a ten-millionth part, and his demonstration is
perfectly satisfactory. Indeed, if the attraction were
absorbed by the bodies it passes through, it would no
longer be proportional to their masses ; it would be
relatively weaker for large than for small bodies, since
it would have a greater thickness to traverse. The
attraction of the Sun for the Earth would therefore be
relatively weaker than that of the Sun for the Moon,
and a very appreciable inequality in the Moon's motion
would result. We must therefore conclude, if we adopt
Lesage's theory, that the total surface of the spherical
molecules of which the Earth is composed is, at the
most, the ten-millionth part of the total surface of the
Earth.

Darwin proved that Lesage's theory can only lead
exactly to Newton's law if we assume the corpuscles
to be totally devoid of elasticity. The attraction
exercised by the Earth upon a mass i at a distance i
will then be proportional both to S, the total surface
of the spherical molecules of which it is composed, to
f, the velocity of the corpuscles, and to the square
root of p, the density of the medium formed by the
corpuscles. The heat produced will be proportional
to S, to the density p, and to the cube of the
velocity v.

But we must take account of the resistance ex-
perienced by a body moving in such a medium. It
cannot move, in fact, without advancing towards certain
collisions, and on the other hand retreating before

NEW MECHANICS AND ASTRONOMY. 245

those that come from the opposite direction, so that
the compensation realized in a state of repose no longer
exists. The calculated resistance is proportional to S,
to p, and to v. Now we know that the heavenly bodies
move as if they met with no resistance, and the pre-
cision of the observations enables us to assign a limit
to the resistance.

This resistance varying as Spv, while the attraction
varies as S Jpv, we see that the relation of the resist-
ance to the square of the attraction is in inverse ratio
of the product Sv.

We get thus an inferior limit for the product Sv.
We had already a superior limit for S (by the absorp-
tion of the attraction by the bodies it traverses). We
thus get an inferior limit for the velocity v, which must
be at least equal to 24.10" times the velocity of light.

From this we can deduce p and the amount of heat
produced. This would suffice to elevate the tempera-
ture 10^ degrees a second. In any given time the
Earth would receive io^° as much heat as the Sun
emits in the same time, and I am not speaking of
the heat that reaches the Earth from the Sun, but of
the heat radiated in all directions. It is clear that
the Earth could not long resist such conditions.

We shall be led to results no less fantastic if, in
opposition to Darwin's views, we endow Lesage's
corpuscles with an elasticity that is imperfect but
not nil. It is true that the vis viva of the corpuscles
will not then be entirely converted into heat, but the
attraction produced will equally be less, so that it
will only be that portion of the vis viva converted
into heat that will contribute towards the production
of attraction, and so we shall get the same result. A

246 SCIENCE AND METHOD.

judicious use of the theorem of virial will enable us
to realize this.

We may transform Lesage's theory by suppressing
the corpuscles and imagining the ether traversed in
all directions by luminous waves coming from all
points of space. When a material object receives a
luminous wave, this wave exercises upon it a mechani-
cal action due to the Maxwell-Bartholi pressure, just as
if it had received a blow from a material projectile.
The waves in question may accordingly play the part
of Lesage's corpuscles. This is admitted, for instance,
by M. Tommasina.

This does not get over the difficulties. The velocity
of transmission cannot be greater than that of light,
and we are thus brought to an inadmissible figure for
the resistance of the medium. Moreover, if the light
is wholly reflected, the effect is nil, just as in the
hypothesis of the perfectly elastic corpuscles. In
order to create attraction, the light must be partially
absorbed, but in that case heat will be produced. The
calculations do not differ essentially from those made
in regard to Lesage's ordinary theory, and the result
retains the same fantastic character.

On the other hand, attraction is not absorbed, or
but very slightly absorbed, by the bodies it traverses,
while this is not true of the light we know. Light
that would produce Newtonian attraction would re-
quire to be very different from ordinary light, and
to be, for instance, of very short wave length. This
makes no allowance for the fact that, if our eyes were
sensible to this light, the whole sky would appear
much brighter than the Sun, so that the Sun would
be seen to stand out in black, as otherwise it would

NEW MECHANICS AND ASTRONOMY. 247

repel instead of attracting us. For all these reasons,
the light that would enable us to explain attraction
would require to be much more akin to Rontgen's
X rays than to ordinary light.

Furthermore, the X rays will not do. However
penetrating they may appear to us, they cannot pass
through the whole Earth, and we must accordingly
imagine X' rays much more penetrating than the
ordinary X rays. Then a portion of the energy of
these X' rays must be destroyed, as otherwise there
would be no attraction. If we do not wish it to be
transformed into heat, which would lead to the pro-
duction of an enormous heat, we must admit that it
is radiated in all directions in the form of secondary
rays, which we may call X" rays, which must be much
more penetrating even than the X' rays, failing which
they would in their turn disturb the phenomena of
attraction.

Such are the complicated hypotheses to which we
are led when we seek to make Lesage's theory tenable.

But all that has been said assumes the ordinary
laws of Mechanics. Will the case be stronger if we
admit the new Dynamics ? And in the first place, can
we preserve the Principle of Relativity ? First let us
give Lesage's theory its original form, and imagine
space furrowed by material corpuscles. If these
corpuscles were perfectly elastic, the laws of their
collision would be in conformity with this Principle
of Relativity, but we know that in that case their effect
would be nil. We must therefore suppose that these
corpuscles are not elastic ; and then it is difficult to
imagine a law of collision compatible with the Prin-
ciple of Relativity. Besides, we should still get a

248 SCIENCE AND METHOD.

considerable production of heat, and, notwithstanding
that, a very appreciable resistance of the medium.

If we suppress the corpuscles and return to the
hypothesis of the Maxwell-Bartholi pressure, the
difficulties are no smaller. It is this that tempted
Lorentz himself in his Memoire to the Academy of
Sciences of Amsterdam of the 25th of April 1900.

Let us consider a system of electrons immersed in
an ether traversed in all directions by luminous waves.
One of these electrons struck by one of these waves
will be set in vibration. Its vibration will be syn-
chronous with that of the light, but there may be a
difference of phase, if the electron absorbs a part ol
the incident energy. If indeed it absorbs energy, it
means that it is the vibration of the ether that keeps
the electron in vibration, and the electron must ac-
cordingly be behind the ether. An electron in motion
may be likened to a convection current, therefore
every magnetic field, and particularly that due to the
luminous disturbance itself, must exercise a mechani-
cal action upon the electron. This action is very
slight, and more than that, it changes its sign in the
course of the period ; nevertheless the mean action
is not nil if there is a difference of phase between
the vibrations of the electron and those of the ether
The mean action is proportional to this difference,
and consequently to the energy absorbed by the
electron.

