Einstein and the Universe: A popular exposition of the famous theory

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Title: Einstein and the Universe: A popular exposition of the famous theory
Author: Nordmann, Charles
Language: English
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EINSTEIN AND THE UNIVERSE

_A Popular Exposition of the Famous Theory_

_By_ CHARLES NORDMANN
_Astronomer to the Paris Observatory._

_Translated by_ JOSEPH McCABE

_With a Preface by the Rt. Hon._
THE VISCOUNT HALDANE, O.M.

T. FISHER UNWIN LTD.
LONDON: ADELPHI TERRACE

_First published in English April 1922_
_Second Impression June 1922_

_All rights reserved_

PREFACE

A distinguished German authority on mathematical physics, writing
recently on the theory of Relativity, declared that if his publishers
had been willing to allow him sufficient paper and print he could have
explained what he wished to convey without using a single mathematical
formula. Such success is conceivable. Mathematical methods present,
however, two advantages. Their terminology is precise and concentrated,
in a fashion which ordinary language cannot afford to adopt. Further,
the symbols which result from their employment have implications
which, when brought to light, yield new knowledge. This is deductively
reached, but it is none the less new knowledge. With greater precision
than is usual, ordinary language may be made to do some, if not a great
deal, of this work for which mathematical methods are alone quite
appropriate. If ordinary language can do part of it an advantage may
be gained. The difficulty that attends mathematical symbolism is the
accompanying tendency to take the symbol as exhaustively descriptive
of reality. Now it is not so descriptive. It always embodies an
abstraction. It accordingly leads to the use of metaphors which
are inadequate and generally untrue. It is only qualification by
descriptive language of a wider range that can keep this tendency in
check. A new school of mathematical physicists, still, however, small
in number, is beginning to appreciate this.

But for English and German writers the new task is very difficult.
Neither Anglo-Saxon nor Saxon genius lends itself readily in this
direction. Nor has the task as yet been taken in hand completely, so
far as I am aware, in France. Still, in France there is a spirit and a
gift of expression which makes the approach to it easier than either
for us or for the Germans. Lucidity in expression is an endowment which
the best French writers possess in a higher degree than we do. Some of
us have accordingly awaited with deep interest French renderings of the
difficult doctrine of Einstein.

M. Nordmann, in addition to being a highly qualified astronomer and
mathematical-physicist, possesses the gift of his race. The Latin
capacity for eliminating abstractness from the description of facts is
everywhere apparent in his writing. Individual facts take the places
of general conceptions, of _Begriffe_. The language is that of
the _Vorstellung_, in a way that would hardly be practicable in
German. Nor is our own language equal to that of France in delicacy of
distinctive description. This book could hardly have been written by an
Englishman. But the difficulty in his way would have been one as much
of spirit as of letter. It is the lucidity of the French author, in
combination with his own gift of expression, that has made it possible
for the translator to succeed so well in overcoming the obstacles
to giving the exposition in our own tongue this book contains. The
rendering seems to me, after reading the book both in French and in
English, admirable.

M. Nordmann has presented Einstein’s principle in words which lift
the average reader over many of the difficulties he must encounter in
trying to take it in. Remembering Goethe’s maxim that he who would
accomplish anything must limit himself, he has not aimed at covering
the full field to which Einstein’s teaching is directed. But he
succeeds in making many abstruse things intelligible to the layman.
Perhaps the most brilliant of his efforts in this direction are
Chapters V and VI, in which he explains with extraordinary lucidity
the new theory of gravitation and of its relation to inertia. I think
that M. Nordmann is perhaps less successful in the courageous attack he
makes in his third chapter on the obscurity which attends the notion
of the “Interval.” But that is because the four-dimensional world,
which is the basis of experience of space and time for Einstein and
Minkowski, is in itself an obscure conception. Mathematicians talk
about it gaily and throw its qualities into equations, despite the
essential exclusion from it of the measurement and shape which actual
experience always in some form involves. They lapse on that account
into unconscious metaphysics of a dubious character. This does not
destroy the practical value of their equations, but it does make them
very unreliable as guides to the character of reality in the meaning
which the plain man attaches to it. Here, accordingly, we find the
author of this little treatise to be a good man struggling with
adversity. If he could make the topic clear he would. But then no one
has made it clear excepting as an abstraction which works, but which,
despite suggestions made to the contrary, cannot be clothed for us in
images.

This, however, is the fault, not of M. Nordmann himself, but of a phase
of the subject. With the subject in its other aspects he deals with the
incomparable lucidity of a Frenchman. I know no book better adapted
than the one now translated to give the average English reader some
understanding of a principle, still in its infancy, but destined, as I
believe, to transform opinion in more regions of knowledge than those
merely of mathematical physics.

_Haldane_

CONTENTS

PREFACE BY THE RT. HON. VISCOUNT HALDANE, O.M. pp. 5-8

INTRODUCTION pp. 13-15

CHAPTER I
THE METAMORPHOSES OF SPACE AND TIME

_Removing the mathematical difficulties—The pillars
of knowledge—Absolute time and space, from
Aristotle to Newton—Relative time and space,
from Epicurus to Poincaré and Einstein—Classical
Relativity—Antinomy of stellar aberration and the
Michelson experiment_ pp. 17-31

CHAPTER II
SCIENCE IN A NO-THOROUGHFARE

_Scientific truth and mathematics—The precise function
of Einstein—Michelson’s experiment, the Gordian knot
of science—The hesitations of Poincaré—The strange,
but necessary, Fitzgerald-Lorentz hypothesis—The
contraction of moving bodies—Philosophical and
physical difficulties_ pp. 32-52

CHAPTER III
EINSTEIN’S SOLUTION

_Provisional rejection of ether—Relativist
interpretation of Michelson’s experiment—New aspect
of the speed of light—Explanation of the contraction
of moving bodies—Time and the four dimensions of
space—Einstein’s “Interval” the only material
reality_ pp. 53-72

CHAPTER IV
EINSTEIN’S MECHANICS

_The mechanical foundation of all the
sciences—Ascending the stream of time—The speed of
light an impassable limit—The addition of speeds
and Fizeau’s experiment—Variability of mass—The
ballistics of electrons—Gravitation and light as
atomic microcosms—Matter and energy—The death of
the sun_ pp. 73-100

CHAPTER V
GENERALISED RELATIVITY

_Weight and inertia—Ambiguity of the Newtonian
law—Equivalence of gravitation and accelerated
movement—Jules Verne’s projectile and the principle
of inertia—Why rays of light are subject to
gravitation—How light from the stars is weighed—An
eclipse as a source of light_ pp. 101-123

CHAPTER VI
THE NEW CONCEPTION OF GRAVITATION

_Geometry and reality—Euclid’s geometry and
others—Contingency of Poincaré’s criterion—The
real universe is not Euclidean but Riemannian—The
avatars of the number π—The point of view of the
drunken man—Straight and geodetic lines—The new
law of universal attraction—Explanation of the
anomaly of the planet Mercury—Einstein’s theory of
gravitation_ pp. 124-147

CHAPTER VII
IS THE UNIVERSE INFINITE?

_Kant and the number of the stars—Extinct stars and
dark nebulæ—Extent and aspect of the astronomical
universe—Different kinds of universes—Poincaré’s
calculation—Physical definition of the infinite—The
infinite and the unlimited—Stability and
curvature of cosmic space-time—Real and virtual
stars—Diameter of the Einsteinian universe—The
hypothesis of globes of ether_ pp. 148-159

CHAPTER VIII
SCIENCE AND REALITY

_The Einsteinian absolute—Revelation by
science—Discussion of the experimental bases of
Relativity—Other possible explanations—Arguments in
favour of Lorentz’s real contraction—Newtonian space
may be distinct from absolute space—The real is a
privileged form of the possible—Two attitudes in
face of the unknown_ pp. 160-172

CHAPTER IX
EINSTEIN OR NEWTON?

_Recent discussion of Relativism at the Academy
of Sciences—Traces of the privileged space
of Newton—The principle of causality,
the basis of science—Examination of M.
Painlevé’s objections—Newtonian arguments and
Relativist replies—M. Painlevé’s formulas of
gravitation—Fecundity of Einstein’s theory—Two
conceptions of the world—Conclusion_ pp. 173-185

INTRODUCTION

This book is not a romance. Nevertheless.... If love is, as Plato says,
a soaring toward the infinite, where shall we find more love than in
the impassioned curiosity which impels us, with bowed heads and beating
hearts, against the wall of mystery that environs our material world?
Behind that wall, we feel, there is something sublime. What is it?
Science is the outcome of the search for that mysterious something.

A giant blow has recently been struck, by a man of consummate ability,
Albert Einstein, upon this wall which conceals reality from us. A
little of the light from beyond now comes to us through the breach he
has made, and our eyes are enchanted, almost dazzled, by the rays. I
propose here to give, as simply and clearly as is possible, some faint
reflex of the impression it has made upon us.

Einstein’s theories have brought about a profound revolution in
science. In their light the world seems simpler, more co-ordinated,
more in unison. We shall henceforward realise better how grandiose and
coherent it is, how it is ruled by an inflexible harmony. A little of
the ineffable will become clearer to us.

Men, as they pass through the universe, are like those specks of dust
which dance for a moment in the golden rays of the sun, then sink into
the darkness. Is there a finer or nobler way of spending this life than
to fill one’s eyes, one’s mind, one’s heart with the immortal, yet so
elusive, rays? What higher pleasure can there be than to contemplate,
to seek, to understand, the magnificent and astounding spectacle of the
universe?

There is in reality more of the marvellous and the romantic than there
is in all our poor dreams. In the thirst for knowledge, in the mystic
impulse which urges us toward the deep heart of the Unknown, there is
more passion and more sweetness than in all the trivialities which
sustain so many literatures. I may be wrong, after all, in saying that
this book is not a romance.

I will endeavour in these pages to make the reader understand,
accurately, yet without the aid of the esoteric apparatus of the
technical writer, the revolution brought about by Einstein. I will try
also to fix its limits; to state precisely what, at the most, we can
really know to-day about the external world when we regard it through
the translucent screen of science.

Every revolution is followed by a reaction, in virtue of the rhythm
which seems to be an inherent and eternal law of the mind of man.
Einstein is at once the Sieyès, the Mirabeau, and the Danton of the
new revolution. But the revolution has already produced its fanatical
Marats, who would say to science: “Thus far and no farther.”

Hence we find some resistance to the pretensions of over-zealous
apostles of the new scientific gospel. In the Academy of Sciences M.
Paul Painlevé takes his place, with all the strength of a vigorous
mathematical genius, between Newton, who was supposed to be overthrown,
and Einstein. In my final pages I will examine the penetrating
criticisms of the great French geometrician. They will help me to fix
the precise position, in the evolution of our ideas, of Einstein’s
magnificent synthesis. But I would first expound the synthesis itself
with all the affection which one must bestow upon things that one would
understand.

Science has not completed its task with the work of Einstein. There
remains many a depth that is for us unfathomable, waiting for some
genius of to-morrow to throw light into it. It is the very essence
of the august and lofty grandeur of science that it is perpetually
advancing. It is like a torch in the sombre forest of mystery. Man
enlarges every day the circle of light which spreads round him, but
at the same time, and in virtue of his very advance, he finds himself
confronting, at an increasing number of points, the darkness of the
Unknown. Few men have borne the shaft of light so deeply into the
forest as has Einstein. In spite of the sordid cares which harass us
to-day, amid so many grave contingencies, his system reveals to us an
element of grandeur.

Our age is like the noisy and unsubstantial froth that crowns, and
hides for a moment, the gold of some generous wine. When all the
transitory murmur that now fills our ears is over, Einstein’s theory
will rise before us as the great lighthouse on the brink of this sad
and petty twentieth century of ours.

CHARLES NORDMANN.

EINSTEIN AND THE UNIVERSE

CHAPTER I

THE METAMORPHOSES OF SPACE AND TIME

“Have you read Baruch?” La Fontaine used to cry, enthusiastically.
To-day he would have troubled his friends with the question “Have you
read Einstein?”

But, whereas one needs only a little Latin to gain access to Spinoza,
frightful monsters keep guard before Einstein, and their horrible
grimaces seem to forbid us to approach him. They stand behind strange
moving bars, sometimes rectangular and sometimes curvilinear,
which are known as “co-ordinates.” They bear names as frightful as
themselves—“contravariant and covariant vectors, tensors, scalars,
determinants, orthogonal vectors, generalised symbols of three signs,”
and so on.

These strange beings, brought from the wildest depths of the
mathematical jungle, join together or part from each other with a
remarkable promiscuity, by means of some astonishing surgery which is
called _integration_ and _differentiation_.

In a word, Einstein may be a treasure, but there is a fearsome troop
of mathematical reptiles keeping inquisitive folk away from it; though
there can be no doubt that they have, like our Gothic gargoyles, a
hidden beauty of their own. Let us, however, drive them off with the
whip of simple terminology, and approach the splendour of Einstein’s
theory.

Who is this physicist Einstein? That is a question of no importance
here. It is enough to know that he refused to sign the infamous
manifesto of the professors, and thus brought upon himself persecution
from the Pan-Germanists.[1] Mathematical truths and scientific
discoveries have an intrinsic value, and this must be judged and
appreciated impartially, whoever their author may chance to be. Had
Pythagoras been the lowest of criminals, the fact would not in the
least detract from the validity of the square of the hypotenuse. A
theory is either true or false, whether the nose of its author has the
aquiline contour of the nose of the children of Sem, or the flattened
shape of that of the children of Cham, or the straightness of that of
the children of Japhet. Do we feel that humanity is perfect when we
hear it said occasionally: “Tell me what church you frequent, and I
will tell you if your geometry is sound.” Truth has no need of a civil
status. Let us get on.

[1] Albert Einstein, born in 1879, is a German Jew of Würtemberg.
He studied in Switzerland, and was an engineer there until 1909,
when he became professor at Zurich University. In 1911 he passed to
Prague University, in 1912 to the Zurich Polytechnic, and in 1914 to
the Prussian Academy of Science. He refused to give his name to the
manifesto in which ninety-three professors of Germany and Austria
defended Germany’s war-action.—Trans.

* * * * *

All our ideas, all science, and even the whole of our practical life,
are based upon the way in which we picture to ourselves the successive
aspects of things. Our mind, with the aid of our senses, chiefly
ranges these under the headings of time and space, which thus become
the two frames in which we dispose all that is apparent to us of the
material world. When we write a letter, we put at the head of it the
name of the place and the date. When we open a newspaper, we find the
same indications at the beginning of each piece of telegraphic news.
It is the same in everything and for everything. Time and space, the
situation and the period of things, are thus seen to be the twin
pillars of all knowledge, the two columns which sustain the edifice of
men’s understanding.

So felt Leconte de Lisle when, addressing himself to “divine death,” he
wrote, in his profound, philosophic way:

Free us from time, number, and space:
Grant us the rest that life hath spoiled.

He inserts the word “number” only in order to define time and space
quantitatively. What he has finely expressed in these famous and superb
lines is the fact that all that there is for us in this vast universe,
all that we know and see, all the ineffable and agitated flow of
phenomena, presents to us no definite aspect, no precise form, until it
has passed through those two filters which are interposed by the mind,
time and space.

The work of Einstein derives its importance from the fact that he has
shown, as we shall see, that we have entirely to revise our ideas
of time and space. If that is so, the whole of science, including
psychology, will have to be reconstructed. That is the first part of
Einstein’s work, but it goes further. If that were the whole of his
work it would be merely negative.

Once he had removed from the structure of human knowledge what had been
regarded as an indispensable wall of it, though it was really only a
frail scaffolding that hid the harmony of its proportions, he began to
reconstruct. He made in the structure large windows which allow us now
to see the treasures it contains. In a word, Einstein showed, on the
one hand, with astonishing acuteness and depth, that the foundation of
our knowledge seems to be different from what we had thought, and that
it needs repairing with a new kind of cement. On the other hand, he has
reconstructed the edifice on this new basis, and he has given it a bold
and remarkably beautiful and harmonious form.

I have now to show in detail, concretely, and as accurately as
possible, the meaning of these generalities. But I must first insist on
a point which is of considerable importance: if Einstein had confined
himself to the first part of his work, as I have described it, the part
which shatters the classical ideas of time and space, he would never
have attained the fame which now makes his name great in the world of
thought.

The point is important because most of those—apart from experts—who
have written on Einstein have chiefly, often exclusively, emphasised
this more or less “destructive” side of his work. But, as we shall
see, from this point of view Einstein was not the first, and he is not
alone. All that he has done is to sharpen, and press a little deeper
between the badly joined stones of classical science, a chisel which
others, especially the great Henri Poincaré, had used long before him.
My next point is to explain, if I can, the real, the immortal, title of
Einstein to the gratitude of men: to show how he has by his own powers
rebuilt the structure in a new and magnificent form after his critical
work. In this he shares his glory with none.

* * * * *

The whole of science, from the days of Aristotle until our own, has
been based upon the hypothesis—properly speaking, the hypotheses—that
there is an absolute time and an absolute space. In other words, our
ideas rested upon the supposition that an interval of time and an
interval of space between two given phenomena are always the same, for
every observer whatsoever, and whatever the conditions of observation
may be. For instance, it would never have occurred to anybody as long
as classical science was predominant, that the interval of time, the
number of seconds, which lies between two successive eclipses of the
sun, may not be the fixed and identically same number of seconds for an
observer on the earth as for an observer in Sirius (assuming that the
second is defined for both by the same chronometer). Similarly, no one
would have imagined that the distance in metres between two objects,
for instance the distance of the earth from the sun at a given moment,
measured by trigonometry, may not be the same for an observer on the
earth as for an observer in Sirius (the metre being defined for both by
the same rule).

“There is,” says Aristotle, “one single and invariable time, which
flows in two movements in an identical and simultaneous manner; and if
these two sorts of time were not simultaneous, they would nevertheless
be of the same nature.... Thus, in regard to movements which take place
simultaneously, there is one and the same time, whether or no the
movements are equal in rapidity; and this is true even if one of them is
a local movement and the other an alteration.... It follows that even
if the movements differ from each other, and arise independently, the
time is absolutely the same for both.”[2] This Aristotelic definition
of physical time is more than two thousand years old, yet it clearly
represents the idea of time which has been used in classic science,
especially in the mechanics of Galileo and Newton, until quite recent
years.

It seems, however, that in spite of Aristotle, Epicurus outlined the
position which Einstein would later adopt in antagonism to Newton. To
translate liberally the words in which Lucretius expounds the teaching
of Epicurus:

“Time has no existence of itself, but only in material objects, from
which we get the idea of past, present, and future. It is impossible
to conceive time in itself independently of the movement or rest of
things.”[3]

[2] _Physics_, bk. iv, ch. xiv.

[3] _De Natura Rerum_, bk. i, vv. 460 ff.

Both space and time have been regarded by science ever since Aristotle
as invariable, fixed, rigid, absolute data. Newton thought that he was
saying something obvious, a platitude, when he wrote in his celebrated
Scholion: “Absolute, true, and mathematical time, taken in itself and
without relation to any material object, flows uniformly of its own
nature.... Absolute space, on the other hand, independent by its own
nature of any relation to external objects, remains always unchangeable
and immovable.”

The whole of science, the whole of physics and mechanics, as they are
still taught in our colleges and in most of our universities, are based
entirely upon these propositions, these ideas of an absolute time and
space, taken by themselves and without any reference to an external
object, independent by their very nature.

In a word—if I may venture to use this figure—time in classical
science was like a river bearing phenomena as a stream bears boats,
flowing on just the same whether there were phenomena or not. Space,
similarly, was rather like the bank of the river, indifferent to the
ships that passed.

From the time of Newton, however, if not from the time of Aristotle,
any thoughtful metaphysician might have noticed that there was
something wrong in these definitions. Absolute time and absolute
space are “things in themselves,” and these the human mind has always
regarded as not directly accessible to it. The specifications of space
and time, those numbered labels which we attach to objects of the
material world, as we put labels on parcels at the station so that
they may not be lost (a precaution that does not always suffice), are
given us by our senses, whether aided by instruments or not, only when
we receive concrete impressions. Should we have any idea of them if
there were no bodies attached to them, or rather to which we attach the
labels? To answer this in the affirmative, as Aristotle, Newton, and
classical science do, is to make a very bold assumption, and one that
is not obviously justified.

The only time of which we have any idea apart from all objects is the
psychological time so luminously studied by M. Bergson: a time which
has nothing except the name in common with the time of physicists, of
science.

It is really to Henri Poincaré, the great Frenchman whose death has
left a void that will never be filled, that we must accord the merit of
having first proved, with the greatest lucidity and the most prudent
audacity, that time and space, as we know them, can only be relative. A
few quotations from his works will not be out of place. They will show
that the credit for most of the things which are currently attributed
to Einstein is, in reality, due to Poincaré. To prove this is not in
any way to detract from the merit of Einstein, for that is, as we shall
see, in other fields.

This is how Poincaré, whose ideas still dominate the minds of
thoughtful men, though his mortal frame perished years ago, expressed
himself, the triumphant sweep of his wings reaching further every day:

“One cannot form any idea of empty space.... From that follows the
undeniable relativity of space. Any man who talks of absolute space
uses words which have no meaning. I am at a particular spot in
Paris—the Place du Panthéon, let us suppose—and I say: ‘I will come
back _here_ to-morrow.’ If anyone asks me whether I mean that I
will return to the same point in space, I am tempted to reply, ‘Yes.’ I
should, however, be wrong, because between this and to-morrow the earth
will have travelled, taking the Place du Panthéon with it, so that
to-morrow the square will be more than 2,000,000 kilometres away from
where it is now. And it would be no use my attempting to use precise
language, because these 2,000,000 kilometres are part of our earth’s
journey round the sun, but the sun itself has moved in relation to the
Milky Way, and the Milky Way in turn is doubtless moving at a speed
which we cannot learn. Thus we are entirely ignorant, and always will
be ignorant, how far the Place du Panthéon shifts its position in space
in a single day. What I really meant to say was: ‘To-morrow I shall
again see the dome and façade of the Panthéon.’ If there were no
Panthéon, there would be no meaning in my words, and space would
disappear.”

Poincaré works out his idea in this way:

“Suppose all the dimensions of the universe were
increased a thousandfold in a night. The world would
remain the same, giving the word ‘same’ the meaning
it has in the third book of geometry. Nevertheless,
an object that had measured a metre in length will
henceforward be a kilometre in length; a thing that
had measured a millimetre will now measure a metre.
The bed on which I lie and the body which lies on it
will increase in size to exactly the same extent. What
sort of feelings will I have when I awake in the morning,
in face of such an amazing transformation? Well,
I shall know nothing about it. The most precise
measurements would tell me nothing about the revolution,
because the tape I use for measuring will have
changed to the same extent as the objects I wish to
measure. As a matter of fact, there would be no
revolution except in the mind of those who reason as
if space were absolute. If I have argued for a moment
as they do, it was only in order to show more clearly
that their position is contradictory.”

It would be easy to develop Poincaré’s argument. If all the objects in
the universe were to become, for instance, a thousand times taller, a
thousand times broader, we should be quite unable to detect it, because
we ourselves—our retina and our measuring rod—would be transformed
to the same extent at the same time. Indeed, if all the things in
the universe were to experience an absolutely irregular spatial
deformation—if some invisible and all-powerful spirit were to distort
the universe in any fashion, drawing it out as if it were rubber—we
should have no means of knowing the fact. There could be no better
proof that space is relative, and that we cannot conceive space apart
from the things which we use to measure it. When there is no measuring
rod, there is no space.

Poincaré pushed his reasoning on this subject so far that he came to
say that even the revolution of the earth round the sun is merely a
more convenient hypothesis than the contrary supposition, but not a
truer hypothesis, unless we imply the existence of absolute space.

It may be remembered that certain unwary controversialists have tried
to infer from Poincaré’s argument that the condemnation of Galileo was
justified. Nothing could be more amusing than the way in which the
distinguished mathematician-philosopher defended himself against this
interpretation, though one must admit that his defence was not wholly
convincing. He did not take sufficiently into account the agnostic
element.

Poincaré, in any case, is the leader of those who regard space as a
mere property which we ascribe to objects. In this view our idea of
it is only, so to say, the hereditary outcome of those efforts of our
senses by means of which we strive to embrace the material world at a
given moment.

It is the same with time. Here again the objections of philosophic
Relativists were raised long ago, but it was Poincaré who gave them
their definitive shape. His luminous demonstrations are, however, well
known, and we need not reproduce them here. It is enough to observe
that, in regard to time as well as space, it is possible to imagine
either a contraction or an enlargement of the scale which would be
completely imperceptible to us; and this seems to show that man
cannot conceive an absolute time. If some malicious spirit were to
amuse itself some night by making all the phenomena of the universe a
thousand times slower, we should not, when we awake, have any means of
detecting the change. The world would seem to us unchanged. Yet every
hour recorded by our watches would be a thousand times longer than
hours had previously been. Men would live a thousand times as long, yet
they would be unaware of the fact, as their sensations would be slower
in the same proportion.

When Lamartine appealed to time to “suspend its flight,” he said a
very charming, but perhaps meaningless, thing. If time had obeyed his
passionate appeal, neither Lamartine nor Elvire would have known and
rejoiced over the fact. The boatman who conducted the lovers on the Lac
du Bourget would not have asked payment for a single additional hour;
yet he would have dipped his oars into the pleasant waters for a far
longer time.

I venture to sum up all this in a sentence which will at first sight
seem a paradox: in the opinion of the Relativists it is the measuring
rods which create space, the clocks which create time. All this was
maintained by Poincaré and others long before the time of Einstein, and
one does injustice to truth in ascribing the discovery to him. I am
quite aware that one lends only to the rich, but one does an injustice
to the wealthy themselves in attributing to them what does not belong
to them, and what they need not in order to be rich.

There is, moreover, one point at which Galileo and Newton, for all
their belief in the existence of absolute space and time, admitted
a certain relativity. They recognised that it is impossible to
distinguish between uniform movements of translation. They thus
admitted the equivalence of all such movements, and therefore the
impossibility of proving an absolute movement of translation.

That is what is called the Principle of Classic Relativity.

* * * * *

An unexpected fact served to bring these questions upon a new plane,
and led Einstein to give a remarkable extension to the Principle
of Relativity of classic mechanics. This was the issue of a famous
experiment by Michelson, of which we must give a brief description.

It is well known that rays of light travel across empty space from
star to star, otherwise we should be unable to see the stars. From
this physicists long ago concluded that the rays travelled in a medium
that is devoid of mass and inertia, is infinitely elastic, and offers
no resistance to the movement of material bodies, into which it
penetrates. This medium has been named ether. Light travels through it
as waves spread over the surface of water at a speed of something like
186,000 miles a second: a velocity which we will express by the letter
=V=.

The earth revolves round the sun in a veritable ocean of ether, at a
speed of about 18 miles a second. In this respect the rotation of the
earth on its axis need not be noticed, as it pushes the surface of the
globe through the ether at a speed of less than two miles a second.
Now the question had often been asked: Does the earth, in its orbital
movement round the sun, take with it the ether which is in contact with
it, as a sponge thrown out of a window takes with it the water which it
has absorbed? Experiment—or rather, experiments, for many have been
tried with the same result—has shown that the question must be
answered in the negative.

This was first established by astronomical observation. There is in
astronomy a well-known phenomenon discovered by Bradley which is
called aberration. It consists in this: when we observe a star with a
telescope, the image of the star is not precisely in the direct line of
vision. The reason is that, while the luminous rays of the star which
have entered the telescope are passing down the length of the tube,
the instrument has been slightly displaced, as it shares the movement
of the earth. On the other hand, the luminous ray in the tube does
not share the earth’s motion, and this gives rise to the very slight
deviation which we call aberration. This proves that the medium in
which light travels, the ether which fills the instrument and surrounds
the earth, does not share the earth’s motion.

Many other experiments have settled beyond question that the ether,
which is the vehicle of the waves of light, is not borne along by the
earth as it travels. Now, since the earth moves through the ether as a
ship moves over a stationary lake (not like one floating on a moving
stream), it ought to be possible to detect some evidence of this speed
of the earth in relation to the ether.