I cannot here enter into the details of the calcula-
tions. I will merely state that the final result is an
attraction between any two electrons varying in the
inverse ratio of the square of the distance, and pro-
portional to the energy absorbed by the two electrons.

NEW MECHANICS AND ASTRONOMY. 249

There cannot, therefore, be attraction without ab-
sorption of light, and consequently without production
of heat, and it is this that determined Lorentz to
abandon this theory, which does not differ funda-
mentally from the Lesage-Maxwell-Bartholi theory.
He would have been still more alarmed if he had
pushed the calculations to the end, for he would have
found that the Earth's temperature must increase 10^'
degrees a second.

IV.

Conclusions.

I have attempted to give in a few words as com-
plete an idea as possible of these new doctrines ; I
have tried to explain how they took birth, as other-
wise the reader would have had cause to be alarmed
by their boldness. The new theories are not yet
demonstrated — they are still far from it, and rest
merely upon an aggregation of probabilities suffi-
ciently imposing to forbid our treating them with
contempt. Further experiments will no doubt teach
us what we must finally think of them. The root of
the question is in Kaufmann's experiment and such
as may be attempted in verification of it.

In conclusion, may I be permitted to express a
wish? Suppose that in a few years from now these
theories are subjected to new tests and come out trium-
phant, our secondary education will then run a great
risk. Some teachers will no doubt wish to make room
for the new theories. Novelties are so attractive, and
it is so hard not to appear sufficiently advanced ! At
least they will wish to open up prospects to the chil-

250 SCIENCE AND METHUD.

dren, who will be warned, before they are taught the
ordinary mechanics, that it has had its day, and that
at most it was only good for such an old fogey as
Laplace. Then they will never become familiar with
the ordinary mechanics.

Is it good to warn them that it is only approximate ?
Certainly, but not till later on ; when they are steeped
to the marrow in the old laws, when they have got
into the way of thinking in them, and are no longer
in danger of unlearning them, then they may safely
be shown their limitations.

It is with the ordinary mechanics that they have to
live ; it is the only kind they will ever have to apply.
Whatever be the progress of motoring, our cars will
never attain the velocities at which its laws cease to
be true. The other is only a luxury, and we must not
think of luxury until there is no longer any risk of
its being detrimental to what is necessary.

BOOK IV.
ASTRONOMICAL SCIENCE.

I.

THE MILKY WAY AND THE THEORY
OF GASES.

The considerations I wish to develop here have so
far attracted but little attention from astronomers. I
have merely to quote an ingenious idea of Lord
Kelvin's, which has opened to us a new field of re-
search, but still remains to be followed up. Neither
have I any original results to make known, and all
that I can do is to give an idea of the problems that
are presented, but that no one, up to this time, has
made it his business to solve.

Every one knows how a great number of modern
physicists represent the constitution of gases. Gases
are composed of an innumerable multitude of mole-
cules which are animated with great velocities, and
cross and re-cross each other in all directions. These
molecules probably act at a distance one upon another,
but this action decreases very rapidly with the distance,
so that their trajectories remain apparently rectilineal,
and only cease to be so when two molecules happen
to pass sufificiently close to one another, in which case
their mutual attraction or repulsion causes them to
deviate to right or left. This is what is sometimes
called a collision, but we must not understand this

254 SCIENCE AND METHOD.

word collision in its ordinary sense ; it is not necessary
that the two molecules should come into contact, but
only that they should come near enough to each other
for their mutual attraction to become perceptible.
The laws of the deviation they undergo are the same
as if there had been an actual collision.

It seems at first that the orderless collisions of this
innumerable dust can only engender an inextricable
chaos before which the analyst must retire. But the
law of great numbers, that supreme law of chance,
comes to our assistance. In face of a semi-disorder
we should be forced to despair, but in extreme disorder
this statistical law re-establishes a kind of average or
mean order in which the mind can find itself again.
It is the study of this mean order that constitutes the
kinetic theory of gases ; it shows us that the velocities
of the molecules are equally distributed in all directions,
that the amount of these velocities varies for the dif-
ferent molecules, but that this very variation is subject
to a law called Maxwell's law. This law teaches us
how many molecules there are animated with such and
such a velocity. As soon as a gas departs from this
law, the mutual collisions of the molecules tend to
bring it back promptly, by modifying the amount
and direction of their velocities. Physicists have
attempted, and not without success, to explain in this
manner the experimental properties of gases — for
instance, Mariotte's (or Boyle's) law.

Consider now the Milky Way. Here also we see
an innumerable dust, only the grains of this dust are
no longer atoms but stars; these grains also move
with great velocities, they act at a distance one upon
another, but this action is so slight at great distances

THE MILKY WAY. 255

that their trajectories are rectiHneal ; nevertheless, from
time to time, two of them may come near enough
together to be deviated from their course, like a comet
that passed too close to Jupiter. In a word, in the eyes
of a giant, to whom our Suns were what our atoms
are to us, the Milky Way would only look like a
bubble of gas.

Such was Lord Kelvin's leading idea. What can
we draw from this comparison, and to what extent is
it accurate ? This is what we are going to enquire
into together ; but before arriving at a definite con-
clusion, and without wishing to prejudice the question,
we anticipate that the kinetic theory of gases will be,
for the astronomer, a model which must not be
followed blindly, but may afford him useful inspira-
tion. So far celestial mechanics has attacked only
the Solar System, or a few systems of double stars.
It retired before the aggregations presented by the
Milky Way, or clusters of stars, or resoluble nebulae,
because it saw in them only chaos. But the Milky
Way is no more complicated than a gas ; the statistical
methods based upon the calculation of probabilities
applicable to the one are also applicable to the other.
Above all, it is important to realize the resemblance
and also the difference between the two cases.

Lord Kelvin attempted to determine by this means
the dimensions of the Milky Way. For this purpose
we are reduced to counting the stars visible in our
telescopes, but we cannot be sure that, behind the
stars we see, there are not others which we do not
see ; so that what we should measure in this manner
would not be the size of the Milky Way, but the scope
of our instruments. The new theory will offer us other

256 SCIENCE AND METHOD.

resources. We know, indeed, the motions of the stars
nearest to us, and we can form an idea of the amount
and direction of their velocities. If the ideas ex-
pounded above are correct, these velocities must follow
Maxwell's law, and their mean value will teach us, so
to speak, what corresponds with the temperature of
our fictitious gas. But this temperature itself depends
upon the dimensions of our gaseous bubble. How, in
fact, will a gaseous mass, left undisturbed in space,
behave, if its elements are attracted in accordance
with Newton's law? It will assume a spherical shape ;
further, in consequence of gravitation, the density will
be greater at the centre, and the pressure will also
increase from the surface to the centre on account of
the weight of the exterior parts attracted towards the
centre ; lastly, the temperature will increase towards
the centre, the temperature and the pressure being
connected by what is called the adiabatic law, as is
the case in the successive layers of our atmosphere.
At the surface itself the pressure will be nil, and the
same will be true of the absolute temperature, that is
to say, of the velocity of the molecules.

Here a question presents itself I have spoken of
the adiabatic law, but this law is not the same for all
gases, .since it depends upon the proportion of their
two specific heats. For air and similar gases this pro-
portion is 1.41 ; but is it to air that the Milky Way
should be compared ? Evidently not. It should be
regarded as a monatomic gas, such as mercury vapour,
argon, or helium — that is to say, the proportion of the
specific heats should be taken as equal to 1.66. And,
indeed, one of our molecules would be, for instance, the
Solar System ; but the planets are very unimportant

THE MILKY WAY. 257

personages and the Sun alone counts, so that our
molecule is clearly monatomic. And even if we take
a double star, it is probable that the action of a foreign
star that happened to approach would become suffi-
ciently appreciable to deflect the general motion of
the system long before it was capable of disturbing
the relative orbits of the two components. In a word,
the double star would behave like an indivisible atom.

However this may be, the pressure, and consequently
the temperature, at the centre of the gaseous sphere
are proportional to the size of the sphere, since the
pressure is increased by the weight of all the over-
lying strata. We may suppose that we are about at
the centre of the Milky Way, and, by observing the
actual mean velocity of the stars, we shall know what
corresponds to the central temperature of our gaseous
sphere and be able to determine its radius.

We may form an idea of the result by the following
considerations. Let us make a simple hypothesis.
The Milky Way is spherical, and its masses are dis-
tributed homogeneously : it follows that the stars
describe ellipses having the same centre. If we sup-
pose that the velocity drops to nothing at the surface,
we can calculate this velocity at the centre by the
equation of vis viva. We thus find that this velocity
is proportional to the radius of the sphere and the
square root of its density. If the mass of this sphere
were that of the Sun, and its radius that of the ter-
restrial orbit, this velocity, as is easily seen, would be
that of the Earth upon its orbit. But in the case we
have supposed, the Sun's mass would have to be
distributed throughout a sphere with a radius 1,000,000
times as great, this radius being the distance of the

(1,777) 17

258 SCIENCE AND METHOD.

nearest stars. The density is accordingly lO^' times
as small ; now the velocities are upon the same scale,
and therefore the radius must be lO* as great, or l,ooo
times the distance of the nearest stars, which would
give about a thousand million stars in the Milky Way.

But you will tell me that these hypotheses are very
far removed from reality. Firstly, the Milky Way is
not spherical (we shall soon return to this point) ; and
secondly, the kinetic theory of gases is not compatible
with the hypothesis of a homogeneous sphere. But if
we made an exact calculation in conformity with this
theory, though we should no doubt obtain a different
result, it would still be of the same order of magni-
tude : now in such a problem the data are so uncertain
that the order of magnitude is the only end we can
aim at.

And here a first observation suggests itself Lord
Kelvin's result, which I have just obtained again by
an approximate calculation, is in marked accordance
with the estimates that observers have succeeded in
making with their telescopes, so that we must conclude
that we are on the point of piercing the Milky Way.
But this enables us to solve another question. There
are the stars we see because they shine, but might
there not be dark stars travelling in the interstellar
spaces, whose existence might long remain unknown?
But in that case, what Lord Kelvin's method gives us
would be the total number of stars, including the dark
stars, and as his figure compares with that given by
the telescope, there is not any dark matter, or at least
not as much dark as there is brilliant matter.

Before going further we must consider the problem
under another aspect. Is the Milky Way, thus con-

THE MILKY WAY. 259

stituted, really the image of a gas properly so called ?
We know that Crookes introduced the notion of a
fourth state of matter, in which gases, becoming too
rarefied, are no longer true gases, but become what he
calls radiant matter. In view of the slightness of its
density, is the Milky Way the image of gaseous or of
radiant matter? It is the consideration of what is
called Xh^ free path of the molecules that will supply
the answer.

A gaseous molecule's trajectory may be regarded
as composed of rectilineal segments connected by
very small arcs corresponding with the successive
collisions. The length of each of these segments is
what is called the free path. This length is obviously
not the same for all the segments and for all the
molecules ; but we may take an average, and this is
called the mean free path, and its length is in inverse
proportion to the density of the gas. Matter will be
radiant when the mean path is greater than the
dimensions of the vessel in which it is enclosed, so
that a molecule is likely to traverse the whole vessel
in which the gas is enclosed, without experiencing a
collision, and it remains gaseous when the contrary
is true. It follows that the same fluid may be radiant
in a small vessel and gaseous in a large one, and this
is perhaps the reason why, in the case of Crookes'
tubes, a more perfect vacuum is required for a larger
tube.

What, then, is the case of the Milky Way? It is
a mass of gas of very low density, but of very great
dimensions. Is it likely that a star will traverse it
without meeting with any collision — that is to say,
without passing near enough to another star to be

26o SCIENCE AND METHOD.

appreciably diverted from its course? What do we
mean by near enough ? This is necessarily somewhat
arbitrary, but let us assume that it is the distance
from the Sun to Neptune, which represents a deviation
of about ten degrees. Supposing, now, that each of
our stars is surrounded by a danger sphere of this
radius, will a straight line be able to pass between
these spheres ? At the mean distance of the stars of
the Milky Way, the radius of these spheres will sub-
tend an angle of about a tenth of a second, and we
have a thousand million stars. If we place upon the
celestial sphere a thousand million little circles with
radius of a tenth of a second, will these circles cover
the celestial sphere many times over? Far from it.
They will only cover a sixteen-thousandth part. Thus
the Milky Way is not the image of gaseous matter,
but of Crookes' radiant matter. Nevertheless, as there
was very little precision in our previous conclusions,
we do not require to modify them to any appreciable
extent.

But there is another difficulty. The Milky Way is
not spherical, and up to now we have reasoned as
though it were so, since that is the form of equilibrium
that would be assumed by a gas isolated in space.
On the other hand, there are clusters of stars whose
form is globular, to which what we have said up to
this point would apply better. Herschel had already
applied himself to the explanation of their remarkable
appearance. He assumed that the stars of these
clusters are uniformly distributed in such a way that
a cluster is a homogeneous sphere. Each star would
then describe an ellipse, and all these orbits would be
accomplished in the same time, so that at the end of

THE MILKY WAY. 261

a certain period the cluster would return to its original
configuration, and that configuration would be stable.
Unfortunately the clusters do not appear homogene-
ous. We observe a condensation at the centre, and
we should still observe it even though the sphere were
homogeneous, since it is thicker at the centre, but it
would not be so marked. A cluster may, therefore,
better be compared to a gas in adiabatic equilibrium
which assumes a spherical form, because that is the
figure of equilibrium of a gaseous mass.