One of the devices that may be imagined for the purpose is the
following. We know that the earth turns on itself from west to east,
and travels round the sun in the same way. It follows that in the
middle of the night the revolution of the earth round the sun means
that Paris will be displaced, in the direction from Auteuil toward
Charenton, at a speed of about thirty kilometres a second. During the
day, of course, it is precisely the opposite. Paris changes its place
round the sun in the direction from Charenton toward Auteuil. Well, let
us suppose that at midnight a physicist at Auteuil sends a luminous
signal. A physicist receiving this ray of light at Charenton, and
measuring its velocity, ought to find that the latter is =V= + 30
kilometres. We know that, as a result of the earth’s motion, Charenton
recedes before the ray of light. Consequently, since light travels
in a medium, the ether, which does not share the earth’s motion, the
observer at Charenton ought to find that the ray reaches him at a less
speed than it would if the earth were stationary. It is much the same
as if an observer were travelling on a bicycle in front of an express
train. If the express travels at thirty metres a second and the cyclist
at three metres a second, the speed of the train in relation to the
cyclist will be 30-3 = 27 metres a second. It would be _nil_ if
the train and the cyclist were travelling at the same rate.

On the other hand, if the cyclist were going toward the train, the
speed of the train in relation to him would be 30 + 3 = 33 metres a
second. Similarly, when the physicist at Charenton sends out a luminous
message at midnight, and the physicist of Auteuil receives it, the
latter ought to find that the ray of light has a velocity of =V= +
30 kilometres.

All this may be put in a different way. Suppose the distance between
the observer at Auteuil and the man at Charenton were exactly twelve
kilometres. While the ray of light emitted at Auteuil speeds toward
Charenton, that town is receding before it to a small extent. It
follows that the ray will have to travel a little more than twelve
kilometres before it reaches the man of science at Charenton. It will
travel a little less than that distance if we imagine it proceeding in
the opposite direction.

Now the American physicist Michelson, borrowing an ingenious idea
from the French physicist Fizeau, succeeded, with a high degree of
accuracy, in measuring distances by means of the interference-bands of
light. Every variation in the distance measured betrays itself by the
displacement of a certain number of these bands, and this may easily be
detected by a microscope.

Let us next suppose that our two physicists work in a laboratory
instead of between Charenton and Auteuil. Let us suppose that they
are, by means of the interference-bands, measuring the space traversed
by a ray of light produced in the laboratory, according as it travels
in the same direction as the earth or in the opposite direction. That
is Michelson’s famous experiment, reduced to its essential elements
and simplified for the purpose of this essay. In those circumstances
Michelson’s delicate apparatus ought to reveal a distinctly measurable
difference according as the light travels with the earth or in the
opposite direction.

But no such difference was found. Contrary to all expectation, and
to the profound astonishment of physicists, it was found that light
travels at precisely the same speed whether the man who receives it is
receding before it with the velocity of the earth or is approaching
it at the same velocity. It is an undeniable consequence of this that
_the ether shares the motion of the earth_. We have, however, seen
that other experiments, not less precise, had settled that _the ether
does not share the motion of the earth_.

Out of this contradiction, this conflict of two irreconcilable yet
indubitable facts, Einstein’s splendid synthesis, like a spark of light
issuing from the clash of flint and steel, came into being.

CHAPTER II

SCIENCE IN A NO-THOROUGHFARE

It would be foolish to pretend that we can penetrate the most obscure
corners of Einstein’s theories without the aid of mathematics. I
believe, however, that we can give in ordinary language—that is to
say, by means of illustrations and analogies—a fairly satisfactory
idea of these things, the intricacy of which is usually due to the
infinitely subtle and supple play of mathematical formulæ and equations.

After all, mathematics is not, never was, and never will be, anything
more than a particular kind of language, a sort of shorthand of thought
and reasoning. The purpose of it is to cut across the complicated
meanderings of long trains of reasoning with a bold rapidity that is
unknown to the mediæval slowness of the syllogisms expressed in our
words.

However paradoxical this may seem to people who regard mathematics as
_of itself_ a means of discovery, the truth is that we can never
get from it anything that was not implicitly inherent in the data which
were thrust between the jaws of its equations. If I may use a somewhat
trivial illustration, mathematical reasoning is very like certain
machines which are seen in Chicago—so bold explorers in the United
States tell us—into which one puts living animals that emerge at the
other end in the shape of appetising prepared meats. No spectator could
have, or would wish to have, eaten the animal alive, but in the form
in which it issues from the machine it can at once be digested and
assimilated. Yet the meat is merely the animal conveniently prepared.
That is what mathematics does. By means of a marvellous machinery the
mathematician extracts the valuable marrow from the _given facts_.
It is a machinery that is particularly useful in cases where the wheels
of verbal argument, the chain of syllogisms, would soon be brought to a
halt.

Does it follow that, properly speaking, mathematics is not a science?
Does it follow at least that it is only a science in so far as it is
based upon reality, and fed with experimental data, since “experience
is the sole source of truth.” I refrain from answering the question, as
I am one of those who believe that everything is material for science.
Still, it was worth while to raise the question because many are too
much disposed to regard a purely mathematical education as a scientific
education. Nothing could be further from the truth. Pure mathematics
is, in itself, merely an abbreviated form of language and of logical
thought. It cannot, of its own nature, teach us anything about the
external world; it can do so only in proportion as it enters into
contact with the world. It is of mathematics in particular that we may
say: _Naturæ non imperatur nisi parendo._

Are not Einstein’s theories, as some imperfectly informed writers have
suggested, only a play of mathematical formulæ (taking the word in the
meaning given to it by both mathematicians and philosophers)? If they
were only a towering mathematical structure in which the _x_’s
shoot out their volutes in bewildering arabesques, with swan-neck
integrals describing Louis XV patterns, they would have no interest
whatever for the physicist, for the man who has to examine the nature
of things before he talks about it. They would, like all coherent
schemes of metaphysics, be merely a more or less agreeable system of
thought, the truth or falseness of which could never be demonstrated.

Einstein’s theory is very different from that, and very much more than
that. It is based upon facts. It also leads to facts—new facts. No
philosophical doctrine or purely formal mathematical construction ever
enabled us to discover new phenomena. It is precisely because it has
led to such discovery that Einstein’s theory is neither the one nor the
other. That is the difference between a scientific theory and a pure
speculation, and it is that which, I venture to say, makes the former
so superior.

Like some suspension bridge boldly thrown across an abyss, Einstein’s
theory rests, on the one side, on experimental phenomena, and it leads,
at the other side, to other, and hitherto unsuspected, phenomena, which
it has enabled us to discover. Between these two solid experimental
columns the mathematical reasoning is like the marvellous network of
thousands of steel bars which represent the elegant and translucent
structure of the bridge. It is that, and nothing but that. But the
arrangement of the beams and bars might have been different, and the
bridge—though less light and graceful, perhaps—still have been able
to join together the two sets of facts on which it rests.

In a word, mathematical reasoning is only a kind of reasoning in a
special language, from experimental premises to conclusions which are
verifiable by experience. Now there is no language which cannot in some
degree be translated into another language. Even the hieroglyphics of
Egypt had to give way before Champollion. I am therefore convinced that
the mathematical difficulties of Einstein’s theories will some day be
replaced by simpler and more accessible formulæ. I believe, indeed,
that it is even now possible to give by means of ordinary speech an
idea, rather superficial perhaps, but accurate and substantially
complete, of this wonderful Einsteinian structure which ranges all the
conquests of science, as in some well-ordered museum, in a new and
superb unity. Let us try.

* * * * *

We may resume in the few following words the story of the origin, the
starting-point, of Einstein’s system.

1. Observation of the stars proves that interplanetary
space is not empty, but is filled with a special medium,
ether, in which the waves of light travel.

2. The fact of aberration and other phenomena seems to prove
that the ether is not displaced by the earth during its
course round the sun.

3. Michelson’s experiment seems to prove, on the contrary, that
the earth bears the ether with it in its movement.

This contradiction between facts of equal authority was for years
the despair and the wonder of physicists. It was the Gordian knot of
science. Long and fruitless efforts were made to untie it until at last
Einstein cut it with a single blow of his remarkably acute intelligence.

In order to understand how that was done—which is the vital point of
the whole system—we must retrace our steps a little and examine the
precise conditions of Michelson’s famous experiment.

I pointed out in the preceding chapter that Michelson proposed to study
the speed of a ray of light produced in the laboratory and directed
either from east to west or west to east: that is to say, in the
direction in which the earth itself moves, at a speed of about eighteen
miles a second, as it travels round the sun, or in the opposite
direction. As a matter of fact, Michelson’s experiment was rather more
complicated than that, and we must return to it.

Four mirrors are placed at an equal distance from each other in the
laboratory, in pairs which face each other. Two of the opposing
mirrors are arranged in the direction east-west, the direction in
which the earth moves in consequence of its revolution round the sun.
The other two are arranged in a plane perpendicular to the preceding,
the direction north-south. Two rays of light are then started in the
respective directions of the two pairs of mirrors. The ray coming from
the mirror to the east goes to the mirror in the west, is reflected
therefrom, and returns to the first mirror. This ray is so arranged
that it crosses the path of the light which goes from north to south
and back. It interferes with the latter light, causing “fringes of
interference” which, as I said, enable us to learn the exact distance
traversed by the rays of light reflected between the pairs of mirrors.
If anything brought about a difference between the length of the two
distances, we should at once see the displacement of a certain number
of interference-fringes, and this would give us the magnitude of the
difference.

An analogy will help us to understand the matter. Suppose a violent
steady east wind blew across London, and an aviator proposed to cross
the city about twelve miles from extreme west to east and back: that is
to say, going with the wind on his outward journey and against it on
the return journey. Suppose another aviator, of equal speed, proposed
at the same time to fly from the same starting-point to a point twelve
miles to the north and back, the second aviator will fly both ways at
right angles to the direction of the wind. If the two start at the same
time, and are imagined as turning round instantaneously, will they both
reach the starting-point together? And, if not, which of them will have
completed his double journey first?

It is clear that if there were no wind, they would get back together,
as we suppose that they both do twenty-four miles at the same speed,
which we may roughly state to be 200 yards a second.

But it will be different if, as I postulated, there is a wind blowing
from east to west. It is easy to see that in such circumstances the
man who flies east to west will take longer to complete the journey.
In order to get it quite clearly, let us suppose that the wind is
travelling at the same speed as the aviator (200 yards a second). The
man who flies at right angles to the wind will be blown twelve miles
to the west while he is doing his twelve miles from south to north.
He will therefore have traversed _in the wind_ a real distance
equal to the diagonal of a square measuring twelve miles on each
side. Instead of flying twenty-four miles, he will really have flown
thirty-four in the wind, the medium in relation to which he has any
velocity.

On the other hand, the aviator who flies eastward will never reach his
destination, because in each second of time he is driven westward to
precisely the same extent as he is travelling eastward. He will remain
stationary. To accomplish his journey he would need to cover _in the
wind_ an infinite distance.

If, instead of imagining a wind equal in velocity to the aviator (an
extreme supposition in order to make the demonstration clearer), I had
thought of it as less rapid, we should again find, by a very simple
calculation, that the man who flies north and south has less distance
to cover in the wind than the man who flies east and west.

Now take rays of light instead of aviators, the ether instead of
the wind, and we have very nearly the conditions of the Michelson
experiment. A current or wind of ether—since the ether has
been already shown to be stationary in relation to the earth’s
movement—proceeds from one to the other of our east-west mirrors.
Therefore the ray of light which travels between these two mirrors,
forth and back, must cover a longer distance in ether than the ray
which goes from the south mirror to the north and back. But how are we
to detect this difference? It is certainly very minute, because the
speed of the earth is ten thousand times less than the velocity of
light.

There is a very simple means of doing this: one of those ingenious
devices which physicists love, a differential device so elegant and
precise that we have entire confidence in the result.

Let us suppose that our four mirrors are fixed rigidly in a sort of
square frame, something like those “wheels of fortune” with numbers on
them that one sees in country fairs. Let us suppose that we can turn
this frame round as we wish, without jerking or displacing it, which is
not difficult if it floats in a bath of mercury. I then take a lens and
observe the permanent interference-fringes which define the difference
between the paths traversed by my two rays of light, north-south and
east-west. Then, without losing sight of the bands or fringes, I
turn the frame round a quarter of a circle. Owing to this rotation
the mirrors which were east-west now become north-south, and _vice
versa_. The double journey made by the north-south ray of light
has now taken the direction east-west, and has therefore suddenly
been lengthened; the double journey of the east-west ray has become
north-south, and has been suddenly shortened. The interference-fringes,
which indicate the difference in length between the two paths, which
has suddenly changed, must necessarily be displaced, and that, as we
can calculate, to no slight extent.

Well, we find no change whatever! The fringes remain unaltered. They
are as stationary as stumps of trees. It is bewildering, one would
almost say revolting, because the delicacy of the apparatus is such
that, even if the earth moved through the ether at a rate of only three
kilometres a second (or ten times less than its actual velocity), the
displacement of the fringes would be sufficient to indicate the speed.

* * * * *

When the negative result of this experiment was announced, there was
something like consternation amongst the physicists of the world.
Since the ether was not borne along by the earth, as observation had
established, how could it possibly behave as if it did share the
earth’s motion? It was a Chinese puzzle. More than one venerable grey
head was in despair over it.

It was absolutely necessary to find a way out of this inexplicable
contradiction, to end this paradoxical mockery which the facts seemed
to oppose to the most rigorous results of calculation. This the men of
science succeeded in doing. How? By the method which is generally used
in such circumstances—by means of supplementary hypotheses. Hypotheses
in science are a kind of soft cement which hardens rapidly in the open
air, thus enabling us to join together the separate blocks of the
structure, and to fill up the breaches made in the wall by projectiles,
with artificial stuff which the superficial observer presently mistakes
for stone. It is because hypotheses are something like that in science
that the best scientific theories are those which include least
hypotheses.

But I am wrong in using the plural in this connection. In the end
it was found that one single hypothesis conveniently explained the
negative result of the Michelson experiment. That is, by the way, a
rare and remarkable experience. Hypotheses usually spring up like
mushrooms in every dark corner of science. You get a score of them to
explain the slightest obscurity.

This single hypothesis, which seemed to be capable of extricating
physicists from the dilemma into which Michelson had put them, was
first advanced by the distinguished Irish mathematician Fitzgerald,
then taken up and developed by the celebrated Dutch physicist Lorentz,
the Poincaré of Holland, one of the most brilliant thinkers of our
time. Einstein would no more have attained fame without him than Kepler
would without Copernicus and Tycho Brahe.

Let us now see what this Fitzgerald-Lorentz hypothesis, as strange as
it is simple, really is.

But we must first glance at a preliminary matter of some importance. A
number of able men have declared—after the issue, let it be said—that
the result of the Michelson experiment could only be negative _a
priori_. In point of fact, they argue (more or less), the Classic
Principle of Relativity, the principle known to Galileo and Newton,
implies that it is impossible for an observer who shares the motion
of a vehicle to detect the motion of that vehicle by any facts he
observes while he is in it. Thus, when two ships or two trains pass
each other,[4] it is impossible for the passengers to say which of the
two is moving, or moving the more rapidly. All that they can perceive
is the relative speed of the trains or ships.

[4] It is assumed that the ship is not rolling or pitching, and that
there is no vibration in the train.

The men of science to whom I have referred say that, if Michelson’s
experiment had had a positive result, it would have given us the
absolute velocity of the earth in space. This result would have been
contrary to the Principle of Relativity of classical philosophy and
mechanics, which is a self-evident truth. Therefore the result could
only be negative.

This is, as we shall see, ambiguous. There is, if I may say so, a flaw
in the argument which has escaped the notice even of distinguished men
of science like Professor Eddington, the most erudite of the English
Einsteinians. It was he who organised the observations of the solar
eclipse of May 29, 1919, which have, as we shall see, furnished the
most striking verification of Einstein’s deductions.

In the first place, if Michelson’s experiment had had a positive
result, what it would have indicated is the velocity of the earth in
relation to the ether. But, for this to be an absolute velocity, the
ether would have to be identical with space. This is so far from being
necessary that we can easily conceive a space—to put it better, a
discontinuity—between two stars that contains no ether and across which
neither light nor any other known form of energy would travel.

When Eddington says that “it is legitimate and reasonable,” that it is
“inherent in the fundamental laws of nature,” that we cannot detect any
movement of bodies in relation to ether, and that this is certain “even
if the experimental evidence is inadequate,” he affirms something which
would be evident only if space and ether were evidently identical. But
this is far from being the case. If Michelson’s experiment had had
a positive result, if we had detected a velocity on the part of the
earth, should we have discovered a velocity in relation to an absolute
standard? Certainly not. It is quite possible that the stellar universe
which is known to us, with its hundreds of thousands of galaxies which
it takes light millions of years to cross, may be contained in a sphere
of ether that rolls in an abyss which is devoid of ether, and is sown
here and there with other universes, other giant drops of ether, from
which no ray of light or anything else may ever reach us. It is, at
all events, not inconceivable. And in that case, assuming that the
ether has the properties attributed to it by classic physics, even if
we had detected the movement of the earth in relation to it, we should
not have discovered an absolute movement, but at the most a movement
in relation to the centre of gravity of our particular universe,
a standard which we could not refer to some other which would be
absolutely stationary. The Classical Principle of Relativity would not
be violated.

Hence, whatever may have been said to the contrary, the issue of
Michelson’s experiment might, in these hypotheses, be either positive
or negative without any detriment to Classical Relativism. As a matter
of fact, it was negative, so nothing further need be said. Experiment
has pronounced, and it alone had the right to pronounce.

These distinctions were not unknown to Poincaré, and he wrote: “By the
real velocity of the earth I understand, not its absolute velocity,
which is meaningless, but its velocity in relation to the ether.”
Therefore the possibility of the existence of a velocity discoverable
in relation to the ether was not regarded as an absurdity by Poincaré.
He said: “Any man who speaks of absolute space uses a word that has no
meaning.”

It is worth while noticing that in all this the development of
Poincaré’s ideas betrays a certain hesitation. Speaking of experiments
analogous to those of Michelson, he said:

“I know that it will be said that we
are not measuring its absolute velocity, but its velocity
in relation to the ether. That is scarcely satisfactory.
Is it not clear that, if we conceive the principle in
this fashion, we can make no deductions whatever from it?”

From this it is evident that Poincaré, in spite of himself and all his
efforts to avoid it, was disposed to find the distinction between space
and ether “scarcely satisfactory.”

I must admit that Poincaré’s own argument seems to me not wholly
satisfactory, or at least not convincing. “Nature,” says Fresnel,
“cares nothing about analytical difficulties.” I imagine that it cares
just as little about philosophical or purely physical difficulties. It
is hardly an incontestable criterion to suppose that a conception of
phenomena is so much nearer to reality the more “satisfactory” it is to
us, or the better it is found adapted to the weakness of the human
mind. Otherwise we should have to hold, whether we liked or no, that
the universe is necessarily adapted to the categories of the mind;
that it is constituted with a view to giving us the least possible
intellectual trouble. That would be a strange return to anthropocentric
finalism and conceit! The fact that vehicles do not pass there, and
that pedestrians have to turn back, does not prove that there are no
such things as no-thoroughfares in our towns. It is possible, even
probable, that the universe also, considered as an object of science,
has its no-thoroughfare.

Clearly one may reply to me that it is not the universe that is adapted
to our mind, but the mind that has become adapted to the universe in
the evolutionary course of their relations to each other. The mind
needs in its evolution to adapt itself to the universe, in conformity
with the principle of minimum action formulated by Fermat: perhaps the
most profound principle of the physical, biological, and moral world.
In that respect the simplest and most economical ideas are the nearest
to reality.

Yes, but what proof is there that our mental evolution is complete and
perfect, especially when we are dealing with phenomena of which our
organism is insensible?

* * * * *

Experiment alone has proved, and had the right to prove, that it
is impossible to measure the velocity of an object relatively to
the ether. At all events, this is now settled. After all, since it
is evidently in the very nature of things that we cannot detect an
absolute movement, is it not because the velocity of the earth in
relation to the ether is an absolute velocity that we have been unable
to detect it? Possibly; but it cannot be proved. If it is so—which is
not at all certain—it is in the last resort _experience_, the
one source of truth, which thus tends to prove, indirectly, that the
ether is really identical with space. In that case, however, a space
devoid of ether, or one containing spheres of ether, would no longer be
conceivable, and there can be nothing but a single mass of ether with
stars floating in it. In a word, the negative result of Michelson’s
experiment could not be deduced _a priori_ from the problematical
identity of absolute space and the ether; but this negative result does
not justify us in denying the identity _a posteriori_.

Let us return to our proper subject, the Fitzgerald-Lorentz hypothesis
which explains the issue of the Michelson experiment, and which was
in a sense the spring-board for Einstein’s leap. The hypothesis is as
follows.

The result of the experiment is that, whereas when the path of a ray
of light between two mirrors is transverse to the earth’s motion
through ether, and it is then made parallel to the earth’s motion,
the path ought to be longer, we actually find no such lengthening.
According to Fitzgerald and Lorentz, _this is because the two mirrors
approached each other in the second part of the experiment_. To put
it differently, _the frame in which the mirrors were fixed contracted
in the direction of the earth’s motion, and the contraction was such in
magnitude as to compensate exactly for the lengthening of the path of
the ray of light which we ought to have detected_.

When we repeat the experiment with all kinds of different apparatus, we
find that the result is always the same (no displacement of the
fringes). It follows that the character of the material of which the
instrument is made—metal, glass, stone, wood, etc.—has nothing to
do with the result. Therefore all bodies undergo an equal and similar
contraction in the direction of their velocity relatively to the
ether. This contraction is such that it exactly compensates for the
lengthening of the path of the rays of light between two points of the
apparatus. In other words, the contraction is greater in proportion as
the velocity of bodies relatively to the ether becomes greater.

That is the explanation proposed by Fitzgerald. At first it seemed to
be very strange and arbitrary, yet there was, apparently, no other way
of explaining the result of Michelson’s experiment.

Moreover, when you reflect on it this contraction is found to be
less extraordinary, less startling, than one’s common sense at first
pronounces it. If we throw some non-rigid object, such as one of those
little balls with which children play, quickly against an obstacle,
we see that it is slightly pushed in at the surface by the obstacle,
precisely in the same sense as the Fitzgerald-Lorentz contraction.
The ball is no longer round. It is a little flattened, so that its
diameter is shortened in the direction of the obstacle. We have much
the same phenomenon, though in a more violent form, when a bullet
is flattened against a target. Therefore, if solid bodies are thus
capable of deformation—as they are, for cold is sufficient of itself
to concentrate their molecules more closely—there is nothing absurd or
impossible in supposing that a violent wind of ether may press them out
of shape.

But it is far less easy to admit that this alteration may be exactly
the same, in the given conditions, for all bodies, whatever be the
material of which they are composed. The little ball we referred to
would by no means be flattened so much if it were made of steel instead
of rubber.

Moreover, there is in this explanation something quite improbable,
something that shocks both our good sense and that caricature of
it which we call common sense. Is it possible to admit that the
contraction of bodies always exactly compensates for the optic effect
which we seek, whatever be the conditions of the experiment (and
they have been greatly varied)? Is it possible to admit that nature
acts as if it were playing hide-and-seek with us? By what mysterious
chance can there be a special circumstance, providentially and exactly
compensating for every phenomenon?

Clearly there must be some affinity, some hidden connection, between
this mysterious material contraction of Fitzgerald and the lengthening
of the light path for which it compensates. We shall see presently
how Einstein has illumined the mystery, revealed the mechanism which
connects the two phenomena, and thrown a broad and brilliant light upon
the whole subject. But we must not anticipate.

The contraction of the apparatus in Michelson’s experiment is extremely
slight. It is so slight that if the length of the instrument were equal
to the diameter of the earth—that is to say, 8,000 miles—it would be
shortened in the direction of the earth’s motion by only six and a half
centimetres! In other words, the contraction would be far too small to
be in any way measurable in the laboratory.

There is a further reason for this. Even if Michelson’s apparatus were
shortened by several inches—that is to say, if the earth travelled
thousands of times as rapidly as it does round the sun—we could not
detect and measure it. The measuring rods which we would use for the
purpose would contract in the same proportion. The deformation of any
object by a Fitzgerald-Lorentz contraction could not be established by
any observer on the earth. It could be discovered only by an observer
who did not share the movement of the earth: an observer on the sun,
for instance, or on a slow-moving planet like Jupiter or Saturn.

Micromegas would, before he left his planet to visit us, have been
able to discover, by optical means, that our globe is shortened by
several inches in the direction of its orbital movement; supposing
that Voltaire’s genial hero were provided with trigonometrical
apparatus infinitely more delicate than that used by our surveyors
and astronomers. But when he reached the earth, Micromegas, with
all his precise apparatus, would have found it impossible to detect
the contraction. He would have been greatly surprised—until he met
Einstein and heard, as we shall hear, the explanation of the mystery.

I have, unfortunately, neither the time nor the space—it is here,
especially, that space is relative, and is constantly shortened by the
flow of the pen—to give the dialogue which would have taken place
between Micromegas and Einstein. Perhaps, indeed, if we are to be
faithful to the Voltairean original, the dialogue would have been very
superficial, for—to speak confidentially—I believe that Voltaire
never quite understood Newton, though he wrote much about him, and
Newton was less difficult to understand than Einstein is. Neither did
Mme. du Châtelet, for all the praise that has been lavished upon her
translation of the immortal _Principia_. It swarms with meaningless
passages which show that, whether she knew Latin or no, she did not
understand Newton. But all this is another story, as Kipling would say.

The movement of the apparatus in the ether varies in speed according to
the hour and the month in which the Michelson and similar experiments
are made. As the compensation is always precise, we may try to
calculate the exact law which governs the contraction as a function
of velocities, and makes it, as we find, a precise compensation for
the latter. Lorentz has done this. Taking =V= as the velocity of
light and _v_ as the velocity of the body moving in ether, Lorentz
found that, in order to have compensation in all cases, the length of
the moving body must be shortened, in the plane of its progress, in the
proportion of

( _v_² )
1 to √(1 ———————— ).
( V² )

If we take by way of illustration the case of the orbital movement of
the earth, where v is equal to thirty kilometres, we find that the
earth contracts in the plane of its orbit in the proportion

( 1 )
√(1 ——————————— ).
( 100,000,000 )

The difference between these two numbers is ¹/₂₀₀,₀₀₀,₀₀₀, and
the two hundred millionth part of the earth’s diameter is equal to 6½
centimetres. It is the figure we had already found.

This formula, which gives the value of the contraction in all cases,
is elementary. Even the inexpert can easily see the meaning of it. It
enables us to calculate the extent of contraction for every rate of
velocity. We can easily deduce from it that if the earth’s orbital
motion were, not 30 kilometres, but 260,000 kilometres a second, it
would be shortened by one-half its diameter in the plane of its motion
(without any change in its dimensions in the perpendicular). At that
speed a sphere becomes a flattened ellipsoid, of which the small
axis is only half the length of the larger axis; a square becomes a
rectangle, of which the side parallel to the motion is twice as small
as the other.

These deformations would be visible to a stationary spectator, but they
would be imperceptible to an observer who shares the movement, for the
reason already given. The measuring rods and instruments, and even the
eye of the observer, would be equally and simultaneously altered.

Think of the distorting mirrors which one sees at times in places of
amusement. Some show you a greatly elongated picture of yourself,
without altering your breadth. Others show you of your normal height,
but grotesquely enlarged in width. Try, now, to measure your height and
breadth with a rule, as they are given in these deformed reflections in
the mirror. If your real height is 5 feet 6 inches, and your real width
2 feet, the rule will, when you apply it to the strange reflection of
yourself in the glass, merely tell you that this figure is 5 feet 6
inches in height and 2 feet in breadth. The rule as seen in the mirror
undergoes the same distortion as yourself.