But, you will say, these clusters are much smaller
than the Milky Way, of which it is even probable that
they form a part, and although they are denser, they
give us rather something analogous to radiant matter.
Now, gases only arrive at their adiabatic equilibrium
in consequence of innumerable collisions of the mole-
cules. We might perhaps find a method of reconciling
these facts. Suppose the stars of the cluster have just
sufficient energy for their velocity to become nil when
they reach the surface. Then they may traverse the
cluster without a collision, but on reaching the surface
they turn back and traverse it again. After traversing
it a great number of times, they end by being deflected
by a collision. Under these conditions we should still
have a matter that might be regarded as gaseous. If
by chance there were stars in the cluster with greater
velocities, they have long since emerged from it, and
have left it never to return. For all these reasons it
would be interesting to examine the known clusters
and try to get an idea of the law of their densities and
see if it is the adiabatic law of gases.

But to return to the Milky Way. It is not spherical,
and would be more properly represented as a flattened

262 SCIENCE AND METHOD.

disc. It is clear, then, that a mass starting without
velocity from the surface will arrive at the centre with
varying velocities, according as it has started from the
surface in the neighbourhood of the middle of the disc
or from the edge of the disc. In the latter case the
velocity will be considerably greater.

Now up to the present we have assumed that the
individual velocities of the stars, the velocities we
observe, must be comparable to those that would be
attained by such masses. This involves a certain
difficulty. I have given above a value for the dimen-
sions of the Milky Way, and I deduced it from the
observed individual velocities, which are of the same
order of magnitude as that of the Earth upon its orbit ;
but what is the dimension I have thus measured ? Is
it the thickness or the radius of the disc ? It is, no
doubt, something between the two, but in that case
what can be said of the thickness itself, or of the
radius of the disc ? Data for making the calculation
are wanting, and I content myself with foreshadowing
the possibility of basing at least an approximate
estimate upon a profound study of the individual
motions.

Now, we find ourselves confronted by two hypo-
theses. Either the stars of the Milky Way are
animated with velocities which are in the main
parallel with the Galactic plane, but otherwise dis-
tributed uniformly in all directions parallel with
this plane. If so, observation of the individual
motions should reveal a preponderance of components
parallel with the Milky Way. This remains to be
ascertained, for I do not know that any systematic
study has been made from this point of view. On the

THE MILKY WAY. 263

other hand, such an equiHbrium could only be pro-
visional, for, in consequence of collisions, the molecules
— I mean the stars — will acquire considerable velocities
in a direction perpendicular to the Milky Way, and
will end by emerging from its plane, so that the
system will tend towards the spherical form, the only
figure of equilibrium of an isolated gaseous mass.

Or else the whole system is animated with a common
rotation, and it is for this reason that it is flattened,
like the Earth, like Jupiter, and like all rotating
bodies. Only, as the flattening is considerable, the
rotation must be rapid. Rapid, no doubt, but we
must understand the meaning of the word. The
density of the Milky Way is 10^^ times as low as the
Sun's ; a velocity of revolution v'lo'-^ times smaller
than the Sun's would therefore be equivalent in its
case from the point of view of the flattening. A
velocity 10^- times as slow as the Earth's, or the
thirtieth of a second of arc in a century, will be a
very rapid revolution, almost too rapid for stable
equilibrium to be possible.

In this hypothesis, the observable individual motions
will appear to us uniformly distributed, and there will
be no more preponderance of the components parallel
with the Galactic plane. They will teach us nothing
with respect to the rotation itself, since we form part
of the rotating system. If the spiral nebulae are other
Milky Ways foreign to ours, they are not involved
in this rotation, and we might study their individual
motions. It is true that they are very remote, for if
a nebula has the dimensions of the Milky Way, and
if its apparent radius is, for instance, 20", its distance
is 10,000 times the radius of the Milky Way.

264 SCIENCE AND METHOD.

But this does not matter, since it is not about the
rectilinear motion of our system that we ask them for
information, but about its rotation. The fixed stars,
by their apparent motion, disclose the diurnal rotation
of the Earth, although their distance is immense.
Unfortunately, the possible rotation of the Milky
Way, rapid as it is, relatively speaking, is very slow
from the absolute point of view, and, moreover, bear-
ings upon nebula; cannot be very exact. It would
accordingly require thousands of years of observation
to learn anything.

However it be, in this second hypothesis, the figure
of the Milky Way would be a figure of ultimate
equilibrium.

I will not discuss the relative value of these two
hypotheses at any greater length, because there is a
third which is perhaps more probable. We know that
among the irresoluble nebulae several families can be
distinguished, the irregular nebulae such as that in
Orion, the planetary and annular nebulae, and the
spiral nebulae. The spectra of the first two families
have been determined, and prove to be discontinuous.
These nebulae are accordingly not composed of stars.
Moreover, their distribution in the sky appears to
depend upon the Milky Way, whether they show a
tendency to be removed from it, or on the contrary
to approach it, and therefore they form part of the
system. On the contrary, the spiral nebulae are
generally considered as independent of the Milky
Way : it is assumed that they are, like it, composed
of a multitude of stars ; that they are, in a word,
other Milky Ways very remote from ours. The work
recently done by Stratonoff tends to make us look

THE MILKY WAY. 265

upon the Milky Way itself as a spiral nebula, and this
is the third hypothesis of which I wished to speak.

How are we to explain the very singular appear-
ances presented by the spiral nebula, which are too
regular and too constant to be due to chance? To
begin with, it is sufficient to cast one's eyes upon one
of these figures to see that the mass is in rotation, and
we can even see the direction of the rotation : all the
spiral radii are curved in the same direction, and it is
evident that it is the advancing wing hanging back
upon the pivot, and that determines the direction of
the rotation. But that is not all. It is clear that
these nebulae cannot be likened to a gas in repose,
nor even to a gas in relative equilibrium under the
domination of a uniform rotation ; they must be
compared to a gas in permanent motion in which
internal currents rule.

Suppose, for example, that the rotation of the central
nucleus is rapid (you know what I mean by this word),
too rapid for stable equilibrium. Then at the equator
the centrifugal force will prevail over the attraction,
and the stars will tend to escape from the equator,
and will form divergent currents. But as they recede,
since their momentum ot rotation remains constant
and the radius vector increases, their angular velocity
will diminish, and it is for this reason that the advan-
cing wing appears to hang back.

Under this aspect of the case there would not be
a true permanent motion, for the central nucleus
would constantly lose matter which would go out
never to return, and would be gradually exhausted.
But we may modify the hypothesis. As it recedes,
the star loses its velocity and finally stops. At that

266 SCIENCE AND METHOD.

moment the attraction takes possession of it again and
brings it back towards the nucleus, and accordingly
there will be centripetal currents. We must assume
that the centripetal currents are in the first rank and
the centrifugal currents in the second rank, if we take
as a comparison a company in battle executing a
turning movement. Indeed the centrifugal force must
be compensated by the attraction exercised by the
central layers of the swarm upon the exterior layers.

Moreover, at the end of a certain length of time,
a permanent status is established. As the swarm
becomes curved, the attraction exercised by the
advancing wing upon the pivot tends to retard the
pivot, and that of the pivot upon the advancing wing
tends to accelerate the advance of this wing, whose
retrograde motion increases no further, so that finally
all the radii end by revolving at a uniform velocity.
We may nevertheless assume that the rotation of the
nucleus is more rapid than that of the radii.

One question remains. Why do these centripetal
and centrifugal swarms tend to concentrate into radii
instead of being dispersed more or less throughout,
and why are these radii regularly distributed ? The
reason for the concentration of the swarms is the
attraction exercised by the swarms already existing
upon the stars that emerge from the nucleus in their
neighbourhood. As soon as an inequality is produced,
it tends to be accentuated by this cause.

Why are the radii regularly distributed ? This is
a more delicate matter. Suppose there is no rotation,
and that all the stars are in two rectangular planes in
such a way that their distribution is symmetrical in
relation to the two planes. By symmetry, there would

THE MILKY WAY. 267

be no reason for their emerging from the planes nor
for the symmetry to be altered. This configuration
would accordingly give equilibrium, but it would be an
unstable equilibrium.

If there is rotation on the contrary, we shall get
an analogous configuration of equilibrium with four
curved radii, equal to one another, and intersecting at
an angle of 90°, and if the rotation is sufficiently
rapid, this equilibrium may be stable.

I am not in a position to speak more precisely. It
is enough for me to foreshadow the possibility that
these spiral forms may, perhaps, some day be ex-
plained by the help only of the law of gravitation and
statistical considerations, recalling those of the theory
of gases.

What I have just said about internal currents shows
that there might be some interest in a systematic
study of the aggregate of the individual motions.
This might be undertaken a hundred years hence,
when the second edition of the astrographic chart of
the heavens is brought out and compared with the
first, the one that is being prepared at present.

But I should wish, in conclusion, to call your
attention to the question of the age of the Milky Way
and the nebulae. We might form an idea of this age
if we obtained confirmation of what we have imagined
to be the case. This kind of statistical equilibrium of
which gases supply the model, cannot be established
except as a consequence of a great number of col-
lisions. If these collisions are rare, it can only be
produced after a very long time. If actually the
Milky Way (or at least the clusters that form par<-
of it), and if the nebulae have obtained this equilibrium,

268 SCIENCE AND METHOD.

it is because they are very ancient, and we shall get an
inferior limit for their age. We shall likewise obtain a
superior limit, for this equilibrium is not ultimate and
cannot last for ever. Our spiral nebula; would be com-
parable to gases animated with permanent motions.
But gases in motion are viscous and their velocities
are finally expended. What corresponds in this case
to viscidity (and depends upon the chances of collision
of the molecules) is exceedingly slight, so that the
actual status may continue for a very long time, but
not for ever, so that our Milky Ways cannot be ever-
lasting nor become infinitely ancient.

But this is not all. Consider our atmosphere. At
the surface an infinitely low temperature must prevail,
and the velocity of the molecules is in the neighbour-
hood of zero. But this applies only to the mean
velocity. In consequence of collisions, one of these
molecules may acquire (rarely, it is true) an enormous
velocity, and then it will leave the atmosphere, and
once it has left it, it will never return. Accordingly
our atmosphere is being exhausted exceedingly slowly.
By the same mechanism the Milky Way will also lose
a star from time to time, and this likewise limits its
duration.

Well, it is certain that if we calculate the age of
the Milky Way by this method, we shall arrive at
enormous figures. But here a difficulty presents itself
Certain physicists, basing their calculations on other
considerations, estimate that Suns can have but an ephe-
meral existence of about fifty millions of years, while
our minimum would be much greater than that. Must
we believe that the evolution of the Milky Way began
while matter was still dark? But how have all the

THE MILKY WAY. 269

stars that compose it arrived at the same time at the
adult period, a period which lasts for so short a time ?
Or do they all reach it successively, and are those that
we see only a small minority as compared with those
that are extinct or will become luminous some day ?
But how can we reconcile this with what has been said
above about the absence of dark matter in any con-
siderable proportion ? Must we abandon one of the
two hypotheses, and, if so, which ? I content myself
with noting the difficulty, without pretending to solve
it, and so I end with a great mark of interrogation.
Still, it is interesting to state problems even though
their solution seems very remote.

II.

FRENCH GEODESY.*

Every one understands what an interest we have in
knowing- the shape and the dimensions of our globe,
but some people would perhaps be astonished at the
precision that is sought for. Is this a useless luxury ?
What is the use of the efforts geodesists devote to it ?

If a Member of Parliament were asked this question,
I imagine he would answer : " I am led to think that
Geodesy is one of the most useful of sciences, for it is
one of those that cost us most money." I shall
attempt to give a somewhat more precise answer.

The great works of art, those of peace as well as
those of war, cannot be undertaken without long
studies, which save many gropings, miscalculations,
and useless expense. These studies cannot be made
without a good map. But a map is nothing but a
fanciful picture, of no value whatever if we try to
construct it without basing it upon a solid framework.
As well might we try to make a human body stand
upright with the skeleton removed.

Now this framework is obtained by geodetic meas-

* Throughout this chapter the author is speaking of the work of his
own countrymen. In the translation such words as "we" and "our"
have been avoided, as far as possible ; but where they occur, they must
be understood to refer to P'rance and not to England.

FRENCH GEODESY. 271

urements. Therefore without Geodesy we can have
no good map, and without a good map no great
public works.

These reasons would no doubt be sufficient to justify
much expense, but they are reasons calculated to con-
vince practical men. It is not upon these that we
should insist here ; there are higher and, upon the
whole, more important reasons.

We will therefore state the question differently :
Can Geodesy make us better acquainted with nature ?
Does it make us understand its unity and harmony?
An isolated fact indeed is but of little worth, and the
conquests of science have a value only if they prepare
new ones.

Accordingly, if we happened to discover a little
hump upon the terrestrial ellipsoid, this discovery
would be of no great interest in itself It would
become precious on the contrary if, in seeking for the
cause of the hump, we had the hope of penetrating
new secrets.

So when Maupertuis and La Condamine in the
eighteenth century braved such diverse climates, it
was not only for the sake of knowing the shape of our
planet, it was a question of the system of the whole
World. If the Earth was flattened, Newton was
victorious, and with him the doctrine of gravitation
and the whole of the modern celestial mechanics.
And to-day, a century and a half since the victory
of the Newtonians, are we to suppose that Geodesy
has nothing more to teach us ? We do not know
what there is in the interior of the globe. Mine
shafts and borings have given us some knowledge
of a stratum one or two miles deep — that is to say,

272 SCIENCE AND METHOD.

the thousandth part of the total mass ; but what is
there below that?