Hence it is that, even if the globe of the earth had the fantastic
speed which we suggested above, its inhabitants would have no means
of discovering that they and it were shortened by one-half in the
plane east to west. A man 5 feet 6 inches in height, lying in a large
square bed in the direction north-south, then changing his position to
east-west, would, quite unknown to himself, have his length reduced
to 2 feet 9 inches. At the same time he would become twice as stout
as before, because previously his breadth was orientated from east to
west. But the earth travels at the rate of only thirty kilometres
a second, and its entire contraction is only a matter of a few
centimetres.

In contrast with the earth’s velocity, the speed of our most rapid
means of transport is only a small fraction of a kilometre a second.
An aeroplane going at 360 kilometres an hour has a speed of only 100
metres a second. Hence the maximum Fitzgerald-Lorentz contraction of
our speediest machines can only be such an infinitesimal fraction of an
inch that it is entirely imperceptible to us. That is why—that is the
only reason why—the solid objects with which we are familiar seem to
keep a constant shape, at whatever speed they pass before our eyes. It
would be quite otherwise if their speed were hundreds of thousands of
times greater.

All this is very strange, very surprising, very fantastic, very
difficult to admit. Yet it is a fact, if there really is this
Fitzgerald-Lorentz contraction, which has so far proved the only
possible explanation of the Michelson experiment. But we have already
seen some of the difficulties that we find in entertaining the
existence of this contraction.

There are others. If all that we have just said is true, only objects
which are stationary in the ether would retain their true shapes, for
the shape is altered as soon as there is movement through the ether.
Hence, amongst the objects which we think spherical in the material
world (planets, stars, projectiles, drops of water, and so on), there
would be some that really are spheres, whilst others would, on account
of the speed or slowness of their movements, be merely elongated or
flattened ellipsoids, altered in shape by their velocity. Amongst the
various square objects, some would be really square, while others,
travelling at different speeds relatively to the ether, would be rather
rectangles, shortened on their longer sides owing to their velocity.
And it is supposed that we would have no means of knowing which of
these objects moving at different speeds are really shaped as we think
and which are shaped otherwise, because, as the Michelson experiment
proves, we cannot detect a velocity relatively to the ether.

This we utterly decline to believe, say the Relativists. There are too
many difficulties about the matter. Why speak persistently, as Lorentz
does, of velocities in relation to the ether, when no experiment can
detect such a velocity, yet experiment is the sole source of scientific
truth? Why, on the other hand, admit that some of the objects we
perceive have the privilege of appearing to us in their real shape,
without alteration, while others do not? Why admit such a thing when it
is, of its very nature, repugnant to the spirit of science, which is
always opposed to exceptions in nature—science deals only with general
laws—especially when the exceptions are imperceptible?

That was the state of affairs—very advanced from the point of view of
the mathematical expression of phenomena, but very confused, deceptive,
contradictory, and troublesome from the physical point of view—when
“at length Malherbe arrived” ... I mean Einstein.

CHAPTER III

EINSTEIN’S SOLUTION

Einstein’s first act of intelligent audacity was that, without
relegating the ether to the category of those obsolete fluids, such as
phlogiston and animal spirits, which obstructed the avenues of science
until Lavoisier appeared—without denying all reality to ether, for
there must be some sort of support for the rays which reach us from the
sun—he observed that, in all that we have as yet seen, there is always
question of velocities relatively to the ether.

We have no means whatever of establishing such velocities, and perhaps
it would be simpler to leave out of our arguments this entity, real
or otherwise, which is inaccessible and merely plays the futile and
troublesome part of fifth wheel to the electro-magnetic chariot in the
progress of physicists along the ruts of their difficulties.

The first point is then: Einstein begins, provisionally, by omitting
the ether from his line of reasoning. He neither denies nor affirms its
existence. He begins by ignoring it.

We will now follow his example. We shall no longer, in the course of
our demonstration, speak about the medium in which light travels. We
shall consider light only in relation to the beings or material objects
which emit or receive it. We shall find that our progress becomes at
once much easier. For the moment we will relegate the ether of the
physicists to the store of useless accessories, along with the suave,
formless, vague—but so precious artistically—ether of the poets.

* * * * *

Shortly, what does Michelson’s experiment prove? Only that a ray of
light travels at the surface of the earth from west to east at exactly
the same speed as from east to west. Let us imagine two similar guns in
the middle of a plain, both firing at the same moment, in calm weather,
and discharging their shells with the same initial velocity, but one
toward the west and the other toward the east. It is clear that the two
shells will take the same time to traverse an equal amount of space,
one going toward the west and the other toward the east. The rays of
light which we produce on the earth behave in this respect, as regards
their progress, exactly as the shells do. There would therefore be
nothing surprising in the result of the Michelson experiment, if we
knew only what experience tells us about the luminous rays.

But let us push the comparison further. Let us consider the shell fired
by one of the guns, and imagine that it hits a target at a certain
spot, and that, when it reaches the target, the residual velocity of
the shell is, let us say, fifty metres a second. I imagine the target
mounted on a motor tractor. If the latter is stationary the velocity of
the shell in relation to the target will be, as we said, fifty metres a
second at the point of impact. But let us suppose that the tractor and
the target are moving at a speed of, for instance, ten metres a second
toward the gun, so that the target passes to its preceding position
exactly at the moment when the shell strikes it. It is clear that the
velocity of the shell relatively to the target at the moment of impact
will not now be fifty metres, but 50 + 10 = 60 metres a second. It
is equally evident that the speed will fall to 50-10 = 40 metres a
second if (other things being equal) the target is travelling away from
the gun, instead of toward it. If, in the latter case, the velocity
of the target were equal to that of the shell, it is clear that the
relative velocity of the shell would now be _nil_.

So much is clear enough. That is how jugglers in the music-halls can
catch eggs falling from a height on plates without breaking them. It is
enough to give the plate, at the moment of contact, a slight downward
velocity, which lessens by so much the velocity of the shock. That is
also how skilled boxers make a movement backward before a blow, and
thus lessen its effective force, whereas the blow is all the harder if
they advance to meet it.

If the luminous rays behaved in all respects like the shells, as they
do in the Michelson experiment, what would be the result? When one
advances very rapidly to meet a ray of light, one ought to find its
velocity increased relatively to the observer, and lessened if the
observer recedes before it. If this were the case, all would be simple;
the laws of optics would be the same as those of mechanics; there
would be no contradiction to sow discord in the peaceful army of our
physicists, and Einstein would have had to spend the resources of his
genius on other matters.

Unfortunately—perhaps we ought to say fortunately, because, after all,
it is the unforeseen and the mysterious that lend some charm to the
way of the world—this is not the case. Both physical and astronomical
observation show that, under all conditions, when an observer advances
rapidly toward luminous waves or recedes rapidly from them, they still
show always the same velocity relatively to him. To take a particular
case, there are in the heavens stars which recede from us and stars
which approach us; that is to say, stars from which we recede, or which
we approach, at a speed of tens, and in some cases hundreds, of miles a
second. But an astronomer, de Sitter, has proved that the velocity of
the light which reaches us is, for us, always exactly the same.

Thus, up to the present it has proved quite impossible for us, by any
device or movement, to add to or lessen in the least the velocity with
which a ray of light reaches us. The observer finds that the rate of
speed of the light is always exactly the same relatively to himself,
whether the light comes from a source which rapidly approaches or
recedes from him, whether he is advancing toward it or retreating
before it. The observer can always increase or lessen, relatively to
himself, the speed of a shell, a wave of sound, or any moving object,
by pushing toward or moving away from the object. When the moving
object is a ray of light, he can do nothing of the kind. The speed of a
vehicle cannot in any case be added to that of the light it receives or
emits, or be subtracted from it.

This fixed speed of about 186,000 miles a second, which we find always
in the case of light, is in many respects analogous to the temperature
of 273° below zero which is known as “absolute zero.” This also is, in
nature, an impassable limit.

All this proves that the laws which govern optical phenomena are not
the same as the classic laws of mechanical phenomena. It was for
the purpose of reconciling these apparently contradictory laws that
Lorentz, following Fitzgerald, gave us the strange hypothesis of
contraction.

* * * * *

But we shall now find Einstein showing us, in luminous fashion, that
this contraction is seen to be perfectly natural when we abandon
certain conceptions—perhaps erroneous, though classical—which ruled
our habitual and traditional way of estimating lengths of space and
periods of time.

Take any object—a measuring rod, for instance. What is it that settles
for us the apparent length of the rod? It is the image made upon our
retina by the two rays that come from the two ends of the rod, and
which reach our eye _simultaneously_.

I italicise the word, because it is the key of the whole matter. If the
rod is stationary before us, the case is simple. But if it is moved
while we are looking at it, the case is less simple. It is so much
less simple that before the work of Einstein most of our learned men
and the whole of classic science thought that the instantaneous image
of an object that was not subject to change of shape was necessarily
and always identical, and independent of the velocities of the object
and the observer. The whole of classical science argued as if the
spread of light was itself instantaneous—as if it had an infinite
velocity—which is not the case.

I stand on the bank by the side of a railway. On the line is a handsome
Pullman car, in which it is so pleasant to think that space is
relative, in the Galileian sense of the word. Close to the line I have
two pegs fixed, one blue, the other red, and they exactly mark the
ends of the coach and indicate its length. Then, without leaving my
observation-post on the bank, my face turned towards the middle of
the coach, I give orders for the coach to be drawn back and coupled
to a locomotive of unheard-of power, which is to carry the coach past
me at a fantastic speed, millions of times faster than the speed any
mere engineer could provide. Such is the potential superiority of
the imagination over sober reality! I assume further that my retina
is perfect, and is so constituted that the visual impressions will
remain on it only as long as the light which causes them. These
somewhat arbitrary suppositions count for nothing in the essence of the
demonstration. They are only for the sake of convenience.

Now for the question. Will the coach (which I assume to be of some
rigid metal), as it passes before me at full speed, seem to me to
be exactly the same length as it did when it was at rest? To put it
differently, at the moment when I see its front end coincide with the
blue peg I had planted, shall I see its back end coincide at the same
time with the red peg? To this question Galileo, Newton, and all the
supporters of classic science would reply _yes_. Yet according to
Einstein the answer is _no_.

Here is the simple proof, as we deduce it from Einstein’s general idea.

I am, recollect, on the edge of the track, at an equal distance from
both pegs. When the front end of the coach coincides with the blue peg,
it sends toward my eye a certain ray of light (which, for convenience,
we will call the front ray), and this coincides with the luminous ray
coming to me from the blue peg. This front ray reaches my eye _at
the same time_ as a certain ray that comes from the back end of the
coach (which we will call the back ray). Does the back ray coincide
with the ray which comes to me from the red peg? Clearly not. The front
ray leaves the front end of the coach at the same speed as the back ray
leaves the back end; as any observer in the coach would find who cared
to try the Michelson experiment on them. But the front end of the coach
is receding from me while the back end is approaching me. Hence the
front ray travels toward my eye more slowly than the back ray, though
I cannot perceive this, as, when they reach me, I find that they both
have the same velocity. Hence the back ray, which reaches my eye at the
same time as the front ray, must have left the back end of the coach
later than the front ray left the front end of the coach. Therefore,
when I see the front end of the coach coincide with the blue peg, I
at the same time see the back end of the carriage _after_ it has
passed the red peg. Therefore the length of a coach travelling at full
speed, and such as it appears to me, is shorter than the distance
between the two pegs, which indicated the length of the coach at rest.
Q.E.D.

Very little attention is needed for any person to understand this
argument, though its elementary simplicity has not been attained
without difficulty. It is part of Einstein’s mathematical argument and
of his conception of simultaneity.

* * * * *

It follows that the coach, or, in general, any object, seems to
be contracted in virtue of its velocity, and in the direction of
that velocity, relatively to the spectator. The same thing happens,
obviously, if the observer moves in relation to the object, because we
can know only relative velocities, in virtue of the Classical Principle
of Relativity of Newton and Galileo.

In this new light the Lorentz-Fitzgerald contraction becomes
intelligible, or at least admissible. The contraction, thus considered,
is not the cause of the negative result of the Michelson experiment: it
is an effect of it. It is now quite clear, and we see that there was
something wrong with the classical way of estimating the instantaneous
dimension of objects.

Certainly the fact that luminous rays, starting out from their sources
at different speeds, should have the same speed when they reach our
eye, is strange. It upsets our habitual way of looking at things. If
I may venture to use a comparison simply for the purpose of provoking
reflection, not at all in the way of explanation, we have here
something analogous to what happens with the bombs of aviators. Bombs
of a given type, whether released at a height of 5,000 or of 10,000
metres, which therefore have very different downward velocities at
5,000 metres from the ground, have always the same residual velocity
when they reach the ground. This is due to the moderating and
equalising influence of the atmospheric resistance, which prevents the
speed from increasing indefinitely, and makes it constant when it has
attained a certain value.

Must we suppose that there is round our eye and round objects a sort of
field of resistance which sets a similar limit to the light? Who knows?
But perhaps such questions have no meaning for the physicist. He can
know nothing about the behaviour of light except when it leaves its
source or when it reaches the eye, whether armed with instruments or no.
He cannot learn how it behaves during its passage across the
intermediate space, in which there is no matter.

Indeed, the more deeply we study the new physics, the more we see that
it derives almost all its strength from its systematic disdain of all
that is beyond phenomena, all that cannot fall under experimental
observation. It is because it is solely based upon facts (however
contradictory they may be) that our proof of the necessary contraction
of objects owing to their velocity relatively to the observer is so
strong.

* * * * *

We must understand the profound significance of the Fitzgerald-Lorentz
contraction. This apparent contraction is by no means due to the
movement of objects relatively to the ether. It is essentially the
effect of the movements of objects and observers relatively to each
other, or relative movements in the sense of the older mechanics.

The greatest relative velocities to which we are accustomed in our
daily life are less than a few kilometres a second. The initial
velocity of the shell fired by “Bertha” was only about 1,300 metres
a second. For movements so slow as this the Relativist contraction
is entirely negligible. Hence, as the classical mechanics had never
observed such contraction, it regarded the shapes and dimensions of
rigid objects as independent of systems of reference.

It was very nearly true; and that makes all the difference between
true and false. To say that 999,990 + 9 = 1,000,000, is to say
something that is very nearly true, and is therefore false. When it
was discovered that the earth was round no change was made in their
procedure by architects. They continued to build as if the direction
indicated by the plumb-line was always parallel to itself. In the same
way those who make our locomotives and aeroplanes will not have to
consider the forms of the machines as dependent on their velocities.
What does it matter? The practical point of view is not, and cannot be,
that of science except indirectly. So much the worse if there is no
indirect influence, or if it is slow in coming.

Some years ago, however, we discovered things which move at speeds,
relatively to us, of tens or hundreds of thousands of kilometres a
second; the projectiles of the cathode rays and of radium. In this case
the Relativist contraction is very considerable. We shall see how it
has been observed.

But let us first recapitulate what we have seen. Objects seem to alter
their shape in the direction of their movement and not in the direction
perpendicular to this. Therefore their forms, even if they be composed
of an ideal and perfectly rigid material, depend on their velocity
relatively to the observer. This is the essentially new point of view
which Einstein’s “Special Relativity” superimposes upon the Relativity
of classical mechanics and philosophers. For these the absolute
dimensions of a rigid object or a geometrical figure were not absolute;
it was only the _relations_ of these dimensions which were real.

The new point of view is that these relations are themselves relative,
because they are a function of the velocity of the observer. It is
a sort of Relativity in the second degree, of which neither the
philosophers nor the classic physicists had dreamed.

Spatial relations themselves are relative, in a space which is already
relative.

In the case of our Pullman car and the two pegs which mark its length
when it is stationary, an observer situated in the carriage would find
the distance between the two pegs shortened as he passes them. The
coach would seem to him longer than the distance between the pegs. I
who remain beside the pegs observe the contrary. Yet I have no means of
proving to the passenger that he is wrong. I see quite plainly that the
ray of light which comes from the back peg runs behind the coach, and
has therefore, relatively to it, a speed of less than 186,000 miles a
second. I know that this is the reason for the passenger’s error, but I
have no means of convincing him that he is wrong. He will always say,
and rightly: “I have measured the speed at which this ray reaches me,
and I have found it 186,000 miles a second.” Each of us is really right.

In very rapid motion a square would seem to the observer a rectangle;
a circle would appear to be an ellipse. If the earth travelled some
thousands of times faster round the sun, we should see it elongated,
like a giant lemon suspended in the heavens. If an aviator could fly
at a fantastic speed over Trafalgar Square, in the direction of the
Strand—and if the impressions on his retina were instantaneous—he
would see the Square as a very flattened rectangle. If he flew in a
diagonal line about it, he would find it shaped like a lozenge. If the
same aviator flew across a road on which fat cattle were being driven
to the slaughter-house, he would be astonished, for the beasts would
seem to him extraordinarily lean, while there would be no change in
their length.

The fact that these alterations of shape owing to velocity are
reciprocal is one of the most curious consequences of all this. A man
who could pass in every direction amongst his fellows at the fantastic
speed of one of Shakespeare’s spirits—let us put it at about 170,000
miles an hour, though there would be no limit—would find that his
fellows had become dwarfs only half as large as himself. Would he have
become a giant, a sort of Gulliver amongst the Lilliputians? Not in
the least. Such is the justice of the scheme of earthly things that he
himself would seem a dwarf to the people whom he thought smaller than
himself, and who are quite sure of the contrary.

Which is right, and which wrong? Both. Each point of view is accurate,
but there are only personal points of view.

Again, any observer whatever will only see things that are not
connected with him as smaller—never larger—than the things which are
connected with his movement. If I might venture to relieve this sober
exposition by a reflexion rather less austere than is usual in physics,
I would say that the new system affords a supreme justification of
egoism, or, rather, of egocentricism.

It is the same with time as with space. By similar reasoning to that
which has shown us how the distance of things in space is connected
with their velocity relatively to the observer, it can be shown that
their distance in time likewise depends upon this.

It would be useless to reproduce here the whole of the Einsteinian
argument as to duration. It is analogous to that which we have used
in regard to length, and even simpler. The result is as follows. The
time expressed in seconds which a train takes to pass from one station
to another is shorter for the passengers on the train than for us who
watch it pass, though our watches may be just the same as theirs.[5]
Similarly, all the gestures of men who are on moving vehicles will seem
to a stationary observer slowed down, and therefore prolonged, and vice
versa. But the velocity would, as in the case of variation in length,
have to be fantastic to make these variations in time perceptible.

[5] The best definition of the second that can be given is the
following: it is the time which light takes to cover 186,000 miles
in empty space and far from any strong gravitational field. This
definition, the only strict definition, is further justified by the
fact that there is no better means of regulating clocks than luminous
or Hertzian (which have the same speed) signals.

It is not less true that the time between the birth and the death of
any creature, its life, will seem longer if the creature moves rapidly
and fantastically relatively to the observer. In this world, where
appearance is almost everything, this is not without importance, and it
follows that, philosophically speaking, to move on is to last longer;
but for others, not for oneself; just as others may seem to me to last
longer. A striking, a profound, an unforeseen justification of the
words of the sage: immobility is death!

* * * * *

Formerly, before the Einsteinian _hegira_, before the Relativist
Era opened, everybody was convinced that the portion of _space_
occupied by an object was sufficiently and explicitly defined by its
dimensions—length, breadth, and height. These are what are called
the three _dimensions_ of an object; just as we speak, to use a
different expression, of the longitude, latitude, and altitude of each
of its points, or as we speak in astronomy of its right ascension,
declination, and distance.

It was quite understood that we had, in addition, to indicate the
epoch, the moment, to which these data correspond. If I define the
position of an aeroplane by its longitude, latitude, and altitude,
these indications are only correct for a certain moment, because the
aeroplane is moving relatively to the observer, and the moment also
must be indicated. In this sense it has long been known that space
depends upon time.

But the Relativist theory shows that it depends upon time in a much
more intimate and deeper manner, and that time and space are as closely
connected as those twin monsters which the surgeon cannot separate
without killing both.

The dimensions of an object, its shape, the apparent _space_
occupied by it, depend upon its velocity: that is to say, upon the
_time_ which the observer takes to traverse a certain distance
relatively to the object. Here we have _space_ already depending
upon _time_. In addition, the observer measures the time with
a chronometer, the seconds of which are more or less accelerated
according to his velocity.

Hence it is impossible to define space without time. That is why we
now say that time is the fourth dimension of space, or that the space
in which we live has four dimensions. It is remarkable that there were
able men in the past who had a more or less clear intuition of this.
Thus we find Diderot, in 1777, writing in the _Encyclopédie_, in
the article “Dimension”:

“I have already said that it is impossible to conceive more than three
dimensions. A learned man of my acquaintance, however, believes that
one might regard duration as a fourth dimension, and that the product
of time by solidity would be, in a sense, a product of four dimensions.
The idea may not be admitted, but it seems to be not without merit, if
it be only the merit of originality.”

It was algebra, undoubtedly, that gave rise to the idea of a space with
more than three dimensions. Since, in point of fact, lines or spaces of
one dimension are represented by algebraical expressions of the first
degree, surfaces or spaces of two dimensions by formulæ of the second
degree, and volumes or spaces of three dimensions by expressions of the
third degree, it was natural to ask oneself if formulæ of the fourth
and higher degrees are not also the algebraical representation of some
form of space with four or more dimensions.

The four-dimensional space of the Relativists is, however, not quite
what Diderot imagined. It is not the product of time by extension, for
a diminution of time is not compensated in it by an increase of space.
Quite the contrary. Take two events, such as the successive passage
of our Pullman car through two stations. For a passenger in the car
the distance between the two stations, measured by the length of the
track covered, is, as we saw, shorter than for a person who is standing
stationary beside the line. The time between passing through the two
stations is likewise less for the first observer. The number of seconds
and fractions of seconds marked by his chronometer is smaller for him,
as we saw.

In a word, distance in time and distance in space diminish
simultaneously when the velocity of the observer increases, and both
increase when the velocity of the observer lessens.

Thus velocity (velocity relatively to the things observed, we must
always remember) acts in a sense as a double brake lessening durations
and shortening lengths. If a different illustration be preferred,
velocity enables us to see both spaces and times more obliquely, at an
increasingly sharp angle. Space and time are therefore only changing
effects of perspective.

Can we conceive space of four dimensions? That is to say, can we
imagine or visualise it? Even if we cannot, it proves nothing as
regards the reality of such space. During ages no one conceived such
a thing as the Hertzian waves, and even to-day we have no direct
sense-impression of them. They exist none the less. As a matter of
fact, we find it difficult to conceive space of three dimensions. If it
were not for our muscular changes, we should know nothing about it. A
paralysed and one-eyed man, that is to say, a man without the sensation
of relief which we get from binocular vision—and even this is, in the
first place, a muscular sensation—would, with his single eye, see all
objects on the same plane, as on the drop-scene of a theatre. He could
have no perception of three-dimensional space.

I believe there are people who can form an idea of four-dimensional
space. The successive appearances of a flower in its various phases of
growth, from the day when it is but a frail green bud until the time
when its exhausted petals fall sadly to the ground, and the successive
changes of its corolla under the influence of the wind, give us a
globular image of the flower in four-dimensional space.

Are there any who can see all this together? I believe that there are,
especially amongst good chess-players. When a skilful player plays
well, it is because he can take in with a single glance of his mental
eye the whole chronological and spatial series of moves that may follow
the first move, with all their effects on the board. He _sees the
whole series simultaneously_.

The words I have italicised look contradictory. That is because we are
in a province where it is all but impossible to express the fine shades
of things in words. One might just as well attempt to define verbally
all that there is in a symphony of Beethoven. “The translator is a
traitor.” If there is any truth in the proverb, it is because words are
the organ of translation.

* * * * *

We have reached a point in our gradual progress into Relativist physics
where we have before our eyes merely a battlefield strewn with corpses
and ruins.

We had regarded time and space as hooks solidly fastened to the wall
behind which lurks reality, and on these we hang our floating ideas of
the material world, just as we hang our coats on the rack. Now they
lie, torn down and crumpled, amongst the rubbish of ancient theories,
victims of the hammer-blows of the new physics.

We knew quite well, of course, that the souls of men were inscrutable
to us, but we did think that we saw their faces. Now, as we approach
them, we find that it is only masks we saw. The material world, as
Einstein shows it to us, is a sort of masked ball, and, by a deceptive
irony, it is we ourselves who have made the black velvet masks and the
gay costumes.

Instead of revealing reality to us, space and time are, according to
Einstein, only moving veils, woven by ourselves, which hide it from us.
Yet—strange and melancholy reflection—we can no more conceive the
world without space and time than we can observe certain microbes under
the microscope without first injecting colouring matter into them.

Are time and space, then, merely hallucinations? And, if so, what
_is_ real?

No. Once the Relativist has thrown down the tottering ruins, he begins
to reconstruct. Behind the veils, now torn down and trodden under foot,
a new and more subtle reality is about to appear.

If we describe the universe in the usual way, in separate categories
of space and time, we see that its aspect depends upon the observer.
Happily, it is not the same when we describe it in the unique category
of the four-dimensional continuum in which Einstein locates phenomena,
and in which space and time are inseparably united.

If I may venture to use this illustration, time and space are like
two mirrors, one convex, the other concave, the curvature of which
is accentuated in proportion to the velocity of the observer. Each
of these mirrors gives us, separately, a distorted picture of the
succession of things. But this is fortunately compensated for by the
fact that, when we combine the two mirrors so that one reflects the
rays received by the other, the picture of the succession of things is
restored in its unaltered reality.

The distance in time and the distance in space of two given events
which are close to each other both increase or decrease when the
velocity of the observer decreases or increases. We have shown
that. But an easy calculation—easy on account of the formula given
previously to express the Lorentz-Fitzgerald contraction—shows that
there is a constant relation between these concomitant variations of
time and space. To be precise, the distance in time and the distance in
space between two contiguous events are numerically to each other as
the hypotenuse and another side of a rectangular triangle are to the
third side, which remains invariable.[6]

[6] In the geometrical calculus or representation that may be
substituted for this the hypotenuse of the triangle is the distance in
time, each second being represented by 300,000 kilometres.

Taking this third side for base, the other two will describe, above
it, a triangle more or less elevated according as the velocity of the
observer is more or less reduced. This fixed base of the triangle, of
which the other two sides—the spatial distance and the chronological
distance—vary simultaneously with the velocity of the observer, is,
therefore, a quantity independent of the velocity.

It is this quantity which Einstein has called the _Interval_
of events. This “Interval” of things in four-dimensional space-time
is a sort of conglomerate of space and time, an amalgam of the two.
Its components may vary, but it remains itself invariable. It is the
constant resultant of two changing vectors. The “Interval” of events,
thus defined, gives us for the first time, according to Relativist
physics, an impersonal representation of the universe. In the striking
words of Minkowski, “space and time are mere phantoms. All that exists
in reality is a sort of intimate union of these entities.”

The sole reality accessible to man in the external world, the one
really objective and impersonal thing which is comprehensible, is the
Einsteinian _Interval_ as we have defined it. The _Interval_
of events is to Relativists the sole perceptible part of the real.
Apart from that there is something, perhaps, but nothing that we can
know.

Strange destiny of human thought! The principle of relativity has, in
virtue of the discoveries of modern physics, spread its wings much
farther than it did before, and has reached summits which were thought
beyond the range of its soaring flight. Yet it is to this we owe,
perhaps, our first real perception of our weakness in regard to the
world of sense, in regard to reality.

Einstein’s system, of which we have now to see the constructive
part, will disappear some day like the others, for in science there
are merely theories with “provisional titles,” never theories with
“definitive titles.” Possibly that is the reason of its many victories.
The idea of the _Interval_ of things will, no doubt, survive all
these changes. The science of the future must be built upon it. The
bold structure of the science of our time rises upon it daily.

It must, in fine, be clearly understood that the _Einsteinian
Interval_ tells us nothing about the absolute, about things in
themselves. It, like all others, shows us only relations between
things. But the relations which it discloses seem to be real and
unvarying. They share the degree of objective truth which classic
science attributed, with, perhaps, unfounded assurance, to the
chronological and spatial relations of phenomena. In the view of the
new physics these were but false scales. The Einsteinian Interval alone
shows us what can be known of reality.

Einstein’s system, therefore, takes pride in having lifted for all
future time a corner of the veil which conceals from us the sacred
nudity of nature.