Of all the extraordinary voyages dreamed of by
Jules Verne, it was perhaps the voyage to the centre of
the Earth that led us to the most unexplored regions.

But those deep sunk rocks that we cannot reach,
exercise at a distance the attraction that acts upon
the pendulum and deforms the terrestrial spheroid.
Geodesy can therefore weigh them at a distance, so to
speak, and give us information about their disposition.
It will thus enable us really to see those mysterious
regions which Jules Verne showed us only in imagi-
nation.

This is not an empty dream. By comparing all the
measurements, M. Faye has reached a result well
calculated to cause surprise. In the depths beneath
the oceans, there are rocks of very great density, while,
on the contrary, beneath the continents there seem
to be empty spaces.

New observations will perhaps modify these con-
clusions in their details, but our revered master has, at
any rate, shown us in what direction we must push
our researches, and what it is that the geodesist can
teach the geologist who is curious about the interior
constitution of the Earth, and what material he can
supply to the thinker who wishes to reflect upon the
past and the origin of this planet.

Now why have I headed this chapter French
Geodesy? It is because, in different countries, this
science has assumed, more perhaps than any other,
a national character ; and it is easy so see the reason
for this.

There must certainly be rivalries. Scientific rivalries

FRENCH GEODESY. 273

are always courteous, or, at least, almost always. In
any case they are necessary, because they are always
fruitful.

Well, in these enterprises that demand such long
efforts and so many collaborators, the individual is
effaced, in spite of himself of course. None has the
right to say, this is my work. So the rivalry is not
between individuals, but between nations. Thus we
are led to ask what share France has taken in the
work, and I think we have a right to be proud of
what she has done.

At the beginning of the eighteenth century there
arose long discussions between the Newtonians, who
believed the Earth to be flattened as the theory of
gravitation demands, and Cassini, who was misled b)'
inaccurate measurements, and believed the globe to
be elongated. Direct observation alone could settle
the question. It was the French Academy of Sciences
that undertook this task, a gigantic one for that
period.

While Maupertuis and Clairaut were measuring a
degree of longitude within the Arctic circle, Bouguer
and La Condamine turned their faces towards the
mountains of the Andes, in regions that were then
subject to Spain, and to-day form the Republic of
Ecuador. Our emissaries were exposed to great
fatigues, for journeys then were not so easy as they
are to-day.

It is true that the country in which Maupertuis'

operations were conducted was not a desert, and it is

even said that he enjoyed among the Lapps those soft

creature comforts that are unknown to the true Arctic

navigator. It was more or less in the neighbourhood
(1,777) 18

274 SCIENCE AND METHOD.

of places to which, in our day, comfortable steamers
carry, every summer, crowds of tourists and young
English ladies. But at that date Cook's Agency did
not exist, and Maupertuis honestly thought that he
had made a Polar expedition.

Perhaps he was not altogether wrong. Russians
and Swedes are to-day making similar measurements
at Spitzbergen, in a country where there are real ice-
packs. But their resources are far greater, and the
difference of date fully compensates for the difference
of latitude.

Maupertuis' name has come down to us considerably
mauled by the claws of Dr. Akakia, for Maupertuis
had the misfortune to displease Voltaire, who was
then king of the mind. At first he was extravagantly
praised by Voltaire ; but the flattery of kings is as
much to be dreaded as their disfavour, for it is followed
by a terrible day of reckoning. Voltaire himself learnt
something of this.

Voltaire called Maupertuis " my kind master of
thought," " Marquess of the Arctic Circle," " dear
flattener of the world and of Cassini," and even, as
supreme flattery, " Sir Isaac Maupertuis " ; and he
wrote, " There is none but the King of Prussia that
I place on a level with you ; his sole defect is that he
is not a geometrician." But very soon the scene
changes ; he no longer speaks of deifying him, like
the Argonauts of old, or of bringing down the council
of the gods from Olympus to contemplate his work,
but of shutting him up in a mad-house. He speaks
no more of his sublime mind, but of his despotic pride,
backed by very little science and much absurdity.

1 do not wish to tell the tale of these mock-heroic

FRENCH GEODESY. 275

conflicts, but I should like to make a few reflections
upon two lines of Voltaire's. In his Discours sur la
Moderation (there is no question of moderation in
praise or blame), the poet wrote :■ —

Vous avez confirme dans des lieux pleins d 'ennui
Ce que Newton connut sans sortir de chez lui.

(You have confirmed, in dreary far-off lands,
What Newton knew without e'er leaving home.)

These two lines, which take the place of the hyper-
bolical praises of earlier date, are most unjust, and
without any doubt, Voltaire was too well informed
not to realize it.

At that time men valued only the discoveries that
can be made without leaving home. To-day it is
theory rather that is held in low esteem. But this
implies a misconception of the aim of science.

Is nature governed by caprice, or is harmony the
reigning influence ? That is the question. It is when
science reveals this harmony that it becomes beauti-
ful, and for that reason worthy of being cultivated.
But whence can this revelation come if not from the
accordance of a theory with experience ? Our aim
then is to find out whether or not this accordance
exists. From that moment, these two terms, which
must be compared with each other, become one as
indispensable as the other. To neglect one for the
other would be folly. Isolated, theory is empty and
experience blind ; and both are useless and of no
interest alone.

Maupertuis is therefore entitled to his share of the
fame. Certainly it is not equal to that of Newton,
who had received the divine spark, or even of his

276 SCIENCE AND METHOD.

collaborator Clairaut It is not to be despised, how-
ever, because his work was necessary ; and if France,
after being outstripped by England in the seventeenth
century, took such full revenge in the following cen-
tury, it was not only to the genius of the Clairauts,
the d'Alemberts, and the Laplaces that she owed
it, but also to the long patience of such men as
Maupertuis and La Condamine.

We come now to what may be called the second
heroic period of Geodesy. France was torn with
internal strife, and the whole of Europe was in arms
against her. One would suppose that these tre-
mendous struggles must have absorbed all her ener-
gies. Far from that, however, she had still some left
for the service of science. The men of that day
shrank before no enterprise — they were men of faith.

Delambre and M^chain were commissioned to
measure an arc running from Dunkirk to Barcelona.
This time there is no journey to Lapland or Peru ;
the enemy's squadrons would close the roads. But
if the expeditions are less distant, the times are so
troublous that the obstacles and even the dangers
are quite as great.

In France Delambre had to fight against the ill-
will of suspicious municipalities. One knows that
steeples, which can be seen a long way off, and ob-
served with precision, often serve as signals for
geodesists. But in the country Delambre was working
through, there were no steeples left. I forget now
what proconsul it was who had passed through it and
boasted that he had brought down all the steeples
that raised their heads arrogantly above the humble
dwellings of the common people.

FRENCH GEODESY. 277

So they erected pyramids of planks covered with
white linen to make them more conspicuous. This
was taken to mean something quite different. White
Hnen ! Who was the foolhardy man who ventured
to set up, on our heights .so recently liberated, the
odious standard of the counter-revolution ? The
white linen must needs be edged with blue and red
stripes.

Mechain, operating in Spain, met with other but
no less serious difficulties. The Spanish country
folk were hostile. There was no lack of steeples,
but was it not sacrilege to take possession of them
with instruments that were mysterious and perhaps
diabolical ? The revolutionaries were the allies of
Spain, but they were allies who smelt a little of the
stake.

"We are constantly threatened," writes Mechain,
"with having our throats cut." Happily, thanks to
the exhortations of the priests, and to the pastoral
letters from the bishops, the fiery Spaniards con-
tented themselves with threats.

Some years later, Mdchain made a second expedi-
tion to Spain. He proposed to extend the meridian
from Barcelona to the Balearic Isles. This was the
first time that an attempt had been made to cross a
large arm of the sea by triangulation, by taking
observations of signals erected upon some high moun-
tain in a distant island. The enterprise was well
conceived and well planned, but it failed nevertheless.
The French scientist met with all kinds of difficulties,
of which he complains bitterly in his correspondence.
" Hell," he writes, perhaps with some exaggeration,
" hell, and all the scourges it vomits upon the earth—

278 SCIENCE AND METHOD.

storms, war, pestilence, and dark intrigues — are let
loose against me ! "

The fact is that he found among his collaborators
more headstrong arrogance than good-will, and that
a thousand incidents delayed his work. The plague
was nothing ; fear of the plague was much more
formidable. All the islands mistrusted the neighbour-
ing islands, and were afraid of receiving the scourge
from them. It was only after long weeks that
M^chain obtained permission to land, on condition of
having all his papers vinegared — such were the anti-
septics of those days. Disheartened and ill, he had
just applied for his recall, when he died.

It was Arago and Biot who had the honour of
taking up the unfinished work and bringing it to a
happy conclusion. Thanks to the support of the
Spanish Government and the protection of several
bishops, and especially of a celebrated brigand chief,
the operations progressed rapidly enough. They were
happily terminated, and Biot had returned to France,
when the storm burst.

It was the moment when the whole of Spain was
taking up arms to defend her independence against
France. Why was this stranger climbing mountains
to make signals ? It was evidently to call the French
army. Arago only succeeded in escaping from the
populace by giving himself up as a prisoner. In his
prison his only distraction was reading the account
of his own execution in the Spanish newspapers. The
newspapers of those days sometimes gave premature
news. He had at least the consolation of learning
that he had died a courageous and a Christian death.

Prison itself was not safe, and he had to make his

FRENCH GEODESY. 279

escape and reach Algiers. Thence he sailed for Mar-
seilles on an Algerian ship. This ship was captured
by a Spanish privateer, and so Arago was brought
back to Spain, and dragged from dungeon to dun-
geon in the midst of vermin and in the most horrible
misery.

If it had only been a question of his subjects and
his guests, the Dey would have said nothing. But
there were two lions on board, a present the African
sovereign was sending to Napoleon. The Dey
threatened war.

The vessel and the prisoners were released. The
point should have been correctly made, since there was
an astronomer on board ; but the astronomer was sea-
sick, and the Algerian sailors, who wished to go to
Marseilles, put in at Bougie. Thence Arago travelled
to Algiers, crossing Kabylia on foot through a thousand
dangers. He was detained for a long time in Africa
and threatened with penal servitude. At last he was
able to return to France. His observations, which he
had preserved under his shirt, and more extraordinary
still, his instruments, had come through these terrible
adventures without damage.

Up to this point, France not only occupied the first
place, but she held the field almost alone. In the
years that followed she did not remain inactive, and
the French ordnance map is a model. Yet the new
methods of observation and of calculation came
principally from Germany and England. It is only
during the last forty years that France has regained
her position.

She owes it to a scientific officer, General Perrier,
who carried out successfully a truly audacious enter-

28o SCIENCE AND METHOD.

prise, the junction of Spain and Africa. Stations were
established upon four peaks on the two shores of the
Mediterranean. There were long months of waiting
for a calm and clear atmosphere. At last there was
seen the slender thread of light that had travelled
two hundred miles over the sea, and the operation had
succeeded.

To-day still more daring projects have been con-
ceived. From a mountain in the vicinity of Nice
signals are to be sent to Corsica, no longer with a
view to the determination of geodetic questions, but
in order to measure the velocity of light. The dis-
tance is only one hundred and twenty-five miles, but
the ray of light is to make the return journey, after
being reflected from a mirror in Corsica. And it must
not go astray on the journey, but must return to
the exact spot from which it started.

Latterly the activity of French Geodesy has not
slackened. We have no more such astonishing
adventures to relate, but the scientific work accom-
plished is enormous. The territory of France beyond
the seas, just as that of the mother country, is being
covered with triangles measured with precision.

We have become more and more exacting, and
what was admired by our fathers does not satisfy
us to-day. But as we seek greater exactness, the
difficulties increase considerably. We are surrounded
by traps, and have to beware of a thousand unsuspected
causes of error. It becomes necessary to make more
and more infallible instruments.

Here again France has not allowed herself to be
outdone. Her apparatus for the measurement of bases
and of angles leaves nothing to be desired, and I would

FRENCH GEODESY. 281

also mention Colonel Defforges' pendulum, which
makes it possible to determine gravity with a pre-
cision unknown till now.

The future of French Geodesy is now in the hands
of the geographical department of the army, which
has been directed successively by General Bassot and
General Berthaut. This has advantages that can
hardly be overestimated. For good geodetic work,
scientific aptitude alone is not sufficient. A man
must be able to endure long fatigues in all climates.
The chief must know how to command the obedience
of his collaborators and to enforce it upon his native
helpers. These are military qualities, and, moreover, it
is known that science has always gone hand in hand
with courage in the French army.

I would add that a military organization assures
the indispensable unity of action. It would be more
difficult to reconcile the pretensions of rival scientists,
jealous of their independence and anxious about what
they call their honour, who would nevertheless have
to operate in concert, though separated by great
distances. There arose frequent discussions between
geodesists of former times, some of which started
echoes that were heard long after. The Academy
long rang with the quarrel between Bouguer and
La Condamine. I do not mean to say that soldiers
are free from passions, but discipline imposes silence
upon over-sensitive vanity.

Several foreign governments have appealed to
French officers to organize their geodetic depart-
ments. This is a proof that the scientific influence of
France abroad has not been weakened.