CHAPTER IV

EINSTEIN’S MECHANICS

When Baudelaire wrote:

I hate the movement that displaces lines,

he thought only, like the physicists of his time, of the static
deformations which have been known as long as there have been men
to observe them. What we have seen about Einsteinian time and space
has taught us that there must be, in addition to these, kinematic
deformations, to which every material object, however rigid it seems,
is liable.

Movement, therefore, displaces lines much more than Baudelaire
supposed, even the lines of the hardest of marble statues. This kind of
deformation, which is pleasant rather than hateful, since it brings us
nearer to the heart of things, has upset the whole of mechanics.

Mechanics is at the foundation of all the experimental sciences,
because it is the simplest, and because the phenomena it studies are
always present—if not exclusively present—amongst the phenomenal
objects of the other sciences, such as physics, chemistry, and biology.

The converse of this is not true. For instance, there is not a single
phenomenon in chemistry or biology in which one has not to study bodies
in movement, objects endowed with mass and giving out or absorbing
energy. On the other hand, the peculiar aspects of a biological,
chemical, or physical phenomenon, such as the existence of a difference
of potential, an oxidation, or an osmotic pressure, are not always
found in the study of the movements of a ponderable mass and of the
forces which act upon and through it.

Compared with mechanics, the sciences of physics, chemistry, and
biology have, in the order in which we name them, objects of increasing
complexity and generality, or, to put it better, of decreasing
universality. These sciences are mutually dependent in the way that the
trunk, branches, leaves, and flowers of a tree are. They are to some
extent related to each other as are the various parts of the jointed
masts on which military telegraphists fix their antennæ. The lower part
of the mast, the larger part, sustains the whole; but it is the upper
parts which bear the delicate and complicated organs.

The object of the great synthetists in science has always been, and
is, to reduce all phenomena to mechanical phenomena, as Descartes
attempted. Whether these attempts are well-grounded or no, whether
they will some day succeed or are condemned _a priori_ to
failure because physico-biological phenomena involve elements that
are essentially incapable of reduction to mechanical elements, is a
question that has been, and will continue to be, much discussed. But,
however thinkers may differ on that point, they are agreed on this: in
all natural phenomena, in all phenomena that are objects of science,
there is the mechanical element—exclusive in some, the principal
element in others.

All this leads to the conclusion that whatever modifies mechanics,
modifies at the same time the whole structure of ideas founded
thereon—that is to say, the other sciences, the whole of science, our
entire conception of the universe. But we are now going to see that
Einstein’s theory, as a direct effect of what it teaches in regard to
space and time, completely upsets the classical mechanics. It is in
this way, particularly, that it has shaken the rather somnolent frame
of traditional science, and the vibration is not yet over.

In approaching the Einsteinian mechanics we shall have the pleasure of
passing from ideas of time and space that are rather too exclusively
geometrical and psychological to the direct study of material
realities, of _bodies_. Here we can compare theory and reality,
the mathematical premises and the substantial verifications; and we
shall be pleased to see what the facts, given in experience, have to
say on the matter. We shall be able to make our choice, with informed
minds and sound criteria, between the old and the new ideas.

In a word, if I may use this illustration, as long as we were dealing
with ideas of space and time—which are empty frames in themselves,
vases that would interest us chiefly by the liquids they contain—we
were rather like the young men who have to choose a _fiancée_
solely by the description of her which has been given them. We are
now going to see with our own eyes, and see at work the two aspirants
to our affection: classical science and Einstein’s theory. We shall
see both of them take up the paste of facts, and we shall be able to
compare the delicious dishes which they respectively make from it for
the nourishment of the mind.

Theories have no value except as functions of facts. Those which, like
so many in metaphysics, have no real criterion by which we may test
them, are all of the same value. Experience, the sole source of truth
of which Lucretius said long ago:

unde omnia credita pendent,

or the material facts, is going to judge Einstein’s system for us.

* * * * *

The result of the Michelson experiment, the impossibility of proving
any velocity of the earth in relation to the medium in which light is
propagated, amounts to this: we have no means whatever of detecting
a speed higher than that of light. This consequence of the Michelson
experiment will be better understood, perhaps, if we put it in a
tangible form. Here is an illustration that will serve our purpose.

In some astronomical novel an imaginary observer is supposed to recede
from the earth at a speed greater than that of light—at 300,000 miles
a second, let us say—yet to keep his eyes (armed with prodigious
glasses) steadily fixed on this little globe of ours.

What will happen? Evidently, our observer will see the train of earthly
events in inverse order, because in the course of his voyage he will
catch up in succession the luminous waves which left the earth before
him. The farther away they are, the longer it must be since they left
the earth. After a time our man, or our superman, will witness the
Battle of the Marne. He will first see the field strewn with the
dead. Gradually the dead men will rise and join their regiments, and
presently they will be seen in groups in Gallieni’s taxis, which will
travel backwards at full speed to Paris, arriving in the midst of a
population that is extremely anxious about the issue of the struggle,
and the soldiers will, naturally, be unable to give them any news. In
a word, our observer will, if he recedes from the earth at a speed
greater than that of light, see terrestrial events happening as if he
were _ascending_ the stream of time.

It would be very different if the observer remained stationary, and
the earth receded from him at a speed of 300,000 miles a second. What
would happen then? It is clear that in this case our observer will see
terrestrial events, not in inverse order, but as they are: except that
they would seem to him to take place with majestic slowness, because
the rays of light which leave the earth at the end of some particular
event will take a much longer time to reach him than the rays which
left the earth at the beginning of the event.

In sum, the phenomena observed by him being essentially different in
the two cases, our imaginary observer would be able to say whether it
is he who is receding from the earth or the earth that is receding
from him; to detect the real movement of the event through space. This
means, of course, movement relatively to the medium of the propagation
of light, not necessarily, as we saw, movement in relation to absolute
space.

The experiment we have imagined could not very well be carried out
with the actual resources of our laboratories. We cannot attain these
fantastic speeds, and even if we could the observer would not
distinguish much. But we have chosen a colossal instance, and the
results of it would be colossal, as there would be question of nothing
less than a reversal of the order of time.

If we were to use more modest means, the results will be more modest,
but according to the older theories they ought to be recorded in our
instruments. But the Michelson experiment—a miniature version of what
we have just described—shows that the differences we should expect
are not observed. Therefore the premise we laid down—that there can
be velocities greater than that of light in empty space—does not
harmonise with reality. Hence this velocity of light is a wall, a limit
that cannot be passed.

* * * * *

Now let us see what follows. There is at the base of classical
mechanics, as it was founded by Galileo, Huyghens, and Newton, and as
it is taught everywhere, a principle which is in the long run, like
all the principles of mechanics, grounded upon experience. It is the
principle of the composition of velocities. If a boat, which makes
ten miles an hour in smooth water, sails down a river which flows at
five miles an hour, the speed of the boat in relation to the bank will
be, as we may find by actual measuring, equal to the sum of the two
speeds, or fifteen miles an hour. This is the rule of the addition of
velocities.

In a more general way, if a body starts from a state of rest, and
under the action of some force takes on in a second the velocity
=V=, what will it do if the action of the force is prolonged for
another second? According to classical mechanics it will take on the
velocity =2V=.[7] Let us imagine an observer who is travelling
at the velocity =V=, yet thinks he is at rest. It will seem to
him, at the end of the first second, that the body is at rest (because
it has the same velocity as the observer). In virtue of the Classical
Principle of Relativity, the apparent movement of the body must be the
same for our observer as if the rest were real. This means that at the
end of the second second the relative velocity of the body in reference
to the observer will be =V=, and, as the observer already has the
velocity =V=, the absolute velocity of the body will be =2V=.
In the same way it will be =3V= at the end of three seconds,
=4V= at the end of four seconds, and so on. Could it increase
indefinitely if the force continues to act long enough? Classical
mechanics says “yes.” Einstein says “no,” because there cannot be a
greater velocity than that of light.

[7] As an example of an identical force acting during periods of time
successively equal to 1, 2, or 3, we may take three guns of the same
calibre, but of lengths equal to 1, 2, and 3, and of which the charges,
or rather, their propulsive forces, are identical and constant. It is
found that the initial velocities of the shells are, in relation to
each other, 1, 2, and 3.

We have imagined an observer who has the velocity V relatively to us,
and who believes that he is at rest. For him the body observed was
likewise at rest at the beginning of the second second, because its
velocity was the same as that of the observer. From the fact that the
apparent movement of the body is for the observer, during the second
second, the same as it was for us during the first, classical mechanics
concluded that its velocity doubles during the second second. It did
not know what Einstein has now taught us: that the time and space of
this observer are different from ours.

What is a velocity? It is the space traversed in the course of a
second. But the space thus measured by our moving observer, which he
believes to be of a certain length, is in reality, for us who are
stationary, smaller than he thinks, because the rules he uses are, as
Einstein has shown, shortened by velocity without his perceiving it.
Therefore the velocities are not added together in equal proportions
and indefinitely for a given observer, as classical mechanics
maintained.

Under the action of the same force, the old mechanics said, a body
will always experience the same acceleration, whatever be the velocity
already acquired. Under the action of the same force, the new mechanics
says, the motion of the body will be accelerated less and less in
proportion to its velocity.

Take, for instance, some movable object having, relatively to me, a
velocity of 200,000 kilometres a second. Let us place an observer on
this object. The observer will then start, in the same direction and
under the same conditions as we have done, a second movable object,
which will thus have, _relatively to him_, a speed of 200,000
kilometres. The Relativist says that the resultant velocity of the
second object relatively to us will not be, as the classical addition
of velocities would make it, 200,000 + 200,000 = 400,000 kilometres a
second. It will be only 277,000 kilometres a second. What the second
moving observer took to be 200,000 kilometres (because his measuring
rod was shortened owing to velocity) was really only 77,000 of our
kilometres. How is it possible to calculate that? Simply by using the
formula of Lorentz which I gave in Chapter II, which gives us the value
of the contraction due to velocity. We then easily find that, if we
have two velocities, _v_ and _v_₂, and if we call the
resultant _w_, classical mechanics stated that

_w_ = _v_₁ + _v_₂

The Einstein mechanics says that this is not correct, and that what we
really have (C being the velocity of light) is

_v_₁ + _v_₂
_w_ = —————————————
( _v_₁_v_₂ )
(1 + —————————)
( C² )

I apologise for again introducing—it shall be the last time—an
algebraical formula into my work. But it spares me a large number of
words, and it is so simple that every reader who has even a tincture of
elementary mathematics will at once see its great significance and the
consequences of it.

The formula expresses in the first place the fact that the resultant
of the velocities, however great it may be, cannot be greater than
the speed of light. It conveys also that, if one of the component
velocities is that of light, the resultant velocity must have the same
value. It means, in fine, that in the case of the slight velocities
we have to do with in actual life (that is to say, when the component
velocities are much smaller than that of light) the resultant is
very nearly equal to the sum of the two components, as the classical
mechanics says.

The classical mechanics was, we must remember, founded upon experience.
We understand how, in those circumstances, Galileo and his successors,
dealing only with relatively slowly moving bodies, reached a principle
which seemed to be true for them, but is only a first approximation.

For instance, the resultant of two velocities, each equal to a hundred
kilometres a second (which is far higher than any velocities obtainable
by Galileo and Newton), amounts to, not 200 kilometres, but 199·999978
kilometres. The difference is scarcely twenty-two millimetres in 200
kilometres! We can quite understand that the earlier experimenters
could not detect differences even less minute than that.

* * * * *

Amongst the verifications of the new law of composition of velocities
we may quote one, the outcome of an early experiment of the great
Fizeau, which is very striking.

Imagine a pipe full of some liquid, such as water, and a ray of light
travelling along it. We know the speed of light in water: it is much
lower than in air or in empty space. Suppose, further, that the water
is not stationary, but flows through the pipe at a certain speed.
What will be the velocity of the ray of light when it leaves the pipe
after traversing the moving liquid? That was what Fizeau, with many
variations of the conditions of the experiment, tried to ascertain.

The velocity of light in water is about 220,000 kilometres a second.
There is question here of so rapid a propagation that there is a great
difference between the law of addition of the old classical mechanics
and of Einsteinian mechanics. Now the results of Fizeau’s experiment
are in complete harmony with Einstein’s formula, and are not in harmony
with that of the older mechanics. Many observers, including, recently,
the Dutch physicist Zeeman, have repeated Fizeau’s experiment with the
greatest care, but the result was the same.

When Fizeau made the experiment in the last century, attempts were made
to interpret his results in the light of the older theories. This,
however, led to very improbable hypotheses. Fresnel, for instance,
trying to explain Fizeau’s results, had been compelled to admit that
the ether is partially borne along by the water as it flows, and that
this partial displacement varies with the length of the luminous waves
sent through, or that it is not the same for the blue as for the red
waves! A very startling deduction, and one very difficult to admit.

The new law of composition of velocities given to us by Einstein, on
the other hand, immediately and with perfect accuracy explains Fizeau’s
results. They are opposed to the classical law.

The facts, the sovereign judges and criteria, show in this case that
the new mechanics corresponds to reality; the earlier mechanics does
not, at least in its traditional form. Here is something, therefore,
which enables us to see at once the profound truth (scientific truth
being what is verifiable), the beauty, of the doctrine of Einstein:
something which shows us, superbly, how a scientific, a physical,
theory differs from an arbitrary and more or less consistent
philosophical system.

Experience, the supreme judge, decides in favour of the Einsteinian
mechanics against the older mechanics. We shall see further examples;
and we shall not find a single case in which the verdict is the other
way.

* * * * *

Let us turn now to a different matter. The new law of composition
of velocities and the resistance of a velocity-limit equal to that
of light may be expressed in a different language from that we have
hitherto used. Up to this we have spoken only of velocities and
movements. Let us see how these things look when we at the same time
examine the particular qualities of the moving objects, of bodies, of
matter.

Everybody knows that the characteristic feature of matter is what we
call inertia. If matter is at rest, a force is needed to set it in
motion. If it is in motion, it needs a force to stop it. It needs
one to accelerate the movement and one to alter the direction. This
resistance which matter offers to the forces which tend to modify its
condition of rest or movement is what we call _inertia_. But
different bodies may offer a different degree of resistance to these
forces. If a force is applied to an object, it will give it a certain
acceleration. But the same force applied to another object will, as a
rule, give it a different acceleration. A race-horse making a supreme
effort will get along much more quickly under a small jockey than under
a man of fifteen stone. A draught-horse will run more quickly if the
cart it draws is empty than if it is full of goods. You can start a
perambulator with a push that would be useless in the case of a heavy
truck.

When a locomotive with a few coaches suddenly starts, the velocity
imparted to the train during the first second is what we call its
acceleration. If the same locomotive starts, in the same conditions,
with a much longer train, we see that the acceleration is less. Hence
the idea, introduced into science by Newton, of the _mass_ of
bodies, which is the measure of their inertia.

If in our example the locomotive produces in the second case an
acceleration only half as great, we express this by saying that the
mass of the second train is double that of the first. If we find that
the acceleration produced by the locomotive is the same for three
trucks loaded with wheat as for a single truck loaded with metal, we
see that the two trains are equal in mass.

In a word, the masses of bodies are conventional data defined by the
fact that they are proportional to the accelerations caused by one
and the same force. To put it differently, the mass of a body is the
quotient of the force which acts upon it by the acceleration given to
it. Poincaré used to say picturesquely: “Masses are coefficients which
it is convenient to use in calculations.”

If there is one property of bodies which comes within the range of our
senses, a property of which every man has some sort of instinct or
intuition, it is _mass_. Yet careful analysis shows us that we are
unable to define it otherwise than by disguised conventions. Poincaré’s
definition seems paradoxical in its admission of powerlessness. But it
is correct. Mass is only a “coefficient,” a conventional outcome of our
weakness!

Nevertheless, something remained upon which we thought we could
base, if not our craving for certainty—genuine men of science gave
up the idea of certainty long ago—at least our desire for accuracy
of deduction in our classification of phenomena. We believed in
the constancy of mass, of this convenient and so clearly defined
_coefficient_.

Here again, unfortunately, we have to recant—or, perhaps, we should
say fortunately, as there is no pleasure like that of novelty.

The older mechanics taught us that mass is constant in one and the
same body, and is therefore independent of the velocity which the body
acquires. From which it followed, as we have already explained, that,
if a force continues to act, the velocity acquired at the end of a
second will be doubled at the end of two seconds, tripled at the end of
three seconds, and so on indefinitely.

But we have just seen that the velocity increases less during
the second second than during the first, and so on, continuously
diminishing until, when the velocity of light is attained, that of the
moving body can increase no further, whatever force may act upon it.

What does that mean? If the velocity of a body increases less during
the second second, it must be because it offers an increasing
resistance to the accelerating force. Everything happens as if its
inertia, its mass, had changed! Which amounts to saying that _the
mass of bodies is not constant: it depends upon their velocity, and
increases with an increase of velocity_.

In the case of feeble velocities this influence is imperceptible.
It was because the founders of classical mechanics, an experimental
science, had experience only of relatively feeble velocities that they
found that mass was _perceptibly_ constant, and believed they
might conclude that it was _absolutely_ constant. In the case of
greater velocities that is not so.

Similarly, in the case of feeble velocities, in the new mechanics as
well as the old, bodies perceptibly oppose the same resistance of
inertia to the forces which tend to accelerate their movement as to
those which tend to alter the direction, to give a curve to their
trajectories. In the case of great velocities that is not so.

Mass, therefore, increases rapidly with velocity. It becomes infinite
when the velocity equals that of light. No body whatever can attain or
surpass the velocity of light, because, in order to pass that limit, it
would need to overcome an infinite resistance.

In order to make it quite clear, let us give certain figures which
show how mass varies with velocity. The calculation is easy, thanks to
the formula which we have previously seen, giving the values of the
Fitzgerald-Lorentz construction.

A mass of 1,000 grammes will weigh an additional two grammes at the
velocity of 1,000 kilometres a second. It will weigh 1,060 grammes
at the velocity of 100,000 kilometres a second; 1,341 grammes at the
velocity of 200,000 kilometres a second; 2,000 grammes (or double)
at the velocity of 259,806 kilometres a second; 3,905 grammes at the
velocity of 290,000 kilometres a second.

* * * * *

That is what the new theory tells us. But how can we verify it?
It would have been impossible only fifty years ago, when the only
velocities known were those of our vehicles and projectiles, which
then did not rise, even in the case of shells, above one kilometre a
second. The planets themselves are far too slow for the purpose of
verification. Mercury, for instance, the swiftest of them, travels at a
speed of only a hundred kilometres a second, which is not enough.

If we had at our disposal no higher velocities than these, we should
have no means of settling which was right, the classical mechanics
with its constancy of mass or the new mechanics with its assertion of
variability.

It is the cathode rays and the Beta rays of radium which have provided
us with velocities great enough for the purpose of verification. These
rays consist of an uninterrupted bombardment by small and very rapid
projectiles, each of a mass less than the two-thousandth part that of
an atom of hydrogen, and charged with negative electricity. They are
the _electrons_.

The cathode tubes of radium give out a continuous bombardment of these
minute projectiles, charged, not with melinite, but electricity:
far smaller than the shells of our artillery, but animated with
infinitely greater initial speeds. The velocity of “Bertha’s” shells is
contemptible in comparison.

But how was it possible to measure the speed of these projectiles?

We know that electrified bodies act upon each other. They attract or
repel each other. Now our electrons are charged with electricity.
If, therefore, we put them in an electric field, between two plates
connected at the edges by an electrical machine or an induction coil,
they will be subjected to a force that will cause them to change
their direction. The cathode rays, in other words, will change their
direction under the influence of an electric field. The amount of
diversion will depend upon the speed of the projectiles and upon their
mass; that is to say, upon the resistance of inertia which the mass
opposes to the causes which tend to divert it.

But this is not all. The electric charges borne by the projectiles are
in movement, even rapid movement. Now, electricity in movement is an
electric current, and we know that currents are diverted by magnets or
magnetic fields. Therefore the cathode rays will be diverted by the
magnet. This diversion will, like the former, depend upon the velocity
and the mass of the projectile; but not quite in the same way. Other
things being equal, the magnetic diversion will be greater than the
electrical diversion, if the velocity is high. As a matter of fact, the
magnetic diversion is due to the action of the magnet on the current.
It will be greater in proportion to the intensity of the current; and
the current will be more intense in proportion to the height of the
velocity, since it is the movement of the projectile which causes the
current. On the other hand, the trajectory of our little projectiles
will be less influenced by the electrical attraction in proportion as
the velocity of the projectile is great.

Hence it is easy to see that when we subject a cathode ray to the
action of an electric field, then to that of a magnetic field, we may,
by comparing the two deviations, measure at one and the same time the
velocity of the projectile and its mass (related to the known electric
charge of the electron).

In this way we find enormous velocities, rising from a few tens of
kilometres to 150,000 kilometres a second, and even more. As to the
Beta rays of radium, they are still more rapid. In cases they attain
velocities not far short of that of light, and higher than 290,000
kilometres a second. Here are just the velocities we need in order to
test whether or no mass increases with them.

* * * * *

In order to understand clearly the progress of the experiments, it
remains to say a few words about the curious phenomenon of electrical
inertia which is called _self-induction_. When we want to set up
an electric current, we find a certain initial resistance which ceases
as soon as the current begins. If afterwards we want to break the
current, it tends to maintain itself, and we have just the same trouble
to stop it as to stop a vehicle in motion. It is a matter of daily
experience. Sometimes the trolley of a tramcar leaves for a moment the
wire which conducts the current, and we then see sparks. Why? There
was a current passing from the wire to the trolley, and if the trolley
breaks away from the wire for a moment, leaving an interval of air
which obstructs the passage of electricity, the current will not stop.
It has been set going, as it were, and it leaps the obstacle in the
form of a spark. This phenomenon is what we call self-induction.

Self-induction—or “self” as the electrical workers call it—is a real
inertia. The surrounding medium offers resistance to the force which
tends to establish an electric current, and to that which tends to stop
a current already set up; just as matter resists the force which tends
to cause it to pass from rest to movement, or from movement to rest.
There is, therefore, a real electrical inertia as well as mechanical
inertia.

But our cathodic projectiles, our electrons, are charged. When they
begin to move, they start an electric current; when they come to
rest, the current ceases. Besides mechanical inertia, then, they
must also have electrical inertia. _They have, so to speak, two
inertias; that is to say, two inert masses, a real and mechanical
mass, and an apparent mass due to the phenomena of electro-magnetic
self-induction._ By studying the two deviations, electric and
magnetic, of the Beta rays of radium or of the cathode rays, it is
possible to determine the respective parts of each of these masses in
the total mass of the electron. The electro-magnetic mass due to the
causes which we have explained varies with the velocity, according to
certain laws which we gather from the theory of electricity. Hence, by
observing the relation between the total mass and the velocity, we can
see what part belongs to the real and invariable mass and what to the
apparent mass of electro-magnetic origin.

The experiment has been made repeatedly by physicists of distinction.
The result of it is surprising: the real mass is _nil_, and
the whole mass of the particle is of electro-magnetic origin. Here
is something that is calculated to modify entirely our ideas of the
essence of what we call matter. But that is another story.

Physicists then asked themselves—this is what we were coming to, after
clearing the way of various difficulties—whether the relation between
the mass and the velocity of the cathodic projectiles was the same as
that which we found in virtue of the Principle of Relativity.

The result of the experiments is absolutely clear and consistent, and
some of them have dealt with Beta rays corresponding to a mass-value
ten times greater than the original mass. This result is: mass
varies with velocity, and in exact accord with the numerical laws of
Einstein’s dynamics.

Here is a new and valuable experimental confirmation. This in
turn tends to show that classical mechanics was merely a rough
approximation, valid at the most only for the comparatively slight
velocities with which we have to deal in the very restricted course of
daily life.

Thus the mass of bodies, the Newtonian property which was believed
to be the very symbol of constancy, the equivalent of what loyalty
to treaties is in the moral order of things, is now merely a small
coefficient, variable, undulating, and relative to the point of view.
In virtue of the reciprocity which we have described, when there is
question of contraction due to velocity, the mass of an object
increases in the same way, not only if the object is displaced, but if
the observer is displaced, and without any other observer, connected
with the object, being able to detect the difference.

For instance, a measuring rod that moves at a velocity of about
260,000 kilometres a second will not only have its length shortened
by one-half, but will have its mass doubled at the same time. Hence
its density, which is the relation of its mass to its volume, will be
quadrupled.

The physical ideas which were believed to be most solidly established,
most constant, most unshakeable, have been uprooted by the storm of
the new mechanics. They have become soft and plastic things moulded by
velocity.

* * * * *

Further confirmations of the new formula, quite independent of the one
we have just described, have recently been provided by physicists. One
of the most astonishing of these is given in spectroscopy.

As is well known, when we cause a ray of sunlight, admitted through
a narrow slit, to pass through the edge of a glass prism, the ray
expands, as it issues from the prism, like a beautiful fan, the
successive blades of which consist of the different colours of the
rainbow. When we examine closely this coloured fan, we notice certain
fine discontinuities, narrow lines or gaps, in which there is no light.
They look like cuts made with a pair of scissors in our polychrome fan.
They are the dark lines of the solar spectrum. Each of these lines,
or each group of them, corresponds to a special chemical element, and
serves to identify this, whether in our laboratories or in the sun and
the stars.

It was explained long ago that these lines are due to electrons which
revolve rapidly round the nuclei of the atoms. Their sudden changes of
velocity give rise to a wave (like those caused in water when you drop
a pebble into it) in the surrounding medium, and this is one of the
characteristic luminous waves of the atom. It reveals itself in one
of the lines of the spectrum. The Danish physicist Bohr has recently
developed this theory in detail, and has shown that it accurately
explains the various spectral lines of the different chemical elements.
These, I may note, differ from each other in the number and arrangement
of the electrons which revolve within their atoms.

Now Sommerfeld has argued as follows. The electrons which gravitate
near the centre of an atom must have a higher velocity than those
which revolve in its outer part; just as the smaller planets, Mercury
and Venus, revolve round the sun far more rapidly than the larger
planets, Jupiter and Saturn. It follows if Lorentz and Einstein are
right that the mass of the interior electrons of the atoms must be
greater than that of the exterior electrons: appreciably greater, as
the former revolve with enormous velocities. We can calculate that,
in those conditions, each line in the spectrum of a chemical element
must in reality consist of a number of fine lines joined together. This
is precisely what Paschen afterwards (1916) found. He discovered that
the structure of the fine lines is strictly such as Sommerfeld had
predicted. It was an astonishing confirmation of an hypothesis: a proof
of the soundness of the new mechanics.

But that is not all. We know that the X-rays are vibrations analogous
to light, the same in origin, but consisting of much shorter waves, or
waves with a far higher frequency. Hence, while light comes from the
external electrons of the miniature solar system which we call an atom,
the X-rays come from the most rapid electrons—those nearest to the
centre. It follows that the special structure of the fine lines, due
to the variation of the mass of the electron with its velocity, must
be much more marked in the case of the X-rays than in the case of the
spectral lines of light. This, again, was confirmed by experiment.
The figures expressing the observed facts correspond exactly with the
calculations of the new mechanics, as regards the predicted variation
of mass with velocity.

It is therefore settled that the phenomena which take place in the
microcosm of each atom are subject to the laws of the new mechanics,
not the old, and that, in particular, masses in motion vary as the new
mechanics demands.

Experience, “sole source of truth,” has given its verdict.

We are now very far from the ideas which were once prevalent. Lavoisier
taught us that matter can neither be created nor destroyed. It remains
always the same. What he meant was that mass is invariable, as he
proved by means of scales. Now it appears that, perhaps, bodies have no
mass at all—if it is entirely of electro-magnetic origin—and that, in
any case, mass is not invariable. This does not mean that Lavoisier’s
law has now no meaning. There remains something that corresponds to
mass at low velocities. Our idea of matter is, however, revolutionised.
By matter we particularly meant mass, which seemed to us to be at once
the most tangible and most enduring of its properties. Now this “mass”
has no more reality than the time and space in which we thought we
located it! Our solid realities were but phantoms.

The reader must pardon me for whatever difficulties he finds in this
exposition. The new mechanics opens out to us such strange new horizons
that it is worth far more than a rapid and superficial glance. If
you want to see a vast prospect in an unexplored world, you must not
hesitate to do some rough climbing, however breathless it may leave you
for the time.

* * * * *

There is, in fine, another fundamental idea of mechanics, that of
_energy_, which takes on a new aspect in the light of Einstein’s
theory: an aspect which, in turn, is largely justified by experiment.

We saw that a body charged with electricity and in motion makes a
certain resistance to interference, on account of the electrical
inertia which is known as self-induction. Calculation and experiment
show that, if we reduce the dimensions of a body that is charged with
a certain quantity of electricity, without altering the charge, the
electrical inertia increases. As a matter of fact, in our hypotheses,
and if the inertia is entirely electro-magnetic in origin, the
electrons are now merely a sort of electric trails moving in the
propagating medium of electrical and luminous waves which we call ether.

The electrons are no longer anything in themselves. They are merely,
in the words of Poincaré, a sort of “holes in ether,” round which the
ether presses much as a lake makes eddies which check the progress of a
boat.

In that case, however, the smaller the holes in the ether are, the
more important will be the agitation of the ether round them; and,
consequently, the greater will be the inertia of the “hole in ether”
which represents the corpuscle under investigation. What will
follow? We know from measurements we have made that the mass of the
tiny sun of each atom, the _positive nucleus_, round which the
planet-electrons revolve, is greater than that of an electron. If this
mass and the corresponding inertia are electro-magnetic in origin, it
follows that the positive nucleus of the atom is much smaller than the
electron.

Let us consider the atom of hydrogen, the lightest and simplest of
the gases. We know that it consists of one planet only, one single
negative electron revolving round the minute central sun, the positive
nucleus. We know also that the mass of the electron is two thousand
times as small as that of the hydrogen atom. It follows, as we can
calculate, that the _positive nucleus_ must have a radius two
thousand times smaller than that of the electron. Now, the experiments
of the English physicists have proved that the large Alpha particles of
the radium emanation can pass through hundreds of thousands of atoms
without being appreciably diverted by the positive nucleus. We conclude
that the latter is in reality much smaller than the electron, as theory
predicted.

All this irresistibly compels us to think that the inertia of the
various component parts of atoms—that is to say, of all matter—is
exclusively electro-magnetic in origin. There is now no matter. There
is only electrical energy, which, by the reactions of the surrounding
medium upon it, leads us to the fallacious belief in the existence of
this substantial and massive something which hundreds of generations
have been wont to call “matter.”

And from all this it also follows, by calculation and by the simple and
elegant reasoning of Einstein, of which I here convey only the faintest
adumbration, that mass and energy are the same thing, or are at least
the two different sides of one and the same coin. There is, then, no
longer a material mass. There is nothing but energy in the external
universe. A strange—in a sense, an almost spiritual—turn for modern
physics to take!

According to all this the greater part of the “mass” of bodies must be
due to a considerable and concealed internal energy. It is this energy
which we find gradually dissipated in radio-active bodies, the only
reservoirs of atomic energy which have as yet opened externally.

If this is true, if energy and mass are synonymous, if mass is merely
energy, it follows that free energy must possess the property of mass.
As a matter of fact, light, for instance, has mass. Careful experiments
have shown that when a ray of light strikes a material object, it
exerts upon it a pressure which has been measured. Light has mass;
therefore it has weight, like all masses. When we come to consider the
new form given by Einstein to the problem of gravitation, we shall see
a further and beautiful proof that light has weight.

We can calculate that the light received from the sun by the earth in
the space of a year is rather more than 58,000 tons. It seems very
little when one thinks of the formidable weight of coal that would be
needed to maintain our globe at the temperature at which the sun keeps
it—in the event of a sudden extinction of our luminary.

The reason for the difference is that, when we produce heat from a
certain amount of coal, we use only a small proportion of its total
energy, its chemical energy. Its intra-atomic energy is inaccessible
to us. It is a pity, as otherwise we should need only a few ounces of
coal to supply heat for a whole year to all the towns and workshops
of England! How many problems that would simplify! When humanity
emerges from the ignorance and the clumsy barbarism in which it lives
to-day—that is to say, in some hundreds of centuries—this will be
accomplished. Yes, it will one day be done. It will be a glorious
spectacle, one in which we may justly rejoice in advance.

Meantime, our sun, like all the other stars, like every incandescent
body, loses its weight in proportion as it radiates. But this happens
so slowly that we need not fear to see it disappear at some early date,
like the ephemeral things which die because they gave themselves too
freely.

* * * * *

To finish with Einstein’s mechanics, let me reproduce a very suggestive
application of these ideas about the identity of energy and mass.

There is in chemistry a well-known elementary law which is called
“Prout’s Law.” It states that the atomic masses of all the elements
must be whole multiples of the mass of hydrogen. Since hydrogen has the
lightest atoms amongst all known bodies Prout’s Law started from the
hypothesis that all the atoms are built up of a fundamental element,
the atom of hydrogen. This supposed unity of matter seems to be more
and more confirmed by the facts. On the one hand, it is proved that the
electrons which come from different chemical elements are identical. On
the other hand, in the transformation of radio-active bodies we find
heavy atoms simplifying themselves by successively emitting atoms of
helium gas. Lastly, the great British physicist Sir Ernest Rutherford
showed in 1919 that by bombarding the atoms of nitrogen gas, in
certain circumstances, by means of radium emanation, we can detach
hydrogen atoms from them. This experiment, the importance of which has
not been fully realised—it is the first instance of transmutation
really effected by man—also tends to prove the soundness of Prout’s
hypothesis.

Yet, when we accurately measure and compare the atomic masses of the
various chemical elements, we find that they do not strictly conform to
Prout’s Law. For instance, while the atomic mass of hydrogen is 1, that
of chlorium is 35·46, which is not a whole multiple of 1.

But we can calculate that, if the formation of complex atoms from
hydrogen upwards is accompanied, as is probable, by variations of
internal energy, as a consequence of the radiation of a certain amount
of energy during the combination, it necessarily follows (since the
lost energy has weight) that there will be variations in the mass of
the body composed, and these will explain the known departures from
Prout’s Law.

* * * * *

In our somewhat hurried and informal excursion into the bush of the new
facts which confirm the mechanics outlined by Lorentz and completed
by Einstein our progress has been rather difficult. It is because,
since we could not use terminology and technical formulæ which would
be unsuitable in this work, we have had to be content with bold and
rapid moves into the districts we wished to reconnoitre. Perhaps they
have sufficed to enable the reader to understand what a revolution
in the very bases of science, what an explosion amidst its age-old
foundations, the brilliant synthesis of Einstein has caused. New light
now streams upon all who slowly climb the slopes of knowledge: upon all
who, wisely renouncing the desire to know “why,” would at least learn
the “how” in many things.

A little before his death, foreseeing, with the intuition of genius,
that a new era opened in mechanics, Poincaré advised professors not to
teach the new truths to the young until they were steeped to the very
marrow of their bones in the older mechanics.

“It is,” he added, “with ordinary mechanics that their life is
concerned: it is that alone that they will ever have to apply. Whatever
speed our motor-cars may attain, they will never reach a speed at
which the old mechanics ceases to be true. The new is a luxury, and
we must think of luxuries only when it can be done without injury to
necessaries.”

I would appeal from Poincaré’s text to the man himself. For him this
luxury, the truth, was a necessary. On the day in question, it is true,
he thought of the young. But do men ever cease to be children? To that
the master, too early taken from us, would have replied, in his grave,
smiling manner: “Yes—at all events, it is better to suppose so.”

CHAPTER V

GENERALISED RELATIVITY

We are now on the threshold of the great mystery of gravitation.

In the preceding chapter we saw how Einstein brought under one
magnificent law both the slow movements of massive objects and the far
more rapid movements of light. They had hitherto been separate and
anarchic provinces of the universe. We now know that the same laws
govern mechanics and optics. If for a time it appeared otherwise, it
was because at velocities which approach that of light the lengths and
masses of objects experience in the eyes of the observer an alteration
which is imperceptible at familiar speeds. It is in its power of
synthesis that Einstein’s mechanics is so splendid. Thanks to it, we
perceive more unity, more harmony, more beauty, than formerly in this
astounding universe, in which our thoughts and our anxieties are so
ephemeral.

The theory of Relativity, however, has up to the present not touched a
phenomenon that is fundamental, essential, ubiquitous in our cosmos. I
mean gravitation, the mysterious property of bodies which rules the
tiny atom no less than the most gigantic star, and directs their paths
in majestic curves.

The universal attraction which, as far as earth is concerned, we call
weight was a kind of steep-cliffed island in the sea of phenomena,
something unrelated to the rest of natural philosophy.

The Einsteinian mechanism, as we have described it up to now, passed
by this island, taking no notice of it. For that reason it was, in
this form, known as “the theory of Special Relativity.” In order to
convert it into a perfect instrument of synthesis, the phenomenon of
gravitation had to be introduced. It is thus that Einstein crowned his
work, and his system assumed the form which is well called “the theory
of General Relativity.”

Einstein has drawn gravitation from its “splendid isolation,” and has
annexed it, docile and vanquished, to the triumphal chariot of his
mechanics. He has, moreover, given Newton’s famous law a more correct
form, and experiment, the supreme judge, has declared this the only
just form.

How he did this, by what subtle and powerful chain of reasoning, by
what calculations based upon facts, I will now endeavour to tell;
and I will again do my best to avoid the network of barbed wire of
mathematical terminology.

Why did Newton, followed by the whole of classical science, believe
that gravitation, the fall of bodies, did not belong to the mechanics
of which he formulated the laws? Why, in a word, did he regard
gravitation as a force or—to use a vaguer but more general term—an
action which prevents heavy bodies from changing their positions
_freely_ in space?

_Because of the principle of inertia._ This principle, the
foundation of the whole Newtonian mechanics, may be expressed thus: a
body which is not acted upon by any force maintains its velocity and
direction unchanged.

Why do we equip steam-engines with the heavy wheels which we call
“fly-wheels,” which work nothing? Because the principle of inertia is
certainly nearly true. When the engine experiences a sudden and sharp
check, or an acceleration, the fly-wheel serves to keep it steady.
Driven by the speed it has acquired, and driving the engine in its
turn, it tends to preserve its velocity, and it prevents or modifies
accidental checks or accelerations. The principle is therefore based
upon experience, especially on the experiments of Galileo, who verified
it by rolling balls down planes inclined at different angles.

For instance, we find that a ball set in motion on a highly polished
horizontal plane keeps its direction, and would preserve its velocity
if the resistance of the atmosphere and the friction of the plane did
not gradually reduce it to zero. We find that, in proportion as we
reduce the friction, the ball tends to maintain its speed so much the
longer.

Newton’s principle of inertia is based upon a number of these
experiments. It is by no means in the nature of a self-evident
mathematical truth. This is so true that ancient thinkers believed,
contrary to classical mechanics, that the movement ceases as soon as
the cause of it is removed. Certain of the Greek philosophers even
thought that all bodies travel in a circle, if nothing interferes with
them, because the circular is the noblest of all movements.

We shall see later how the principle of inertia of Einstein’s
generalised mechanics has a strange affinity to this idea, and at the
same time to the curious declination, the _clinamen_, which the
great and profound Lucretius attributed to the free path of the atoms.
But we must not anticipate.

* * * * *

This belief, that an object left freely to itself and not acted upon
by any force preserves its velocity and direction, cannot pretend to
be more than an experimental truth. But the observations on which it
is based, especially those of Galileo, but any that may be imagined by
physicists, could not possibly be conclusive, because in practice it is
impossible to protect a moving body from every external force, such as
atmospheric resistance, friction, or other.

I am aware that Newton grounded his principle on astronomical as
well as terrestrial observations. He noticed that, _apart from any
attraction by other celestial bodies_, and as far as we can see, the
planets seem to maintain their direction and velocity relatively to the
vault of heaven. But Relativists think that the words I have italicised
in the preceding sentence, which reflect Newton’s idea, really beg
the question. His argument assumes that the planets do not circulate
_freely_; that they are governed in their motions by a force which
he called universal attraction.

We shall see how Einstein came to think that this is not a force,
and in that case the issue of the argument is very different.
However that may be, the classical principle of inertia is a truth
based upon (imperfect) experience, and it is therefore subject to
the constant control of facts. All that we can say about it is that
practically—that is to say, approximately—it harmonises with what we
find.

Newton did not regard it as such, not as a more or less precise
approximation, but as a strict truth. That is why, when he saw that
the planets do not travel in straight lines but in curved orbits, he
concluded—which is a _petitio principii_—that they were subject
to a central force, gravitation. That is why heavy bodies did not seem
to him amenable to the mechanical laws which he had formulated for
bodies left freely to themselves. That is why, in a word, Newton’s law
of gravitation and his laws of dynamics are two distinct and separate
things.

The great genius, the mind which had no equal, was nevertheless human.
The immortal Descartes put forward strange statements and very occult
hypotheses (about the pineal gland and animal spirits), after he had
expressly resolved to affirm nothing that he did not perceive clearly
and distinctly. In the same way Newton, after laying down as his
principle _Hypotheses non fingo_, put the hypotheses of absolute
time and space at the very basis of his mechanics. At the basis of
his masterly theory of gravitation he put the hypothesis—which is
_a priori_ easier to admit—that there is a special force of
gravitation.

These are weaknesses which the greatest of men do not escape. They
ought to make us admire all the more the finer aspects of their work.
So deep is the furrow ploughed by these great students of the unknown
that, even when it is not straight, it takes two centuries and a half
before men dream of inquiring afresh whether Newton’s distinction
between purely mechanical and gravitational phenomena was just.

It is the signal distinction of Einstein that he successfully
accomplished this: that, after erasing many things which were supposed
to be finally settled, he blended mechanics and gravitation in a superb
synthesis, and enabled us to see more clearly the sublime unity of the
world.

* * * * *

To tell the truth—let us premise this before we go further into the
profound and marvellous truths of General Relativity—it is _a
priori_ evident that Newton’s law of universal attraction can no
longer be considered satisfactory.

It says: _Bodies attract each other in direct proportion to their
masses and in inverse proportion to the square of their distances._
What does that mean? We saw that the mass of a body varies with its
velocity. When, for instance, we introduce the mass of our planet into
calculations which involve Newton’s law, what precisely do we mean? Do
we mean the mass which the earth would have if it did not revolve round
the sun? Or do we mean the larger mass which it has in virtue of its
motion? This motion, however, is not always of the same speed, because
the earth travels in an ellipse, not a circle. What value shall we give
to this variable mass in the calculation? That which corresponds to
perihelion or aphelion, the period when the earth travels most rapidly
or most slowly? Moreover, ought we not also to take into account the
velocity of translation of the solar system, which in turn increases or
diminishes according to the season?

Again, under Newton’s law what shall we make the distance from the
earth to the sun? Is it to be the distance relatively to an observer on
the earth or on the sun, or to a stationary observer in the middle of
the Milky Way who does not share the motion of our system across it?
Here again we shall have different values in each case, because spatial
distances vary, as we saw with Einstein, according to the relative
velocity of the observer.

Hence Newton’s law is, in spite of its simple and artistic form,
ambiguous and far from clear. I am aware that the differences I have
just noted are not very important, but our calculations show that
they are by no means negligible. Einsteinians therefore regard it
as indisputable, apart from the considerations which we shall see
presently, that Newton’s law, in its classical form, is obscure, and
must be modified and completed.

These preliminary remarks will serve to at least put us in the frame of
mind that is required of iconoclasts; and in science the iconoclasts
are often the makers of progress. The particular idols at which we
are preparing to deal a few audacious blows are the conception of the
Newtonian law and gravitation.

Laplace wrote, in his _Exposition du Système du Monde_: “It
is impossible to deny that nothing is more fully proved in natural
philosophy than the principle of universal gravitation in virtue of
mass and in inverse proportion to the square of the distance.” Nothing
can better show us than this sentence of the great mathematician the
importance of the step taken by Einstein when he, as we shall see,
improved what had been regarded as the very type, the most perfect
example, of scientific truth: the famous Newtonian law.

* * * * *

Gravitation, or weight, has this in common with inertia, that it is
a quite general phenomenon. All material objects, whatever may be
their physical and chemical condition, are both inert (that is to say,
according to their mass they resist forces which tend to displace them)
and heavy (they fall when they are left to themselves). But it
is a strange thing, noted by Newton, though he did not realise
the significance of it—he regarded it merely as an extraordinary
coincidence—that the same figure which defines the inertia of a body
also defines its weight. This figure is the mass of the body.

Let us return to the illustration which I used in a previous chapter
in dealing with Einstein’s mechanics. If two trains drawn by two
similar locomotives start in the same conditions, and if the velocity
communicated to the first train at the end of a second is double that
communicated to the second, we conclude that the inertia, the inert
mass, of the second train (leaving out of account the friction with the
rails) is twice as great as that of the first. If we afterwards weigh
our two trains, we find that the weight of the second is similarly
twice as great as that of the first.

This experiment, though crude enough in our illustration, has been made
with great precision by physicists, who used delicate methods which we
need not describe here. The result was the same. The inert mass and
the weight of bodies are exactly expressed by the same figures. Newton
saw in this a mere coincidence. Einstein found in it the key to the
hermetically sealed and inviolate dungeon in which gravitation was
isolated from the rest of nature. Let us see how.

There is one remarkable feature of weight or gravitation: whatever be
the nature of the objects, they always fall at the same speed (apart
from atmospheric resistance). This is easily proved by causing a number
of different objects to fall, in the same period of time, down a long
tube in which a vacuum has been created. They all reach the bottom of
the tube at the same time. A ton of lead and a sheet of paper will, if
they are launched into the void simultaneously from the summit of a
tower, reach the ground simultaneously, with a velocity the acceleration
of which is, near the ground, 981 centimetres a second. This fact was
known to Lucretius. Two thousand years ago that profound and immortal
poet wrote:

Nulli, de nulla parte, neque ullo
Tempore, inane potest vacuum subsistere rei,
Quin sua quod natura petit concedere pergat.
Omnia quapropter debent per inane quietum
Æque ponderibus non æquis concita ferri.[8]

[8] _De Natura Rerum_, bk. ii, vv. 235-40.

Now if weight were a _force_ analogous to electrical attraction,
to the propulsion of a locomotive, or even to the propulsive action
of a charge of powder, this ought not to be the case. The velocities
which it communicates to different masses ought to be different from
each other. The two trains of unequal mass in our illustration receive
unequal accelerations from the same locomotive. Nevertheless, if a
great trench suddenly opened before them, they would fall into it with
the same velocity.

From this it is only one step to conclude that gravitation is not a
force, as Newton thought, but simply a property of space in which
bodies move freely. Einstein took this step without hesitation.

Imagine the cable of the lift in some colossal skyscraper suddenly
breaking. The lift will fall with an accelerated movement, though
less rapidly than it would in a vacuum, on account of the atmospheric
resistance and the friction of the cage of the apparatus. But let us
suppose, further, that the electrical engine which works the lift has
its commutator reversed at the same time, and this accelerates the
fall to such an extent that the velocity of the descent increases 981
centimetres in every second. It would be quite easy for our engineers
to carry out this experiment, though the interest of it has not up to
the present seemed great enough to justify it. But we have the right,
when it is necessary to make a subject clear, to say with the poet:

An thou wilt, let us dream a dream.

Let us suppose our dream fulfilled. The lift falls from above with
precisely the accelerated velocity of an object falling in a vacuum.

If the passengers have kept cool enough in their giddy rush downward to
observe what happens, they will notice that their feet cease to press
against the floor of the lift. They can imagine themselves like La
Fontaine’s charming and poetic princess:

No blade of grass had felt
The light traces of her steps.

Our passengers’ purses will, even if they are full of gold, no longer
be heavy in their pockets—which may give them a momentary anxiety. If
their hats are released from their hands, they will remain suspended in
the air beside them. If they happen to have scales with them, they will
notice that the pans remain poised at equal height, even if various
weights are put in one pan. All this is because the objects, as a
natural effect of their weight, fall toward the ground with the same
velocity as the lift itself. Their weight has disappeared.

* * * * *

Jules Verne described this state of things in the projectile which he
imagined taking his heroes from the earth to the moon, at the moment
when the romantic projectile reaches the “neutral point”: that is to
say, the point where it leaves the earth’s sphere of gravitation, but
has not yet entered that of the moon. We might add that Jules Verne
perpetrated a few little scientific heresies in connection with his
projectile. In particular, he forgot that, in compliance with what is
most conspicuously evident in the principle of inertia, the unfortunate
passengers ought to have been flattened like pancakes against the
bottom of the projectile when the charge was fired. He also wrongly
supposed that objects ceased to have weight in the projectile only at
the point where it was exactly between the two spheres of attraction,
that of the earth and that of the moon.

But let us overlook these trifles and return to the admirable
illustration he has prophetically provided for our convenience in
explaining Einstein’s system.

Let us take the projectile when it begins to fall freely toward the
moon.[9] It is evident that from this point onward, until it lands on
the moon, it will behave exactly like the lift which we have described.

[9] It is obvious that we assume the projectile to be without rotation:
that is to say, the Columbia cannon must not, in our hypotheses, be
rifled. This is indispensable, for if the projectile turned, there
would be centrifugal effects which would greatly complicate both the
phenomena and our argument.

During this fall upon the moon the passengers, if they have
miraculously escaped being flattened at the start, will see the various
objects about them suddenly deprived of their weight, floating in the
air, and, at the slightest shake, adhering to the walls or the vaulted
roof of the projectile. They will feel themselves extraordinarily
light, and they will be able to make prodigious leaps without any
effort. This is because they and all the objects about them fall
toward the moon with the same velocity as the projectile. Hence the
disappearance of weight or gravitation, which vanish as if spirited
away by some magician. The magician is the properly accelerated
movement, the unimpeded fall of the observers.

In a word, to get rid of the apparent effects of gravitation in any
place whatever it is enough for the observer to acquire a properly
accelerated velocity. That is what Einstein calls the “principle
of equivalence”: equivalence of the effects of weight and of an
accelerated movement. The one cannot be distinguished from the other.

Let us imagine Jules Verne’s projectile and its unfortunate passengers
transported a long distance from the moon, the earth, and the sun,
to some deserted and glacial region of the Milky Way where there is
no matter, and so remote from the stars that there is no longer any
weight or attraction. Let us suppose that our projectile is abandoned
there, and motionless. It is clear that in these circumstances there
will be no such thing as high or low—no such thing as weight—for the
passengers. They will find themselves relieved of every inconvenience
of weight. They may, if they choose, stand on the inner wall of the
upper part of the projectile or on the floor, as it was when they were
falling upon the moon.

Now let us suppose that the wizard Merlin quietly approaches and,
fastening a cord to the ring on the top of the projectile, begins to
drag it with a uniformly accelerated movement. What will happen to the
passengers? They will notice that they have suddenly recovered their
weight, and that they are riveted to the floor of the projectile,
much as they were drawn to the surface of our planet before they left
it. Indeed, if the motion of Merlin is accelerated 981 centimetres a
second, they will have exactly the same sensations of weight as they
had on the earth.

They will notice that if they throw a plate into the air at a given
moment, it will fall upon the floor and be broken. “This is,” they
will think, “because we are again subject to weight. The plate falls
in virtue of its weight, its inert mass.” But Merlin will say: “The
plate falls because, on account of its inertia, it has retained the
increasing velocity which it had at the moment when it was thrown.
Immediately afterwards, as I drew the projectile with an accelerated
movement, the ascending velocity of the projectile was greater than
that of the plate. That is why the bottom of the projectile, in its
accelerated ascending course, knocked against the plate and broke it.”

This proves that the weight or gravitation of a body is
indistinguishable from its inertia. Inert mass and heavy mass are not,
as Newton supposed, two things which happen by some extraordinary
coincidence to be equal; they are identical and inseparable. The two
things are really one.

And we are thus led to believe that the laws of weight and the laws
of inertia, the laws of gravitation and those of mechanics, must be
identical, or must at least be two modalities of one and the same
thing: much as the full face and the profile of the same man are the
same face seen under two different angles.

Even if the travellers in the projectile—who look rather like
guinea-pigs—peep out of the window and see the cord that is drawing
them, it will not alter their illusion. They will believe that they are
at rest and floating at a point of space where weight has been
restored: that is to say, in the language of the experts, at a point of
space where there is a “gravitational field.” This phrase is analogous
to the familiar “magnetic field,” which refers to a part of space in
which there is magnetic action, a part in which the needle of the
compass has a definite direction imposed upon it.

In sum, we can at any point replace a gravitational field, or the
effects of weight, by a properly accelerated movement of the observer,
and vice versa. There is a complete equivalence between the effects of
weight and those of an appropriate movement.

* * * * *

This now enables us to establish very simply the following fundamental
fact, unknown only a few years ago, but now brilliantly proved by
experiment: _Light does not travel in a straight line in those parts
of the universe where there is gravitation, but its path is curved like
that of heavy objects._

We showed in one of the preceding chapters that in the four-dimensional
continuum in which we live, which we might call “space-time” but which
we more simply call the universe, there is something that remains
constant, identical for observers who move at given and different
velocities. It is the “Interval” of events.

It is natural to suppose that this “Interval” will remain identical
even if the velocity of the observers changes—even if it is
accelerated like the velocity of the lift in our illustration, or of
Jules Verne’s projectile, during their fall.

In point of fact, if something in the universe is an _invariant_,
as physicists say, or invariable, for the observers who move at
different speeds, this something must _naturally_ remain the same
for a third observer whose velocity changes gradually from that of
the first to that of the second observer, and who is therefore in a
state of uniformly accelerated movement. From this we deduce certain
consequences of a fundamental character.

In the first place, one thing is evident, and is unanimously admitted
by physicists: in a vacuum, and in a region of space where there is no
force acting and no such thing as weight, light travels in a straight
line. That is certain for many reasons—in the first place, on the
mere ground of symmetry, because in a region of isotropic vacuum a ray
which is uninfluenced will not depart from its rectilinear path in any
direction whatever. That is evident, whatever hypothesis we adopt as
to the nature of light, and even if, like Newton, we suppose that it
consists of ponderable particles.

Admitting that, let us now suppose that at some point in the universe
where there is weight—at the moon’s surface, for instance—there is a
remarkable gun which can fire a ball that has and retains (along its
whole path) the velocity of light.

The trajectory of this ball will be very extensive, on account of its
great velocity, yet curved toward the surface of the moon on account
of its weight. As we may make our choice in the field of hypotheses,
there is nothing to prevent us from supposing that the ball is of such
a nature as to disclose its path by a faint luminous trail. There were
projectiles of this character during the Great War.

As the ball advances, it also falls every second toward the moon’s
surface, to the same extent as any other projectile would which was
fired at any velocity whatever, or had no velocity. All objects near
the surface of the ground (in a vacuum) fall at the same vertical
velocity, and this is independent of their motion in the horizontal
direction. That is, in fact, the reason why the paths of projectiles
are the more curved the less initial speed they have.

Seen from the windows of Jules Verne’s projectile (which is itself
falling toward the moon), the trajectory of the ball will seem to
the passengers to be a straight line, because it falls with the same
velocity as they.

Now let us suppose that a luminous ray, from the flame of the gun,
starts at the same time and in the same direction as the ball. This
luminous ray will obviously be rectilinear for the passengers in the
projectile, because light travels in a straight line when there is
no weight. Consequently, since it has the same form, direction, and
velocity as the luminous ball, the passengers will see the ray of light
coincide in its whole course with the trajectory of the ball.

It further follows that the “Interval” (both in time and space) of
the luminous ray and of the ball is, and remains, zero. Now this
“Interval” must remain the same, whatever be the velocity of the
observer. Hence, if Jules Verne’s projectile ceases to fall, and is
stopped at the moon’s surface, its passengers will continue to see the
luminous ray coincide at every point with the trajectory of the ball.
This trajectory is, as they now notice, curved on account of weight.
Therefore, the luminous ray is similarly curved in its path on account
of weight.