Her hydrographic engineers also supply a famous

282 SCIENCE AND METHOD.

contingent to the common work. The chart of her
coasts and of her colonies, and the study of tides, offer
them a vast field for research. Finally, I would
mention the general levelling of France, which is
being carried out by M. Lallemand's ingenious and
accurate methods.

With such men, we are sure of the future. Work for
them to do will not be wanting. The French colonial
empire offers them immense tracts imperfectly explored.
And that is not all. The International Geodetic Asso-
ciation has recognized the necessity of a new measure-
ment of the arc of Quito, formerly determined by La
Condamine. It is the French who have been entrusted
with the operation. They had every right, as it was
their ancestors who achieved, so to speak, the scientific
conquest of the Cordilleras. Moreover, these rights
were not contested, and the French Government
determined to exercise them.

Captains Maurain and Lacombe made a preliminary
survey, and the rapidity with which they accomplished
their mission, travelling through difficult countries, and
climbing the most precipitous peaks, deserves the
highest praise. It excited the admiration of General
Alfaro, President of the Republic of Ecuador, who
surnamed ^them los hombres de hierro, the men of
iron.

The definitive mission started forthwith, under the
command of Lieutenant-Colonel (then Commandant)
Bourgeois. The results obtained justified the hopes
that had been entertained. But the officers met with
unexpected difficulties due to the climate. More than
once one of them had to remain for several months at
an altitude of 13,000 feet, in clouds and snow, without

FRENCH GEODESY. 283

seeing anything of the signals he had to observe, which
refused to show themselves. But thanks to their per-
severance and courage, the only result was a delay,
and an increase in the expenses, and the accuracy of
the measurements did not suffer.

GENERAL CONCLUSIONS.

What I have attempted to explain in the foregoing
pages is how the scientist is to set about making a
selection of the innumerable facts that are offered to
his curiosity, since he is compelled to make a selection,
if only by the natural infirmity of his mind, though a
selection is always a sacrifice. To begin with, I ex-
plained it by general considerations, recalling, on the
one hand, the nature of the problem to be solved, and
on the other, seeking a better understanding of the
nature of the human mind, the principal instrument in
the solution. Then I explained it by examples, but
not an infinity of examples, for I too had to make
a selection, and I naturally selected the questions
I had studied most carefully. Others would no
doubt have made a different selection, but this matters
little, for I think they would have reached the same
conclusions.

There is a hierarchy of facts. Some are without
any positive bearing, and teach us nothing but them-
selves. The scientist who ascertains them learns
nothing but facts, and becomes no better able to
foresee new facts. Such facts, it seems, occur but
once, and are not destined to be repeated.

There are, on the other hand, facts that give a large

GENERAL CONCLUSIONS. 285

return, each of which teaches us a new law. And
since he is obliged to make a selection, it is to these
latter facts that the scientist must devote himself.

No doubt this classification is relative, and arises
from the frailty of our mind. The facts that give but
a small return are the complex facts, upon which a
multiplicity of circumstances exercise an appreciable
influence — circumstances so numerous and so diverse
that we cannot distinguish them all. But I should
say, rather, that they are the facts that we consider
complex, because the entanglement of these circum-
stances exceeds the compass of our mind. No doubt
a vaster and a keener mind than ours would judge
otherwise. But that matters little ; it is not this
superior mind that we have to use, but our own.

The facts that give a large return are those that we
consider simple, whether they are so in reality, because
they are only influenced by a small number of well-
defined circumstances, or whether they take on an
appearance of simplicity, because the multiplicity of
circumstances upon which they depend obey the laws
of chance, and so arrive at a mutual compensation.
This is most frequently the case, and is what com-
pelled us to enquire somewhat closely into the
nature of chance. The facts to which the laws of
chance apply become accessible to the scientist, who
would lose heart in face of the extraordinary com-
plication of the problems to which these laws are not
applicable.

We have seen how these considerations apply not
only to the physical but also to the mathematical
sciences. The method of demonstration is not the
same for the physicist as for the mathematician. But

286 SCIENCE AND METHOD.

their methods of discovery are very similar. In the
case of both they consist in rising from the fact to the
law, and in seeking the facts that are capable of
leading up to a law.

In order to elucidate this point, I have exhibited
the mathematician's mind at work, and that under
three forms : the mind of the inventive and creative
mathematician ; the mind of the unconscious geome-
trician who, in the days of our far-off ancestors or in
the hazy years of our infancy, constructed for us our
instinctive notion of space ; and the mind of the youth
in a secondary school for whom the master unfolds the
first principles of the science, and seeks to make him
understand its fundamental definitions. Through-
out we have seen the part played by intuition and
the spirit of generalization, without which these
three grades of mathematicians, if I may venture
so to express myself, would be reduced to equal
impotence.

And in demonstration itself logic is not all. The
true mathematical reasoning is a real induction,
differing in many respects from physical induction,
but, like it, (proceeding from the particular to the
universal. All the efforts that have been made to
upset this order, and to reduce mathematical induction
to the rules of logic, have ended in failure, but poorly
disguised by the use of a language inaccessible to the
uninitiated.

The examples I have drawn from the physical
sciences have shown us a good variety of instances of
facts that give a large return. A single experiment of
Kaufmann's upon radium rays revolutionizes at once
Mechanics, Optics, and Astronomy. Why is this? It

GENERAL CONCLUSIONS. 287

is because, as these sciences developed, we have recog-
nized more clearly the links which unite them, and
at last we have perceived a kind of general design of
the map of universal science. There are facts com-
mon to several sciences, like the common fountain
head of streams diverging in all directions, which may-
be compared to that nodal point of the St. Gothard
from which there flow waters that feed four different
basins.

Then we can make our selection of facts with more
discernment than our predecessors, who regarded
these basins as distinct and separated by impassable
barriers.

It is always simple facts that we must select, but
among these simple facts we should prefer those that
are situated in these kinds of nodal points of which
I have just spoken.

And when sciences have no direct link, they can
still be elucidated mutually by analogy. When the
laws that regulate gases were being studied, it was
realized that the fact in hand was one that would give
a great return, and yet this return was still estimated
below its true value, since gases are, from a certain
point of view, the image of the Milky Way ; and these
facts, which seemed to be of interest only to the
physicist, will soon open up new horizons to the
astronomer, who little expected it.

Lastly, when the geodesist finds that he has to turn
his glass a few seconds of arc in order to point it upon
a signal that he has erected with much difficulty, it is
a very small fact, but it is a fact giving a great return,
not only because it reveals the existence of a little
hump upon the terrestrial geoid, for the little hump

288 SCIENCE AND METHOD.

would of itself be of small interest, but because this
hump gives him indications as to the distribution of
matter in the interior of the globe, and, through that,
as to the past of our planet, its future, and the laws of
its development.

THE END.

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