This shows that light does not travel in a straight line, but falls,
under the influence of gravitation, like all other objects. The reason
why this was never known before, and it was always thought that light
travels in a straight line, is that on account of the enormous velocity
of light its trajectory is only very slightly curved by weight.

That is easy to understand. At the earth’s surface, for instance,
light must fall (like all other objects) with a velocity equal to
981 centimetres at the end of a second. Now by the end of a second
a luminous ray has travelled 300,000 kilometres. Suppose we could
observe a horizontal luminous ray 300 kilometres long near the earth’s
surface—a very far-fetched supposition—during the thousandth part of
a second, which it will take the ray to pass from one observer to the
other, it will fall to the extent of only about the five-thousandth of
a millimetre.

We can understand how it was that a luminous ray that deviates only to
this imperceptible extent from its initial direction in the course of
three hundred kilometres was always considered rectilinear.

Is there no means of verifying whether light is or is not bent out of
its path by gravitation? There is such a means in astronomy, as we
shall now see.

* * * * *

It is impossible to detect the curvature of a luminous ray travelling
from one point to another on the earth’s surface, mainly because weight
on the earth is too slight to bend the ray much. A further reason is
that our planet is so ridiculously small that we cannot follow the
light over a sufficient distance.

But what cannot be done on this little globule of ours, the entire
diameter of which light can cover in the twenty-fifth of a second, may
possibly be done in the gigantic laboratory of celestial space. We
have, almost within our reach—a mere matter of 93,000,000 miles away,
that is to say—a star on which weight is twenty-seven times greater
than on the earth. We mean the sun. On the sun a body left to itself
falls 132 metres in the first second. Its fall is twenty-seven times as
rapid as on the earth.

Hence, near the sun, light will be much more bent out of its path by
gravitation. The deviation will be all the greater from the fact that
the sun is 800,000 miles in diameter, and a luminous ray needs a much
longer time to cover this distance than to travel the length of the
earth’s diameter. Hence gravitation acts upon the ray of light during a
much longer time than upon a ray that reaches the earth, and it will be
all the more curved.

Take a luminous ray that comes from a star at a great distance behind
the sun. If it reaches us after passing near to the sun, it will behave
like a projectile. Its path will no longer be rectilinear. It will be
slightly curved toward the sun. In other words, the ray will deviate
from a straight line, and the direction it has when our eyes receive it
on the earth is a little different from the direction it had when it
left the star. It has been diverted.

Calculation shows that this deviation, though very slight, can be
measured. It is equal to an angle of a second and three-quarters: an
angle which the delicate methods of our astronomers are able to measure.

Certainly such an angle is very far from considerable, for it takes
324,000 angles of one second to make a right angle. In other words,
an angle of one second is that at which we should see the two ends of
a rod, a metre in length, fixed in the ground, at a distance of 206
kilometres. If our eyes were sharp enough to see a man of normal height
standing 200 kilometres away from us, our glance, in passing from his
head to his feet, would have a very small angle of deviation. Well,
this angle accurately represents the deviation experienced by the light
that comes to us from a star when it has passed close to the golden
globe of the sun.

Minute as this angle is, the methods of the astronomer are so delicate
and precise that he can determine it. The tiny measurement is by no
means to be despised. Disdain of the men who devote themselves to
such refined subtleties is very much out of place, because our modern
science has been revolutionised by this measurement. Einstein is right,
and Newton wrong, because we have been able to measure this minute
angle and establish the curvature of light.

A great difficulty arose when we wished to verify this. How can we
observe in full daylight a ray of light that comes to us from a star
and passes close to the sun? It cannot be done. Even if we use the
most powerful glasses the stars on the farther side of the sun are
completely drowned in its blaze—to speak more correctly, in the light
which is diffused by our atmosphere.

To say the truth—if we may venture upon a parenthetic remark at this
juncture—night has taught us much more than day about the mysteries
of the universe. In literary symbolism, in politics, the light of day
is the very symbol of progress and knowledge: night is the symbol of
ignorance. What folly! It is a blasphemy against night, the sweetness
of which we ought rather to venerate. I do not refer to its romantic
charm, but to the mighty progress in knowledge which it has enabled us
to make.

Midnight is not merely the hour of crime. It is also the hour of
prodigious flight toward remote worlds. During the day we see only one
sun: by night we see millions of suns. The blinding veil which the
sunlight draws across the heavens may be woven of the most brilliant
rays, but it is none the less a veil, for it makes us as blind as the
moths which, in a strong light, can see no further than the tips of
their wings.

In order to solve our problem, therefore, we have to observe in
complete darkness stars which are nevertheless near the edge of
the sun’s disk. Is that impossible? No. Nature has met our need by
providing total eclipses of the sun which may at times be seen from
various stations on the earth. At those times the bright disk is hidden
for a few minutes behind the disk of the moon. Midday is turned into
midnight. We see stars shine out close to the masked face of the sun.

* * * * *

Fortunately, a total eclipse, visible in Africa and South America, was
due on May 29, 1919, shortly after Einstein had, on the strength of an
argument like that we have just expounded, announced the deviation of
the light of the stars when it passed the sun.

Two expeditions were organised by the astronomers of Greenwich and
Oxford. One proceeded to Sobral, in Brazil, the other to the small
Portuguese island Principe, in the Gulf of Guinea. Some of the English
astronomers were rather sceptical about the issue. How could we, until
it was proved, admit that Newton was wrong, or had at least failed
to formulate a perfect law? But this _was_ proved, and very
decisively, by the observations.

These observations consisted in taking a certain number of photographs
during the few minutes of total eclipse of the stars near the sun. They
had been photographed with the same instruments some weeks before, at
a time when the region of the sky in which they shine was visible at
night and far from the sun. As everybody knows, the sun passes
successively, in its annual course, through the different
constellations of the zodiac.

If the light of the stars which were photographed were not bent out
of its path in passing the sun, it is clear that their distances
ought to be the same on the plates exposed during the eclipse as on
the negatives taken during the night some time previously. But if the
light from them were bent out of its course during the eclipse by the
gravitational influence of the sun, it would be quite otherwise. The
reason is as follows. When the moon rises on one of our plains, it is
not round, as everybody will have noticed, but flattened at top and
bottom, somewhat like a giant tangerine lifted above the horizon for
some magic supper. The moon has, of course, not ceased to be round. It
merely seems to be flattened because the rays which come from its lower
edge, and have to pass through a thick stratum of the atmosphere before
they reach us, are bent toward the ground by the refraction of the
denser atmosphere much more than are the rays coming from the moon’s
upper edge, which pass through a less dense mass of air. Our eyes see
the edge of the moon in the direction from which its rays come to us,
not in the direction from which they started. That is why the lower
edge of the moon seems to us to be raised higher above the horizon than
it really is. This deviation is due to refraction.

In the same way a star situated a little to the east of the sun (the
rays in this case being curved by weight, not by refraction) will seem
to us further away from it. It will look as if it were further east
than it really is. Similarly, a star to the west of the sun will seem
to us still further from the sun’s western edge.

Hence the stars on either side of the sun will, if Einstein is right,
be more widely separated from each other in the negatives taken during
the eclipse. In their normal position, on the photographs taken during
the night, they will seem nearer to each other.

This is precisely what was found when the photographs taken at Sobral
and Principe were studied with the aid of the micrometer. Not only was
it thus proved that the light of the stars is bent out of its path by
the sun, but it was found that the deviation had exactly the extent
which had been predicted by Einstein. It amounts to an angle of one
second and three-quarters (1″·75) in the case of a star that is quite
close to the sun’s disk, and the angle decreases rapidly in proportion
to the distance of stars from the sun. It was a great triumph for the
theory of Einstein, and for the first time it gave us some connecting
link between light and gravitation.

On the preceding page I compared the curvature of light owing to its
weight with the deviation that is caused by atmospheric refraction.
As a matter of fact, there were astronomers who wondered whether the
agreement between Einstein’s theory and the results obtained during the
eclipse was not merely a coincidence: whether the deviation that was
recorded was not due to refractive action by the sun’s atmosphere.

It seems impossible to admit this. Sometimes we see comets passing
quite close to the surface of the sun during their journey through
space. Their movement would be considerably disturbed if the sun’s
atmosphere were refractive enough to account for the deviations
observed at Sobral and Principe. Perturbations of cometary orbits of
this nature, near the sun, have never been recorded. The only possible
interpretation, therefore, is that the phenomena are due to the effect
of weight upon light.

Thus the light of the stars, weighed in a balance of the most
exquisite delicacy, has given us a decisive confirmation of Einstein’s
theoretical deductions. By its fruit we know the tree.

CHAPTER VI

THE NEW CONCEPTION OF GRAVITATION

Does the universe conform to the laws of geometry? It is a question
that has been much discussed by philosophers and scholars, but the
deviation of light owing to its weight now enables us to approach it
with confidence.

In our schools we are taught a magnificent series of geometrical
theorems, all solidly interconnected, the principal of which were
created by the great Greek genius, Euclid. That is why classical
geometry is known as Euclidean geometry. Its theorems are based upon a
certain number of axioms and postulates, though these are really only
affirmations or definitions.

The most important of these definitions is: “A straight line is the
shortest distance between two points.” That seems to schoolboys quite
simple, because they know that the youth who amuses himself by running
in a zigzag on the racing track will be the last to reach the tape; and
at the sports ground one is not in a mood or has not time to bother
about the validity of the axioms of geometry. What is the precise
meaning of this definition of a straight line? There has been a great
deal of discussion of that point. Henri Poincaré has written a number
of fine and profound pages on it, yet his conclusions are not entirely
without an element of uncertainty.

In practice we all know what we mean by a straight line: it is the
line that we make by means of a good ruler. But how do we know that a
ruler is good and correct? By holding it up before the eye, and seeing
that both ends of it and all the intermediate points in its edge merge
together when we look along it. That is how a carpenter tells if a
board is smoothly planed. In a word, in practice we mean by a straight
line the line which is taken by the eye of the rifleman looking along
his sights.

All this amounts to saying that a straight line is the direction
in which a ray of light travels. However we look at the matter, we
always come back to the same point—to say that the edge of an object
is straight means that the delimiting line coincides in its whole
length with a ray of light.[10] We may therefore say that practically a
straight line is the path followed by light in a homogeneous medium.

[10] It goes without saying that in all this we assume that the
luminous ray travels in a homogeneous medium.

And that gives rise to a question. Is the world in which we live, the
universe, in conformity with Euclid’s geometry? Is it Euclidean?

It must be understood that Euclid’s geometry is not the only one
that has been created. In the nineteenth century there were bold
and profound mathematicians—Riemann, Bolyay, Lobatchewski, even
Poincaré—who founded new and different and rather strange geometries.
They are just as logical and coherent as the classical geometry of
Euclid, but they are based upon different axioms and postulates—in a
word, different definitions.

For instance, “parallels” are said to be two straight lines, being in
the same plane, which can never meet. The geometry which we learned
in our boyhood says: “Through a given point there can be only one
straight line parallel to a given straight line.” This is said to be
Euclid’s postulate. Riemann, however, does not admit this and wishes
to replace it by: “Through a given point there cannot be any straight
line parallel to a given straight line”—that is to say, any line which
never meets it. Upon this Riemann founds a quite consistent system of
geometry.

Who will venture to say that Euclid’s geometry is true and that of
Riemann false? As theoretical ideal constructions they are both equally
true.

* * * * *

A question that we may legitimately ask is: Does the real universe
correspond to the classical geometry of Euclid or to that of Riemann?

It was long believed that it corresponded to Euclid’s geometry.
Poincaré himself, speaking of Euclid’s system, said:

“It is, and will remain, the most convenient, (1)
because it is the simplest; (2) because it agrees
very well with the properties of natural solids,
the bodies with which our limbs and our eyes are
concerned, and out of which we make our measuring
instruments.”

When people used to say in earlier ages that the earth is flat,
they argued pretty much as Poincaré does: “This theory is the most
convenient, (1) because it is the simplest; (2) because it agrees very
well with the properties of the natural objects with which we are in
contact.” But when men came into touch with more remote objects, when
navigators and astronomers multiplied these remote objects, the idea of
a flat earth ceased to be the most convenient, the simplest, and the
best suited to the facts of experience. Then appeared the idea that the
earth is round, and this was found infinitely more convenient, simpler,
and better adapted to the material universe.

“Convenience,” which Poincaré makes a criterion of scientific truth, is
a contingent and elastic thing. A point of view may be convenient in
London and not in Bedford. A theory may be convenient in an area of a
hundred yards and no longer convenient for an area of a hundred million
miles.

The hypothesis of a flat earth has been replaced by the theory of
the earth’s rotundity. The stationary earth has been replaced by a
revolving globe. In the same way, it seems that in our time Euclid’s
geometry must give way to another as a _convenient_ representation
of the real world.

Can there be, in our universe, our space, a parallel to a straight
line? That is to say, is it true that two straight lines being in the
same plane will never meet? The real meaning of the question is: Is it
impossible for two luminous rays, travelling in empty space and being
in what (for each fraction of the rays) we will call the same plane,
ever to meet? _The answer to this question is in the negative._

As these two luminous rays are bent out of their paths in space by the
gravitation of the stars, and as they are differently affected in this
way because they are at different distances from the stars, it follows
necessarily that they will cease to be parallel (in the Euclidean sense
of the word) and will finally meet; or at least that they cease to
realise the first condition of parallelism—coexistence—in the same
local plane.

In a word, if we consider the matter, not within the ridiculously
limited field of experiment in the laboratory, but in the vast field
of celestial space, the real universe is not Euclidean, because in it
light does not travel in a straight line.

Kant regarded the truths—to be accurate, the deductive
affirmations—of the Euclidean geometry as “synthetic judgments _a
priori_,” or self-evident propositions. As we have seen, Kant was
wrong, not only from the point of view of theoretical geometry, but
also from the point of view of real geometry. The etymology of the word
“geometry” (which means “measuring the earth”) is enough of itself to
show that it was originally, and chiefly, a practical science. That is
a sufficient justification for our asking which geometry is most in
accord with the real universe.

Gauss, a profound thinker, asked the question long ago, in the last
century, and he made certain delicate experiments to measure if the sum
of the angles of a triangle is really equal to two right angles, as
the Euclidean geometry says. With this view he took a vast triangle,
the apices of which were formed by the highest peaks of three widely
separated mountains. One of them was the famous Brocken. With his
assistants he took simultaneous sights of each peak in relation to
the other two, and he found that the sum of the three angles of the
triangle only differed from 180 degrees to an extent that might be put
down to error in observation.

There were many philosophers who ridiculed Gauss and his experiments.
With the _a priori_ dogmatism that one so often encounters amongst
these people they said that his measurements, even if they had had
a different result, would have proved nothing to the detriment of
Euclid’s theorems, but would merely have shown that some disturbing
cause bent the luminous rays between the three apices of the triangle.
This is true, but it does not matter.

If Gauss had found that the sum of the angles of the triangle in
question was larger than two right angles, it would have proved that
real geometry is not the geometry of Euclid. The question which Gauss
asked was profound and reasonable. The philosophers who ridiculed it
might have been challenged to define real straight lines, natural
straight lines, in any other terms than those of the passage of light.

Gauss did not find the sum of the angles different from two right
angles because his measurements were not sufficiently precise. If they
had been much more rigorous, or if he could have used a much larger
triangle—with the earth, Jupiter in opposition, and another planet as
its apices—he would have found a considerable difference.

The real universe is not Euclidean. It is only approximately Euclidean
in those parts of space where light travels in a straight line: that is
to say, in the parts which are far from any gravitational mass, such as
that in which, on an earlier page, we left Jules Verne’s projectile.

There are many other reasons why the universe, in consequence of
gravitation, does not conform to the laws of Euclid’s geometry.

For instance, in the Euclidean geometry the extent of the circumference
has a well-known proportion to its diameter, and this is indicated by
the Greek letter π. This proportion, expressing how many times the
diameter is contained in the circumference, is equal to 3·14159265
... etc., but I pass over the rest, as π has an infinite number of
decimals. We then ask: In practice is the proportion of circumferences
to their diameters really equal to the classic value of π? For
instance, is this precisely the proportion of the earth’s circumference
to its diameter?[11] Einstein says that it is not, and he gives us the
following proof. Imagine two very clever and quick and wizard-like
surveyors setting out to measure the circumference and diameter of the
earth at the Equator. They both use the same scales of measurement.
They begin measuring at the same moment, and they start from the same
point on the Equator. But one goes westward and the other eastward,
and their speeds are equal, and such that the one who goes westward
keeps up with the earth’s rotation, and thus sees the sun all day
long stationary at the same height above the horizon. In music-halls,
for instance, one sometimes sees an acrobat walking on a rolling ball
and keeping to the top of the ball, because the pace of his steps is
exactly equal and contrary to the displacement of the spherical surface.

[11] We are, of course, imagining the earth as perfectly circular,
without irregularities.

A stationary observer in space—on the sun, let us say—would thus see
our surveyor who is going westward, stationary right opposite to him.
On the other hand, the surveyor who goes eastward will seem to him to
go round the earth, and twice as quickly as if he had remained at the
starting-point.

When each of our surveyors, both going at the same speed, has finished
his task of measuring the round of the earth, will they both have the
same result? Evidently not. As the super-observer in the sun will see,
the yard of the surveyor who travels eastward is shortened by velocity
in virtue of the Fitzgerald-Lorentz contraction. On the other hand,
the yard of the surveyor who travels westward does not experience this
contraction, as the super-observer on the sun, in reference to whom he
remains stationary, would see.

Consequently the two surveyors reach different figures for the earth’s
circumference, the one who travels westward finding a result a few
yards less than that of the other. Yet it is obvious that when they
proceed to measure the earth’s diameter, travelling at the same speed,
the two observers will reach the same figure for it.

Hence the π which expresses the proportion of the earth’s circumference
to its diameter on the ground of actual measurement differs according
as the measurer travels in the direction of the earth’s rotation or
in the opposite direction. Therefore, as the real values of π are
different, they cannot be the unique and quite definite figure of
classical geometry. Therefore the real universe does not conform to
this geometry.

These differences, in the illustration we have given, are due to the
earth’s rotation. From the standpoint of gravitation the earth’s
rotation has centrifugal effects which modify the centripetal influence
of weight. We have seen, moreover, that for the surveyor whose speed
equals that of the earth’s rotation the value of π is smaller than for
the observer whose speed seems to be double that of the rotation. Thus
the effects of weight being the reverse of those of rotation, or of
centrifugal force, it follows (it would be just as easy to prove this
as the preceding) that the effect of weight is to give π something less
than its classical value.

In a word, in the universe real circumferences traced upon gravitating
masses, such as stars, are, in proportion to their diameters, less than
they are in the Euclidean geometry.

The difference is generally very slight, it is true. But there
_is_ a difference. If we put a mass of a thousand kilogrammes
in the centre of a circle that is ten metres in diameter, the figure
π will differ in reality from its Euclidean value by less than
one-thousand-million-billionth.

In the neighbourhood of such formidable masses of matter as the stars
are, the difference may be far greater, as we shall see. This is the
origin of the divergences between Newton’s law of gravitation and that
of Einstein: divergences which observation has settled in favour of the
latter. But we will not anticipate.

* * * * *

We showed in a previous chapter that the real universe of the
Relativists is a four-dimensional continuum—not three-dimensional, as
classic science thought—and that in this continuum distances in time
and space are relative. The only thing that has a value independent
of the conditions of observation—that has an absolute, or at least
objective, value—is what we called the “Interval” of events, the
synthesis of the spatial and chronological data.

Yet, in spite of its four dimensions, the universe, as we discussed it
in connection with the Michelson experiment and the Special Relativity
which this discloses, was nevertheless a Euclidean continuum, in which
the classical geometry was verified, and light travelled in a straight
line. As we have just seen, we have to recant this. The universe not
only has four dimensions, but it is not Euclidean.

With what geometry does the universe accord best—or most conveniently,
to use the language of Poincaré? Probably that of Riemann. When we take
the compasses and draw a small circle on a sheet of paper spread on
the table, the radius of the circle is found by the distance between
the points of the compasses, and the circle is Euclidean. But if we
draw the circle on an egg, the fixed point of the compasses being
stuck in the top of the egg, and again get the radius by the distance
between the points, the circle we have now drawn is not Euclidean.
The proportion of the circumference to the radius as thus defined is
smaller than π, just as it is smaller than π when the circle is traced
round a massive star.

Well, there is the same difference between the non-Euclidean real
universe and a Euclidean continuum as there is between our flat sheet
of paper and the surface of the egg, taking into account the fact that
these surfaces have only two dimensions while the universe has four.

Two-dimensional space may be flat like the sheet of paper or curved
like the surface of the egg. By leaving the sheet of paper flat or
rolling it up we can make the geometry of the figures drawn on it
correspond with or differ from the Euclidean geometry. In just the same
way space with more than two dimensions may or may not be Euclidean.

As a matter of fact, the universe is, as we saw, only approximately
Euclidean in those regions which are remote from all heavy masses. It
is not Euclidean, but curved or warped in the vicinity of the stars;
and the curvature is the greater in proportion as we approach the stars.

Hence the geometry of curved space, as founded by Riemann, seems to be
the best adapted to the real universe. It is the one used by Einstein
in his calculations.

* * * * *

When we sought to prove, on a previous page, that rays of light fall
just as projectiles of the same velocity would, we used the following
argument:

Since the “Interval” of two events is the same for two observers moving
at uniform and different velocities, it is _natural_ to think that
it will be the same for a third observer whose velocity increases from
that of the first to that of the second—that is to say, whose velocity
is uniformly accelerated.

There is, in fact, no reason why the passengers in a train which runs
at a uniform speed of sixty miles an hour should observe an “invariant”
element in phenomena just as do those in another train moving at
half the speed, yet this “invariant” should cease to be such for the
passengers in a third train which passes gradually from the velocity
of the first train to that of the second. To admit the contrary would
be to grant a privileged position in the universe to the first two and
others like them. If there is any estate in the world that has had its
unjust privileges suppressed by the new physics, it is the study of the
material world.

This privilege of observers moving at a uniform velocity would be the
less justified as, if we go to the root of the matter, it is very
difficult to say exactly what a uniform movement is.

What do we mean when we say that a train has a uniform velocity of
sixty miles an hour? We mean that the train has this velocity in
reference to the rails or the ground. But in reference to an observer
in a balloon, or who passes in another train, the velocity has not
the same value, and it may cease to be a uniform velocity. We know
only relative movements, or, to be quite accurate, movements relative
to some material object or other. According to our choice of this
object, this standard of comparison, the same velocity may be uniform
or accelerated. In the long run, it is clear, we should have to have
recourse to Newton’s hypothesis of absolute space to be able to say
whether a given velocity is really uniform or accelerated.

That is the profound reason why the Einsteinian “Interval” of things,
the invariable quantity or “Invariant,” must be the same for all
observers whatever be their velocity, and in particular for observers
moving at velocities equivalent, in a given place, to the effects of
gravitation.

But in that case the inferences we draw from the Michelson experiment,
in regard to the aspect of phenomena for observers in uniform different
movements of translation, no longer suffice to explain to us the
whole of reality. They need to be completed in such fashion that the
universal invariant, the “Interval” of things, remains the same for an
observer who is moving in any way whatever.

If I pass along a street at some unheard-of speed, but with a uniform
motion, its general aspect may, on account of the contraction caused
by my velocity, be a little different from what it would seem to me
if I were stationary.[12] The houses, for instance, will seem narrower
in proportion to their height. Nevertheless the general aspect and
proportions of objects will be much the same in both cases, and they
will have something in common. Thus the gas-lights will seem to me
thinner, but they will be straight.

[12] It goes without saying that we assume the observer to have a
retina with instantaneous impressions.

It will be quite otherwise if the observer’s movements are varied:
if, for instance, we imagine him a drunken giant, reeling about at
a prodigious speed. For such an observer the street will have quite
a new aspect. The gas-jets will no longer be straight, but zigzag,
reproducing in an inverse way the zigzags which he himself makes as he
reels along. This is so true that caricaturists generally represent the
trees and lamp-posts and houses seen by a drunken man by ridiculously
waving lines.

Our observer will be convinced that objects really have the zigzag
forms which he sees, and that the forms change at every step he takes.
Try to tell him that it is he who is dancing, not the objects; that it
is he who is not walking straight, not the dog he has on leash. He will
not believe it—and from the point of view of General Relativity he is
neither more nor less right than you.

Yet there is something in the aspect of the world that must be common
to the drunkard and the drinker of water.

If the whole universe were suddenly plunged in a mass of gelatine
which has set, and one were to squeeze or alter the shape in any way
of this gelatinous mass, there would still be something unchanged in
the coagulated stuff. What is this something? And what is the calculus
to use for it? The answer to these questions was the last stage for
Einstein to cover in order to establish the equations of gravitation
and General Relativity.

* * * * *

Here it was the penetrating genius of Henri Poincaré that indicated the
path. It is very necessary to insist on this, as justice has not been
done in the matter to the great French mathematician.

If all the bodies in the universe were to be simultaneously dilated,
and to an identical extent, we should have no means of knowing it. Our
instruments and our own bodies being similarly dilated, we should not
perceive this formidable historical and cosmic event. It would not
distract us for a moment from the trivialities of the hour.

What is more, not only will it be unrecognisable if worlds are modified
in such a fashion as to alter the scale of lengths and time, but it
would be impossible to distinguish between two worlds, if one single
point of the first corresponds to each point of the second; if to each
object or event of the one world there corresponds one of the same
character, placed exactly in the same position, in the other. Now the
successive and diverse deformations which we impose upon the gelatinous
mass in which we metaphorically enclosed our entire universe in an
earlier paragraph give us precisely indistinguishable worlds from
this point of view. Poincaré has the distinction of first calling our
attention to this and proving that the relativity of things must be
understood in this very broad sense.

The amorphous and plastic continuum in which we place the universe
has a certain number of properties which are exempt from all idea of
measurement. The study of these properties is the work of a special
geometry, a qualitative geometry. The theorems of this geometry have
this peculiarity, that they would still be true even if the figures
were copied by a clumsy draughtsman who made gross errors in the
proportions and substituted irregular and wavy lines for straight lines.

This is the geometry which, as Poincaré ably indicated, must be used
for the four-dimensional and, according to its regions, more or less
Euclidean continuum which is the Einsteinian universe. It is precisely
this geometry which states what there is in common between the forms of
objects seen by the drunken man and those seen by the water-drinker.

It is along this route, or a route analogous to this, that Einstein
at last reached success. The universe being a more or less warped
continuum, he proposed to apply to it the geometry created by Gauss for
the study of surfaces of variable curvature: a geometry generalised by
Riemann. It is by means of this special geometry that we express the
fact that the “Interval” of events is an invariant.

Here is an illustration which will, I think, lead us to the heart of
the problem of gravitation and to the solution of it.

* * * * *

Let us consider a surface of variable curvature—for instance, the
surface of any large district with its hills, mountains, and valleys.
When we travel in this region, we can proceed in a straight line as
long as we are on the level plain. A straight line on a level plain
has the remarkable feature of being the shortest distance between two
points. It has also this peculiarity, that it is the only one of its
kind and its length, whereas we may draw a great number of lines that
are not straight uniting the two points, longer than the straight line
but all of equal length.

But we have reached the hilly district. It is now impossible for us to
follow a straight line from one point to another if there is a hill
between them. Whatever path we take, it will be curved. But amongst the
various possible paths which lead from one point to the other on the
farther side of the hill, there is one—and only one, as a rule—which
is shorter than any of the others, as we could prove by means of a
tape. This shortest path, the only one of its kind, is what is called
the _geodetical_ of the surface covered.

In the same way no vessel can go in a straight line if it is sailing
from Lisbon to New York. It must follow a curved path, because the
earth is round. But amongst the possible curved paths there is a
privileged one which is shorter than the others: the one which follows
the direction of the great circle of the earth. In going from Lisbon
to New York, though they are nearly in the same latitude, vessels
are careful not to head straight westward, in the direction of the
parallels. They sail a little to the north-west, so that when they
reach New York they come from the north-east, having followed pretty
closely a terrestrial great circle. On our globe, as on all spheres,
the _geodetical_, the shortest route between two points, is the
arc of a great circle passing through the two points.

Now the “Interval” of two points in the four-dimensional universe
precisely represents the geodetical, the minimum path of progress
between the two points traced in the universe. Where the universe
is curved, the geodetic is a curved line. Where the universe is
approximately Euclidean, it is a straight line.

I may be told that it is very difficult to imagine as curved a
three-dimensional space, and still more a four-dimensional. I
agree. We have already seen that it is difficult enough to imagine
four-dimensional space even when it is not curved.

But what does that prove? There are many other things in nature which
we cannot visualise or form a mental picture of. The Hertz waves,
the X-rays, and the ultra-violet waves exist all the same, though we
cannot imagine them, or at least only by giving them a visible form
which does not belong to them. It is just one of our human infirmities
that we cannot conceive what we cannot picture to ourselves. Hence our
tendency to—if one may use an inelegant but expressive word—visualise
everything.

Let us therefore return to our geodetics. These we can very well
picture to ourselves, because in the universe, in spite of its four
dimensions, they are lines of only one dimension, like all other lines
that we know.

* * * * *

The existence of geodetics, of shortest-distance lines, will now
beautifully explain to us the connection between inertia and weight,
which did not appear in the Euclidean world of classic science. Hence
the Newtonian distinction between the principle of inertia and the
force of gravitation.

We Relativists find this distinction no longer necessary. Material
masses, like light, travel in a straight line when they are far
from a gravitational field, and in a curved line when they are near
gravitational masses. In virtue of symmetry a free material point can
only follow a geodetic in the universe.

If we now reflect that the force of gravitation introduced by Newton
does not exist—such action at a distance is very problematical—and
that in empty space there are only objects freely left to themselves,
we are driven to the following conclusion, which unites in a simple way
the previously separated sisters, inertia and weight: _Every moving
body freely left to itself in the universe describes a geodetic._

Far from the massive stars this geodetic is a straight line, because
there the universe is almost Euclidean. Near the stars it is a curved
line, because there the universe is not Euclidean. A fine conception,
combining in a single rule the principle of inertia and the law of
gravitation! A brilliant synthesis of mechanics and gravitation,
putting an end to the schism which so long kept them separate and
non-corresponding sciences!

In this bold and simple theory gravitation is not a force. The planets
have curved paths because near the sun, just as in the neighbourhood
of every concentration of matter the universe is curved or warped.
The shortest path from one point to another is a line that only seems
straight to us—poor pygmies that we are—because we measure it with
very small rods and over small distances. If we could follow the line
over millions of miles, and during a sufficient period, we should find
it curved.

In a word—to use an illustration that must be regarded only as an
analogy—the planets describe curved paths because they follow the
shortest path in a curved universe, just as at a sports ground cyclists
have no need to turn the handles when they reach the corner, but pedal
straight on, because the slope of the ground compels them of itself to
turn. In the sports ground, as in the solar system, the curvature is
greater in proportion as the machine is nearer to the inner edge of the
track.

All that now remains is to assign to the universe, to space-time, such
a curvature at its various points that the geodetics will exactly
represent the paths of the planets and of falling bodies, admitting
that the curvature of the universe is caused at each point by the
presence or vicinity of material masses.

In this calculation we have to take into account the fact that the
“Interval”—that is to say, the part of the geodetic between two points
that are very near each other—must be an invariant whoever may be
the observer. In this way the same geodetic will be a curved or even
wavy line for the drunken man we introduced and a straight line for a
stationary observer. The length of the line is the same, whether it
appears straight or curved.

Taking all this into account, and doing prodigies of mathematical
skill of which we have sufficiently indicated the object, Einstein
has succeeded in expressing the law of gravitation in a completely
invariant form.

In calculating, on the ground of Newton’s law, the “Interval” of
two astronomical events—for instance, the successive falls of two
meteorites into the sun—we should find that the “Interval” has not
precisely the same value for observers who are moving at different
velocities.

With the new form given to the law by Einstein the difference
disappears. The two laws, however, differ little from each other, as
was to be expected in view of the accuracy with which astronomers found
Newton’s law verified during a couple of centuries. The improvement
made in Newton’s law by Einstein means, in a word (and to use the old
language of the Euclidean universe), that we consider the law accurate
with the reserve that the distances of the planets from the sun are
measured by a scale which decreases slightly in length as the sun is
approached.

* * * * *

It is surprising that Newton and Einstein agree in expressing the
movements of gravitating stars in an _almost_ identical form,
because their starting-points are very different.

Newton starts from the hypothesis of absolute space, the empirical
laws of the motions of the planets expressed in Kepler’s laws, and
the belief that gravitational attraction is a force proportional to
mass. Einstein, on the other hand, in making his calculations starts
from the conditions of invariance which we indicated. He starts, in a
sense, from the philosophical principle or postulate or impulse to hold
that the laws of nature are invariant and independent of the point of
view—irrelative, if I may use the word.

Einstein even abandons the hypothesis which ascribed the curving of
gravitational paths to a distinct force of attraction. Yet, starting
from a point of view so different from that of Newton, and one that
seems at first less overloaded with hypotheses, Einstein reaches a law
of gravitation which is _almost_ identical with Newton’s.

This “almost” is of immense interest, because it enables us to test
which is the accurate law, that of Newton or that of Einstein. They
give the same results when there is question of velocities that are
feeble in comparison with that of light, but their results differ a
little when there is question of very high velocities. We have already
seen that, near the sun, light itself is bent out of its course in
exact conformity with Einstein’s law, and in a way that Newton’s law
did not predict as such.

But there is another divergence between the two laws. According to the
Newtonian law the planets revolving round the sun describe ellipses
which—neglecting the small perturbations due to the other
planets—have a rigorously fixed position.

Suppose we put on a table a slice of lemon cut through the longer
diameter of the fruit, and imagine that the chief stars, the
northern constellations, are painted on the vaulted roof of the vast
hemispherical room in the middle of which we place our table. The slice
of lemon has very nearly the form of an ellipse, and, if we take one of
the pips to represent the sun, it will stand for the orbit of one of
our planets. Newton’s law says that—after making due corrections—the
planetary orbit keeps a fixed position relatively to the stars as long
as the planet continues to revolve. This means that the slice of lemon
remains stationary.

Einstein’s law says, on the contrary, that the orbital ellipse turns
very slowly amongst the stars while the planet traverses it. This means
that our slice of lemon must turn slightly on the table, in such wise
that the two ends of the lemon do not remain opposite the same stars
painted on the wall.

If we calculate, in virtue of Einstein’s law, the extent to which the
elliptical orbits of the planets must thus turn, we find it so small as
to be impossible of observation except in the case of one planet, the
swiftest of all, Mercury.

Mercury revolves completely round the sun in about eighty-eight days,
and Einstein’s law shows that its orbit must at the same time turn by a
small angle which amounts to forty-three seconds of an arc (43″) at the
end of a century. Small as this quantity is, the refined methods of the
modern astronomer can easily measure it.

As a matter of fact, it had been noticed during the last century that
Mercury was the only one of the planets to show a slight anomaly in its
movements, which could not be explained by Newton’s law. Le Verrier
made prodigious calculations in connection with it, as he thought that
the anomaly might be due to the attraction of an unknown body lying
between Mercury and the sun. He hoped that he would thus discover, by
calculation, an intra-Mercurial planet, just as he had discovered the
trans-Uranian planet Neptune.

But no one ever observed his planet, and the anomaly of Mercury
continued to be the despair of astronomers. Now, in what did the
anomaly consist? Precisely in an abnormal rotation of the planetary
orbit; a rotation which Le Verrier’s calculations showed to be
forty-three seconds of an arc in a century. That is exactly the figure
that we deduce, without using any hypothesis, from Einstein’s law of
gravitation!

It is true that, according to the recent calculations of Grossmann, the
astronomical observations collected by Newcomb give as the recorded
value of the secular displacement of the perihelion of Mercury, not
43″ as Le Verrier believed, but 38″ at the most. The agreement with
Einstein’s theoretical result is, therefore, not perfect (which would
have been extraordinary), but it is striking, and is within the limits
of possible error of observation.

Einstein’s law is just as exact as Newton’s for the slower planets. For
faster bodies, the motion of which can be observed with a higher degree
of precision, Newton’s law is wrong, and Einstein’s triumphs once more.

* * * * *

This improvement of what had been considered perfect—the work of
Newton—is a great victory for the human mind. Astronomy and celestial
mechanics derive additional precision and power of forecast from it. We
can now follow the golden orbs, on the triumphal wings of calculation,
better than we could before, or antedate their movements by centuries.

But there is another test of Einstein’s law of gravitation. If it is
sound, the duration of a phenomenon increases, according to Einstein,
when the gravitational field becomes more intense. It follows that the
duration of the vibration of a given atom must be longer on the sun
than on the earth. The wave-lengths of the spectral lines of the same
chemical element ought to be a little greater in sunlight than in light
which originates on the earth. Recent observations tend to confirm
this, but the verification is less satisfactory than in the case of
Mercury because other causes may intervene to modify the wave-lengths.

On the whole, the powerful synthesis which Einstein calls the theory
of General Relativity, which we have here rapidly outlined, is a lofty
and beautiful mental construction as well as a superb instrument of
exploration.

To know is to forecast. This theory forecasts, and better than its
predecessors did. For the first time it combines gravitation and
mechanics. It shows how matter imposes upon the external world a
curvature or warping of which gravitation is but a symptom: just as the
weeds one sees floating on the sea are but indications of the current
which bears them along.

Whatever modifications it may undergo in the future—for everything in
science is open to improvement—it has shown us a little more of the
harmony that is born of unity in the laws of nature.

But I have sufficiently shown that if I have succeeded in enabling the
reader to understand—to feel, at least—these matters without invoking
the aid of the pure light which geometry pours upon the invisible.

CHAPTER VII

IS THE UNIVERSE INFINITE?

Is the universe infinite? It is a question that men have asked in all
ages, though they have not defined its meaning very accurately. The
theory of Relativity enables us to approach it from a new and subtle
point of view.

Kant—the genial grumbler who found it so horribly monotonous to see
the same sun shining, and the same spring blossoming, every year—took
his stand on metaphysical considerations when he affirmed that space is
infinite, and is sown with similar stars in all parts.

It is, perhaps, better to confine ourselves in such a matter to the
results of recent observation, and close the doors of our debating-room
against the fog of metaphysics. Indeed, the latter would compel us to
define pure space, about which we know nothing—not even if there is
such a thing.

The proof that we know little about it is the fact that the Newtonians
believe in it, while the Einsteinians regard it merely as an
inseparable attribute of material things. They define space by matter;
and they then have to define the latter. Descartes, on the contrary,
defined matter in terms of extension, which is the same thing as
space. It is a vicious circle. It is therefore better to leave Kant’s
metaphysical arguments out of our discussion, and adhere strictly to
experience, to what is measurable.

To simplify matters, we will admit the reality of this continuum in
which the stars float, which is traversed by their radiations, which
common sense calls space. If there were stars everywhere—if they were
infinite in number—there would also be space and matter everywhere.
Newtonians might find this a triumph equally with Einsteinians. Those
who believe in absolute space and those who deny it—Absolutists and
Relativists—would equally rejoice.

It would be fortunate if astronomical observation were to show that
the number of the stars is infinite, and thus the holders of contrary
opinions could both chant a victory in their writings. But what does
astronomical observation actually report?

There are those who deny _a priori_ that the number of the stars
can be infinite. That number, they said, is capable of increase; it is
therefore not infinite, because nothing can be added to the infinite.
The argument is specious, but unsound; although Voltaire himself was
seduced by it. One need not be a great mathematician to see that it
is always possible to add to an infinite number, and that there are
infinite quantities which are themselves infinitely small in comparison
with others. Let us get on to the facts.

If the stellar universe has no limits, there is no visual line drawn
from the earth to the heavens which will not encounter one of the
stars. The astronomer Olbers has said that the whole nocturnal sky
would in that case shine with the brilliance of the sun. But the total
brilliance of all the stars put together is only three thousand times
greater than that of a star of the first magnitude, or thirty million
times less than the light of the sun.

But that proves nothing, as Olbers’ argument is wrong, for two reasons.
On the one hand, there are necessarily a good many extinct or dark
stars in the heavens. Some of them have been closely studied, even
weighed. They betray their existence by periodically eclipsing brighter
stars, with which they revolve. On the other hand, it was discovered
some time ago that celestial space is occupied over large stretches by
dark gaseous masses and clouds of cosmic dust, which absorb the light
of more distant stars. We thus see that the existence of an infinite
number of stars is quite compatible with the poorness of the light of
the heavens at night.

* * * * *

Now let us put on our spectacles—our telescopes, I mean—and turn from
the province of possibility to that of reality, and we shall see that
recent astronomical observation has yielded a number of remarkable
facts which lead irresistibly to the following conclusions.

The number of the stars is not, as was long supposed, limited by the
range of our telescopes alone. As we get further away from the sun, the
number of stars contained in a unity of space, the frequence of the
stars, the density of the stellar population, do not remain uniform,
but decrease in proportion as we approach the limits of the Milky Way.

The Milky Way is a vast archipelago of stars, our sun lying in its
central region. This mass of stars, to which we belong, has, roughly,
the shape of a watch-case, the thickness being only about half the width
of the structure. Light, which travels from the earth to the moon in
little over a second, from the earth to the sun in eight minutes, and
from the earth to the nearest star in three years, needs at least
30,000 years—three hundred centuries—to pass from end to end of the
Milky Way.

The number of stars in the Milky Way is something between 500 and 1,500
millions. It is a small number: scarcely equal to the human population
of the earth, much smaller than the number of molecules of iron in a
pin’s head.

In addition to these we have discovered dense masses of stars, such
as the Magellanic Clouds, the cluster in Hercules, and so on, which
seem to belong to the fringes of our Milky Way—to be suburbs of it,
so to say. These suburbs seem to stretch a considerable distance,
particularly on one side of the Milky Way. The furthest away is,
perhaps, not less than 200,000 light-years from us.

Beyond these, space seems to be deserted, devoid of stars over expanses
which are enormous in comparison with the dimensions of our galactic
universe as we have described it. What is beyond this?

Well, beyond this we find those strange bodies, the spiral nebulæ,
lying like silver snails in the garden of the stars. We have discovered
several hundred thousand of them. Some astronomers believe that these
spiral masses of stars may be annexes of the Milky Way, reduced models
of it. Most astronomers incline to think, for very good reason, that
the spiral nebulæ are systems like the Milky Way, and comparable to it
in their dimensions. If the former view is correct, the entire system
of stars accessible to our telescopes could be traversed by light in
some hundreds of thousands of years. On the second hypothesis the
dimensions of the stellar universe to which we belong must be
multiplied by ten, and light would take at least millions of years to
traverse it.

On the first view the entire stellar universe, in so far as it is
accessible to us, consists of the Milky Way and its annexes: that is to
say, a local concentration of stars, beyond which we can see nothing.
The stellar universe is, in other words, practically limited, or at
least finite.

On the second view our Milky Way is simply one of the myriads of spiral
universes we see. The spiral nebula (with its hundreds of millions of
stars) plays the same part in this vaster universe that a star has in
the Milky Way. We have the same problem as before, but on a vaster
scale: if the Milky Way consists of a concentration of a finite number
of stars, as observation proves, does the accessible universe consist
of a finite number of spiral nebulæ?

Experience has as yet not pronounced on this point. But in my opinion
it is probable that, when our instruments are powerful enough to tackle
such a problem—in several centuries, perhaps—science will answer
“yes.”

If it were otherwise, if the spiral nebulæ were fairly evenly
distributed as we go outward, we can show by calculation that,
attraction being in inverse proportion to the square of the distance,
gravitation would have an infinite intensity in such a universe, even
in the part in which we live. But this is not the case. It follows
that, either the attraction of two masses decreases at great distances
rather more rapidly than in inverse proportion to the square of the
distance (which is not wholly impossible), or that the number of stellar
systems and stars is finite. Personally I favour the second hypothesis,
but it is incapable of proof. In such matters there is always an
alternative, always a way of escaping in accordance with one’s bias,
and there is really nothing that compels us to say that the stars are
finite in number.

* * * * *

Starting from the mean value, as it has been observed, of the proper
motions of the nearer stars, Henri Poincaré has calculated that the
total number of stars in the Milky Way must be about one thousand
million. The figure agrees fairly well with the results of the
star-gauges effected by astronomers by means of photographic plates.

He has also shown that the proper motions of stars would be greater if
there were many more stars than those which we see. Thus Poincaré’s
calculations are opposed to the hypothesis of an indefinite extension
of the stellar universe, as the number of stars “counted” agrees fairly
closely with the number “calculated.” We should add, however, that
these calculations prove nothing if the law of attraction is not quite
the inverse proportion of the square at enormous distances.

On the other hand, if the universe is finite in space as it is
conceived in classic science, the light of the stars, and isolated
stars themselves, would gradually drift away into the infinite,
and the cosmos would disappear. Our mind resents this consequence,
and astronomical observation discovers no trace whatever of such a
dislocation.

In a word, in the space of the “Absolutists” the stellar universe can
only be infinite if the law of the square of distances is not quite
exact for very remote masses; and it cannot be finite except on the
condition that it is ephemeral in point of time.

For Newton, indeed, the _stellar_ universe might be finite within
an infinite universe, because in his view there can be space without
matter. For Einstein, on the contrary, the universe and the material or
stellar universe are one and the same thing, because there is no space
without matter or energy.

* * * * *

These difficulties and obscurities disappear in great part when we
consider space, or space-time, from the Einsteinian standpoint of
General Relativity.

What is the meaning of the sentence, “The universe is infinite”? From
either the Einsteinian, the Newtonian, or the Pragmatist point of view
it means: If I go straight ahead, going on eternally, I shall never get
back to my starting-point.

Is it possible? Newton is compelled to say yes, because in his view
space stretches out indefinitely, independent of the bodies that occupy
part of it, whether the number of the stars is or is not limited.

But Einstein says no. For the Relativist the universe is not
necessarily infinite. Is it therefore limited, fenced in by some sort
of railings? No. It is not limited.

A thing may be unlimited without being infinite. For instance, a man
who moves on the surface of the earth may travel over it indefinitely
in every direction without ever reaching a limit. The surface of the
earth, thus regarded, or the surface of any sphere whatsoever, is
therefore both finite and unlimited. Well, we have only to apply to
space of three dimensions what we find in two-dimensional space (a
spherical surface), to see how the universe may be at one and the same
time finite and unlimited.

We saw that, in consequence of gravitation, the Einsteinian universe
is not Euclidean, but curved. It is, as we said, difficult, if not
impossible, to visualise a curvature of space. But the difficulty
exists only for our imagination, which is restricted by our life of
sense, not for our reason, which goes farther and higher. It is one of
the commonest of errors to suppose that the wings of the imagination
are more powerful than those of reason. If one wants proof of the
contrary, one has only to compare what the most poetic of ancient
thinkers made of the starry heavens with what modern science tells
about the universe.

Here is the way to approach our problem. Let us not notice for the
moment the rather irregular distribution of stars in our stellar
system, and take it as fairly homogeneous. What is the condition
required for this distribution of the stars under the influence of
gravitation to remain stable? Calculation gives us this reply: The
curvature of space must be constant, and such that space is bent like a
spherical surface.

Rays of light from the stars may travel eternally, indefinitely, round
this unlimited, yet finite, universe. If the cosmos is spherical in
this way, we can even imagine the rays which emanate from a star—the
sun, for instance—crossing the universe and converging at the
diametrically opposite point of it.

In such case we might expect to see stars at opposite points in the
heavens, of which one would be the image, the spectre, the “double” of
the other—in the sense which the ancient Egyptians gave to the word.
Properly speaking, this “double” would represent, not the generating
star as it is, but as it was at the time when it emitted the rays which
form the double, or millions of years earlier.

If we observe the original and the double star, the reality and the
mirage, simultaneously from some remote part of the stellar system,
such as our planet, we shall see a great difference between them, since
the “copy” will show us the original as it was thousands of centuries
before. It may, in fact, happen that the second star is more brilliant
than the first, because in the meantime the first has gradually cooled,
and may even be extinct.

* * * * *

It is improbable that we should find many of these phantom-stars,
or virtual stars, luminous and unreal daughters of heavy suns. The
reason is that the rays in their passage through the universe will
generally be diverted by the stars near which they pass. Concentration
or convergence of them at the antipodes of the real star must be rare.
Moreover, the rays are to some extent absorbed by the cosmic stuff they
meet in space. It is, however, not impossible that the astronomers of
the future may discover such phenomena. It is, in fact, not impossible
that we have already observed such things without knowing it.

In any case, what observers have not done in the past they may very
well do in the future, thanks to the suggestions of the new science.
Possibly it is going to have a great effect on observational astronomy
and induce it to furnish brilliant new verifications of theory. There
may be astonishing results, unforeseen by our folly, of the new
conceptions, surpassing in their fantastic poetry the most romantic
constructions of the imagination. Reality, or at least the possible, is
rising to giddy heights that were far beyond the reach of the golden
wings of fantasy.

I spoke on a previous page of the millions of years which light
takes to travel round our curved universe. Starting from the fairly
well-ascertained value of the quantity of matter comprised in the
Milky Way, it is possible to calculate the curvature of the world and
its radius. We find that the radius has a value equal to at least
150,000,000 light-years.

It therefore takes light at least 900,000,000 years, at a speed of
186,000 miles a second, to travel round the universe, assuming that it
consists only of the Milky Way and its annexes. The figure is quite
consistent with the figures we get from astronomical observation for
the dimensions of the galactic system, and also with the much larger
figures which we find if we regard the spiral nebulæ as Milky Ways.

Thus for the Relativist the universe may be unlimited without being
infinite. As to the Pragmatist, who goes straight ahead—who follows
what he calls a straight line, or the path of light—he will get back
in the end to the body from which he started, provided that he has time
enough at his disposal. He will then say that, if that is the nature of
things, the universe is not infinite.

Hence the question of the infinity or finiteness of the universe can
be controlled by experience, and some day it will be possible to prove
whether the whole cosmos and space are Newtonian or Einsteinian.
Unfortunately, it will have to be a very long experience, with various
little practical difficulties to overcome.

We may therefore prefer not to commit ourselves without further
instructions. We may not feel ourselves obliged to choose between the
two conceptions, and we may leave the benefit of the doubt to whichever
of the two is false.

* * * * *

Moreover, there is perhaps a third issue: if not for the Pragmatist, at
least for the philosopher—I mean, seeing that in England physics comes
under the head of “Natural Philosophy,” for the physicist.

Here it is. If all the heavenly bodies we know belong to the Milky Way,
other and very remote universes may be inaccessible to us because they
are optically isolated from us; possibly by the phenomena of the cosmic
absorption of light, to which we have already referred.

But this might also be due to something else which will, perhaps, shock
Relativists, but will seem to Newtonians quite possible. The ether, the
medium that transmits the luminous waves, and which Einstein has ended
by admitting once more (refusing, however, to give it its familiar
kinematic properties), and matter seem more and more to be merely
modalities. We explained this, on the strength of the most recent
physical discoveries, in a previous chapter. There is nothing to prove
that these two forms of substance are not always associated.

Does this not give me the right to think that perhaps our whole visible
universe, our local concentration of matter, is only an isolated clump
or sphere of ether? If there is such a thing as absolute space (which
does not mean that it is accessible to us), it is independent of ether
as well as matter. In that case there would be vast empty spaces,
devoid of ether, all round our universe. Possibly other universes
palpitate beyond these; and for us such worlds would be for ever as if
they did not exist. No ray of knowledge would ever reach us from them.
Nothing could cross the black, dumb abysses which environ our stellar
island. Our glances are confined for ever within this giant—yet too
small—monad.

“Are there, then,” some will cry in astonishment, “things which exist,
yet we will never know them?” Naive pretension—to want to embrace
everything in a few cubic centimetres of grey brain-stuff!

CHAPTER VIII

SCIENCE AND REALITY

We approach the end of our work. Has reality, seen through the prism of
science, changed its aspect with the new theories? Yes, certainly. The
Relativist theory claims to have improved the achromatism of the prism
and by this means improved the picture it gives us of the world.

Time and space, the two poles upon which the sphere of empirical data
turned, which were believed to be unshakeable, have been dislodged
from their strong positions. Instead of them Einstein offers us the
continuum in which beings and phenomena float: four-dimensional
space-time, in which space and time are yoked together.

But this continuum is itself only a flabby form. It has no rigidity. It
adapts itself docilely to everything. There is nothing fixed, because
there is no definite point of reference by means of which we could
distribute phenomena; because on the shores of this great ocean in
which things float there are none left of those solid rings to which
mariners once fastened their vessels.

Up to this point the theory of Relativity well deserves its name. But
now, in spite of it and its very name, there rises something which
seems to have an independent and determined existence in the external
world, an objectivity, an _absolute_ reality. This is the
“Interval” of events, which remains constant and invariable through all
the fluctuations of things, however infinitely varied may be the points
of view and standards of reference.

From this datum, which, speaking philosophically, strangely shares
the intrinsic qualities with which the older absolute time and
absolute space were so much reproached, the whole constructive part
of Relativity, the part which leads to the splendid verifications we
described, is derived.

Thus the theory of Relativity seems to deny its origin, even its very
name, in all that makes it a useful monument of science, a constructive
tool, an instrument of discovery. It is a theory of a new absolute:
the Interval represented by the geodetics of the quadri-dimensional
universe. It is a new absolute theory. So true is it that even in
science you can build nothing on pure negation. For creation you need
affirmation.

The theory of Relativity has won brilliant victories, crowned by the
decisive sanction of facts. We have given some astonishing instances
of these in our earlier chapters. But to say that the theory is true
because it has predicted phenomena that were afterwards verified would
be to judge it from too narrowly Pragmatist a standpoint. It would
also—there is real danger in this—be to close against the mind other
paths where there are still flowers to cull. We will not do that.

It is therefore important, in spite of its successes—nay, on account
of them—to turn the light of criticism upon the foundations of the new
doctrine. Even Cæsar, as he mounted the Capitol, had to listen to
the jokes of the soldiers round his chariot and lower his pride. The
theory of Relativity also, as it advances in all its magnificence along
the Triumphal Way, must learn that it has its limits, perhaps its
weaknesses.

* * * * *

But before we go further into it, before we turn the raw light upon it,
let us make one observation.

Whatever be the obscurities of physical theories, whatever be the
eternal and fated imperfection of science, one thing may be positively
laid down here: scientific truths are the best established, the most
certain, the least doubtful of all the truths we can know in regard
to the external world. If science cannot reveal to us the nature of
things in its entirety, there is nothing else that can do it as well.
The truths of sentiment, of faith, of intuition, have nothing to do
with those of science as long as they remain strictly truths of the
interior world. They are on another plane. But the moment they claim
to be measures of the external world—which would be their only cause
of weakness—they subject themselves to the material reality, to the
scientific investigation of the truth.

It is therefore nonsense to speak of a “bankruptcy of science” as
contrasted with the certainty which other disciplines may give us
respecting the external world. The bankruptcy of one would make all
the others bankrupt. When it is not a question of the intimate oasis
in which the serene realities of sentiment flourish, but of the arid
and imperfectly explored desert of the material world, the scientific
facts are the basis of all constructions. Destroy those and you destroy
everything. If you ram the ground floor of a house and bring it down,
you bring down also the upper stories.

To say the truth, it would seem that nothing here below so much reveals
the mystic presence of the divine as does the eternal and inflexible
harmony that unites phenomena, and that finds expression in the laws of
science.

Is not this science which shows us the vast universe well-ordered,
coherent, harmonious, mysteriously united, organised like a great mute
symphony, dominated by law instead of caprice, by irrefragable rules
instead of individual wills—is this not a revelation?

There you have the only means of reconciling the minds which are
devoted to external realities and those which bow to metaphysical
mystery. To talk of bankruptcy of science—if it means anything more
than to point out human weakness, which is, alas! obvious enough—is
really to calumniate that part of the divine which is accessible to our
senses, the part which science reveals.

* * * * *

In sum, the whole Einsteinian synthesis flows from the issue of the
Michelson experiment, or at least from a particular interpretation of
that issue.

The phenomenon of stellar aberration proves that the medium which
transmits the light of the stars to our eyes does not share the motion
of the earth as it revolves round the sun. This medium is known to
physicists as ether. Lord Kelvin, who was honoured by being buried in
Westminster Abbey not far from the tomb of Newton, rightly regarded the
existence of interstellar ether as proved as fully as the existence of
the air we breathe; for without this medium the heat of the sun, mother
and nurse of all terrestrial life, would never reach us.

In his theory of Special Relativity, Einstein, as we saw, interprets
phenomena without introducing the ether, or at least without
introducing the kinematic properties which are usually attributed to
it. In other words, Special Relativity neither affirms nor denies the
existence of the classic ether. It ignores it.

But this indifference to or disdain of the ether disappears in the
theory of General Relativity. We saw in a previous chapter that the
trajectories of gravitating bodies and of light are directly due, on
this theory, to a special curvature and the non-Euclidean character of
the medium which lies close to massive bodies in the void—that is to
say, ether. This, therefore, though Einstein does not give it the same
kinematic properties as classic science did, becomes the substratum
of all the events in the universe. It resumes its importance, its
objective reality. It is the continuous medium in which spatio-temporal
facts evolve.

Hence in its general form, and in spite of the new kinematic attitude
which is ascribed to it, Einstein’s general theory admits the objective
existence of ether.

Stellar aberration shows that this medium is stationary relatively to
the orbital motion of the earth. The negative result of Michelson’s
experiment tends, on the contrary, to prove that it shares the earth’s
motion. The Fitzgerald-Lorentz hypothesis solves this antinomy by
admitting that the ether does not really share the earth’s motion, but
saying that all bodies suddenly displaced in it are contracted in the
direction of the movement. This contraction increases with their
velocity in the ether, which explains the negative result of the
Michelson experiment.

Lorentz’s explanation seemed to Einstein inadmissible on account of
certain improbabilities which we pointed out, and especially because it
assumes that there is in the universe a system of privileged references
which recalls Newton’s “absolute space.” Einstein, taking his stand on
the principle that all points of view are equally relative, does not
admit that there are in the universe privileged spectators—spectators
who are stationary in the ether—who could see things as they are,
whereas these things would be deformed for every other observer.

Then, while preserving the Lorentz contraction and the formulæ in
which it is expressed, Einstein says that this contraction, while it
really exists, is only an appearance, a sort of optical illusion,
due to the fact that the light which shows us objects does not
travel instantaneously, but with a finite velocity. This spread of
light follows laws of such a nature that apparent space and time are
changed in precise accordance with the formulæ of Lorentz. That is the
foundation of Einstein’s Special Relativity.

Hence the two immediate possible explanations of the negative result of
the Michelson experiment are:

1. Moving objects are contracted in the stationary
ether, the fixed substratum of all phenomena. This
contraction is real, and it increases with the velocity
of the body relatively to the ether. That is Lorentz’s
explanation.

2. Moving objects are contracted relatively to any
observer whatsoever. This contraction is only apparent,
and is due to the laws of the propagation of
light. It increases with the velocity of the moving
body relatively to the observer. That is Einstein’s
explanation.

* * * * *

But there is at least one other possible explanation. It introduces
new and strange hypotheses, but they are by no means absurd. Indeed,
it is especially in physics that truth may at times seem improbable.
This explanation will show how we may account for the result of the
Michelson experiment apart from either Lorentz or Einstein.

This third explanatory hypothesis is as follows. Every material body
bears along with it, as a sort of atmosphere, the ether that is
bound up with it. There is, in addition, a stationary ether in the
interstellar spaces; an ether insensible to the motion of the material
bodies that move in it, and which we may, to distinguish it from the
ether bound up with bodies, call the “super-ether.” This super-ether
occupies the whole of interstellar space, and near the heavenly bodies
it is superimposed upon the ether which they bear along. The ether
and the super-ether interpenetrate each other just as they penetrate
matter, and the vibrations they transmit spread independently. When a
material body sends out series of waves in the ether which surrounds
it, these move relatively to it with the constant velocity of light.
But when they have traversed the relatively thin stratum of ether bound
up with the material body, which merges gradually in the super-ether,
they spread in the latter, and it is relatively to this that they
progressively take their velocity.

It is like a boat crossing the Lake of Geneva at a certain speed. About
the middle of the lake it has this speed relatively to the narrow
current which the River Rhone makes there, and then it resumes it
relatively to the stationary lake.

In the same way the luminous rays of the stars, although they come
from bodies which are approaching or receding from us, have the same
velocity when they reach us, and this will be the common velocity which
the super-ether imposes upon them. Thus also, on the other hand, the
stellar rays that reach our telescopes will be transmitted to us by the
super-ether, without the very thin stratum of mobile ether bound up
with the earth being able to disturb their propagation.

These hypotheses explain and reconcile all the facts: (1) the fact of
stellar aberration, because the rays which reach us from the stars are
transmitted to us unaltered by the super-ether; (2) the negative result
of the Michelson experiment, because the light which we produce in the
laboratory travels in the ether that is borne along by the earth, where
it originates; (3) the fact that, in spite of the approach or recession
of the stars, their light reaches us with the common velocity which it
had acquired in the super-ether, shortly after it started.

However strange this explanation may seem, it is not absurd, and it
raises no insurmountable difficulty. It shows that, if the result of
the Michelson experiment is a sort of no-thoroughfare, there are other
ways out of it besides Einstein’s theory.

To resume the matter, we have offered to us three different ways of
escaping the difficulties, the apparent contradictions, involved in
our experience—the antinomy arising from aberration and the Michelson
result—and they are reduced to these alternatives:

1. The contraction of bodies by velocity is real (Lorentz).

2. The contraction of bodies by velocity is only an appearance
due to the laws of the propagation of light (Einstein).

3. The contraction of bodies by velocity is neither real nor
apparent: there is no such thing (hypothesis of super-ether
connected with ether).

This shows that the Einsteinian explanation of phenomena is by no means
imposed upon us by the facts, or is at least not absolutely imposed by
them to the exclusion of any other explanation.

* * * * *

Is it at least imposed by reason, by principles, by the evidential
character of its rational premises, or because it does not conflict
with our good sense and mental habits as the others do?

One would suppose this at first, when one compares it with the teaching
of Lorentz; and, in order to relieve this discussion, I will for the
moment leave out of account the third theory which I sketched, that of
a super-ether.

What seemed most difficult to admit in Lorentz’s hypothesis of real
contraction was that the contraction of bodies was supposed to depend
entirely upon their velocity, not in any way upon their nature; that it
was supposed to be the same for all bodies, no matter what was their
chemical composition or physical condition.

A little reflexion shows that this strange suggestion is not so clearly
inadmissible. We know that the atoms are all formed of the same
electrons, and they differ, and differentiate bodies, only in their
number and arrangement. If, then, the electrons common to all matter
and their relative distances experience simultaneously a contraction
due to velocity, it is natural enough to suppose that the result maybe
the same for all objects. When an iron grating of a given length is
dilated by heat, the extent to which a temperature of a hundred degrees
dilates it will be the same whether it counts ten or a hundred steel
bars to the square yard, provided they are identical.

Hence it is not really here that we find the improbability which caused
Relativists to reject the Lorentz theory. It is in the principles of
the theory. It is because the theory admits in nature a system of
privileged reference—the stationary ether relatively to which bodies
move.

Let us examine this more closely. It has been said that Lorentz’s
stationary ether is merely a resuscitation of Newton’s absolute space,
which the Relativists have so vigorously attacked. That is very far
from the truth. If, as we supposed in the preceding chapter, our
stellar universe is only a giant globe of ether rolling in a space
that is devoid of ether—one of many such globes that will remain for
ever unknowable to man—it is obvious that the drop of ether which
represents our universe may very well be moving in the environing
space, which would then be the real “absolute space.”

From this standpoint the Lorentzian ether cannot be identified with
absolute space. To do so amounts to saying that the space called
“absolute” by Newton does not deserve the name. If Newtonian space is
only the physical continuum in which the events of our universe happen,
it is anything but stationary.

In that case the whole fault one has to find with Newton is that he
used a wrong expression: that he called something absolute which is
merely privileged for a given universe. It would be a quarrel about
grammar; and such things have never succeeded in revolutionising
science.

But the Relativists—at least those impenitent Relativists, the
Einsteinians—will not be content with that. It is not enough for them
that the Newtonian space with all its privileges may not be absolute
space.

Our conception of the universe, as a moving island of ether, is well
calculated to reconcile the pre-eminence of Newtonian space with that
agnosticism which forbids us to hope to attain the absolute. But this
again is not enough for the Einsteinians. What they mean to do is to
strip of all its privileges the Newtonian space on which the structure
of classical mechanics has been reared. They mean to reduce this space
to the ranks, to make it no more than analogous to any other spaces
that can be imagined and which move arbitrarily in reference to it.

* * * * *

From the agnostic, the sceptical, point of view this is a fine and
strong attitude. But in the course of this volume we have so much
admired Einstein’s powerful theoretical synthesis and the surprising
verifications to which it led that we are now entitled to make some
reserves. It is legitimate to call into question even the denials of
doubters, because, after all, they are really themselves affirmations.

We believe that in face of this philosophic attitude of the
Einsteinians—in face of what I should like to call their absolute
relativism—we are justified in rebelling a little and saying something
like this:

“Yes, everything is possible; or, rather, many things are possible, but
all things are not. Yes, if I go into a strange house, the drawing-room
clock may be round, square, or octagonal. But once I have entered the
house and seen that the clock is square, I have a right to say: ‘The
clock is square. It has the privilege of being square. It is a fact
that it is neither round nor octagonal.’

“It is the same in nature. The physical continuum which contains, like
a vase, all the phenomena of the universe, might have, relatively to
me—and as long as I have not observed it—any forms or movements
whatever. But as a matter of fact, it is what it is. It cannot be
different things at the same time. The drawing-room clock cannot at one
and the same time be composed entirely of gold and entirely of silver.

“There is therefore one privileged possibility amongst the various
possibilities which we imagine in the external world. It is that which
has been effectively realised: that which exists.”

The complete relativism of the Einsteinians amounts to making the
universe external to us to such an extent that we have no means of
distinguishing between what is real and what is possible in it, as far
as space and time are concerned. The Newtonians, on the other hand, say
that we can recognise real space and real time by special signs. We
will analyse these signs later.

In a word, the pure Relativists have tried to escape the necessity of
supposing that reality is inaccessible. It is a point of view that
is at once more modest and much more presumptuous than that of the
Newtonians, the Absolutists.

It is more modest because according to the Einsteinian we cannot know
certain things which the Absolutist regards as accessible: real time
and space. It is more presumptuous because the Relativist says that
there is no reality except that which comes under observation. For him
the unknowable and non-existent are the same thing. That is why Henri
Poincaré, who was the most profound of Relativists before the days of
Einstein, used to repeat constantly that questions about absolute space
and time have “no meaning.”

One might sum it up by saying that the Einsteinians have taken as their
motto the words of Auguste Comte: “Everything is relative, and that is
the only absolute.”

Newton, whose spatio-temporal premises Henri Poincaré vigorously
refused to admit, and classical science take up an attitude, on the
contrary, which Newton himself well described when he wrote: “I am
but a child playing on the shore, rejoicing that I find at times a
well-polished pebble or an unusually fine shell, while the great ocean
of truth lies unexplored before me.” Newton says that the ocean is
unexplored, but he says that it exists; and from the features of the
shells he found he deduced certain qualities of the ocean, especially
those properties which he calls absolute time and space.

Einsteinians and Newtonians are agreed in thinking that the external
world is not in our time entirely amenable to scientific research. But
their agnosticism differs in its limits. The Newtonians believe that,
however external to us the world may be, it is not to such an extent as
to make “real time and space inaccessible to us.” The Einsteinians hold
a different opinion. What separates them is only a question of degree
of scepticism. The whole controversy is reduced to a frontier quarrel
between two agnosticisms.

CHAPTER IX

EINSTEIN OR NEWTON?

_Recent discussion of Relativism at the Academy
of Sciences—Traces of the privileged space
of Newton—The principle of causality
the basis of science—Examination of M.
Painlevé’s objections—Newtonian arguments and
Relativist replies—M. Painlevé’s formulæ of
gravitation—Fecundity of Einstein’s theory—Two
conceptions of the world—Conclusion._

What are these “special signs” by which the Newtonian conception of
nature recognises that we are in touch with the privileged space
which Newton called absolute space, and which seemed to him the real,
intrinsic, exclusive frame of phenomena?

These signs or criteria are implicitly at the root of the development
of classic science, but they for a time remained in the shades of the
discussions provoked by Einstein’s theory. Leaving aside for a moment
other, and perhaps less noble, cares, M. Paul Painlevé, addressing
the Academy of Sciences at Paris, has with brilliant success drawn
attention to the arguments, ancient yet ever robust, which constitute
the strength of the Newtonian conception of the world.

Let us from this point speak of the absolute time and space of Newton
and of Galileo as privileged space and privileged time, in order not to
expose our flanks further to the metaphysical objections—not without
justification—which the qualification “absolute” provokes.

Why is classical science, the mechanics of Galileo and Newton, founded
upon privileged space and privileged time? Why do they refer all
phenomena to these unique standards, and consider them adequate to
reality? It is on account of the principle of causality.

The principle may be formulated thus: Identical causes produce
identical effects. That means that the initial conditions of a
phenomenon determine its ulterior modalities. It is briefly a statement
of the determinism of phenomena, and without that science is impossible.

It is, of course, possible to be captious on the point. Conditions
entirely identical with given initial conditions can never be
reproduced or discovered at a different time or in a different place.
There is always some circumstance that will be different; for instance,
the fact that in the interval between the two experiments the Nebula in
Andromeda will have come several thousand miles nearer to us. And we
have no influence on the Nebula in Andromeda.

Happily—this saves the situation—distant bodies have, it seems, only
a negligible influence on our experiments. That is why we can repeat
them. For instance, if we to-day put a gramme of sulphuric acid in ten
grammes of soda-solution (one-tenth), they will in the same period of
time produce the same quantity of sulphate of sodium that they would
have done a year previously in the same conditions of temperature and
pressure; in spite of the fact that meantime Marshal Foch sailed for
the United States.

Thus the principle of causality (like causes, like effects) is always
verified, and never found at fault. It is therefore an empirical truth,
but in addition to this it imposes itself on our mind with irresistible
force. It even imposes itself upon animals. “The scalded cat avoids hot
water,” is proof enough. In any case, not science only but the whole
life of man and animals is based upon it.

It is a consequence of the principle that if the initial conditions of
a movement present a symmetry, this will appear again in the movement.
M. Paul Painlevé insisted strongly on this in the course of the recent
discussion of Relativism at the Academy of Sciences. The principle
of inertia in particular follows from this statement: a body left to
itself far from any material mass will, by reason of symmetry, remain
at rest or travel in a straight line.

It will certainly follow a straight line for a given observer (or for
observers moving with uniform velocities relatively to the first). The
Newtonians say that the space of these observers is privileged.

On the other hand, for another observer who is, relatively to them,
moving at an accelerated velocity, the path of the moving body will be
a parabola, and will no longer be symmetrical. Therefore the space of
this new observer is not privileged space.

It seems to me that the Relativists might reply to this as follows. You
have no right to define the initial conditions for a given observer,
then the subsequent movement for another observer who is moving with
accelerated velocity. If you thus define your initial conditions
relatively to the latter, the moving body at the moment when it is
released is not free for this observer, but falls in a gravitational
field. It is therefore not surprising that the motion produced seems to
him accelerated and dissymmetrical. The principle of causality is not
wrong for either observer.

One might also give a different definition of the privileged system,
saying: it is that relatively to which light travels in a straight
line in an isotropic medium. But in that case the rays from the stars
travel in a spiral for an observer fixed on a turning earth, and the
Newtonians would infer from this that the earth turns relatively to
their privileged space. Einsteinians will reply that the space in which
the rays travel is not isotropic, and that they are diverted from the
straight line in it by the turning gravitational field which causes the
centrifugal force of the earth’s rotation. They will always find an
escape which will leave the principle of causality intact.

It seems difficult, therefore, to give unanswerable proof of the
existence of the privileged system when we start from the principle of
causality. Each party retains its position.

* * * * *

On the other hand, there is evidential value, a keen and convincing
penetration, in the second part of the criticism which M. Painlevé
directs against the principles of Einstein’s theory.

Let us sum up the argument of the distinguished geometrician. You,
he says to the Einsteinians, deny all privilege to any system of
reference whatever. But when you want to deduce, by calculation, the
law of gravity from your general equations, you cannot do it, and
you really do not do it, except by introducing scarcely disguised
Newtonian hypotheses and privileged axes of reference. You only reach
the result of your calculation by sharply separating time and space as
Newton does, and by referring your gravitating moving objects to purely
Newtonian privileged axes, in the case of which certain conditions of
symmetry are realised.

To this fine and profound criticism which M. Painlevé raises may be
added that of Wiechert, who has pointed out various other hypotheses
introduced by Einstein in the course of his calculations.

In a word, Einstein seems not to have kept entirely clear of the
Newtonian premises which he repudiates. He has not the disdain for them
that one would suppose, and he does not hesitate to have recourse to
them occasionally for the purpose of helping out his calculations. That
is rather to pay a little reverence to the idols you have burned.

In reply the Einsteinians will doubtless say that, if they introduce
Newtonian axes in the course of their arguments, it is to make the
results of calculation comparable to the result of experimental
measurements. The axes introduced into their equations have for the
Relativists the sole privilege of being those to which experimenters
refer their measurements. But we must admit that that is no small
privilege.

* * * * *

That is not all. The principle of General Relativity amounts to this:
All systems of reference are equivalent for expressing natural laws,
and these laws are invariant to any system of reference to which they
are related. That means in effect: There are relations between objects
of the material world which are independent of the one who observes
them, and particularly of his velocity. Thus, when a triangle is drawn
on paper, there is something in the triangle which characterises it and
which is identical, whether the observer passes very quickly or very
slowly, or at any speed and in any direction whatever, beside the paper.

M. Painlevé observes, with some reason, that in this form the principle
is a sort of truism. It is a severe verdict, yet it expresses a certain
fact. The real relations of external objects cannot be altered by the
standpoint of the observer.

Einstein replies that it is at all events something to have provided
a sieve by which we may sift the laws and formulæ which serve to
represent the phenomena that have been empirically observed: a
criterion which they must pass before they are recognised as correct.
This is true. Newton’s law, in its classic form, did not meet this
criterion. This proves that it was not quite so obvious. A truth that
was unknown yesterday has become to-day a truism. So much the better.

In expressing one of the conditions which must be satisfied by
natural laws the theory of Relativity at least has what is called in
philosophical jargon a “heuristic” value. But it is none the less true,
as M. Painlevé points out with great force and clearness, that the
principle of General Relativity, considered in this light, would be
unable to provide precise laws. It would be quite consistent with a law
of gravity in which the attraction would be in inverse proportion, not
to the square, but to the seventeenth or hundredth power, or any power
whatever, of the distance.

In order to extract the correct law of gravitation from the
principle of General Relativity we have to add to it the Einsteinian
interpretation of the result of the Michelson experiment—to wit,
that relatively to any observer whatsoever light travels locally with
the same velocity in every direction. We have also to add various
hypotheses which M. Painlevé regards as Newtonian.

To the critical discussion of Relativity which he so brilliantly
presented at the Academy of Sciences M. Paul Painlevé added a valuable
mathematical contribution of which the chief result is the following:
It is possible to excogitate other laws of gravitation than that
offered by Einstein, and all of them will fulfil the Einsteinian
conditions.

The learned French geometrician indicated several of these, especially
one of which the formula differs considerably from that of Einstein,
yet equally and precisely explains the motions of the planets, the
displacement of the perihelion of Mercury, and the deviation of rays of
light near the sun.

This new formula corresponds to a space that is independent of time,
and it does not involve the consequence that Einstein’s formula does
in regard to the shifting toward the red of all the lines in the solar
spectrum. The verification or non-verification of this consequence of
Einstein’s equation, of which we pointed out the difficulties (perhaps
insurmountable) in a previous chapter, thus acquires a new importance.

It is a remarkable thing that many of the formulæ of gravitation
given by M. Painlevé lead to the conclusion, differently from that of
Einstein, that space remains Euclidean even near the sun, in the sense
that measures are not necessarily contracted.

All this light on the astronomical horizon seems like the dawn of a
new era in which observations of unprecedented delicacy will provide
tests that are calculated to give a more precise and less ambiguous
form to the law of gravitation. There are great days—or, rather, great
nights—in store for the astronomer.

* * * * *

As far as the principles are concerned, the controversy will go on. It
must end in something like the following dialogue:

_The Newtonian_: Do you admit that at a point in the universe that
is far away from all material masses a moving object left to itself
must follow a straight line? If so, you recognise the existence of
privileged observers—those for whom the line is straight. For another
observer the line is a parabola. Therefore his point of view is wrong.

_The Relativist_: Yes, I grant it; but in point of fact there
is no point in the universe where there is no influence of distant
material masses. Therefore your moving object left to itself is a mere
fiction, and I am not going to base science upon an unverifiable piece
of imagination. The whole aim of the Relativist is to rid science of
everything that has no experimental significance. As to the observer
who sees the moving object in question describe a parabola, he will
interpret his observation to mean that the object is in a gravitational
field.

_The Newtonian_: You are therefore compelled to admit that far
away from all matter, far from all heavenly bodies, there can be
what you call a gravitational field, that it varies according to the
velocity of the observer, and that it can be very intense in spite of
the distance of the heavenly bodies, and even, at times, increase with
that distance. These are strange and absurd hypotheses.

_The Relativist_: They are strange, but I defy you to prove that
they are absurd. They are less absurd than to localise and set in
motion a point that is isolated and independent of any material mass.

_The Newtonian_: For my part, I can easily imagine a single
material point in the universe having a certain position and a certain
velocity in it.

_The Relativist_: For my part, on the contrary, if such a material
point existed, it would be absurd and impossible to speak of its
position and its motion. It would have neither position nor motion
nor rest. Such things can exist only with reference to other material
points.

_The Newtonian_: That is not my opinion.

_The Impartial Spectator_: In order to know which of you is
right we should need to try an experiment on a material point that is
withdrawn from the influence of the rest of the universe. Can you try
this experiment?

_The Newtonian and the Relativist_ (together): No, unhappily.

_The Metaphysician_ (coming up like the third thief in the fable):
Then, gentlemen, I advise you to return to your telescopes, your
laboratories, and your tables of logarithms. The rest is my affair.

_The Newtonian and the Relativist_ (together): In that case we are
quite sure we shall never learn anything further about it than we know
or believe now.

* * * * *

Meantime, it is impossible to exaggerate the importance of the new
light thrown on the question of Relativity by the intervention of M.
Paul Painlevé at the Academy of Sciences. It will have a lasting and
prodigious echo.

Will Einstein’s fine synthesis be defeated? Shall we see it sink in the
controversies, doubts, and obscurities of which we have given a short
account? I think not.

When Christopher Columbus discovered America, it was all very well to
tell him that his premises were wrong, and that if he had not believed
that he was sailing for the Indies he would never have reached a new
continent. He might have replied, after the style of Galileo: “I
discovered it, for all that.” The method that gives good results is
always a good method.

When we have to plunge into the depths of the unknown to discover
something new, when we have to learn more and better, the end justifies
the means. When he reminds us of optics, mechanics, and gravitation,
now bound up together in a new sheaf, of the deviation of light by
gravity which he foretold against all expectation, of the anomalies
of Mercury which he was the first to explain, and of his improvement
of the Newtonian law, Einstein has the right to say, with some pride:
“There is what I have done.”

It is said that the paths by which he attained all these fine results
are not devoid of unpleasant false turns and quagmires. Well, there are
many ways to Rome and to truth, and some of them are not perfect. The
main thing is to get there. And in this case the truth means ancient
facts brought into a new harmony, and new facts set forth in prophetic
equations and verified in the most surprising manner.

If discussion of principles—if theory, which is only the servant of
knowledge—shrugs its servile and disloyal shoulders a little over
Einstein’s work, at all events experience, the sole source of truth,
has justified him. Brilliant formulæ that Einstein had not foreseen
are now discovered to explain the anomaly of Mercury and the deviation
of light. It is good: but we must not forget that the first of these
correct formulæ, that of Einstein, went boldly in advance of the
verification.

New trenches have been won in the war against the eternal enemy, the
unknown. Certainly we have now to organise them and create more direct
roads to them. But to-morrow we shall have to advance again, to gain
more ground. We shall have, by any theoretical device that we can,
to state other new facts, unknown but verifiable facts. That is what
Einstein did.

If it is a weakness of Einstein’s teaching to deny all objectivity,
all privilege, to any system of reference whatever, while utilising
such a system for the necessities of calculation, it was at all events
a weakness shared by the great Poincaré. To the day of his death he
rebelled energetically against the Newtonian conception. The support of
such a genius, whom one finds involved in all our modern discoveries,
is enough to secure some respect for the Relativist theory.

If we have on the one side Newton and his ardent and persuasive
apologist, equipped with a fine mathematical genius, Paul Painlevé, we
have on the other side Einstein and Henri Poincaré. Even in earlier
history we have Aristotle against Epicurus, Copernicus against the
Scholastics, at the same barricade. It is an eternal war of ideas, and
it may be endless if, as Poincaré believed, the Principle of Relativity
is at the bottom only a convention with which experience cannot quarrel
because, when we apply it to the entire universe, it is incapable of
verification.

It is the fertility of the Einsteinian system which proves that it
is strong and sound. Are the new beings with which it has peopled
science—the discoveries predicted by it—legitimate children? The
Newtonians say that they are not. But in properly ordered science, as
in an ideal State, it is the children that matter, not their legitimacy.

At all events the vigorous counter-offensive of M. Painlevé has driven
back to their lines the over-zealous apostles of the new gospel, who
thought that they had pulverised classic science beyond hope of
recovery. Each side now remains in its positions. There is no longer
any question of regarding the Newtonian conception of the world as a
piece of childlike barbarism. A different conception is now opposed
to it—that is all. The war between them is as yet undecided, and
may remain for ever undecided, as the weapons with which it might be
possible to bring it to an issue are sealed up for ever in the arsenal
of metaphysics.

* * * * *

Whatever may happen, Einstein’s teaching has a power of synthesis and
prediction which will inevitably incorporate its majestic system of
equations in the science of the future.

M. Émile Picard, perpetual secretary of the Academy of Sciences, and
one of the luminous and profound thinkers of our time, has asked if
it is an advance “to try, as Einstein has done, to reduce physics
to geometry.” Without lingering over this question, which may be
insoluble, like all speculative questions, we will conclude with the
distinguished mathematician that the only things which matter are the
agreement of the final formulæ with the facts and the analytic mould in
which the theory casts the phenomena.

Considered from this angle, Einstein’s theory has the solidity of
bronze. Its correctness consists in its explanatory force and in the
experimental discoveries predicted by it and at once verified.

What changes in theories are the pictures we form of the objects
between which science discovers and establishes relations. Sometimes we
alter these pictures, but the relations remain true, if they are based
upon observed facts. Thanks to this common fund of truth, even the most
ephemeral theories do not wholly die. They pass on to each other, like
the ancient runners with their torch, the one accessible reality: the
laws that express the relations of things.

To-day it happens that two theories together clasp the sacred torch.
The Einsteinian and the Newtonian vision of the world are two faithful
reflections of it: just as the two images, polarised in opposite
directions, which Iceland spar shows us in its strange crystal both
share the light of the same object.

Tragically isolated, imprisoned in his own “self,” man has made a
desperate effort to “leap beyond his shadow,” to embrace the external
world. From this effort was born science, and its marvellous antennæ
subtly prolong our sensations. Thus we have in places approached the
brilliant raiment of reality. But in comparison with the mystery that
remains the things we know are as small as are the stars of heaven
compared with the abyss in which they float.

Einstein has discovered new light for us in the depths of the unknown.
He is, and will remain, one of the light-houses of human thought.

_Printed in Great Britain by
Hazell, Watson & Viney, Ld.,
London and Aylesbury._

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